Introduction to Optics 2nd ed - F. Pedrotti, L. Pedrotti (Prentice-Hall, 1993) WW.pdf

Second Edition

Introduction to Optics FRANK L.. PEDROTTI, S ..J .. Marquette University Milwaukee. Wisconsin Vatican Radio, Rome

LENO S. PEDROTD Center for Occupational Research and Development Waco, Texas Emeritus Professor of Physics Air Force Institute of Technology Ohio

Prentice-Hall International, Inc.

PHYSICAL CONSTANTS Speed of light Electron charge Electron rest mass Planck constant Boltzmann constant Permittivity of vacuum Permeability of vacuum

c = 2.998 X 108 mls e 1.602 x 10- 19 C m 9.109 x 10- 31 h 6.626 X 10-34 J8 k 1.3805 X 10- 23 J/K Eo = 8.854 x /-LO = 41T X

xvii

Preface

Optics is today perhaps the most area of both theoretical and applied physics. Since the 1960s the parallel emergence and of fiber optics, and a variety of semiconductor sources and detectors have revitalized the field. The need for a variety of texts with different and is therefore apparent, both for the student of optics and for the laborer in the field who needs an occasional review of the basics. With Introduction to we propose to teach introductory modern optics at an intermediate level. for several of the final (19, which are written at a somewhat the text assumes as background a good course in introductory physics, at the level usually given to physics and engineering majors, and at least two semesters of calculus. The book is written at the level of understanding appropriate to the average sophomore major, who has the necessary physics and mathematics prerequisites as a freshman. the traditional areas of college optics, as wen as several rather new ones, the text can be 11.""",,,,,11 ther a half- or a full-year course. We believe that the and today warrant readjustment of curricula to provide for a full year of program. For those who are familiar with the first edition, it may be the major changes introduced in this second edition. Two new ch:aJ)ters 11.".11",,, with laser-beam characteristics and nonlinear have been added. The new laser chapter now appears, together with the two earlier laser toward the end of the book, where the three function as a unit. In addition, the has been

xix

greatly expanded and moved to a later chapter. Several new sections have been introduced. They are Ray and The Thick Lens (Chapter 4), Effect (Chapter 8), and Evanescent Waves 20). Worked examples are now within the text, and 175 new problems have been added to the chapter exercises. Specific features of the text, in terms of coverage beyond the traditional areas, include extensive use of 2 X 2 matrices in dealing with ray and multiple thin-film interference; three devoted to a chapter on the eye, induding laser treatments of the eye; and individual chapters on holography, coherand Fresnel equaence, fiber optics, Fourier optics, nonlinear tions. A final chapter a brief introduction to the optical constants of dielectrics and metals. We have attempted to make many of the more specialized chapters independent of the others so that can be omitted without detriment to the remainder of the book. This should be helpful in shorter versions of the course. traditional Organization of the material in three major parts follows lines. The first part of the book deals with geometrical as a limiting form of wave optics. The middle develops wave optics in detail, and the final treats topics generally referred to as modern optics. In the first I presents a brief historical review of the theories of light. including wave, and photon descriptions. In Chapter 2, we describe a variety of common sources and detectors of as well as the radiometric and units of measurement that are used throughout the book. In this and the remainder of the text, the rationalized MKS system of units is 3 reviews the geometrical optics covered by inthe usual reflection and refraction relations for mirphysics courses, rors and lenses. Chapter 4 shows how one can extend paraxial optics to of amicomplexity through the use of 2 x 2 matrices. Also in this we include an introduction to the ray-tracing that are widely applied computer programming. Chapter 5 presents a semiquantitative treatment of third-order aberration theory. Chapter 6 discusses the of geometrical optics and aberration to apertures and to several devices: the prism, the camera, the "",>ni.""" m:u;rOlscope. and the telescope. The of the eye as the final in many optical systems is in a separate chapter (7). This explains the functions and the defects of the eye and discusses some of the treatments of these of laser light. defects that make use of the with two chapters The next section of the text introduces wave or physical (8 and 9) that discuss the wave and the superposition of waves. Interference nh.>nnlmp,n" are then treated in Chapters 10 and II, the second dealing with both Michelson and Fabry-Perot interferometers in some detail. Although the of coherence is handled in general terms in discussions, it receives a more and treatment in Chapter 12. After a brief explanation of Fourier series and the Fourier integral, the chapter deals with both temporal and spatial coherence and presents a quantitative discussion of partial coherence. Chapter i 3 presents, as a tion of interference, an introduction to holography, including some current aPlpli(;atiions. 14 and 15 treat the of We first give a mathematical 2 x 2 matrices to the electric field vector (Chapter 14), before in detail the mechanisms responsible for the production of p0larized light (Chapter 15). Thus Chapter 14 uses matrices to describe the various modes of and types of without reference to the physics of its u,,",,,,,,,,,nUlIU'll';U the order of these can be we feel this choice is more effective. Diffraction is discussed in the following three chapters ]7, 18). Since an adequate treatment of Fraunhofer diffraction is too long for a we have included a separate chapter (17) on the diffraction grating and

Preface

instruments following the discussion of diffraction in Chapter 16. Fresnel diffraction is then taken up in 18. The final chapters are generally more demanding in mathematical sophistication. 19 2 x 2 matrices to treat reflectance of thin films. Chapter 20 derives the Fresnel equations in an examination of reflection from both dielectric and metallic surfaces. The basic elements of a laser and the basic characteristics of laser are treated in Chapter 21, followed by a rather chapter (22) that describes the features of laser beams. The and mode structure of laser beams are dealt with here in a 21 and 22 are best taken in sequence, and together with Chapter 23, an essay on laser applications, form a suitable unit for a minicourse on lasers. The other chapters in this final part of the book are self-contained in the sense that no sequence is ,",U'''P'''''' 24 presents a survey of the basic features of fibers with special attention to communication applications. Thus of bandwidth, allowed 25 introduces the and mechanisms of attenuation and distortion are treated here. of Fourier in a discussion of optical data nr"C'"·",,, Chapter 26 presents a variety of effects under the umbrella of nonlinear The final chapter (27) considers the propagation of a light wave in both dielectric and metallic media and shows how the optical constants arise. Each of the 27 chapters contains a limited bibliography related to the chapter contents and referred to at times within the text square brackets. In addition, at the end of the book, we have included a chronological listing of articles related to optics that have in Scientific American over the I&<;t 40 years or so. It is hoped that this list of excellent articles will prove helpful, to the undergraduate student. This text is intended to be for either one or two semester sequences. The selection of material will on the goals of both teacher and student. As a rough guide, however, a typical one-semester course might include the basic sequenee: Chapter 1 Nature of Light 3 Geometrical Optics 6 Optical Instrumentation 8 Wave Equations 9 Superposition of Waves 10 Interference of Light 12 Coherence 13 Holography 15 Production of Polarized 16 Fraunhofer Diffraction ]8 Fresnel Diffraction 21 Laser Basics As a further aid to selection, those sections that could be omitted in abbreviated versions of the course are marked with an asterisk. See the Contents. We wish to thank the many teachers who have inspired us with an interest in optics and in teaching and the many students who have motivated us to teach with and For their very helpful of portions of the for the first """""V,,,, we are indebted to Hugo James Tucci, Hajime Sakai, Arthur H. Guenand Thomas B. Greenslade. For their suggestions for improving the second ediwhich we have considered very we wish to thank the team of reviewers selected by Prentice Hall: Joel Blatt, Florida Institute of Technology; James Ore-

Preface

xxi

gon Institute of Technology~ Harry Daw. New Mexico State University; Edward University of Delaware; and Daniel Wilkins, of Nebraska. We are grateful to Leno M. Pedrotti for his critical reading of most of the new material added in this second edition. For his review and in the chapter on the eye, we are also to acknowledge and thank Dr. Michael Pedrotti, 0.0. We also wish to thank Lawson for her sketch of Einstein that graces page I. Finally, we express our tude to the editorial and production staff of Prentice HaiL In particular. we are indebted to our editors, Holly Hodder for the first edition and Ray Henderson for the second and to our production editor, Kathleen Lafferty, who helped us with both editions of this text.

Frank L. Pedrotti, S.l. Leno S. Pedrotti

xxii

Preface

0-13-016973-0

This edition may be sold only in those countries to which it is consigned by Prentice-Hall International. It is not to be and it is not for sale in the U.S.A., Mexico, or Canada.

© 1993, 1987 by Prentice-Hall, Inc. A Simon & Schuster Company Englewood Cliffs, NJ 07632

AU rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher.

Printed in the United States of America 1098765432

ISBN 0-13-016973 0 Prentice-Hall International (UK) Limited, London Limited, Prentice-Hall of Australia Prentice-Hall Canada Inc., Toronto Prentice-Hall Mexico Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc., Tokyo Simon & Schuster Asia Pte., Ltd., "''''',,"u,vu.c Ediwra Prentice-Hall do Brasil, Ltda., Rio de Janeiro New Jersey Prentice-Hall, Inc., Englewood

on nts

list of Tables

xv

Physical Constants xix

1

Nature of light

.,

Introduction 1 A Brief Problems 6 References 7

2

2

Production and Measurement of light Introduction

8

8 8

2-1 2-2 2-3

2-4 *2-6

15 Blackbody ~""'''lQ'''V'' Sources Radiation Detectors of Radiation 23 Problems 28 References 29

17

vii

3

Geometrical Optics 3-1 3-2 3-3

34 3-5 3-6 3-7

3-8 3-10 3-11

4

Matrix Methods in Paraxial Optics 4-1 4-2 4-3 4-4

4-5 4-7 4-8

4-9 4-10 *4-U

.5

5-3 5-4 5-5 5-6

5-7

Contents

62

Introduction 62 62 The Thick Lens 65 The Matrix Method The Translation Matrix 66 The Refraction Matrix 66 The Reflection Matrix 67 Thick-Lens and Thin-Lens 68 Ray-Transfer Matrix 71 Significance of System Matrix Elements 72 Location of Cardinal Points for an Optical System 74 Examples the System Matrix and Cardinal Points 77 Ray Tracing 79 Problems 84 References 86

Aberration Theory 5-1 *5-2

viii

30

Introduction 30 Huygens' Principle 31 Fennat's 34 of Reversibility 36 Reflection in Plane Mirrors 37 Refraction through Plane Surfaces 38 Imaging by an Optical System 40 Reflection at a Spherical Surface 43 Refraction at a Spherical 47 Thin Lenses 50 Velrgeince and Refractive Power 55 Newtonian Equation for the Thin Lens 57 Problems 58 References 60

87

Introduction 87 Ray and Wave Aberrations 88 Third-Order Treatment of Refraction at it Spherical Interface 89 Spherical Aberration 93 Coma 96 Astigmatism and Curvature of Field 98 100 Distortion Chromatic Aberration 102 Problems 106 References 107

6

Opticailnstrumentation 6-1 6-2 6-3 6·4 6-5 6-6

Introduction Stops,

8

The Camera 125 Simple Magnifiers and Eyepieces 135 139 Problems 146 References 149

9

151

151 Introduction 152 Biological Structure of the Optical of the 153 155 Functions of the Eye Errors of Refraction and Their for Ocular Defects Laser 163 169 Problems References 171

Wave Equations 8-1 8-2 8-3 8-4 8-5 8-6 8-7 *8-8

Waves

Introduction

9-3 9-4 9-5

Contents

158

172

172 One-Dimensional Wave 172 174 Hannonic Waves Complex Numbers 177 Waves as Complex Numbers 178 Plane Waves 180 Waves 181 Electromagnetic Waves Effect 183 184 186 References

Superposition 9-1 9-2

109

~sms

Optics of the Eye 7-1 7-2 7-3 7-4

., 09

109 and Windows

178

187

187

SUiJenDOSiti(J~n ~nciple

187 lDerocIsttion of Waves of the Same 188 Random and Coherent 191 Standing Waves 192 Phase and Group Velocities 194 197 199 References ix

10

Interference of light 10-1 10-2 10-3 10-4 10-5 10-6 *10-7

200

Introduction 200 Two-Beam Interference 200 Young's Double-Slit Experiment 205 Double-Slit Interference with Virtual Sources Interference in Dielectric Films 211 of Equal Thickness 2J 5 Newton's 217 Film-Thickness Measurement by Problems 221

209

219

223

11

Optical Interferometry 11-1 11-2 11-3 *114 *11-5 *11-6 *U-7 *11-8

12

Coherence 12-1 12-2 12-3 *124 12-5 12-6

13

224

Introduction 224 The Michelson 225 Applications of the Michelson Interferometer 228 Variations of the Michelson Interferometer 230 Stokes Relations 232 Multiple-Beam Interference in a Parallel Plate 233 Fabry-Perot Interferometer 236 Fringe Profiles: The Function 238 Power Free Spectral 242 Problems 244 References 246

Introduction 247 Fourier Analysis 248 Fourier Analysis of a Finite Hannonic Wave 251 Train Temporal Coherence and Natura] Line Width Partial Coherence 254 Coherence 259 Spatial Coherence Width 260 Problems 263 References 265

Holography 13-1 13-2 13-3 134

Contents

253

266

Introduction 266 Conventional versus Holographic Photography Hologram of a Point Source 267 Hologram of an Object Hologram Properties 273 White-Light Holograms 273

266

Appli(;ati1ons of Holography

274

References

14

Matrix Treatment of Polarization 14-1 14-2

1

Production of Polarized light IS-I

15-2 15-3 15-4 15-5 *15-6 *15·7

16

17

323 323

Introduction Diffraction from a Slit 324 Spreading 329 Circular Rectangular 330 Resolution 335 Double-Slit Diffraction 338 Diffraction from Many Slits 341 346 348

The Diffraction Grating 17-1 17-2 17-3 *17-5 Contents

298

298 Polarization by ...:.. I",..·,t'"'' Absorption 298 Polarization Reflection from Dielectric Surfaces by 303 Polarization with Two ..,..",. ..~'u Indices 306 Double Refraction 310 Optical Activity 313 Photoelasticity 317 Problems 319 322

Fraunhofer Diffraction 16·1 16-2 16-3 16-4 16-5 16-6

280

Introduction 280 Mathematical Representation of Polarized Light: 281 Jones Vectors Mathematical Representation of Polarizers: Jones Matrices Problems References 297

349

Introduction 349 The Grating Equation 349 Free Spectral Range of a Grating of a Grating 352 Resolution a Grating Types of 355 xi

*17-6 *17-7 *17-8 *17-9

18

Fresnel Diffraction 18-1 18-2 18-3 184 18-5 18-6 18-7 18-8 18-9 18-10

19

356 359 Grating Replicas Interference Gratings 359 361 Grating Instruments Problems 363 References 365

366

Introduction 366 Fresnel-Kirchhoff Diffraction Integml 367 Criterion for Fresnel Diffraction 369 The Obliquity Factor 370 Fresnel Diffraction from 370 Phase Shift of the Diffracted Light 374 The Fresnel Zone Plate 374 Fresnel Diffraction from Apertures with Rectangular Symmetry 376 The Cornu Spiral 378 Applications of the Cornu Spiral 382 Babinel's Principle 388 388 390

Theory of Multilayer Films Introduction 19-1 19-2 19-3 194 19-5

20

20-4 *20-5 *20-6 *20-7

xii

Contents

391

Tr.ansfer~atrix

392 Reflectance at Nonnal Incidence Antireflecting Films IreIlecl[mg Films High-Reflectance Layers 402 Problems 405 References 406

Fresnel Equations 20-1 20-2

391

396

398 401

407

Introduction 407 The Fresnel Equations 407 External and Internal Reflections 411 Phase on 414 Conservation of 417 Evanescent Waves 419 Complex Refractive Index 420 Reflection from ~etals 422 Problems 423 References 425

21

laser Basics 21-1 21-2 21-3 21-4 21-5

22

22-222-3 22-4 22-5 22-6 *22-7

427 434

456

Introduction 456 Three-Dimensional Wave Equation and Electromagnetic Waves 457 Phase Variation of Spherical Waves Along a Transverse Plane 459 Basis for Defining Laser-Beam Mode Structures 459 Gaussian Beam Solution for Lasers 461 Spot Size and Radius of Curvature of a Gaussian Beam 464 Laser Propagation through Arbitrary Optical Systems 468 Higher-Order Gaussian Beams 476 480 Problems References 483

laser Applications 23-1 23-2 23-3

24

Introduction 426 Einstein's Quantum Theory of Radiation Essential Elements of a Laser 431 Simplified Description of Laser Operation Characteristics of Laser Light 440 Laser Types and Parameters 451 Problems 453 References 454

Characteristics of laser Beams 22-1

23

426

484

Introduction 484 Lasers and Interaction 485 Lasers and Information 491 More Recent Developments 496 Problems 498 500 References

Fiber Optics 24-1 24-2 24-3 24-4 24-5 24-6 24-7

Contents

501 Introduction 501 501 Applications Communications System Overview 504 Bandwidth and Data Rate 505 Optics of Propagation 507 Allowed Modes Attenuation 509 Distortion 512 519 Problems 521 References

502

xiii

25

fourier Optics 25-1 25-2

26

Nonlinear Optics and the Modulation of light 26-1 26-2 26-3 264 26-5 26..6 26-7 26-8

27

Introduction Optical Data Imaging and Processing Spectroscopy Problems 539 References 540

Introduction 541 The Nonlinear Medium 542 543 Second Harmonic Generation Frequency 546 The Pockels 547 551 553 The Effect 556 Nonlinear Optical Phase Conjugation Problems 565 References 567

Optical Properties of Materials

27-1 27-2 27-3 27-5 27-6

559

568

Introduction 568 Polarization of a Medium 568 Propagation of Waves in a Dielectric Conduction in a Metal 575 Propagation of Skin Depth 577 Plasma Frequency 578 Problems 579 References 580

Suggestions for further Reading

581

Articles on Optics from Scientific American

Answers to Selected Problems Index

xiv

Contents

541

597

585

581

List of Tables

Table 2-1 Table 3-1 Table 4-1 Table 4-2 Table 4-3 Table S-I Table 6-1 Table 6-2 Table 7-1 Table 14-1 Table 14-2 Table 15-1 Table 15-2 Table IS-3 Table 18-1 Table 19-1 Table 19-2 Table 21-1 Table 21-2

Radiometric and Photometric Tenns 10 Summary of Gaussian Mirror and Lens Fonnulas 54 Summary of Some Simple Ray-Transfer Matrices 70 Cardinal Point Locations in Terms of System Matrix Elements 77 Meridional Ray-Tracing 82 of Optical Glasses 104 Fraunhofer Lines 120 Standard Relative Apertures and (rradiance Available on Cameras 128 Con.<;tants of a Schematic Eye 154 Summary of Jones Vectors 288 Summary of Jones Matrices 293 Refractive Indices for Several Materials 308 Specific Rotation of Quartz 3/4 Refractive Indices for Quartz 3/6 Fresnel 380 Refractive Indices for Several Coating Materials 400 Reflectance of a Stack 403 Comparison of Linewidths 441 Laser Parameters for Severa] Common Lasers 452

Table 23-1 Table 23-2 Table 23-3 Table 24-1 Table 26-1 Table 26-2 Table 26-3 Table 26-4

A Classification of Laser Applications 485 Laser Types in Medicine 489 Medical Fields Involved with Lasers 490 Cbaracterization of Several Optical Fibers 506 Linear and Nonlinear Processes 546 Linear Electro-optic Coefficients for Representative Materials Kerr Constant for Selected Materials 553 554 Verdet Constant for Selected Materials

xvi

list of Tables

549

1

Nature of Light

INTRODUCTION

The evolution in our understanding of the physical nature of light forms one of the most fascinating accounts in the history of scien("'e. Since the dawn of modern science in the sixteenth and seventeenth centuries, light has been pictured either as particles or waves-incompatible models-each of which enjoyed a period of prominence among the scientific community. In the twentieth century it became clear that somehow light was both wave and particle, yet it was precisely neither. For some time this perplexing state of affairs, referred to as the wave-particle duality, motivated the greatest scientific minds of our age to find a resolution to apparently contradictory models of light. The solution was achieved through the creation of quantum electrodynamics, one of the most successful theoretical structures in the annals of physics. In what follows, we will be content to sketch briefly a few of the high points of this developing understanding. I Certain areas of physics once considered to be disciplines apart from optics-electricity and magnetism, and atomic physics-are very much involved in this account. This alone suggests that the resolution achieved also constitutes one of the great unifications in our understanding of the physical world. The final result is that light and subatomic particles, like electrons, are both considI A more in-depth historical account may be found, for example, in [lJ (see References at the end of the chapter).

1

ered to be manifestations of matter or energy under the same set of formal principles. In the seventeenth century the most prominent advocate of a particle theory of light was Isaac Newton, the same creative giant who had erected a complete science of mechanics and gravity_ In his treatise Optics, Newton clearly regarded rays of light as streams of very small particles emitted from a source of light and traveling in straight lines. Although Newton often argued forcefully against positing hypotheses that were not derived directly from observation and experiment, here he adopted a particle hypothesis, believing it to be adequately justified by the phenomena. Important in his considerations was the observation that light can cast sharp shadows of objects, in contrast to water and sound waves, which bend around obstacles in their paths. At the same time, Newton was aware of the phenomenon now referred to as Newton's rings. Such light patterns are not easily explained by viewing light as a stream of particles traveling in straight lines. Newton maintained his basic particle hypothesis, however, and explained the phenomenon by endowing the particles themselves with what he called "fits of easy reflection and easy transmission," a kind of periodic motion due to the attractive and repulsive forces imposed by material obstacles. Newton's eminence as a scientist was such that his point of view dominated the century that followed his work.

A BRIEF HISTORY

Christian Huygens, a Dutch scientist contemporary with Newton, championed the view (in his Treatise on Light) that light is a wave motion. spreading out from a light source in all directions and propagating through an all-pervasive elastic medium called the ether. He was impressed, for example, by the experimental fact that when two beams of light intersected, they emerged unmodified, just as in the case of two water or sound waves. Adopting a wave theory. Huygens was able to derive the laws of reflection and refraction and to explain double refraction in calcite as well. Within two years of the centenary of the publication of Newton's Optics, the Englishman Thomas Young performed a decisive experiment that seemed to demand a wave interpretation, turning the tide of support to the wave theory of light. It was the double-slit experiment, in which an opaque screen with two small, closely spaced openings was illuminated by monochromatic light from a small source. The "shadows" observed formed a complex interference pattern like those produced with water waves. Victories for the wave theory continued up to the twentieth century. In the mood of scientific confidence that characterized the latter part of the nineteenth century, there was little doubt that light, like most other classical areas of physics, was well understood. We mention a few of the more significant confirmations. In 1821 Augustin Fresnel published results of his experiments and analysis, which required that light be a transverse wave. On this basis, double refraction in calcite could be understood as a phenomenon involving polarized light. It had been assumed that light waves in an ether were necessarily longitudinal, like sound waves in a fluid, which cannot support transverse vibrations. For each of the two components of polarized light, Fresnel developed the Fresnel equations, which give the amplitude of light reflected and transmitted at a plane interface separating two optical media. Working in the field of electricity and magnetism, James Clerk Maxwell synthesized known principles in his set of four Maxwell equations. The equations yielded a prediction for the speed of an electromagnetic wave in the ether that turned 2

Chap. 1

Nature of Light

out to be the measured speed of light. suggesting its electromagnetic character. was viewed as a particular region the electromagnetic specFrom then on, trum of radiation. The experiment (1887) of Albert Michelson and Edward M()rle~v which attempted to detect optically the earth's motion the and the special theory of relativity (1905) of Albert Einstein were of monumental importance. Together led inevitably to the conclusion that the assumption of an ether was superfluous. The with transverse vibrations of a wave in a fluid thus vanished. If the nineteenth century served to place the wave theory of light on a firm foundation, this foundation was to crumble as the century came to an end. The we mention wave-particle controversy was resumed with vigor. some of the key events along the way. Difficulties in the wave theory seemed to at show up in situations that involved the interaction of light with matter. In the very dawn of the twentieth Max Planck announced at a meeting of the German that he was able to derive the correct blackbody radiation the curious assumption that atoms emitted light in only by energy chunks rather than in a continuous manner. Thus and quantum mechanics were born. According to Planck, the energy E of a quantum of netic radiation is proportional to the of the v, E

hv

O-I)

where the constant of proportionality, PlaflCk's constant, has the very small value of 6.63 x 10- 34 J-s. years later, in the same year that published his theory of relativity, Albert Einstein offered an explanation of the photoelectric the emission of electrons from a metal surface when irradiated with light. Central to as a stream of whose explanation was the conception of (1 Then in 1913 the Danish related to frequency Planck's of radiation in his explanation of Bohr once more incorporated the the emission and absorption processes of the hydrogen atom, providing a physical basis for understanding the hydrogen spectrum. Again in the photon model of came to the rescue for Arthur Compton, who explained the scattering of from electrons as particlelike collisions between photons and electrons in which both energy and momentum were conserved. indicated that light All such victories for the photon or particle model of could be treated as a particular kind matter, possessing both energy and momenwho saw the other side of the In 1924 he pubtum. It was Luis de are endowed with wave properties. his speCUlations that subatomic that a particle with momentum p had an associated waveHe suggested, in length of h (1 p where h was, Planck's constant. confirmation of de Broglie's hypothesis appeared the years 1927-1928, when Clinton Davisson and Lester Thomson in performed Germer in the United and Sir ments that could only be interpreted as the diffraction of a beam of electrons. Thus the wave-particle duality came full circle. Light behaved like waves in propagation and in the phenomena of interference and diffraction~ it could, however, also behave as particles in its interaction with matter, as in the photoelectric effect. On the other hand, electrons usually behaved like particles, as in the exposed to a beam of electrons; in other situations like scintillations of a were found to behave Hke waves, as in the diffraction produced by an electron

A Brief History

3

Photons and electrons that behaved both as and as waves seemed at first an contradiction, since and waves are very different entities extent through the reflections of Niels Gradually it became clear, to a Bohr and especially in his principle of that photons and electrons were neither waves nor particles, but more complex than either, In to explain it is natural that we appeal to As it turns out, however, the models like waves and '''~JUJ:',JIV''''~' of a photon or an electron is not exhausted by either model. In cerin other situations, particlelike tain wavelike attributes may attributes stand out. We can appeal to no simpler model that is adequate to handle all cases. Quantum mechanics, or wave mechanics, as it is often called, deals with all more or less localized in space, and so describes both light and matter, Combined with special relativity, the momentum p, A., and speed v for both material particles and photons are by the same equations:

p=

(1-3)

c

he

h A.=-= P pe2 v=-= E

(1-4) (1-5)

eQllati!on:s, m is the rest mass and E is the total energy, the sum of rest-mass and kinetic energy, that is, the work done to accelerate the particle from ..,.."..""" ..",rl speed. The relativistic mass is given by ym, where y is the ratio simply ~mv2, . The proper expression for kinetic energy is no but mc 2(y I), The relativistic expression for kinetic energy tmv 2 for

v
(1-6)

A.=!.!=he

(1-7)

pc 2 v=-=c E

0-8)

P

E

while nonzero rest-mass particles like electrons have a of c, Eq. (1 shows that zero rest-mass particles like photons must travel with the constant speed c. The energy of a photon is not a function of its speed but of its frein (1-1) or in Eqs. 0-6) and (1-7), taken Notice quency. as that for a photon, because of its zero rest mass, there is no distinction between its total energy and its kinetic energy. The following example illustrates the pn~ce,all1'~ equations.

2

help of

4

This discussion is not meant to be a condensed tutorial on relativistic mechanics, but, with the (1-3) to (1-8), a summary of some basic relations thai unify of mailer and

Chap. 1

Nature of

Example An electron is accelerated to a energy of 2.5 MeV. Determine its relativistic momentum, de Broglie wavelength, and speed. Also determine the same properties for a photon having the same energy as the electron. Solution The electron's total energy E must be the sum of its rest mass energy and its kinetic energy, = 0.511 MeV

+

2.5 MeV

3.011 MeV

or E

= 3.011

X

1(1' eV

X

(1.602

X

10- 19 J/eV) = 4.82

The other quantities are then calculated in order. From p

From

=

1.58

X

X

10- 13 J

(1-3):

21

10- kg-m/s

(1-4):

A = 41.8

X

10- 12 m

41.8 pm

From Eq. (1-5):

v = 2.95 For the photon, with m

lOS mJs

0, we get instead From p

From

X

= 1.61

X

10

(1-6):

21

(1-7):

A

= 0.412 pm

From Eq. (l v

=

c

3.00

X

108 m/s

Another important distinction between electrons and is that electrons obey Fermi statistics whereas photons obey Bose statistics. A consequence of Fermi be in statistics is the restriction that no two electrons in the same interacting the same physical properties. Bose statistics the same state, that is, have no such prohibition, so that identical photons with the same energy and momentum can occur together in large numbers. Because light beams can possess so many similar photons in proximity, the granular structure of the beam is not ordinarily experienced, and the beam can be adequately represented by a continuous ""U'IaOiU"' •• .., wave. From this of fields appear as a A profound consequence of the wave nature of is embodied in the Heisenberg principle of indeterminacy. As a result of this principle, particles do not predicts probabilities. obey deterministic laws of motion. Rather, the Wave functions are associated with the particles through the fundamental wave equaor. the square of the wave tion of quantum mechanics. The wave amplitudes to these provide a means of expressing the probability that a particle will be found within a region of space during an interval of time. Thus the irradiance (power/area) of these waves at some intercepting surface, also proportional to the square of the wave amplitudes, provides a measure of this probability. so When numbers of particles are involved, probabilities of light at a location is proportional to the number of photons that the irradiance ""~_'V"",", through the location per second. 2 Ee n (photons/m -s) = hv

A Brief

0-9) 5

In this way, the interference and diffraction patterns explained by waves can be interpreted as manifestations of particles. V<>r'tu'!I", wave amplitudes predict the probabilities of their locations in the same patterns. In the theory called quantum electrodynamics, which combines the pnncllpl(~ of quantum mechanics with those of special relativity, are assumed to interact only with An for example, is of both absorbing and emitting a photon, with a probability that is proportional to the square of the charge. There is no conservation law for photons as there is for the charge associated with particles. In this the duality reconciled. Essential distinctions between and electrons are removed. Both are considered subject to the same principles. Through this unification, light is viewed as basically just another form of matter. Nevertheless. the aspects of "Slr'firl", and wave of light remain, justifying our use of one or the other description when The wave description of light will be found adequate to describe most of the optical phenomena treated in this text.

PROBLEMS 1-1. Calculate the de of (a) a golf ball of mass 50 g moving at 20 mls and (b) an electron with kinetic energy of 10 eV. 1-2. The threshold of of the human eye is about 100 photons per second. The of around 550 nm. For this determine eye is most sensitive at a the threshold in watts of power. 1-3. What is the energy, in electron volts, of light photons at the ends of the visible spectrum, that is, at of 400 and 700 nm? 1-4. Determine the wavelength and momentum of a photon whose energy the restmass energy of an electron. 1-5. Show that the rest-mass energy of an electron is 0.511 MeV. 1-6. Show that the relativistic momentum of an electron, accelerated a potential difference of I million can be conveniently as 1.422 MeV/c, where c is the speed of light. 1-7. Show that the of a photon, measured in angstroms, can be found from its the convenient relation, energy, measured in electron 12,400

E(eV)

1-8. Show that the relativistic kinetic energy,

Ek

mc 2(')'

-

I)

reduces to the classical , when t; ~ c. Find (a) 1-9. A proton is accelerated to a kinetic energy of 2 billion electron volts it'> momentum; (b) its de wavelength; (c) the wavelength of a with the same total energy. 1·10. Solar radiation is incident at the earth's surface at an average of 1353 W/m2 on a surface normal to the rays. For a mean of 550 nm, calculate the number of photons falling on 1 cm 2 of the surface each second. 1-11. Two parallel beams of radiation with different deliver the same power to equivalent surface areas normal to the beams. Show that numbers of photons striking the surfaces per second for the two beams are in the same ratio as their wavelengths.

6

Chap. 1

Nature of light

IIEFERENCES [I] Ronchi, vasco. The Nature Cambridge: Harvard '"nIP..."ru Press~ 1970~ [2] Hoffmann, Banesh. The Strange Story of the Quantum. New York: Dover Publications, 1959. vuantum Mechanics. New York: Charles Scribner's [3J

Feinberg, Gerald. "Light." In Lasers and Light, pp. 4-13. San Francisco: W. H. Freeman and Publishers, 1969. Newton. [5] Cantor, G. N. Optics N.H.: Manchester University Press, 1983. [6] John. Concepts ofClassicai San Francisco: W. H. Freeman and Company Publishers, 1958. Appendix G.

Chap. 1

References

1

2

Production and Measurement of Light

INTRODUCTION

Electromagnetic radiation may vary in (or frequency) and in is summarized in the electromagnetic Classification due to variation in spectrum. Variations in are described in more precise physical terms, which have developed in the areas caned radiometry and photometry. Sources and detectors of electromagnetic radiation can be classified on the basis of their spectral range and the strength of produced (sources) or (detectors). These COllSldlerations are essential to the production and measurement of electromagnetic and are discussed in this chapter.

2-1 ELECTROMAGNETIC SPECTRUM

An electromagnetic disturbance that propagates through space as a wave may be monochromatic. that characterized for practical purposes by a single wavelength, wavelengths, either disor polychromatic, in which case it is repre..<;entoo by

crete or in a continuum. The distribution of energy among the various constituent waves is called the spectrum the radiation, and the spectral implies a dependence on wavelength. Various regions of the spectrum are referred to by particular names, such as radio waves, cosmic light, and ultraviolet or detected. Most of radiation, because of differences in the way they are 8

the common are given in Figure 2-1, in which the electromagnetic is displayed in terms of both wavelength (A) and frequency (v). The two quantities are related, as with all wave motion, through the wave vel()(.'ity (c):

c = Av 1) The radiation in Figure 2-1 is assumed to propagate in free space, for 3 x lOS m/s. Common units for wavelength shown are which, approximately, c m), the nanometer (1 nm m), and the micromethe angstrom (1 A = ter (1 p,m = 10- 6 m). The regions ascribed to various types of waves, as shown, are not precisely bounded. Regions may overlap, as in the case of the continuum from on the manner in which the X-rays to gamma rays. The choice of label will I

1m)

I

to- 15

I

(Hz)

3 X 102"

I

I GAMMA! RAYS I

10-14

3 X 1022

I

I 10-12

-

I

I X-RAYS

10-10

1A 1 nm

3 X 10:11)

I

I

I

I

I I I

3 X 10'8

I

I

V

,(Vacuum)

10-8

380nm

B G

1 I'm

10-8

X 10'4

I

OPTICAL

IIR 10-4

I

I

710nm

I

I

to-2

0 3 X 10'2

I

t (Far) 1 em

~ I

I

Ill-WAVES I (radar)

spectrum Y

3 X 1010

I I I (UHF-TV)'

1m

I

3 X lOS

(VHF-TVI,

IFM-radioll 102

I

I

3 X lOS

(AM-radio) I

1 km

I

10"

l,WAVES RADIO

3 X 10"

I l000km

tOS

I I

I

3 X 102

(ACpowerl'

I

lOS

I I I

3

I

Figurel-l Electromagnetic spectrum, arranged by in meters and frequency in hertz. The narrow portion by I he visible spectrum is high-

Sec. 2-1

Electromagnetic Spectrum

9

radiation is either produced or used. The narrow range of tromagm~tic waves from approximately 380 to 770 nm is of n ..cvl ..... n,n sensation in the the human eye and is properly referred to as which includes the spectrum of colors from red (short-wavelength end) is bounded by the invisible ultraviolet and Intl"nrl"£1 ....'oi."..." as shown. The three regions taken the gion of the eJectromagnetic in a textbook of

2-2 RADIOMETRY

Radiometry is the science of measurement of radiation. In the discussion we present the radiometric or physical terms used to characterize the energy content of radiation. Later we briefly discuss some of the more common principles used in the instruments to measure radiation. Many radiometric however we include here only approved Interterms have been introduced and national System (SI) units. These terms and units are summarized in Table 2-1. 1 TABLE 2-1

RADIOMETRIC AND PHOTOMETRIC TERMS

Term Radiant energy Radianl energy den.<;ily

Symbol (units)

Symbol (units)

Term

Q. (J = W-s)

w, (JIm')

Radiant flux Radiant exitance Irradiance

IP.(W) M, (W/m2) Ec (W/m2)

Radiant intensity

Ie (W/sr)

Luminous energy Luminous energy

We

IP, dQc/tit Me = dcP./dA E. =

Luminous flux Luminous exitance Illuminance

Defining equation

Qt, (Im-s) (talbot) We

(Im-s/m')

cPo (1m) Mv (lm/m2) Eu (1m/m 2 )

w" = dQ./dV

cPo = dQv!dt Mv = dcPv/dA E. = dIP./dA

or (Ix)

Radiance

L, (W/sr-ml )

I.

Le

Luminous intensity

dcPe/dw

cos

e

1< (Im/sr)

I. = dcPJdw

or (cd) Luminance

Lv

Lv = dlv/dA cos

e

Abbreviations: J, joule; W, watt; m, meter; 1m, lumen; Ix, lux; sr, steradian; cd, candela.

Radiometric quantities appear either without or with the subscript e (electromagnetic) to them from similar terms, to be described afterwards. The terms radiant energy, (J radiant energy density, We (J/m 3), and radiant (W watts J/s), need no further explanameasured in watts per square meter, may be tion. Radiant flux density at a either emitted (scattered. a in which case it is called radiant exilance, Me. or onto a in which case it is called irradiance, The radiant flux (
d
I

10

(2-2)

The introduction "in the abstract" of so many new units, some rarely used and others misused, is P""""UU; !C""U"!'UC"W""Y. Table 2-1 is meant to serve as a convenient summary that can be re-

2

Production and Measurement of light

2-2 The radiant intensity is the flux the cross section dA per unit of solid Here the solid angle dw dA/r 2 •

where sr = steradian. The radian intensity Ie a radiating W of power uniformly in all directions, for example, is 4le l41r Wlsr, since the total surrounding solid angle is 471" sr. law of radiation a point source, in The familiar Figure 2-3, is now apparent by calculating the irradiance of a point source on a spherical surface surrounding the point, of solid angle 471" sr and surface area 41rr2. Thus E

= e

41"

(2-3)

A

2-3 Illustration of the inverse square law. The flux leaving a point source within any solid is distributed over IIllOlct:,,,.ngIY larger areas, producing an irradiance decreases inversely with the square of the distance.

The describes the intensity per unit of projected area, perpendicular to the specified direction, and is given by Le

dA cos 0

dw(dA cos 0) (W/m2-sr)

(2-4)

The importance of the radiance is in the following Suppose a plane or reflector is diffuse, by which we mean that it radiates uniformly in all directions. The is measured for a fixed solid angle defined by the fixed aperture Ap at some distance r from the radiating surface, shown in 2-4. The aperture might be the input aperture of a detecting instrullr<;a.:>I.l'UIl'l', an the flux that so enters. When viewed at 0 0°, along the nora certain I (0) is observed. As the aperture is of radius r, the 0, the cross section pn:selnte:d by the surface decreases in such a way I (0)

I (0) cos 0

(2-5)

a relation called Lambert's cosine law. If the radiance is determined at each angle 0, it is found to be constant, because the intensity must be by the projected

Sec. 2-2

Direction of IIi ewing Radiating

surface A

\

Ap

\ \ \

Normal

I I I I Projected surface A cos 6

I

I

Figure 2-4 Radiant flux collected a direction making an 6 with the normal to the radiating surface. area of the surface is shown shaded.

area A cos 9 such that the cosine dependence cancels: 1(9)

1(0) cos 9

1(0)

Acos9

Acos9

A

--'--'---- = - -

constant

Thus when a

(or reflecting) surface has a radiance that is of the the surface is said to be perfectly diffuse or a Lambertiml surface. We show next that the radiance has the same value at any point along a ray in a uniform, nonabsorbing medium. Figure illustrates a narrow radiation in such a medium, including a central ray and a bundle of VU1.UU.f', rays that pass through the elemental areas dAI and dAz situthe beam. The central ray makes of 91 and 92 ,

6, Central ray

dw,

2-5 Geometry used to show the invariance of the radiance in a uniform, lossless medium.

relative to the area normals, as shown. The cos 921r 2 , where cos 92 represents the projection of area central ray. According to Eq. (2-4), the radiance LJ at dAI is

=

dw l = normal to the by

= ------:-'-----

a similar ar,g;UITlenl, in which we reverse the roles of dA] dw 2(dA 2 cos ( 2)

(dAI cos

For a

(2-7)

in the (2-8)

ITledium, the power associated passing the bundle of rays remains constant, that so that we can conclude from Eqs. (2-7) and (2-8) that Ll = . It •." ..".,.,", that the radiance

thr,"'.. .,'h

12

Chap. 2

Production and Measurement of light

of the beam is also the radiance of the source, at the initial point of the beam, or LI = L2 = Lo. Suppose, referring to Figure 2-6, that we wish to know the quantity of radiant power reaching an element of area dA 2 on surface S2 due to the source element dA 1 on surface SI. The line joining the elemental areas, of length r12, makes angles of 01 /

/

Figure 2·6 General case of the iIlwnination of one surface by another radiating surface. Each elemental radiating area dA I contributes to each elemental irradiated area dA 2 •

S1

and O2 with the respective normals to the surfaces, as shown. The radiant power is d 2 <1>12, a second-order differential because both the source and receptor are elemental areas. By Eq. (2-7) or Eq. (2-8), ?h

d 2 '¥12

_

LdA 1dA 2 cos 0 1 cos O2

-

2

r 12

and the total radiant power at the entire second surface due to the entire first surface is, by integmtion, <1>12

=

ff

Al A2

L cos 01

c~~ lJz dA]

dA 2

(2-9)

12

By adding powers mther than amplitudes in this integmtion, we have tacitly assumed that the radiation source emits incoherent radiation. We shaH say more about coherent and incoherent radiation later.

2-3 PHOTOMETRY

Radiometry applies to the measurement of all radiant energy. Photometry, on the other hand, applies only to the visible portion of the optical spectrum. Whereas radiometry involves purely physical measurement, photometry takes into account the response of the human eye to radiant energy at various wavelengths and so involves psycho-physical measurements. The distinction rests on the fact that the human eye, as a detector, does not have a "flat" spectral rsponse; that is, it does not respond with equal sensitivity at all wavelengths. If three sources of light of equal radiant power but radiating blue, yellow, and red light, respectively, are observed visually, the yellow source will appear to be far brighter than the others. When we use photometric quantities, then, we are measuring the properties of visible radiation as they appear to the normal eye, rather than as they appear to an "unbiased" detector. Since not all human eyes are identical, a standard response has been determined by the International Commission on Illumination (CIE) and is reproduced in Figure 2-7. The relative response or sensation of brightness for the eye is plotted versus wavelength. showing that peak sensitivity occurs at the "yellow-green" wavelength of 555 nm. Actually the curve shown is the luminous efficiency of the eye for photopic vision, that is, when adapted for day vision. For lower levels of illumination, when adapted for night or scotopic vision, the curve shifts toward the green, peaking at 510 nm. It is interesting to note that human color sensation is a function of iUumiSec. 2-3

Photometry

13

Figure 2·7 CIE luminous efficiency curve. The luminous flux to I W of radiant power at any wavelength is by the product of 685 1m and the luminous at the same (A) = 685V(A) for each watt of radiant power.

nation and is totally absent at lower levels of illumination. One confirm this is to compare the color of stars, as appear visually. to graphic images on color film using a suitable time exposure. dramatic way to human color on illumination is to LIn"",." 35-mm color slide of a scene onto a screen with a low current in the projector At sufficiently low currents, the scene appears black and white. As the current is increased, the full in the scene gradually emerge. On the other tense radiation may be visible beyond the of the CIE curve. The .."",""'."". . an intense laser beam 694.3 nm from a ruby laser is easily seen. Even the infrared radiation around 900 nm from a gallium-arsenide semiconductor laser can be seen as a red. Radiometric are now related to photometric quantities the luminous efficiency curve of Figure 2-7 in the way: Corresponding to a rawavelength of 555 nm, where the luminous ...rrl""'nr'v diant flux of 1 W at the is maximum, the luminous flux is defined to 685 1m. Then, for at A = 610 nm, in the range where the luminous is 0.5 or 50%,1 W ofradiant flux would only 0.5 x 685 or 3421m luminous flux. The curve shows thai again at A 510 nm, in the blue-green, the brightness has dropped to 50%. Photometric units, in terms of their definitions, parallel radiometric units. is amply demonstrated in the summary and comparison provided in Table 2-1. In analogous are related by the following eOllatlon: photometric unit

K(A) x raCllornetrlC unit

10)

where K (A) is called the luminous efficacy. If V (A) is the luminous efficiency, as on the CIE curve, then 14

Chap. 2

Production and Measurement of light

K(A)

= 685V(A)

(2-11)

Photometric terms are preceded by the word luminous and the corresponding units are subscripted with the letter v (visual); otherwise the symbols are the same. Notice that the SI unit of luminous energy is the talbot, the unit of luminous incidence is the lux (Ix), and the unit of luminous intensity is the candela (cd). Notice also the distinction between the analogous terms irradiance (radiometric) and illuminance (photometric). Example A light bulb emitting 100 W of radiant power is positioned 2 m from a surface. The surface is oriented perpendicular to a line from the bulb to the surface. Calculate the irradiance at the surface. If all 100 W is emitted from a red bulb at A = 650 run, calculate also the illuminance at the surface. Solution irradiance Ee = PIA = 100 W/41r (2 ftl From the CIE curve, V (650 run) =

== 2 W/m2

o. 1. Thus

illuminance E" = K(A) x irradiance = 685 V(A) x E. E" = 685 x 0.1 x 2 = 137 Im/m2 or lux Thus, whereas a radiometer with aperture at the surface measures 2 W/m2, a photometer in the same position would be calibrated to read 137 Ix. When the radiation consists of a spread of wavelengths, the radiometric and the photometric terms may be functions of wavelength. This dependence is noted by preceding the term with the word spectral and by using a subscript A or adding the A in parentheses. For example, spectral radiam flux is denoted by
2-4 BLACKBODY RADIA nON

A blackbody is an ideal absorber: An radiation falling on a blackbody, irrespective of wavelength or angle of incidence, is completely absorbed. It follows that a blackbody is also a perfect emitter: No body at the same temperature can emit more radiation at any wavelength or into any direction than a blackbody. Blackbodies are approached in practice by blackened surfaces and by tiny apertures in radiating cavities. An excenent example of a blackbody is the surface formed by the series of sharp edges of a stack of razor blades. The array of blade edges effectively traps the incident light, resulting in almost perfect absorption. The spectral radiant exitance MA of a blackbody can be derived on theoretical grounds. It was first so derived by Max Planck, who found it necessary to postulate quantization in the process of radiation and absorption by the blackbody. The result of this calculation [I] is given by MA =

Sec. 2-4

Blackbody RadiatiC!n

2:~2 ChclAk~ _ I)

(2-12)

15

where the physical constants c, and k represent the Planck constant, speed of light in vacuum, and the Boltzmann constant, respectively. When the known values of these constants are used, the result is _ 3.745

M" -

1

X lOll (

el431l81AT _

) 2 1 (W/m -/-Lm)

where A is in micrometers and T is in Kelvin. The quantity MJ. is plotted in Figure 2-8 for different temperatures. The spectral radiant exitance is seen to increase

,

10E1 I

t

I I

I I I ,

,

I I I

I I

I I

..,'"c: e ';("E ::: ::t :;; I

.-1\l '"E

--

~ -3:

~-

'"a.

(/)

5E1

I I I I

I I

I

I

I

I

,t 1

I I

I

Wavelength (lIm)

Figure 2-8 Blackbody radiation spectral distribution at four Kelvin temperatures. The vertical dashed lines mark the visible spectrum, and the dashed curve connectthe of the four curves illustrates the Wien displacement law (5£7 = 5 x 107).

absolute temperature at each wavelength. The peak also shifts toward shorter with increasing falling into the visible spectrum (dashed vertical lines) at T = 5000 and 6000 K. The variation of , the wavelength at which MA peaks, with the temperature can be found by differentiating M" with to A and setting this equal to zero. The result is the Wien displacement /aW,2 by he

T = 5k

3

2.88 x 10 (!-Lm-K)

(2-13)

2 Although Wien's law is often found written in this form, the number 5 is an approximation to 4.965 that. when used, gives a more accurate Wien constant of 2.898 x 103

16

Chap. 2

Production and Measurement of

and is indicated in 2-8 by dashed curve. If, on the tral exitance of 12) is integrated over all wavelengths. tance or area under the blackbody radiation curve at temperature T is M

ur

exi(2-14)

known as the Stefan-Boltzmann law, with u as the Stefan-Boltzmann constant, equal to 5.67 x 10- 8 >N",n-_IK The from real surfaces is always less than that the blackbody or Planckian source and is accounted for quantitatively by the em".',''''''' E. Distinguishing now between the radiant exitance M of a measured and that of a blackbody Mbb at the same temperature, we define M

(2-15)

Mbb

If the radiant of the blackbody and the specimen are compared in various narrow intervals, a emissivity is calculated, which is not in general a constant. In those special cases where the emissivity is independent of wavelength. the is said to be a In this instance spectral exitance of the is proportional to that of the blackbody and curves are the same except a constant factor. The radiation a heated tungsten wire, for example, is close to that of a graybody with E 0.4-0.5. Blackbody radiation is used to establish a color scale in terms of absolute temperature The color temperature of a specimen of is then the temperature of the with the closest energy distribution. In this way, a flame can said to have a color of 1900 K, whereas the sun has a kal color of 5500 K.

2-5 SOURCES OF OPTICAL RADIATION Sources of may be natural, as in the case of sunlight and or as in the case of incandescent or lamps. from various sources may as monochromatic, spectf'dlline, or continuous. The way in which also be energy is in the radiation determines the color of the light and, consequently, the color of surfaces seen under the Jight. Anyone who has used a camera is aware the actual color response of depends on the of light used to illuminate the subject. The following brief survey of sources of light cannot hope to be comprehensive; rather it is intended to direct attention to an extensive area of practical int;nrt1n",_ lion. For the purposes of this limited survey, we classify a number of sources as lows: A. Sunlight, skylight B. Incandescent sources 1. Blackbody sources 2. Nernst glower and 3. Tungsten ."'"....." .. C. Discharge lamps 1. Monochromatic and spectral sources 2. High-intensity sources a. Carbon arc b. Compact short arc Sec. 2-5

Sources of Optical Radiation

11

c. Flash d. Concentrated zirconium arc 3. Fluorescent lamps D. Semiconductor light-emitting diodes (LEOs) E. Coherent source-laser Daylight is a combination of sunlight and skylight. Direct from the sun has a spectral distribution that is clearly from that of skylight, which has a predominantly blue hue. A plot spectral solar is given in Figure 2.4r--~-----c-r------------r------------r----------~

Ultraviolet

Visible

i=vtrAt....""tri.,,1

1.6

sun

Sea-level sunlight 1M = 1 air mass)

., t)

<: II)

'i'i

.~ E ts.,

0.8

a.

(/)

0.0 0.2

0.8

1.4 Wavelength (,urn)

2.0

2.6

Figure 2-9 Solar irradiation above the atmosphere and on a horizontal surface at sea level: clear day, sun at zenith.

Extraterrestrial solar radiation indicates that the sun behaves approximately as a of 6000 K at its center and K at edge, but the blackbody with a radiation at the earth's surface is modified by absorption in the earth's atmosphere. The annual average of total irradiance just outside the earth's atmosphere is the solar constant, 1350 W/m 2 • Although solar radiation is not routinely used as a xenon lamps, with appropriate light source in the laboratory, provide an excellent artificial source for solar simulation and are commerArtificial optical sources that use light produced by a material heated to incandescence by an electric current are caned incandescent lamps. Radiation arises from the de-excitation of the atoms or molecules of the material after have been thermally excited. The energy is emitted over a brood continuum wavelengths. Commercially available blackbody sources consist of equipped with a small hole. Radiation from the small hole has an emissivity that is essentially constant and equal to unity. Such sources are available at operating temperatures from that of liquid nitrogen 196°q to 3QOO0 c. Incandescent sources particularly useful in the infrared include the Nernst glower. This source is a cylindrical tube or rod of refractory maby an electric current and useful from the terial (zirconia, yttria, thoria), 18

Chap. 2

Production and Measurement of Light

ible to around 30 /-Lm. Nemst glower behaves like a with an "'.."''',, .. greater than 0.75. When the material is a rod of bonded silicon carbide, the source is a globar, approximating a graybody with an average emissivity of 0.88 (see Figure 2-10). The filament lamp is the most popular source for optical instrumentain the and into the infrared region. tion designed to use continuous in a wide variety of filament configurations and of bulb and The lamp is base shapes. The filament is in coil or ribbon form, the ribbon providing a more

Typical spectral irradiance from bare element per 1O-mm2 area

o

2

4

6

8

10

12

14

16

18

20

22

24

26

28

Wa\lelength Illm)

Figure 2·10 Globar jnfrared source, providing continuous usable emission from 1 10 over 25 /-Lm al a temperature variable up to 1000 K. The source is a 6.2-mm diameter silicon carbide resistor. (Oriel Corp., General Stratford, Conn.)

form surface. The bulb is usually a envelope, quart2 is used for higher-temperature operation. Radiation over the spectrum emissivities approaching unity for tightly coiled filaments. that of a graybody, Lumen output depends both on the temperature and the electrical power input (wattage). During operation, tungsten gradually evaporates from the filament and deposits on the inner bulb surface, leaving a dark film that can decrease the flux outthe life of the This also weakens the put by a<; much as 18% resistance. The presence of an filament and increases its pressure, to slow nitrogen or argon, introduced at around 0.8 down the evaporation. More recently problem ha<; been minimized by the addia halogen vapor (iodine, bromine) to the gas in the quartz-halogen or tUl1!llst'en-natOIl€'n lamp. The halogen vapor functions in a cycle to reacts with the deposited to form the gas the bulb free of tungsten. tungsten which then at the hot filament to the ......E>"."''' and free the iodine for repeated operation. A spectra] curve for a 100-W quartz-halogen filament source is given in 2-11. The lamp approxisource, a useful continuum from 0.3 to 2.5 /-Lm. mates a 3000-K In arc lamps, an arc between two heats the Sec. 2-5

Sources of Optical Radiation

19

Typical spectral irradiance

c. 3.0

.. §

~ ~ 2.5

c:.c

...E' e e

ii I Ii

2.0 1.5

e·!!! "V ::i1 E 1.0

~

0.5 0.2

0.3

0.4

0.5

0.6

0.7

O.B

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.B

Wavelength (lAm)

2·11 irradiance from a IOO-W quartz halogen lamp, providing continuous radiation from 0.3 to 2.5 p.m. (Oriel Corp., General Catalogue. Stratford, Conn.)

electrodes to

in an atmosphere of argon, providing a spectral distributhat of lamps at 3100 K. The discharge lamp for its radiation output on the dynamics of an electrical discharge in a gas. A current is passed through the ionized gas between two sealed in a glass or quartz tube. (Glass envelopes absorb ultraviolet radiation below about 300 nm, whereas quartz transmits down to about 180 nm.) An acc:ele:ratles pl!p('trnr\..: sufficiently to ionize the vapor atoms. The source of the electrons may be a heated cathode (thermionic emission), a strong field applied at the cathode (field or the impact of positive ions on the cathode (secondary emission). De-excitation of the excited vapor atoms provides a release of energy in the form of radiation. and high-current operation generally results in a continuous output, in addition to lines characteristic of the vapor. At lower and current, lines appear, and the background continuum When lines are desired, as in monochromatic sources, is to operate at low temperature, pressure, and current. The sodium arc lamp, for example, radiation almost completely confined to a narrow "yellow" band due to the lines at 589.0 and 589.6 nID. The mercury tube is often used to provide, with the help of isolating monochromatic radiation at wavelengths of 404.7 and nm (violet). 546.1 nm and 577.0 and 579.1 nm (yellow). Other gases or vapors may be used to lines of other desired wavelengths. For the highest spectral purity, of the gas are used. When high intensity rather purity is desired, other designs bepprh~t~ the oldest source of this kind is the carbon arc, still widely used in and motion The arc is formed between two carbon rods in air. A 200-A arc lamp may have a peak luminance of 1600 . The source has a distribution close to that of a graybody at 6000 K. A wide range of spectral outputs is possible by using different materials in the core of the carbon rod. When the arc is enclosed in an atmosphere of vapor at pressure, the lamp is a short-arc source and the radiation is divided between line and continuous See 2-12 for a sketch of this type of and its The most of these lamps, designed to operate from 50 W to 25 kW, are mercury arc lamp, with comparatively weak background radiation but strong spectral lines and a good source of ultraviolet; the xenon arc lamp, with practically continuous radiation from the near-ultraviolet

20

Chap. 2

Production and Measurement of Ught

2-12 High-intensity, compact shortarc light source. (a) Compact arc (b) Lamp installed in housing, showing back reflector and focusing system. (The Eating Corp.)

through the visible and into the and the mercury-xenon arc lamp, providing essentially the mercury spectrum but with xenon's contribution to the continuum and its own strong spectral emission in the 0.8- to l-lLm range. As mentioned the color quality of the xenon lamp is similar to that of sunlight at 6OOO-K color temperature. emission curves for Xe and lamps are shown in 2-13 and 2-14. The hydrogen and deuterium arc lamps are ideal TOO Xenon-cathode tip

90

90

80

80

70

70

60 50 40

30 20 10 0 b-~__~__~__~~L-~__~__J -_ _~~L-~__~__J -_ _~_ _b-~__~__-L~O 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 t050 1 tOO 1150 Wavelength (oml

Figure 2-13 Spectral emission for xenon compact arc lamp. (Canrad-Hanovia, Inc.)

Sec. 2-5

Sources of Optical Radiation

21

100

I

100

I

I

Mercury-xenon

90

cathode tip

eo

eo

70

70

60

eo

50

50

40

40

30

30

20

20

10

o

"

,-"U

~

I

m __

I

~

.u

'-+..I

~~~~~~~~~~D~~

"-'I~ __

10

0

~~

Wavelength (nm)

arc lamp. (Canrad-Hanovia. Inc.)

emission for

for ultraviolet spectroscopy because produce a radiance with a continuous background in the ultraviolet region. Figure 2-15 shows the typical spectral output of a deuterium lamp, which produces a line-free continuum from below 180 nm to 400 nm. tubes represent a high output source of visible and near infrared tion. produced by a discharge of stored electrical energy through a gas-filled tube. The gas is most often xenon. The photoflash in contrast, provides highintensity, short-duration illumination by the rapid combustion of metallic (aluminum or zirconium) foil or wire in a pure oxygen atmosphere. When a high-intensity point source of radiation is desired in optical instrumentation. a useful lamp is the concentrated zirconium arc lamp, with wattages ,,,n',,",,,,, from 2 W to 300 W. Zirconium vapor is formed by evaporation from an oxide-

w g to

1;;

~:c ~ E

V .~ f6

0.02

'i5

0,01

0.008 0.000

0.004 0.003 0.002 0.001L-__ 200

~

____- L____ 300

~

____L -__~____~____~____~~~~~~

400

500

600

Wavelength (nm)

Figure 1-15 Spectral emission for deuterium aTe lamp at 50 W. (Oriel Corp., General Catalogue, Stratford, Conn.)

22

Chap. 2

Production and Measurement of light

700

coated cathode in an argon atmosphere. Radiation originates both from the incandescence of a molten cathode surface and from the excited zirconium vapor and argon gas. It is viewed through a small 0.13 to 2.75 mm in in a .. ,............. anode, and through an optically flat window sealed into the tube. The specdistribution approximates that of a gray body source. The familiar fluorescent lamps use low-pressure, low-current electrical in mercury vapor. ultraviolet radiation from excited mercury atoms is converted to visible light by fluorescence in a phosphor on inside of the glass-envelope surface. Spectral outputs depend on the particular used. "Daylight" lamps, for use a mixture of zinc beryllium and tungstate. A very different type of source is the low-intensity diode , ..........., .. a solid-state device employing a p-n junction in a semiconducting crystal. The device is hermeticaUy sealed in an optically centered package. When a small bias voltage is applied in the optical energy is the recombination of electrons and holes in the vicinity of the junction. LEDs include the infrared GaAs with a output wavelength near 900 nm. and the visible SiC device, with peak output at 580 nm. LEDs provide narrow emission bands, as evident in 2-16. Solid solutions of similar cOln~oof1ld materials produce output in a of spectral regions when the alloy is varied. 120 100

.~ r:::

.E'"

SO

r::: 0

"iii

'E.,

.'" >-

~~

60 40

a:

20 0 915

895 Emission wavelength 111m I

tlgure 2-16 Spectral output from a GaAs light·emitting diode.

The laser is a very important modem source of coherent and extremely monochromatic radiation. of very intensity. Lasers emit in the ultraviolet. visible, and of the spectrum. Because of the central role instrumentation, they are treated in a chapter. that lasers play in

2-6

nJC"rll:l"'TIf'U:U::!

OF RADIA TION Any device that produces a measurable physical response to incident radiant energy is a detector. The most common detector is, of course, the eye. Whereas the eye n ..'~",,""'C a and subjective response, the detectors discussed here provide role played by the eye a quantitative and response. In view of the in human vision, it is treated in another Sec. 2-6

Detectors of Radiation

The most common detectors may be classified as follows: A. Thermal detectors 1. Thermocouples and thermopiles 2. Bolometers and 3. Pyroelectric 4. Pneumatic or Golay B. Quantum detectors 1. and photomultipliers 2. Photoconductive 3. Photovoltaic 4. Photographic When the primary measurable response of a detector to incident radiation is a rise in temperature, the device is a thermal detector. The receptor is usually a blackened metal strip or flake, which absorbs at all wavelengths. Such a device, in which an in at a of two dissimilar metals or semiconductors a is caned a thermocouple (Figure 2-17a). When the

(b)

2-17 (a) Thermocouple made of dissimilar materials (dark and light lines) joined at T, and T2 , where a difference in temperaproduces an emf between terminals VI and V2 • (b) made of couples in series. Radiation is absorbed at the junctions TJ in thermal contact with a black absorber and thermally insulated from the junctions T2 •

effect is enhanced by using an array of such the device is caned a thermopile (Figure 2-17b). Thermal detectors also bulk devices that .."""·.....""..1 to a rise in temperature by a in resistance. Such an instrument may employ as its sensitive element either a metal (bolometer) or, more commonly, are used a semiconductor (thermistor). TypicaHy, two in adjacent arms of a bridge circuit, one of which is to the incident radiation. The imbalance in the circuit, due to the is indicated by a galvanometer deflection or current. In the the temperature change causes a change in the surface of certain materials, like lithium tantalate or triglycine sulfate (TGS) that exhibit the The detector behaves like a capacitor whose of the The Golay cell measures instead the thermal of a gas. Heat absorbed by a blackened membrane is transmitted to the gas in an airtight chamber. The pressure rise in the gas is usually detected optically, by the deflection of a mirror. A schematic of the cell is shown in Figure 2-18. Thermal detectors are characterized by a slow response to changes in the incident radiation. If the detector is to folIowa changing input signal, such as a detectors are not as desirable as the faster-responding quantum detectors to be next. The speed of response is described by a time constant, a measure of the time to regain equilibrium in output after a change in input. Thus quantum detectors are better suited to high-frequency operation. 24

Chap. 2

Production and Measurement of Ught

Pneumatic chamber

leak

Ballasting reservoir Window

Incident Flexible mirror

Figure 2-18 tor. (Oriel

Absorbing film

pneumatic infrared delecGeneral Catalogue. Stratford.

Quantum detectors respond to the rate of incidence of radiation with rather than to thermal energy. Photons interact tector material. When the measurable effect is the electrons from an illuminated surface, the device is called a photoemissive detector. A photosensitive surphotons that transfer containing alkali metals, absorbs energy to enable some electrons to overcome the work function and escape If the photoemitted electrons are simply coHected by a positive-bithe ased anode in an evacuated rube, enabling a current to be drawn into an external ciris internally amplified detector is called a diode phototube. When the sec:onIlaJ'Y pl"i'fr",,, emission, the detector is a photomultiplier; see Figure 2-19. photoelectrons are so that as a result of a sequence of collisions, each multiplying the current by the addition of secondary elec.".."''''*'~ of electrons becomes available at the output corresponding to

Anode

Cathode

2-19 One type of photomultiplier tube structure. Electrons photoemitted from the zigzag paths cathode are accelerated down the tuhe so as to strike each of the curved surfaces, each time producing additional secon(!lary electrons. The multiplied current is collected at the anode.

Another means of amplification, used in the photocell, aUows the of additional electrons by ionization of the residual gas. In the case of enphotons (A < 550 nm), the of photoemissive detectors is sufficient to allow the counting of individual photons. Such detectors possess superior serlslt.IVlltv in the visible and ultraviolet spectral ranges. For wavelengths in the inover I ILm, phmoemitters are not available and photoconducling detectors are used. In these photons absorbed into thin films or bulk material propairs. Both the negative duce free charges in the form of (electrons) and positive (holes) the electrical conductivity of the ""ljll.'''''. Without illumination, a bias voltage across such a material with high intrinsic produces a small or "dark" current. The presence of illumination and the extra free-charge carriers so lower the resistance of the material, and a photocurrent results. compounds CdS and CdSe are often used in the visible and Figure 2-20), while farther out in the near-infrared region, the compounds PbS (0.8 to 3 ILm) and PbSe (l to 5 ILm) are popular. The most common photo voltaic detector is a p-n junction, the semiconductor photodiode. This device consists of a junction between doped p-type (rich in posiap'\P ..,~t'£\n

Sec. 2-6

Detectors of Radiation

BO

80

~60 :> .~

:f1

~40 20

8000

6000

10.000

Wal/lliength (AI (a)

Wavelength

(AI

(bl

Figure 2-20 of a CdS photocoIldllCting cell. The peak response at 5500 A lhe response of !he human eye. The cell is useful with incandescent, fluorescent, or (b) Spectral response of a CdSe at 7350 cell is sensitive 10 lhe near infrared photoconducling cell with and is useful with incandescent or neon lamps.

live carriers) and doped n- type (rich in negative charge carriers) ..... «t" ..,.« most often Doping involves adding sman amounts of an impurity to the semiconductor to provide either an excess (n- type) or deficiency (p- type) of conduction dectrons. In a narrow region between these materials, a built-in electric field occurs as a consequence of current equilibrium. When photons are absorbed in the vicinity the junction, the electron-hole pairs are separated by the field, causing a The solar ce]] and the photographic expoin the photo voltaic sure meter are perhaps the best known applications of detector. A of the photovoltaic cell, the avalanche diode; provides an internal mechanism of amplification that results in enhanced sensitivity out to around 1.5 p.m. In the region of I to 8 p'm, the semiconductor compounds PbS, PbSe, and PbTe possess a large sensitivity than the thermocouple or the ordinary photovoltaic effect and bolometer. As with other detectors that are to operate at longer wavesensiphotovoltaic detectors are often cooled to enable operation at tivity. Two-dimensional arrays or panels of photodiodes allow the detection of images. Each photodiode, or MOS (metal-ox ide-semiconductor) device, responds to the incident radiation to provide one pixel (picture element) of output. On exposure to light, each of the discrete devices fabricated on a silicon The stored induced in a potential well created by an applied contributed by each pixel, a measure of the local irradiance, is scanned to produce an electronic record of the image. Scanning and readout is performed by charge transfer along rows of such devices (CCD, or charge-coupled devices) or by injection of the charge into the underlying bulk semiconductor material (CID, or charge-injection [2], By sequential reading of the stored the set of x-y coordinates of the original irradiance distribution is ",l",,,fy'nni,,<>

26

Chap. 2

Production and Measurement of light

to form the image. Such photodiode arrays are used in television cameras and have in astronomical and been used to replace photographic graphs. Finally, a detector made popular by the widespread use of the camera is the photographic film or plate. Such photographic emulsions are available with spectral sensitivity that extends from the X-ray region into the near infrared at around 1.2 JLm. The sensitive material is an emulsion of silver halide crystals or grains. An incident photon imparts energy to the valence electron of a halide which can develthen combine with the ion, producing a neutral atom. contains a latent a distribution of reduced oping, the determined by the variations in radiant energy received. The latent "amplified," so to by the action of the developer. The action further free electrons to continue the reduction process, with the latent agent to further action. The density of the silver atoms, image acting as a and thus the opacity of the film, is a measure of both the irradiance and the time of exposure, so that photographic film, unlike other detectors, has the advantage of Iml:-S}:!!mU integration. Even weak radiation can be the cumulative effect of a long exposure. In addition to a knowledge of the spectral range over which a particular detector is effective, it is important to know the actual sensitivity or, more precisely, the responsivity S of the defined as the ratio of output to

S=

input

(2-16)

Input may be radiant flux or Output is almost always a current or voltage. of a detector, it be constant For the responsivity to be a useful over the useful range of the instrument. In other words, the detector, together with associated amplifier and circuits, should provide a response, with output proportional to In general, however, responsivity is not independent of waveCurves of responsivity versus wavelength are provided with commercial detectors. When the responsivity is a function of A, the detector is said to be selective. A nonselective detector is one that depends only on the radiant flux, not on the wavelength. Thermal detectors using a blackened strip as a receptor may be nonsehowever, entrance windows to such devices may wen make them selc~tl.ve. The detectivlty D of a detector is the reciprocal of the minimum power, called the noise equivalent power,
D The minimum detectable power is limited by the noise inherent in the operation of the detector. The noise is that part of the or output not related to the desired input. Many sources of noise including the statistical fluctuations of photons, of current or Johnson or radiation noise, and the thermal inherent in all detectors; the generation and recombination noise due to statistical fluctuations of current carriers in photoconductors; the shot noise due to random emission of electrons in photoemissive detectors; and the noise due to temperature is not useful when it fluctuations in thermal detectors. Mere amplification of a does not distinguish between and noise and results in the same is not in clarifying its ratio, just as the mere magnification of an optical details.

Sec. 2-6

Detectors of Radiation

27

2·1. Calculate the lrec~uenclles of electf()mlilgnletic radiation of ........,""" ...1', a visual sensation in the normal eye. 2·2. A small, monochromatic light source, radiating at 500 nm, is rated at 500 W. (a) If only 2% of its total power is perceived by the eye as luminous power. what is its luminous flux output? (b) If the source radiates uniformly in all determine its radiant and lumi· nom; intensities. (e) If the surface area of the source is 50 cm2 , determine the radiant and luminous exitances. (d) What are the irradiance and illuminance on a screen situated 2 m from the source, with its surfuce normal to the radiant flux? (e) If the screen contains a hole with diameter 5 cm, how much radiant and luminous flux get through? 2·3. (a) A 50-mW He-Cd laser emits at 441.6 nm. A 4-mW He-Ne laser emits at 632.8 om. Using Figure 2-7, compare the relative brightness of the two laser diameter when projected side by side on a white of paper. beams of Assume photopic vision. (b) What power argon laser emitting at 488 om is required to match the brightness of a O.5-mW He-Ne green laser at 543.5 nm, under the conditions of (a)? 2·4. A lamp 3 m directly above a point P on the floor of a room at P an illuminance of 100 Im/m 2 • (a) What is the luminous intensity of the lamp? (b) What is the illuminance at another on the floor, I m distant from P? 2-5. A lot is illuminated at identical at the top of two 30 ft high and in all directions, compare the illumi40 ft apart. Assuming the lamps radiate between them. nance at ground level for points directly under one lamp and 206. A small source of 100 cd is situated at the focal poim of a spherical mirror of 50-em focal length and lO-cm diameter. What is the average illuminance of the beam reflected from the mirror, an overall reflectance of about 80%1 2-7. (a) The sun subtends an of OS' at the earth's surface, where the illuminance is about 105 Ix at normal incidence. Determine the luminance of the sun. (b) Determine the illuminance of a horizontal surface under a hemispherical sky with uniform luminance L. 2-8. A circular disc of radius 20 cm and uniform luminance of IcY' cd/m 2 illuminates a small plane surface area of I cm2 , I m distant from the center of the disc. The small surface is oriented such that its normal makes an angle of 45'" with the axis joining the (0 the circular disc. What is the centers of the two surfaces. The axis is luminous flux incident on the small surfuce? 2-9. Derive the Wien displacement law from the Planck blackbody spectral radiance formula. 2-10. Derive the Stefan-Boltzmann law from the Planck blackbody spectral radiance formula. (Hint: Use a substitution of x = hclAkT to facilitate the integration.) 2-U. The of the solar spectrum falls at about 500 nm. Determine the sun's surfuce tempeI'lltulre. assuming that it radiates like a ..... """"~""'"y. 2·12. (a) At what wavelength does a blackbody at 6000 K radiate the most per unit wavelength? (b) If the blackbody is a I-mm diameter hole in a cavity radiator at this temperature, find the radiated through the hole in the narrow wavelength region 5500-5510 temperature, Amax 550 nm for a blackbody cavity. The temperature 2·13. At a is then increased until its total radiant exitance is doubled. What is the new temperature and the new

28

2

Production and Measurement of light

2-14. What must be the temperature of a with ~m,i!:!:1ivil·v of 0.45 for it to have the same total radiant exitance as a blackbody at 5000 K?

[IJ

[4]

[6] [7] [8]

[91 [10] [II]

[14]

Robert. Basic Concepts in Relativity and Early Quantum Mechanics. New York: John and Sons, 1972. Ch. 4. D. F. Devices Using the Concept." Proceedings the IEEE 63 (Jan. 1915): 38Malacara, Zacarias H., and Morales R. "Light Sources." In Geometrical and Instrumental Optics, edited Daniel Malacara. Boston: Academic 1988. Allen. Geometrical An Introduction. Reading, Mass.: AddisonWesley Publishing Co., 1968. Ch. 6. Budde. W. Physical Detectors of Radiation. Optical Radiation Measurements vol. 4. New York: Academic 1983. and Richard J. Becherer. Radiometry. Optical Radiation Measurements vol. 1. New York: Academic 1979. IES Lighting Handbook. New York: Illuminating Engineering Society, 1981. Handbook of Optics, edited by \Valter G. Driscoll and William Vaughan. by the Optical Society of America. New York: McGraw-Hili Book 1978: "' .... IJII"... F. Jacobs, "Nonimaging Detectors." Jay F. Snell, "Radiometry and PhotomeJ. Zissis and J. "Optical Radiators and Sources." R. H. Detection of and Infrared Radiation. New York: "'ru.;n<'.... _ 1978. Stimson, A. Photometry and I{CU'llOl7U?llrv New York: Wiley-Interscience, 1974. T. H. "Photographic 30. Morehead, Fred F., Jr. U!~t··bn11ttmg Semiconductors." " .....:"''', ... American (May 1967): 108. Clark Jones, R. "How are Detected." In Lasers and Light. ne4UUI,rlXS SciAmerican, pp. 81~88. San Francisco: W. H. Freeman and Pui)lishers, 1968. Fitch, James Marston. "The Control of the Luminous Environment." In Lasers and Light. Readingsfrom Scientific American, pp. 131~39. San Francisco: W. H. Freeman and Company Publishers, 1968.

Chap. 2

References

29

3

Geometrical Optics

INTRODUCTION

The treatment of light as wave motion for a region of approximation in which the wavelength is considered to be negligible compared with the dimensions of the r"I",,,,.nt components of the optical system. This region of approximation is called geometrical When the wave character of the light may not so ignored, the field is known as physical optics. Thus geometrical optics forms a special case of physical in a way that may be summarized as follows: limit {Physical optics} = {geometrical optics} A~O

Since the wavelength of light is very small compared to objects, unrefined observations of the behavior of a light beam passing through apertures or around obstacles in its path could be handled by geometrical optics. Recall that the appearance of distinct shadows influenced Newton to assert that the apparent rectilinear propagation of light was due to a stream of light corpuscles rather than a wave motion. Wave motion longer such as those in water waves and sound waves, was known to give distinct around obstacles. Newton's model of light propagation, therefore, seemed not to allow for the existence of a wave motion with very small wavelengths. There was in fact already evidence of some of even for light waves, in the time of Isaac Newton. The Jesuit Francesco Grimaldi had noticed the fine structure in the edge of a shadow, a 30

structure not explainable in tenns of the of light. This bendof waves the edges of an came to be called diffraction. Within the approximation represented by optics, Ught is understood to travel out from its source along straight or rays. The ray is then simply the which light energy is transmitted from one point to another in an opThe ray is a useful construct, although abstract in the sense that a light in cannot be narrowed down to approach a straight line. A pencil-like laser beam is perhaps the best actual approximation to a ray of light. (When an through which the beam is is made small enough, however, even a beam begins to spread out in a diffraction pattern.) When a light ray traverses an optical system of several homogeneous media in sequence, the optical path is a sequence of Discontinuities in the line occur each time the light is or refracted. The laws of that describe the of the rays are the following: When a ray of light is reflected at an interface dividing ray remains within the plane of incidence, and the two angle of the angle of incidence. The plane of incidence includes the ........._... ray and the normal to the point of incidence. (SnelJ's Law). When a ray of at an intwo uniform media, the transmitted ray within the plane of incidence and the sine of the angle of refraction is directly proportional to the sine of the angle of incidence. l l ..r""'l'ti,nn

in Figure 3-1, in which an incident ray is partially Both laws are reflected and partially transmitted at a plane interface two transparent media.

Figure 3-1 Illustration of the law of reflection and refraction.

sin 8,

3-1 HUYGENS' PRINCIPLE

The Dutch envisioned light as a of emitted from each point of a and propagated in relay fashion the particles of the ether, an elastic medium filling all space. Consistent with his con '. Huygens each of a propagating disturbance as of ~atD1g that contributed to the disturbance an instant Jater. To s W' how his uUlf'U\'-U the laws of geometrical optics, he a Sec. 3-1

Huygens'

Pr.''' .... ,.... I..

fruitful principle that can be stated as follows: Each point on the surfuce of a wave disturbance-the wave front-may be as a source of spherical waves (or wavelets), which themselves progress with the speed of light in the medium and whose envelope at a later time constitutes the new wavefront. Simapplications of the principle are shown in Figure 3-2 for a plane and spherical wave. In each case, AB forms the wave disturbance or wavefront, and A' B I is the new wavefront at a time t later. The radius of each accordingly, ct, where c is the speed of light in the medium. Notice that the new wavefront is tangent to each wavelet at a single point. According to Huygens, the remainder of each wavelet is to be in the of the principle. Indeed, were the remainder of the wavelet to be effective in propagating the Ught disturbance, Huygens could not have the law of rectilinear propagation from his principle. To see this more dearly, refer to Figure 3-3, which shows a spherical wave disturbance originating at 0 and incident upon an aperture with an SS I . According to the notion of rectilinear propagation, the lines OA and OB form the sharp of the shadow to the right of the aperture. Some of the wavelets originate from points of the wavefront (arc SS '), however, overlap into the region of shadow. According to Huygens, however, these are ignored and the new wavefront where the extreme wavelets at ends abruptly at points P and P , , points Sand S I are to the new wavefront. In so the effective-

I 8'

8

Figure 3-2 Illustration of Huygens' principle for plane and spherical waves.

o

B

32

Chap. 3

Geometrical Optics

Figure 3-3 Huygens' construction for an obstructed wavefront.

ness of the overlapping Huygens avoided the possibility of diffraction of the light into the region of geometric shadow. Huygens also ignored the wavefront formed by the back half of the wavelets, since these wavefronts implied a disturbance in the direction. weaknesses in this remedied later by Fresnel and Huygens was to apply his nrln{'lnlp prove the laws of both reflection and refraction, as we show in what follows. 3-4a iUustrates the Huygens construction for a narrow, beam of light to the law of reflection. must be modified to accommodate the case in which a wavefront, such as encounters a plane interface, such as at an angle. Here the angle of incidence of the rays AD, CF relative to the perpendicular PD is OJ. Since points the plane n'
LADX = LIDG = A DIM LIDM = LIDG

A DIG

:.0; = Or

X--------------~~--------~~-------b~~-------------y

(a)

3-4 (a) Huygens' construction

Sec. 3-1

nU'''u~"",.

Principle

to

prove the law of reflection.

33

ni X--------------~~~------~~----~~~---------------y n,

L DIM = 0, L lOF = 0,

sin OJ

=

IK

A

and

OJ

sin 0..

FI

sin Or

OM

-- = -

sin

OG

= -

OM

(J

r

= OM OJ

I I

n, = - = constant n.. (b)

Figure 3-4 (continued) (b) Huygens' construction to prove the law of refraction.

points D, J, and I of the plane interface XYat different times. In the absence of the refracting surface, the wavefront GI is formed at the instant ray CF reaches I. During the progress of ray CF from F to I in time t, however, the ray AD has entered the lower medium, where its speed is, let us say, slower. Thus if the distance DG is ViI, a wavelet of radius v,t is constructed with center at D. The radius DM can also be expressed as DM =

V,I =

DG) = v, ( -;;

(ni) n, DG

(n;!n,)

Similarly, a wavelet of radius Jll is drawn centered at J. The new wavefront KI includes point 1 on the interface and is tangent to the two wavelets at points M and N, as shown. The geometric relationship between the angles Oi and 0" formed by the representative incident ray AD and refracted ray DL, is Snell's law, as outlined in Figure 3-4b. Snell's law of refraction may be expressed as n;

sin 0.

= n, sin 0,

(3-1)

3-2 FERMA T'S PRINCIPLE

The laws of geometrical optics can also be derived, perhaps more elegantly, from a different fundamental hypothesis. The root idea had been introduced by Hero of Alexandria, who lived in the second century B.C. According to Hero, when light is propagated between two points, it takes the shortest path. For propagation between two points in the same uniform medium, the path is dearly the straight line joining the two points. When light from the first point A, Figure 3-5, reaches the second

34

Chap. 3

Geometrical Optics

A

8

3-5 Construction to prove the law of reflection from Hero's principle.

A'

point B after reflection from a plane surface, however, the same principle predicts as follows. 3-5 shows three possible paths from A to B, the law of including the correct one, ADB. Consider, however, the arbitrary path ACB. If point A' is constructed on the perpendicular AO such that AO OA', the right AOC and A'OC are equal. Thus AC A'C and the traveled by the ray of light A to B via C is the same as the distance from to B via C. The shortest distance from A' to B is obviously the straight line A' DB, so the path ADB is the correct choice taken by the actual light ray. Elementary geometry shows that for the this path, 61 = 6,. Note also that to maintain A' DB as a single straight that the plane of the reflected ray must remain within the plane of page. The French mathematician Pierre de Fermat generalized Hero's principle to prove the law of refraction. If the terminal point B lies below the surfuce of a second meolUlm, as in 3-6, the correct path is definitely not the shortest path or of refraction equal to the angle of instraight line AB, for that would make the cidence, in violation of the empiricaJly law of refraction. Appealing to the "economy of nature," Fermat supposed instead that the ray of light traveled the path of least time from A to B, a generalization that included Hero's principle as a special case. If light travels more slowly in the second medium, as assumed in ure 3-6, bends at the interface so as to take a that favors a shorter in the second medium, thereby minimizing the overall transit time from A to B. Mathematically, we are required to minimize the total t

AO

OB

Vi

V,

=-+

n,

n,

J<'igure 3-6 Construction to prove the law of refraction from Fermat's principle.

Sec. 3-2

Fermat's Principle

35

where Vi and v, are the of light in the incident and transmitting respectively. Employing the Pythagorean theorem and the defined in Figure 3-6, we have t = ----

+ ----'---.....:......

Since other choices of path the of point 0 and therefore the distance x, we can the time by setting dtldx = 0: dt

-=

c - x

x

dx

0

from the of incidence and refraction can be conveniently introduced into this condition, giving dt = sin OJ _ sin 0, = 0

dx

Vi

VI

so that v, sin OJ Vi sin Introducing the refractive indices of the media through the relations, V = cln, we arrive at Snell's law, Fermat's principle, like that of Huygens, required refinement to achieve more general applicability. Situations where the actual path taken by a light ray may represent a maximum time or even one of many possible paths, aU requiring equal time. As an example of the latter case, light propagating from one focus to the other inside an ellipsoidal mirror, any of an number of po:sslltlle paths. the is the of all points whose combined from the two foci is a constant, all paths are indeed of equal time. A more precise statement of Fermat's principle, which requires merely an extremum relative to neighboring paths, may be given as follows: The actual taken by a light ray in its propagais such as to make its optical path tion between two given points in an optical in the first approximation, to other paths adjacent to the actual one. With this formulation, Fermat's principle fans in the of problems caned variatioool calculus, a technique that determines the form of a function that is the integral of the remizes a definite integral. In optics, the definite quired for the of a light ray starting to points.· 3-3 PRINCIPLE OF REVERSIBILITY

Refer to the cases of reflection and in 3-5 and 3-6. If the roles of points A and B are interchanged, so that B is the source of light rays, time must predict the same path as determined for the Fermat's principle of original direction of propagation. In general then, any actual ray of in an if reversed in direction, wiH retrace the same path backward. This optical principle will found to very in various applications to dealt with later. I It is of interest to note here that a similar principle, called Hamilton's principle of least action in mechanics that calls for a minimum of the definite integral of the Lagrangian function (the kinetic energy minus the potential energy), an alternative formulation of the laws of mechanics and indeed Newton's laws of themselves.

Chap. 3

Geometrical Optics

3-4 REFLECTION IN PLANE mIRRORS Before the formation of images in a way, we the simplest-and experientially, the most accessible-case of images formed by plane mirrors. In this context it is important to distinguish between specular reflection from a perfectly smooth surface and diffuse reflection from a granular or rough surface. In the former case, all rays of a parallel beam incident on the surface obey the in the law of reflection from a plane surface and therefore reflect as a parallel latter case, though the law of is obeyed locally, the microscopically granular surface results in reflected rays in various directions and thus a diffuse scattering of the originally parallel rays of light. Every plane surface will produce some such since a perfectly smooth surface is not attainable in The treatment that follows assumes the case of specular reH.ection Consider the of a single light ray OP from the xy- plane in Figure 3-7a. By the law of reflection, the reflected ray PQ remains within the plane of incidence, making equal angles with the normal at P. If the path OPQ is resolved it is clear that the direction of ray OP is altered by into its X-, y-, and the only and then in such a way that its z-component is simply reversed. If the direction of the incident ray is described by its (x, y, z), then the reflection causes vector,1-1 1-1 =

~

y, z)

1-2

(x, y, - z)

It follows that if a ray is incident from such a direction as to reflect sequentially from

all three rectangular coordinate planes. as in the "comer reflector" of

1-1 = (x, y, z)

~

1-2 =

3-7b,

-y,

and the ray returns precisely parallel to the line of its original approach. A network of such comer reflectors ensures the exact return of a beam of light, a headJight beam from highway reflectors, for example, or a laser beam from the moon. z

z

Q

.J-f--...,...,.---_.v

x (a)

(b)

Figure 3-7 r ;""corrv-tru of a ray reflected from a plane.

formation in a plane mirror is illustrated in Figure 3-8a. A point object S sends rays toward a plane which reflect as shown. The law of ensures that of SNP and S I NP are equal, so that all reflected rays appear to originate at the image point S', which lies along the normal line SN, and at such a depth that the image distance S' N equals the object distance SN. The eye sees a point image at in exactly the same way as it would see a real point object the there. Since none of the actual rays of light lies below the mirror The image S I cannot be projected on a screen as is said to be a virtual Sec. 3-4

Reflection in Plane Mirrors

31

~ I

I

I

I I

s' la)

(b)

3-8 (e)

Image formation in a

mirror.

(d)

in the case of a real image. All points of an extended object, such as the arrow in 3-8b, are by a plane in similar fashion: Each object point has its image point its normal to the mirror surface and as far below the renectmg surface as the object point lies above the surface. Note that the image does not depend on the position of the eye. Further, the construction of Figure 3-8b size is identical with the glvmg a makes clear that the magnification of unity. In addition, the tmnsverse orientation of object and object, appears in its image. In are the same. A Figure 3-8c, where the mirror does not lie directly the object, the mirror plane may be extended to determine the position of the image as seen by an eye positioned to receive reflected rays originating at the object. Figure 3-8d illustmtes 111".. ulfI..... images of a object 0 formed by two perpendicular mirrors. 11 and lz result from reflections in the two mirrors, but a third 13 results from sequential reflections from both mirrors.

3-5 REFRACTION THROUGH PLANE SURFACES

Consider light ray (1) in 3-9a, incident at angle OJ at a plane interface sepamttwo tmnsparent media characterized, in order, by refractive indices nl and nz. Let the angle of refraction be the ~. Snell's law, which now takes the nl sin 01 =

Hz

sin

reQuirc~s an angle of such that the refmcted ray bends away from the rays 1 and 2, when nz < nl. For nz > nl, on the other normal, as in Figure hand. the refracted my bends toward the normal. The law also requires that my incident normal to the surface (0 1 0), transmitted without change of liirPl'ti ....., «()2 = 0), of the ratio of refractive indices. In 3-9a the three rays shown at a source point S below an and emerge into an upper

38

3

Geometrical

12)

111

n,

s (a)

(b)

s (e)

Figure 3-9 Geometry of rays refracted by a

interface.

medium of lower refractive as in the case of light from water (nJ = 1.33) air (n2 = 1.00). A unique point is not determined by these rays because they have no common intersection or virtual image point below the from which appear to after refraction, as shown by the dashed ext:en:SlOlllS of the refracted rays. For rays making a small angle with the normal approximato the surface, a reasonably image can be located. In the image, the angles of incition, where we allow only such paraxial raysZ to dence and refraction are both small and the approximation, sin 8 is

From Eq.

== tan 8

8 (in radians)

Snell's law can be approximated nl

tan 8 1

== nz tan 82

(3-3)

and taking the appropriate tangents from Figure 3-9b, we have

2 In a paraxial ray is one that remains near the central axis of the image-forming optical system, thus making small with the optical axis.

Sec. 3-5

Refraction through Plane Surfaces

39

The image point occurs at the vertical distance s' below the surface given by

s' = (::)s

(3-4)

where s is the corresponding depth of the Thus objects underwater, viewed from directly overhead, appear to be nearer the surface than they actually are, since angle lh is not small, a in this case Sf = (l/1.33)s = h. Even when the reasoflaDly good image of an underwater is formed the aperture or pupil of the eye admits only a small bundle of rays while forming the Since these rays differ very little in direction, they will appear to originate from approximately the same image point. However, the depth of this image will not be ~ the object depth, as for rays, and in general will vary with the angle of from the that make increasingly angles of incidence with the must, by Snell's law, refract at increasingly larger as shown in me 3-9c. A angle of incidence Be is reached when the angle of refraction law. reaches 90°. Thus from ntprtgr'p

sin Be

(::)Sin 90

or (3-5)

For of incidence BJ > Be. the ray total internal reflection, as shown. This phenomenon is in the transmission of along fibers by a series of total internal reflections, as discussed in Chapter 24. Note that from the phenomenon does not occur unless n I > n2, so that Be can be (3-5). We return to the nature of images tn..'rn""'" by refraction at a plane when from a spherical surface. we deal with such as a case of

3-6 IMAGING BY AN OPTICAL SYSTEM

We discuss now what is meant by an in general and indicate the practical and theoretical factors that render an image less than perfect. In Figure 3-10, let the region labeled "optical system" include any number of reflecting andlor refracting surof any curvature, that may alter the direction of rays leaving an object point O. region may include any number of intervening but we shall assume that each individual medium is homogeneous and isotropic, and so by its point own refractive index. Thus rays spread out radially in all directions from 0, as shown, in real object space, which precedes the first reflecting or refracting Real image space Optical system

3-10 Image formation by an system.

40

Chap. 3

Geometrical

surface of the optical system. The family of spherical surfaces normal to the rays are such that each ray contacting a wavefront reprethe wavefroots, the locus of sents the same time of light from the source. In real object space the rays are diverging and the spherical wavefronts are expanding. now that the optical system redirects these rays in such a way that on leaving the optical system and entering real image space, the wavefronts are contracting and the rays are converging to a common point that we define to be the image point, I. In the spirit of fennat's principle. we can say that since every such ray starts at 0 and ends at I. every such ray requires the same transit These rays are said to isochronous. further. by point, ray will reverse its directhe principle of reversibility, if I is the tion but maintain its path through the optical system, and 0 will be the correspond0 and 1 are said to be conjugate points the optical ing point. The ~J~'-"'" In an ideal system, ray from 0 intercepted by the system-and only these passes I. To image an actual object, this requirement must hold for every object point and its conjugate image point. Non ideal images are formed in practice because of (I) light scattering, (2) aberrations, and (3) diffraction. Some rays 0 do not reach I due to reflection losses at diffuse reflections from and scattering by inhomogeneities in transparent media. Loss of rays by such means merely diof the however, some of these rays are scattered minishes the When the through 1 from nonconjugate object points, degrading the itself cannot produce the one-to-one relationship between object and rays required perfect of all points, we of system aberrations. Such aberrations are treated later. finally, since every optical sysrem intercannot be cepts only a portion of the wavefront emerging from the object, the """.-f"r·tlv sharp. Even if the image were otherwise the effect of a limportion of the wavefront leads to diffraction and a blurred which is to be diffraction limited. This source of images, discussed further in the sections under diffraction, represents a fundamental limit to the sharpness of an image that cannot be entirely overcome. This difficulty rises from the wave nature of light. Only in the unattainable limit of geometrical where A -+ 0, would diffraction effects disappear entirely. that form images are called Cartesian Reflecting or refracting surfaces. In the case of reflection, such surfaces are the conic as shown in points may be figure 3-11. In each of these figures, the roles of object and reversed by the principle of reversibility. Notice that in 3-11 b, the image is virtual. In 3-1 the parallel reflected rays are to form an "at infinity." In each case, one can show that fermat's principle, requiring isochronous rays object and image points, leads to a condition that is equivalent to the geometric definition the corresponding conic section. Cartesian surfaces that perfect by may be more surface that imcomplicated. Let us ask for the equation of the appropriate ages object point 0 at point I, as illustrated in 3-12. There an arbitrary point P with coordinates (x, y) is on the required I. The requirement is that every ray from 0, like OPI, refracts and passes through the I. Another V. By fermat's such ray is evidently OVl, normal to the surface at its vertex pnnCllpl€!, these are isochronous rays. Since the media on either of the refractive indices, the isochronous surface are by rays are not equal in length. The transit time of a ray through a medium of thickness x with index n is x nx t

-:=-

V Sec. 3-6

.rn",,,.,nn

by an Optical System

C

41

la) Ellipsoid

(bl Hyperboloid

lel Paraboloid

Figure 3- n points.

Cartesian reflecting surfuces showing

UJII.lugd"C

and image

v

o~---------.~------~

__

------~~x

Figure 3-12 Cartesian refracting surface which images point 0 at image point I.

equal times imply equal values of the product nx, called the optical path length. In the problem at hand then, Fermat's principle requires that (3-6) where the distances are defined in the first sum of (3-6) becomes no(x 2

3-12. In terms of the (x, }-C()()f(llmltes of

+ y2)112 + n,[y2 + (so +

Si

X)2Jl/2

constant

The constant in the is determined the middle mel'J'ber which can be calculated once the problem is defined. LJ'I.j'U surface is either a hyperboloid (ni > no) or an ellipsoid (no> nl), as shown. Chap. 3

Geometrical

o

(al

(bl

Ie)

Figure 3-13 Cartesian refracting surfuces. (a) Cartesian ovoid images 0 at I by refraction. (b) Hyperbolic surlace images object point 0 at infinity when 0 is at one focus and ni > no. (c) Ellipsoid surface images object point 0 at infinity when 0 is at one focus and no > II;.

The first of these corresponds to the usual case of an object in air. A double hyperbolic lens then functions as shown in Figure 3-14. Note, however. that the aberration-free imaging so achieved applies only to object point 0 at the correct distance from the lens and on axis. For nearby points, imaging is not perfect. The larger the actual object, the less precise is its image. Because images of actual objects are not free from aberrations and because hyperboloid surfaces are difficult to grind exactly, most optical surfaces are spherical. The spherical aberrations so introduced are accepted as a compromise when weighed against the relative ease of fabricating spherical surfaces. In the remainder of our treatment of geometrical optics, we concentrate on the case of spherical reflecting and refracting surfaces with a radius of curvature R. Of course, in the limit R ~ 00, we deal with the special case of a plane surface.

o Figure 3-14 Aberration-free imaging of point object 0 by a double hyperbolic lens.

3-7 REFLECTION AT A SPHERICAL SURFACE

Spherical mirrors may be either concave or convex relative to an object point 0, depending on whether the center of curvature C is on the same or opposite side of the surface. In Figure 3-15 the mirror shown is convex, and two rays of light originating at 0 are drawn, one normal to the spherical surface at its vertex V and the other an arbitrary ray incident at P. The first ray reflects back along itself; the second reflects at P as if from a tangent plane at P, satisfying the law of reflection. The two reflected rays diverge as they leave the mirror. The intersection of the two rays (extended backward) determines the image point I conjugate to O. The image is virtual, located behind the mirror surface. Object and image distances from the vertex are Sec. 3-7

Reflection at a Spherical Surface

43

c

o

Figure 3-15 Reflection at a

surfuce.

shown as sand s', respectively. A perpendicular of height h is drawn from P to the only on the radius of at Q. We seek a relationship between sand s' that curvature R of the mirror. As we shaH see, such a relation is possible only to firstorder approximation of the sines and cosines of the angles made the object and image rays to the spherical surface. This means that in place of the expansion of sinlf'l=lf'l 'T

'T

costp=l

we

3! 2!

+ 5! +

4!

+ ...

(3-8)

~

(3-9)

the first terms only and sin tp

~

tp

and cos tp

1

relations that can be accurate enough if the angle tp is smaH enough. This approximation leads to first-order, or Gaussian, optics, after Karl Friedrich Gauss, who in 1841 developed the foundations of the subject. now to the problem at behand. that two relationships may be obtained from cause the exterior of a triangle equals the sum of its interior are (J

a+tp

and

28

a+a'

which combine to give

-a' =-Ztp

a

Using the small-angle approximation, the their tangents, yielding h s

h = s'

of

(3-10) . (3-10) can be replaced

h

where we have also the axial distance VQ, small when angle Cancellation of h produces the desired relationship,

1 s

1 s'

-2 R

4> is

small.

(3-11)

If the spherical surface is chosen to be concave instead. the center of curvature to would be to the left. For certain positions of the object point 0, it is then find a real image point also to the left of the mirror. In these cases, the resulting geometric relationship analogous to (3-11) consists of terms that are all positive. Chap. 3

Geometrical Optics

It is possible, ernlDlCMfl1! a sign convention, to represent all cases

the

equation 1

-+ S

2 R

Sf

The convention to be used in conjunction with Assume the light from to

(3-12) is as follows:

1. The object distance s is positive when 0 is to the left V, corresponding to a corresponding to a virtual s is negreal object. When 0 is to the 2. The image distance Sf is positive when I is to the left of V, corresponding to a real and negative when I is to the right of V, corresponding to a virtual image. of V, cOlrrel,pclndmg 3. The radius of curvature R is positive when C is to the to a convex mirror, and negative when C is to the left of V, corresponding to a concave mirror. These rules3 can be quickly summarized by noticing that positive object and image distances correspond to real objects and real images and convex mirrors have ....., 0) mirrors. The imas the foeallength of the mirrors. Thus age distance in each case is

f

- !i

{>

concave mirror convex mirror

0, 2 < 0,

(3-13)

and the mirror equation can be written, more compactly, as 1

+ 1

S

s'

1

f

14)

In Figure 3-16c, a construction is shown that allows the determination of the transverse magnification. The object is an extended object of transverse dimension ho • The image of the top of the arrow is located by two rays whose on ray at the vertex must reflect to make equal angles reflection is known. with the axis. The other ray is directed toward the center of curvature along a normal and so must reflect back along itself. The intersection of the two reflected rays occurs behind the mirror and locates a virtual there. Because of the equality of the three shown, it follows that S;

3 While this set of conventions is used. the siudent is cautioned that other schemes ek isl. No one with a continuing involvement in can hope to escape other conventions, nor should the matter be beyond the menlaIIlCAJI""IV of the serious student 10 accommodate.

Sec. 3-7

Reflection at a

Surface

c

leI

(bl

lal

Figure 3-16 Location of focal points (a) and (b) and ronstruction magnification (c) of a spherical mirror.

to

determine

the ratio of lateral image size to cOl'res,DOlfld-

The lateral magnification is defined lateral size,

(3-15)

m

Extending the sign convention to include magnification, we assign a (+) magnification to the case where the image has the same orientation as the object and a magnification when the image is inverted to the object. To a (+) maignm{:auon in the construction of Figure where s' must itself be negative, (3-15) to give general form Si

m

16)

Once the points C and F are image formation a spherical mirror may be determined approximately by methods. 3-17 illustrates several examples that should be examined carefully. The validity of each ray reflection has been established by the discussion above. In each case the of the top of the arrow is located by the intersection of three reflected rays. Example An object 3 cm is placed 20 cm from (a) a convex and (b) a concave spherical mirror, each of 10-cm focal length. Determine the position and nature of the image in each case.

Solution (a) Convex mirror: f = -10 cm and s = +20 cm. I

1

-+s s'

I

f

or

m=

s'

s

s -

f

= _ -~67 =

10)

+0.333 =

= -6.67 cm

~l

The image is virtual (because s I is negative), 6.67 cm to the of the mirror vertex, and is erect (because m is positive) and ~ the size of the object, or 1 cm (b) Concave mirror: f + 10 em and s = + 20 em.

s' Chap. 3

Geometrical Optics

~ s -

f

20 - 10

+20 em

/ (a)

(bl

c

(el

Figure 3-17 Ray diagrams fur mirrors. (a) Realtmage, concave mirror. (b) Virluru concave mirror. (c) Virlual image, convex mirror.

m

$'

20 20

-= --= $

image is real (because $' is positive), 20 cm to the left of the mirror verm is negative) and the same size as the object, or tex, and is inverted happen to be at 2/ 20 cm, the center curva3 cm Image and ture of the

3-8 REFRACTION AT A SPHERICAL SURFACE We turn now to a similar treatment of refraction at a spherical surface, in this case the concave surface Figure 3-18. Two rays are shown emanating from point O. One is an axial ray. normal to the surface at its vertex and so reat P fracted without in direction. The other ray is an arbitrary ray and there to law, (3-17) The two refracted rays appear to emerge from I. In triangle CPO, the exterior angle a Sec. 3-8

Refraction at a Spherical Surface

common intersection, the image 6 1 + cpo In triangle the exte41

c

3-ill Refraction at a s(ilerical surface for which fl2 > fli'

a I fh + 'fl. Approximating for (3-17), we have

rays and substituting for

UUlJ"\l;lIllS for the angles by mSIJeCt:lOn the distance QV in the small angle

~)

HI ( ; _

(J,

and

(3-18)

'fI) =

HI(a neJnei~t

na.",> v ,a

hi -

=

3-

where again

~)

or 8

(3-19)

S'

Employing the same convention as introduced for mirrors (Le., positive distances for real and negative distances for virtual and ......0 .... "') age distance 8 I < 0 and the radius of curvature R < O. If these ",,,,,,,,tii,,,,,, signs are form of understood to to these quantities for the case the refraction may be written as H2 H2+-=....::::...-...:.. 8

becomes a

"'u .........."'"

8'

convex surfaces. When R surface, and 8'

(3-20)

R

= -(::)

- ? 00,

the

!i:ntlerl{,~

surface

8

where 8' is the ammrtemdepth determined previously. For a real (8 > 0), the negative sign in indicates that the image is virtual. The lateral magnifiobject is simply determined by 3-19. cation of an SneIIs law for the ray incident at the vertex and in the smau··an'l!le for angles,

48

Chap. 3

Geometrical

3-19 Construction to determine lateral ma,gnilllcaition at a spherical refracting surface.

The lateral magnification is then

m=

(3-22)

where the negative sign is attached to value corresponding to an inverted image. For the case of a Eq. (3-21) may be incorporated into Eq. (3-22), giving m formed by plane refracting surfaces have the same lateral dimensions and orientation as the object. Example As an extended example of by surfaces, refer to Figure 30 cm from a convex spherical 3-20. In (a), a real object is positioned in surface of radius 5 cm. To the right of the interface, the refractive index is that of water. Before constructing representative rays, we first find the image dis(3-20) and (3-22). tance and lateral magnification of Equation (3-20) becomes 1 1.33 1.33 30+7= 5 R

1

5 em

1<11

fbI

Figure 3·2(} Example of refraction by spherical surfaces. (a) Refraction by a sinI and 2 refer to respherical surface. (b) Refraction by a thick lens. fractions at the first and second surfuces, respectively.

Sec. 3·8

Refraction at a Spherical Surface

49

giving s; +40 cm. The positive sign indicates that image is real and so is located to the of the surface, where real rays of light are refracted. Equation (3-22) becomes

= - (1.33)(+30)

m

1

=

in size to that of the object. 3-20a indicating an inverted image, shows the image as well as several rays, which are now determined. In this example we have that the mediwn to the right of the spherical surface extends far enough so that the image is formed inside it, without further don. Let us suppose now 3-20b) that the second medium is only 10 em forming a thick lens, with a second, concave spherical surface. also of radius 5 cm. The refraction by the first surface of course, unaffected by this change. Inside the lens, therefore, rays are directed as before to form an image 40 cm from the surface. However. rays are intercepted again refracted by the second surface to a different image, as shown. is the convergence of the rays striking the second position of the first image, its location now specifies the appropriate object distance to be for the second refraction. We call the real image for surface (I) a virtual object for surface Then, by the sign convention established previously, we make virtual relative to the second surface, For the second a quantity when Eqs. (3-20) and tion then, Eq. (3-20) becomes I

1.33 --+

$~

-30

or s'

I - 1.33 =----5

+9 cm. The magnification. according to Eq. (3-22), is I

-'----'-'--...:.. = (1)(-30)

m

+2

5

The final is then ~ the lateral of its (virtual) and appears with the same orientation. Relative to the original object, the final is ~ as large and inverted. In of reflecting or surfaces is involved in the individual reflections andJor are conthe processing of a final sidered in the order in which light is actually incident upon them. The object distance of the nth step is determined by image distance for (n - l)st step. If image of (n - I) is not actually formed, it serves as a virtual object for nth step. 3-9 THIN LENSES We now apply the method to discover the thin-lens equation. As in the example of Figure 3-20, two refractions at spherical surfaces are involved. The simplification we make is to the thickness of the lens in comparison with the object image an approximation that is jU!>1ified in most practical situations. At the first refracting surface, of radius R! , nl $1

50

Chap. 3

Geometrical Optics

+

si

(3-23)

surface, of radius R 2 ,

and at the

-n2 +

nl - n2

=

(3-24)

-'---'=

S2

R2

We have assumed that the lens mces the same medium of refractive index nl on both sides. Now the second object in is given by (3-25) where t is the tnu~Kn.ess of the lens. Notice that this relationship produces the correct sign and also when the intermediate image mils inside or to t, the thin-lens approximation, (3-26) When this value added, the terms

Now

S2

is substituted into Eq. (3-24) and Eqs. (3-23) and (3-24) are

nds: cancel and there results

Sl is the original object distance and their and write simply

; + sl,

s~

n2 nl nl

is the final image UJ.,,'''u ........ so we may

(~I - ~J

The focal length of the thin lens is defined as the infinity, or the distance for an image at ,ni',nd·"

!

=

f

(3-27) distance for an object at

(l.. _l..)

n2 - nl nl Rl

(3-28)

R2

Equation (3-28) is called the lensmaker's equation because it predicts the focal fabricated with a given index and radii of curvature and length of a used in a medium of refractive index nl . In most cases, the ambient medium is air, and nl = L The thin-lens equation, in terms of the focal length, is 1

I

I

s

s'

f

-+-=-

(3-29)

Wclvefront analysis for plane wavefionts, as shown in 3-21, indicates that a lens in the middle causes convergence, while one thinner in the middle

F

lal (>0 Figure 3-21 fronts of

Sec. 3-9

Ib) f
Action of {'1)nvf'l">!inl! lens (a) and diverging lens (b) on plane wave-

Thin lenses

51

causes The portion of the wavefront that must pass through the region is relative to the other Converging lenses are characterized by positive focal lengths and diverging lenses by focal as is evident from the figure, where the are rea] and virtual, respectively. Sample ray diagrams for converging (or convex) and (or concave) lenses are shown in Figure 3-22. The thin are best represented. for purposes of ray construction, by a vertical line with suggesting the general of the lens. Ray (I) the of the object is incident parallel to the and converges, in 3-218, the focal or diverges, in Figure 3-22b, as if proceedfrom the focal point. A second ray is simply the inverse of the first. Although may also drawn through the two rays are sufficient to locate center of lenses without of the lens behaves as a lei plate, does not the direction of the incident ray, and because it is thin, that, displaces the ray by a negligible amount. In constructing ray diagrams, for the central ray, every ray refracted by a convex lens bends toward the axis by a concave lens bends away from the axis. From either diaand every ray gram, the angles sub tended by and at the center of lens are seen to be equal. For the real virtual image in (b), it follows that 1lC>'dlj,VO::;

la)

..

(b)

Figure 3-22 cave lens (b).

52

Chap. 3

diagrams for

Geometrical Optics

formation by a convex lens (a) and a con-

and lateral magnification

hi

s'

m=-=It" s In accordance with the sign convention adopted here, a negative sign should be added to this expression. In case (a), .'I > 0, s' > 0, and m < 0 because the image is inverted; in case (b), .'I > 0, .'I' < 0, and m > O. In either case then,

m=

.'I'

(3-30)

Further ray-diagram examples for a train of two lenses are illustrated in Figure 3-23. Table 3-1 and Figure 3-24 provide a convenient summary of image formation in lenses and mirrors.

(2)

la)

(1)

(b)

Figure 3·23 (a) Formation of a virtual image by a two-element train of a convex lellS (I) and concave lens (2). (b) Formation of a real image Rh by a train of two convex lenses. The intermediate image RlI serves as a virtual object V02 for the second lens.

Sec. 3-9

Thin lenses

53

TABLE 3-1

SUMMARY OF GAUSSIAN MIRROR AND LENS FORMULAS

Spherical surfuce

I s

I s'

Plane surface

R 2

-+-

s' =-s

s' s Concave:! > 0, R < 0 Convex: ! < 0, R > 0

m=--

Reflection

III 112 112 III -+-=---

s'

m=

m

s

Refraction Single surface

m = +1

s'

R

1l2 S

+1

Concave: R < 0 Convex: R > 0

I

-+ s s'

I

=-

!

~J

I

Retraction Thin lens

!

m

s'

s

Concave:! < 0 Convex: ! > 0

2F

F

RO

.-----~

c

F

fbI

(a)

3-24 Summary of formation by spherical mirrors and thin lenses. location, nature, and orientation of the image are indicated or suggested. (a) Spherical mirrors. (b) Thin lenses.

Example and describe the intermediate and final produced by a two-lens 3-23a. 15 cm,h 15 ern, system such as the one sketched in and their be 60 ern. Let the object be 25 ern from the first lens, as shown. 54

Chap. 3

Geometrical Optics

Solution The first lens is convex:f]

1 Sl

1 _ 1

+ --; - f SI

or

= + 15 cm,

Sl

,

25 cm. em

SI

37.5 = 25

1.5

s:

Thus the first image is real (because is positive), em to the of the first lens, inverted m is negative), and 1.5 times the size of the object. = -15 em. Since real rays of light diThe second lens is concave: verge from the first real image, it serves as a real object for the second with S2 = 60 37.5 = cm to the left of lens.

-9 22.5

+0.4

Thus the final is virtual (because is negative), 9 em to the left of second erect with respect to its own object 0.4 times its size. The overall magnification is (-1.5)(0.4) = -0.6. Thus the final image is inverted to the original and 6/10 its lateral size. AU these features are exhibited qualitatively in the ray diagram of Figure 3-23a.

3-10 VERGENCE AND REFRACTIVE POWER Another way of interpreting thin-lens equation is useful in certain applications, including optometry. The interpretation is based on two considerations. In the thinlens equation, 1

I 1 =s' f

- +S

(3-31)

notice (I) the of distances in the left member add to give the and (2) the reciprocals of the object and image distances rocal of the focal describe curvature of the wavefronts incident at the lens and centered at the oband positions 0 and I, respectively. A plane wavefront, for example, has a curvature of zero. In 3-25 waves expand from the point 0 and attain a curvature, or vergence, V, by I/s, when intercept the thin lens. On the other hand, once refracted by the lens, the wavefronts contract, in Figure 3-25a, and expand further, in Figure 3-25b, to locate the real and virtual image points shown. The curvature, or vergence, V', of the wavefronts as they emerge from the lens is 1/s '. The change in curvature from object space to space is due to the refracting power P of the given 1/f. With these definitions, Eq. (3-31) may written V

+ V' = P

(3-32)

units terms in Eq. are reciprocal lengths. When the lengths are measured in meters, their reciprocals are said to have units of diopters (D). Thus the

Sec. 3-10

Vergence and Refractive Power

55

2F 0

~-

s

lal

o

1-<1----

s - - - ....

:~,--

-s' ---

~

Figure 3·25 Change in curvature of wavefronts on refraction by a thin lens. (a) Convex lens . (b) Concave lens.

(b)

refracting power of a lens of focal length 20 cm is said to be 5 diopters. This alternative point of view emphasizes the degree of wave curvature or ray convergence rather than object and image distances. Accordingly, the degree of convergence V' of the image rays is determined by the original degree of convergence V of the object rays and the refracting power P of the lens. Eq. (3-32) can also be applied to the case of refmction at a single surface, Eq. (3-20), in which case the refmctive indices in object and image space need not be I. In this event, the power of the refracting surface is (n2 - nl)/R. This approach is useful for another reason. When thin lenses are placed together, the focal length f of the combination, treated as a single thin lens, can be found in terms of the focallengthsfl ,f2, ... of the individual lenses. For example, with two such lenses back to back, we write the lens equations I

I

s.

Sa

I fi

-+-=-

and

I

I

I

-+-=S2 s~ h

Since the image distance for the first lens plays the role of the object distance for the second lens, we may write Sz

=

,

-SI

and, adding the two equations.

1

I

I

I

I

-+-=-+-=SI S~ fl h f Chap. 3

Geometrical Optics

The reciprocals of the individual focal lengths, therefore, add to of the overall focal length f of the pair. In for several thin III I -=-+-+-+ ... f j; /3

Expressed in diopters, the refractive powers simply add.

+ P2 + P3 + ...

PI

P

(3-34)

In a nearsighted eye, the refracted (converging) power of the lens is too so that a real image is formed in front of the retina. the convergence with a number of diverging lenses placed in front an object is cleady focused, an optometrist can the net of the single corrective lens needed by simply adding the diopters of test lenses. In a farsighted eye, the natural converging power of the eye is not and additional of "1A"'''''''i~'''' converging power must be added in the

3-11 NEWTONIAN EQUATION FOR THE THIN When object and distances are measured as by the distances x and x' in equation results, called the lVelWW'nlafn mine two right triangles, each pair constitutes similar represent the lateral magmnc~ulon: h.,

Introducing a

x'

and

X

to the focal points of a lens, form of the thin-lens

::::::-

h.,

f due to the inverted image,

np(J,,,fnJP<

l= x

m

x'

f

(3-35)

Newtonian form of the thin-lens

The two parts of Eq. equation,

xx' :::::: This equation is sOlne1wruat !:lmr,'PI" than ient in certain applications.

(3-36) (3-29) and is found to be more conven-

Figure 3-26 Construction used to derive Newton's equations for the thin lens.

Sec. 3-11

Newtonian

for the Thin lens

51

3-1. Derive an expression for the transit time of a ray of that travels a distance XI through a medium of index NI, a distance X2 through a medium of index N2 • • • • • and a distance Xm through a medium of index N",. Use a summation to express your result. 3-2. Deduce the Cartesian oval for by a surface when the object point is on the optical x- axis 20 em from the surface vertex and its conjugate point lies 10 cm inside the second medium. Assume the refracting medium to have an index of 1.50 and the outer medium to be air. Find the equation of the intersection of the oval with the xy-plane, vvhere the origin of the coordinates is at the object point. Generate a table of (x, y)- coordinates for the surface and plot, together with sample rays. 3-3. A double convex lens has a diameter of 5 em and zero thickness at its A point object on an axis through the center of the lens produces a real image on the opposite side. Both object and image distances are 30 em, measured from a plane bisecting the lens. The lens has a refractive index of 1.52. Using the equivalence of optical through the center and of the lens, determine the thickness of the lens at its center. 3-4. Determine the minimum height of a wall mirror that will permit a 6-ft person to view his or her entire Sketch rays from the top and bottom of the person, and determine the proper placement of the mirror such that the full is seen, regardless of the distance from the mirror. 3-5. A ray of light makes an of incidence of 45° at the center of the top surface of a transparent cube of index 1.414. Trace the raj' through the cube. 3-6. To determine the refractive index of a transparent plate of glass, a microscope is first focused on a tiny scratch in the upper and the barrel position is recorded. Upon further lowering the microscope barrel by 1.87 mm, a focused image of the scratch is seen The thickness is 1.50 mm. What is the reason for the second image, and what is the refractive index of the 3-7. A small source of light at the bottom face of a rectangular slab 2.25 em thick is viewed from above. Rays of light totally internally reflected at the circle of 7.60 em in diameter on the bottom surface. Determine the rell,dCllve the 3-8. Show that the lateral displacement s of a ray of light penetrating a rectangular plate of thickness t is given by

s

t sin '--"-_...2= __

cos 62 of incidence and refraction, respectively. Find the diswhere 6 1 and {Jz are the 1.50, and 6 1 = 50°. pl8!:errlCnt when t = 3 cm, 11 40 em, with 3-9. A meter stick lies along the optical axis of a convex mirror of focal of the meter its nearer end 60 cm from the mirror surface. How is the stick'! 3-10. A glass hemisphere is silvered over its curved surface. A small air bubble in the glass is located on the central axis the hemisphere 5 cm from the plane surface. The radius of curvature of the spherical surface is 7.5 cm, and the glass has an index of 1.50. Looking along the axis into the plane one sees two of the bubble. How do arise and where do they 3-11. A concave mirror forms an on a screen twice as as the object. Both object and screen are then moved to an image on the screen that is three times the size of the object. If the screen is moved 75 cm in the process, how far is the object moved'! What is the focal of the mirror'!

58

Chap. 3

Geometrical

3-12. A sphere 5 cm in diameter has a small scratch on its surface. When the scratch is viewed the glass from a position directly where does the scratch appear and what is its magnification? Assume n = 1.50 the glass. 3-13. (a) At what position in front of a refracting surface must an object be placed so that the refraction produces parallel rays of light? In other words, what is the focal of a single refracting surface? (b) Since real object distances are positive, what does your result for the cases nz > nl and nz < nl? 3-14. A small goldfish is viewed a spherical fishbowl 30 cm in diameter. Determine the apparent position and magnification of the fish's eye when its actual position is (a) at the center of the bowl and (b) nearer to the eye, halfway from center to along the line of Assume that the glass is thin enough so that its effect on the refraction may be ne~~ected 3-15. A small faces the convex spherical glass window of a smalJ water tank. The radius of curvature of the window is 5 cm. The inner back side of the tank is a plane mirror, 25 cm from the window. If the object is 30 cm outside the window, determine the nature of its final neglecting any refraction due to the thin window itself. 3-16. A plano-convex Jens having a focal of 25.0 cm is to be made with ofrefractive index 1.520. Calculate the radius of curvature of the grinding and polishing tools to be used in making this lens. 3-17. Calculate the focal length of a thin meniscus lens whose spherical surfaces have radii of curvature of 5 and 10 cm. The glass is of index 1.50. Sketch both and negative versions of the lens. 3-18. One side of a fish tank is built a large-aperture thin lens made of glass (n 1.50). The lens is equiconvex, with radii of curvature 30 cm. A small fish in the tank is 20 cm from the lens. Where does the fish appear when viewed through the lens? What is its magnification? 3-19. Two thin lenses have focal of -5 and +20 cm. Determine their equivalent focal lengths when (a) cemented togleth(~r and (b) by 10 cm. 3·20. Two identical, thin, plano-convex lenses with radii of curvature of 15 cm are situated with their curved surfaces in contact at their centers. The intervening is filled with oil of refractive index 1.65. The index of the is 1.50. the focal length of the combination. (Hint: Think of the oil layer as an intermediate thin lens.) 3·21. An is made oftwo thin lenses each of +20-mm focal separated by a distance of 16 mm. (8) Where must a small object be positioned so that light from the is rendered parallel by the combination? (b) Does the eye see an image erect relative to the Is it magnified? Use a ray dia:grlllm to answer these by inspection. 3-22. A thin lens and a concave mirror have focal lengths of equal magnitude. An object is 3f /2 from the diverging lens, and the mirror is placed a distance the other side of the lens. Gaussian determine the final image of the system, after two refractions (a) by a three-ray diagram and (b) by calculation. 3-23. A small object is placed 20 cm from the first of a train of three lenses with focal lengths, in order, of 10, 15, and 20 cm. The first two lenses are by 30 cm and the last two by 20 cm. Calculate the final image relative to the last lens and its linear magnification relative to the original object when (a) all three lenses are positive; (b) the middle lens is negative; (c) the first and last lenses are nelmllve_ Provide ray for each case. 3-24. A convex thin lens with refractive index of 1.50 has a focal length of 30 cm in air. When immersed in a certain liquid, it becomes a negative lens with a focal length of 188 cm. Determine the index of the liquid. 3-25. It is desired to project onto a screen an that is four times the size of a brightly illuminated object. A plano-convex lens with n = 1.50 and R = 60 cm is to be used.

3

Problems

59

Employing the Newtonian form of the lens equations, determine the appropriate distance of the and screen from the lens. Is the image erect or inverted? Check your results using the ordinary lens equations. 3-26. Three thin lenses of focal 10 cm, 20 em, and -40 cm are in contact to form a single compound lens. (8) Determine the powers of the individual lenses and that of the unit, in diopters. (b) Determine the vergence of an object 12 cm [rom the unit and that of the resulting Convert the result to an image distance in centimeters. 3-27. A lens is moved along the axis between a fixed object and a fixed image screen. The object and image positions are separated by a distance L that is more than four times the focal length of the lens. Two positions of the lens are found for which an is in focus on the screen, magnified in one case and reduced in the other. If the positions differ distance D, show that the focal length of the lens is = (V - D2)/4L. This is Bessel's method for finding the focal length of a lens. 3-28. An image of an is formed on a SCreen a lens. the lens the object is moved to a new position and the image screen moved until it focused If the two positions are Sl and Sz and if transverse magnifications of the image are MI and M 2 • show that the focal length of the lens is given

f ==

(lIM, - 11M2)

This is Abbe's method for the focal length of a lens. 3-29. Derive the law of reflection from Fermat's minimizing the distance of an arbitrary (hypothetical) ray from a given source point to a given point. 3-30. Determine the ratio of focal lengths for two identical. thin plano-convex lenses when one is silvered on its flat side and the other on its curved side. Light is incident on the unsilvered side. 3-31. Show that the minimum distance between an object and its image, formed by a thin lens, is When does this occur? 3-32. A ray of light traverses successively a series of interfaces, all to one another and separating regions of differing thickness and refractive index. (8) Show that Snell's law holds between the first and last as if the did not exist. (b) Calculate the net lateral displacement of the ray from point of incidence to point of emergence. 3-33. A parallel beam of light is incident on a plano-convex lens that is 4 cm thick. The radius of curvature of the spherical side is also 4 cm. The lens has a refractive index of 1.50 and is used in air. Determine where the light is focused for light incident on each side. 3·34. A spherical interfuce, with radius of curvature 10 cm, separates media of refractive index I and ~. The center of curvature is located on the side of the higher index. Find the focal lengths for incident from each side. How do the results differ when the two refractive indices are interchanged? 3-35. An airplane is used in aerial surveying to make a of ground detail. If the scale of the map is to be I :50,000 and the camera used has a length of 6 determine the proper altitude for the photograph.

[1 J Feynman, Richard P., Robert B. and Matthew Sands. The Lectures on Vol. l. Reading, Mass.: Addison-Wesley Publishing Co., 1975. Ch. 26, 27.

60

3

Geometrical Optics

[2] Smith, F. Dow. "How Images are Formed." In Lasers and Light. Readings from SciAmerican, pp. 59-10. San Francisco: W. H. Freeman and Company P"lhli.,h...", 1968. [31 Winston, Roland. "Norumaging Optics." Scientific American (March 1991): 16. (4] Longhurst, R. S. Geometrical and Physical Optics, 2d ed. New York: John Wiley and 1961. Ch. 1,2. [5] Bruno. Optics. Reading, Mass.: Addison-Wesley Publishing Company, 1957. Ch.1,2.

3

References

61

4

Matrix Methods in Paraxial Optics

INTRODUCTION

This chapter deals with methods of analyzing systems when they become complex, involving a number of refracting and/or reflecting elements in trainlike fashion. Beginning with a description of a thick lens in terms of its cardinal points, the discussion proceeds to an analysis a train of elements by means of mUltiplication of 2 x 2 matrices the elementary refractions or reflections involved in the train. In this way, a system matrix for the entire optical same cardinal points characterizing the system can be found that is related to thick lens. Finally, computer ray-tracing methods for tracing a given ray of light through an optical are described. 4-1 THE THICK LENS

Consider a spherical thick that a lens its optical axis Just when a lens cannot be ignored without leading to serious errors in moves from the of thin to thick dearly depends on the accuracy required. The thick lens can be treated by the methods of Chapter 3. The glass medium is bounded by two spherical refracting surfdces. The image of a given object, formed refraction at the first becomes the object for refraction at the second surface. The object distance for the second surface takes into account the of 62

the lens. The image formed by the second surface is then the final image due to the action of the composite thick lens. The thick lens can also be described in a way that allows graphical determination of images corresponding to arbitrary objects, much like the ray rules for a thin lens. This description, in terms of the so-called cardinal points of the lens, is useful also because it can be applied to more complex optical systems, as will beeome evident in this chapter. Thus, even though we are at present interested in a single thick lens, the following description is applicable to an arbitrary optical system that we can imagine is contained within the outlines of the thick lens. There are six cardinal points on the axis of a thick lens, from which its imaging properties can be deduced. Planes· normal to the axis at these points are caJled the cardinal planes. The six cardinal points (see Figures 4-1 and 4-2) consist of the first and second system focal points (F. and F2 ), which are already familiar; the first and second principal points (H t and Hz); and the first and second nodal points (Nt and N2 ). I

I

-;.J,c'~~--

IH, I

F,

F2

F,

IH2 I I I

I

I

PP,

F2

PP2

(al

Figure 4-1 system.

-,

--":::-.:::-:J...-

Ibl

Illustration of the (a) first and (b) second principal planes of an optical

Figure 4-2 Illustration of the nodal points of an optical system.

A ray from the first focal point F t is rendered parallel to the axis (Figure 4-1a), and a ray paraJlel to the axis is refracted by the lens through the second focal point F2 (Figure 4-1 b). The extensions of the incident and resultant rays in each case intersect, by definition, in the principal planes, and these cross the axis at the principal points, H. and H 2 • If the thick lens were a single thin lens, the two principal planes would coincide at the vertical line that is usually drawn to represent the lens. Principal planes in general do not coincide and may even be located outside the optical system itself. Once the locations of the principal planes are known, accurate ray diagrams can be drawn. The usual rays, determined by the focal points, bend at their intersections with the principal planes, as in Figure 4-1. The third ray usually drawn for thin-lens diagrams is one through the lens center, undeviated and negligibly displaced. The nodal points of a thick lens, or of any optical system, permit the corree-

, These "planes" are actually slightly curved surfaces that can be considered plane in the paraxial approximation.

Sec. 4-'

The Thick lens

63

tion to this ray. as shown in Figure 4-2. Any ray directed toward the first nodal point M emerges from the optical system parallel to the incident ray, but displaced N2 • so that it appears to come from the second nodal point on the The positions of aU six cardinal points are indicated in Figure 4-3. Distances are directed, positive or negative by a sign convention that makes distances directed to the right, positive. Notice that for the thick lens, to the left negative and the distances rand S determine the positions of the principal points relative to the Vl and Vz, while j; and /2 determine focal point positions relative to the principal points. Note carefully that these focal points are not measured from the vertices of the lens.

Figure 4-3 Symbols used to signify the cardinal points and 100000ions for a Ihick lens. Axial include focal points (P), vertices (V). principal points (P). and nodal points (N). Direeted distances separating their corresponding planes are defined in the drawing.

We summarize the basic equations for the thick lens without proof. Although the invo]ve simple a]gebra and they are rather arduous. We shall be content to await the matrix approach in this chapter as a simpler way to justify these equations, and even then some of the work is relegated to the problems. Utilizing the symbols defined in Figure 4-3, the focal length/I is given by - =

t

-=---

(4-1)

and the focal length /2 is conveniently expressed in terms of j; by

n'

(4-2)

n

Notice that the two have the same magnitude if the lens is surrounded by a single ...,.f"'r'
r

and

S

(4-3)

=

The positions of the nodal points are given by n' n

+

nL - n' ) Rtj;andw nL 2

(4-4)

and object distances and lateral magnification are related by

So

+

=landm= Si

ns; n'se

(4-5)

as long as the distances So and Sj, as well as focal lengths, are measured relative to with n = n' :::: principal planes. In the ordinary case of a lens in I, notice that r = v and S = w: First and second points are superimposed Chap. 4

Matrix Methods in Paraxial Optics

over corresponding nodal magnitude, and equations,

Also, first and second focal lengths are equal in

- -I + -1 = So

Si

1

and

Si

m = --

/

(4-6)

So

are valid, with symbols properly reinterpreted. for one negative reare identical with the thin-lens equations. quired by the sign convention, Example Determine the focal lengths and principal points for a 4-cm thick, biconvex lens with index of 1.52 and radii of curvature of 25 cm, when 1.33). lens caps the end of a long cylinder filled with water (n Solution Use the equations for the thick lens in the order given: 1.52

1.33

/1 or fi = - 35.74 cm to

1.52 1 1(+25)

4 (-25)(+25)

left of the first principal plane.

C·;3)(-35.74) = 47.53 cm to the right of the second 1.52 - 1.33 r = (1.52)(-25)

and

= 0.715 cm -2.60 cm

Thus the principal point HI is situated 0.715 cm to the of the left vertex of the lens, and is situated 2.60 cm to the left of the right vertex. 4-2 THE MATRIX METHOD

When the optical system consists of several elements-for example, the four or five lenses that constitute a photographic lens-we need a systematic that facilitates analysis. As long as We restrict our to paraxial rays, this apthe matrix method. We now a treatment image is well handled formation that employs matrices to describe changes in height and angle of a ray as it makes its way by successive reflections and refractions through an optical system. We show in the paraxial approximation, changes in height and direction of a ray can be by linear equations that make this matrix approach possireflections, a ble. By combining matrices that represent individual refractions optical system may be by a matrix, from which the essential properties of the composite optical system may be deduced. The method lends of arbitrary comto techniques for tracing a ray through an optical plexity. Figure 4-4 the progress of a ray through an arbitrary optical system. The ray is described at distance Xo from the first refracting surface in terms of height yo and slope ao relative to the optical axis. Changes in angle occur at each such as at 1 through 5, and at reflection, such as 6. The of the ray changes translations between these We look for Sec. 4-2

The Matrix Method

65

Figure 4-4 be described by

in tracing a ray an optical system. Progress of a ray can in its elevation and direction.

a procedure that win aHow us to calculate the height and slope angle of any in the optical for example, at point T, a distance X7 ror. In other words, given input data Yo, ao at point 0, we wish to of Y7. a7 at point 7 as data.

at

4-3 THE TRANSLATION MATRIX

Lonsl.rler a simple translation of the ray in a medium, as in Figure 4-5. axial progress of the ray be L, as such that at point I, the elevation and direction of the ray are given by "coordinates" YI and at, respectively. dendy, ao and Yl = Yo + L tan ao These equations may be put into an ordered form, where the paraxial approximation tan ao == ao has been used: al

YI = (I)yo al

(O)yo

+ (L)ao + (l)ao

(4-7)

In matrix notation, the two equations are written

[~:J = [~ ~][~:J

(4-8)

EvidenUy, the 2 X 2 ray-transfer matrix represents the of the translation on the ray. The input data (yo, ao) is modified by the ray-tnmster matrix to the correct output data (Yl, al).

o Optical IIxis

4·5 Simple translation of a ray.

4-4 THE REFRACTION MA TRIX

Consider next the refraction of a ray at a spherical surfuce separating media of tive nand n " as shown in Figure 4-6. We need to relate the ray coordinates 66

Chap. 4

Matrix Methods in Paraxial

¢

Optical axis

'"

c

Figure 4-6 surfuce.

Refraction of a ray at a spherical

(y', a ') after refraction to those before refraction, (y, a). Since refraction occurs at a point, there is no change in elevation, and y = y'. The angle a', on the other hand, is by inspection of Figure 4-6

a'

=

(J' - c/J

=

(J' - 1" and a R

=

(J - c/J

=

(J - 1" R

Incorporating the paraxial form of Snell's law, n8 = n'(J'

we have

a'

=

Cn.)(J -

~=

(:,) (a

+~)

or

The appropriate linear equations are then y' = (l)y

+

(O)a

(4-9) or. in matrix form,

(4-10) We use the same sign convention as designed earlier. If the surface is instead concave, R is negative. Furthermore, allowing R -l> 00 yields the appropriate r~frac­ tion matrix for a plane interface. 4-5 THE REFLECTION MA TRIX Finally, consider reflection at a spherical surface, illustrated in Figure 4-7. In the case considered, a concave mirror. R is negative. We need to add a sign convention for the angles that describe the ray directions. Angles are considered positive for all Sec. 4-5

The Reflection Matrix

61

y

~

y'

c

4·7 Reflection of a ray convention for ray

aI

a spherical surface. The inset illustrates the

rays pointing upward, either before or after a angles rays pointing downward are considered negative. The sign convention is summarized in the inset of Figure 4-7. From the geometry of Figure with both 0: and 0: ' positive,

a=6+¢

6

+ Land -R

0:' = 6'

¢ = 6' - L

-R

these relations together with the Jaw of reflection, 6 = 6',

a'=6,+1=6+ 1 R R

O:+R

and the two desired linear equations are y' = (l)y

a'

+ (0)0:

(~)y + (1)0:

(4-11)

In matrix (4-12)

4-6 THICK-LENS AND THIN-LENS MATRICES

We construct now a matrix that the action of a thick lens on a ray of light. For generality, we assume different media on sides of the having refractive indices nand n', as in Figure 4-8. In traversing lens, the ray undergoes two refractions and one translation, for which we have already derived matrices. Referring to Figure 4-8, where we have chosen for simplicity a lens with nn<:ltn"p radii of curvature, we may symbolicaHy,

[~:J = M{~J [~:] = M{~:J 68

Chap. 4

for the first refraction for the translation

Matrix Methods in Paraxial Optics

and

[~:J

=

M{ ~:J

Tellesc:oping these matrix

for the second

r.,1""",,,'tm,n

resuJts in

M 3 M 2 M 1 • Rethe entire thick lens can represented by a matrix M that the multiplication of is associative but not the deorder must be maintained. The individual matrices on the light ray in the same order in which the corresponding optical actions influence the light ray as it traverses the system. the matrix equation any numN translations, reflections, and is given by

[~J = and the

ra"V'-tl1llns~ter

I •••

M2M[~:J

(4-13)

matrix represemmg the entire optical M = MNMN- 1

•••

is

(4-14)

M2MI

this result first to the thick lens of Figure 4-8, whose index is nL and whose paraxial rays is t. The correct approximation for a thin lens is then made by allowing t --,» O. Letting ffi a refraction matrix and f'J a translation matrix, the matrix for the lens is, by Eq. (4-14), the composite matrix

M or

1

M= n'R2

[: :]

1 - nL nLR I

~J

15)

We simplify immediately for the case where t is negligible (t 0) and where the lens is surrounded by the same medium on either side (n = n '). Then M

Figure 4-8

Sec. 4-6

(4-16)

Progress of a ray through a thick lens.

Thick-lens and Thin-lens Matrices

69

TABLE 4-1

SUMMARY Of SOME SIMPLE RAY-TRANSfER MATRICES

Translation matrix: M =

[~

Refraclion matrix, interface:

M=

[n ~ n' :] Rn'

n'

I+RI:convex

1- R I : concave

Refraction matrix, interface:

----n

n'

Thin-lens matrix:

(+" : convex 1- f I : concave

S!ilerical mirror matrix:

I+RI:convex 1- R ) : concaw

10

Chap. 4

Matrix Methods in Paraxial Optics

When multiplied together,

element in the first column. second row, may be expressed in terms of the lens, by the lensmaker's formula, 1

f so that the thin-lens ray-transfer

is

(4-18)

f

is taken as positive for a convex lens and negative for a concave lens. together with those previously derived are summarized reference in Table 4-1.

4-7 SYSTEM

combining appropriate individual matrices in the proper order, any by a single 2 x 2 14), it is possible to we call the system matrix.

to which

Example Find the system matrix for the thick lens of Figure 4-8, whose matrix multiplication is expressed (4-15), and specify the thick lens choosing RI = 45 em, R2 em, t = 5 em, TIL = 1.60, and n

Solution

M=

[~

or M = [

50 The elements of this cOlmposite symbol ic form

M

I1ly-t:rarlsh~r

matrix, usually

23 24

2:]

1;00

17 16

reh~rrt~d

to in the

[~~J

describe the relevant of the optical system, as we will see. Be aware that the particular values of the matrix elements of a system depend on the IOCam)fl the my at input and output. In the case of the thick lens just calculated, the input was chosen at the left surface of the lens, and the output was chosen at sumce. If of these is moved some distance from the the an initial and a final translation matrix incorporating

Sec. 4-7

System

71

these distances. The matrix elements and the system matrix now rep,re81ents this enlarged "system." In any case, the determinant of the system matrix has a very useful property: AD - Be

DetM

=

TIo

nt

(4-19)

where TIo and flJ are the refractive indices of the initial and final media the optical "",•..,,,,,. The proof of this follows upon noticing first that the determinant of all the individual in Table 4-1 have values of n/n' or and then making use the that the determinant of a product of matrices is equal to the product of the determinants. Symbolically, if M = MIM2M3 ••• MN , then Det (M) =

(4-20)

In forming this product, using determinants of ray-transfer aU intermediate refractive indices cancel, and we are left with the ratio no/nf, as stated in Eq. (419). Most often, as in the case the thick-lens example, TIo and nt refer to and Det (M) is unity. The expressed by Eq. (4-19) is in '-..'-'-'lUIlli'> the correctness of the calculations that produce a system matrix.

4-8 SIGNIFICANCE OF SYSTEM MA TRIX EU:MJ:NJ"S We examine now the implications that follow when each of the matrix elements is zero. In symbolic we from Eq. (4-13), (4-21) which is equivalent to the

relations Yt

Ayo

at

CYo

+ +

Baa Dao

(4-22)

1. D = O. In this case, at = Cyo. independent of ao. Since yo is this means that all rays a point in the input plane will have the same angle at at the output plane, of their angles at input. As shown in Figure 4-9a, the with the first focal plane the optical system. 2. A = O. This case is much like the previous one. Here Yt Bao implies that Yt is independent of Yo, so that all rays departing the input plane at the same angle, regardless of altitude, arrive at the same altitude Yt at the output plane. As shown in the output thus as the second focal plane. 3. B O. Then Yt Ayo, independent of aa. Thus aU rays from a point at height yo in the input arrive at the same point of height Yt in the output plane. The points are then related as object and points, as shown in 4-9c, and the input and output planes to conjugate planes for the

theorem can be verified for the product of two matrices and generalized by induction can be found in any standard textbook on mato the product of any number of matrices. Formal trices and determinants, for E. T. Browne, Introduction to the Theory of Determinants and Matrices Hill: of North Carolina, 1958).

72

Chap. 4

Matrix Methods in Paraxial

y,

Axis

Optical elements f - o l - - - - Optical_ system

system

(a)

(b)

Output plane

Optical elements f-ol-_ _ _ _ Optical _ _ _-Jo-i

system

Output plane

Ie)

(d)

4-9 illustrating the of the vanishing of system matrix elements. (a) When D = 0, the corresponds to the first focal plane of the optical system. (b) When A output plane corresponds to the second of the optical system. (c) When 8 0, the output is the image corIJu~~te to the input plane, and A is the linear (d) When 0, a bundle of rays al lhe plane is parallel at the output and D is the angular magnification.

optical system. Furthermore. since A = y,/Yo. the matrix element A represents the magnification. 4. C = O. Now Cif Dao. independent yo. This case is analogous to case 3, with replacing ray Input rays, all of one direction, now produce output rays in some other direction. Moreover, D = atlao is the angular magnification. A which C = 0 is sometimes called a admits rays into its objective "telescopic system," because a from its eyepiece. and outputs parallel rays for Example We illustrate case 3 by an example. We place a small at a distance of 16 cm from the left end of a plastic rod with a polished spherical end of 4 cm, as indicated in 4-10. The refractive index of the plastic is Output plane

Input plane

R

= 4cm

n

Sec. 4-8

I

1.50

Figure 4-10 Schematic defining an example for ray-transfer matrix methods.

Significance of System Matrix Elements

13

1.50 and the object is in air. Let the unknown image be formed at the output reference plane, a distance x from the spherical cap. We desire to determine the distance x and the lateral magnification. The system matrix consists of the product of matrices, to (I) a translation :11 in air from to the a refraction at the &-pherical and (3) a translation :12 in plastic to the image. Remembering to take the matrices in reverse order, we have M == :12 rtlt:1 I

M ==

I X][ 1 0][1 16] [o 4(1.50) 1.50 0 1 1

1 - 1.50

1

or

with the unknown quantity x incorporated in the elements. ACCOl'diIIJ?, to this when B = the output plane is the image plane, so that the distance is by setting

16

2x 3

o

or x

24cm

Further, the linear magnification is then given by the value of element A:

m

A

x

= 1-12

-I

We conclude that the

occurs 24 cm inside the rod, is inverted, and has as the object. This illustrates how the system matrix can nPrtnrlm the customary job of finding locations and sizes, although this may usually done more quickly by using the Gaussian formulas derived earlier.

4-9 LOCATION OF CARDINAL POINTS FOR AN OPTICAL SYSTEM nrc''''''' .... '''' of an optical system can be deduced from the elements of the ra\i'-tr:aru,fer matrix, it foHows that relationships must exist between the matrix A, C, and D and the cardinal of the system. In 4-11, we Figure 4-3 by defining distances the six cardinal relative to the input and planes that define the limits of an optical system. The focal points are located at distances /1 and j; from the principal and at distances p and q from the reference input and from the and output planes, the rand s the principal points and the v and w locate the nodal points. Distances measured to the right of their reference pJanes are considered positive and to the left, negative. The principal and nodal points often occur outside the optical system, that is, outside the region by the input output planes.

74

Chap. 4

Matrix Methods in Paraxial Optics

Output plane

J Optical t I r+---r-f>oi

.>0---+-) system , I

I I

I I t-r-....l I ~v--..l I

I I I lIC~11:\1Ii1UU."~

for the six cardinal points of an sysand are also shown.

associated with the nodal

We now the relationships between the distances defined in Figure 4-11 and the system matrix elements. Consider 4-12a, which distances p. r. and II, as they are by the of the first point and the of the ray are (Yo, and output coorfirst plane. Input 0). Thus the ray equations, Eqs. (4-22), become for this ray

ao)

Bao

+

y, and

o For small

+

Dao

or

yo

=

-(~)ao

(4-23)

Figure 4-12a shows that

-p where the negative sign indicates that FI is located a distance p to the left of the input plane. Incorporating Eq. D C

p

Similarly, ao =

), and thus

Ii =

=

II =

+

-(Ayo

ao AD

BC

= AD

C

(no)! n, C

D

not

C

n,C

----

4-12b, one can similarly discover Using and s. together those just tances q.

Sec. 4-9

(4-25)

the n ....'nn'.. UJ,,,.ou,,,,,, r can be ex-

(4-24) and

.....".("'~.t1 in terms of p and II :

- B

C

= Det (M) =

C

Finally, with the help of

Bao)

ao

location of Cardinal Points for an Optical System

(4-26) the output disfor P,/I, and r, are

15

n,

no Y,

Yo

F,I I

r.....-j

r-- P I-

I f,

I

Output plane

Input plane

lal IPP2

I I Yo

Output plane

Ibl

Input plane

4-12 (a) Construction used to relate dislancellp, r. and/lto matrix elements. (b) Construction used to relate distances q. s. to matrix elements. (c) Construction used 10 relate distances v and w to matrix elements.

Output plane

lei

listed in Table 4-2. With the help of Figure 4-

may also be determined. For example, for small a=

=

Cyo

(4-27)

v

'Where the indicates that the ray axis. Input and output rays make the same (4-22), with ao = at a, a

the nodal plane distances v and w a,

+ Da or

mh'~·rt ..

1- D

a

=--C

(4-28)

Combining Eqs. (4-27) and (4-28),

v

16

Chap. 4

Matrix Methods in Paraxial

D - 1 C

(4-29)

TABLE 4-2 CARDINAL POINT LOCATIONS IN TERMS OF SYSTEM MATRIX ELEMENTS D

p=C A

q =-C r=

D-

Located relative 10 input (l) and output (2) reference planes

c ]- A

s=--

C D- I v=-C

w=

nolnJ

A

C

f,=p-r

ji=q-s

Located relative to principal planes

C I C

Similarly, one can show that

w=

(no/nf) C

A

(4-30)

using the fact that Del (M) AD - BC = no/nf. These results are also included in Table 4-2. The relationships listed there can be used to establish the following useful generalizations: 1. Principal points and nodal that is, r v and s = w, when the initial and final media have the same refractive indices. of an optical system are in magnitude 2. First and second focal when initial and final media have the same refractive points is the same as the of nodal 3. The separation of the points, that is, r s

4-., 0

....,...""'I'Ir...... ..,

AND CARDINAL

USING THE SYSTEM

POINTS

As an example, consider an optical system that consists of two thin lenses in air, separated by a distance as shown in Figure 4-13. The lenses have focal lengths of /A and Is, which may be positive or negative. If and output reference

I....---L-----o..;

system consisting of two in air, separated by a distance L.

Sec. 4-10

Examples

the System Matrix and Cardinal Points

77

planes are located at the the system matrix includes two thin-lens 5£A and 5£B, and a translation matrix
[-i :][: :][-i ~]

M

L

IA

M

Li.!

(4-31)

We may "derive" the equivalent focal length of any such system, measured relative to principal planes, by simply reading out the C element. SinceI"" II/cl, 1

IlL

-=-+

I""

-

(4-32)

IB IAIe

Furthermore, the first principal and nodal points coincide at a distance given by l)/C from the first lens, and the second principal and nodal points coincide at a given by (1 A)/C from the second lens. Thus

(D

r

=v =

(j:)L

and

S

w

(;:)L

(4-33)

Example Let us apply these results to the case a Huygens eyepiece, which consists of two positive, thin lenses separated by a distance L equal to the average of their focal Suppose JA 3.125 cm and IB 2.083 cm, L = 2.604 cm and I"" = 2.5 cm, by Eq. (4-32). Incidentally, the ftlo.~ ... it". power of this given by 25/1, is therefore lOx. From Eq. (4-33), we conclude that r = + 3 .125 cm and S = -2.083 cm. The optical system, together with its cardinal points and sample rays, is shown roughly to scale in Figure 4-14. The incident rays determine an image location between the which acts as a virtual object for the optical system. An ",... a~J:;"'u, tual image (not shown) is formed by the rays the as seen by an eye looking into the is discussed further in Chapter 6. 2

Figure 4-14

78

Chap. 4

construction for a Huygens eyepiece, using cardinal points.

Matrix Methods in Paraxial Optics

Example As a final let us find the cardinal points and sketch a ray dla,gralm for the hemispherical glass lens shown in Figure 4-15. The radii of curvature of 1.50 . are RI = 3 cm and Rz --7> 00, and the lens in air has a refractive

/'

..-

Figure 4-15

construction for a hef1l1is~!herical lens, using cardinal points.

Solution The matrix, for surfaces of the lens, is then M =

and output reference

at the two

or

with Det (M) = I

The relations in Table 4-2 then give of p = -6 cm, q = 4 em, r = 0, s - 2 -6 em, = 6 em. and nodal points eOllnell
4-11 RA Y TRACING The assumption of paraxial rays greatly simplifies the description of the progress of rays of light through an optical system, because trigonometric terms do not appear in the equations. For many purposes, this treatment is In rays of

Sec. 4-11

Ray Tracing

79

light to an in an optical system are. in usually rays in the near neighborhood of the optical axis. If the quality of the image is to be improved, aberrations that from however, ways must be found to reduce the the presence of rays deviating, more or less, from this ideal assumption. To determine the actual path of individual rays of light through an optical system, each ray must be traced, independently, only the laws of reflection and refraction toit was formerly gether with geometry. technique is called ray tracing done by hand, graphically, with ruler and compass, in a step-by-step process through an accurate sketch of the optical system. Today, with the help of computers, in a ray's altitude and the necessary calculations yielding the is done more easily and quickly. Graphics techniques are used to actually draw the optical system and to trace the ray's progress through the optical system on the monitor. 3 av-traclIlll! procedures, such as the one to be described here, are often limited to meridional rays, that is, rays that pass through the optical axis of the system. Since the law of refraction requires that rays remain in the plane of incidence, a meridional ray remains within the same meridional plane throughout its trajectory. Thus the treatment in terms of meridional rays is a two-dimensional treatment,4 greatly the geometrical relationships required. contributing rays and reto the image that do not pass through optical axis are called quire three-dimensional geometry in their calculations. The added complexity is no problem for the computer, once the ray-tracing program is written. Analysis of ous aberrations, such as aberration, astigmatism, and coma (described in the following require knowledge of the progress of nonparaxial rays and skew rays. The design a complex lens system, such as a photographic lens with four or five elements, is a combination of science and skill. By alternating ray tracing with small changes in the positions, focal lengths, and curvatures of the " .... t<>r,,.,, involved and in refractive of the the of the lens system is graduaUy optimized. For our purposes, it will be sufficient to show how the appropriate ""II'I for meridional ray tracing can be developed and how they can be repeated in fashion to follow a ray through any number of surfaces that constitute an optical system. The technique is well adapted to reiterative loops handled by programs. Figure 4-16 shows a single, representative step in the ray-tracing analysis. By incorporating a convention, the equations developed from this diagram can The ray selected made to apply to any ray and to any spherical refracting Ignrlatc::5 (or passes through) point an a with the optical axis. The ray passes the optical axis at 0 and then intersects the surface at p. where it is refracted into a medium of index n', cutting the axis again at I. The of incidence () and refraction () / are related by Snell's law. Points 0 and 1 are conjugate points with distances sand s 1 from the surface vertex at V. The radius of the surface is also shown, passing through the center of curvature at C. Other points and lines are added to in developing the necessary geometrical is the same as that used previously in this chapter. DisThe sign tances to the left of the vertex V are negative and to the right, positive. If we use

An of commercial software available, well-adapted for academic purposes, !Uld inexis BEAM2, a product of Stellar Software, Berkeley, California. It is IBM PC, XT. AT, !Uld rnrr..""ihl .. !Uld supports CGA, EGA, VGA, !Uld Hercules graphics adapters. two dimensions arc those of the page on which we have been our ray diagf"dms. Without emphasizing this, we have been using meridio11ll1 rays in all our diagrams. 3

80

Chap. 4

Matrix Methods in Paraxial Optics

-h A

a

4·16 refraction at a spherical surfilce. The defines the symbols and shows the geometrical relationships that lelw to rav..,tra<;m~ equ;atiolllS for a meridional ray.

from left to right, their as their measured above the axis are An important quantity in the calculations, also subject to this convention, is the the perpendicular distance from the vertex to the ray, as shown. parameters for the ray are assumed to he its "',..""u ....." '.";:}ILaU•• .., D. 4-16 shows that the "object (11s1tanc:e JLJll'LUj""'~•.,

h s = D - -tan a

(4-34)

Also, in

.

Q

sma = -

-s

(4-35)

In

.

sml)

=

a+

R

In

.

sma =

a R

Eliminating the length a from the last two equations, we get sin I)

Q R

. + sma

Snell's law at P: nsinl) Sec. 4·11

Ray

n'sinl)' 81

In

8

8'

a

a'

(4-38)

The Q parameter for the refracted ray is shown in Figure 4-17a. Analogous to the relations just found, we see that

In LCMV:

a'

sin(-a') =

R

In LPLC: a'

sin 8'

R

(a)

(b)

Figure 4-17 (a) Geometrical relationship of refracted-ray parameters with the distance . (b) Geometrical relationships iIIuSlrating the transfer between Q and a after one refraction and before the next.

TABLE 4-3 MERIDIONAL RAY-TRACING EQUATIONS (INPUT; n, n', R, a, h, D) paralJel to axis: a

General case

D-

s

-.~

Q = () = sin-I il' _

u

• -sm

a'

h

Plane Surfuce: R ~

Q

sin a ()

sin- 1

tan a Q = -s sin a

h

(~ + sin a)

_1(nsin8) -n'

()'-8+a

= R(sin

()f

sin a')

a' Q'

()'-()+a R(sin()'

sin a')

n sin- 1 - -n' sin a

a'

Q'

, s'

sin

a'

so' = • sin a'

Transfer: Input: t Q=Q'+tsina' a =a' n = tt' Input: new n', R Return: to calculate ()

82

Chap. 4

00

h D---

s

tan a

(~ + sin a)

0

Matrix Methods in Paraxial

s' ==

sin a'

when a' is eliminated, there results

As

Q'

R(sin8'

(4-39)

sinO")

In /:::,ITV:

sine -a ') =

s'

or

-Q' sin a'

=--

(4-40)

The relevant describing the first are included in Table 4-3 under of a, Q, the first column for the general case. The lead to new and s (now which prepare the next refraction in the The geometrical transfer to the next surface, at distance t from the first, is shown in Figure in /:::'V2 MVI, 4-17b,

or

Qi + t sin 0'2

(4-40 Thble shows how the equations must be modified for two cases: (1) when the ray is parallel to the axis and (2) when the surface is plane, with an infinite radius of curvature. Q2

=

Example Do a ray trace for two rays through a Rapid landscape photographic lens of

......"u,..."",. The rays enter the lens from a distant object at altitudes of I and 5 mm above the optical axis. The specifications (all in rum) are as follows: RI = -120.8 6 nl = 1.521 R2 = -34.6 2 n2 = 1.581 R3 = -96.2 3 n3 = 1.514 R", = -51.2 Solution Since the rays are parallel to the axis, the second column of Table 4-3 is used to calculate the progress of the ray. These can be tabulated as follows: Input:

Results: ra:y at h "" I

Results: ray at II

5

First surface:

n I, n' 1.521 a 0 h = ] or 5 R -120.8 Second surface: 1=6 n = 1.581 R = -34.6

Q a' = 0.1625" s' = -352.66 Q' 1.0000

Q 5 a' = 0.8128" s' = -352.53 Q' 5.0010

Q= a s' = Q' =

1.0170 0.2202" -264.59 1.0170

Q = 5.0861 a' 1.1041" s' = -264.03 Q' = 5.0876

Q = 1.0247 a' = 0.2030" s· = -289.26 Q' = 1.0247

Q = 5.1261 = 1.0178" -288.58 Q' = 5.1260

Third surfuce: 1 2 n = 1.514 R = -96.2

Final surface; t = 3

n

I R = -51.2

Sec. 4-11

Ray

Q 1.0353 a' = -0.2883" s' '" 205.72 Q' .." 1.0353

a' s'

Q a

I

=

S' =

Q'

5.1793 1.4520" 203.91 5.1672

83

Thus the two rays intersect the opdcal axis at 205.72 and the tina I surface, missing a common focus by 1.8 mm.

mm beyond

4·1. A biconvex lens of 5 cm thickness and index 1.60 has surfaces of radius 40 cm. If this lens is used for objects in water. with air on its opposite side, determine its effective focal and sketch its focal and points. 4·2. A double concave lens of glass with n 1.53 has surfaces of 5 D (diopters) and 8 D, respectively. The lens is used in air and has an axial thickness of 3 cm. (a) Determine the position of its focal and principal planes. (b) Also find the position of the relative to the lens center, to an at 30 cm in front of the first lens vertex. (c) Calculate the paraxial distance assuming the thin-lens approximation. What is the percent error involved? 4-3. A biconcave lens has radii of curvature of 20 cm and 10 cm. Its refractive index is 1.50 and its central thickness is 5 cm. Describe the of a I-in. tall object, situated 8 cm from the first vertex. 4-4. An equiconvex lens surfaces of radius 10 cm, a central thickness of 2 cm, and a refractive index of 1.61 is situated between air and water (n = I object 5 cm high is placed 60 cm in front of the lens surface. Find the cardinal for the lens and the position and size of the image formed. 4-5. A hollow, of radius 10 cm is filled with water. Refraction due to the thin walls is for rays. (a) Determine its cardinal points and make a sketch to scale. (b) Calculate the position and magnification of a small object 20 cm from the (c) your analytical results by appropriate rays on your sketch. 4-6. rays enter the plane surface of a hemisphere of radius 5 cm and refractive index 1.5. (a) Using the system matrix representing the hemisphere, determine the exit elevation and angle of a ray that enters parallel to the axis and at an elevation of I.cm. (b) the system to a distance x beyond the hemisphere and find the new matrix as a function of x. (c) the new system matrix, detennine where the ray described above crosses the optical axis. 4·7. Using Figure 4-12b and c, verify the expressions in Table 4-2 for the distances q,/2, s, and w. 4-8. A lens has the following specifications: RI == + 1.5 cm = Rh d (thickness) = 2.0 cm, nl = 1.00, n2 = ] .60, n3 1.30. Find the principal points the matrix method. Include a sketch, roughly to scale, and do a ray for a finite of your choice. 4·9. A positive thin lens of focal length 10 cm is separated by 5 cm from a thin negative lens of focal -10 cm. Find the equivalent focal length of the combination and the position of the foci and principal planes the matrix approach. Show them in a sketch of the opticaJ system, roughly to scale, and use them to find the image of an arbitrary object placed in front of the system. 4-10. A lens 3 em thick along the axis has one convex face of radius 5 em and the also convex, of radius 2 cm. The former face is on the left in contact with air and the other in contact with a liquid of index 1.4. The refractive index of the glass is 1.50. Find the positions of the foci, principal planes, and focal of the system. Use the matrix approach.

84

Chap. 4

Matrix Methods in Paraxial

4-11. (a) Find the matrix for the of a thin lens of focal the lens. input plane at 30 em in front of the lens and output plane at 15 em (b) Show that the matrix elements the locations of the six cardinal points as would be for a thin lens. (e) Why is B = 0 in this case? What is the special meaning of A in this case? crystal ball has a refractive index of 1.50 and a diameter of 8 in. 4-12. A the matrix approach, determine the location of its principal (b) Where will sunlight be focused by the crystal ball? 4-13. A thick lens presents two concave each of radius 5 cm, to incident The lens is I em thick and has a refractive index of 1.50. Find (a) the matrix for the lens when used in air and (b) its cardinal points. Do a ray diagram for some object. 4-14. An achromatic doublet consists of a crown glass positive lens of index 1.52 and of thickness 1 em, cemented to a flint lens of index 1.62 and of thickness 0.5 cm. All surfaces have a radius of curvature of 20 cm. If the doublet is to be used in air, determine (a) the system matrix elements for input and output planes 00to the lens (b) the cardinal points; (c) the focal length of the combinathe lensmaker's and the equivalent focal of two lenses in contact. Compare this calculation of I, which assumes thin lenses, with the previous value. 4-15. the optical system of 4-15 to include an object space to the left and an space to the right of the lens. Let the new input plane be located at distance :s in object space and the new output at distance s' in image space. (a) Recalculate the system matrix for the system. (b) Examine element B to determine the relationship between and image distances for the lens. Also determine the general relationship for the lateral magnification. From the results of (b), calculate the distance and lateral for an object 20 em to the left of the lens. (d) What information can you find for the by setting matrix elements A and D to zero? 4-16. Find the system matrix for a Cooke camera lens (see Figure 6-18a). enterfrom the left encounters six surfaces whose radii of curvature are, in tum, r] to r6. The thickness of the three lenses are, in tum, II to 13, and the refractive indices are nl to n3. The first and second air between lens surfaces are d l and dz . Sketch the lens system with its cardinal points. How far behind the last surface rays? must the film plane occur to focus Data: rl = 19.4 mm r2 = 12S.3 mm r3 = -57.S mm r4 = IS.9 mm rs 311.3 mm r6 -66.4 mm

I,

= 4.29 mm = 0.93 mm

13

3.03 mm

II

d l = 1.63 mm d2 = 12.90 mm

nl

1.6110

n,

1.5744

n3

1.6110

4·17. Process the product of matrices for a thick as in Eq. (4-15), without """U"'ll1~ the n = n' and t O. Thus find the general matrix elements and D for a thick lens. in tenns of the matrix elements for a 4-18. Using the cardinal point locations thick lens (problem 4-17), that II and fz are given by I) and 4-19.

the cardinal point locations thick lens 4-17), and

Chap. 4

Problems

in terms of the matrix elements for a by

4-20. Write a computer program that incorporates (4-34) to (4-41) for ray tracing through an arbitrary number of refracting, spherical surfaces. The program should allow for the special cases of rays from fur-distant objects and for surfaces of refraction. 4~21. Trace two rays through the hemispherical lens of Figure 4-15. The rays originate from the same object point, 2 cm above the optical axis and an axial distance of 10 cm from to the axis and the other makes an angle of -200 the first surface. One ray is with the axis. 4-22. Trace a ray originating 7 mm below optical axis and 100 mm distant from a doublet. The ray makes an angle of + 50 relative to the horizontal. The doublet is an equiconvex lens of radius 50 mm, index 1.50, and central thickness 20 mm, followed by a matched meniscus lens of radii -50 mm and -87 mm, index 1.8, and central thickness 5 mm. Determine the final values of s, a, and Q. 4-23. Trace two rays, both from far-distant through a Protor photographic lens, one at altitude of 1 mm and the other at 5 mm. Determine where and at what angle the rays including an intercross the optical axis. The specifications of this four-element mediate air space of 3 mm, is as follows, with distances in mm: R, = 17.5

5.8 R2 RJ = 18.6 R4 = -12.8

18.6 Rs R6= -14.3

II

= 2.9

nl

= 1.6489

12

1.3

n2

1.6031

13 =

3.0

n3

4

1.1

n. = 1.5154

Is

1.8

n,

1.6112

REFERENCES [1]

[2]

[4] £5] [6]

[8] [9]

W. H. A, and M. H. Freeman. Optics. 9th ed. Boston: Butterworth Publishers, 1980. Ch. 8,9, 19. Kingslake, Rudolf. Lens Design Fundamentals. New York: Academic 1978. Ch. 2,3,7. Smith, ¥hirren J. "Image Formation: Geometrical and Physical Optics." In Handbook of edited by Walter G. Driscoll and William Vaughan. New York: McGraw-Hill Book Company, 1978. Brouwer, William. Matrix Methods in Optical Instrument Design. New York: W. A. Benjamin, 1%4. A., and J. M. Burch. Introduction to Matrix Methods in Optics. New York: John Wiley and Sons, 1975. Atneosen, Richard, and Richard "Learning Optics with Optical Design Soft242-47. ware." American Journal of Physics 59 (March Nussbaum, Allen. Geometric Optics: An Introduction. Reading, Mass.: Addison-Wesley Publishing Company, 1968. Ch. 2-4. Nussbaum, Allen, and Richard A. Contemporary Optics jor Scientists and Engineers. Englewood CliffS, N.J.: Prentice-Hall, 1976. Ch. I. Blaker, J. Warren. Geometric Optics: The Matrix Theory. New York: Marcel Dekker, 1971.

Chap. 4

Matrix Methods in Paraxial Optics

5

Aberration Theory

INTRODUCTION

The paraxial formulas developed earlier for image formation by spherical reflecting and refracting surfaces are, of course, only approximately correct. In deriving those rays both near to the opequations, it was necessary to assume paraxial rays, that tical axis and making small angles with it. Mathematically, the power expansions for the sine and cosine functions, by 5

sin x = x cos x

3!

x2 1- -

2!

+ -x - ... 5!

+

4!

were accordingly approximated by their first terms. To the extent that these firstorder approximations are valid, Gaussian optics implies exact imaging. The inclusion of higher-order terms in the derivations, however, predicts increasingly larger from imaging with increasing These are reto as "aberrations." When the next term involving x 3 is included in the aphave proximation for sin x, a third-order aberration theory results. The been studied and classified the German mathematician Ludwig von Seidel and are referred to as third-order or Seidel aberrations. For monochromatic light, there are five Seidel aberrations: spherical coma, astigmatism, curvature offield, ffI

and distortion. An additional aberration. chrOl1Ultic aberration, results from the The details wavelength dependence of the imaging properties of an optical of aberration are too formidabJe to treat in this We include here a brief, quantitative of how aberrations follow from a thirdorder trealment and a qualitative description of each aberration, with typical procedures for its elimination.

5-1 RA V AND WAVE ABERRATIONS The departure from paraxial may be described quanltitativeJy eral ways. In 5-1 two wavefronts are shown system. Wavefront W I is a spherical wavefront the Gaussian, or paraxial, approximation that an image at I. Wavefront W 2 is an example of the actual wavefront, an envelope whose represents an exact solution of the optical This shape could be deduced by precisely a sufficient number of rays the optical system by the methods described in Section 4-11. Rays from adjacent points A and B. being normal to their respective wavefronts, do not intersect the paraxial image plane at the same point. The "miss" the optical axis, represented by the distance LI, is called the longitudinal while miss IS, measured in the image plane, is called the transverse, or aberration. These are ray aberrations. Alternatively, the aberration may be described in terms of the deviation of the deformed wavefront from the ideal at various distances from the optical ax is. At the location of point shown in Figure 5-1, the wave aberration is given by the distance AB. Notice that rays from both at their point 0 of tangency on the optical axis, reach the same image point I. Rays from intermediate points of the actual wavefront between 0 and B intersect imscreen at other around /, producing a image, the result aberration. The maximum ray aberration thus the size of the blurred The ultimate goal of is to reduce the ray aberrations until they are comparable to the blurring due to diffraction itself.

Optical system

l

o Paraxial image plane

Figure 5-1

IlIuslration of ray and

Wdve

aberrations.

Lateral ray aberrations corresponding to the wave aberration AB may once the variation in AB with aperture dimension y is known. Referring to life the angle a between actual and ideal rays from a point P of the W8VClrol1ll. at elevation y, is the same as the angle between wavefront tangents at P. The wavehaving been shaped by the optical system, exist in image space with 88

5

Aberration

T"',~~.~,

y

I

I

ro--b.~

I II

~---#~--~--------------~~--~~~~r-~z

Detail

Figure 5-2 Construction used to relate the ray aberrations by and b, to the wave aberration a. The detail shows how to relate a do in wave aberration to a change dy in the aperture dimension.

live index n2. The detail of then shows that the incremental wave aberration (]a, expressed as an optical path length in image space, is (]a

naCo: dy)

(5-1)

local curvature of wavefront at P. The lateral The derivative da/dy describes ray aberration by due to the rays from the neighborhood of P may then be approximated by

b = Y

O:S'

=

s'

(]a

(5-2)

n2

where s' is the paraxial image distance from the wavefront and from (5-1). Similarly, along the other transverse yz- axes in the plane of the page,

0:

has been taken to the

s'da

= n2 dx

(5-3)

The longitudinal ray aberration b z is related to

aberration by by (5-4)

y 5-2 THIRD-ORDER TREATIIIENT OF REFRACTION AT A SPHERICAL INTERFACE

Let us solve now the case of refraction from a single spherical prove the approximation to include "third-order" angle bitrary ray PQ from an axial object point P is refracted by a sptlencal "'........"".v. at that separates media of refractive indices nl and n2. an axial image at I. To a first approximation, the optical path •...,Ui15..1" and POI are identical. according to Fermat's principle. Aberration corltnlt)Utc~ to the formation of the image because, beyond a first approximation. ray differ for different points Q along the spherical surfuce. Thus we define atQas a (Q) = (PQI - POI)opd

Sec. 5-2

Third-Order Treatment of Refraction at a

Interface

89

Figure 5-3 Refraction of a ray at a spherical surface.

where opd

'I1UII'-'cllI,,:>

More precisely,

the optical a(Q) = (nit'

+

(nls

+ n2st)

Employing the cosine law, the lengths t' and f' may be exactly of the defined in Figure = R2

t"2

Approximating cos

= R2

tP

e;xrlre;~:"e;lI

+ (s + R)2 + R) cos 4> + (s' R)2 + 2R(s' R) cos 4>

4> == 1

2!

(5-7) (5-8)

+

hiR, we have costP=l-

Introducing

in terms

4> by cos

and with

(5-6)

10)

(5-10) into Eqs. (5-7) and

s{

I

+

[h2(R

+

and rearranging

s)

I)

RS2

i'

12)

Next, representing the quantities enclosed in square brackets by x in Eq. 11) and in braces may be approximated using the hit1lnrr.i",,1 '""'~.... ..".vu

x' in Eq. (5-12), the square roots of the (1

+ x)1/2 ==

1

+

x

2

8

(5-13)

Thus

8

x 8

90

Chap. 5

Aberration Theory

14) 15)

When all terms of

than h4 are discarded, there remains

t=S[1 +~-~

+ s) _ h4(R + sf] 24R3 S 2 8R2 S 4

h4(R

16)

[I + --'---::--"-

€'

(5-17)

These expressions for € and t' are introduced into Eq. (5-6), and after some rearranging, the result is

h22[

a(Q)

s

The first term in a first-order approximation to the aberration and is accordingly zero since the quantity within brackets vanishes by Fermat's priII1CII)le. In mct, this quantity to zero reproduces the Gaussian formula for by a I There remains the third-order "hF'..Nltlt1.n resented by the term in . When h is small enough, the rays are paraxial, and the by this term may be negligible. In any case, since the brackets independent of h, we have shown that theory a wave aberration that is proportional to the fourth power of the aperture h. measured from or (5-19) where c constant of proportionality. This is the principal calculation for points. We will use this in generalizing calculation to include off-axis imaging. In this way, the other Seidel aberrations will also appear. The a (Q) we have calculated as a difference in ODI:lCaLl-Dam between and actual rays must correspond to the wave AB of 5-1. The deviation AB of the actual from the ideal spherical wavefront is a function of the distance from the optical axis at which the ray the wavefront and is to as spherical aberration. spherical aberration in more detail, we wish to show how third-order aberrations arise. To do this. we to consider the of rays whose case of an off-axis object point. Shown in Figure 5-4 are two limits are by an aperture En P serving as the entrance pupil. An axial pencil from on-axis point 0 forms an image at and around the "''''''''vi., I. This will be affected by spherical aberration, as discussed earl"l..i! ......... ;~_l"l by the displacement y of the extreme rays of the pencil. is symmetrical about the axis OCI, where C is the center of curvature of refl'RCting surface. Also shown is an oblique pencil of rays at the off0' This is certainly not symmetrical about the axis 01; in the absence of the En P, its axis of symmetry would be line 0' CI ' . It is from this axis that the displacement y' of the rays of the would have to be measured to determine the degree of aberration 19). Notice that such from the axis of is much 1 The

ingat

Sec. 5-2

same quantity appears in processing the h4 term and is also set (5-18).

to zero there in arriv-

Third-Order Treatment of Refraction at a Spherical Interface

91

l'

--

I

5-4 Comparison of axial and oblique of rays from an defined by pasentrance aperture E"P.

of the oblique an oblique of rays due to off-axis points is far more susceptible to aberration than corresponding axial points. The position of the aperture is critica1 in determining the magnitude of y' and is harmful in this respect when at the center of curvature, C. (In this one may recall the use of symmetrical lenses or lens such as the doublemeniscus objective. where the aperture is placed midway between them.) Consider then the off-axis pencil of rays from object point ure 5-5. The function a '(Q) for Q on the """',,,"+..nn pre~.sed as a'(Q)

Cp'4

(PQP' - PBP

(5-20)

In . (5-20) we the elevation of the ray , to the axis PBP' and consider points B, 0, and Q to in a vertica1 plane approximating the wavefront at O. It can be shown that this approximation does not affect the results of third-order "N,,...... i,,.,, theory. We have also use of Eq. (5-] 9) and the QlSltan(;e p '. A section of the plane that includes the relevant points and

-E~~p'-_--

c

,h'

s

Detail

Figure 5-5 Imaging of off-axis poinl P. Aberration at an arbitrary poinl Q on the wavefront may be related to the symmetry axis PBP' or the optical axis oes. The detail shows a frontal view of a portion of a wavefront

92

Chap. 5

Aberration

l".a~.~.

distances p', b, and r is also shown in Figure 5-5 (detail). In a similar manner, we m~ for the wavefront point 0,

=

a'(O)

If the point Q is m~ be previously.

r",t,~ ....."11

as the

' - PBP')opd = c(BOt

= cb 4

(5-21)

to the optical axis OC, an off-axis aberration function a(Q) between the axial aberrations at Q and 0 found (5-22)

A n",I"U'I<,

the

to the

OP.uTl,,,t .. ii ....

detail shown in Figure 5-5, we have

+ b2 +

2rb cos 8

.........,.,.r'n for p' into Eq. (5-22) gives

and

() + 2r 2 b 2 + 4r 3 b

a(Q)

From similar OB b is axis. This may be

cos ()

+

4rb 3 cos ()

(5-23)

OBC and SCP' in Figure 5-5, we see that the distance to the height h' of the paraxial image P' above the optical p"""rP<1"'~f1 by b = kh' (5-24)

where k is the proportionality constant. When b in Eq. (5-23) is replaced by kh', we have, lumping all constants into term-by-term coefficients, a(Q)

+1

cos ()

+ +

(5-25)

The C coefficients in are subscripted by numbers that specify the powers delJenderlce on h', r, and cos 8, respectively. For example, the C coefficien t I term h cos 8, where h' is to the first power, r is cubed, and cos () is to the first power. The individual terms describe wavefront aberrations that contribute to the total aberration at the These terms comprise the five or as follows: aberration h

cos ()

h

()

h h

coma

curvature of field cos 8

distortion

Each aberration is characterized its dependence on h' (departure from axial of and 8 (symmetry around the axis). Notice that the first term for aberration agrees with (5-19), derived axial imaging, where h the """,.. h"r"" We now briefly describe each of these aberrations in terms of their visual effeets and indicate some means that are to reduce them.

''''''5'''5'' r

5-3 SPHERICAL ABERRADON The aberration known as spherical aberration results from the first , in Eq. (5-25). It is the only term in the third-order wave a(Q) that not depend on h f. Thus spherical aberration even for and image Sec. 5-3

Spherical Aberration

93

points, as illustrated for a single lens in Figure 5-6a. The paraxial image point I is distinct from axial points, such as due to rays refracted by lens aperprotures. The axial miss distance due to rays from the extremities of the the distance IG vides the usual measure of longitudinal spherical in the paraxial image plane measures the corresponding transverse spherical aberration. These quantities also depend on the distance. When E is to the left of I, as shown for the case of a positive the spherical is positive; for a to the right of I, and the spherical aberration is negnegative lens, E ative. At some intermediate point between E and I, a "best" focus is attained in practice. The broadened image there is called, descriptively, the "circle of least confusion." Eqs. (5-2) and (5-4) for lateral and longitudinal the corre"'1)<.,.,...n.1;:. spherical ray aberrations may be as foHows:

s'oo b y =-Hz

and r

Hz

o I

lal

(bl

Figure 5-6 aberration of a lens, producing in (a) different 00 the lens aperture. tances and in (b) different focal lengths.

dis-

Example

Axially collimated light enters a glass rod its end, a convex, spherical rod has a index of 1 Determine surface of radius 4 cm. The ray aberrations for entering at an the longitudinal and lateral aperture height of I cm. 18), with object distance s very

Solution According to Eq. remains

a To calculate

-

~4[;~

1,

~r]

one needs the derivative 00/ dh: 00 dh

3[HZ (.ls' _!)2]

h 2

S'

R

The image distance s', also the focal length of the surface, is v"',"""," equation,

I 00

94

Chap. 5

there

1.6

0.6

+ -s' = - 4

Aberration Theory

, or s = 10.667 cm

from the

Then da/dh can be calculated:

1

da dh =

-2

)2]

I. 6 (1 1 10.67 10.67 - 4

10.667

bz

-0.00183] J)

10.~7 (-0.0122)

= s; by

= -0.0122 cm -0.130 cm

5-6b shows spherical aberration when the object is at infinity. Various circular zones of the lens about the axis different focal lengths, so that/is a focal of the lens is due to the intersecfunction of aperture h. The tion of paraxial rays for which h ~ O. This focal length is given by the lensmaker's formula, ]

- = (n

(5-26)

/

n and radii of curvature rl and r2. when used in air. for a thin lens of refractive From Eq. it is obvious that a given/may result from different combinations of rl and r2. Various choices of the radii of curvature, while not changing the focal may have a large effect on the degree of spherical aberration of the lens. ure 5-7 illustrates the "bending," or in shape. of a lens as its radii of curvature vary but its focal remains fixed. A measure of this bending is the Codby dington shape factor u,

u = -=--...:..

(5-27)

where the usual sign convention for rl and r2 is assumed. For example, a thin lens of n 1.50 and / = 10 cm may result from an equiconvex lens of u 0 (rt = 10, r2 lOa lens of u + 1 (rl 5 a meniscus lens of u = + 2 (rl 3.33, r2 = to cm). These as well as their mirror with negative factors, are shown in 5-7.

-2

- 1

0

:Figure 5-7 of a lens into various versions the same focal The Coddington shape below each version serves 10 classify Ihem.

+2

+1

The spherical of a spherical refracting surface is given in Eq. (5-18). A thin lens combines two such each of which contributes to the total aberration. The total longitudinal spherical aberration of a thin lens with focal length/and index n, where ,~/, is the distance for a ray at elevation h, s~ is the image distance, and p

s' s'

s

+s

is given by

+ 4(n +

l)pu

+ (3n + Sec. 5-3

Spherical Aberration

2)(n -

I)p2

+

n

(5-28)

95

One can show further (problem 5-11), that minimum (but not ration results when the bending is such that 2(n 2

a=

spherical aber-

I)

29)

n+2 P

Notice that for an object at infinity, a .;;: 0.7 for a lens of refractive index n = 1.50. This shape factor is dose to that of the plano-convex lens with a = + 1. Accordingly, optical systems often employ plano-convex lenses (with the convex side facing the parallel rays) to spherical aberration. In a minimum in spherical is associated with the condition of equal refraction by each of the two calling to mind the case of minimum deviation in a prism. When lenses are used in combination, the possibility of canceling spherical aberration arises the fact that positive and lenses produce spherical aberration of sign. A common application of this is found in the cemented "dOUblet" lens.

5-4 COMA Coma is represented by the term t C3th 1 r3 cos (J, which indicates an off-axial aberration (h I 0) that is nonsymmetrical about the optical axis (J constant) and r. 5-8a illustrates the due to a increases with the vertical, or tangential, fan of parallel rays refracted by a single lens. Each circular called the coma/ic circle. Zone rays lying in zone of the lens forms a circular the tangential fan shown form an image at the top of each comatic circle, whereas zone rays lying in a sagittal in the horizontal plane, form an image at the bottom of each comatic circle. other fan of rays forms that complete the comatie circle. The combination of all such comatic circles, which increase in radius as the zone radius is the cometlike figure shown in Figure glvmg this aberration its name. In effect, each zone produces a different magnification, so that due to central rays is not equal to he due to extreme rays. like "rnlPr."", may occur as a quantity (he> he) or a negative

'*

(he

'*

<

he). Without the usual paraxial approximation-restricting rays to those making small with the axis-one can show that for a small object near the axis, any ray from an object point that is at a interface must satisfy the Abbe sine condition,

nh sin (J

+

n' h I sin (J'

=0

Here hand h' are object and image size, respectively, and the angles IJ and IJ' are of the rays in optical media n and n I, These quantities the slope are in Figure When (5-30) is rearranged to express the lateral magnification, the condition can be written

h'

m=-,;=

n sin IJ n'sin(J'

To coma, the lateral magnification resulting from refraction by lens must be the same. Thus coma is absent when, for all values of IJ, sin IJ

-:--;;; = constant Slnu

96

Chap. 5

Aberration

an zones of a

\

I I

I

:I

I

(al

2R.

\

h"

(b)

n sin if> '" n' sin /fl'

h h'

"

h

PC

-h' = P'C

lel Figure S-S (a) Coma due to a tangential fan of rays. When all such azimuthal funs are considered. each image point in the becomes the lop of a comatie circle of image points. (b) Formation of a comatie image from a series of comatic circles. The shape of the comatic image is such thai its maximum extension is three times the radius of the comatic circle formed by rays from the outer zone of the lens. The between the dashed lines is 60·, (c) Nonparaxial rays from object point P near axis form an image al P', to the Abbe sine condition, The condition follows from Snell's law and the relationships given in the figure.

The of a found useful in spherical is also Eq. (5-27), which results in useful in reducing coma. The Coddington minimum spherical aberration, is close to that producing zero coma, so that both aberrations may be significantly reduced in the same lens by proper '--""""'"t1e;. can show that coma is absent in a lens when (T

=

(2n2 - n I) n+1

(5-31)

For the example of the lens considered with n = 1.50 and a value of (T close to the value of (T infinity, Eq. which yielded minimum spherical aberration. A lens or optical spherical aberration and coma is said to be aplanatic.

Sec. 5-4

Coma

91

5-5 ASTIGMATISM AND CURVATURE OF FIELD Aplanatic optics is still susceptible to two closely related aberrations whose wave aberration terms can be combined to give h (J + The term produces astigmatism, and the which is symmetrical about the is called curvature offield. Both increase similarly with the tance of the object and with the aperture of the refracting surface. and b illustrates the astigmatic images of an off-axis point P due to a tangential fan of rays through the section tt' and a sagittal fan of rays through the section ss I of a lens. Since these fans of rays focus at different distances from the lens, the two images are line images. shown as T and S for the tangential and sagittal fans, respectively. The focal line T lies in the sagittal plane, and the focal line S falls in the tangential If a screen held perpendicular intermediate will be elliptical in to the principal ray is moved from S to shape. Approximately midway between S and the focus will be circular, the of least confusion. The locus of the line images S and T for various object points P are paraboloidal surfaces, as illustrated in Figure 5-9c. The deviation between the two along any principal ray from a given point measures the magnisquare of tude of the astigmatism for this object, approximately proportional to the distance from the optical axis. When the T surface fans to the left of the S suras shown. the astigmatic difference is taken as positive; otherwise it is negalive. If points like P fall a circle in an object perpendicular to the optical axis, the corresponding line images in the T surface merge into a well-focused image circle. In the S surface, however, the image of the circle will not be sharp, having everywhere the width of the S focal line. On the other hand, object radial in the circle produce radial only in the S surwhere the elongated radial merge to produce well-focused radial lines. Thus if the object plane contains both circular and radial elements, the image distance for a good focus will be different for each type of element, with a compromise image somewhere between. From the of view of the elimination of astigmatism rprllll'r= that the tangential and surfaces be made to coincide. When the curvatures of these surfaces are changed by altering lens or spacings so that they coincide, the resulting surface is caned the Petzval surfoce. In this focal for an aplanatic system, point are formed. If the surface is curved, then, astigmatism has been the aberration called curvature of field remains. record sharp under these the fUm must be shaped to fit the PetzvaI surface. A Petzval surface can be determined for any optical system, even when the T and S surfaces do not coincide. Unlike the T and S surfaces, the Petzval surface is unaffected lens bending or placement and depends only on the refractive indices and focal of the lenses involved. In third-order theory, the PetzvaI is always situated three times farther from the T surface than from the S surface and lies on the side of the S surface opposite to that of the T surface. For example, two lenses will have a flat Petzval surface, eliminating curvature of field, if

ndi + ndi. In general

Petzval surface for a number of thin lenses in air """"'''"''''

2: 98

Chap. 5

0

Aberration Theory

1

(5-32)

p

s lei

Cd)

where Rp is the radius of curvature of the Petzval surface. Field by this lens, but artificial field flattening may condition cannot be accomplished for a be accomplished by use of an stop positioned as in In this arrangement, oblique chief rays, now determined by the "Y'"~... t,,n·'" lens center. The Sand T surfaces then appear oPl)Ositellv surface of least confusion is flat, as shown. This inexpensive meth(1d flattening the field has been used in simple box cameras. In more difficult situations, other requirewhere the Petzval condition cannot be satisfied without The lens helps to ments, a low-power lens is sometimes used near the image counteract curvature of field without otherwise seriously compromising image quality. Finally, according to fifth-order aberration theory, the T and S surfaces may acthe optitually be made to come and intersect at some distance cal axis. The result is less average astigmatism over the compromise focal plane. is designed to take advantage of this. The anastigmat camera 5-6 DISTORTION

The last of the five monochromatic Seidel aberrations, even if all the others cos e. Even have been eliminated, is distortion, represented by the term as points, distortion shows up as a variation in the though object points are points at different distances from optical axis. If lateral magnification for the magnification increases with distance from the the rectangular grid of Figwill have an image as shown in Figure 5- lOb, descripure 5- lOa, serving as tively called pincushion distortion. On the other hand, if magnification decreases with distance from the the appears as in 5-IOc, with barreL distortion. The image in either case is but distorted. Such distortion is often augelements effectively actmented due to the limitation of ray bundles by stops or refer to Figure 5-1Ia. Shown there is the image of ing as stops. To see this of rays are each liman off-axis point, formed a single lens. Two when located (1) at some distance from the lens and (2) ited by the aperture near the lens. As the aperture approaches the lens, it a shorter path to the effective object to image distance is greater-hence the lens. It will be seen that lateral magnification is smaller-for position I. This decrease in lateral magnification due to the position is more noticeable as the object point recedes further from the so that the image suffers from distortion. The effect of placing the on the image side of the lens can also be seen from the same figure rays and the played and image. Now the distance is and pincushion distortion apratio of effective

Cal

Ib)

Ie)

Figure 5-10 of a square grid (a) pm(:usnllOfl distortion (b) and barrel distortion (e) due to nonuniform ma:gnilticaltions.

100

Chap. 5

Aberration Theory

(a)

(el

pears in the image. When the aperture stop is placed at the position of the lens, such distortion does not occur. Also, a symmetric doublet with a central stop, combining both effects, is free from distortion for unit magnification. Photographs of the effects of stop location on distortion are reproduced in Figure 5-11 b, c, and d.

5-7 CHROMATIC ABERRA nON The final aberration to be discussed is not one of the Seidel aberrations, which are all monochromatic aberrations. Neither our first-order (Gaussian or paraxial) approximations nor the third-order theory sketched briefly in the preceding sections took into account an important fact of refraction: the variation of refractive index with wavelength, or the phenomenon of dispersion. Because of dispersion, an additional chromatic aberration (C.A.) appears, even for paraxial optics, in which images formed by different colors of light are not coincident. In terms of the monochromatic third-order aberrations of Eq. (5-25), we could introduce chromatic effects by considering the wavelength dependence of each of the coefficients of the terms. The chromatic aberration of a lens is simply demonstrated by Figure 5-12a. Since the focal length f of a lens depends on the refractive index n of the glass, f is also a function of wavelength. The figure shows convergence of parallel incident light rays by the lens to distinct focal points for the red and violet ends of the visible spectrum. Notice that a cone of violet light will form a halo around the red focus at R. If the incident light contains all wavelengths of the visible spectrum, intermediate colors focus between these points on the axis. Just as for a prism, greater refraction of shorter wavelengths brings the violet focus nearer the lens for the positive lens shown. Figure 5-12b illustrates chromatic aberration for an off-axial object point and displays both longitudinal chromatic aberration and lateral chromatic aberration. Notice that if longitudinal chromatic aberration were absent, the lateral chromatic aberration could be interpreted as a difference in magnification for different colors. The longitudinal chromatic aberration of a convex lens may easily be comparable to its spherical aberration for rays at widest aperture .

.. R I I I

~fv- .. i

I I 1

i i

~fR--..o--il

Longitudinal C.A.

(al

Lateral C.A.

(bl

Figure 5-12 Chromatic aberration (exaggerated) for a thin lens, illustrating the effect on the focal length (a) and the lateral and longitudinal misses (b) for red (R) and violet (V) wavelengths.

Chromatic aberration is eliminated by making use of mUltiple refracting elements of opposite power. The most common solution is achieved with the achromatic doublet, consisting of a convex and concave lens, of different glasses, cemented together. The focal lengths and powers of the lenses differ, through shaping of their surfaces, to produce a net power of the doublet that may be either positive or

102

Chap. 5

Aberration Theory

5·1.3 Achromatic doublet, consisting of (I) crown glass equiconvex lens cemented to (2) a negative flint glass lens. Notation for the four radii of curvature are shown.

negative. The dispersing powers of the components are, through appropriate selection of glasses, in inverse proportion to their powers. The result is a compound lens over a significant portion of the that has a net focal length but reduced visible spectrum. We consider next the quantitative details of this design. The general of the achromatic doublet is shown in Figure 5-13. The powers of the two lenses for by the Fraunthe yellow center of the visible /wfer wavelength, AD = 587.6 nm, are Pw Pw

=

I flD -

I

(nw -

r:J t)(~ r21

= (nw

-I

(nw -

= (n2D

I)KI I)K2

(5-34)

where the radii of curvature are designated in Figure 5-13. Here nD to the refractive index of each glass for the D Fraunhofer line, and we have introduced conas an abbreviation for the curvatures. We have already shown with stants KI and (4-32) that the of a doublet, with lens separation is given by I

f

j;

L

+

/liz

(5-35)

and (5-36) For a cemented doublet of thin lenses, L = 0, and the powers of the ply additive:

are sim(5-37)

Incorporating Eqs. (5-33) and (5-34), P

=

I)KI

(nl

+

(5-38)

power is independent of Chromatic aberration is absent at the wavelength AD if wavelength, or (ap jaA)D = O. Applied to (5-38), this condition is ap

(5-39)

aA

The variation of n with A in the neighborhood of AD may be approximated the red and blue Fraunhofer wavelengths, 656.3 nm and AF = 486.1 nm, respectively:

an aA

Sec. 5-7

Chromatic Aberration

AF

Ac

(5-40)

103

The

constant for the

may be introduced by

nr""""no

the terms of

(5-39) as K anlD I

K

aA

(AF - AC)VI

(:: - !) =

iJn2D 2

iJA -

where we have used Eqs. as the of the

11,,"nP#1

-(A-F--==-A-c-W-z

(5-42)

and (5-34) as well as a constant V, power (see pages 119-120) and by

FJ'<:lrwr'""JP

V

1 A

- I nF

(5-43)

nc

(5-39), the condition for the absence of

Substituting Eqs. (5-41) and into chromatic aberration may be written as

+ V1 P2D = 0 Combining Eqs. (5-37) and (5-44), the powers of the individual elements may be exn ..~·''''~.t1 in terms of the desired power of the combination: V2

P ID =

The K curvature factors

(5-45)

(5-33) and (5-34) may then be calculated

PJ(lrlrf"<;;'<;;'f-il

and

nlD

K2 = --==--

(5-46)

nZD -

Finally, from the values , the four radii of curvature of the lens faces may be determined. For of construction, the crown lens (1) may be chosen to be equiconvex. In addition the curvature of the two must match at their interface. The radii of curvature thus satisfy (5-47) In the design of an achromatic doublet. the three of refraction for each of the glasses to be are taken from manufacturer's specifications, like those presented in Table 5-1. One also inputs the desired overall of the achroTABLE 5-1

SAMPLE OF OPTICAL GLASSES

Type

Borosilicate crown Borosilicate crown Light barium crown Dense barium crown Dense flint Flint Dense flint Dense flint Fused silica

104

Chap. 5

code

V

nc

nD

nF

~

IOV

nD - t nF - nc

656.3 nm

587.6 nm

486.1 nm

517/645 520/636 573/574 638/555 617/366 620/380 689/312 805/255 458/678

64.55 63.59 57.43 55.49 36.60 37.97 31.15 25.46 67.83

1.51461 1.51764 1.56956 1.63461 1.61218 1.61564 1.68250 1.79608 1.45637

1.51707 1.52015 1.57259 1.63810 1.61715 1.62045 1.68893 1.80518 1.45846

1.52262 1.52582 1.57953 1.64611 1.62904 1.63198 1.70462 1.82771 1.46313

Aberration Theory

mat. In the series of calculations to the that is sequence. For crown designing an achromat of focal length 15 cm, radii of curvature given by

four radii of curvature, a calculation (5-46), and are employed in and 617/366 flint are used in these equations lead to lenses with

rll = 6.6218 cm

-6.6218 cm

rl2 rZI r22

with these each of the

=

-6.6218 cm cm

Eqs. (5-33) and (5-34) permit the calculation of focal lengths for wavelengths. In this case we find

I-;r~I"nhn'tp..

I, 6.3653 em 6.3961 em 6.2966 em

I I 1.0575 em 11.147 em 10.8485 em

15.0000 em 15.007 em 15.007 em

For a thin achromatizing renders focal lengths (nearly) equal, syslongitudinal and aberration at the same time. In a thick lens or an tem of lens combinations, the second principal for different wavelengths may not coincide as they do in a thin lens. When this is the case, equal focal for do not two measured as they are from their respective principal lead to a focal point on the axis, and chromatic aberration remains (Figure 5-14a). If the focal lengths for red and blue light are made such that they a single focus (Figure 5-14b), the difference inf8 andfR reof lateral magnifications, and chromatic aberration results in a mains. Thus the condition for removing lateral chromatic aberration is the coincifor the two wavelengths. dence of the

lal

(b)

Figure 5-14 Doublet with se<:ond principal planes separated for red and blue (a) focal lengths result in residual chromatic aberration. (b) Equal foci result in residual lateral chromatic aberration.

Another for zero chromatic aberration results if one uses two separated lenses (L =1= 0) of same glass (nl n2 n). The condition (JP jaA = 0 applied to Eq. (5-36) now

ap aA Sec. 5-7

Chromatic Aberration

K2L] = 0 105

the

and canceling iJnl iJA, there remains (5-48)

which is the same result derived for a double-lens in the following ""':1UL''''' Thus two lenses of the same material, separated by a distance to the average of their focal lengths. zero chromatic aberration for the wavelength at the focal1engths are calculated.

PR (5-18). 5-1. Carry oot the 5-2. If image and surfdce-in addition to ~""'~'J'''''' Eq. (3-20)-also the relation lis' = + 0/R). show that (a) s' = -(ndn2)s, and aberration in Eq. (5-18) vanishes. (b) a(Q) for (c) Show that a(Q) also vanishes for s' R and for rays intersecting with the cal surfaee vertex. Such image points are caned apla1llltic poilUS. (d) Find the points for a spherical surface of + 8 cm separating two media of refractive indices 1.36 and 1.70, respectively. 5-3. A collimated beam is incident on the plane side of a plano-convex lens of index 1.50, diameter 50 mm, and radius 40 mm. Find the spherical wave aberration and the 101llgltuolnal and transverse spherical ray aberrations. 5-4. Show that for a concave a calculation like that done for a retlract.mg surface a third-order aberration of a

=

!t..(!s 4R

where R is the magnitude of the radius of curvature. 5-5. Using the result of problem 5-4. determine the wave aberration, transverse """......Irion and aberration for a spherical mirror of 2-m focal length and 50-em diameter, when it forms an image of a distant object. 5-6. A uses a spherical mirror with a 3-m focal length and an aperture given (a) the results of problem 5-4, determine the magnitude of the spherical wave aberration for the telescope. (b) If a correcting plane of refractive index lAO were installed to correct the aberration, what would be the required difference in thickness between the center and edge of the plate7 an image of an axial point a +4.0-diopter lens with a diameter of a longitudinal spherical aberration of + 1.0 cm. If the object is 50 em from the lens. determine (a) the transverse spherical aberration and (b) the diameter of the blur circle in the paraxial focal 5-8. Determine the longitudinal and lateral ray aberration for a thin lens of n I YI == + 10 cm, and r2 = 10 cm due to rays parallel to the axis and through a wne or radius h = I em. 5-9. the equation for spherical aberration of a thin lens, given in .v",E."'"..... "u. spherical ray aberration of a lens as a function of ray by the longitudinal ray aberration as a function of ray for h = 0, 1, 2, 36 cm and 3, 4, and 5 em. The lens has a refractive index of 1.60 and radii YI r2 18 cm. The incident light rays are to the optical axis.

106

Chap. 5

Aberration Theory

thin lens of index 1.50 and radius 15 cm forms an of an axial 25 cm in front of the lens and for rays through a zone of radius h = 2 cm. Determine the longitudinal and lateral spherical ray aberration. problem 5-8.) 5-11. Show that if L (lIs:') - (lIs;), dLld(7 = 0 the condition for minimum aberration: pnIlIlNl"",,'y

n+2 5-12. A lens of index L50 and focal length 30 cm is "bent" to produce Coddington shape mctors of 0.700 and 3.00. Determine the corresponding radii of curvature. 5-13. A thin lens offocallength 20 cm is designed to have minimal spherical aberplane, 30 cm from the lens. If the lens index is 1 determine its ration in its radii of curvature. 5-14. A thin, plano-convex lens with l-m focal length and index 1.60 is to be used in an orientation that less spherical aberration while focusing a collimated beam. Prove that the proper orientation is with incident on the side ing the Coddington shape factor for cach orientation with the value spherical aberration. 5-15. A lens is needed to focus a beam of light with minimum spherical aberration. The focal length is 30 cm. If the glass has a refractive index of determine (a) the required Coddington factor and (b) the radii of curvature of the how do these lens. (c) If the lens is to be used instead to produce a collimated answers 5-15 when the lens is to reduce coma. 5-16. Answer 5-17. A 20-cm focal length positive lens is to be IlSed as an inverting that it simply inverts an without altering its size. What radii of curvature lead to minimum spherical aberration in this application? The lens refractive index is 1.50. 5-17 when the lens is to reduce coma. 5-18. Answer 5-19. It is desired to reduce the curvature of field of a lens of 20-cm focal made of crown (n 1.5230). For this purpose a second lens of flint glass (n 1.7200) is added. What should be its focal length? Refractive indices are for sodium light of 589.3 nm. 5-20. A doublet objective is made of a cemented positive lens (111 1.5736, II = 3.543 negative lens (n2 1 = 5.391 cm). (a) Determine the radius of their Petzval SUITdce. (b) What focal length for the negative lens a flat Petzval surface? 5-21. Design an achromatic doublet of 517/645 crown and 620/380 flint overall focal of 20 cm. Assume the crown glass lens to be Determine the radii of curvature of the outer surfaces of the lens. as well as its resultant focal length for the D, C, and F Fraunhofer lines. 5-22. an achromatic doublet of 5-cm focal length using 638/555 crown and 8051255 flint glass. Determine (a) radii of curvature; (b) focal lengths for D, and F Fraunhofer lines; (c) powers and dispersive powers of the individual elements. (d) Is (5-44) satisfied? 5-23. Design an achromatic doublet of -lO-cm focal 573/574 and 689/312 glasses. Assume the crown glass lens to be eqllic()flcave Determine (a) radii of curva(b) individual focal for the Fraunhofer D (c) the ture of the lens of the lens for the Fraunhofer and F lines. overall focal

[I] Martin, L. C. Technical Optics, Vol. 2, 2d ed. London: Sir Isaac Pitman & Ch. V and Appendix V. Chap. 5

References

1960.

107

[2]

14] 15] [6] [7]

[9] 110) [11] [12]

108

W. H. and M. H. Freeman. 9th ed. Boston: Butterworth Publishers, 1980. Ch. 18. Welford, W. T. Geometrical Optics. Amsterdam: North-Holland Publishing Company, 1962. Ch. 6. Welford, W. T. Aberrations Systems. Boston: Adam Hilger 1986. Guenther, Robert D. Modern Optics. New York: John Wiley and 1990. Appendix 5-B. Fundamentals. New York: Academic Press, 1978. Rudolf. Lens A. E. Applied Optics and Optical Design. New York: Dover Publications, 1957. Smith, Wcmen J. "Image Formation: Geometrical and Physical Optics." In Handbook of Optics, edited by \¥.lIter G. Driscoll and William Vaughan. New York: McGraw-Hill Book Company, 1978. R. E. "Geometrical Optics." In Geometrical and Instrumental Optics, edited Daniel Malacara. Boston: Academic 1988. Brouwer, William. Matrix Methods in Optical Instrument Design. New York; W. A. Benjamin. 1964. An Introduction. Reading, Mass.: Addison-Wesley Nussbaum, Allen. Geometric t'UIJ'lISIIIUg Company, 1968. Ch. 7, 8. Are Formed." Scientific American (Sept. 1968): 59-70.

Chap. 5

Aberration Theory

6

Optical Instrumentation

INTRODucnON The principles of developed are in this in order to discuss several practical optical instruments. 'I:'he discussion begins with an practical imporintroduction to the operation of stops, pupils, and windows, of tance to instrumentation. The optical instruments treated in the following articles then include the the camera, the eyepiece, the microscope, and the tele-

6-1 STOPS, PUPILS, AND WINDOWS We have been studying ways to trace rays through an optical system using the by-step application of Gaussian formulas, matrix methods, and ray tracing. Howray from an object point, directed toward or into an optical sysever, not every Depending on location of the object point and tem, survives to the final many of these rays are blocked by the limiting apertures of lenses and the ray intentionally inserted into the optical system. In this mirrors or by actual section, we wish to concentrate on the effects of such spatial limitations of light beams in an optical system. The dealt with are often inserted into an optical system to can be purposes. We have seen (Chapter how 109

used to modify the effects of spherical aberration. astigmatism. and distortion. In other applications apertures may be introduced to produce a sharp border to the age, like the outline we see looking into the eyepiece of an optical instrufrom undesirable light scatment. Apertures may also be used to shield the tered from optical components. In any case, are inevitably present because every lens has a finite diameter that effectively puts an aperture into the system. system influences its in ImThe presence of apertures in an portant ways. We have learned how reducing the useful of a single lens can reduce spherical aberration. An image fonned by the lens is obviously also reduced in In this chapter we examine how the depth of field (or focus) of the lens increases as its effective diameter becomes smaller. In the study of diffraction (Chapter 16) we examine the effect of lens diameter on the resolution of the image. All these properties are also influenced by the appearance of apertures in more complex optical instruments. In addition, apertures playa role in limiting the field of view, the extent of the object field that appears in the final The variety in complexity of practical optical is obviously very placement of apertures with reflecting and refracting large. In this introduction we must limit ourselves to the definition and explanation of the terms by this kind of analysis and to their application in a few simple optical "',,,,,,..,,,

Image Brightness: Aperture Stops and Pupils Aperture Stop (AS). The stop of an optical system is the actual optical component that limits the the maximum cone of rays from an axial object point that can be processed by the entire system. Thus it controls the of the image. The diaphragm of a camera and the iris of the human eye are "v'........~I"" of Another example is the telescope, in which the first, or objective, on lens determines how much light is accepted by the telescope to form a final the retina of the eye. In this simple case, the objective lens is the aperture stop of the optical system. However, the aperture stop is not always identical with the first component of an optical system. For to 6-Ia. As shown, the the extreme (or margilUll) rays aperture stop (AS) in front of the lens that can be accepted by the lens. But if the object 00' is moved nearer to AS, at some point the lens rim becomes the limiting aperture. (This is just the interof a line drawn from the rim through the of the apersection with the ture stop AS.) In this case, the subtended by the lens at 0 becomes smaller than the angle subtended by the and we designate the lens as the aperture stop. Entrance Pupil (E"P). The entrance pupil is the limiting that the light rays " looking into the optical from the~object. In Figure 6- I a, this is simply the stop so in this case, AS and E"P are identical, To see that this is not the case, look at Figure 6-1 b, where the aperture stop is behind the lens (a rear stop), as in most cameras. Which component now limits the cone of light rays? It is that component whose limit rays from 0 to their smallest relative to the Looking into the optical object directfy, but sees the AS through the lens. In other words, space, one sees the due to AS is image fonned by the the dashed line the effective marked P. Since rays from 0, directed toward this virtual aperture, make a this virtual aperture serves smaller angle than rays directed toward the lens as the entrance pupil for the Notice that rays from directed toward the of are in fact refracted by the lens so as to pass through the 110

6

Optical Instrumentation

lal AS

lens

Ibl

fel

Figure 6·1 Limitation of light rays by various combinations of positive lens and diaphragm.

of the real

This must be the case, since AS and EIIP are, by edges of EnP are the of the edges of AS. O"P'''',.,,,I rule: The entrance pupil is the image of the con· the imaging elements it. I When the con-

I "Preceding" is used in the sense that light must pass those always use light rays directed from left to right, we can simply say. "by all

Sec. 6-1

Stops, Pupils, and Windows

elements first. If we elements to its left."

111

trolling aperture is the first such element (afront stop), it serves itself as the entrance pupil. Another example, in which an aperture placed in front of the lens functions as the AS for the system, is shown in 6-1c. It is from 6-1 a in that the is inside the length of the lens. the aperbecause it, not the lens, limits the system rays to their ture is the AS for the smallest angle with the axis. Furthermore, it is the EnP of the system because it is the first element encountered by the light from the object. Exit Pupil (E"P). We have described the EnP of an optical as the from the If one image of the AS one sees by looking into the optical looks the optical from space, another image of the AS can be seen that appears to limit the output beam size. This image is called the exit pupil of the optical system. Thus the exit pupil is the image of the controlling aperture stop formed by the imaging elements following it to the right of it, in our The 6-1 b is automatically the for the because it is the last rear stop in of the the exit pupil is the optioptical component. to our cal conjugate of the AS; ExP and AS are conjugate planes. It follows that ExP is also conjugate with P. In Figure 6-1 a, Ex P is the real image of 6it is the virtual image. Notice that in each case, rays the entrance also or when intersect the edges of the exit pupil. In a a screen held at the position of the exit pupil opening of the aperture If the system of some optical instrument, the exit pupil is matched in position and to the pupil of the eye. Notice further that if the screen is moved closer to the lens, it intercepts a sharp II' of the object 00 I. The exit pupil is seen to limit the solid angle of rays forming each point of the

Chief Ray. The chief, or ray is a ray from an point that of the entrance pupil. Given the passes through the axial point, in the stop and the exit pupil, this nature of the entrance pupil with both the ray must also (actually or when extended) through their axial points. Verify this systems of Figure 6-1. Notice that the chief ray in the pencil of behavior in all rays from the axial point 0 coincides with the optical adding to our collection of new concepts that arise from a consideration of """'tlllr,,,", in optical systems, we consider a system slightly more complex than those of Figure 6-1, Of course, the single lens in those systems could each TPr"'pc,pnt an entire system whose ray paths are determined by its cardinal points. However. in 6-2, we specify a optical consisting of two lenses L 1 and L2 with an A between as shown. The first question to be answered is: Which element serves as the AS for the whole The answer to this question is not always obvious. It can be answered, however, by determining which of the actual elements in the system-in this case, L 1, or L2-has an entrance pupil that confines rays to their smallest as seen from the point. To decide which candidate nr"·"Pln ..., ture, it is necessary to find the entrance pupil for each by .... _,..,.,..,.., part of the optical system lying to its left: L 2:

112

Chap. 6

By ray or by calculation, the (as if light went from right to left), is (magnification) are shown. Optical Instrumentation

of lens . Both

formed by LI location and size

A;

A'1 I

I

I

I

I A

I

I

I I

I

L2' I

I I

0'

_-----j--

L1

I I

-:;:-~.J

L2

I I I I

I I

6-2 Limitation of light rays in an tive lenses and a diaphragm.

~:.---'-

system cOTlsislting of two

The of aperture A backward through L I is virtual and is shown as A:. L 1: Since lens L I is the first element, it acts as its own entrance pupil. A:

The three entrance pupils, L 2', L I, and A:, are next viewed from the axial point O. Since A ( subtends the smallest angle at 0, we conclude that aperture A is the aperture stop of the "y~, .....,,,. Once the AS is identified, it is imaged through the optical elements to its case, aperture A is imaged L 2 to A 2' . to find the exit pupil. In The chief ray, together with the two marginal rays, is drawn from each of the extreme points 0 and 0 I of the object. Notice that it passes (actually or when extended) through the centers of AS and its conjugate EnP and The chief ray the at A, at A ; it is virtual), and at The bundle of dent rays from either object point 0 or 0' , by the size of the entrance pupil A: • just makes it through the exit pupil At The image formed by L 1 is shown as II '; the final image (not shown) is virtual, since the rays from either 0 or O· diverge on L2.

of View: Field Stops and Windows. In describing the lions of a cone of rays from an axial object point, we have seen that entrance and exit pupils are related to the stop and so govern the brightness of the image. of view handled by the The controliing elApertures also determine ement in this connection is called the field stop, and it is related to an entrance window and an exit window in the same way that the stop is related to entrance and exit pupils. The simplest experience of a limitation in the field of view is that of looking such as a window. The of the window determine how through an much of the outdoors we can see. This field of view can be described in terms of the viewed, or in terms of the angular extent of the wi~­ lateral dimensions of the dow, relative to the line of sight. one can talk about the field of view in terms of the object or in terms of the image (on the retina, in this example). Sec. 6-1

PupilS, and Windows

113

To see how an aperture restricts the field of view-using diagrams that could well be to the case of window and eye just discussed-look at Figure 6-3. In (a), the optical is a A placed in of a single planes are also shown. from an axial point 0 are limlens. Object and ited in angle by the aperture and focused by the lens at point 0'. The same is true for the off-axis point T and its T'. In both cases, the lens is large enough to intercept the entire cone of rays. If the object plane is uniformly bright and the aperture is a circular hole, then a circle of radius O'T' is uniformly illuminated in the Lens

T'

----------~--~--~O'

Image plane

Object plane (a)

U' T' 0' Image

V Object plane

Ib)

EnPl

a

E"W

L1

1

lAS

1 Ie)

Figure 6-3 to the same system, diagrams (a) and (b) illustrate both the way in which an aperture the field of view and the process by vignelling. Diagram (c) is an of a more complex system, showing the al1gular field of view in object and image space.

However, if one considers object below T, the top rays from such points, passing through the aperture, miss the lens. Such a point, U, is shown in part (b) of the the same optical as (a), but redrawn for clarity. It is chosen such that the chief or central ray of the bundle just the top of lens. point U receives only about half as much About half the beam is lost so that light as points 0' and T'. Thus the circular image to dim as its inI

114

Chap. 6

Optical Instrumentation

creases. This partial of the outer portion of the image by the for object points is vignetting. vignetting may the image of a point object appear astigmatic. Finally, object point V is chosen such that all its rays through the the lens The lateral field of processed as the optical OU if one considers the field of view as consisting of circle of all object points that produce image points having at least half the maximum irradiance near the center. One can then also define the angular. field of view as the angle the enmade with the axis by the chief ray at the center of the aperture trance pupil. The lens in this case, acts both as the field stop and the entrance quantities may be described window. In other, more complex, systems, the as follows. Field Stop (FS). The field stop is the that controls the field of view formed by chief rays. As seen from the center the enby the solid its image) subtends the smallest angle. When the trance pupil, the field stop of the field of view is to be sharply delineated, the field stop should be plane so that it is sharply focused along with the final image. A of such a field stop is the opening directly in front of the film that view an <>np·.. t1l1rp image in a camera. Intentional limitation of the is desirable when either far-off axis imaging is of unacceptable quality due to aberraor when vignetting reduces the illumination in the outer portions of

Entrance Window The entrance window is the image the field stop formed by all optical preceding it as the entrance pupil is an image of the aperture stop). The entrance window the lateral dimensions of the object to be viewed, as in the viewfinder of a camera, and its diameter nptprn"ln,f'''' the angular field of view. When the is located in an in the plane, where it the plane, the entrance window field imaged by the optical system. lateral Oimensions of the Exit Window (Ex W). The exit window, to the exit pupil, is the of the field stop formed by all optical following it. To an observer in space, the exit window seems to limit the area of the image in the same way as an outdoor scene appears limited by the window a room.

In 6-3c, the role of field stops and entrance and exit windows is shown in a more complex consisting of two lenses and two apertures. The first is the AS of the and, as we have seen, is related to an entrance pupil, image in L 1, and an exit pupil, its image in L 2. The second aperture is the field with corresponding images through the lenses: the entrance window to in object space can then the left and the exit window to the right. The field of be by a, the angle subtended by the entrance window at the center the entrance pupil. Similarly, the view in image space can be described by a', the subtended by the exit window at the center of the exit pupil. We see that of the field imaged by the system is effectively determined by the enthe Notice that, since W trance window actually, the size of the field they are conjugate planes. Thus the beam that and Ware both images of the the entrance window the field stop and the exit window. The Summary of Terms is provided as a convenient reference to a subject that and experience with many examples, to master. Sec. 6-1

Stops, Pupils, and Windows

115

SUMMARY OF TERMS

Brightness Aperture stop: Entrance pupil: Exit pupil:

The real aperture in an optical system that limits the size of the cone of rays by the system from an axial point. The image of the aperture stop formed by the any) that it. The image of the aperture stop formed by the optical elements any) that follow it.

Field of view Field stop:

The real aperture that limits the angular field of view formed by system. an Entrance window: The image of the field stop formed by the optical elements (if any) that it. Exit window: The image of the field stop formed by the optical elements (if any) that follow it.

6-2 PRISMS

Angular Deviation of a Prism. The top half of a double-convex, spherical lens can form an image of an axial object point within the paraxial approx6-4. If the lens surfaces are flat, a prism is formed. and imation, as shown in paraxial rays can no longer produce a unique point. It is nevertheless helpful in some cases to think of a prism as functioning approximately like one-half of a convex lens.

:ngurc 6-4

due to half of a convex lens approximates the action of a

In the following we derive the relationships that describe exactly the progress of a single ray of light through a prism. The bending that occurs at each face is deteris a function of the refractive index of mined by Snell's law. The degree of the prism and is, a function of the of the incident light. The with wavelength is called dispersion and variation of refractive index and is discussed later. For the present we assume monochromatic light, which has its own characteristic refractive index in the prism medium. The relevant angles de6-5. scribing the progress of the ray through the prism are defined in of incidence and refraction at each face are shown relative to the normals con{) structed at the point of intersection with the light ray. The total angular

I<'igllrc 6·5

Progress of an arbitrary ray a prism.

116

Chap. 6

Optical Instrumentation

of the ray due to the action of the prism as a whole is the sum of the angular deviations 6 1 and 62 at the first and faces, respectively. Snell's law at each prism face requires that (6-1) sin 8 = n sin 8: sin 82

n sin

(6-2)

and inspection will show that the following geometrical relations must hold between the angles:

6.

8;

(6-3)

=lJz-tH

(6-4)

=

81

B

180 - 8; -

A

8:

180 - A

+ 82

(6-5) (6-6)

The two members of (6-5) follow because the sum of the of a triangle is 1800 and because the sum of the of a quadrilateral must be 3600 • Notice that formed by the normals with the prism the A and B and the two right sides constitute such a quadrilateral. Using (6-1) through (6-6), a programmable calculator or computer may easily be to perform the sequential operations that determine the of deviation, /). that the prism A and refractive index n are given, then the stepwise calculation for a ray incident at an angle 8. is as follows:

8: = sin-I

(6-7)

/)1

81

8;

(6-8)

82

A -8i

(6-9)

(Jz

-

sin-I (n

(2)

/) = 8 1 + 82

8: -

=

(6- 10)

8i

(6-11)

The variation of deviation with angle of incidence for A = and n = 1.50 is occurs for (}I = 23°. shown in Figure 6-6. Notice that a minimum tion by a prism under the condition of minimum is most often utilized in practice. We may argue rather neatly that when minimum deviation occurs, the ray of light passes symmetrically through the prism. making it unnecessary to subscript angles, as shown in Figure 6-7. Suppose this were not the case, and minimum deviation occurred for a nonsymmetrical case, as in 6-5. Then if the ray were refollowing the same path backward, it would have the same total deviation as the forward ray, which we have supposed to be a Hence there would be of incidence, (}. and (}2, producing minimum deviation, contrary to expetwo rience. The geometric relations simplify in this case: From Eq.

1),

/) = 21J

21J'

(6-12)

and from Eq. (6-6), A

(6-13)

21J'

Together these allow us to write (}' Sec. 6-2

Prisms

A 2

and

(}

/)+A 2

(6-14) 111

c;

'" :E c::

0 .;:;

.;; '"

'"

25

"0

'0

'"c

Ci

<:

20

15

20

40

80

60

Angle of incidence (deg)

Figure 6-6 Graph of total deviation versus angle of incidence for a light ray through a prism with A = 30" and n = 1.50. Minimum deviation occurs for an angle of 23°.

Figure 6-7 Progress of a ray through a prism under the condition of minimum deviation.

so that Eq. (6-1) becomes

. (A S)

. (A)"2

+ - = nsm sm - 2 or

n=

sin[(A + 8)/2] sin(A/2)

(6-IS)

Eq. (6-1S) provides a method of determining the refractive index of a material that can be produced in the form of a prism. Measurement of both prism angle and minimum deviation of the sample determines n. An approximate form of Eq. (6-1S) follows for the case of small prism angles and, consequently, small deviations. Approximating the sine of the angles by the angles in radians, we may write (A

n ==

+ S)/2 A/2

or S

==

A(n -

I),

minimum deviation, small A

(6-16)

For A = ISO, the deviation given by Eq. (6-16) is correct to within about 1%. For A = 30°, the error is about S%.

118

Chap. 6

Optical Instrumentation

n A-~~:;:::::!::- Red Green Viol!?!:

200

1600

Figure 6-8 Typi,al normal dislDCn;ion curve and consequent ,olor white light

refracted through a

1-.. (nm)

Dispersion.

The minimum deviation of a monochromatic beam through a (6-] in terms of the refractive refracimplicitly by tive however. on wavelength, so that it would be better to write nIl for this quantity. As a the total deviation is itself wavelength dependent, means that various components of the incident light are separated twrmal dispersion curve and the nature of on refraction from the the resulting color separation are shown in 6-8. shorter wavelengths have larger refractive indices and. therefore. smaller in the prism. The disConsequently violet light is deviated most in refraction through the nP'·...1l'.n indicated in the of 6-8 is called "normal" When the of wavelengths medium has excitations that absorb within the range of the dispersion curve, the curve is monotonicaHy decreasing, as but has a positive slope in the wavelength region of the absorption. When this occurs, the term anomalous dispersion is used, although there is nothing anomait. (This is further in Section The normal dispersion lous but varies somewhat for different An empirical recurve shown is lation that approximates the curve, introduced by Augustin Cauchy, is

B C A+-+ + ... A2

{6-17}

where A, B, C, ... are constants to be fitted to the data of a particular material. Often the first two terms are sufficient to provide a in which case knowledge of n at two distinct wavelengths is sufficient to determine of A and B that represent The dispersion, defined as dn/dA, is then approximately, using Cauchy's formula, dn/dA = It is important to dispersion from Although prism matedeviation at a given the dispersion or rials of large n need not be correspondingly large. Figure 6-9 separation of neighboring depicts extreme cases illustrating the distinction. Historically, dispersion has been characterized by using wavelengths of light near the middle and ends of the

Small deviation Large dispersion

Extreme cases the dispersion S; for three wavelengths and the deviation {j for the intermediate wavelength.

Figure 6-9

Sec. 6-2

Prisms

119

visible spectrum, called Fraunhofer lines. These lines were among those that apin the studied by J. von Fraunhofer. Their wavelengths, towith indices, are in Table 6-1. The F and C dark lines are due to absorption by hydrogen atoms, and the D dark line is due to absorption by the sodium atoms in the sun's outer atmosphere. 2 Using the thin prism at minimum TABLE 6-1

fRAUNHOfER LINES

n (nm)

OJaracterizalion

486.1 589.2 656.3

F, blue D, yellow

C, red

Crown

Flint glass

1.5286 1.5230 1.5205

1.7328 1.7205 1.7076

deviation for the D for the ratio of angular spread of the F and C wavelengths to the deviation of the D wavelength, as suggested in Figure 6-9, is

2C fj

This measure of power,

nD

ratio of dispersion to deviation is defined as the dispersive

(6-18)

nD

Using Table 6-1, we calculate the dispersive power of crown glass to be is, while that of flint glass is , more than twice as great. The reciprocal of the dispersive power is known as the Abbe number.

Prism Spectrometers. An analytical instrument employing a prism as a dispersive together with the means of measuring the prism angle and the angles of deviation of various wavelength components in the incident light, is called a prism spectrometer. Its essential components are shown in Figure 6-10. Light to be is focused onto a narrow slit S and then collimated by lens Land rewhich typically rests on a rotatable platform. Rays of light by the prism corresponding to each wavelength component emerge mutually parallel after refraction by the prism and are viewed by a telescope focused for infinity. As the scope is rotated around the prism table, a focused image of the sUt is seen for each

Figure 6-10

Essentials of a spectrometer.

2 Because the sodium D line is a doublet (589.0 and 589.6 nm). the more monochromatic d line of helium at 587.56 nm is often preferred to characterize the center of the visible spectrum. The green line of mercury at 546.07, lying nearer to the center of the luminosity curve (Figure 2-7), is also used.

120

Chap. 6

Optical Instrumentation

wavelength component at its corresponding angular deviation. The deviation 0 is measured to the telescope position when the slit without the in place. When the instrument is used for visual observations without the capability of angular displacement of the spectra] "lines," it is called a spectroscope. If means are provided for recording the spectrum, for example, with a photographic film in the focal plane the telescope objective, the instrument is called a spectrograph. When the prism is made of some type of its wavelength range is limited by the absorption of glass outside the visible To extend the of spectrograph farther into the ultraviolet, for prisms made from quartz (Si02) and fluorite (CaF2) have been used. Wavelengths extending further into the infrared can be handled by prisms made of salt (NaCI, KCl) and sapphire (AhOJ).

Chromatic Resolving Power. If the wavelength between two components of the light incident on a prism is aHowed to diminish, the ability of prism to resolve them will ultimately fail. The resolving power of a prism spectograph thus an important performance which we shall evaluate in this section. Imagine two lines on a photographic film in a prism spectrograph. The lines are images of the slit, so that precise wavelength measurements the entrance slit should be as narrow as possible consistent with of illumination of the film. Even with the narrowest slit widths, however. the spectra] line is found to possess a width. directly traceable to the limitation that edges of the collimating lens or prism impose on the light beam. phenomenon is therefore due to the diffraction of light, treated an irreducible width due to diffraction, as flA delater. Since the line images creases and the lines approach one another, a point is reached where the two lines appear as one, and the limit of resolution of the instrument is realized. No amount of magnification of the images can proouce a higher resolution or enhancement of the to discriminate between such closely lines. 6-1 in which a monochromatic parallel beam of Consider cident on a prism, such that it fills the prism face. Employing Fermat's ray FTW is isochronous with ray GX, since they begin and end on the same plane wavefronts, GF and XW, respectively. Their optical can be equated to give FT

+ TW

= nb

T

x

G Ill)

T

A+ A A

Sec. 6-2

Prisms

Figure 6-11 Constructions used to delennine chromatic power of a (a) Refraction of monochromatic light. (b) Refraction of two wavelength components separated t:.A.

121

where b is the base of the prism and n is the index of the prism, corresponding to the wavelength A. If a second neighboring wavelength component A' is now also in the incident such that A' - A AA, the component A' will be associated with a different refractive n For normal persion, will be a small, quantity. The wavefronts for the two components, shown in Figure 6-11 b, are thus separated by a small angular difference t.w: and are accordingly focused at different points in the focal plane of the telescope objective. Fermat's principle, applied to the second component A', gives

FT

+ TW

-

as

(n - An)b

Subtracting the last two equations. we conclude

or, introducing

bAn

(6-19)

b(~~) AA

(6-20)

dispersion.

=

as

Equation (6-20) now relates path difference to the wa'/ele:nglln difference ll.A. The angular difference t.w: can also introduced, using

(!!.)

A = As = (dn) ll.A a d d dA

(6-21)

where d is beam width. We appeal now to Rayleigh's criterion, which determines the limit of resolution of the diffraction-limited line images. This criterion is explained and used in the later treatment of diffraction, where it is shown that the minimum separation Aa of the two wavefronts, that the are barely resolvable, is given by

t::.a =

A d

(6-22)

Combining Eqs. (6-21) and

A d

(~)(~~)

or the minimum wavelength ser1anltiGIn permissible for resolvable (ll.A)min = b

(dJ~/dA)

is (6-23)

The resolving power provides an alternate way of describing the resolution limit instrument. By definition, power

b dn dA where we have incorporated (6-23), Since dispersion is limited by glass, resolving power be improved by increasing the b. this technique soon requires large and prisms. The dispersion dn/dA may be for from the formula for the prism material,

(6-17). Example Determine the resolving power and minimum resolvable wavelength difference for a made from flint with a of 5 cm.

122

Chap. 6

Optical Instrumentation

Solution We can "'",,"UI''''' an approximate average value of the dispersion for A = 550 nm by "' .....w ........

l1n

nF

AA

AF

Thus the

L 7328 - 1.7205 486 - 589

nD

1.19

X

10- 4 nm- I

power is

Wi = b (:)

(0.05

x

109 nm)(1.19

= 5971

x

The minimun resolvable wavelength difference in then A (AA)min = Wi

5550

A

= 5971

around 550 nm is

1A

Although grating spe:ctroglllplt1S powers, they are generally more wasteful of Furthermore, they pr(J(im:e 01""".. _",..""".. images of the same wavelength component, which can be when interpreting records. These instruments are discussed later. Prisms with Applications. Prisms may be combined to duce achromatic overall behavior, that is, the net for two given wavelengths may be made zero, even though the deviation is not zero. On the other the direct vision accomplishes zero deviation for a particular wavelength while at the same time dispersion. Schematics involving combinations of two prism types are shown in Figure 6-12. The of prisms in 6-12a, combined so that one prism cancels the of the other, can reversed so that the is additive, providing double dispersion.

---4--~_ _~~~~~~~~;:::=~1<~ ~2

Ib) Direct vision prism for wavelength

(al Achromatic prism

6-12

Nondispersive and

n011,(1... "i"ti,,,p

>

11

~

prisms.

useful in spectrometers is one that produces a constant as they are observed or detected. One example is the PellinBroca in Figure 6-13. A of light enters at fuce AB and at fuce AD, making an of 90° with the incident direction. The dashed lines are merely added to in analyzing the operation of the only one will refract at structure. Of the incident that conforms to the case deviation, as with to the prism base AE. At face BC total reflection occurs to direct beam into the prism section ACD, where it traverses under the condition of minimum deviation. Since the prism section BEe serves as a mirror, the passes effectively with deviation through sections wa'veH~ml~ms

Sec. 6-2

Prisms

123

8

A

c

6·13 deviation.

F

Pellin-Broca prism of constant

(bl

(8)

Ie)

(d)

Figure 6-14 Image manipulation by refracting Dove prism. (c) Penta cross

(a) Right-angIe prism. (b) (d) Porro

ABE and ACD, which together constitute a prism of 6(t apex angle. In use, the spectral line is observed or recorded at F, the focal point of L. Thus an observtelescope may be mounted. Instead, the on its prism table wavelengths in the (or about an axis normal to the page), and as it incident beam successively meet the condition of incidence for minimum deviation, producing path with focus at F. The rotation may be caliin terms of W3'Vell,-nQ brated in terms of Reflecting Prisms. Total internally reflecting are frequently used in optical systems, both to alter the direction of the optical and to change the orientation of an course, prisms alone cannot When used in conjunction with elements, the light incident on the prism is first collimated and rendered normal to the prism face to avoid aberrations in the image. Plane mirrors may substitute for the reflecting the prism's reflecting faces are to of contamination, and the process of total internal reflection is capable of reflectivity. The stability in the angular relationship of prism faces may be an important in some applications. Some examples of in use are illustrated in 6-14. The Porro prism. Figure 6-14d, of two right-angle prisms, oriented in such a way that the face of one prism is to receive the incident light and the face of the second prism is partially revealed to output the refracted The prism halves are in the to its action. Images are inverted in both vertical and horizontal directions by the so that the Porro prism is commonly used in binoculars to produce erect

6-3 THE CAMERA The simplest type of camera is the pinhole camera. illustrated in Figure 6-15a. Light rays from an object are admitted into a light-tight box and onto a photographic film through a tiny pinhole, which may be with any simple means of shuttering,

1111

Figure 6-15

(bl

a pinhole camera.

of black An is projected on the wall of is lined with a As stated earlier. an image point is determined ideally when every ray from a object point, each by the optical system, intersects at point. A pinhole does no and actually blocks out most of rays from object point. Because of the of the pinhole, every point in the image is intersected only by rays that originate at approximately the same point of the object. as in Figure 6-15b. Alternatively. every object point sends a of rays to the screen, which are limited by the small pinhole and so form a

Sec. 6-3

The Camera

125

small circle of light on screen, as in Figure 6-1Sa. The overlapping of these circles of light due to every point maps out an image whose sharpness aer;leoclS on the diameter of the individual circles. If they are too large, image is blurred. improves in clarity, until a certain Thus, as the pinhole is reduced in size, the pinhole size is reached. As the pinhole is reduced further, the images of each object point actually grow larger again due to diffraction, with consequent degradation of the Experimentally, one finds that optimum pinhole size is around 0.5 mm when the pinhole-to-film distance is around 25 cm. The pinhole itself must be accurately formed in as thin an aperture as possible. A pinhole in aluminum foil, supported by a larger aperture, works well. The primary advantage of a pinhole camera (other than its elegant simplicity!) is that, since there is no focusing involved, all are in focus on the screen. In other words, the depth of field of the camera is unlimited. The primary disadvantage is that, since the pinhole admits so little of the available light, exposure times must long. The pinhole camera is not useful in freezing the action of moving objects. The pinhole-to-film of the and the field of view. As while not critical, does affect the this distance is the angular seen by the film is larger, so that more of the scene is with corresponding decrease in size of any feature of scene. Also, the image circles decrease in size, producing a clearer image.

Figure 6-16 Simple camera.

If the pinhole aperture is opened sufficiently to accommodate. a ('Cu''''''rmlno lens, we have the basic elements of the ordinary camera (Figure 6-16). The mo!>t immediate benefits of this modification are (1) an increase in the brightness of the image due to the of all the rays of light from each object point onto its conjugate image point and (2) an increase in sharpness of the image, also due to the focusing power of the lens. The lens-to-film distance is now critical and depends on the object distance and lens focal length. For distant objects, the film must be situated in the focal plane of the lens. For closer objects, the focus falls beyond the film. Since the film plane is fixed, a focused is by the lens to be moved farther from the film, that by "focusing" the camera. The extreme possible position of the lens determines the nearest distance of objects that can be handled by the camera. "Close-ups" can be managed by changing to a lens with shorter focal length. Thus the focal length of the lens determines the subject area received by the film and the size. In general, image size is proportional to focal length. A wide-angle lens is a short focal-length lens with a large field of view. A telephoto lens is a long lens, providing magnification at the expense of subject area. The telephoto lens avoids a correspondingly "long" camera by using a positive lens, separated from a second negative lens of shorter focal length, such that the combination remains positive.

126

Chap. 6

Optical Instrumentation

Also to the operation of the camera is size of its which admits to the film. In most cameras, the aperture is variable and is coordinated with the exposure time (shutter speed) to determine the total exposure of the film to plane (irradiance E" in light from the scene. The light power incident at the and inwatts per square depends directly on (1) the area of the of the image. If, as in 6the aperture is circular on the with diameter D and the energy of the light is assumed to be distributed uniformly over a corresponding circle of diameter d, then oc - - - - ' - - - -

(6-25)

Figure 6·17 Illumination of image. The aperture (not shown) determines the useful diameter D of the lens.

The we can write

as in Figure 6-17, is proportional to the focal length of the lens, so

(~r

(6-26)

The lID is the relative aperture of Ilstop), which we symbolize by the letter

caned f number or

oc

A

D

but unfortunately, usually identified by the 4-cm focal length that is stopped down to an ture of A 4/0.5 8. This aperture is The irradiance is now Ee

1 0:-

(6-27)

II A. For example, a lens of of 0.5 cm has a relative aperreferred to by photographers as (6-28)

Most cameras provide selectable apertures that sequentially change the irradiance at of 2. The correspoinding f numbers then form a geometric sea ries with ratio , as in Table 6-2. numbers correspond to smaller film depends on the product of irraexposures. Since the total exposure diance (J/m 2-s) and time (s), a desirable total exposure may be met in a variety of ways. Accordingly, if a particular film is described by an ASA numis perfectly exposed by light from a scene with a shutter speed of sb s by any other combiand a aperture of 118, it will also be nation that the same total exposure, for by a shutter in shutter cuts the total exposure s and an aperture of I /5.6. The in half, but opening the aperture to the nextllstop doubles the exposure, leaving no in net exposure. The particular combination of shutter and relative aperture chosen for an total exposure is not always The shutter speed must be of course, to capture an action shot without blurring the image. The choice of relative Sec. 6-3

The Camera

121

TABLE &-2

STANDARD RELATIVE APERTURES AND IRRADIANCE AVAILABLE ON CAMERAS

A

[number

(A

E, I

1.4 2

2.8 4 5.6 8 II 16

22

2 4 8 16

32 64

128 256 512

Eo Eo/2 E o/4 Eo/8 E o/16 E o/32 E o/64 E o/128 E o/256 Eo/512

aperture also another property of the the depth offield. To define this which shows an axial object point at quantity precisely, we utilize Figure imaged at distance on the other side. All objects in the plane are focused in the plane, lens aberrations. Objects both closer to and farther from the send bundles of rays that focus farther from and closer to the image plane, respectively. Thus a flat film, situated at distance sb from the lens, intercepts circles of confusion corresponding to are small enough, the resultant such points. If the diameters of these is still Suppose the largest acoeptable diameter is d, as shown, are suitably "in focus." such that all images within a distance x of the precise The depth of field is then said to be the interval in object spaoe conjugate to the interval sb x to So + x, as shown. Notice that although the interval is ",mrlrnptri;f' about So in spaoe, the of field interval is not about So.

Figure 6-18 Construction mu:,u"'''''15 depth of field. Object and image spaces are not shown to the same scale.

The near-point and SI and S2, of the depth of field can be determined onoe the allowable blurring d is chosen and the lens is "rw·f',i-.P
128

Chap. 6

Optical Instrumentation

so that x

(6-29)

D

It is then required to find, from the lens equation, the object distance

SI corresponding to image distance sb + x and the object distance S2 corresponding to image distance sb - x. After a moderate amount of algebra, one finds

where the aperture is A

Sl

=

82

=

+ Ad) /2 + Ad80

(6- 30)

so/(f - Ad) fz - Adso

(6- 31)

So/(f

= / / D. the depth of field, depth of field =

82 -

81,

2 Adso(so - f) /2 /4 _ A 2 d 2 2

can be expressed as (6-32)

80

Acceptable values of the circle diameter d depend on the quality of the photograph desired. A slide that will be projected or a negative that will be enlarged requires better original detail and hence a smaller value for d. For most photographic work, d is of the order of thousandths of an inch. As an example, let d = 0.04 mm. A 5-cm focal length lens with / /16 aperture used to focus on an object 9 ft away will focus all objects from around 5 ft to 30 ft to an acceptable degree. Most cameras are equipped with a depth-of-field scale from which values of SI and 82 can be read, once the object distance and aperture are selected. According to Eg. (6-32), depth of field is greater for smaller apertures (larger f numbers), shorter focal lengths, and longer shooting distances. The camera lens is called upon to perform a prodigious task. It must provide a large field of view, in the range of 35° to 65° for normal lenses and as large as 120° or more for wide-angle lenses. The image must be in focus and reasonably free from aberrations over the entire area of the film in the focal plane. The aberrations that must be reduced to an acceptable degree are, in addition to chromatic aberration, the five monochromatic aberrations: spherical aberration, coma, astigmatism, curvature of field, and distortion. Since a corrective measure for one type of aberration often causes greater degradation in the image due to another type of aberration, the optical solution represents one of many possible compromise lens designs. The labor involved in the design of a suitable lens that meets particular specifications within acceptable limits has been reduced considerably with the help of computer programming. Human ingenuity is nevertheless an essential component in the design task, since there are many more than one optical solution to a given set of specifications. The demands made upon a photographic lens cannot all be met using a single element. Various stages in solving the lens design problem are illustrated in Figure 6-19a, from the single-element meniscus lens, which may still be found in simple cameras, to the four-element Tessar lens. The use of a symmetrical placement of lenses, or groups of lenses, with respect to the aperture is often a distinctive feature of such lens designs. In such placements, one group may reverse the aberrations introduced by the other, reducing overall image degradation due to factors such as coma, distortion, and lateral chromatic aberration. The multiple-element lens in a modern 35-mm camera is shown in the cutaway photo (Figure 6-19b).

Sec. 6-3

The Camera

129

Single meniscus lens

Achromatic double meniscus

Cooke triplet

Tessar

Petzval

(sl

(b)

Figure 6-19 (a) Camera lens design. (b) Cutaway view of a modem 35-mm Olympus Corp., Woodcamera, revealing the multiple element lens. bury. N.Y.)

130

6

Optical Instrumentation

6-4 SIMPLE MAGNIRERS AND EYJIEP/JIECf:S

The simple is a lens used to read small print, in which case it is often called a rp~j""T.'" glass, or to assist the eye in examining any small detail in a real object. It is a simple convex lens but may be a doublet or a triplet, thereby providing for principle of the sip/pie magnifier. A small Figure 6-20 illustrates the object of dimension h, when examined by the unaided eye, is assumed to be held at t,~~ I -..;;...::::--I I I

I

- -- -- -

-- --.......-..-

I

I I I I I

h (al

I~

6-20

Operation of a simple magnifier.

the near point of the normal of distinct vision-position (a), 25 cm from the eye. At this subtends an angle ao at the eye. To project a on the is inserted and the object is moved closer to (b), where it is at or just the focal point of the lens. In this position, the lens forms a virtual subtending a larger angle aM at is defined to be the ratio the eye. The angular magnification 3 of the aMIao. In the paraxial approximation, the may be represented by their tangents, giving 25

ao

=

f

If the image is viewed at infinity, s

25

M

s

at ,nll"''''''

f

At the other extreme, if the virtual s' = - 25 cm, a nd from the thin -lens

(6-33)

is viewed at the nearpoint of the eye, then ""'I~I'lLl'U",

s = --"-25 + f

giving a magnification of

M =

25

f

+

1

at normal near

(6-34)

The actual angular magnification then on the particular viewer, who will move the simple magnifier until the virtual is seen comfortably. For small focal lengths, Eqs. (6-33) and do not differ and in magnifica3 When viewing virtual with instruments, the may be al great d'··.........,."... even "al infinity," when rays entering the eye are paraDe!. In such cases, latenl! magnificati S IlIso'apinfinity and are not very usefuL The more COIlvenienl magnification is cI ~.'i:t measure of the image size formed on the retina and is used to describe when ey,. s are in.. volved, as in microscopes and telescopes.

Sec. 6-4

Simple Magnifiers and

131

tions, is most often used. Simple may have magnifications in the range of to lOx, although the achievement of higher magnifications usually ""-"AU'''''' a lens for aberrations. In when magnifiers are used to aid the eye in viewing images formed by prior components of an optical system, they are oculnrs. or eyepieces. The for example, serves as real formed by the objective lens of a that is viewed by the eyepiece, whose magnification contributes to the overall magnification of the instrument. To quality images. the ocular is corrected to some extent for aberrations and, in particular, to reduce transverse chromatic aberration. To accomplish this improvement, two lenses are most often used. We showed earlier that the effective focal to principal planes!) of two thin lenses, separated by a (llsltanc:;e I

I

/

/1

-=-+

L

(6-35)

h

""",r"",p"t the individual focal lengths of the formula, for lenses made of the same glass,

1. = (n

-

= (n

1)(_1 _ _ I )

/.

RII

the lensmaker's

(6-36)

RI2

and (n -

0(_1 - _I) R21

where the

nrp'.""',nn"

1

f

= (n

I)Kz

(6-37)

R22

in parentheses involving the radii of curvature of the lens surconstants K, and K 2 , respectively. (6-36)

(n

I)KI

+

(n - 1)K2 - L(n -

1)2

(6-38)

To correct for transverse aberration. we that the effective focal length of the remain independent of refractive index,4 or

From Eq. (6-38),

This condition is met,

th"",.",tr...",

L

when the lenses are

"""',
1

----+---1)

Kz(n-

or more simply, when (6-39) This condition is valid independent of the lens shapes, leaving the choice of shapes as latitude for other aberrations. • Some longitudinal chromatic aberration remains because tbe principal planes of the system do nol coincide. Refer back to 5-14 and the related discussion.

132

Chap. 6

Optical Instrumentation

El Exit

Fl

pupil

6-21

Huygens eyepiece.

Both the Huygens and Ramsden eyepieces, Figures 6-21 and incorporate by. (6-39), that is, plano-convex are separated the design feature 6-21, the focal by half the sum of their focal lengths. In the diagram of is approximately 1.7 times the of the eye length of the field lens, or ocular, EL. The image "observed" by the ev(~pu::ce is in this case a lens, since its virtual between the virtual object (VO) for lenses. The field lens then forms a real image (RI) that is When the real in the focal plane of the eye viewed at infinity by the eye located at the exit pupil. Note that the If cross hairs or a with a scale piece cannot be used as an is used with the eyepiece to make possible quantitative measurements, then to be in of RI, convefocus with the the cross hairs must be placed in the focal or stop placed there (Figure The image attached to the of the cross hairs does not share in the image quality provided by the as a whole, however, because the eye lens alone is involved in forming the image. This is not a problem in the eyepiece, Figure 6-22, in which both the primary and intermediate just in front of the field lens. In f and, according to the lenses have the same by f. Ideally, rays emerge from the eyepiece parallel to one another, magnified image at infinity, when the real object, RO, falls at the IJV"""LI' first lens. A reticle is at this A disadvantage of this the surface of the lens is then also in including dust and '"''''''''i5''''' lens separation slightly smaller than the reticle is in focus at a

RO

Fie,dT stop

El

Fl 6-22

Sec. 6-4

Simple Magnifiers and

Exit pupil

Ramsden eyepiece.

133

front of the lens, as shown in the ray 6-23. With a lens separation somewhat less than I, however, the on L that corrects for transverse chromatic aberration is somewhat violated. A modification of the Ramsden "''''''',,'£OP that almo~1 eliminates defects is the Kellner eyepiece, which rethe Ramsden eye lens with an achromatic doublet. Other eyepieces have also rlp,,,,,,,p.rl to achieve higher and wider fields. Ramsden eyepiece

Huygenian eyepiece

I

I

Eye lens

Eye lens

I

t:;::==F==¢~- Retaining ring

Field lens

Retaining ring

6-23 Construction of Huygens and Ramsden eyepieces.

A eyepiece uses two lenses having focal lengths of 6.25 cm and 2.50 em, respectively. Determine their optimum in chromatic their equivalent focal length, and their magnification when an image at infinity. Solution The optimum separation is given by

+ h)

L = !(fl

=

h6.25 +

cm

The Pi1.,,.,,,lp,,r focal length is found from 1

which

1

L

1

]

I

-=-+---=--+-I II h 1t/2 6.25 2.50 (6.25)(2.50) I = 3.57 cm. The angular maJgnilhea1lion is M=25=~=7 I 3.57

In one usually desires an exit pupil that is not much greater than the size of the pupil of the eye, so that radiance is not lost. Recall that the exit pupil is an image of the entrance pupil as formed by the ocular and that the ratio of entrance to exit pupil diameters equals the magnification. Since the entrance elements in the optical of the pupil is telescope), this requirement places a limit on the objective lens, in a ing power of the and, thus, a lower limit on its focal length. of an eyepiece, assuming its aberrations are The important within limits for a application, include the following: '''-O'''-J

134

Chap. 6

Optical Instrumentation

magnification, 25 / f, where f is the focal length in centimeters. Available values are 4x to 25X, to focal lengths of 6.25 to 1 cm. 2. Eye that is, the distance from eye lens to exit pupil. Available eye:pieces have eye reliefs in the range of 6 to 26 mm. 3. Field-of-view or size of the primary that the eVfmu~ce can cover, in the range of 6 to 30 mm. 1.

6-5 MICROSCOPES

The magnification of small objects by the magnifier is increased further by the compound microscope. In its simplest form, the instrument lenses, an objective lens of small focal length that faces the consists of two and a magnifier functioning as an eyepiece. The eyepiece "looks" at the real where the object lies outformed by the Referring to Figure side the length 10 of the objective, a real I is formed within the microscope tube. After coming to a focus, the light rays continue to the eyepiece, or ocuEyepiece

--

Objective

"I

Figure 6-24

fomlalion in a compound mi(:roscope.

lar lens. For visual the intermediate is made to occur at or just focal point of the The eye positioned near the then inside the sees a virtual image, inverted and as shown. The objective lens functions as the entrance pupil of the optical The image of the objective formed by the eyepiece is then the exit pupil, which locates the position of maximum radiant energy density and thus the optimum position for the entrance pupil of the eye. A is placed at the position of the intermefunctioning as a field diate The eye then sees both in focus the field of view a sharply defined boundary. If a camera is attached to the a real image is required. In this case the intermediate must be located outside the ocular focal length,

Total Magnification. When the final image is viewed by the eye, the may be defined as in the case of the simple magnimagnification of the fier. Thus the angular magnification for an image viewed at infinity is M Sec. 6-5

25

(6-40)

135

where leff (in cm) is the effective focal length of the two lenses, separated by a distance d and given by Eq. (4-32).

1

lid

- =- +- - !err j;, !e lo!e

(6-41)

Substituting Eq. (6-41) into Eq. (6-40), M = 25( Ie

+ j;, -

d)

(6-42)

j;,fe

Using the thin-lens equation, however, we may express the ratio of image to object distance for the objective lens by (6-43) where we have used the fact that ing Eq. (6-43) into Eq. (6-42),

s,;

= d - Ie, evident in the diagram. Incorporat(6-44)

M=

showing that the total magnification is just the product of the linear magnification of the objective multiplied by the angular magnification of the eyepiece when viewing the final image at infinity. The negative sign indicates an inverted image. Making use of Newton's formula for the magnification of a thin lens, Eq. (3-35),

x;

L

10

10

m=-=-

(6-45)

where L represents the distance between the objective image and its second focal point. as shown. The magnification of the microscope may then be expressed, perhaps more conveniently, as M

=

-(~~)(~)

(6-46)

In many microscopes, the length L is standardized at 16 cm. The focal lengths Ie and /0 are themselves effective focal lengths of multielement lenses, appropriately corrected for aberrations.

Example A microscope has an objective of 3.8-cm focal length and an eyepiece of 5-cm focal length. If the distance between the lenses is 16.4 cm, find the magnification of the microscope.

Solution L = d -

10 - Ie

= 16.4 - 3.8 - 5

= 7.6 cm

and

M= _(25)(L) = _(25)(7.6) = Ie j;, 5 3.8

-10

Numerical Aperture. To collect more light and produce brighter images, cones of rays from the object, intercepted by the objective lens, should be as large as possible. As magnifications increase and the focal lengths and diameters of the objective lenses decrease corespondingly, the solid angle of useful rays from the ob136

Chap. 6

Optical Instrumentation

Figure 6-25 Microscope illustrating the increased light-gathering power of an oilimmersion lens.

o

ject also decreases. In the useful light rays originating at the object of the point 0, passing through a thin cover glass and then air to the first lens L, make a half-angle of aa on the right of the optical axis. Due to refraction at the rays a larger than aa do not reach the a coupling transparent fluid lens. This limitation is somewhat relieved by whose index matches as closely as that of the glass. On the left of the optical Typaxis in the diagram, a layer of oil is used, and a larger half-angle a o is ically, the cover index is 1.522 and the oil index is 1.516, providing an excellent match. The capability of the objective lens is thus increased by increasing the refractive index in object space. A measure of this capability is the called the numerical aperture, product of half-angle and refractive N. A. = n sin a The numerical aperture is an invariant in the case of air,

(6-47)

space, due to Snell's law. That

in

N. A. = n" sin au = sin a:

and when an oil-immersion

OOJ'eCllVe

is used, •

•

N. A. = ng sin a o = nu sm a 0

I

The maximum value of the numerical when air is used is unity, but when object space is filled with a fluid of index n, the maximum numerical aperture may be increased up to the value of n. In practice, the limit is around 1.6. The numerical "T\j~rtl" ..p is an alternative means of defining a relative aperture or of describing how "fast" a lens is. As shown previously, brightness is inversely proportional to brightness is proportional to (N. the square of the f- number. Here also, design also because it limits the The numerical aperture is an resolving power and the depth of focus of the lens. The resolving power is proportional to the numerical aperture, whereas the depth of focus is inversely proporin the tional to (N. A.f. Most microscopes use objectives with numerical approximate range of 0.08 to 1.30. Biological specimens are covered with a cover of 0.17 or 0.18 mm thickness. For with numerical over the cover has increasquality, since it introduces a large degree of spherical ing influence on the aberration when oil immersion is not involved. Thus a biological objective compenIn contrast, a metallurgical obsates for the aberration introduced by a cover is designed without such compensation. may be classified broadly in relation to the corrections introduced into their For low magnifications, Sec. 6-5

Microscopes

131

with focal lengths in the range of 8 to 64 mm, achromatic objectives are generally for the Fraunhofer C (red) used. Such objectives are chromatically at the Fraunhofer D (sodium and F (blue) wavelengths, and spherically In yellow) For magnifications, lenses with focal the nmge of 4 to 16 mm incorporate some which together with the the visual spectrum. When the correcelements provide better correction throughout the visual the objectives are said to be tion is nearly apochromatic. Since correction is more crucial at magnifications, apochromats of 1.5 to 4 mm. For even are usually objectives with lengths in the higher magnifications, the objective is Modern techniques and materials have objectives that eseliminate field curvature over the useful portion of the field. With ultraviolet immersion it is customary to the oil with and the optical elements with fluorite and quartz because of their higher transmissivity at short wavelengths. This discussion should make it dear thai microscopes today are of an objective or an ev(~ml~ce is directly related to the other optical elements in the instrument, often including a relay lens within the tube of the microscope as well. Thus it is generally not possible to objectives and eyepieces between different model microscopes without deterioration of the 6-26 illustrates the optical in a standard '(,Tn",'"""" and the detailed " .. r.J"_'f'• ..'c~"'....,.., of light rays through the instrument.

Standard microscope illustrating illumination. Carl Zeiss, Thol'tJwoud. N. Y.)

lal

138

6

Instrumentation

Final image

Real intermediate image

Exit pupil of objective

Specimen

Condenser diaphragm

Field diaphragm

Light source

Imaging beam path

Illuminating beam path (bj

6-26

(continued)

6-6 TELESCOPES

leI,escopes may be broadly classified as refracting or according to whether lenses or mirrors are used to produce the There are, in addition, catadioptric systems combine and reflecting surfuces. may also be distinguished by the erectness or inversion of the image and by a visual or photographic means of observation. Refracting Telescopes. 6-27 and 6-28 show two respectively, inverted and erect The telescope in Figure 6-27 is often referred to as an astronomical telescope since

Sec. 6-6

Telescopes

139

Objective

Figure 6-27

ASlronomical

Objective

6-28 Galilean

telc~scc'pe.

sion of in the images produced creates no difficulties. The Galilean illustrated in Figure 6-28, produces an erect image by means of an focal length. In either case, nearly rays of light from a distant are collected by a positive objective which forms a real image The objective lens, being larger in diameter than the pupil of the in its focal eye, permits the collection of more light and makes visible point sources such as stars that not be detected. The lens is a doublet, is obcorrected for chromatic aberration. The real image formed by the served with an represented in the figures as a lens. serves as a real object ate image, located at or near the focal point of the (RO) for the ocular in the astronomical telescope and a object (VO) in the In either case, the light is by the eyepiece in case of the Galilean order to produce or light rays. An eye near the ocular at but with an angular magnification given by the ratio of the views an sub tends the angle a at the unaided eye and the angles a '/ a, as shown. The angle a' at the From the two right triangles formed by the intermediate it is evident that the angular magnification is image and the optical M

a' a

=_f Ie

The negative sign is introduced, as usual, to indicate that the is inverted in Figure 6-27, where Ie > 0, and is erect in Figure 6-28, where Ie < O. In 140

Chap. 6

Optical Instrumentation

case, the length L of the telescope is given by L

fo+Je

(6-49)

convepermitting a short Galilean telescope, a circumstance that makes this nient in the opera The astronomical telescope may be modified to produce an erect image by the insertion of a third positive lens \\-hose function is simply to invert the intermediate image, but this lengthens the telescope by at least four times the focal length of the additional lens. Image inversion may also be achieved without additional length, as in binoculars, through the use of discussed previously. The objective of either telescope functions as the entrance pupil, whose image in the ocular is the exit pupil, as shown. In the astronomical the exit pupil is just outside the and is to match the size' of the of the eye. A telescope should produce an exit pupil at sufficient distance from the eyepiece to produce a comfortable eye relief. Greater ease of observation is also achieved if the exit pupil is a little larger in diameter than the eye pupil, allowing for some relative motion between eye and Notice that in the Galilean telescope the exit pupil falls inside the eyepiece, where it is to the eye. This represents a disadvantage of the Galilean telescope, to a restriction in the field of view. Notice also that a stop with reticle can be employed at the locawhereas no such arrangetion of the intermediate image in the terrestrial ment is in conjunction with the Galilean The of the exit pupil Dex is simply related to the diameter of the objective lens Dobj through the anmagnification, as follows. Since the exit pupil is the image of entrance pupil formed by the eyepiece, we may write for the linear, transverse either (6-50) or, employing the Newtonian form of the magnification,

/0

(6-51)

lens) from the focal point of the eyewhere x is the distance of the object \~~'I~" and (6-51), or /0. Combining (6-48),

so that M

Thus the diameter of the bundle of parallel rays filling the objective lens is greater the exit by a factor of M than the diameter of the bundle of rays that pass pupil. It should be pointed out that the image is not, therefore, brighter by the same image increases by the same proportion, however, because the apparent size of factor M. The brightness of the image cannot be than the brightness of the in fact, it is less due to inevitable light losses due to reflections from lens surfaces. Binoculars 6-29) afford more comfortable allowing both eyes to remain active. In addition, the use of Porro or other prisms to proSec. 6-6

141

Figure Cutaway view of binoculars revealing compound objeclive and ocular lenses Carl

duce erect final images also lenses to be greater than the interpupillary distance, enhancing effect produced by binocular The X for binoculars means that the angular magnification M produced is 6x and the of the lens is 30 mm. . (6-52), we conclude that for this of binoculars is 5 mm, a match for the normal pupil diameter, For night viewing, when the pupils are somewhat larger, a of 7 producing an exit pupil diameter of 7 mm, would be nr~·tpr'
for the 6 x 30 binoculars just deof 15 cm and a field lens (eyepiece)

Determine the eye relief and field of scribed. Assume an objective focal diameter of 1.50 cm.

Solution The focal length of the ocular is found from em The eye relief is the distance of the next pupil from the eyepiece. Since the exit pupil is the of the objective formed by the the eye relief is the image distance s , given by S'

~=~ s - f L-/e

2.92 cm

The field of view from the subtends both the object on one side and the field lens of the '-''I''4JUoA-'-' on the other. Thus, for objects at a standard distance of 1000 yd,

0='2 s

142

6

Optical Instrumentation

or

h

sO

=

L

=

-'----:..::.--.!.

15

+ 2.5

ft at 1000 yd

Reflection Telescopes. Larger-aperture objective lenses provide greater light-gathering power and resolution. Large homogeneous lenses are difficult to produce without optical defects, and their weight is difficult to support. These probas well as the elimination of chromatic are solved by using curved, reflecting surfaces in place of lenses. The largest telescopes, like the Hale 200-in. reflector on Mount Palomar. use such mirrors. Such large reflecting telescopes are usually to examine very faint astronomical objects and use the power of exposed over long time intervals, in observations. Several basic for reflecting telescopes are shown in Figure 6-30. In the Newtonian (a), a parabolic is used to focus accurately all parallel rays to the same primary focal point, J;,. Before a plane mirror is used to divert the converging rays to a secondary focal point, Is, near the body of the telescope, where an is located to view the image. The use of a parabolic mirror avoids both chromatic and spherical aberration, but coma is for off-axis points, <:pvprphl limiting the useful field of view. In the 200-in. Hale telescope, the flat mirror can be dispensed with and the rays allowed to converge at their primary focus.

la)

(b)

(e)

Sec. 6-6

Telescopes

Figure 6-30 Basic for telescopes. (a) Newtonian telescope. (b) Cassegrain (c) Gregorian telei.copoe.

143

This is enough so that built platform situated just behind the obstruction pJaced inside the telescope rpt'!tur,'" waves contributing to the image. In secondary mirror is hyperboloidal convex mirror an aperture in the conveniently viewed or recorded. The hvt)f'rinnlnirl between the primary and secondary focal hyperboloid. Such accurate imaging is concave ellipsoidal, as in the Gregorian tClt~'+C.)[le focal points of this tel(~sc(}pe

be mounted on a specially 6-31). Of course, any section of the incident light 6-30b), the design from the primary focus, where it is surface permits imaging function as the foci of the the mirror is 6-3Oc). The primary and of the ellipsoid.

Hale (200-in.) showing prime-focus cage and reflecting surmirror. (California Institute of

The Schmidt Telescope. celebrated catadioptric telcscope is due to a design of Bernhardt ;:,cltlmun to remove the spherical aberration of a primary spherical mirror correcting plate at 6-32. A conthe aperture of the telescope. To cave primary reflector in (a) receives small U\AIIU""" rections. Each bundle enters at the !>",'ytllrp bundle may be considered an ture of the primary mirror. Since the and astigmatism do not enter axis, there are no off-axis points and into the aberrations of the system. When the bundles each bundle consists of paraxial rays that focus at the same distance a distance to the mirror. The locus of such imits focal or half the radius of curvature the dashed line. when age points is then the spherical surface as shown in (b), aberration occurs, which produces a the bundles arc mirror relative to the opshorter focus for rays reflecting from the outer t"",n"'''I"n'l' plate, to be tical axis of the bundle. Schmidt at the aperture, whose function focus of all zones to the mOlcatea in (c). The shape in same point on the spherical focal is designed to make the fOCal agree with the focal point the the usual choice. The resultof a zone whose radius is 0.707 of the

144

6

Optical Instrumentation

Aperture (al

Aperture (b)

Schmidt correcting plate

Aperture

Figure 6-32 The Schmidt optical system.

(c)

ing Schmidt optical system is therefore highly corrected for coma, astigmatism, and spherical aberration. Because the correcting plate is situated at the center of curvature of the mirror, it presents approximately the same optics to parallel beams arriving from different directions and so permits a wide field of view. Residual aberrations are due to errors in the actual fabrication of the correcting plate and because the plate does not present precisely the same cross section, and therefore the same correction, to beams entering from different directions. One disadvantage is that the focal plane is spherical, requiring a careful shaping of photographic films and plates. Notice also that with the correcting plate attached at twice the focal length of the mirror, the telescope is twice as long as the telescopes described previously, Figure 6-30. Nevertheless, the Schmidt camera, as it is often called, has been highly successful and has spawned a large number of variants, including designs to flatten the field near the focal plane.

Sec. 6-6

Telescopes

145

6-1. An object measures 2 em high above the axis of an optieal system consisting of a 2-cm aperture stop and a thin convex lens of 5-cm focal length and 5-em aperture. The object is 10 cm in front of the lens and the stop is 2 em in front of the lens. Determine the position and size of the entrance and exit pupils, as well as the Sketch the chief ray and the two extreme rays through the optieal system, from top of object to its conjugate point. 6-2. Repeat problem 6-1 for an object 4 em high, with a 2-em aperture stop and a thin convex lens of 6-em foeal length and 5-em aperture. The object is 14 em in front of the lens and the stop is 2.50 em behind the lens. 6-3. Repeat problem 6-1 for an object 2 em with a 2-em aperture stop and a thin convex lens of 6-cm foeal length and 5-cm aperture. The object is 14 em in front of the lens and the stop is 4 em in front of the lens. 6-4. An optieal system, centered on an optieal axis, consists of (left to I. Source plane 2. Thin lens L\ at 40 cm from the source plane 3. Aperture A at 20 em further from L\ 4. Thin lens L2 at 10 em farther from A 5. plane Lens L\ has a focal length of 40/3 em and a diameter of 2 cm; lens ~ has a focal of 20/3 em and a diameter of 2 cm; aperture A has a centered circular opening of 0.5-em diameter. (8) Sketch the system. plane. (b) Find the location of the (c) Locate the aperture stop and entrance pupil. (d) Locate the exit pupil. (e) Loeate the field stop, the entrance window, and the exit window. (t) Determine the angular field of view. 6-5. Plot a curve of total deviation angle versus entrance angle for a prism of apex and refractive index 1.52. 6-6. A parallel beam of white light is refracted by a 60° prism in a position of mini1 and blue mum deviation. What is the angular separation of ptnpru'na red (n (l light? 6-7. (8) Approximate the Cauchy constants A and B for crown and flint using data for the C and F Fraunhofer lines from Table 6-1. Using these constants and the Cauchy relation approximated by two terms, calculate the refractive index of the D Fraunhofer line for each ease. Compare your answers with the values given in the table. (b) Calculate the dispersion in the vicinity of the Fraunhofer D line for each glass, using the Cauchy relation. (c) Calculate the chromatic power of crown and flint prisms in the vicinity of the Fraunhofer D line, if each prism base is 75 mm in length. Also calculate the minimum resolvable wavelength interval in this region. 6-8. An equilateral of dense barium erown is used in a spectroscope. Its refractive index varies with wavelength, as given in the table: nm

n

656.3 587.6 486.1

1.63461 1.63810 I.646U

(8) Determine the minimum angle of deviation for sodium light of 589.3 nm. (b) Determine the dispersive power of the prism.

146

Chap. 6

Optical Instrumentation

(c) Determine the Cauchy constants A and B in the wavelength region; from the Cauchy find the of the prism at 656.3 nm. (d) Determine the minimum base of the prism if it is to resolve the hydrogen doublet at 656.2716- and 656.2852-nm wavelengths. Is the practical? 6-9. A prism of 6ft' refracting IDlllowmg angles of minimum deviation when 38"33'; F line, 39"12'. Determine measured on a spectrometer: C the power of the prism. 6·10. The refractive indices for certain crown and flint glasses are Crown: Flint:

6·U.

6·12.

6·13.

6-14.

6-15.

6-16.

6-17.

6-18.

nc

nc = 1

nv nv

=

1

=

I

nF nF

= 1.536 = 1.648

The two are to be combined in a double that is a direct vision prism for the D The angle of the flint is 5". Determine the required angle of the crown and the angle of between the C and the F rays. Assume that the prisms are thin and the condition of minimum deviation is satisfied. An achromatic thin for the C and F Fraunhofer lines is to be made using the crown and flint glasses described in Table 6-1. If the crown glass prism has a prism angie of 15", determine (a) the prism for the flint glass and (b) the resulting "mean" deviation for the D line. A perfectly or Lambertian, surfuce has the form of a square, 5 cm on a side. This object radiates a total power of 25 W into the forward directions that constitute half the total solid of 4'lT. A camera with a 4-cm focal length lens and down to f /8 is used to photograph the when it is placed 1 m from the lens. (a) Determine the radiant radiant and radiance of the object. Table 2-1.) (b) Determine the radiant flux delivered to the film. (c) Determine the irradiance at the film. the behavior of Eq. (6-32), the dependence of the depth of field on aperture, focal length, and object distance. With the help of a calculator or rornn,'ltpr program, generate curves showing each deJ>en;oen.ce. A camera is used to photograph three rows of students at a distance 6 m away, focusing on the middle row. Suppose that the image or blur circles due to object points in the first and third rows is to be kept smaller than a typical silver grain of the emulsion, say I J-Lm. At what object distance nearer and further than the middle row does an unacceptable blur occur if the camera has a focal of 50 mm and is down to an f /4 setting? A telephoto lens consists of a combination of two thin lenses focal lengths of +20 cm and -8 cm, The lenses are by a distance of 15 cm. (a) Determine the focal length of the combination, distance from negative lens to film plane, and size of a distant object sub tending an angle of 2" at the camera. (b) Determine the same quantities and the position of the planes, as well, using the matrix approach of Chapter 4. A 5-cm focal length camera lens withf /4 is focused on an object 6 ft away. If the maximum diameter of the circle of confusion is taken to be 0.05 mm, determine the depth of field of the photograph. The sun subtends an angle of 0.5" at the earth's where the illuminance is about lOS Ix at normal incidence. What is the illuminance of an image of the sun formed by a lens with diameter 5 cm and focal 50 cm? (a) A camera uses a convex lens of focaJ 15 cm. How an image is formed on the film of a 6-ft-tall person 100 ft away? (b) The convex lens is by a telephoto combination of a 12-cm focal length convex lens and a concave lens. The concave lens is situated in the position of the lens, and the convex lens is 8 cm in front of it. Whal is the required focal of the concave lens such that distant form focused on

Chap. 6

Problems

141

the same film plane? How much larger is the of the person using this telelens? 6-19. The tens on a 35-mm Qlmera is marked "50 mm, I: 1.8." (a) What is the maximum aperture diameter? (b) Starting with the maximum aperture setting, the next three f numbers that would allow the irradiance to be reduced to ~ the precedlmJ); at each successive stop. (C) What aperture diameters correspond to numbers? (d) If a picture is taken at maximum aperture and at s, what exposure time at each of the other openings provides equivalent total eXlIOSllre!,'! 6-20. The by Eq. (6-33) is also valid for a double-lens eyepiece if the eOlllVlHelU focal length given by Eq. (6-35) is used. Show that the magnification of a double-lens designed to satisfy the condition for the elimination of chromatic for an image at infinity,

{1 +

M = 12.5\];'

I

6-21. A is made of two thin plano-convex each of 3-cm focal length and spaced 2.8 cm Find (a) the equivalent focal and (b) the magnifying formed at the near point of the eye. power for an 6-22. The of a microscope has a focal length of 0.5 cm and forms the intermediate image 16 cm from its second focal point. (a) What is the overall magnifiQltion of the microscope when an eV~~Djl~ rated at lOX is used? (b) At what distance from the objective is a point 6-23. A homemade compound microscope has, as ..hiP£";"'" thin lenses of focal I cm and 3 cm, respectively. An is situated at a distance of 1.20 cm from the If the virtual DrcKlu.ced the is 25 em from the eye, (a) the magnifying power of the .... i/...."'.,..""~ and (b) the separation of the le~s. when placed 25 cm apart, form a compound microscope ""'gllJll......".lI is 20. If the focal of the lens representing the eYi~pu~ is 4 em. determine the focal length of the other. 6-25. A level contains a graticule-a circular glass on which a SQlle has been etched-in common focal plane of objective and so that it is seen in focus with a distant If the telescope is focused on a pole 30 m away, how much of the post falls between millimeter marks on the The focal length of the is 20 cm. 6-26. A pair of binoculars is marked "1 X 35." The focal of the is 14 em, and the diameter of the field lens of the eyepiece is 1.8 cm. Determine (a) the angular magnification of a distant (b) the focal length of the ocular. the diameter of the exit pupil, (d) the eye relief, and (e) the field of view in terms of feet at 1000 yd. 6-27. (a) Show that when the final image is not viewed at infinity, the angular magnification of an astronomical may be expressed by M = m""Jobj

s" where moe is the linear magnification of the ocular and s" is the distance from the ocular to the final (b) For such a two converging lenses with focal of 30 cm and magnifiQltion when the image is viewed at and 4 em, find the when the image is viewed at a near point of 25 em. 6-28. The moon sub tends an of 0.5" at the objective lens of a terrestrial tele:scc.pe. The focal lengths of the and ocular lenses are 20 cm and 5 cm, Find

148

Chap. 6

Optical Instrumentation

6-29.

6-30.

6-31.

6-32.

6-33.

the diameter of the image of the moon viewed the telescope at near point of 25 cm. An opera uses an objective and of + 12 cm and -4.0 cm, Determine the length separation) of the instrwnent and its magnifying power for a viewer whose eyes are focused (a) for infinity and (b) for a near of 30 cm. An astronomical is used to a real of the moon onto a screen 25 cm from an ocular of 5-cm focal length. How far must the ocular be moved from its normal position? (8) The Ramsden of a telescope is made of two lenses of focal length 2 cm each and also separated by 2 cm. Calculate its magnifying power when viewing an image at infinity. (b) The objective of the telc~CI)pe is a 30-cm positive with a diameter of 4.50 cm. Calculate the overall magnification of the tele:scc~pe. (c) What is the position and diameter of the exit pupil? (d) The diameter of the field lens is 2 cm. Determine the angle defining the field of view of the tele:SC()'pe. Show that the magnification of a Newtonian telescope is by when the the ratio of objective to ocular focal lengths, as it is for a refracting image is furmed at infinity. telescope has a focal length of 12 ft. The primary mirror of a The secondary mirror, which is convex, is 10 ft from the primary mirror along the principal axis and forms an image of a distant object at the vertex of the primary mirror. A hole in the mirror there permits viewing the with an of 4-in. focal length, placed just behind the mirror. Calculate the focal length of the convex mirror and the angular magnification of the instrument.

[1] Horne, D. F. Optical Instruments and Their Applications. Bristol, England: Adam

[2] [3]

[6]

[7]

[9] [IO} [II] [121 [13]

Hilger 1980. Goodman, Douglas S. "Basic Optical Instruments." In Geometrical and Instrumental Optics, edited by Daniel Malacara. Boston: Academic Press, 1988. Brouwer, William. Matrix Metlwds in Oplicallnstrument Design. New York: W. A. 1964. Benford, James and Harold E. Rosenberger. "Microscope and Eyepieces." In Handbook of edited by \\alter G. Driscoll and William Vaughan. New York: McGraw-Hill Book Company, 1978. A. Photographic Optics, 15th ed. New York: Focal Press, 1974. Horne, D. F. Optical Production Technology. New York: Crane, Russak and Company, 1972. R. Lens Design Fundamentals. New York: Academic 1978. McLaughlin, R. B. Special Methods in Light Microscopy. London: Microscope Publications Ltd., 1977. W. J. Modern Optical New York: McGraw-Hill Book Company, 1966. Nussbawn, Allen. Geometric Optics: An Introduction. Reading, Mass.: Addison-Wesley Publishing Co., 1968. Miles V. Optics. New York: John Wiley and Sons, 1970. Ch. 4.1: Radiometry and Photometry. Kirkpatrick, Paul. Microscope." Scientific American (Mar. 1949): 44. George. "Eye and Camera." ScientifIC American (Aug. 1950): 32.

Chap.S

References

149

[14] Wilson, Albert G. 'The Schmidt." Scientific American (Dec. 1950): 34. [15] Muller, Erwin W. "A New Microscope." Scientific American (May 1952): 58. [16] McClain, Edward, Jr. 'The 6OO-Foot Radio " Scientific American (Jan. 19(0): 45. [l7] Albert. "A High-Resolution Electron " Scientific American (Apr. 1971): 26. [IS] Price, William H. "The Photograjilic Lens." Scientific American (Aug. 1976): 72. (19) Quale, Calvin. "Acoustic Microscope." American (Oct. 1979): 62. [20J Bahcall, J. and L. Spitzer, Jr. "The Space Telescope." American (July 1982): 40. [21] Howells, Malcolm R., Janns and William Sayre. II iCJro!>co~>es." Scientific American (Feb. 1991): 8S.

150

Chap. 6

Optical Instrumentation

7 Incident radiation

mm

20 mm

Optics of the Eye

INTRODUCTION In this chapter we acquaint ourselves with the optics of the eye. we examine the structure and functions of the eye. Following this we note the errors of refraction in a defective eye and indicate the usual corrective optics. we describe capsulotomyseveral current procedures-radial keratotomy and laser light of a irradiance and wavelength is used to restore visual acuity in less-than-perfect eyes. The eyes, in conjunction with the brain, constitute a truly remarkable biooptical system. Consider briefly the distinctive characteristics of this system. It forms of a continuum of at distances of a foot to infinity. It scans a scene as as the overhead sky or focuses on detai1 as minute as the head of a pin. It adapts itself to an extraordinary range of intensities, from the barely visible flicker of a candle miles away on a dark night to sunlight so bright that the optical image on the retina causes serious solar bum. It distinguishes between subtle shades of color, from deep purple to deep red. Most importantly for us, functioning as a unique spain space, accurately mapping out our threetial sense organ, it localizes dimensional world. " , n . " , ...,.,..,

151

7

shape of the lens is controlled by the ciliary connected by fibers (zonules) to the periphery of the lens. While the muscles are relaxed, the lens assumes its flattest shape, providing the least refraction of light rays. In this state, the eye is focused on distant objects. When the muscles are the shape of the lens becomes increasingly curved, providing increased refraction of light. In this "strained" the eye is focused on The lens is itself a complex, onionlike mass of tissue, held by an elastic Due to the rather intricate laminar structure of fibrous refractive index of the lens is not homogeneous. Near the center or core of lens (on the index is about 1.41; near it falls to about 1.38. the After its final refraction by the light enters the posterior chamber or the vitreous humor, a transparent jellylike substance whose refractive index humor, essentially structureless, (1.336) is again close to that of water. The contains small particles of cellular debris that are referred to asjloaters. They derive their name from the manner in which they are Seen to float in one's field of view, when one looks or squints at a white ceiling, for "'A."Hfl''''. After traversing the vitreous humor, rays reach their terminus at the inner as "net." The retina, or net, is rear layer of the eye, the retina, literally dotted with an overlapping pattern of cells called rods and cones. The long, thin rods, numbering over 100 million, are located more densely toward the periphery of the retina. They are exceedingly sensitive to dim light, yet they are unable to distinguish between colors. The wider cones, under 10 million in number, cluster preferentially near the center of the a 3-mm diameter region called the cones are sensitive to the macula. In sharp contrast to the response of the light and color but do not function well in dim Linked to the photorecells are three distinct types of nerve cells (amocrine, bipolar, horizontal) that transmit the visual impulse to the nerve. The optic nerve is the main trunk line that carries visual information from the retina to the brain, completing the reIllal:KaOle process we call vision. In addition to the key optical components encountered by light traveling along the axis of vision, the eye contains other components that should be mentioned. As in 7-1. the eye is covered with a white coating, the sclera, that forms the supporting framework of the eye. Just inside the sclera lies the choroid, about four-fifths of the eye toward the back and containing most of the blood vessels that nourish the eye. The choroid, in turn, serves as the backing for the all-important net that houses the rods and cones. At the center of the located somewhat above the optic nerve, is the region of OY'p·"tp<;:t visual acuity. When it is required that one see and detailed informa[Joln--wrme removing a small splinter with a for eXllffilme-lme .........,,,,,,,, ....... .1 so that light coming from the area of int,,,, ....<;:t about 200 p;m in diameter. in located at the point of exit of the nerve, is COlmlllletelV to light. This spot, devoid of any receptors, is appropriately

7-2 OPTICAL REPRESENTATION

THE EYE

As we seen, the normal biological eye is a near spheroid, some 22 mm from cornea to retina. The optical surfaces that provide the bulk of the 1'",'''''''''' the interface, the aqueous-lens vitreous int,,,,,t',,,,,, Overall, the eye can be represented, quite tive lens of focal length equal to 17 mm in the relaxed state Sec. 1-2

Representation of the Eye

153

f-of--------22.38 m m - - - - - - - - . J f-of-----------20mm------~~

2.38 mm 1.96 mm ~---------15mm--------1-~

3.6mm

Figure 7-2 Representation of H. V. Helmholtz's schematic eye I, as modified by L. Laurance. for definition of symbols, refer to Table 7-1. (Adapted with permission from Mathew Alpern, "The Eyes and Vision," Section 12 in Handbook of Optics, New York: McGraw-Hili. 1978.)

TABLE 7-1

CONSTANTS OF A SCHEMATIC EYE

Optical surface or element Cornea Lens (unit) front surface Back surface Eye (unit) Front focal plane Back focal plane Front principal plane Back principal plane front nodal plane Back nodal plane Anterior chamber Vitreous chamber Entrance pupil Exit pupil

symbol SI L S2 S3

F F' H H'

N N'

Distance from corneal vertex (mm)

Refractive index

+8" 1.45 +3.6 +7.2

+ l()b -6

Refractive power (dioplers)

+41.6 +30.5 +12.3 +20.5 +66.6

-13.04 +22.38 +1.96 +2.38 +6.96 +7.38 1.333 1.333

AC VC

En P E,P

Radius of curvature of surface (mm)

+3.04 +3.72

SOURCE: Adapted with pennission from Mathew Alpern, "The and Vision," Table I, Section 12, in Handbook of Optics, New York: McGraw-Hili Book r_.__ ._ •.. 1978. "The cornea is assumed to be infinitely Ihin. Value is for the relaxed eye. for the tensed or fully accommodated eye, the radius of curvalure of the fronl surlace is changed to +6 mm.

b

154

Chap. 7

Optics of the

14 mm in the tensed state (near vision). In an attempt to """'"''''0''''' the optical powschematic eyes have been While still an apers of the eye more proximation, a schematic eye presents a fairly valid of the true (but complex) biological eye. A schematic eye (after H. V. Helmholtz and L. L44""""U"""_J that rer:,re~;en:[S a living, biological eye with accuracy is shown in the refracting are shown, as are the cardinal state. For the fully a whole. The schematic eye shown corresponds to tensed eye, the front of the lens its curvature from a radius of R = 10 mm to R 6 mm. By way of summary and in conjunction with Figure 7-2, Table 7-1 lists the optical surfuces, their distances from the corneal vertex on the several radii of curvature, indices of and refracting powers of the optical surfaces related to the cornea and Note carefully that the values for the refractive of various parts of the eye, as wen as radii of curvature of surfaces, may not agree with values of the biological eye itself. When taken as a whole, however, the values that describe the eye do fuithfully represent the optical of a living, biological eye.

7-3 FUNCTIONS OF THE EYE To operate as an effective the eye must form a of an external object or scene, either or nearby, in bright as weB as dim light. To achieve efficient the eye takes advantage of special functions. To see obclosely and fur away, the eye accommodates. To process light of varying hrlohfn""~<;: the eye To sense the spatial orientation of three-dimensional scenes, the eyes make use of vision. To form a faithful, image of the external object, the eye relies on its visual acuity. In what follows, we discuss each of these visual functions in somewhat more detaH.

Accommodation. on the distance of the or scene from the eye, the lens accommodates-tenses or relaxes-appropriately to fine-focus the image on the retina. For a distant object, the ciliary muscle attached to the lens relaxes and the lens assumes a flatter its of curvature and, consequently, its focal As the object moves closer to the eye, the ciliary muscle tenses or contracts, or bulging the lens and in decreased radii of curvature and a shorter focal length. The smaller the radii of curvature and focal length, the higher the or bending power of the lens, precisely the condition needed to closer into sharp focus. In the before the normal aging process the lens of its elasticity and itself-accommodation produces fuithful retinal images of objects from distant points (infinity) to nearby points about one foot away. The near point point of accommodation) recedes from the eye with advancing age, at a nr..:,uu\n of 7 to 10 cm from the eye for a increasing to 20 to 40 cm for a middleaged adult, and extending to as fur as 200 cm in later years. For presbyopia (loss of accommodation) sets in during the early 4Os, "'O'n<>''''' for reading glasses to restore the near point to a comfortable position near 25 cm or so. Adaptation. The of the eye to respond to light that range from very dim to very bright, a range of light intensities that differ by an astonishing fuctor of about 105 , is referred to as adaptation. The amount of light (flux or photon Sec. 1-3

Functions of the Eye

155

number) that enters first of all by the iris, with its adjustable aperture, the pupil. of pupil diameter (from 8 mm down to 2 mm) cannot of itself account for the enormous range of intensities processed by the eye. The remarkable of eye is traced, in fact, to the photoreceptors in the the rods and cones, and to particular sensitivity to light. The key dient seems to be a called visual pigment, contained in both the rods and cones. The rods, low-level light signals (scotopic vision), contain pigment of only one kind, called visual purple. The cones, sensitive to 1ight signals of and variable composition (photopic vision), each contain one of three different kinds of visual pigment. The numerous, thin rods are multiply conit possible for anyone of a hundred rods or so to actinected to nerve fibers, vate a nerve The less numerous, wider cones in the macular region, by contrast, are individually connected to nerve fibers, and thus individually activated. The activation of nerve very heart of the vision process itself-depends on chemical that occur in the visual pigment contained in the rods and cones. When light falls on of the visual pigment changes from a dark state to a clearer state. a sort of bleaching process, The change in state of the visual in the rods or cones is transformed into an electrical output or nerve These "''''r>n'.r><> the optic nerve and on to the stimulating signal. When the loreceptor cells become insensitive to in the rods must occur

Stereoscopic Vision. The ability to depth or bf object."i accurately in a three-dimensional field is vision. In the optic nerves from the two eyes come near the brain. From the optic chiasma, nerve fibers originating in right half of each eye extend to the right half of the brain. Nerve fibers in the left half each eye terminate in the left half of the brain. Thus, even though half of the brain recelvl~s an image from both eyes. the brain forms but a The fusion the brain of two distinct images into a single image is referred to as vision. Never~ theless, the slight differences between the two images from the left and provide the basis for stereoscopic vision in humans. It should be that even monocular vision is not without some depth This is due to clues like parallax, shadowing, and the particular of To have proper binocu]ar vision without double must full at corresponding points on each retina. when the eyebaHs move appropriate1y to focus on an 156

Chap, 7

Optics of the Eye

age to fall on the fovea centralis of each eye. Most individuals are either right-eyed or left-eyed, indicating a dominance of one eye over the other. To determine which is your dominant eye, try the following simple test. Hold a pencil in front of you at eye With both eyes open, line the pencil up with the vertical of a picture, door, or window across the room. Holding the pencil fixed, close one eye at a time. Whichever eye is open when the pencil remains lined up with the reference is your dominant eye. The brain records the message seen by the dominant eye, while other.

Visual Acuity. The to see and to perceive differences in spatial orientation of objects is related to visual This ability depends directly upon resolving power of the eye or its minimum of resolution of two closely spaced objects or points. speaking, visual acuity is defined as the reciprocal of the minimum angle of resolution. Operationally, assessment of resolving power or visual acuity of the eye is measured in different ways. discrimination is referred to as minimum separable the smallest angle by a black bar on a white background is caned minimum and the smaHest subtended by block letSince most of us, ters that can be read (on an eye chart) is called minimum at one time or another, are required to read eye charts in a vision test, we limit our discussion of visual to resolving associated with minimum legible resolution. The nature of the eye chart owes its existence to a Dutch ophthalmologist, Herman Snellen. According to the letters on the eye chart are constructed so that the overall block size of a letter, from top to bottom, or side to side, subtends an angle of 5' of arc at the test The detailed lines within a such as the vertical bar in the letter T or the horiwntal bar in the letter H, are all constructed so that the width of each "bar" subtends an angle of I' of arc at the test distance. The two choices of angle grew out of the best data available to Snellen on the minimum resolution of the eye. For the normal eye could just deresolve a letter that subtended 5' of arc at 20 ft, with l' of arc contained in case the eye is considered "normal," and tails of the letter. See Figure 7-3. In visual acuity is referred to as "20/20 vision."

v

H

o

T

H

T

o

v

T

H

T

v

H

o

v

T

20

20

Figure 7-3 Construction of a Snellen eyechart letter H to measure visual acuity. The lOp portion of Ihe shows a section of an eye chart (reduced) containing the teller H and several other letters.

I

I

I I I

1----

Sec. 7-3

FUnctions of the

157

To detect defects in visual acuity, Snellen letters of different sizes are also included on the eye chart. For example, a very large letter may be such that its subof 5' and l' of arc hold for a test distance say, 300 ft. Other letters tense are constructed of appropriate size, subtending angles of 5' and l' for other selected distances, such as 200 ft, 100 ft, 80 ft, and so on, down to 15 or even 10 ft. Then, when the letters are read by a test subject at a test distance of 20 ft, visual acuity is measured in terms of the Snellen The numerator of the Snellen fraction expresses the fixed testing and the denominator expresses the distance at which the smallest readable letter subtends 5' of arc overall. For example, if the of 5' at 300 ft is readable by a test large block letter E that subtends an acuity is reported as 20/300. A Snellen subject seated 20 ft from the letter, fraction of 20/300 means that the test subject sees reading at a distance of 20 feet what the normal eye reads as well at a distance of 300 ft. While normal vision is 20/20, visual readings as good as 20/15 are not uncommon. 7-4 ERRORS OF REFRACTION AND THEIR CORRECTION

errors of refraction of the eye to three well-known defects in vision: nearsightedness (myopia), farsightedness (hyperopia), and astigmatism. The first two are traceable, for the most part, to an abnormally shaped eyeball, axially too long or too short. Either deviation from normal length the ability of the combined refracting cornea and lens, to form a clear image of objects at both remote and positions. third defect, astigmatism, is due to unequal or asymmetric curvatures in the corneal surfuce, rendering impossible the simuHaneous focusing of all light incident on the eye. Whether the errors of refraction occur singly or in some combination (as usually do), are generally correctable with appropriately external (eyeglasses). As a point of reference for judging the departure of defective vision from the norm, refer to the normal eye depicted to the left in 7-4. With accommodation, the normal eye forms a distinct image of objects located anywhere between its fur point (F.P.) at infinity and its near point (N.P.), nominally a distance of 25 cm for the young adult. When normal eye is at (distant vVII""'''''J. parallel light enters the relaxed eye and forms a distinct 7-4a). When focused at the near point, diverging light enters the tensed eye (fully accommodated) brought to sharp focus on the retina (Figure 7-4b). and is Myopia. When compared with the normal eye, a myopic. eye or nearsighted eye is commonly found to be longer in axial distance-from cornea to retina-than the usual, span of 22 mm. As a consequence, and as in the myopic eye forms a sharp image of distant trated objects in/ront of the retina, and, of course, a blurred image at the retina. Distinct retinal images are not formed with the unaccommodated myopic eye until the object moves in from infinity and reaches the myopic far point, the most distant point for clear vision 7-4d). From the far point with <>n'''·.........i,,'.. ac<::ornrnlodation. the eye sees quite clearly, even at closer than the normal near point (Figure Since angular magnification of detail increases with proximity to the eye, the myopic eye enjoys "superior" vision of objects held close to the eye. (It can be an advantage for a watchmaker to be myopic, therefore, at least during working hours!) In short then, the nearsighted person has a contracted, drawn-in field of vision, a less-remote fur and a closer near point than a person with normal vision. While the more proximate near point might serve as an advantage, the less-remote far point is a distinct disadvantage and calls for correction.

158

Chap. 7

Optics of the

Normal Eye

Myopic Eye (corrected)

Myopic Eye (unaided)

Distant object Distant object

----i>-----i

Lens (negative power)

----------

M.F.P. No accommodation {al

No accommodation

If)

(e)

No accommodation (d)

Distinct at'~ ,,~U~l

.

Myopic

F,P.

.

M.F.P.

Normal N,P.

Lens

Full accommodation

accommodation (b)

(e)

Figure 74 A comparison of normal and myopic vision, with the eye lens is not shown. The abbreviations read as follows: == normal near ooint: M.N.P. mvooic near

....

tB

(g)

correction. Note that refraction far point;

Partial accommodation

Myopic vision is routinely corrected spectacles of dioptic power and near point outward (diverging that move the myopic far to normal positions. Figure 7-4f shows the corrected vision for distant objects. Note that as far as the optics of the eye itself is concerned, light from distant objects apto originate at its own myopic far point. illustrates the ,t""f'''''' for corrected near vision under accommodation. Light from an object at the normal near point appears now to originate at a point somewhat closer in than the true near point of the myopic eye. To gain some into the of negative lens power required to correct myopic consider the following eXl:lml,le.

Example A myopic person (without astigmatism) has a far point of 100 cm and a near point of 15 cm. (a) What correction should an optometrist prescribe to move the myopic far point out to infinity? (b) With this correction, can the myopic person also read a book held at the normal near point, 25 cm from the Solution (a) tion, with s ~

to 00

and

S'

1

and 100 cm, one has

= 1

1

-:; + s' = f or

1 00

use I 100

+

the thin-lens equa-

1

f

This f 100 cm. Accordingly, the corrective should have a focal length of - 100 cm. The optometrist would prescribe spectacles 1.00 D). with a correction of -1.00 diopter (b) Referring to 7-4g again using the equation with f = - 100 cm and s = 25 cm, the virtual s I is found from

I

I

-:; + S'

I

=

f

1 or 25

1

+~

1

-100

Solving s = - 20 cm. Thus the virtual of the print held at s 25 cm is formed by the at a distance of 20 cm in front of the eye. the myopic person can see clearly objects brought in as close as 15 cm from the eye. the virtual image of the print formed by the lens at 20 cm is seen without difficulty. In met, using O/s) + (l/s') = 1/ f, with s' = 15 cm (myopic near point for person) = - 100 cm, solving s, one finds that objects can be brought in as close as 17.6 cm from the eye and still be seen clearly. I

Hyperopia. The farsighted, or hyperopic, eye is commonly shorter than normal. Whereas the longer-than-normal myopic eye has too much convergence in its "optical and a to correct its the shorter-than-normal hyperopic eye has too little convergence and a converging lens to increase refraction. The drawings in Figure 7-5, in analogy with those in Figure 7-4, illustrate the defects and correction associated with the farsighted eye. In from a distant object enters the relaxed eye and focuses behind the blurred vision. The point behind the retina as the hyperopic far point. Figure 7-5b shows that the hyperopic eye is In Figure it is clear must (and can) accommodate to see distant objects that for distinct vision, the hyperopic near point is farther away than the normal near point. Consequently, located closer than the hyperopic near point would be out of focus, even with fuJI aocommodation. The two end-point corrective meas160

Chap. 7

Optics of the

Hyperopic Eye (corrected)

Hyperopic Eye (unaided) Blurred

Lens (positive power)

Distinct

No accommodation

==H.F.P.

lal No accommodation (dl

Partial accommodation

Distinct

Nearby object at N.N.P.

r

Ibl Distinct

H.N.P.

N.N.P.

accommodation

(e)

Full accommodation

Ie! Figure 7-5 Hyperopic vision with correction. The abbreviations read as follows: RN.P. = hyperopic near N.N.P. normal near RF.P. far point.

ures, with the appropriate lens in place, are indicated in Figure and e. The corrected eye now sees distant objects clearly, without accommodation, and at the normal near it sees objects clearly, with full accommodation. Let's see how an optometrist might calculate the spectacle power required to correct hyperopic vision. Consider the following example. Example A farsighted person is tive power is required for brought in to the normal near

to have a near point at 150 cm. What correcto enable this to see objects 25 em from the

Solution Referring to 7-5e and use of the thin-lens eQlliatlon with s = 25 em and s = -150 cm, one can solve for the focal required lens as follows: I

I

I

s

s'

-+

Calculation gives f = + 30 cm. Since the power in diopters is given by 1/ f (meters), we determine a power of 3.3 D. With spectacles shaped to a power of + 3.3 diopters, the hyperopic can now see clearly in as close as 25 em from the eye. Sec. 7-4

Errors of Refraction and Their Correction

161

Astigmatism. The astigmatic eye suffers from uneven curvature in the elements, most significantly, the cornea. Generally speaksurface of the the radii curvature of the corneal surface in two meridional planes (those containing the are unequal. Such asymmetry leads to different refractive distances from the powers and, consequently, to focusing of light at of course, in blurred vision. If the two are orthogonal to cornea, is referred to as one another, one and the other vertical, say, the regular astigmatism, a condition that is correctable with appropriate spectacles. If the two are not orthogonal, a rather rare called irregular astigmatism, the surface anomaly is not so easily is treated with cylindrical surfaces ground on the back surfuce of the spectacle lens. Assume, for example, that the refractive power in the vertical meridian of the t than the power in the horizontal meridian. This situacornea is tion means that the surface is more sharply curved in the vertical meridian and that details in an object are brought to a focus nearer the cornea than are horizontally oriented details. Consider a cylindrical surface with in the vertical meridian. has no curvanegative power of I the surface has no power in the meridian. If this surture along its face is included in the spectacle design, it would cancel the distortion introduced by the cornea and equalize powers in both meridians. As a result, vertical and scene are formed at the same from the horizontal details in the cornea, and does not occur. mixed with myopia or For most us, blurred vision is a result of hyperopia. If myopic astigmatism is present, for vision is faulty on two the astigmacounts. The myopia itself causes an overall blurring of distant in one meridian tism compounds the considerably more than another. for both defects is accomplished with sphero-cylindrical to correct for lenses, spherical surfuces to correct for myopia and cylindrical astigmatism. When n ....'tnt'1nM ... "'t" corrective eyeglasses for I'n.l£hlt"......' hyperopic O'Ia',unlO'lti identify three numbers. tism, the three written in prescription format, might be -2.00 For hyperopic

0'I!:.t"on10'1h!!:.m

-l.00

x 180

the prescription might read

+2.00

-1.50

x 180

The first number refers to the power, the required power in of the spherical surfuces on the lens that correct for the overall myopia or hyperto the cylinder power, the power of the opia. The second number to correct cylindrical surface on the back surface of the for astigmatism. The third number refers to the orientation of the cylinder is to be vertical, horizontal, or somespecifying whether the axis of the where in between. In optometric notation, the horizontal axis is referred to as the 180" axis, or simply "x 180," and the vertical axis as "x 90." Figure 7-6 indicates the optical conditions associated with the corrective script ions just cited for both myopic and hyperopic astigmatism. For the case of myopic astigmatism, Figure 7-6a, the corneal surface is evidently less sharply the horizontal meridian (power 45.00 D) than in the vertical (power 46.00 The myopic correction, always measured in the meridian of least refractive power, is the horizontal meridian. The found in this instance to be -2.00 (X ISO), is determined to be 1.00 D. correction, with cylinder axis 162

Chap. 7

OptiCS of the

Cornea fronta I view

900 I

,43.50 0

1BOO-

o

18O"--ffi--4200 0 I I

Myopic astigmatism

Hyperopic astigmatism

la!

Ib)

'·6 O:mditions of myopic and with corrective spectacle prescriptions. (a) Refraction in the 180" meridian yields -2.00 D of myopia. The eyeglass pre1.00 X 180. (b) scription is R,: -2.00 Refraction in the ISO° meridian +2.00 D of hyperopia. The prescnpllon is R,: +2.00 ··1.50 XISO.

With the appropriate cylindrical surface ground on the rear of the lens. the meridian from 46.00 D to correction of -1.00 D reduces the power in the 45.00 equalizing the refracting powers in the two meridians and negating the corneal astigmatism. for tlv,,,,,,r·nn.'" Figure 7-6b shows a comparable condition and astlgrnal]Srn. Note that a sphere power correction of +2.00 D is needed to correct for the hyperopia, and a cylinder power correction of 1.50 D is needed along the the refractive power in the two orthogonal vertical meridian (X 180) to meridians. 7-5

THERAPY FOR OCULAR DEFECTS Not all ocular problems can be corrected optically with appropriate eyeglasses, as are the structural defects we have just discussed. There are organic disorders that require treatments, often involving surgery. The as discussed in Chapter 21, has as a powerful tool in the operating room, used successfully in the treatment of major ocular defects. Laser beams, both pulsed and continuous wave, are currently used to treat retinal bleeding, macular degeneration, retinal detachments, and blockage glaucoma, a disease of the eye intraocular membranes. the eye and leading to characterized by increased fluid pressure blindness, is treated with argon (A) and neodymium:yttrium aluminum garnet (Nd: YAG) lasers. The treatment consists of repairing structural flaws in the eye that cause pressure, the laser beam to open up blocked ducts or to create new canals for better drainage between the chambers of the eye. The laser is also cause of blindness. To neuused effectively to treat diabetic retinopathy. the beam is used to place tralize such organic disorders, the thermal energy in the thousands of burns or welds at the back of the thereby preventing the harmful growth or rupture of new, unwanted blood vessels (neovascularization). The thermal content of a laser so where coagulation is required, is also used to tack or weld retinas that fall away from the choroid at eral positions. This procedure, in fact, was the first successful use of lasers in a clinical environment. Lasers with wavelength emission in the deep ultraviolet. principally the exlasers, offer great for reshaping the eyeball to correct myopic vision (radial keratotomy) and for removing cataracts. an organic disorder characterized by poc:ke:ts of cloudy or opaque discoloration in the lens tissue of the eye, sig,nificantly degraded vision. the most use of therapy involves the Nd: YAG laser in a procedure that internal ocular membranes. Following cataract surgery-removal of the cataractous lens and replaceSec. 1-5

laser Therapy for Ocular Defects

163

ment with a pla<>tic implant lens-certain membranes that help hold the new in become and effectively block along the axis of vision. In a cora beam is rective procedure to as posterior focused to a point near the membrane. Due to the enormous power densities dielectric breakdown of the optical medium, followed by acoustical shock waves, occurs. The overpressures associated with the latter the me:mt}ralne, allowing light to pass and vision to be restored. The procedures we described under the headings of radial keratotomy and posterior capsulotomy are in the sections that follow. treated in

Radial Keratotomy. The eye-shaping procedure to as radial keratotomy introduces radial cuts in the cornea of the elongated, myopic eyeball 7-7). After the cuts have healed, the cornea thereby reducing the axial length of the eye. As a normal, or vision is restored. This radical procedure, done mostly with a surgical in the hands of a skilled ophthalmologist, had its beginning in the Soviet Union in 1972. As the story is told, a was engaged in a schoolyard fight in rather myopic Soviet lad, Boris Moscow. In the usual exchange, a hard punch struck one of the thick lenses he was shattering it into fragments. Some of the shards of glass, in one of those freak accidents of embedded themselves in the lad's cornea in a somewhat regular, radial enough but still not penetrating the cornea. A Sodid not viet ophthalmologist, Svyatoslav N. Fyodorov, who treated the for restored vision in the eye, even though the cuts were suhold out much perficial. Astonishingly, though, as the corneal surface healed with aU its scars, the cornea flattened out and most of the myopia The lad saw better than of what he witnessed, deRecognizing the to replicate, under controlled conditions, what nature and a fist had accomplished so haphazardly. This he did, many in the procedure that has come to be known as radial keratotomy. This very procedure has been by unusual in the United States. It has been (and is now) successfully nevertheless, with a surgical blade. Short-term results have been Myopic eye

Blurred vision Myopic eye

Before I

I I

I I

I I

I

k-Cornea""""'"

II

II

II Flattening -J>.lk-

Eyeball

After

II

(a)

(bl

Figure 7-7 Removing (a) Frontal view showing

164

Chap. 7

of the Eye

by reshaping the cornea with radial keratotomy. cuts. (b) Side view before and afler cuts.

insufficient time has elapsed to guarantee beneficial, long-term results. Indeed, in some cases, myopia has been cured but unfortunately, with As a the is currently out on an experimental basis. As far as the laser is concerned, the challenge in radial keratotomy lies in usa powetful laser, sharply focused, in place of the surgical blade. Since the cornea abundant amounts of water, it has a very high absorption coefficient for infrared radiation at 10.6 p.m (carbon dioxide CO2 laser) and ultraviolet radiation below 400 nm. Figure 7-8 indicates the absorption of laser light at ","",...,un wavelengths in 1 mm of both water (dashed curve) and hemoglobin (solid curve). Note that at 10.6 #Lm, the absorption is 100%, the claim is strongly absorbed by the cornea. (Incidentally, note also that CO2 laser that the argon ion and frequency-doubled Nd: YAG laser radiation near 500 nm is strongly absorbed in hemoglobin, making one an candidate for pm)to··coaglUallVe treatments.) The powerful laser beam, almost totally absorbed the corneal tissue, etches into the cornea, creating a neat cut or trough about the of the laser beam. The surgical blade currently in radial is around 50 to 100 #Lm wide. CO2 can be focused down to less than 50 #Lm in width and excimer lasers, even narrower. 7-9 shows a histologic section (X 100) of a laser cut made on the cornea of a cow with a laser. With average power of 0.5 W focused to a beam width below 50 #Lm, the after 20 passes, made an incision of about 60 #Lm wide and over 400 #Lm I

I

Hemoglobin

I

__ -------1-,.. I

I 90

I I

BO

1

/"

I

I

'IE

E 60

.5: c

o

1

15

40

I

,-

I~

<1:

I~

1

: 1 I

10

IE ::t.

I I

I

20

I

I

I I I I I I

Ig .....

I 30

:

I

I I

10_.

115.

e-

jg

a I~

I

11

50

+=l

1

Water

I I I

1-

I~ I~ O 1-

IE

l~

I

I

70

I I

L

I

I

I

:

:I

I I

I

o~~~~~~~==~====~~~~~ 0.4

0.5

0.6

O.B

1.0

2.0

4.0

10.0

Wavelength of light (lim)

Figure 7·8 Absorption of laser light in ocular tissue for several important lasers. The percentage absorption is for I mm of water (dashed curve) and I mm of l1en~ogloblin (solid curve). Note that radiation at 10.6 Ilffi is almost totally absorbed in the cornea, whereas thai at 1.06 Ilffi is almost tolaUy transmitted.

Sec. 1-5

laser T

~.~,~ . .

....

for Ocular Defects

165

Histoi(}gic section (X roO) of laser

Smallest scale divi(Reprinted from Ophno. 2 (Febmary 1981): I.)

If.

and the beam power conthe width (50 /Lm) not beyond ","~"'.c"'" the number and nature byproduct of astigmacorrection with eyeglasses

or "pt:rrrianient procedure, identified unwanted, opacified Nd: YAG laser emits a
":'UnP'LAfnu. If,t't1''rl''',,t

Incident radiation

view of Nd:YAG laser

166

Chap. 7

of the

Prior to laser surgery, the opacified membrane was removed invariably with .","""",,,,,. intravention by instruments. In addition to the trauma involved with the operation, invasive surgery of the eyeball is always attended by possible introduction of foreign bodies and increased risk of By contrast, the laser surgery, on an outpatient basis in a matter of minutes, is neither traumatic nor infectious. Noninvasive laser intervention grew largely from the successful procedures by Daniele Aron-Rosa and coworkers at Franz Fankhauser and at the UniverTrousseau Hospital in Paris sity Eye Clinic in Bern, Switzerland. Posterior capsulotomy, as currently carried out both in the United States and abroad, focuses a short pulse of 1 laser radiation at the target. The typical delivered by a Q-switched has an energy of I to 4 mJ and a pulse of several nanoseconds. With a mode-locked only the pulse energy is about the same, but the pulse length is much tens of picoseconds in duration. With such short times, the laser power, even for energies as small as reaches the megawatt range and higher.

Determine the (power per unit area) delivered by a Nd:YAG laser of energy 4 mJ and pulse length of I ns when the pulse is focused on to a tiny spot of 30 J,Lm. Solution The irradiance

E e

=

P

is calculated directly as follows:

where P

A

3

4

X I X

10- J 10- 9 S

=

4

X

1(1' W

and -"-----'...0.-_ _ _" -

= 7.065

X

10- 10

Thus

E

e

4xH1'W ------:----::

= 7.065 x

=

5.7

X 10 15

W/m2

or

Ee = 5.7

X

1011

As the example shows, when such high powers are focused at the or power denin tiny spots ranging from 5 to 50 J,Lm in diameter, of 10 12 W/cm 2 are Power densities of this are accompanied by very high electric that cause, a dielectric breakdown of optical tissue and, second, the formation of a plasma. The explosive growth of the plasma rise to a strong shock wave that radially outward, mechanically rupturthe taut, opacified membrane. Neither the laser which moves on toward the retina, or the shock wave causes elsewhere in the eye. is shown in Figure 7 -11. The beam inAn expanded view of these on the focusing lens has been by prior optics so that its beam divergence is very small (the beam is highly collimated). Because of this and the ability to focus highly collimated, coherent beams onto very small the bcam converges to a "point" near the the tric that ultimately to mechanical rupture of the membrane. Figure 7-12 shows a typical Nd:YAG laser used in membrane rupture surgery. Figure 7 - shows the orientation of ophthalmologist, and laser during a Sec. 7-5

laser Therapy for Ocular Defects

167

Beam energy diffused on

the retina

f'igure 7~1I Focusing laser 10 mies_ The larger the beam convergellce lary aperture-me larger the beam di'ier!!CllCC laser on the retina,

posterior capsuloto17°)-limiled the pupilthe less the of

Typical Nd: YAG laserlamp system used 10 perfonn a (Reproduced by from FlIITn"'W,Wn,'_ November 1984,)

168

of the

Figure 7-13 Laser is used in ('mlll"""",n,, the

membrane. lions. November 1984,)

nnllll",lmir

7-1.

modaled schematic eye is 8 mm. fraction can be neglected), bounded other. determine the refractive power fraClive power is by convention that cOlrre:spc,ndmR 7-2. Consider the un accommodated radii of curvature and effective refractive Table 7-1 (3) Calculate its focal length and (b) Calculate its focal and rounded on both sides with fluid of (c) Calculate its focal length and of thickness 3,6 mm. (The matrix [cc.nm(IU(lS this

7-3.

slil

I K(;Pf()(lU,ce(l

of curvature for the unaccoma thin surface (whose own reside and aqueous humor on the surface. (Note that the stated rethe shorter of its focal lengths.) as an isolated unit, having for the schematic eye in

sur-

it as a thick lens, 4 may well be applied to

nrc>hl,f'm

values for refractive indices and sermnifi(m "Ipmp"t<: from the schematic unaccommodated eye given in Table 7-1 determine the distance behind the cornea where an is focused at infinity and (b) an at 25 cm from the eye, Use the Gaussian formation by a surchain of calculations. face in a assume that the fully accommodated eye differs in the following of the lens is more sharply curved, a radius of +6 mm, remains at - 6 mm, As a reII'1(,,'p;:;;;r'\;; to 4.0 mm, and the distance the thickness of the lens from cornea to the front surface of

7

Problems

169

74. Use the matrix to find the system matrix for the unacrommodated schematic eye of Table 7-1 and Figure 7-2. (8) Determine the four matrix elements of the system matrix, where the system extends from the first refraction at the cornea to the final refraction at the second lens surface. (b) From the matrix elelmellts, determine the first and second focal points and the first and second relative to the corneal surface. Compare with the dislance 7-5. You have been asked to a Snellen eye chart for a test distance of 5 ft. The chart is to include rows of letters to test for visual acuities of 20/300 (same as 5/75), 20/60, and 20/15. Determine the size of the block letter and letter for each row of leaters. hvrler(lOic person has no astigmatism but has a near point of 125 cm. Correction that this person see objects al the normal near point (25 cm) (8) What is the power of the corrective lens?

enable focusing of a distant object on the retina? (b) Will the corrective 7-7. A person has a far point of 50 cm and a near point of 15 cm. What power eyeglasses is needed to correct the far point? Using the eyeglasses, what is the person's new near point? 7-8. From an examination of 7-8. determine which lasers would be suitable for (a) of vessels on the retina; (b) thermal cutting of corneal .tV""<:IrIO of light energy in the vitreous chamber without absorption during pascornea, aqueous humor, lens, and associated blood vessels. 7·9. Consider each of the following speclacle prescriplions and describe the refractive errors that are involved: (8) I.SO, I axis 180 (b) -2.00 (c) +2.00 (d) +2.00, 1.50, axis 180 7-10. The laser used to make the corneal incision shown in Figure 7-9 has an average power of 5 Wanda beam of 2.2 mrad. After from the laser, it is sent through a 5x beam and then focused onto the cornea by a 3.3-cm focal length germanium lens. (a) used as a lens material? is discussed, it is shown that the angular (b) In beam of an expanded laser beam is equal to the divergence of the incident beam divided the beam factor. What then is the beam dilaser beam after through the 5 x beam expander? "nrU'nliirr,,,tp formula, D = 14>, where 1 is the focal length. and 4> is the (c) beam of the beam. determine the diameter D of the focal spot on the cornea. to Section 21-4 for a discussion of this equation.) of the focused CO 2 laser beam on the (d) What is the power cornea? 7-11. For the """tpr',nr ......~"'u "vn'" surgery described in this the following data are typical: Laser: Nd:YAG Wavelength: 1.06 pm Pulsewidth: 10 I1S per pulse: 10 m] Beam divergence at lens: 0.1 mrad Power of lens: 20 D (8) What is the average power Nd:YAG laser beam at the opaque membrane (b) What is the 5pm size of the ", ..",hiI'm 7-lOc.) in the interior of the

110

Chap. 7

Optics of the

(c) Assuming that none of the incident power is what is the beam irradiaoce on target? 7-12. When working with lasers, one is admonished to "never stare into the beam." With low-power helium-neon (He-Ne) lasers (milliwatt level), one be tempted to consider this caution somewhat loosely. Even the power in a laser beam may he fairly small, the focusing action of the eye can concentrate the Jaser power onto the retinal surface to create a irradiance level. To iHustrate this consider a 4mW He-Ne laser that emits a collimated beam of 7 mm diameter at 632.S nm, with a of 1.5 mrad. beam (a) Calculate the irradiance of the in W/cm\ just outside the laser. (b) Assume that the entire beam enters a dark-adapted eye (pupil diameter of 7 mm) that is staring at the beam and focused at infinity. Treating the eye as a simple thin lens of 17 mm focal determine the irradiance of the fucused spot on the retinal surface. (Refer to problem 7-tOe for in calculating the size of the focused spot.) (c) By what factor does the focusing action of the eye increase the irradiance? What do you conclude about the seriousness of the admonition? 7-13. Laser protective eyewear filters are used by personnel who laser beams. The optical density (OD) for a filter is the formula OD (Ep/MPE), where is the energy density (J/cm2 ) in the laser pulse and MPE is "maximum permissible exposure" to the eye, also in units of density. Suppose you are to determine an appropriate filter for use with a Nd:YAG (L06p.m) that emits single laser pulses of SO mJ. For safety reasons, this laser is rated as a Class IV laser with an MPE of 5.0 x 10- 6 J/cm2 • (a) Determine the energy density in J/cm 2 of the SO-m1 laser a 7-mm diameter laser pulse. (b) What should be the OD of the protective filter that absorbs 1.06 p.m radiation?

[I]

[2] [3] [4]

[7] [8]

[10]

Mathew. "The and Vision." [n Handbook oj Optics, edited by Walter G. Driscoll and William New York: McGraw-Hili Book 1975. Duke-Elder, and D. Abrams. Ophthalmic and Refraction. Vol. 5 of Systems of Ophthalmology, edited by S. Duke-Elder. St. Louis: C. V. Mosby Company, 1970. Michaels, D. D. Visual Optics and 2d ed. S1. Louis: C. V. Mosby Co!mPlany 1980. Rubin, M. L Optics for Clinicians, 2d ed. Fla.: Triad Scientific Publishers, 1974. Fincham, W. H. A., and M. H. Freeman. Optics, 9th ed. Boston: ButterWorth Publishers, 19S0. Ch. 20. Richard P., Robert B. Leighton, and Matthew Sands. The Feynmo.n Lectures in Physics, vol. 1. Mass.: Addison-WesJey Company, 1975. Ch. 36. Sperry, R. W. "The and the Brain." Scientific American (May 1956): 48. Milne, L J., and M. J. Milne. "Electrical Events in Vision." Scientific American (Dec. 1956): 113. Michael, Charles R. "Retinal Pre",... !,,,;,,,O of Visual " Scientific American (May (969): 104. D. H., and M. L Wolbarsht. with Lasers and Other Optical Sources: A Comprehensive Handbook. New York: Plenum Press, 19S0.

Chap. 7

References

111

8

Wave Equations

INTRODUCTION

In this we develop mathematical but concentrate on the most useful case, the wave. Hannonic wave functions are then specified further to represent electromagnetic waves, which include light waves. Results from electromagnetism describing the physics of electromagnetic waves are borrowed to enable a determination of the energy by such waves.

8-1 ONE-OIMENSIONAI.. WAVE EQUATION

The most form of a traveling wave, and the differential equation it satlst1eS, can be determined in the following wdy. Consider first a one-dimensional wave pulse of arbitrary shape, described by y I = f), fixed to a coordinate system Of (x', y'), as in Figure 8-la. Consider next that the 0' system, together with the pulse, moves to the right along the x-axis at uniform speed v relative to a fixed coordinate system, 0 (x, y), as in Figure 8-1 b. As it moves, the pulse is assumed to maintain its shape. Any on the pulse, such as P, can be described of two coordinates, x or x', where x I = X vI. The y -coordinate is identical in either

112

y'

---------------+------------~x'

o·

(ill Stationary wave pulse

---------------1-------------+------------------~--------~x'

{bl Wave pulse translating at constant speed

8·1

Translating wave

"'· .....C.,' .. From the point of view of the stationary coordinate system, then, has the mathematical form y

If the moves to the may write

=

I(x'}

y'

I(x

of v must be

the

y=

:t

vt)

mov-

VI) T'P"l'PT'"p£1

so that in uP.,,,.,,.,, we

(8-0

as the general form of a traveling wave. Notice that we have assumed x x' at t = O. The original shape of the y' I(x '), not vary but is found sim- . ply translated the x-direction the amount vI at time t. The is any function whatsoever, so that for example. y = A sin (x - vt)

y = A(x

+ vtf

y all represent traveling waves. Only the first. however, represents the important case of a periodic wave. We wish to find next the partial differential equation that is satisfied by all such periodic waves, of the function I. y is a function of two variables, x and t, we use the rule of partial differentiation and write y

= I(x')

where x'

Sec. 8-1

One-Dimensional Wave

x ± vI

l;;\fUC""'''

113

so that

ax' fiJx = 1 and ax' fiJt the

rule, the space derivative is

oy ax Repeating the

±v

nrrV"J>,t1.

afax' ax' ax

to find the second

y a (a ) ox ax

ax'

U"'llV"lllV"'.

a(ay/ax) ax' ax' ox

Similarly, the time derivatives are found:

ax' at

ax =

I

at

o (Oy) at

ot

of - ax'

+v-

........:...-"-'---'-

ax'

ox' at

Combining the results for the two second sional differential wave equation,

0

ax' np,c ..",.n,p
a'2j ax'l

= v2 --

we arrive at the one-dimen-

(8-2) Any wave of the form of (8-1) must satisfY the wave of the physical nature of wave itself. Thus, to determine whether a function of x and t a wave, it is sufficient to show that it is of the general form of (8-1) or that it satisfies the wave (8.2).

B-2 HARMONIC WAVES Of special importance are harmonic waves that involve the sine or cosine functions,

y

A ~~s[k(x ± vt)]

(8-3)

where A and k are constants that can be varied without clliilnging aeter of the wave. These are periodic waves, representing ",,,,vV'c" themselves endlessly. Such waves are often generated by un(:ian[lpe:d dergoing simple harmonic motion. More important, the sine and functions that a of terms like together form a complete set of those in Eq. can be to represent any actual wave form. Such a series of terms is called a fourier series and is treated further in Section 12-1. Thus combinations of harmonic waves are potentially capable of representing more complicated wave forms, even a series rectangular pulses or square waves. Since sin x cos (x only difference between the and cosine functions is a relative translation of 1T /2 radians. It is sufficient in what '''IJ
114

Chap.S

Wave "'~'U<>"'UI

y

y

x

constant

t

x = constant

la)

(bl

Figure 8-2 Exlension of a sine v.ave in space and lime. (a) Sine wave at a fixed lime. (b) Sine wave al a fixed point.

by 271". Symbolically,

the sine function is

A sin k[(x + A) + vt]

A sin [k(x + vt) + 271"]

A sin (kx + kA + kvt)

A sin

or

It follows that kA

+

kvt

+

so that the propagation constant k contains

re-

garding the wavelength. (8-4)

k

Alternatively, if the Wdve is viewed from a fixed position, as in temporal unit called the period T. riodie in time with a the wave form is exactly reproduced, so that A

k[x

+

vet

+

Tn

A sin [k(x

+

VI)

+

or

A sin (kx

+ kvt + kvT)

A sin (kx

+ kvt +

and we have an that relates the period T to the propClearly. kvT agation constant k and Wdve velocity v. The same information is included in the. relation V

where we have used and frequency v,

vA

(8-5)

(8-4) together with the reciprocal relation between period T

T

(8-6)

of wave parameters are often used. The combination w = 271"v Related is caned and the of the K 1/A is called wave With these it is easy to show the equivalence of the following common forms for ""rnl£1,nh" Y

A

Y = A

y=A

Sec. 8-2

Harmonic Waves

(x ± vt)]

(8-7)

~:[ 271"(~ ± f) ]

(8-8)

sin

± wI)]

(8-9) 115

In any case, the argument of the sine or cosine, which is an angle that depends on space and time, is called the phase, (fl. For example, in Eq. (8-7), (fl

= k(x ± vI)

(8-10)

When x and 1 change together in such a way that (fl is constant, the displacement y = A sin (fl is also constant. The condition of constant phase evidently describes the motion of a fixed [Xlint on the wave form, which moves with the velocity of the wave. Thus if (fl is constant, d£p

= 0 = k(dx

± vdl)

and dx dt

- = +v

confirming that v represents the wave velocity, which is negative when (fl == k (x + vt) and [Xlsitive when (fl = k (x - vI). In any of the wave equations, Eqs. (8-7) to (8-9), notice that under initial conditions x = 0 and t = 0, y = 0 if the sine function is used and y = A if the cosine function is used. As pointed out previously, both situations could be handled by either the sine or cosine function if an angle of 90° is added to the phase. In general, to accommodate any arbitrary initial displacement, some angle (flo must be added to the phase. For example, Eq. (8-7) with the sine function becomes y = A sin [k(x ± vt)

+

(flo]

Now suppose our initial boundary conditions are such that y = Yo when x = 0 and t = o. Then

y = A sin from which the required initial phase angle (flo

(flo (flo

= sin-I

= yo can be calculated as

(~)

The wave Eqs. (8-7) to (8-9) can be generalized further to yield any initial displacement, therefore, by the addition of an initial phase angle (flo to the phase. In many cases, the precise phase of the wave is not of interest. Then (flo is set equal to zero for simplicity.

Example A traveling wave propagates according to the expression y (x, t) = 0.35 sin (371"X - 10 71"t

+ ~)

where x is in meters and t is in seconds. Determine the wavelength, frequency, velocity, and initial phase angle. Also find the displacement at x = 10 em and t = o. Solution

By comparison with Eq. (8-9), k = 371" and w = Hhr. Thus 271" 2 A=-=-m k 3

w and f=-=5Hz 271"

The initial phase (x = 0, t = 0) is 71"/4. The velocity of the wave may be found from v = Af = (2/3) 5 = 3.33 mls in the positive x-direction (due to

116

Chap. 8

Wave Equations

the negative sign in the phase). One can also set the phase cp = 10m + 7T / 4 equal to a constant. so that

37TX -

dcp = 37T dx - IOrr dt = 0 or v

= dx/dt

= lOrr/37T = +3.33 m/s. Furthermore, the displacement

y(O.l,O) = 0.35 sin (0.37T

+~)

= +0.346 m

8-3 COMPLEX NUMBERS In many situations it proves to be convenient to represent harmonic waves in complex-number notation. To this end, we first review briefly the forms in which we may write a complex number and their most useful relationships. A complex number i is expressed as the sum of its real and imaginary parts,

£=a+ib

(8-11)

where a = Re (£)

b = 1m (£)

and

are real numbers and i = v=I. The form of the complex number given by Eq. (8-11) can also be cast into polar form. Referring to Figure 8-3, the complex number i is represented in terms of its real and imaginary parts along the corresponding axes. The magnitude of £, symbolized by I£ I, also called its absolute value or modulus, is given by the Pythagorean theorem as (8-12) 1m

--------~~--~~~Re

a

Figure 8-3 Graphical representation of a complex number along real (Re) and imaginary (1m) axes.

Since from Figure 8-3, a = £ by

Ii I cos e and b £ = lil(cos

= Ii I sin

e, it is also possible to express

e + i sin 0)

The expression in parentheses is, by Euler's formula, ei/J

= cos

e+

i sin

e

(8-13)

so that (8-14) where

e= Sec. 8-3

Complex Numbers

tan-I

(~)

(8-15) 111

+i

-1--1--+1

Figure 8-4

The complex conjugate i* is simply the complex number i with i Thus if i a + ib,

i*

a - ib

A.....pA"''''~U

!i I e- 1fJ

or i*

by -i. (8-16)

the asterisk is used to denote the complex conjugate. A very useful minitheorem is that the of a complex number with its conjugate the square of its absolute value. Using the polar form,

ii*

lil2

(!i!eifJ)(lile- i8 )

it will be helpful to list the values of ,using Euler's frequently special cases. These are given in them quickly. a mnemonic device to assist in

(8-17) '~¥'~nl~

(8-13), 8-4, together with

8-4 HARMONIC WAVES AS COMPLEX NUMBERS

Euler's formula, it is possible to express a harmonic wave

y

(8-18)

where Re (y )

A cos (kx

wi)

(8-19)

1m (y )

A sin (kx - WI)

(8-20)

and

Expressed in of Eq. (8-18), the harmonic wave function thus both sine and cosine waves as its real and imaginary parts. Calculations employing the complex form implictIy carry correct results for both sine and cosine waves. At any point in such calculations, appropriate for either form can be extracted sides of the Because the taking the real or the imaginary parts of mathematics with functions is usually simpler than with trigonometric functions, it is often convenient to deal with harmonic waves written in the form of Eq. (8-18). 8-5

_It ............

WAVES

the wave equation further so that it can represent a along any direction in space. Since an arbitrary direction involves the three spatial x, y, and z, we the wave displacement by 1/1 rather than y; for '-''''''U'l'''-',

1/1 = A sin (kx

wt)

(8-21)

Equation (8-21) a traveling wave moving along the + x-direction. At fixed time (for simplicity we take t o ) , the spatial extent of this wave is

1/1 118

Chap. 8

Wave '-"I'"''''''''

A sin kx

(8-22)

When x constant, the q; = Joe = constant. Thus the of constant constant phase are the family of planes given in Figure 8-5. The constitute the wavefroms of the disturbance. Evidently, then, the wave displacement given by '" is the same for all points of a wavefront. The wave disturbance at an arbitrary in space, defined by the vector r in 8-6a, is therefore the same as the poi nt x the xwhere x = r cos 8. Eq. may then be written as '" A sin (kr cos 8)

v

z Figure 8-S Plane waves along x-axis. Surfaces of constant phase are the planes = constant. The waves penetrate the planes x a. x = h. and x c at the points shown.

x

v

y

.t1

/ (

/

/

/

/

/1 I

I I

1

I I

)

z

z lal

(b)

FIgUre 8-6 Generalization of the plane wave to an arbitrary direction. The wave direction is given the vector k along the x-axis in (a) and an arbitrary direction in (b).

Sec. 8-5

Plane Waves

119

Some simplification results if the propagation magnitude 211"1 A has already been determined in Eq. (8-4), is now considered to be a vector quantity, pointing in the direction of propagation. Then kr cos (J k . r, and the harmonic wave of Eq. (8-21) becomes

t/J

=

A sin (k

. r - rut)

(8-23)

In this form, Eq. (8-23) can represent plane waves propagating in any arbitrary direction by k, as shown in Figure 8-6b. In the general case,

xk" + yky + zl<.

k .r

where (k.. , ky, kz) are components of the propagation direction and (x, y, z) are the components of the point in space where the displacement t/J is evaluated. The harmonic 'Wave equation is now a three-dimensional wave equation that also be in complex as (8-24) The partial differential equation satisfied by such three-dimensional waves is a generalllzaltlOn of . (8-2) in the form

as can be verified by computing the second partial of t/J from Eq. (8-24). The 'Wave is often written more compactly by sect/J, them as operators, spatial the 'Wave

(ax + a~2 + ::z)t/J 2

and defining the

so that

ope:ratc)r in parentheses as the La,pUlClll'n operator.

. (8-25) becomles simply

V'lt/J

=

~2 v

(8-26)

8-6 SPHERICAL WAVES Harmonic wave disturbances emanating from a point source in a homogeneous medium travel at equal rates in all directions. Surfaces constant phase, that is. 'Wavefronts. are then spherical centered at the source. Such 'Waves also be represented the harmonic wave equations for waves, one modification: amplitude must be divided by the distance r to give

t/J

=

(~) e,lk'r-wt}

(8-27)

The spherical wave, as it propagates from the source, decreases in amplitude, in contrast to a plane wave for which amplitude is constant. If the amplitude at distance r from the source is Air, then the irradiance (W/m2) of the wave there is proportional to (Alr}2, and we see that we are simply describing the familiar inverse law of propagation for spherical wave disturbances. Notice that in this case

180

Chap. 8

Wave Equations

the meaning A must be carefully described. we cannot tude to become infinite at the source point, as r approaches zero. must correspond to the amplitude of the wave at unit distance (r source.

the amplivalue of A 1) from the

8-7 ELECTROMAGNETIC WAVES

can represent any type wave disturThe harmonic wave equations discussed so bance that in a sinusoidal manner. This includes, waves on a string, water waves, and sound waves. The equations apply to a situation as soon as the physical significance of the displacement l/J is identified. The quantity l/J may refer to vertical displacements of a string or pressure variations due to a sound Pf()P3:gatlmg in a gas. For waves that can the propal/J for either of the electric or fields that together the wave. Figure a plane wave traveling in some arbitrary direction. From Maxwell's equations, which describe such waves, we k:now that the harmonic variations of the electric and magnetic fields are always perpendicular to one another and to the direction given by k, as suggested by the orthogonal set of axes in Figure variations may be described by the hannonic wave equation in the fonn

E

Eoei(k .• -w.)

(8-28)

B

Boei (k.r-w1)

(8-29)

where E and B represent the electric and magnetic fields, respectively, and Eo and

Do are their amplitudes. Both components of the wave travel with the same propagation vector k and frequency wand thus with the same and speed. Furthermore, electromagnetic theory us that the field are Eo cBo, where c is the speed wave. At any specified time and place,

E = cB In free space, the velocity c is

(8-30)

by I

c=---

(8-31)

8-7 Plane elel:tromalgneltic wave. The elel:tric field E, magnetic field B, and propagation vector k are mutually lV'rnf'n:c';C',

Sec. 8-7

Electromagnetic Waves

181

where the constants Eo and j.Lo are, respectively, the permittivity and permeability of vacuum. Measured values for constants, Eo = 8.8542 X 10- 12 (C-s)2/kg-m3 1 and j.Lo = 47T X 10- kg-m/(A-s)2, provide an indirect method of determining the 2.998 x 108 m/s. speed of light in free space and yield a value of c Such a wave, of course, the of energy. The energy .-1"""".1" (J/m 3) with the electric field in free space is (8-32) and the energy density associated with the magnetic

in free space is

UB

These expressions, easily derived for the static electric field of an and the static field of an idea] solenoid, are generally valid. Incorporating Eqs. (8-30) and (8-31) into either of the Eqs. (8-32) or (8-33), u£ and UB are shown to be equal. For example, starting with Eq. (8-33),

(!£)2 =

_1 2J.Lo c

(8-34)

The energy of an electromagnetic wave is therefore divided equally between its conis the sum stituent electric and magnetic fields. The total energy U

=

UE

+ UB

2UB

or (8-35) COlnslder next the rate at which energy is transported by electromagnetic wave, or its power. In a time l1t, the energy transported through a cross section of area A (Figure 8-8) is the energy associated with the volume l1Vof a volume of length c l1t. Thus power

---'----'- = uc A

=

(8-36)

Figure 8-8 Energy Ilow of an electromagnetic wave. In lime AI, the energy enclosed in the rectangular volume flows across the surface A.

or the power t""'n"t"rr~>t1 per unit area, S, is

S

(8-37)

uc

We now express the energy density u in terms of E and B, as follows, making use of Eqs. (8-31) and (8-35): =

182

Chap.S

Wave

t;;1..I'IJ<1LIU"''''

= EocEB

(8-38)

Inserting this result into Eq. (8-37),

S = Eoc 2 EB

(8-39)

The power per unit area, S, when assigned the direction of propagation, is called the Poynting vector. Since this direction is the same as that of the cross product of the orthogonal vectors, E and B, we can write, finally

S =

Eoc

2

E x B

(8-40)

Because of the rapid variation of the electric and magnetic fields, whose frequencies are 1014 to lOIS Hz in the visible spectrum, the magnitude of the Poynting vector in Eq. (8-39) is also a rapidly varying function of time. In most cases a time average of the power delivered per unit area is all that is required. This quantity is called the irradiance, Ee.

E.

=
± wt»

(8-40

where the angle brackets denote a time average and we have expressed the fields as sine functions of the phase. The average of the functions sin 2 () or cos 2 () over a period is easily shown to be exactly!, so that

Ee = ~~C2 EoBo Ee = ~EocE6 Ee =

(8-42)

1(:JB~

The alternative forms of Eq. (8-42) are expressed for the case of free space. They apply also to a medium of refractive index n if Eo is replaced by n 2 Eo and c is replaced by the velocity cI n. Notice that these changes leave the first of the alternate forms invariant.

Example A laser beam of radius I mm carries a power of 6 kW. Determine its average irradiance and the amplitude of its E and B fields. Solution The average irradiance E = power

area

e

=

6000

7T (l0- 3)2

From Eq. (8-42),

Eo

=

(2 Ee)1

/2 =

EoC

9 )]112

[2(1.91 x 10 EoC

= 1.20 X 106 VIm

and, from Eq. (8-30),

Eo 1.20 Bo = - = C

X

c

106

= 4.00 X 10- 3 T

8-8 DOPPLER EFFECT The fumiliar Doppler effect for sound waves has its counterpart in light waves, but with an important difference. Recall that when dealing with sound waves, the apparent frequency of a source increases or decreases depending on the motion of both source and observer along the line joining them. The frequency shift due to a movSec. 8-8

Doppler Effect

183

source is based physically on a in transmitted The frequency due to a moving observer is based physically on the change in speed of the sound waves relative to the observer. The two effects are physically distinct and described different equations. are also essentially different from the case of waves. The difference between the Doppler effect in sound and light waves is more than the difference in wave speeds. Whereas sound waves propagate through a medium, light waves in vacuum. As soon as medium of propis removed, there is no a physical basis for distinction between moving observer and moving source. There is one relative between them that determines the frequency shift in the Doppler effect light. derivation of the UOIDpII~r effect for light requires the of special out here. The result [1] is by

-: R

A'

(8-43)

1+c

where A' is the Doppler-shifted wavelength and v is the relative velocity between source and observer. The sign of v is positive when they are one anWhen v ~ c, this equation is approximated by

A'

v c

A

(8-44)

The effect is especially when used to speed of astronomical sources emitting radiation. The red shift is the shift in wavelength of such radiation toward longer wavelengths, due to a relative speed of the source away from us. Doppler broadening of spectral lines another '"",nno"o,,' application in which atoms of a gas with both increases and decreases in frequency to their random motion toward and away from the observer making spectroscopic measurements.

Example a distant galaxy characteristic lines of oxygen specthat the wavelengths are shifted from their values as measured us,,,nnnIH"·V sources. In particular, the line expected at 513 nm shows up at 525 nm. What is the speed of relative to the earth? Solution

Here, A

= 513 525 513 v

nm and A'

=

525 nm. Thus,

(8-44),

v

c

=

-0.0234«::

Since the apparent A is larger the earth with a speed

-7020 km/s

U"'~U'~U"''I

less), the galaxy is moving away UXlmal[elY 7020 km/s.

8-1. A pulse of the form y = ae- bx2 is formed in a rope, where a and b are constants and x is in centimeters. Sketch this pulse. Then write an equation that the moving in the negative direction at 10 em/s,

184

Chap. 8

Wave Equations

g...2. A transverse wave pulse, described y=

4

is initiated at t = 0 in a stretched (a) Write an equation for the of 2.5 mls in the negative x-direction. (b) Plot the pulse at I = 0, I 2, and I 5 seconds. g...3. Consider the following mathematical p".~rp<""';I"'I" where distances are in meters: L I) = A sin 2 (t + z)j

=

2. 1)=A(x3. y(x, t) = A/(Bx 2 I) (a) Which qualify as waves? Prove your conclusion. (b) If they qualify, give the and direction of the wave ""I,"t'"hl 84. ]f the following represents a wave, determine its """tv"',,, (ITI8jgnitudle and direction), where distances are in meters. y

x - lOt

g...S. A harmonic traveling wave is in the negative z-direction with an amplitude (arbitrary units) of 2, a wavelength of 5 m, and a period of 3 s. Its at the origin is zero at time zero. Write a wave equation for this wave (a) that exhibits di-

8-6.

g...7.

8-8.

8·9.

g...1O.

rectly both wavelength and (b) that exhibits directly both constant and velocity; (c) in form. (a) Write the equation of a harmonic wave traveling along the x-direction at t = 0 if it is known to have an of 5 m and a wavelength of 50 m. (b) Write an for the disturbances at I = 4s if it is in the negative x-direction at 2 m/s. For a harmonic wave by y 10 sin (628.3x - 6283/), with x and y in centime(c) conters and t in seconds, determine (a) wavelength; (b) stant; (d) angular frequency; period; (0 velocity; (g) amplitude. Use the constant phase condition to determine the velocity of each of the following waves in terms of the constants A, B, C, and D. Distances are in meters and time in seconds. Verify your results dimensionally. (a) fey, t) = A(Y 1)2 (b) f(x, t) = A(Rx + Ct + (e) fez, I) = A exp + BC 2t 2 - 2BCZ1) A harmonic wave "a"\OIl"'!,; in the +x-direction has, at t = 0, a of 13 units at x = 0 and a of -7.5 units at x = 3A/4. Write the equation for the wave at t = O. (a) Show that if the maximum positive displacement of a sinusoidal wave occurs at angle qJo is distance XQ centimeters from the origin when t = 0, its initial given by

where the W:I've;lf'.r1PIn A is in centimeters. (b) Determine the and sketch the wave when A 10 cm and Xo = 0, ~, ~, 5, and ~ cm. (<:) What are the ",...,,,·...,..... i,!>f ... initial phase angles for (b) when a cosine function is used instead? g...ll. By finding ",nr...,.,.,ri!>tp nlr~';l{)rlJ'; lor k . r, write describing a sinusoidal plane wave in three rlin""... ~ir",~ displaying wavelength and velocity, if propagation is (a) along the

Chap.S

Problems

185

the line x = y, z 0; perpendicular to the planes x + y + z = constant. S-12. Show that if i is a complex number. (al Re (2) = (i + (b) 1m (Z) (i (c) cos (j = (ell! + (d) sin (j = (eil! 8-13. Show that a wave function, in complex form, is shifted in (a) by 'IT/2 when multiplied by i and (b) by 'IT when multiplied by I. 8-14. Two waves of the same amplitude, and frequency travel in the same reof space. The resultant wave may be written as a sum of the individual waves,

=

",(y, I) = A sin (ky + WI) + A sin (ky With the help of complex

WI

+ 'IT)

show that

",{y, t)

2A cos (ky) sin (WI)

8-15. The energy flow to the earth associated with sunlight is about 1.4 kW/m'. Find the maximum values of E and B for a wave of this power density. 8-16. A light wave is traveling in of index 1.50. If the electric field of the wave is known to be 100 VIm, find the amplitude of the IIIi:1gm;uo.; field and (b) the average of the Poynting vector. 8-17. The solar constant is the radiant flux (irradiance) from the sun at the earth's surface and is about 0.135 W/cm2 • Assume an average wavelength of 700 nm for the sun's radiation which reaches the earth. Find (a) the amplitude of the E- and B..fields; (b) the number of photons that arrive each second on each square meter of a solar panel; (c) a harmonic wave equation for the E-field of the solar inserting all constants numerically. 8-18. (a) The from a 220-W lamp uniformly in all directions. Find the irradiance of these optical electromagnetic waves and the amplitude of their E-field at a distance of 10 m from the lamp. Assume that 5% of the lamp energy is converted to (b) a 2000-W laser beam is concentrated by a lens into a cross-sectional area of about I x 10- 6 cm2 • Find the irradiance and amplitudes of the E- and B-fields there. 8-19. Show that the amplitude of a cylindrical wave must vary inversely with 8-20. Show that (8-44) for the Doppler effect follows from Eq. (8-43) when v «i c. 8-21. How fast does one have to approach a red traffic light to see a green So that we all get the same answer, say that a red is 640 nm and a good green is 540 nm. 8-22. A quasar near the limits of the observed universe to date shows a that is 4.80 times the wavelength emitted by the same molecules on the earth. If the Doppler effect is for this what does it determine for the 8-23. Estimate the broadening of the 706.52-nm line of helium when the gas is at 1000 K. Use the root-mean-square velocity of a gas molecule given by Vrnu

V

3RT M

where R is the gas constant, T the Kelvin temperature, and M the molecular

[IJ Resnick, Robert. Basic Concepts in Relativity and John and 1972. Ch. 2. [2] Hecht, and Alfred Zajac. Optics. Company, 1974. Ch. 2. [3] Ghatak. Ajoy K. An Introduction to Modern pany, 1972. Ch. J.

186

Chap.S

Wave Equations

Quantwn Mechanics. New York: Mass.: Addison-Wesley New York: McGraw-Hili Book Com-

9

t

0, T. 2 T.. ..

Superposition of Waves

INTRODUCTION

8 equations describing waves wavelength, and frewere developed. Quite to deal with situations in two or more such waves arrive at the same in space or exist together along the same direction. Several important cases the combined effects of two or more harmonic waves are treated in this chapter. The first case deals with the superof harmonic waves of differing amplitudes and but with the same The analysis shows that the resultant is another harmonic wave having the same frequency. This leads to an important difference between the irradiance attainable from randomly phased and coherent harmonic waves. The chapter a harmonic wave next treats standing waves that result from the with its counterpart. We end with a discussion of the modulating of a group of harmonic waves differing in an application that immediately to the important case of dispersion. " ' . . .Lt. ....." .

9-1 SUPERPOSITION PRjrNC::~IPI,E

To explain combined effects of waves successfully one must ask "1";;....llU...'UIJI What is net 1/1 at a point in space where waves with independent di~,piflceme:nts 1/12 exist together? In most cases of correct answer 181

is given the superposition principle: The resultant displacement is the sum of the separalte displacements of the constituent waves: (9-1)

this the resultant wave amplitude and power density (W/m2) can be calculated and by measurement. In this way, the superposition principle has been determined to be valid for all kinds waves. The same principle can be stated more formally as follows. If 1/11 and 1/12 are insolutions of wave equation, _ 1

iPI/I

- v 2 at2 then the linear combination,

1/1 = al/ll + bl/J2 a and b are constants, is also a solution. The superposition of electromagnetic (em) waves may be <>... ,,,..,,,,,,,,,,·11 in terms of their electric or magnetic fields by the vector equations, E

+

and

H = HI + H2

In general, the orientation of the electric or magnetic fields must be taken into account. The superposition of waves at a point where their electric fields are othogonat, for example, does not yield the same result as the case in which they are paralA more forma] accounting of the vector nature of E in the superposition of two em waves will be taken in the next chapter. The case of orthogonal E waves is considered in detail in the on the polarization of light. For the present, we treat electric fields as quantities. This treatment is strictly valid for cases where the individual E vectors are parallel; it is often applied in cases where they are nearly parallel. The treatment is valid also for cases of light, in two orthogonal components. The scalar thewhich the E field can be represented ory applies to each component and its parallel counterpart in the superposing waves, and thus to the entire wave. N0I1Iin.ear effects for which the superposition principle does not predict all the amplitude interacts with matter. observed results can occur when light of very densities, using laser light, has facilitated The possibility of study and use such effects, nonlinear optics an important branch of modern optics. An introduction to this subject is taken up in Chapter 26.

9-2 SUPERPOSIDON OF WAVES OF THE SAME FREQUENCY The first case of superposition to be considered is the situation in which two harmonic waves of the same frequency combine to form a resultant wave disturbance. We the two waves to differ in amplitude and Beginning with a wave in the form E Eo sin (k . r + (11t + q:>o) where an initial stant because we

angle q:>o is added for generality, we set k . r to examine waves at a fixed in space. Thus

Eo sin «(11t +

E

to a con(9-2)

where the constant phase

a = k .r 188

9

Superposition of Waves

+ q:>o

(9-3)

waves,

Two

mtlersc;ctlln~

az

at a fixed point, may differ in phase by

k . (r2

al

rl)

+ ('P02

- 'POI)

by the first term) and an initial phase difference second term). The time variations of the ern waves at the given point pyr\rp~:~pll by

(wi sin (wt

By

SU~lefiJOSiti(ln

+ + az)

(9-4) (9-5)

J)Jrinc;iplle, the resultant electric field ER at the point is sin (WI + al) + E02 sin (WI + a2)

O'r..""'r..""· ...1' ",!pnht"

sin (A

for the sum of two angles,

+ B)

A cos B

+ cos A sin B

and re(:ollrlbinirlg terms, cos a2) sin WI

+ (Eol

sin al

+ E02 sin a2) cos wi

(9- 6)

a moment, notice that if we picture each of the component ~raphjlcallv as phasors by plotting magnitude and phase Fr:::_ •••• _ ta) as if they were vectors, a resultant, or sum, is and phase a. From Figure 9-1 b, the components of the resultant are cos a =

cos al

+ E02 cos az

and sin a and a defined by

In terms of the comes

en" a sin wt +

graphical technique, Eq. (9-6) besin a cos wi

or sin (wt

+ a)

(9-7)

is another harmonic wave of the same freto the constituent waves

We conclude that quency w, with

y

EO! cos "'I (a)

Eoo. cos "'2 (b)

9-1 Phasor diagrams for the of two harmonic waves. (a) Adding two harmonic waves. (b) Phasor components .

Sec. 9-2

•"<> ..nr",,.t.nn of Waves of the Same Frequency

189

by the phasor diagram, Figure 9-1. The cosine law may be applied to yielding an expression for

9-la, (9-8)

and from Figure 9-1 b, the phase angle is clearly given by

= -"--------=--.::..

tan a

(9-9)

As with vectors, the graphical procedure could be extended to accommodate any number of component waves of the same frequency, as shown in Figure for four such waves. The diagram makes apparent the proper Eqs. (9-8) and (9-9) for N such harmonic waves: N

tan a

(9-10) ;=1

I

/

Eo /

I

I<'igure 9-2 Phasor diagram for four harmonic waves of the same frequency. SUrlClJlOsi:tion pmduces a resultant wave of the same frequency, with amplitude Eo and phase a .

I

and by the Pythagorean

thp,nrpl11

(~

(~ EOi cos

+

sin

aJ

(9-10

Eq. II) may profitably be cast into a form that of the (9-8). Expanding each term, law in

(~EO/ sin

aJ

N

N

2:

sin 2 ai

+

N

EOt Eo) sin a/ sin aj

2

(9-12)

i=1

N

N

cos aiY =

ai

+

2

N

cos a( cos a}

(9-13)

The first term of the right members is the sum of the squares of the individual terms the series in the left members. The double sums represent all the cross products, exduding---by the use of notation j > i-the self-products already accounted for in the first term and also avoiding a duplication of products already tallied by the factor 2. Adding (9and (9-13), N

N

ail

+

2

2:

j>i

190

Chap. 9

Superposition of Waves

N

Eo£o} (cos at cos a)

+ sin al sin a)

The in parentheses are equivalent to unity in the first term and are equivalent to cos (a; - aj) in the so that, finally N

=

(9-14) j>II=1

Summarizing, the sum of N harmonic waves of identical monic wave the same frequency, amplitude and by Eq. (9-10),

rreQU(~ncy

is again a har(9-11) or (9-14)

Example Determine the result of the superposition EI = 7 (wI + Tr/3), E2 12 cos (wt Tr/5).

the following harmonic waves: = 20 sin (WI +

+ Tr/4), and

Solution To make all phase angles consistent, first change the cosine wave to wave: :::: 12 (wt + Tr/4 + Tr/2) 12 sin (wI + Eq. (9-11),

E5 =

[7 sin

(~)

+ 12 sin

e:)

+ 20 sin

+ [7 cos

(~)

(~)

r

+ 12 cos

e:)

+ 20 cos

(~)

r

or E~ = 26.303 2 + 11.195 2 and Eo 28.6. The same result can be found usEq. (9-14), which would take the

E~

+ 122 + 202 + 2 [7 +7

X

12 cos

X

20 cos (

e: -

~)

~

+ 12

X

20 cos

(~

3:)

J

The phase of the resulting harmonic wave is found (9-10). Since the sums forming the numerator and denominator have already been calin the first part, we have tan a

26.303 11.195

and

a

1.17 (radians)

Thus the resulting harmonic wave can be expressed as 28.6 sin (WI

+ 1.17) or 28.6 sin (WI + 0.372Tr)

9-3 RANDO'" AND Cll'HE.RENT SOURCES The effort in achieving the of Eq. (9-14) pays immediate dividends in us to distinguish rather neatly two cases of superposition: (l) the case of N randomly phased sources of amplitude and frequency, where N is a number, and (2) the case of N coherent sources of the same type. In the first instance, if phases are random, the differences (a; - aj) are also random. The sum of cosine terms in Eq. 14) then approaches zero as N increases, because terms are equally divided between positive and fractions from - 1 to + 1. This leaves N

Sec. 9-3

Random and Coherent Sources

191

because there are N sources equal amplitude. Thus for N randomly phased sources, the squares of the individual amplitudes add up to produce the square the resultant amplitude. Recalling that the irradiance (W/m2) is proportional to the square of the amplitude of electric field, we can that the resultant but randomly phased sources is the sum of the individirradiance of N ual irradiances. On other hand, if the N sources are coherent, and in phase, so that all a are equal, then (9-14) becomes N

N

E~

E~

+

2

2:

N

EOiEoj

j>i 1=1

since all cosine factors are unity. The side should be recognizable as the amplitudes. more square of the sum of the N

EOi

)2

(9-16)

the individual amplitudes simply add to produce a resultant = NEOI rather than VNE o1 ., as before. Thus the resultant irradiance of N identical coherent sources, radiating in with each is N 2 times the irradiance of the individual sources. Notice that in this case the result does not require that N be a large number. We conclude that the irradiance of 100 coherent inthan the more usual case of sources, for example, is 100 times 100 random sources. If E is interpreted as the amplitude a compressional as well. wave, result holds for sound

9-4 STANDING WAVES Another important case of superposition arises when a wave exists in both forward and reverse directions the same medium. condition occurs most frequently when the forward wave a reflection at some point along its path, as in Figure Let us assume for the moment an ideal situation in which none of the energy is lost on reflection nor absorbed by the transmitting medium. This permits us to write both waves with the same amplitude. Forward and reverse waves are, = Eo sin (fa: wI) (9-17)

E2 =

sin (fa:

+ wi)

(9-18)

~ '-

Source

'-----

~

- E2 (a)

T

2' I - - -....--~'-------Jf__---------l.... x

t

0, T, 2 T, ...

Ib)

192

Chap,9

.n.:.rn,,.,,,'i'.n,n

of Waves

I"igure 9-3 Standing waves. (a) A standing wave situation occurs when a wave and its reflection exist along the same medium. A phase shift (not shown) occurs on reflection. (b) Resultant displacement of a standing wave, shown at various instants. The solid lines represent the maximum displacement of the wave. The displacement al the nodes (N) is always zero.

The resultant wave in the medm!m,

is

+ WI) + sin (kx

[sin

=

- WI)]

(9-19)

It is expedient in this case to define

ex

kx

+ wi

and

f3

kx

wi

and employ the trigonometric identity sin Applied to

ex + sin f3

2 sin

Hex + (3) cos Hex

(3)

'-"~'-'J to the

(9-19), this

sin kx) cos WI

(9-20)

which represents a wave, plotted in 9-3b. Interpretation is facilitated by regarding the quantity in as a space-dependent amplitude. At any point x along the medium, the oscillations are given by

A (x) cos wI where A(x) = sin kx. There exist values of x for which A (x) = 0 for aU t. These values occur whenever

=

0, and thus

EN

sin kx

or kx

27TX

nm,

m

0, ±1, ±2, ...

or (9-21) Such points are wave and are separated by half a wavelength. At various wave will appear as sine waves, like those shown in Although their amplitudes vary with time, aU pass has its maximum value at aU points when through zero at the fixed nodal cos wI = ±l, or when wI

Thus the outer em/el()Oe of the

"UlIIUlIl!;

wave occurs at times

T 3T 0, 2' T, 2 , ... where T is the There are also periodic times when the standing wave is everywhere zero, cos wi 0 t T /4, 3T /4 , .... Unlike traveling waves, standing waves transmit no energy. All the energy in the wave goes into the oscillations between nodes, at which points forward and reverse waves cancel. since mirrors are not perfect reflectors and the medium some of the wave energy, wave amdecreases x. the source continues to replace lost energy, the amplitude also decrea.'ics with time. In this case, the two waves do not cancel exactly at the nodes nor do add to the maximum of 2Eo at the antinodes, points halfway between nodes. The wave will then be found to include a traveling wave COlmoom~nt that carries energy to the mirror and back. Introduction of a between the waves of Eqs. (9-17) and (9-18), such as would be leads to a phase angle component in the sine and Nodes will then be displaced from the posiSec. 9-4

Standing Waves

193

tions shown in but their separation remains ),,/2. Times at which the form is everywhere zero or everywhere at its maximum displacement also change. The principal features of the standing wave, however, remain unaffected.

9-5 PHASE AND GROUP VEI..OCrrlES Yet another case of superposition, with important applications in optics, is that of waves of the same or comparable amplitude but differing in frequency. Differences ""l.f\JIL.U_}' imply in wavelength and velocity. The of sevsuch waves, with wave crests moving at different speeds, exhibits periodically large and small resultant amplitudes. A point where individual crests are coincident, yielding the maximum net amplitude, is itself a location changing with time and th"" ..p~n..p pol,sellSl[lg its own speed. Between such points of maximum response, at any time, there appear locations of minimum amplitude due to the juxtaposition constituent waves more or less out phase. These features in the resuleven in the case of two component waves, as will be made tant wave are in what follows. Let the two waves in frequency and wave be by

(9-22) (9-23)

Eo cos (k1x - Wit)

= Eo cos (k 2x

wzt)

The superposition of these waves, which are traveling together in a given medium, is

Making use of the

identity

cos a

+

cos (3

==

2 cos 4{a

+ (3) cos 4{a

- (3)

(9-24)

and identifying

we have cos

(9-25) Now let Wp

(9-26)

2

and Wg

2

W2

kg

k, - k2

2

Then ER = 2Eo cos (kpX - wpt) cos

194

Chap. 9

Superposition of Waves

Wgt)

(9-27)

Equation (9-28) represent'> a product of two cosine waves. The first possesses a frequency Wp and propagation constant that are, the averages of the frequencies and propagation constants of the component waves. The second cosine factor represents a wave with frequency Wg and propagation constant kg that are smaller by comparison, since of the values are taken in (9functions may appear like those of Figure 927). With Wp ~ W g , plots of the 4a. calculated at the same point Xo. The cosine function may be considered as a fraction that ranges between + 1 and - I for various t. Such a fraction mUltiplying the rapidly varying function reduces displacement proportionately. E

+1

\

"

/

\

I

\

\

I

\

I

\

\

\

\

\

I

\

I

\

\

\

\

I \

I

(II)

E[<

<;;:-- +2Eo \

,,-

\ \

I

\

I

f

I

\

\ \

\ \

\

I

\

/

I

\

~\!

I

T\~

I

I

I

\ \

I

I

I

---2Eo

"

",-

\

I

\

I

\ \

\

I

I

I

\

\

\

\, ~//

/

I

I

~\j

II

I

J\~ f

/

\

~\

I

I

\

\).;~

I I

I

\

I

~\ \

I

I I

I

/

\

\

\ \

\

I

\

I --'

/

I

\

\,

Ib)

Sec. 9-5

t

Phase and Group Velocities

94 (a) plots of the cosine factors of (9-28) at x = Xo, where wp :P (b) Modulated wave representing Eq. at x Xo.

195

The overall effect is that the low-frequency wave serves as an envelope modulating the high-frequency wave, as shown in Figure 9-4b. The dashed lines depict the enof the resulting wave disturbance. Such a wave disturbance exhibits the nomenon of beats. Because the square of displacement of the wave at any is a measure of its radiant flux density, the energy delivered by the traveling sequence of pulses in Figure 9-4b is itself pulsating at a beat frequency, Wh. The figure shows that the beat frequency is twice the frequency of the modulating envelope, or (9-29) From Eq. (9-29) we see that the beat is simply the difference frequency for the two waves. In the case of sound, this is the usual beat frequency heard when two tuning forks are made to vibrate simultaneously, equal to the difference in fork frequencies. in the The discussion has immediate application to nomenon dispersion. Due to dispersion, light components of different wavelengths travel with different through a refractive medium. Even so-called monochromatic light possesses a spread of wavelengths, however narrow, about the average. Any two wavelength components of such a light beam, moving through a (9-22) and (9-23) and thus produce a dispersive medium, can be represented by resultant like the one pictured in 9-4b. The velocity the wave as well as that of the envelope can be found from the general relation for velocity, v = vA

(9-30)

k

The velocity of the higher-frequency wave, from velocity,

(9-26), is then the phase (9-31)

where the final member is an approximation in the case WI == W2 = wand kg == k neighboring frequency and wavelength components in a continuum. On the other hand, the velocity of the envelope, called the group velocity, is dw dk

(9-32)

assuming that the between frequencies and propagation constants and phase velocity Vp = w/k need not be are small. Now group velocity Vg the same. If Vp > V 8• the high-frequency waves would appear to have a velocity to the right relative to the envelope, also in motion. These waves, which can be produced by an oscilloscope, would seem to at the right node and be generated at the left node of each pulse. If Vp < Vg, their relative motion would, of course, be reversed. When Vp = Vg, the high-frequency waves and envelope would move together at the same rate, without relative motion. The relation between group and phase velocities can be found as follows. Substituting Eq. (9-31) into Eq. (9-32) and performing the differentiation of a product, Vg

dw d dk = dk (kvp)

(9-33)

196

Chap. 9

Superposition of Waves

When the velocity of a wave does not depend on wavelength, in a nondispersive medium, dVp/dk O. and phase and group velocities are equal. This is the case propagating in a vacuum, where Vp Vg = c. In dispersive media, hn'wp'upr c/n, where the refractive index n is a function of A or k. Then n n(k), and -c When incorporated into group velocities,

(~~)

(9-33), we have an alternate relation between phase and (9-34) (9-34) can be reformulated as

Again,

(9-35) In regions of normal dispersion, dn/dA < 0 and Vg < Vp. These derived here for the case of two wave holds in general for a number of waves with a narrow range of frequencies.' Their sum can be both by a velocity, the average velocity the individual waves, the and by the group velocity, the velocity the modulating wave form itself. latter determines the speed with which energy is transmitted, it is the directly measwaves. When waves are modulated to contain informaurablespeed of modulation (AM) of radio waves, we may the group tion, as in less than the velocity the velocity as the signal velocity, which is carrier waves. When light consisting of a number of harmonic waves extending over a range of frequencies, are transmitted through a dispersive medium, the of the group is the velocity of the pulses and is different from the electron is repreindividual harmonic waves. In wave mechanics wave that can be decomposed into a number of harsented by a monic waves with a range of wavelengths. The measured velocity of the electron is that is, the group velocity of the constituent waves. the velocity of the wave

9-1. Two

by

waves are (3x

5 Eo and 4/)2 + 2

E2

= -:-:----..::....-.::----::+2

with x in meters and t in seconds. (a) Describe the motion of the two waves. (b) At what instant is their superposition everywhere zero'! (c) At what point is their superposition always zero? 9-2. (a) Show in a phasor the following two harmonic waves:

E.

= 2 sin wi

and

E2

= 7 sin

(wt + ~)

(b) Determine the mathematical expression for the resultant wave. I Fourier analysis for dealing with the superposition of many harmonic components. This subject is introduced in Chapter 12.

9

Problems

197

9-3. Find the resultant of the superposition of two harmonic waves in the form

+ a)

E = Eo sin (WI

with amplilUdes of 3 and 4 and phases of 30" and 90", respectively. Both waves have a period of I s. 9-4. Two waves traveling together along the same line are given by

~J

YI = 5 sin [ wt

+

Y2 = 7 sin [ wI

+ ~]

Write the resultant wave equation.

9-5. Plot and write the equation of the superposition of the following harmonic waves: EI = sin (wt - 10°), E2 period of each is 2 s.

= 3 cos (wt +

lO
9-6. One hundred antennas are putting out identical waves, given by E = 0.02 sin (WI

+

EO) Vim

The Wdves are brought together at a point. What is the amplitude of the resultant when (a) all waves are in phase (coherent sources) and (b) the Wdves have random phase differences.

9-7. Two plane waves of the same frequency and with vibrations in the z-direction are given by

"'(x, ",(y,

I) = 4 sin [201

= 2 sin

I)

+

(~)x + 7T]

[20t + (~)y + 7T]

Write the resultant wave equation expressing their superposition at the point x = 5 and y = 2.

9-8. Beginning with the relation between group velocity and phase velocity in the form Vg

==

Vp -

A (dv/dA)

(a) express the relation in terms of nand wand (b) determine whether the group velocity is greater or less than the phase velocity in a medium having a normal dispersion. 9-9. The dispersive power of glass is defined as the ralio (nF - nc)/(nv - I), where C, D, and F refer to the Fraunhofer wavelengths, A, = 6563 A. AD = 5890 A, and AF = 4861 A. Find the approximate group velocity in glass whose dispersive power is -do and for which nD = 1.50. 9-10. The dispersion curve of glass can be represented approximately by Cauchy's empirical equation, n = A + B/A2. Find the phase and group velocities for light of 500-nm wdvelength in a particular glass for which A = 1.40 and B = 2.5 X 106 N. 9-11. The dielectric constant K of a gas is related 10 its index of refraction by the relation K = n 2• (a) Show that the group velocity for waves traveling in the gas may be expressed in terms of the dielectric constant by Vg

=

~ [I

- ;K :]

where c is the speed of light in vacuum. (b) An empirical relation giving the variation of K with frequency is K

198

Chap. 9

Superposition of Waves

=I +

[A/(w~ - w 2 )]

where A and CUD are constants for the gas. If the second tenn is very small compared to the first, show that Vg

C

[1 --,-:::----::-::

can be ..",..,...._"li:PL! as

9-12. (a) Show that group

(b) Find the group velocity for waves in a medium, for which vp = A + BA, where A and B are constants. Interpret the result. 9-13. Waves on the ocean have different velocities, depending on their depth. Long wavewaves, traveling in the ocean, have a speed given approximately by 1/2

v where g is the acceleration of face have a velocity

wa'veltmgth waves, corresponding to

SUf-

1/2

V=

where p is the and T is the surface tension. Show that the group for velocity and the group velocity for short wavelong wavelength waves is 4their length waves is ~ their phase vel,OClltV. 9-14. A I~r emits a monochromatic beam of wavelength A, which is reflected nonnally from a plane mirror, at a speed v. What is the beat between the incident and reflected 9-15. Standing waves are produced by the superposition of the wave y

= 7 sin

y in cm; t in s)

and its reflection in a medium whose absorption is negligible. For the resultant wave, find the amplitude, length of one loop, velocity, and period. 9-16. A medium is disturbed an oscillation described by y = 3 sin

(~~) cos (507rt)

(x, y in cm; tins)

(a) Determine the amplitude, frequency, wavelength, speed, and direction of the comwaves whose superposition this result. • (b) What is the internodal distance? (c) What are the displacement, velocity. and acceleration of a particle in the medium atx = 5 cm and t = 0.22 s? 9-17. Express the waves of and (9-18) in the complex In this fonn. show that the sUf,er!Jositic)n of the waves is the standing wave given by Eq.

Optics. ",,,,,a.nul>, Mass.: Addison-Wesley Publishing [ll Hecht, and Alfred Company, 1974. Ch. 7. New York: McGraw-Hili Book Com[2) Ghatak, Ajoy K. An Introduction to Modern pany, 1972. Ch. I.

9

References

199

10 High Low

High Low

High

low Substnlte

Interference of Light

INTRODUCTION Like standing waves and beats, treated in the chapter, the phenomenon of interference depends on the superposition of two or more individual waves under When interest primarily in the rather strict conditions that will soon be light waves, due to their superpoeffects of enhancement or diminution of When ensition, these effects are usually said to be due to the hancement, or constructive interference, and ence, conditions alternate in a spatial display, the pattern of fringes, as in the double-slit int,,,rt.,r,,,,nrp may lead to the enhancement of one or color at the expense of the others, in which case interference colors are pr<)(IUlCtxl, and soap films. The simplest explanation of these phenomena can be ~U"U;;~~I dertaken by treating light as a wave motion. In this and following ct\llpters such applications, considered under the of are presented.

10-1 TWO-BEAM INTERFERENCE We consider first the interference oftwo waves, re~)rel;enteCl take into account the vector property of the electric source and reunite after both waves typically originate from a 200

U"V..,"UI

different paths. The direction of travel the waves need not be the same when they come however, so that whereas maintain the same frequency, they generally do not have the same propagation vector k. Accordingly, we may express the wave equations by r

wi

E02 cos (k2 . r

wi

cos (k l

E2

•

+ + e2)

(10-1)

(10-2)

to produce defined by position vector r, the waves field is by the principle of superposition,

At some general point a disturbance whose

+& Now EJ and & are rapidly functions with optical frequencies of the order of 1014 to 1015 Hz for light. Thus both EI and Ez average to zero over very short time intervals. Measurement of the waVes by their effect on the eye or some other light detector depends on the energy of the light beam. The radiant power density, or irradiance, Ee (W/m2), measures the time average of the square of the wave amplitude. Unfortunately, the standard symbol for irradiance, for subscript, is the same as that for the electric field. To avoid confusion we use, temporarily, the symbol I for irradiance. (10-3)

1=

Thus the resulting irradlance at P is given by

= EOc(E;)

I

Eoc(Ep' Ep)

eoc«EJ +

or (10-4)

I

By (10-3), the first two terms correspond to the irradiances of the individual waves, II and 12 • The last term depends on an interaction of the waves and is called term, 112 • We may then write the (10-5) If light behaved without interference, like classical particles, we would then I = II + 12 • presence of the third term 112 is indicative of the wave nature of light, which can produce enhancement or diminution of the irradiance through interand are orthogonal. so that dot product vanference. Notice that when ishes, no interference results. When the electric fields are parallel, on the other hand, the interference term makes its maximum contribution. Two beams of unpolarized light produce interference because each can be resolved into orthogonal off with similar components of the other components of E that can then be II E2 (E, parallel beam. Each component produces an interference term with to Consider the interference term, 112

where

and

are

=

(10-6)

Eqs. (10-1) and

. Eoz cos (k l • r can be of two

"'J\~J"'''uu,~

+

the cosine

a == kl . r + EJ Sec. 10-1

wi

Two-Beam Interference

Their dot product, cos interpreted as

difference

and {3201

so that

EOI . E02 cos

WI)

(0: -

COS ({3

Expanding and multiplying the cosine factors, we arrive at

(EI . Ez)

Em· E02l cos

0:

+ sin 0: sin {3 (

cos {3 (cos 2 wi)

+ (cos

0:

sin (3 + sin

0:

wI)

cos (3)(sin wI cos wi)]

where time averages are indicated for each time-dependent of complete cycles, one can show that

wt)

Over any number

=!

(sin2 wt)

and

0

(sin wt cos WI) Thus

. Ez) = ~Eol • E02 cos

(0: -

(3)

or

.r

(EI • Ez)

+

where the expression in brackets is the phase difference between in Eqs. (10-1) and

B

(k 1

-

k 2)

•

r

+

(101 -

(10-7) and

, as given (10-8)

102)

Combining Eqs. (10-6), (10-7), and (10-8), EoC &1 •

Em cos fi

(10-9)

Similarly, the irradiance tenns II and 12 of Eg. (10-5) can be shown to yield II (10-10) EoC (E 2) 1 -- 2"1EoC£201 and

h

(l0-11)

In the case &1 II their dot product in (10-9) is identical with the product of their magnitudes. These may be .....".""".£1 in tenns of 11 and 12 by the use of Eqs. (10-10) and (10-11), in

2v'J;J;. cos

fi

(10-12)

so that we may write, finally,

I

IJ

+ 12 + 2v'J;J;. cos fi

(10-13)

Notice that once we have made the assumption that the are !-' 0 or cos B < 0 in Eq. ence term either augments or diminishes the sum of the individual irradiances II and 12 , leading to constructive or interference, respectively. On the other hand, if the initial phase difference - E2) in Eq. (10-8) varies the waves are said to be cos fi becomes a factor whose average is zero. Even though interference is always occurring, no pattern Ullp......... ' " ' " .

202

10

Interference of light

can be sustained long enough to be Thus some degree of coherence. that {cos i» 0, is necessary to observe In particular, if the two waves originate from independent sources, such as incandescent bulbs or gas-discharge lamps, the waves will be mutually incoherent. sources, even though independent, can possess sufficient mutual for to be observed over (kl - k 2) • r. As short periods of time. The other term in cos i) the point of observation given by r cos i) takes on alternating maximum and minimum values and interference separated. To be more specific, when cos i) interference yields the maximum irradiance

'*

(10-14) This condition occurs whenever the difference 6 = where m is any inI, destructive interference yields or zero. On the other hand, when cos 6 the minimum, or background,

+ 12 that occurs whenever 6 = (2m + l min

( 10-15)

II

of irradiance I versus interference is comThen, Eqs. (10-14) and

in Figure to-la, exhibits periodic plete, that is, cancellation is complete, when II (10-15) give lma.

4/0

and

0

I min

linin - - - - - L_ _ _ _

~

__

_L~

-11"

__

0

~

_ _ _ L_ _

~

______

II

+

12

~6

11"

(a)

(b)

10-1 Irradiance of interference fringes as a function trast is enhanced in (b), where the background irradiance lmin

shown in Figure 1O-lb, now contrast, also caned with quantity

bener contrast. A measure of by the between 0 and I, is

. 1m.. frmge contrast = I

max

Sec. 10-'

Two-Beam Interference

con-

h.

+

I min I ' mm

203

In the experimental it is therefore usually desirable to arrange that the beams have same amplitudes. Another useful form of (10-13), for the case of interfering beams of equal amplitude, is found by W'rlhrlO'

10 + 10 + 2~ cos

I and then

= 2/0(1 + cos I)

identity

use of the

I The

I)

+ cos

I)

2 cos

(%)

2

interfering beams is then

for two

I

(~)

4/0

00-16)

Notice that energy is not conserved at each point of the superposition, that I =1= 2/0 , but that over at least one spatial period of the fringe pattern la\' 2/0 • This situation is typical of interference and diffraction phenomena: If the power density falls below the average at some point", it rises above the average at other points in such a way that total pattern satisfies the principle of energy conservation. Example beams with parallel electric fields are given by

= 2 cos

( kl . r -

wI

5 cos ( k2 . r - wI

~)

(kV/m)

+~)

(kV/m)

+

Let us determine the irradiance contributed each beam at a point where their path due to mutual We have I

rlitt",rpnr'p

is zero.

4E"0c(2000Y = 5309 W/m2

2 !

(5000)2:::: 33,180 W/m2

2

cos I) To find the

2\1'(5309 x 33180) cos

(~

contrast near the region of superposition we must

= II +

UUI..-tUi:tt<;:;

12 + 2\,1iJ;. ::::: 5309 + 33180 +

65,034 W/m2 II + 12 - 2\,1iJ;. = 5309 + 33180 11,945 W/m2 The contrast is then given by (lmax . fnnge contrast

Irnin)!(lmax + I min ), or 65,034

= 65,034 +

II 11

If the amplitudes of the two waves were equal, then the fringe contrast would be I.

204

10

Interference of light

0.690

4/0 , I min

0, and

fO-2 YOUNG'S DOUBU-SUT EXPERIMENT

The decisive experiment performed by Thomas Young in 1802 is shown schematically in Figure 10-2. Monochromatic light is first allowed to pass through a single small hole in an aperture in order to approximate a single point source S. The light spreads out in spherical waves from the source according to Huygens' principle and is allowed to fall on the two closely spaced holes, SI and Sz, in an aperture. The holes thus become two coherent sources of light, whose interference can be observed on a screen some distance away. If the two holes are equal in size, light emanating from the holes have comparable amplitudes, and the irradiance at any point of superposition is given by Eq. (10-16). For observation points, such as P, on the screen a distance s from the aperture, the phase difference 8 between the two waves arriving must be determined to calculate the resultant irradiance there. Clearly, if S2P - SI P = rnA, the waves will arrive in phase, and maximum irradiance or brightness results. If S2 P - SIP = (m + !)A, the requisite condition for destructive interference or darkness is met. Practically speaking, the hole separation a is much smaller than the screen distance s, allowing a simple expression for the path distance, S2P - SIP. Using P as a center, let an arc SIQ be drawn of radius SIP, so that it intersects the line S2P at Q. Then SzP - SI P is equal to the segment !J., as shown. The first approximation is to regard arc SI Q as a straight-line segment that forms one leg of the right triangle, S] S2 Q. If 6 is the angle between the aperture and SI Q, !J. = a sin 6. The second approximation identifies the angle e with the angle between the optical axis OX and the line drawn from the midpoint 0 between holes to the point P at the screen. Observe that the corresponding sides of the two angles (J are related such that OX 1- SI S2. and OP is almost exactly perpendicular to S] Q. The condition for constructive interference at a point P on the screen is then, to a very good approximation, (10-17) whereas for destructive interference,

!J.

= (m

+ 1)A

= a

sin

e

(10-18)

y p

./

s

--

...-

...\

e

......- ...-

...- . /

""

1 y

J

Figure 10-2 Schematic for Young's double-slit experiment. The holes S, and S2 are usually slits, with the long dimensions extending into the page.

Sec. 10-2

Young's Double-Slit Experiment

205

where m is zero or of integral value. The irradiance on the screen, at a point determined by the angle e, is found using Eq. (10-16) and the relationship between path difference t1 and phase difference j),

The result is I = 410 cos 2

(:t1)

= 410 cos2

For point'> P near the optical axis, where y sin e == tan e == y/ s, so that I = 410 cos

2

~

(7Ta ~in e)

s, we may approximate further:

(:~)

(10-19)

By allowing the cosine function in Eq. (10-19) to become alternately ± I and 0, the conditions expressed by Eqs. (10-17) and (10-18) for constructive and destructive interference are reproduced. Arguing now from Eq. (10-17) for bright fringe positions in the form mAs Ym = - - , a

m = 0,1,2, ...

(10-20)

we find a constant separation between irradiance maxima, corresponding to successive values of m, given by

t1y =

As a

00-21)

with minima situated midway between them. Thus fringe separation is proportional both to wavelength and screen distance and inversely proportional to the hole spacm = +3

5

m+ 1 2

2

m= +2 m+ 1

3

2

m

=

2

+1 1

m+ 1 2

I .....- - - - - - - E - - - - - - - l m

=

m-

2

0, y = 0

.!.

=

2

_

1

2

m =-1

m-!

=

2

_3

2

m =·-2

m-.!. 2

m=-3

206

Chap. 10

Interference of light

=

_5

2

Figure 10-3 lrradiance versus distance from optical axis for double-slit fTinge pattern. The order of the interference pattern is indicated by m. with integral values of m determining positions of fringe maxima.

ing. Reducing the hole spacing expands the fringe pattern formed by each color. separation provides a means of determining the waveMeasurement of the length of the light. The hole, used to secure a of spatial coherence, may be eliminated if laser light. both highly monochromatic and coherent, is used to illuminate the double slit. In the observational just described, fringes are observed on a screen placed perpendicular to the optical axis at some distance from the as indicated in Figure 10-3. maxima coincide with integral orders of m, and minima fall halfway oel[W€~n Example Light from a narrow slit passes through two and parallel slits, 0.2 mm are seen on a screen 1 m away, with a separation of mm. How does the irradiance at the screen vary, if the contribution of one slit alone is lo? What is the wavelength of the light? Solution

to

I = 4/0

(10-19), I = 4/0

[way/As]. In this case,

[w(0.0002)y/(658 x 10- 9)(1)]

(955y)

From Eq. (10-21),

A=

(0.0002)(3.29 x 10- 3 )/1 =

nm

An alternative way to view the formation of bright (B) of constructive interference and dark (D) positions of destructive ure 10-4. The crests and of waves from S, and are shown approaching the screen. directions marked B, wave crests (or wave valleys) from both slits producing maximum irradiance. Along directions marked waves are seen to be out of step by half a wavelength, and tP."tpr,f>nc'p results. Obviously, should be present in all the space the holes, where light from the holes is allowed to interfere, though the irradiance is greatest in D

D

/ / / /

/ "-

"

\

\ \

\ \

\

Figure 10-4 and dark inlerference are by lighl from two coherent sources. Along directions where crests (solid circles) from S, intersect crests from S2. (B) resullS. Along directions where creslS meet valleys (dashed circles). dark.ness (0) resullS.

Sec. 10-2

Young's Double-Slit

FYln~r·irn.l'!nt

201

the forward

rlir",,,tinn

If we

two coherent point sources of light radiating in by Eq. (10-17) for bright fringes, - SIP = mA

00-22)

" .... fu"'.., in the space surrounding the holes. To we may advantage of the inherent symmetry in the ar10-5, the intersection of several bright fringe with a plane that the two sources is shown, each surface COlrre!spo·ndmg gral value of order m. surfaces are hyperbolic, since is n""'r,,,,,,hf condition for a of hyperbolic curves with parameter m. Inasmuch as the yaxis is an axis of symmetry, the corresponding bright fringe surfaces are gell1eral:ed by rotating the pattern about the y-axis. One should then be to the intercept of surfaces with the plane of an observational screen anywhere in the vicinity. In particular, a screen placed perpendicular to the OX axis, as 10-2, intercept.", hyperbolic arcs that appear as straight-line near the circular whereas a screen perpendicular to the OY axis shows centered 011 the axis. Because the fringe system extends throughout the space surrounding the two sources, the fringes are said to be nonlocalized.

y

-------------------------n+-----+-----------~~-----x

10-5 fringe surfaces for two coherent point sources. The distances from SI and S, 10 any fringe point P differ by an number of ,""",,"If"ln,,1I1~ The surfaces are generated by rotating the pattern about the

The holes S I, and S2 of Figure 10-2 are usually narrow with their long sides perpendicular to the page in 10-2) to iIlumore fully the interference pattern. The of array of point sources the each sel producing its own described, is simply "'.VIU;:,"..... the pattern parallel to the fringes, without reThis is true even when two points a source slit are not mutually coherent. 208

10

Interference of Ught

10-3 DOUBLE-SUT INTERFERENCE WITH VIRTUAL SOURCES tprtpr'''l1c'p fringes may sometimes appear in arrangements when only one light source is present. It is possible, through reflection or to produce virtual images acting or with the actual source, behave as two coherent sources that can produce an figures 10-6 to 10-8 illustrate three such These examples are not only of some historic importance; they us with the of ways fringe may apespecially when the coherent light of a laser pear in optical is used. In Figure 10-6, interference fringes are due to the superposition of light at the screen that originates at the actual source S and, by reflection, also originates effectively from its virtual source S' below the surmce of the plane mirror MM' Where the direct and reflected beams strike the screen, fringes will appear. The position of bright is given by the double slit (10-20), where a is twice the of source S above the mirror plane. The arrangement is known as Lloyd's mirror. If the screen contacts the mirror at M " the at their intersection is found to be dark. Since at this point the optical-path difference between the two interfering beams one should a fringe. The

y

m

mAS =-

a

s s'

Screen

10-6 Interference with Lloyd's mirror. Coherent sources are the source S and its virtual image, S'.

111

121

1<1gure 10-7 Interference with Fresnel's mirrors. Coherent sources are the two virtual of point source S, formed in the two plane mirrors MI and M z . Direel light from S is not allowed 10 reach the screen.

Sec. 10-3

Double-Slit Interference with Virtual Sources

209

iii

=

2da(n

11

Y", I

I I I I

T 1: S

I! 2

---

I

I ~

I

~SS~I~~~=t==~~-----~--------------~~ 1 I 1

1/

S2t"-- - - I

Figure 18-8 Interference with Fresnel's biprism. Coherent sources are the virtual SI and S2 of source S, formed by refraction in the two halves of the

contrary result is by requiring a phase shift of '1T for the glass reflection.' Another closely related arrangement is Fresnel's mirrors, Figure 10-7. Interference occurs between the light reflected from each of two mirrors MI and M 2 • inclined at a small relative angle O. Two rays reflected from each are shown labeled as (I) from M, and (2) from Interference appear in the region of overlap. Interference occurs between the two coherent virtual images S, and acting as sources. Once the virtual image separation a is related to the tilt angle (J and to the distance d from actual source to the intersection of the mirrors at 0, the fringe pattern may again be described by (10-20). The screen is shown at tance D from point O. 10-8 shows Fresnel's biprism, which refracts from a small source S in a way that it to come from two coherent, virtual sources, SI and S2. Extreme rays for refraction at the top and bottom halves are shown. Interference fringes are seen in the overlap region on the screen. In practice, the prism angle 0: is very smaH, of the order of a One the rays (shown) passes through tasmon, making equal entrance and with the two sides and the condition for minimum deviation. this ray the deviation angle Sit! is given by = 0: (n - I). The geometry of this particular ray provides a means of approximately determining the virtual source separation a in terms of prism index n and angle 0:: a

2d5m = 2d0: (n

are then described by

J)

(10-23)

(10-20), as usual.

I A theoretical explanation for phase changes on reflection results from an analysis based OIl Maxwell's equations and requires identification of the slate of of the light. The subject is treated in Chapter 20.

210

Chap. 10

Interference of light

10-4 INTERFERENCE IN DI":LECl1FlIC FILMS

The familiar appearance of on the surface of oily water and soap films and the beautiful often seen in mother-of-pearl, peacock and butterfly of light in single or thin surface are associated with the exists a variety of situations in which such intransparent material. t.,. ..r"''''''nr'.,. can take place, affecting the nature of the pattern and the under which it can be Variables in the the size ~npi'tr'RI width of the source and shape and reflectance of the p

Transparent film

figure 10-9 Double-beam interference from a film. Rays reflected from the top and bottom together surfaces of the film are at P by a lens.

Substrate

LOnSij[]er the case of a film material bounded by ,","'"lA''''' be formed by an oil slick, a metal oxide layer, or an "".''''''r<>t,>rI ing on a flat, substrate (Figure 10-9). A beam of light incident on face at A divides into reflected and refracted portions. This separation of light into two preliminary to and interference, is referred to as amplitude division, in contrast to a situation like double slit, in which is said to occur by division. The refracted beam reflects again at the film-substrate interfuce B and leaves the film at C, in the same direction beam may reflect internally at C and as the beam reflected at A. Part of continue to multiple reflections within the film layer until it has lost its beams emerging the top surintensity. There will thus exist with rapidly diminishing amplitudes. Unless the reflectance of the face, film is a good approximation to the more complex situation of multiple (MclICm 11-5) is to consider the first two emerging beams. The two parallel beams the film at A and C can be brought together by a £''''r,,,,,,ro'1'IO' lens, the eye, for example. The two beams at P superpose and Since the two beams travel different paths from point A onward, a relative ference develops that can produce constructive or destructive interference at P. The in the case of normal is the additional path optical path index of the film. Thus length ABC traveled by the refracted ray times the t:.

= n(AB +

BC)

n(2t)

(10-24)

where t is the film For example, if 2m '\0. the wavelength of the light in vacuum, the two interfering bearns, on the of optical path difference and produce constructive However, an additional would be in changes on reflection, must be phase difference. due to the phenomenon of considered. Suppose that nf > no and nf > n.. In often no = n. because the media bounding the film are identical, as in the case of a water film (soap bubble) in from a lower index no toward a air. Then the reflection at A occurs with light Sec. 10-4

Interference in Dielectric Films

2n

higher index nj, a condition usually called externo.l reflection. The reflection at on the other hand, occurs for going from a higher index nf toward a index ns , the of internol reflection. A relative shift of 'iT" occurs between the externaIiy and internally reflected so that equivalently, an additional path ift"·.,p,,f'P of A/2 is introduced between the two beams. The net optical path difference between the beams is then A + A/2, which them out of and destructive interference results at P. If, both reflections are external (no < nf < or if both reflections are (no > nf > n s ), no relative phase difference due to reflection needs to be taken into account. In that case, constructive interference occurs at P. A frequent use such single-layer films is in the production of antireflecting rflJnn10_~ on optical surfaces. In most cases, the enters the film from air, so that no L Furthermore, if n, > nJ, no relative shift between the two reflected beams occurs, and the optical-path difference determines the type of ence to be If the film thickness is Af /4, where Af is the wavelength of the light in the then 2t AJ!2 and the optical path 2nft = Ao/2, since Ao Destructive interference occurs at this wavelength and to some extent at neighboring wavelengths, which means that the light reflected from such a film is the incident spectrum minus the wavelength region around Ao. If the incident light is the reflected light is colored. Extinction of a and Ao is in the visible of the by films of A/4 thickness of course, more effective if the amplitudes of the two reflected beams are In general, all one can say is that for constructive interference the two amplitudes add (being in phase), and for destructive interference the amplitudes subtract (being exactly out of phase). For the difference to be zero, that for destructive interference to be complete, the amplitudes must be We show later (Chapter 19) in the case of normal inCldlence. the coefficient (or ratio of reflected to incident electric amplitudes) is given by

n

r

+n

(10-25)

where the relative index n = nz/'h. The amplitudes of electric field reflected internally and externally from the film of Figure 10-9 are then equal, assuming a nonabsorbing film, if the relative indices are for these cases, that if

no

tlf

or

tlf

=

(10-26)

Since usually no = I, the requirement that reflected beams be of equal amplitude is met choosing a film whose refractive index is square root of the substrate's refractive index. A suitable film material for the application mayor may not and some compromise is made. For example, to reduce the of lenses employed in instruments handling white light, the film of A/4 is mined with a A in the center of the visible spectrum or wherever the detection system is most sensitive. In the case of the eye, this is the portion near 550 nm. Assuming n 1.50 for the lens, ideally tlf = 1.22. The nearest practical film material with a matching is MgF20 with n 1.38. The beneficial loss reflected light near the middle of the spectrum results in a predominance of the and red ends of the spectrwn, so that the coatings appear purple in reflected light. As another example, consider a multilayer stack of alternating high-low index dielectric films (Figure 10-10). If each film has an optical thickness of AJ!4, a Iitt1e analysis shows that in this case all emerging beams are in phase. Multiple reflections 212

Chap. 10

Interference of Light

High

Low High

Low High

low

10-10 Multilayer dielectric mirror of alternating high and low index. Each film is Al4 in optical thickness.

Substrate

in the region of Ao the total intensity and the quarter-wave stack stacks can be designed to satisfy experforms as an efficient mirror. Such tinction or enhancement of reflected light over a portion of the spectrum detail in Chapter 19. than the single film. They are treated in Returning now to the single-layer film, we want first to generalize the conditions for constructive and destructive interference by calculating the optical path difrays are not normal. to-II illustrates a ray inciference in the case dent on a film at an OJ. The phase at points C and D between to the optical path between AD and ABC. emergmg beams is After points C and D are reached, the respective beams are parallel and in same medium, so that no further phase difference occurs. To assist in the calculation, foot of the altitude BG in the point G is shown midway between A and C at E and F are by constructing the perpenisosceles triangle The optical path difdiculars GE and GF to the ray paths AB and ference between the beams is then !J.

nAAB

+ BC)

no(AD)

n,

B

figure 10-11

Sec. 10-4

~mll!1e-lnlm

interference with light incident at

Interference in Dielectric films

l!rhilrnlt"1i

angle 6,.

213

where nf and no are the refractive indices of film and external medium, as shown. It is helpful to break the distances AB and BC into parts, in

a=

[n,(AE

+

+

+

(10-27)

The quantity in square brackets vanishes, as we now show. By Snell's law,

no sin 8;

nf

sin 6,

(10-28)

(A~)sin 8,

(10-29)

In addition, by inspection, AE = AG sin 0, =

and AC sin 8i

AD

From

(10-30) (10-30) and (10-28),

00-29) and incorporating, in turn, 2AE = AC sin 6,

AD

(s~

( 1

)

sm 6i

AD

(~)

so that (10-31)

noAD

which was to be proved. There remains then, from Eq. (10-27),

a

= n,(EB

+

(10-32)

The length EB is related to the film thickness t by EB = t cos 8" so we have, finally,

a

2n,t cos 0,

(10-33)

The path difference a is economically by (10-33) in terms of the of not the of incidence, which of course can be recovered through Snell's law, (10-28). Notice that for normal incidence, 6; 8, = 0 and a 2nJt, as expected. The corresponding phase difference is l) k a = (2'lT /..\0) a. The net phase difference must also take into account possible phase differences that arise on reflection, as discussed previously. Nevertheless, if we call ap the optical path dlfference given by Eq. (10-33) and the equivalent path difference arising from phase change on reflection, we can state quite generally the conditions for constructive interference:

ap + ar

= rnA

destructive interference:

a + aT

(m

(10-34)

and p

+

where m = 0, I, 2, .... If, for example, constructive interference results between the two sing~e beam incident at angle 6;, the same condition will for all bel;lms ''''.d'''''''''' at tHe same angle. This is possible if the source is an extended source, in Figure 10_1'2. Independent point sources S" S2, and SJ are shown, all contributing to the intensity of the light at P. Since these sources are noncoherent, interference is sustained only between pairs of reflected rays from the same source. If the lens aperture becomes too small to admit two such beams, such as (a) and (b) from SI, no interference is detected. This may happen, for example, if the film thickness and, therefore, the spatial separation of two interfering beams-such as (a) and 214

Chap. 10

Interference of light

Figure 10-12 Interference by a dielectric film with an extended source. or inclination are focused by a lens.

while the pupil of viewing the reflected light is limited these virtual fringes do not They are in size. Without a focusing called locaJizedfringes because they are, so to speak, localized at infinity. Recall that nonJocalizedJringes (Figure 10-5) are, in contrast, formed everywhere. Fringes 10-12 are also to as Haidinger or fringes of formed as in equal inclination, since they are formed parallel incident beams an extended parallel rays from the various source source. If a different inclination is point'> are incident on the film at a different angle, reflect as parallel rays from the film at a and an focus at some other point where accon,ClUlons expres,<;ed (10-34) and (10-35). of equal inclination just are not possible if source is a point or is very small, since every ray of light from the source to the film must, in that case, arrive at a different angle of (Figure 10-13). of a different kind are nonetheless formed. Since rays are reflected to any point P from the virtual sources S, and ,this may be two film surfaces as if they originated at considered an instance of the two-point source pattern already discussed in connec10-5. Real, nonlocalized are formed in the space above the tion with is clearly visible on a screen film. If the source of light is a laser. the vicinity of the film. The condition for interference is placed anywhere in where the slit separation is the ...."'un."''' that of the two-source interference between virtual sources S 1 and Sz. In 10-13, S I and Sz are approximately by refraction in the film.

10-13 Interference by a dielectric film with a point source. Real, nonlocali:l.ed appear as in the two-point source Figure 10-5. Refraction has been

10-5 FRINGES

EQUAL THICKNESS thickness t, the optical path difference a 2 nft cos 6, varies even without variation in the angle of if the direction of incident light is say at normal or dark fringe will be associ-

If the film is of

Sec. 10-5

Fringes of Equal Thickness

215

ated with a particular thickness for which .:l satisfies the condition for constructive or destructive interference, respectively. For this reason, fringes produced a varifor able-thickness film are called fringes of equal thickness. A typical these is shown in Figure 1O-14a. An extended source is used in conjunction with a beam splitter set at an angle of 45° to the incident The beam splitter in position enables light to strike the film at normal incidence, while at

Source

--=I==:::JFilm Ib}

lal

Figure 10-14 Interference from a shaped film, producing localized equal thickness. (a) Viewing asse:mbly. wedge formed with two microscope slides.

the same time providing for the transmission of part of the reflected light into the detector (eye), often called Fizeau fringes, are seen localized at the film, I and from which the rays diverge. normal incidence, cos (J, .:l 2t7fl. Thus the condition for and dark fringes, (10-34) and (10-35), is

+

mA. { (m + !)A,

bright dark

(10-36)

.:lr is either or 0, depending on there is or is not a relative phase shift of 1T between the rays reflected from the top and bottom surfaces of the film. One way of forming a suitable wedge for experimentation is to use two dean, glass microscope slides, wedged at one end by a thin spacer, perhaps a as in 10-14b. resulting air layer between the slides shows fringes when are illuminated by monochromatic For this the two reflections are from to air (internal reflection) and from air to (external reflection), so that.:l, in Eq. (10-36) is A/2. As t increases in a linear fashion along the length of the slides from I = 0 to t = d, (10-36) is satisfied for consecutive orders of m, and a series of alternating bright and dark will be seen by reflected light. These are virtual, fringes and cannot be pre'lectecl onto a screen. and white light is incident at If the extended source of Figure 10-14a is the some angle on a film of variable thickness, as in Figure 10-15, the film may appear in a of colors, like an oil slick after a rain. Suppose that in a small region of the film the thickness is such as to produce constructive interference for wavelengths

Figure 10-15 Imerference an film illuminated by an extended source. Variations in film thickness, as well as angle of incidence, determine the wavelength region reinforced by interference.

216

Chap. 10

Interference of Light

in the red portion of the spectrum at some order m. If the wavelengths at which constructive interference occurs again for orders m + I and m - 1 are outside the This can occur for low orders ble spectrum, the reflected light appears and therefore for thin films.

10-6 NEWTON'S RINGS Since Fizeau fringes are of equal their contours directly reveal in the thickness of the film. 1O-16a shows how this cirany cumstance can be put to practical use in determining the quality of the surface of a lens. for example. in an arrangement in which the Fizeau fringes have come formed between the spherical surto be referred to as Newton's rings. An air fuce and an optically flat surface, is illuminated with normally incident monochromatic light, such as a sodium lamp or a mercury with a filter, to isolate one of its spectral lines. Equal-thickness contours for a perfectly spherical surfuce. and therefore the viewed, are concentric circles around the point of contact with 0 and the path between reflected rays is the optical flat. At that point, I as a result of The center of the fringe pattern thus appears and (10-36) gives m 0 for the order of destructive interference. Irregularities in the surfdce of the lens show up as distortions in the concentric ring pattern. This can also be used as an optical means of measuring the of curvarelation exists between the rm of the ture of the lens surfuce. A air...film 1m, and the curmth-order dark the IO-16b and making vature R of the air film or the lens surface. Referring to use of the Pythagorean theorem, we have R2 =

r,! +

(R

110)2

or

/"

.--

---

/

" "-

/

~~~>' Viewing ... microscope

;'

" \. , \

I

\

I

\ I I

I

Beam splitter

I I \

I

I

\

f

Lens

Optical flat la)

Ibl

}'igure 10·16 (a) Newton's apparatus. Interference fringe.,> of equal thickness are by the air wedge between lens and flat. (b) Es.'>Cnlial geomelry for of Newton's rings.

Sec, 10-6

Newton's

211

(al

Ibl

Figure 10·17 Newlon's in (a) rellected lighl and (b) transmilled light are complementary. (from M. M. FrancOI1, and J. C. Thrierr. Atlas of Optical Phenomenon, Plate 9. Berlil1: Spril1ger·Verlag, 1962.)

The radius of the mth dark ring is measured and the corresponding thickness of air is determined from the interference condition of . (10-36). Thus R can be found. A little thought should convince one that light transmitted through the optical Hat will also show circular interference fringes. As shown in Figure 10-17, the pattern differs in two important respects from the reflected light. the show poor contrast, because the two transmitted beams with largest amplitudes have quite different values and result in incomplete cancellation. Second, the center the fringe pattern is bright rather than dark, and the entire fringe system is complementary to the system by reHection. Example A plano-convex lens (n = 1.523) of ~ diopter power is placed, convex surface down, on an optically flat surface. Using a traveling microscope and sodium (A 589.3 nm), interference fringes are observed. Determine the radii of the first and tenth dark Solution In this case, ll., = A/2, so that . 00-36) leads to an air-film thickness at the mth dark ring given by 1m mA/2nf. Since the film is air, n, = I and 1m = mA/2. The ring radii are given by Eq. (10-37). On neglecting the very small term in t~, this is r ~ 2Rtm • The radius of curvature of the convex surface of the lens is found from the lensmaker's equation: I - = (n

f

With f

= 8 m,

n "'" 1.523, and R2 =

r~

2Rtm = 2R(

-? 00,

m;)

(1)(4.184)(589.3

mRA X

10

9

)

= (10)(4.184)(589.3 x 10- 9 )

or rl 218

Chap. 10

1.57 mm and rIO = 4.97 mm. Interference of Ught

4. t 84 m. Then

this gives R

2.466

=

X

10-6

24.66 x 10- 6 m'

It is ironic that the phenomenon we been tip.:l'f'ihi ..... mately the wave nature of light, should be known as Newton's after one who championed the corpuscular theory of light. Probably the measurement of the wavelength of light was made Newton, using this technique. Consistent with his corpuscular theory, however, Newton interpreted this quantity as a measurement of distance between the "easy fits of reflection" of corpuscles.

10-7 RI..M-THICKNESS MEASUREMENT BY INTERFERENCE of equal provide a optical means for measuring thin A sketch of one possible is shown in Figure Suppose the film F to be measured has a thickness d. The film has been deposited on some substrate S. Monochromatic light is channeled from a light source LS through a fiberwhich transmits one optics light pipe LP to a right-angle to reflection, each is beam to a flat mirror M and the transmitted the beam splitter into a microscope MS, where they are allowed to interfere. Equivalently, the beam reflected from the mirror M can be considered to arise from its virtual image M'. The virtual mirror M' is constructed by imaging M through the beam-splitter reflecting plane. This construction makes it clear that the interference pattern results from interference due to the air film between the at M' and the F. In practice, mirror M can be moved toward or away from the beam splitter to equalize optical path lengths and can be tilted to make M' more or less parallel to the film surface. Furthermore, beam splitter and mirror assembly form one unit that can be attached to the microscope in place of its lens. When M' and the film surface are not parallel, the usual the microscope, which has been Fizeau fringes due to a wedge will be seen prefocused on the film. The light beam striking the film is allowed to cover the of the film so that two fringe systems are seen side by side, corresponding to air Figure 1O-19a shows a films that differ by the required thickness at their typical photograph of fringe made through a The transla-

M

s Sec. 10-7

d

Figure 10-18 Film-thickness measurement. Interference fringes produced by light reflected from the film surface and substrate allow a de· termination of the film thickness d.

Film-Thickness Measurement by Interference

219

181

----------~,~--------

10-19 (a) Photograph of inlerference produced by the arrangement shown In H1-I8. The lroughlike evident in the interference pattern was made by ~.g'I"V'''''''5 the film over a thin, straight wire. of one side of the trough shown in The pattern shifts by an al Ihe film {photo by

(bl

a means of determining d, as ( 10-34),

tion of one fringe relative to follows. For normal incidence, bright

where t represents the thickness of ness now changes by an amount t::.l and we have

= d,

the

point. If the air-film thickof interference m ac-

2n t::.t where we have set n for an air ample, the order of any lates by one whole fringe. For a shift of the by t::.m

the thickness t by for exI, that the fringe pattern transof magnitude t::.x (Figure 1O-19b)

d= be measured with a stable microSince both spacing x and fringe shift t::.x scope-or ITom a photograph like that of 1O-19-the film thickness d is determined. When using monochromatic nel shift of fringe is ambiguous because a shift t::.x = for will look exactly like a shift t::.x 1.5x. This may be two ways. If the shift is more than one fringe width, this situation is viewing white-light formed in the same way. The that form the white-light fringes creates a pattern whose center at = 0 is unique, as an unambiguous index of fringe location. The shift of fringe is then seen and can be combined with the monochromatic mC.L"urement of t::.x described previ220

10

Interference of light

ously. A second method is to prepare the film so that its edge is not sharp but tails fringe of one set can be followed down the film edge off gradually. In this case into the corresponding of the second set, as in 10-19. If the film cannot be provided a gradually tailing edge, a film of silver, example, can step in metal film will usube over both the film and substrate. be somewhat but total step will be same as the thickness of the film to be measured. A one-to-one correspondence between individual fringes of be made visually. each set can

to-I. Two beams

parallel electric fields are described by

EI = 3 sin (kl .r

WI

+ ~)

(k2 . r

WI

+ ~)

E2

4 sin

with amplitudes in kV/m. The beams interfere at a where the phase difference due to path is TI" /3 (the first beam having the longer path). At the point of superposicalculate (a) the irradiances II and h of the individual beams; (b) the irradiance 112 due to their (c) the net (d) the fringe visibility. to-2. Two harmonic waves with amplitudes of 1.6 and 2.8 interfere at some point on a screen. What contrast or visibility results there if their electric field vectors are parallel and if they are perpendicular? to-3. The ratio of the amplitudes of two beams forming an interference fringe is . What is the fringe contrast? What ratio of amplitudes produces a fringe contrast ofO.5? 10-4. (a) Show that if one beam of a two-beam interference setup has an irradiance of N times that of the other beam, the visibility is by

2VN V=N+l

(b) Determine the beam irradiance ratios for visibilities of 0.96, 0.9, 0.8, and 0.5. 10-5. A mercury source of light is positioned behind a glass filter, which allows transmission of the 546.I-nm green from the source. The light is allowed to pass a narrow, horizontal slit positioned I rum above a flal mirror surfuce. Describe qualitatively and quantitatively what appears on a screen I m away from the slit. light that consists of two wavelengths. One wavelength is 10-6. Two slits are illuminated known to be 436 nm. On a screen, the fourth minimum of the 436-nm light coincides with the third maximum of the other light. What is the wavelength of the unknown light? 10-7. In a Young's experiment, narrow double slits 0.2 mm apart diffract monochromatic light onto a screen 1.5 m away. The distance between the fifth minima on either side of the zeroth-order maximum is measured to be 34.73 mm. Determine the wavelength of the light. 100S. A quasi-monochromatic beam of light illuminates Young's double-slit setup, generala fringe pattern a 5.6-mm separation between consecutive dad: bands. The distance between the plane containing the apertures and the plane of observation is 10m, and the two slits are separated 1.0 mm. Sketch the experimental arrangement. is an initial single slit What is the wavelength of the light?

Chap. 10

Problems

221

10-9. In an interference experiment of the Young type, the distance between slits is 0.5 mm, and the wavelength of the light is 600 nm. (a) If it is desired to have a fringe spacing of 1 mm at the screen, what is the proper screen distance? (b) If a thin plate of glass (n = 1.50) of thickness 100 microns is placed over one of the slits. what is the lateral fringe displacement at the screen? (e) What path difference corresponds to a shift in the fringe pattern from a peak maximum to the (same) peak half-maximum? 10-10. White light (400 to 700 nm) is used to illuminate a double slit with a spacing of 1.25 mm. An interference pattern falls on a screen 1.5 m away. A pinhole in the screen allows some light to enter a spectrograph of high resolution. If the pinhole in the screen is 3 mm from the central white fringe. where would one expect dark lines to show up in the spectrum of the pinhole source? 10-11. Sodium light (589.3 nm) from a narrow slit illuminates a Fresnel biprism made of glass of index 1.50. The biprism is twice as far from a screen on which fringes are observed as it is from the slit. The fringes are observed to be separated by 0.03 cm. What is the biprism angle? 10-12. The small angle between two-plane. adjacent reflecting surfaces is determined by examinmg the interference fringes produced in a l'resnel mirror experiment. A source slit is parallel to the intersection between the mirrors and 50 cm away. The screen is I m from the same intersection, measured along the normal to the screen. When illuminated with sodium light (589.3 nm), fringes appear on the screen with a spacing of 0.5 mm. What is the angle? 10-13. The prism angle of a very thin prism is measured by observing interference fringes as in the Fresnel biprism technique. The distances from slit to prism and from prism to eye are in the ratio of I : 4. Twenty dark fringes are found to span a distance of 0.5 cm when green mercury light is used. If the refractive index of the prism is 1.50, determine the prism angle. 10-14. Light of continuously Ydfiable wavelength illuminates normally a thin oil (index of 1.30) film on a glass surface. Extinction of the reflected light is observed to occur at wavelengths of 525 and 675 nm in the visible spectrum. Determine the thickness of the oil film and the orders of the interference. 10-15. A thin film of MgF2 (11 = 1.38) is deposited on glass so that it is antireflecting at a wavelength of 580 nm under normal incidence. What wdvelength is minimally rellected when the light is incident instead at 45°? 10-16. A nonreflecting, single layer of a lens coating is to be deposited on a lens of refractive index 11 = 1.78. Determine the refractive index of a coating material and the thickness required to produce zero reflection for light of wavelength 550 nm. 10-17. Remember that the energy of a light beam is proportional to the square of its amplitude. (a) determine the percentage of light energy reflected in air from a single surface separating a material of index 1.40 for light of A = 500 nm. (b) When deposited on glass of index 1.60, how thick should a film of this material be in order to reduce the reflected energy by destructive interference? (e) What is then the effective percent reflection from the film layer? 10-1S. A soap film is formed using a rectangular wire frame and held in a vertical plane. When illuminated normally by laser light at 632.8 nm, one sees a series of localized interference fringes that measure 15 per cm. Explain their formation. 10-]9. A beam of white light (a continuous spectrum from 400 to 700 nm, let us say) is incident at an angle of 45° on two parallel glass plates separated by an air film 0.001 cm thick. The reflected light is admincd into a prism spectroscope. How many dark "lines" are seen across the entire spectrum? 10-20. Two microscope slides are placed together but held apart at one end by a thin piece of lin foil. Under sodium light (589 nm) normally incident on the air film formed be-

222

Chap. 10

Interference of light

10-22.

10-23.

10-24.

10-25.

10-26.

tween the one observes 40 fringes from the edges in contact to of the tin foil. Determine the thicknes.<; of the foil. of are in contact one side and held apart by a wire 0.05 mm in parallel to the edge in contact and 20 cm distant. filtered green mercury light, directed normally on the air film between plates. interference fringes are seen. Calculate the of the dark fringes. How many dark appear between the and the wire? Show that the separation of the virtual sources producing interference from a film of index n and uniform thickness t, when illuminated a point source, is 21/n. Assume the film is in air and light is incident at near-normal incidence. Newton's rings are formed between a spherical lens surface and an optieal flat. If the tenth ring of green (546.1 is 7.89 mm in what is the radius of curvature of the lens surface? Newton's are viewed both with the space between lens and optical flat empty and filled with a liquid. Show that the ratio of the radii observed for a particular order is very nearly the square root of the refractive index. A Newton's ring is illuminaled light with two wavelength components. One of the wavelengths is 546 nm. If the eleventh bright ring of the 546-nm fringe system coincides with the tenth ring of the other, what is the second wavelength? What is the radius at which takes and the thickness of the air film there? The spherical surface has a radius of I m. A pattern found using an interference ITI;,r,.",,,,.,,,np objective is observed to have a spacing of I mm. At a eertain point in the the are observed to shift laterally by 3.4 mm. If the illumination is green of 546.1 nm, what is the dimension of the "step" in the film that caused the shift?

and Gerald Pincus. "Optical Interference Coatings." .scienl'jf!cAmeri[ I] Baumeister, 58. can (Dec. [2] Fincham, W. H. A., and M. H. Freeman. 9th ed. London: Butterworths, 1980. Ch.14. Miles V. New York: John Wiley and 1970. Ch. 5. [41 Hecht, and Alfred Zajac. Reading, Mass.: Addison-Wesley Publishing Company, 1974. Ch. 9. Ghatak, K. An Introduction 10 Modern New York: McGraw-Hili Book Company, 1972. Ch. 4. [6] Richard Robert B. Leighton, and Matthew Sands. The Feynman Lectures 1975. Ch. vol. I. Reading, Mass.: Addison-Wesley Publishing

13]

[7] Longhurst. R. S. Geometrical and Physical Optics, 2d ed. New York: John Wiley and Sons, 1967. Ch. 7, 8.

10

References

223

11

s

Optical Interferometry

INTRODUCTION An instrument to exploit the interference of and the fringe n<>tlt#"rr,., that from optical path differences, in any of a V'driety of ways, is called an opdescription of instrument should reflect the tical interferometer. This wide variety designs and uses of interferometers. Applications extend also to acoustic and radio waves, but here we are interested in the optical interferometer. In we discuss chiefly the Michelson and the Fabry-Perot interferometers this of many and only a To achieve interference between two coherent beams of an interferometer divides an initial beam into two or more parts that tmvel diverse optical paths and then reunite to produce an interference pattern. One criterion for broadly classiinitial beam is sepafying interferometers distinguishes the manner in which portions of the same wavefront of a rated. Wavefront division interferometers coherent beam of light. as in the case of Young's double slit, or adaptations like using Uoyd's mirror or Fresnel's biprism. Amplitude-division interferometers use some type of beam splitter that divides the initial beam into two parts. The Michelson interferometer is of this type. Usually the beam splitting is managed a semireflecting or dielectric film; it can also occur by frustmted total ina or by means of douternal reflection at the interface of two prisms ble refraction or Another means of classification distinguishes between

224

use of the interference of two as in the case and that with U"'''''I'''''' beams, as in Fabry-Perot interferometer. 11-1 THE MICHEl.SON INTERFEROMETER

Michelson interferometer, first introduced by Albert MICht:lscm a vital role in the development modern physics. waS used, for example, to experimental the validity of the special theory of relativity, to detect and measure hyperfine structure in line to meaSure the tidal the moon on the earth, and to a substistand.ard for the meter in terms of of light. himself piomuch of this work. l1-1a. From A of the Michelson interferometer is shown in an extended source of light S, a beam 1 of light is split by beam (8S) by means a semitransparent surface metallic or film, deposited on The interferometer is of the amplitude-splitting type. Reflected beam 2 and transmitted beam 3, of roughly equal amplitudes, continue to fully .....t1",,,'h ...,o mirrors M 2 and M 1, where their directions are reversed. On to the beam splitter, beam 2 is now transmitted and 3 is reflected the film so that come again and leave ter as beam 4. The useful aperture of this double-beam interferometer is such that all rays M 1 and M2 will be or so. Thus beam 4 includes rays that traveled different and will demonstrate interference. At least one of mirrors is equipped with tilting screws that allow surface of M 1 to be made perpendicular to that M 2. One of the mirrors is movable

/"

, / / , )c

,

s (1)

----+---\-''r__ s'

M1 (41

(al

(bl

Figure 11-1 (a) The Michelson interferometer. (b) Equivalent optics for the Michelson interferometer.

Sec. 11-1

The Michelson Interferometer

225

along of the beam means of an accurate track and mI4::;romele£ screw. In this way the difference between the optical paths of beams 2 and 3 can be gradually varied. Notice that beam 3 traverses the beam splitter three whereas beam 2 traverses it only once. In some applications, where white light is used, it is that the paths of the two beams be made equal. Although this can be at one by appropriately increasing the distance of M 2 from the correction would not suffice at wavelength because of the dispersion of the glass. To compensate for all wavelengths at once, a plate C made of the Same material and dimensions as BS is inserted parallel to BS in the path of beam 2. Any in optical paths can be removed allowing the the optical through the thickness of its glass plate. The actual interferometer in ll-la possesses two optical axes at angles to one another. A simpler but equivalent optical having a of source S and mirror cal can be displayed working with virtual M I via reflection in the BS mirror. These positions are most simply found by regarding the assembly including S, M 1, and I and 3 of 11-1a as rotated counterclockwise 90° about the point of intersection of the beams with the BS mirror. geometry is shown in Figure II-lb. The new position of source plane is S', and the new position of the mirror M I is M I'. Light from a point Q on the source plane S I then effectively reflects from both mirrors M 2 and M I', shown parallel and with an optical path difference of d. The two reflected of object point Q. beams appear to come from the two virtual images, Q: and Since the S I' and S2' of the source plane in the mirrors must be separated by twice the mirror separation, the distance between Q: and is 2d, and the optical path between the two beams emerging from the interferometer is

t:.p = 2d cos

(1

(11-0

where the angle (J measures the inclination of the beams relative to the optical axis. 0 and t:.p = 2d. We expect this result, since, if one mirror For a normal beam, (J is further from than the other by a d, the extra traversed beam longer route includes distance d twice, once before and once after reflection. in addition, t:. mA, so that the two beams interfere it follows that they will do so repeatedly for every A/2 translation of one of the mirrors. The optical system of Figure II-Ibis now equivalent to the case of interference due to a parallel air film, illuminated by an source. Virtual fringes of inclination be seen looking into beam splitter ray 4, with eye or a telescope at infinity. that the two int,p'rh'rirlO beams are of equal amplitude, the irradiance of the fringe system of circles concentric with the optical axis is given, as in (10-16), by 1 = 410

where

(~)

(11-2)

phase difference is (11-3)

The net optical difference is t:. + t:." as usual. A 1T phase between the two beams occurs because beam 2 experiences two external reflections

226

11

Optical Interferometry

but beam 3 PYr"""'IP"',""''':' only one,' For dark fringes, then,

A

2d cos fJ + 2

or, more simply, 2d cos

e

mA,

dark fringes

0,1,2",.

m

the center of the fringe ~)'s­ is dark, then its order, given by

If d is of such magnitude that normal rays tern satisfy Eq. (11-4), that is, the center

2d A

mmax

(11-4)

(11-5)

is a large integer. Neighboring dark in outwards from the center of the pattern, as cos fJ clp,"t"p"'~''':' from its maximum value of 1. This ordering of fringes may be inverted for another integer p with each fringe of order m, p

m

mmax

01-6) to replace m in

pA

2d A

(11-6)

m

we arrive at

= 2d(1 - cos fJ),

0, I,

p

dark fringes

(11-7)

zero and the neighboring fringes increase in where now the central fringe is of order, outward from the center. 11-2 illustrates the relationship between orders m and p for the arbitrary case where mmax 100. Equation (11-4) or (11-7) indicates that, as d is varied, a particular point in the pattern (fJ = constant) will correspond to gradually changing values of m or p. Integral values occur whenever the point coincides with a dark this means that as d is varied, fringes of the pattern appear to shrink toward the center, where they disappear, or else expand outward from where seem to originate, depending on whether the optical path difference is or increasing. The motion of the fringe pattern thus reverses as one of the mirrors is moved continually way, 01-4) rethrough the point of zero path difference. Viewed in m

99

m=98 2

p

m

=

91

p = 3

m = mm .. = 100 p

= mm".

- m =0

Figure 11·2

Alternate nrrl,erinl~~ of

1 This conclusion assumes dielectric coatings of index less than that of The is made to allow discussion of the fringe pattern in a concrete (and common) situation. It does not affect the validity of results like Eq. (11-8), for example, because measurements on the net motion of the pattern, nol on precisely where it is dark and where it is

Sec. 11-1

The Michelson Interferometer

227

quires an increase in the angular separation t:J:) of a the mirror spacing d becomes since

small fringe interval Am as

Allm 2d sin (J This means that the are more widely separated optical path are small. In fact, if d = 11./2, then from . (1 J m cos (J, and the entire field of view encompasses no more than one fringe! For a mirror translation the number Ilm of fringes passing a point at or near the center of pattern is, according to Eq. (11-4),

Ilm

=

2

(11-8)

A

an experimental way of either A when Ild is the micrometer translation screw when A is known.

"' ...15,15""·...,.

known or Example

are observed due to monochromatic light in a Michelson ter. the movable mirror is translated by 0.73 mm, a shift of 300 fringes is observed. What is the of the What displacement of the of index 1.51 and 0.005 mm fringe system takes when a flake of thickness is placed in one arm of the (Assume that the beam is normal to the Solution

Using

2 Ild Ilm

A

the of Ild

=

( 11-8), 300

4.87 x

mm

487 nm

inserted, one arm is effectively lengthened by a path difference - na"t, so that

Ilm or

=

fringes.

f f-2 APPLICATIONS OF THE MICHELSON INTERFEROMETER The Michelson interferometer is easily adaptable to the measurement of thin films, a technique essentially the same as that in the chapter. It is also easily adaptable to the of the index of refraction of a gas. An evacuable cell with plane, parallel windows is interposed in the path of beam Figure n -la, and is filled with a gas at a pressure and temperature for which its index of refraction is desired. The system established under these conditions is monitored as the gas is pumped out of the A count of the net fringe shift is related to the change in optical path during the replacement of the gas by vacuum. If the actual length of the cell is accurately known to be the change in optical path is given by

Ild

228

Chap. 11

nL - L

Optical Interferometry

L (n - 1)

01-9)

and using

(11-8), it follows that the index can be determined from

n- 1

(11-10)

Consider another direct application of the Michelson interferometer, the determination of wavelength difference between two closely spaced components of a "1A""'u,u, "line," A and A'. Each wavelength forms its own system of fringes according to . (11-4). Suppose we view circular systems near their center, so that cos 0 1. Then for a path difference d of the interferometer, the product mA is that mA = m' A'. When the the pattern appears sharp, whereas when the fringes of one system in the of observation lie midway between the fringes of the second system, the pattern appears rather uniform in brightness, or "washed out." The mirror movement M required between consecutive coincidences is to the wdvelength IlA as follows. At one coinwhen are "in " the of the two systems corresponding to A and A' must be related by

m

m' +N

where N is an integer. If the optical path difference at this time is d 1 , then from (l1-4),

(1l-11) Let the optical path difference be increased to found. Then

when the next coincidence is

m = m' + (N + I) or

2d2 A

subtracting . (11-11) from dz - dl, we find

A'

+N +

(11-12) and by

A' - A =

"/r.t •.""

the mirror movement

01-13)

)..At

2ad

Now since A and A ' are very close, the components can be approximated by IlA

(11-12)

1

difference of the two unresolved

=

2M

(1l-14)

This technique is often employed in an optics laboratory course to measure the wavelength difference of 6 A between the two components of the yellow "line" of sodium. All the discussion of the from a Michelson interferometer has been in terms of virtual of equal inclination. We have assumed that mirrors M 1 and M2 are precisely perpendicular, or what amounts to the same thing, precisely parallel in the equivalent optical system of Figure II-lb. If the alignment is such that the air space between M l' and M 2 in Figure 11-1 b is a of equal thickness may be seen localized at the mirrors. These

Sec. 11-2

Applications of the Michelson Interferometer

229

oriented

to the line that intersection of M l' and M 2. they will be in a way thai can be shown to be fringes appear hyperbolic arcs. Again, if the source is small, then real, in the light emerging from the interferometer, if formed by the two virtual imappear without effort when the inages of the source in M I' and M 2. These have already distense, coherent light of a laser is used. These chapter, where we treated the fringes that cussed in the illumination of a film. 1 a photograph showing the can be formed of equal thickness a candle flame when situated in distortion of one arm of a Michelson interferometer. produce variations in optical length by the index of the air.

If the wedge is of large

Deformation of fringes of equal thickness in the of a candle flame. (From M. M. Francon, and J. C. Thoerr, Phenomenon, Plate 12, 1962.)

11-3 VARIA nONS OF

MICHELSON

'NJrEl1rF£ROME:~1"E,R

Although there are many ways in which a beam of may be split into two and reunited after traversing diverse paths, briefly two variations that A slight modification can be considered adaptations of the Michelson by Twyman and Green is shown in 11-4a. of using an extended a lens L I, source, this interferometer uses a point source so that all rays enter the interferometer parallel to the optical axis, or cos () = 1. The parallel rays emerging from the interferometer to a focus by lens L 2 at where the eye is placed. The circular of inclination no longer appear; in their place are seen of equal reveal imperlength. When no fections in the optical system that cause variations in the uniform illudistortions appear in the plane wavefronts are of high quality, this mination is seen near P. If the interferometer can be used to test the optical quality of another component, such as 230

Chap. 11

Optical Interferometry

'-r--""""" M2 '-r-_..,.......M2 Ml

L1 BS

s s

(bl

la)

fIgure H-4 (a) IWvtnlm-,jl interferometer. (b) Twyman-Green interferometer used in the of a prism and a lens (inset).

a situated as shown in Figure 11-4b. Any SUrfdce imperfection or internal variation in refractive index shows up as a distortion of the fringe Lenses mirror M 1 is a are for aberrations in the same way, once convex spherical surface that can reflect the refracted rays back along as sugges~ted in the inset of 11-4b. A more radical sketched in 11 is the Mach-Zehnder ferometer. The beam of roughly parallel light is divided into two beams at beam splitter BS. Each beam is totally reflected by mirrors M 1 and M2, and the beams are made again by the semitransparent mirror M3. Path of beams 1 and 2 around the rectangular and through the of the beam splitters are identical. This interferometer has used, for in ",,,,,·nrt,,n,,rn,,.. research, where geometry of air flow around an object in a wind tunnel is revealed through local variations of pressure and refractive index. A wintest chamber, into which the model and a flow of air is introis placed in 1. An identical chamber is in path 2 to of optical paths. The air-flow pattern is by the fringe pattern. For such applications the must be constructed on a rather large scale. An advantage of the Mach-Zehnder over the Michelson interferometer is that, by approsmall rotations of the the fringes may be made to appear at the so that both can be viewed or In the fringes appear on the mirror and so cannot be seen in focus at the same time as a test object placed in one of its arms. M3

Ml

:?f--------:;;?/t-=::::: (11

(1

+ 2)

(2)

BS

Sec. 11-3

(21

M2

Figure II-S

Variations of the Michelson Interferometer

Mach-Zehnder

inter~meter.

231

The Michelson, Twyman-Green, and Mach-Zehnder interferometers are all two-beam interference instruments that operate by division of amplitude. We turn instrument, the interfernow to an important case of a ometer. Before discussing the however. it will be necessary to examine the phenomenon of multiple reflections from a parallel tl"dIlsparent plate.

11-4 STOKES RELA nONS We with an argument due to Sir Stokes, which yields information conof reflected and portions of a wavefront incerning the cident on a refracting surface, as in 11-6a. Let Ei the amplitude of the incident light. We define reflection and transmission by

t

r

EI

(11-15)

so that at the E, is divided into a reflected part, Er rEI, and a transmitted part, as shown. For a ray incident from the second we define similar which we with notation, r' and t'. According to the ......;n";'"I,,, of ray reversibility. the situation shown in Figure 11-6b must also be 11-6b, valid. In however, two rays incident at the interface, as in in a reflected and a ray, all of which are shown, with approin Figure 11-6c. We conclude that the situations in Figure 11-6b and c must be physical1y so that we can write

=

E,

+t

and

o

(r't

+

or

(11 16)

ft'

r

-r/

(11-17)

Stokes relations between amplitude Equations (11 16) and ( 11-17) are coefficients for angles of incidence through Snell's law. (11-17) states that the amplitUdes of reflected beams for rays incident from either direction

n,

la)

(el

(b)

Stokes relations.

11-6 figures used in

2

of the

232

We will have occasion later to also use reflectance (R) and transmittance (T), defined as the ratio irradiances. Although R = r\ T t\ as explained in Section 20-4.

Chap. 11

*

Optical Interferometry

are the same in magnitude but differ by a 1T phase shift. This clearer if (11-17) is written in the equivalent form, r I. This result agrees with the preFresnel equalions, treated in Chapter 20. Both the dictions of the more establish the that the theory and such as Lloyd's on the interface from the side of higher velocshift occurs for the ray or lower index. This wave phenomenon has its analogy in the of waves from the fixed end of a rope. Both of the Stokes relations will be needed in the discussion that follows.

11-5 MULTIPLE-BEAM INTERFERENCE IN A PARALLEL PLATE

We return now to the problem reflections from a parallel plate, in a two-beam approximation. Consider the mUltiple reflections the narrow beam of light of amplitude and angle of OJ, as shown in Figure 11-7. The .."t1",...t,nn and transmission amplitude are rand t at an ",vt",,.,,,,, and r I and t I at an reflection. amplitude of each segment of the beam can be assigned by multiplying the previous the appropriate reflection or transmission coefficient, beginning with the incident wave of amplitude Eo and working progressively the train of reflections. Multiple parallel beams emerge from the top and from the bottom of the plate. Multiple-beam ;nt.",..b,..".~,...p takes when either set is focused to a point by a lens, as shown for the multiple beams are transmitted beam. Having originated from a single Further, if the incident beam is near normal, the beams are brought toparallel. with their E vibrations We consider the of the .."fl",...t",i beams from the top of the (10-33), the phase between successive reflected beams According to is given by (11-18) 111

(21

(31

(4)

(5)

(6)

11-7 Multiply reflected and transmitted beams in a

Sec. 11-5

MUltiple-Beam Interference in a Parallel Plate

233

Here nf is the refractive index of the and t is its thickness. If the incident ray is expressed as Eoe iw/. the successive ..",fi,,,,.,t.,.! rays can be expressed by appropriately modifying both the amplitude and of the initial wave. Referring to Figure 117, these are

= (It'r

(lI'r

(It'r and so on. A little inspection of wave can be written as

\O\.jL'''UVU''

shows that the Nth such reflected 01-19)

a form that holds good for all but E 1 , which never traverses the plate. When these may be written as waves are superposed, therefore, the resultant

ER =

2

EN = rEoe iw' +

tI

N~I

a bit, we have

ER = Eoeiwt[r

+ It'r

r

The summation is now in the form of a ge
2

=l+x+

+ ...

N=2

where

x

r

1x 1 < I, the series converges to the sum S

1/( 1

x). Thus

. ( r + ------, It'r ER =Eoe lWt 1 r HU",""',/'i

use next of the Stokes relations, Eqs. (11-16) and (11 (7),

After

of the resultant beam is proportional to the square of the ampliso we 1 12 , or which is itself

After prclCelssirlg the product of the bracketed terms and 2 cos 234

11

Interferometry

(j

== (e i6 + e-- iIl )

use of the identity,

there (11 or, in terms of irradiance, (11-21 ) where Ii represents the irradiance of the incident beam, and we have used the proportionality

=

01-22)

A similar treatment of the transmitted beams leads to

resultant transmitted

(11-23) Equation (11-23) also follows (llore directly by combining Eq. (11 with the relation II/ + IT Ii, by the conservation of energy for nonabsorbing A minimum in irradiance occurs, according to (11-21), when cos /) = I, or when (11-24) this must also be the condition for a transmission A study of 11-7. orthe """'I.,.....'v.,,-, the set of reflected shows that in the case of a reflection minimum, the with one another but second reflected beam and all subsequent beams are in with the first beam. Since net reflected exactly out of vanishes, is a perfect cancellation of the first beam with the sum of all the remaining beams. The two-beam approximation works well, then, if the amplitude of second beam is close to the amplitude of the first beam. Our show that their ratio is (1

= 1 -

which is

to unity when is small. For normal incidence on of index 0.04. Thus 96% of the occurs between the first two reflected beams alone, and the two-beam treatment is well justified. Reflection maxima occur, in the other extreme, when cos S - 1, or

n = 1.5, r2

/) =

(m

1T,

+

and

A In this case, Eqs. (1

2nfl cos 0,

= (m +

DA

(11-25)

and (11

II/

[(l ~:2)2}i

(I (11-27)

Sec. 11-5

Multiple-Beam Interference in a Parallel Plate

235

It is easily verified that IR + Ii. of condition for a transmission minimum occurs when cos 5 does give the maximum reflected intensity.

(t

shows that the . (11-26)

- t, so that

11-6 FABRY-PEROT INTERFEROMETER

Fabry-Perot interferometer makes use of the plane parallel to produce an interference pattern the multiple ted light. This instrument, probably the most adaptable of all been used, for example, in precision wdvelength measurements, analysis of hyperfine spectral line structure, determination of refractive indices of gases, and the calibration of the standard meter in terms of wavelengths. Although simple in structool in a ture, it is a high-resolution instrument that has proved to be a variety of applications. A typical arrangement is shown in Figure 11-8. Two thick glass or plates are used to enclose a plane parallel "plate" of air between them, which forms the medium within which the beams are multiply The important cn ..rn,.,.~c of the glass plates are therefore the inner ones. Their surfaces are generally polished to a flatness of better than and coated with a highly reflective layer of sHver or aluminum. Silver films are most useful in the visible region of the spectrum, but for appHcations below 400 their reflectivity off sharply around 400 nm, so nm, aluminum is usually used. Of course the films must be thin to be partially Optimum thicknesses for silver are around 50 nm. The outer surfaces of the glass plate are purposely formed at a small angle relative to the patinner faces (several minutes of arc are sufficient) to eliminate spurious terns that can arise from the glass as a parallel plate. The spacing, or thllcKIles:s, t of the air layer, is an important parameter of the i ... tprl-", .._ ometer, as we shall see. When the spacing is fixed. the instrument is often 1'"""1".'...."11 to as an etalon. \..u",..."",.:Jll1'15

--+-------3. p

8,

11-8

Klhrv-lt'en~I

interferometer.

Consider a narrow, monochromatic beam from an extended source point S making an angle (in air) of (J, with respect to the optical axis of the system, as in 11-8. The beam produces mUltiple coherent beams in the interferomeset of parallel rays are brought at a point P in the foter, and the cal plane of the L. The nature of the at P is detercos (J,. mined by the path difference between successive parallel beams, a Using nf = 1 for air, the condition for brightness is 21 cos

(J/

rnA

(11-28)

beams from different points of the source but in the same plane and making and also arrive at P. the same angle (J/ with the axis satisfy the same path

236

Chap. 11

Optical Interferometry

With t fixed, . (1] angles Of, and the fringe is the familiar concentric of of inclination. as shown in When a collimating lens is used between Figure 11-9a, every set of parallel beams the etalon must arise from the same source point. A one-to-one COitTCSpOnloellce then exists between source and 11-9b screen points. The screen may be the illustrates another arrangement, in small. Collimated light in this 0 is shown) and comes to a focus instance reaches the at a fixed at a light detector. As the spacing t is the detector records the interference If the SOl.lCe light consists of two pattern as a function of time in an the two is either a douwavelength components, for ble set of circular plot of resultant irradiance I versus the 11-9b.

fa)

fbI

Figure 11-9 (a) Fabry-Perut interf~Tometer. used with an extended source and a fixed plate A circular the one shown may be photographed al the screen. (Photo from Francon. and J. C. Thrierr, Atlas Phenomenon, Plate 10, Berlin: 1962.) (b) Perol interferometer, used wilh a poinl Variable plate A delector al the focal of the second lens to a ploHer 10 like the one shown, for a translation order of imerference.

Sec. 11-6

Fabry-Perot Interferometer

231

11-7 FRINGE PROFILES: THE AIRY FUNCTION

The variation of the irradiance in the pattern of the Fabry-Perot as a function of the phase or path is called fringe profile. The sharpness of the fringes is, of course, important to the ultimate resolving power of the instrument. The irradiance by the resultant of the transmitted beams has already been found in Eq. (11-23) and is repeated here:

-]1

h = [-1_(,--1___r-=..2)_2

+ r4

cos li

f

Using the trigonometric identity cos li == 1 - 2 and simplifying a

(%)

the transmittance T, or Airy function, can be

T

IT

(11

Ii

The square-bracketed which is a function of the ..",fiPl't,....n called the coefficient of finesse by Fabry:

was

F

(11-30)

Equation (11 known as be more compactly as

formula for the transmitted irradiance, can then

T

01-31)

The F is a sensitive function of reflection coefficient as r varies from 0 to 1, F varies from 0 to infinity. We show that F also represents a certain measure of fringe contrast, written as the ratio (II

From the Airy formula, Eq. (11-31), I/O + F) when sin li/2 = ± 1. Thus

1 when sin li/2

1 - 1/(1 + F) I/O + F) = F

0, and

=

(t 1-33)

The fringe profile may be plotted once a value of r is chosen. Such a plot, for several = 1 at li = m(2?T), choices of r, is given in Figure 11-10. For each curve, T = and T Tm1n 1/(1 + F) at li (m + D2?T. Notice that Tmax = 1 of r this V"d)ue as r t. The transmitand that Tmin is never zero but tance sharply at higher values of r as the approaches integral mUltiples of 2?T, remaining near zero for most of the region between fringes. As r increases even more to an attainable value of 0.97, for example, F increases to 1078 and the fringe widths are less than a third of their V"dlues at half-maximum for r = The of these fringes is to be compared with the broader cos 2 (li/2) dependence on the from a Michelson interferometer, which have a 11-10 by the dashed nor(Eq. 01-2». These are also shown in malized to the same maximum value. 238

Chap. 11

Optical Interferometry

1.0 0.9 0.8 c: 0

0.7

'fic:

.a

0.6

«:;;

0.5

I

f

0.4

\ \ \ \

I I

c:: fl

c:

·st::'" '"c:

'" ~

I I

, I

I

0.3

\ \

I I

I

0.2

I

\

I

\

I I

,

I I

0.1

I

I

I I

I

\

2(m

+ lhr

2(m

+ 2hr

2(m +311,.

Phase difference (rad)

Figure 11-10 Fabry-Perot fringe profile. A of transmittance or Airy function versus phase difference for selected values of reflection coefficient. Dashed lioes represent comparable fringes from a Michelson interferometer.

'1'1-8 RESOLVING POWER When two wavelength components are nr"'"p"t the Fabry-Perot interferometer gives a double set of circular oclon.gm'g to one of the wavelengths. A detector scanning across the particular order m of the int,p..t'~..""nl'''' a plot for two wavelength COITIJ)4Jne:nts Figure 11-11a. Although the two are shown c"r'''T!>tpl" two, which follows the dashed line between wavelengths are very close the are very two separate peaks in the separalion (aA)min that can be by depends on one's ability to detect the dip in the measured pattern between peaks. According to a resolution criterion due to Rayleigh, the dip caused by identical may not be less as it to the than about 20% of the maximum irradiance. Rayleigh's diffraction images with secondary maxima (Chap. 16). cannot be applied here. it will be sufficient for our purposes to approximate that criterion by that the crossover point be not more than half the maximum of either individual peak. The phase in terval 8e (Figure 11-11 b) for the • "".AU ""'"" and half-maximum values of T can found from . (11

. 8e 1 sm- = - 2 Vi Sec. 11-8

Resolving Power

239

T

T

(al

(bl

Figure l1-n (a) Scan of two wavelength components in a Fabry-Perot fringe pattern. (b) Application of resolution criterion. The half-width of the at half maximum to the interval15c •

01-34) The phase difference between

two fringe maxima is twice this (~O)min =

The corresponding minimum lows. The phase difference.

(11-35)

\Vavelength 01-18),

ff",,"pn,'p

"",,,nl,,'<>hl!<,,

o For small wavelength

4

Vi

41ft cos 0,

or

may be found as fol-

(±)

the magnitude of ~o is given by cos

el) ~ (±)

= (41ft

~~ e) ~A

Combining with Eq. (1 (~A)min =

A2 --=-----cos

e,

Since at the fringe maxima, (11 we may

more simply,

(11 Here A may be considered to be of the two wavelengths or their average, since are close in value. The resolving power rzJt is defined in general as

(I 240

Chap. 11

Optical Interferometry

When

01-38) is applied to the Rtbry-Perot interferometer, (11-39)

Large resolving powers are, of course, desirable. For the Rtbry-Perot, we see that values occur when the order is large, near the center of the fringe pattern, and coefficients finesse, which to high reflectance. Notice that to center, (11-36) requires that plate separation t be maximize m at as large as possible, giving

2t

(11-40)

A

A Rtbry-Perot interferometer has a I-cm spacing and a reflection coefficient of r = 0.95. For a wavelength around 500 nm, its maximum order of its coefficient of finesse, its minimum resolvable inand its resolving power.

Solution Using

(11-30) and Eqs. (l

mmax --

to 01-40), we find

2t

A-- --"-----:'- = 40,000 -~~--::-=380

5000 0.004 = 1.2 x l(t Good Rtbry-Perot may be expected to have resolving powers of a million or so, and values around 20 million have been used. This represents one to two orders of improvement over the performance of comparable prism and grating instruments. The high-resolution performance of a Rtbry-Perot instrument is ilpattern of the mercury green revealing lustrated by photograph of its fine

n-12

Figure Fabry-Perot obtained with the mercury green line, fine structure. (From M. Cagnet, M. Francon, and J. C. Thrierr, Atlas of Optical Pherwmenon, Plate 10, Berlin: Springer-Verlag, 1962.)

Sec. 11-8

Resolving Power

241

11-9 FREE SPECTRAL RANGE Individual sets of circular fringes, one set produced by each wavelength component, appear simultaneously in a Fabry-Perot interference pattern. Interpretation becomes complicated unless some means are found to limit the range of wavelengths analyzed by the interferometer. For example, consider two wavelengths AI and ,where A2 = AI + For small values of aA the two sets of circular fringes will be dose togemer in each of interference. As !:1A however, the separate. When the separation becomes to the distance between consecutive orders, confusion of orders results. Let us calculate the wavelength difference !:1A such the mth order of A2 falls on the (m + 1) of AI. This is called the free specSince product of m and A corresponds to tral range (fsr) of the the same t and (), (11-36), we can write mA 2

(m

+ l)A,

we find

(11-41) The fsr is the in AI necessary to shift its pattern by the distance consecutive orders. Incorporating Eq. (11-36), with cos (), = 1 near the center of the fringe pattern where resolution is best, we may also write

A2 (aA)fsr

To avoid should have

(11-42)

2t

fringes of one order with those of

aA<

next, then, one

2

Notice that a large order m is detrimental to a large fsr by Eq. (11-41), whereas it is favorable to good resolution by Eq. (11-37). For example, in the preceding high-resolution numerical example, fsr is only 0.125 One would like to maximize the quantity (11-43) The ratio by Eq. (1 thus represents a figure of merit for FabryPerot interferometer and is called, itsfinesse, ~, not to be confused with the CO(~ttil;.ielrltof finesse F, on which it finesse is usually as the ratio of separation of adjacent maxima to the half-width of the individual fringes, as illustrated in Figure 11-13. Let us demonstrate the equivalence of this definition and that of . (II ~43). The phase difference between is 2Tr. The phase width of a fringe at half its maximum irradiance is twice {)c, as calculated in Eq. (11-34): 4

Thus

2 242

Chap. 11

Optical Interferometry

(11-44)

1.0

0.5

----.-- ..... lll/2

21m + 1hr

21m

+ 2hr

2(m

+ 31lT

Phase difference (rOOI

ll-13 The finesse '!J# is the ratio of the ima to their individual width at half-maximum.

M:~'lUilll~I"

of adjacent fringe max-

tal

Red

Green

Violet

(bl

Figure 11-14 (a) Use of a etalon in tandem with a prism spectrograph. (b) Fringed spectral lines formed the system in (a).

Sec. 11-9

Free Spectral Range

243

We conclude that the largest finesse of the interferometer represents the best compromise between the demands of resolution and spectral range. The limitations of this may be overcome in severa] ways. One is to use two etalons in tandem, one of resolution and the of fsr. In this way, it turns out that one can combine both capabilities. Another solution is to follow the etalon with a spectrograph, as shown in 11-14a. Suppose that the source has several well-separated spectral each with its detailed structure. The patterns produced by etalon alone would consist of a confusing superposition of fringes due to each of the constituent wavelengths. If the slit of the spectrograph, opened rather wide, intercepts a wide band through the center of the circular fringe pattern, the prism will perform own spatial separation of wavelengths. Each wavelength interval of the source then appears as a broad image of the slit but with fringe patterns corresponding to each wavelength and its fine structure components, as in 11-14b.

11-1. When one mirror of a Michelson interferometer is translated by 0.0114 cm, 523 are observed to pass the cross hairs of the viewing telescope. Calculate the wavelength of the light. n-2. When looking into a Michelson interferometer illuminated by the 546.1-nm light of mercury, one sees a series of straight-line that number 12 per centimeter. Explain their occurrence. 11-3. A thin sheet of fluorite of index 1.434 is inserted nonnally into one beam of a Michelson interferometer. light of 589 nm, the fringe pattern is found to shift by 35 fringes. What is the thickness of the sheet? 11-4. Looking into a Michelson one sees a dark central disk surrounded by concentric bright and dark rings. One arm of the device is 2 cm than the other, and the wavelength of the light is 500 nm. Determine (a) the order of the central disc and (b) the order of the sixth dark ring from the center. H-S. A Michelson interferometer is used to measure the refractive index of a gas. The gas is allowed to flow into an evacuated cell of length L placed in one arm of the interferometer. The wavelength is A. (8) If N fringes are counted as the pressure in the cell from vacuum to atmospheric pressure, what is the index of refraction n in terms of N, A, and L? (b) How many fringes would be counted if the gas were carbon dioxide (n = 1.00045) for a 100cm cell length, sodium at 589 nm? H-6. A Michelson interferometer is used with red light of wavelength 632.8 run and is adfor a path difference of 20 /Lm. Determine the angular radius of the (a) first (smallest diameter) observed and (b) tenth observed. B-7. A polished surface is examined a Michelson interferometer with the polished surface replacing one of the mirrors. A fringe pattern characterizing the surface contour is observed using He-Ne light 632.8 nm. distortion over the surface is found to be less than ~ the separation at any What is the maximum depth of polishing defects on the surface? n...s. A laser beam from a I-mW He-Ne laser (632.8 nm) is directed onto a film with an incident of 45°. Assume a beam diameter of 1 mm and a film index of 1.414. Determine (a) the amplitude of the E-vector of the incident beam; (b) the angle of refraction of the laser beam into the film; (c) the magnitudes of r' and It', usthe Stokes relations and a reflection coefficient, r = (d) the independent amplitudes of the first three reflected beams and, by comparison with the incident

Chap. 11

Optical Interferometry

the of radiant power density reflected in each; (e) the same information as in for the first two transmitted beams; (f) the minimum thickness of film that would lead to total cancellation of the reflected beams when they are brought a lens. U -9. (8) Using show that amplitudes of the first three reflected and first three transmitted beams from a nonabsorbing glass (n = 1.52) plate. when the incident beam is near normal and of unit amplitude, are given by (I)

(2)

(3)

reflected

0.206

0.198

0.0084

transmitted

0.957

0.041

0.0017

(b) Show as a result that the first two reflected rays produce a fringe contrast or visi-

ll-IO.

11·11.

ll-12.

11-14. 11-15.

II-J6.

U-17.

bility of 0.999, whereas the first two transmitted rays produce a fringe contrast of only 0.085. The plates of a interferometer have a reflectance coefficient of r = 0.99. Calculate the minimwn (a) power and (b) plate separation that will accomplish the resolution of the two of the H-a1rila doublet of the hydrogen spectrwn, whose is 0.1360 at 6563 Po.. A Fabry-Perot interferometer is to be used to resolve the mode structure of a He-Ne laser operating at 632.8 nm. The separation between the modes is 150 MHz. The are by an air gap and have a reflectance (r2) of 0.999. (8) What is the coefficient of finesse of the instrument? (b) What is the resolution reQluined? (e) What plate (d) What is the free range of the instrwnent under these conditions? (e) What is the minimum resolvable interval under these conditions? A Fabry-Perot etalon is fashioned from a slab of transparent material having a high refractive index (n and a thickness of 2 em. The uncoated surfuces of the slab have a reflectance (r2) of 0.90. If the eta Ion is used in the vicinity of wavelength 546 nm, determine the in the interference pattern; (b) the ra(c) the power. tio "",,,<>r<>',,,,,, of a certain doublet is 0.0055 nm at a wavelength of 490 nm. A vari801le-!;oo'ceu ......",.,,_....,·....t interferometer is used to examine the doublet. At what spacing does the mth order of one component coincide with the (m + l)th order of the other? The r2 of a etalon is 60%. Determine the ratio of transmittance of the etalon at maximum to the trasmittance at between maxima. White is passed through a interferometer in the arrangement shown in Figure 11-%, where the detector is a A series of bright bands appear. When mercury light is admitted into the spectroscope slit, 150 of the bright bands are seen to fall between the violet and green lines of mercury at 435.8 nm and 546.1 run, What is the thickness of the etalon? Apply the used to calculate the finesse of a ffit)rv··P~rot the Michelson interferometer: Using the irradiance of Michelson as a function of rilase, calculate (a) the fringe separation; (b) the width at half-maximum; (c) their ratio, the finesse. Assume that in a Mach-Zehnder interferometer mirror M 3 each transmit 80% and reflect 20% of the incident fringe contrast when observing the interference of the two PTTlPN1ino with the fringe contrast that results from the two beams a direction at 90° relative to the first (not For the second case, beam (I) is reflected and beam (2) is transmitted at M 3.

Chap. 11

Problems

245

[1]

P. Optical Interferometry. Orlando, Fla.: Academic 1985. Pierre. "How Light Is Analyzed." Scientific American (Sept. 1968): 72. Smith, F. Dow. "How Images Are Formed." Scientific American (Sept. 1968): 96. Francon, Maurice. Optical Interferometry . New York: Academic Press, 1966. Tolansky, Samuel. An Introduction to Interferometry. New York: John Wiley and ]973. Hernandez, G. Fabry-Perot Interferometers. New York: Cambridge University 1986. J. F., and R. S. Sternberg. The of Optical Spectrometers. London: Chapman and Hall Ltd., 1969. Ch. 7. Robinson, Glen M., David M. Perry, and Richard W. Peterson. "Optical Interferometry of Surfaces." Scientific American (July 1991): 66. Reynolds, 0., John B. DeVe1is, B. Parrent, Jr., and Brian J. Th.om]pson. Physical Optics Notebook: Tutorials in Fourier Optics. Bellingham, Wash.: SPIE Optical 1989. Ch. 22-24. ,",-UIII ..<;;",

[3] [41

[5] [6]

[8]

[9]

246

Chap. 11

Optical Interferometry

12 AMI\~ ~

I

I

1"3

~~

---->.-+l. ..,.-/""':I-
mm_-1"-S................

Coherence

INTRODUCTION

The term coherence is used to describe the correlation between phases of monochrorelationships are, generally speaking, matic mdiations. Beams with random incoherent whereas beams with a constant phase relationship are coherent between interfering beams of light, if they are beams. The requirement of to produce observable fringe patterns, was mentioned in connection with the interrelationship beference mechanisms discussed in Chapter 10. We also discussed 9. There we tween coherence and the net irmdiance of interfering beams in concluded that in the superposition of in-phase coherent beams, amplitudes add whereas in the superposition of incoherent beams, individual irradiances add together. In this chapter, we examine the property of coherence in detail, distinguishing between longitudinal coherence, which is related to the "IJ'.-"'U';U purity of source, and laJeral or spatial which is related to a quantitative measure of partial "n""''',''''''f'''' the size of the source. We also the condition under which most measurements of interference take description of Fourier which we place. We begin our treatment with a will need in this chapter.

247

12-1 FOURIER ANALYSIS

When a number of harmonic waves of the same frequency are added together, even though they differ in amplitude and phase, the result is again a harmonic wave of the given frequency, as shown in Chapter 9. If the superposed waves differ in frequency as weB, the result is periodic but anharmonic and may assume an arbitrary shape, such as that shown in Figure 12-1. An infinite variety of shapes may be """',Tnp',,, in this way. The inverse process of of a given waveform into its harmonic components is called Fourier analysis. fIt)

12-1

Anharmonic function of time

with period T.

The successful decomposition of a waveform into a series of harmonic waves is insured by the theorem of Dirichlet: If 1(1) is a bounded function of period T with at most a finite number of maxima or minima or discontinuities in a period, then the Fourier 00+ 2

I(l)

am

cos mwl

+

bm sin

I)

mwl

converges to I(t) at all points where 1(1) is continuous and to the average of the right and left limits at each point where 1(1) is discontinuous.

In Eq. I), m takes on integral values and w where T is period of the arbitrary f(t). The sine and cosine terms can be interpreted as monic waves with amplitudes of bm and am, respectively. and frequencies of mev. The magnitudes of the coefficients or amplitudes determine the contribution each harmonic wave makes to the resultant anharmonic waveform. IfEq. (12-1) is multidt and over one period the sine and cosine vanish, plied and result is

f

T

lO

Go

2 =T

+

(12-2)

f(tl dl

10

If Eq ( is multiplied throughout instead cos nwl dt, where n is any integer. on the side is and integrated over a period, the only nonvanishing the one including the coefficient am, and one finds

f

T

lO

am

2 T

+

f{t) cos mwt dt

(l

'0

Similarly. mu1tiplying

(12-1) by sin nwl dt and integrating bm

2 T

flll+T

f(t) sin mevt dt

(12-4)

'0

once f(t) is specified, each of the is complete. and the 248

Chap. 12

Coherence

Go,

am, and bm can be calculated,

fIt)

I

T

_L

2

4

T 4

T

2

Figure 12·2 Square wave.

As an example, consider the fourier analysis of the square wave shown in ure 12-2 and represented over a period symmetric with the origin by

OJ J(t)

{

I,

0,

-T/2 < t < -T/4 < t < T/4 < I <

Since the function is even in t, the coefficients bm are found to vanish, and only cosine terms (also even functions of t) remain. From (12-2) and we find

sin so that the series that converges to the square wave of Figure 12-2 as more terms are included in the summation is J(t) =

~ + ~ [(!) sin ('7)] cos mwr

Writing out the first few terms explicitly.

I 2 +2 '1f'

J(t) = -

001

!

3

cos

3w1

+ 1 cos 5

5wt

+ ... )

Notice that the contribution of each successive term decreases because its amplitude decreases. Thus a finite number of terms may represent the function rather well. more the series converges, the fewer are the terms needed for an adt~uate fit. Notice also that some amplitudes may be negative, that some harmonic waves must be subtracted from the sum to accomplish the convergence. Quite fine features in the givenJ(t), such as the corners of the square waves, require waves of smaller wavelength, or higher components, to represent them. Accordingly, if the widths of the waves were allowed to diminish, so that the vidual squares approached spikes, one would expect a contribution from the high-frequency components for an adequate synthesis of the function. With the help of Euler's equation. the Fourier series given in general by (12-1), involving as it does both sine and cosine terms, can be expressed in cOlmplex notation exponential functions. The result is

ene- i"""

J(t) ,,=-00

Sec. 12-1

Fourier Analysis

249

where now the coefficients

= T]f

'O T

ell

+

f(t)e inwl dt

02-6)

10

In cases where we wish instead to a nonperiodic function (cleverly interpreted mathematically as a periodic function whose period T approaches to the series to a Fourier integral. For exinfinity). it is ample, a single pu]se is a nonperiodk function but can be interpreted as a periodic -00 to t +00. It can be shown that the function whose period extends from t now becomes an integral by discrete Fourier (12-7) where the co(:lh(;lerlt 1

g(w) = 2'1t'

J+oc f(t)e/wl dt _00

(12-8)

The Fourier Eq. (12-7), and the expression for its associated coefficient, Eg. (12-8), have a certain of mathematical symmetry and are together reInstead of a discrete spectrum of lre,qU(mCleS ferred to as a Fourier-transform given by the Fourier . 02-6), we are led to a continuous as given by (12-8). In Figure 12-3. a sample discrete set of coefficients, as might be calculated from (12-6), is shown together with a continuous distribution apby the coefficients, such as might result from Eg. 02-8). g(wl

ande"

Figure 12-3 Fourier coefficients of a fUnetion speeify discrete hannonic components of amplitude Cn at frequency w n • The Fourier transfonn of a nooperiodic function requires instead a conlinuous frequency speetrum g(w).

It should be pointed out that if the function to be represented is a function of spatial position x with period L, say, rather than of time t with period T, then in (12-1) through (12-8) T should be replaced by L and the temporal ..r"",~ .."" ......." w = should be by the spatial frequency, k = For eX(lmiJle. the Fourier and ( become f(x) = g(k) =

250

Chap. 12

Coherence

[~oo

dk

] f+ec 2'1t'

-00

f(x)e ib dx

(12-9)

02-10)

12-2 FOURIER ANAL VSIS OF A FINITE HARMONIC WAVE TRAIN The resolution of an infinitely simple: It is the term frequency of the one term of the Fourier wave. In case, all other coefficients vanish. Sinusoidal waves without a beginning or an end are, however, mathematical idealizations. In practice, the wave is turned on and at finite times. The result is a wave train of finite length, as one pictured in 12-4. Fourier of such a wave train must it as a nonperioruc function. Clearly, it cannot be represented by a single sine wave that has no beginning or end. Rather, the various harmonic waves that combine to produce the wave train must be numerous and so selected that they produce exactly the wave train during the time interval it exists and cancel exactly everywhere outside that interval. Evidently, the turning "on" and "off" of the wave adds many other components to that of the wave train itself. The use of the Fourier-transform leads, in to a continuous of What we have said here of a finite wave train is also true of any isolated pulse, regardless of its shape. We consider for simplicity the resolution of a pulse that is, while it at some point, a harmonic wave. The problem must be handled, as by the Fourier integral Eqs. and (12We have placed the origin of the time Figure so that the wave train is symmetrical about it. The wave train has a lifetime of To and a frequency of Wo Thus it may be represented by TO

e -iWOl ,

f(t) {

2

0,

< t <2

(12-1 I)

elsewhere

The frequency spectrum g(w) is calculated from Eq. (12-8), with the specific tionf(t) of Eq. (12-11), g(w) = .I

2'1T

f+'" f(t)eiwl dt

1 2'1T

= -

-00

Integrating, we g{w) --ro/2

g

(w)

=

(I ) [e (W-wo'rrO/ 2 e-iIW-WO'rrO/2] '1TW-Wo 2i i

fltl

J 2-4 Finite harmollic wave train of lifetime To and period 2Tr I Wo. The extension of lhe is = eTo.

eo

Sec. 12-2

Fourier Analvsis of a Finite Harmonic Wave Train

251

or, after

the identity,

== 2; sin x

eiJ(

sin g () w "" _",::",:,....:.!O--"-,,--_....::.:c..:. wo)

12)

Calling u = (-rb/2)(w - wo), we then have g(w) (7o/2'1T)[(sin u)/u]. The function that (sin u)/u. often called simply sinc(u), shows up frequently. It has the as u approaches 0, the approaches a value of l. from 02-12), we condude that 70 2'1T

lim g(w) UI-+WO

02-13)

Furthermore, the sinc function (sin u)/u vanishes whenever sin u = 0, except at u O. the case already described by 02In every other case, sin u = 0 for u = n1T, n ±l, ... ,and so

o

g(w)

when

2n1T w=wo±-70

14)

As w increases from Wo then, g(w) passes periodically zero. The accompanying increase in the u, or of the denominator of (12-12), gradually decreases the amplitude of an otherwise harmonic variation. where the origin of the frequency These results are all displayed in Figure w = Wo. When the amplitude g (w) is spectrum is chosen at its point of squared. the resulting curve is the power spectrum, shown as the solid curve in Figure 12-5. Although frequencies fur from Wo contribute to the power spectrum, the bulk of the energy of the wave train is dearly carried by the frequencies present in the central maximum, of width 41T /70. Notice that the shorter the wave train of ure 12-4, that is, the smaller the lifetime 70, the wider is the central maximum of Figure 12-5. This means that the harmonic waves contributions the half-width of to the actual wave train span a greater frequency interval. We the central maximum, or 2'1T /70, to indicate in a rough way the range dominant frequencies required. This criterion at least preserves the important inverse re1ationg(wl

Iglwl1 2

,

-- ...

.........

~I ,?

3'

.......,

....

~I'?

"I'?

"I ,?

,?

,?

I

I

+

+

3'

3'

3'

3'

+

3'"

-

+ 3'"

Figure 12·5 Fourier tnmsfonn of the finite harmonic wave train of 12-4. The dashed line gives the amplilude of the frequency spectrum and the solid line gives i[S square, the power spectrum. The curves have been normalized to the same maximum amplitude.

252

Chap. 12

Coherence

w

ship with 'To. Accordingly, we write, as a measure of the around Wo required to the harmonic wave of time 'To,

21T

flJ

or

='To

band centered and life-

treQm~ncy Wo

(I2-15) To

15) shows that if To ~ 00, corresponding to a wave train of infinite length. flw ~ O. and a frequency Wo or wavelength Ao to represent the beam, as conwave train. In this idealized case we have a perfectly Sl
12-3 TEMPORAL COHERENCE AND NATURAL UNE WIDTH

there are no monochromatic sources. Sources we call "monochromatic" emit light that can be represented as a sequence of harmonic wave trains of finite length, as in Figure 12-6, each from the others by a discontinuous in phase changes the erratic process which excited atoms in a source undergo transitions between levels, producing brief and random radiation wave trains. A given source can be characterized by an average wave train 'To, called its coherence time. Thus the physical implications of Eq. 02may be summarized as follows: natural width of Hne is inversely to the coherence time of the source. The its coherence time, the more monochromatic the source. The coherence length It of a wavetrain is the of it'> coherent pulse, or

02-16) Eqs. (12-15) and (12-16), the coherence length is I,

J

=

c

flJ

approximating 4f by the cIA, we may also write

of its differential

I,

the

A2

-

!:lA

(12-17)

'filUS the natural line width is

!:lA

A2 I,

(12-18)

Figure 12-6 Sequence of harmonic wavetrains of varying finite or lifelimes T. The wavelrain may be characlerized by an average lifetime, the coherence lime To.

Sec. 12-3

Temporal Coherence and Natural line Width

253

To digress It IS to note that Eq. 02-18) is an expression of the uncertainty principle of quantum mechanics, where a wave pulse is used to represent, an electron. If the coherence it is as the interval Ax within which the particle is to be found-that its in location-and the uncertainty in momentum lip is by the differential of the deBroglie wavelength in the equation p = hI A, the result is IIp = h. The inequality associated with the uncertainty relation arises from the inherent in (12-15). Since the line width of spectral sources can be measured, average coherence times and coherent my be surmised. White light, for example, has a "line width" of around 300 nm, extending from 400 to 700 nm. the average wavelength at 550 nm, Eq.

I, = 300

==

1000 nm

2Aav

a very small coherence indeed. of around a millionth of a centimeter or two "wavelengths" of white light. Understandably, interference fringes by white light are difficult to obtain since the difference in the path of the interfering beams not be than the coherence length for the light. Sodium or mercury ga!;-dischaJ'ge sources are far more monochromatic and coherent. For the green line of mercury at 546 nm may have a line width of around 0.025 nm, ing a coherence length of 1.2 cm. One of the most monochromatic gas-discharge sources is a gas of the krypton 86 isotope, whose orange emission line at 606 nm has a line width of only 0.00047 nm. coherence of this by Eq. (12is 78 Laser radiation has far even the coherence of this gasdischarge source. The short-term stability of commercially CO2 lasers, for example, is such that line widths of around I x 10- 5 nm are attainable at the infrared emission wavelength of 10.6 pm. numbers a coherence length of around II km! Under carefully controlled conditions, He-Ne lasers can this another order of magnitude. Somewhat discouragingly, the common HeNe laser used in instructional laboratories may not have coherence lengths much greater than its cavity length, due to random fluctuations and mirror vibrations. These spurious effects change the cavity length, lead to multimode oscillaaffect the coherence of the laser. Hence the use these tions, and lasers, in holography for still some care in equalizoptical-path

72-4 PARTIAL COHERENCE As out previously, when the difference between two waves is constant, they are mutually coherent waves. In practice, this condition is only approximately and we speak of partial coherence. The is defined more prein what follows. Consider, as in Figure a general situation in which

s

p

Figure 12-7 Interference at P dlle 10 waves from S traveling different The waves are redirecled al S. and S2 by varioos means, including reflection. refraction, and diffraction.

254

Chap. 12

Coherence

interference is produced at P between two beams that originate from a single coherent source S after traveling different paths. Let the two beams be represented in genend by

El(r, t) =

(12-19)

~(r,

{I 2-20)

t)

At point P, each field varies only with ing an extra time interval 'I, we may

and if the lower write at P:

is

where we have set r = 0 and E = 0, for convenience. Except for a multiplicative constant, the irradiance at P is by Jp

= (Ep . Ep*)

«EI

+

+ E~»

where the angle brackets represent time averages. "-'AI'"'''''' Jp

=

(I Ed 2 + I

12 +

+

. Er»

The quantity in parentheses is just the sum of a complex number and its complex conjugate, which is always equal to twice its real part. Thus

Ip

= II + h + 2 Re (E I

02-21)

where II and 12 represent the irradiances of the individual beams and the third term represents interference between them. In Eq. 02-21), we have also assumed that the beams have the same polarization and so have the dot product. The interference term can add to or subtract from the sum of the two beam irradiances, depending on the correlation in phase between the two fields at P. We define, accordingly, a correlation/unction,

r I2 (T) == (El(t)Ef(t + and, dividing by the amplitudes of the fields, a normalized rn,'rpinlulll

(12-22) nmr,llOJ1_

(12-23) The irradiance at P may then be expressed by

Ip = II + h +

2v'i:iz Re ["YdTH

02-24)

The function "Yd'T), now the heart of the interference term, is a function OfT and therefore of the location of point P. We know that the time difference between ."'........ '" to the average coherence time of the source, is to the degree of coherence achieved. We expect that forT> 70, some coherence the two beams will be lost. The dependence of "Y12(7) on 70 is now .... P.·"'~..... sumption that 70 represents a constant coherence time rather than an wave train is shown at the top of Figure 12-8a, with regular dislCOllltiOlui 'l;:Pf1I",."tp.fl by the time interval To. The phase q>(t) of the harmonic wave is .....", ..jt:",.... an arbitrary fashion and is ploued by the solid line below. The wave with h
q> (t + 'T) of the second wave, arriving 7 are also shown as the same function but displaced by the timeT. The cor-

chlllnj:!;C~s

Sec. 12-4

Partial Coherence

255

211

r I

I

'"

'"

c.

I I I

I

I I

I I L

+ 1'1 r

I

I

I

I

I I

I

'PIt

I

I

-...,1' l'

I

I

I

'PIt)

I

I

I I


J::

I

I I

I

I I

I

I

I L

la)

~

rr !-

ITo

-

1'1

Tr+-

21'0

41"0 I

t

31'0

TO

-

.... Ibl

Figure 12·8 (a) Random phase fluctuations ",(t) every 'To of a wave (solid line) and the same phase fluctuations", (t + 1") of the wave (dashed line) at a time 'T earlier. (b) Difference in the phase between the two waves described in (a).

relation function, often called the degree amplitude and out of step by time T, I

of the

out the

E(t)E*(t

Chap. 12

Coherence

+ T)

coherence between two waves of equal by (E(I)E*(I + IEol2

T»

in terms of Eq. (12-25), we have

or

+ -I")

E(t)E*(1

so that "Yd'd =

The time average

Pyrorp<:"",rl

e;""'(eil~(t);P('+T)I}

in this equation may be calculated

(e i!;P(I)-;p(1

+

TH) = -I

IT

T

(12-26)

dt

0

where T is a sufficiently long time. The function q; (I) q; (t + T) in the exponent is pictured in Figure 12-8b and is seen to be a regularly spaced series of rectangular pulses with random falling between -21T and +21T. Consider the first coherence time interval To, in which the pulse function may be by

°< t <

O, { H,

+ T)

q;(t) - q;(t

(To -

(To

<

t

T)

<

To

In successive intervals, the expression is the same, except for the value of H. We may then write the normalized coherence function "Y12 for a large number N of intervals as dt

"Y12

+ similar .ten~s for (N successive mtervaJs

interval N

Integrating over N terms, "Y12 =

(~:)[(TO

'1+

+ ...J

Combining the first terms of each interval and summing the rest, "Y12

=

(~:) [N(TO

'I)

+ 'I

N

e lHj ]

Because of the random nature of Hh the terms in the summation N sufficiently large. Thus only those times during which the waves q;(t) q;(t + 'I)-contribute to the we are left with

(1 -

"YdT)

(12-27)

by

of "Y12, required in

The rea'

COJnCllae-lwnl~n

(1 -

Re ["Y12(T)]

cos WT

02-28)

and so takes on a maximum value of =° lengths), a value of °The when (path difference equals coherence length), and any value between. amplitude of the cosine term in Eq. (12-28) is just the magnitude ofthe degree I

'I

'I

To

of coherence "Y12, that is, 'I

TO

Sec. 12-4

Partial Coherence

(12-29)

257

This quantity sets the limits of the variations in the term and thus conas a function of T. This amplitude, trols the contrast or visibility of the 11'dT) I, is plotted in 12-9. Combining the last three equations, (12-30)

1'dT) = 11'12jeiWT

(l2-31)

70

U ..9 Fringe or of coherence as a function of the difference in arrival times or two waves with coherence time To.

l'

Recalling the empirical expression for

contrast or visibility introduced

v

(12 .. 32)

we may now delineate the following special cases: I. Complete incoherence:

2. Complete coherence:

T

T

~

To

and I1'121 = 0

Ip

II + 12

Ip

210,

v=

--"'--.--'-

for equal

4/0 = 0 and I1'121 = I cos

o

WT

+ 12 + 2 = 410, = II + 12 2 VI:I;. = 0,

lenax = II

for equal beams

I min

for equal beams

4/0 410 3. Partial rnl,pr,,,nr'p' 0

v

1

<

T

<

To

and I

> 11'121 > 0

+ 12 + 2 VI:I;. Re (1'12) Ip 210[1 + Re (1'12)], for equal beams = 2/0(1 + 11'12 D and I min = - 11'12 D Ip = I,

v=

--='~::.J.

4/0

11'121

In aU cases of equal therefore, the visibility V is equal to the tude of the correlation function 11'121, and either one is a measure of the coherence. Fringes corresponding to cases 3 and 2 were depicted in and respectively, in the earlier discussion of interference. 258

12

Coherence

Example In an interference a light is split into two equal-amplitude parts. The two parts are superimposed again after traveling diverse paths. The light is of wavelength 541 nm with a line width of t A, and the path tiifjcprf'ncp is l.50 mm. Determine the visibility of the fringes. How is the visibility modified if the path is doubled?

Solution

The visibility is given by l'

V=l

To

where the ratio of time delay to coherence time is replaced by the correspondto coherence length. In this case, i, A?/t:.A ratio of path (5410 Ay/O A) mm. Thus, V=l

~=

When the path difference is doubled, are incoherent and V = O.

0.49

a > it and T

>

To, so that the beams

SPATIAL COHERENCE of temporal coherence, we have been considering the correlation in between temporally distinct of the radiation field of a source along its of propagation. For this reason, temporal coherence is also called longitudinal coherence. The degree of coherence can be observed by examining the interference fringe contrast in an amplitude-splitting instrument, such as the Michelson lntprt'pr_ ometer. As we have seen, temporal coherence is a measure of the average the constiruent harmonic waves, which depends on the properties of the to as spatial, or source. In contrast, we now turn our attention to what is lateral, coherence. the correlation in Jitase between spatially distinct points of the of coherence is important when using a wavefront-splitting radiation field. This interferometer, such as the double slit. The quality of the interference pattern in the double-slit experiment depends on the of between distinct ,."en".,., of the wave field at the two slits. To sharpen our understanding of the coherence of a wavefield radiating from a spurce, consider the situation depicted in Figure 12-10. Light from a source S passes through a double slit and is also sampled by a Michelson interferometer located nearby. Spatial coherence between wavefront points A and B at the slits is insured as long as the source S is a true point source. In that case all rays emanating from S are on any associated with a set of sJiterical waves that have the same wavefront. Are clear distinguishable then formed on a screen near point PI? The answer, of course, depends on whether the light from S, traveling the two , is temporally as well as spatially coherent. The matter distinct paths SAP) and of temporal coherence a comparison between the path difference a SAP I and the coherence of the radiation. This is to a comdirection of light propagation from the source parison of coherence along any at two wavefronts separated by the same path difference. It is this property of temporal coherence that is measured by the Michelson interferometer. If the path differenCe tJ. is less than the coherence length
Spatial Coherence

259

s

Figure 12-10 Wavefront and amplitude division of radiation from source S, illustrating the practical requirements of and coherence.

(a ;;;:;; [,), interference fringes are poorly or absent altogether. In pracof course, S is an extended source, so that rays reach A and B from many points of the source. In ordinary (nonlaser) sources, light emitted by different points of a source, wen over a wavelength in separation, is not correlated in phase and so lacks coherence. Thus the spatial coherence of light at the slits depends on how the source resembles a point source of light, either in extension or in its actual coherence properties. We wish to show now that iftwo source SI and as in 12-11, are separated by a distance s and if light of wavelength A from these sources is observed at a distance r away, there will be a region of high spatial coherence of dimension given by

(12-33) point sources at the observation point P. Acwhere 11 is the angle subtended by cepting this result for the moment and combining it with the temporal or longitudinal coherence length It, we conclude that there exists at any point in the radiation field of a real light source a of space in which the light is coherent. reIs and longitudinal of It relative to the gion lateral dimensions source and thus occupies a volume or roughly {;It around the point P. It is from this volume .that any interferometer must accept radiation if it is to produce observable interference fringes.

12-11 Lateral region of coherence I" due to two independent point sources.

12-6 SPATIAL COHERENCE WIDTH Consider now the spatial coherence at points PI and P2 in the radiation field of a quasi-monochromatic extended source, simply represented two mutually incoher12-12). We may think ent points A and B at the edges of the source 260

12

Coherence

p

Screen

B

Figure 12·12 from each of two sources A and Breach P, and P2 in the radiation field and are allowed to interfere at the screen. In s ~ e and 8 are approximately equal.

of and as two slits light to a screen, where interference fringes may be viewed. Each point source, alone, then produces a set of double-slit interference on the screen. When both sources act together, however, the systems overlap. If the fringe systems overlap with their maxima and minima is highly visible, and the ra<.111atlon falling the fringe the two incoherent sources is considered highly coherent! When the fringe systems are relatively displaced, however, so that maxima of one faU on the minima of the other, composite pattern is not visible and the radiation is considered incoherent. Suppose that source B is at the position of source A, or that the UJ",.........,,, 12-12 is zero. The at the screen then coincide and to the fringes of a single point source. A maximum in the interference pattern occurs at P if P lies on the perpendicular bisector of the two slits. In this condition, - API

=0

If source B is moved below A, the fringe systems separate until, at a certain distance s, where

A 2

BPI = fl = -

the in the system at P due to source B is replaced by a minimum, and the composite fringe pattern disappears. If the angle 6 represents the angular separation of the sources from the plane of the slits, then from the fl l6, where l is the distance between slits, where r is the distance to the sources. It fol1ows that and 6 ==

sf rA or s = 2l r

A fl=2

(12-34)

When the distance AB is considered instead to be a continuous array of point sources, the individual fringe systems do not complete cancellation until the value s in (12-34). If extreme spatial extent AB of the source reaches twice definition is assured. Reare separated by an amount s < rAj f, then garding this result as describing instead the maximum slit separation f, given a source dimension s, we have for the spatial coherence width fs, fs

rA

A

< - ==s 6

02-35)

As fs is restricted to smaller fractions of this value, the fringe contrast is correspondingly improved. According to this moving the source B even farther should bring the system into coincidence again, so that the degree of coherence I'}'121 between PI and is a periodic function. In a more complete mathematical argument, Sec. 12-6

Spatial Coherence Width

261

the extended source is by a continuous array of elemental emitting areas rather than by two point sources. Results show that outside the coherence width given by 02-35), the fringe visibility, while oscillatory, is negligible. According to a general known as the Van Cittert-Zernike theorem [1], a of the of coherence versus separation l of points PI and is the same as a plot of the diffraction due to an aperture of the same size and shape as the extended source. Such patterns for rectangular and circular sources are discussed in 16. The of is apparent in the case of double-slit experiment, where an extended source is used together with a single slit to render 12-13. We may the light striking double slit reasonably coherent, as in now use Eq. to determine how small the must be to ensure coherscreen. The two SI and must faU ence and the production of fringes at within the coherence width is due to the primary slit of width s.

12·13

Young's double-slit setup. Slits

SI and S2 must fall within the lateral coherence width I, due to the single-slit source.

Let the source-to-slit distance be 20 cm, the slit separation 0.1 mm, and the wavelength 546 nm. Determine the maximum width of the primary or slit. Solution

Eq.

s

< rA _ (0.2)(546 x 10-9 ) is -

I

X

10-4

I 1 - . mm

Now suppose that the source slit in the example is made 1.] mm in width and that the separation between slits SI and S2 is adjustable. When the are very close together (a ~ Is). they fall within a high coherence region and the fringes in the interference pattern appear sharply defined. As the are moved farther of coherence I'Yill decreases and the contrast to deapart, the When the slit separation a reaches a value of 0.1 mm, I'Ylzi = 0 and the fringes disappear. Evidently an experimental determination of this slit separation could be used to deduce the size s of the extended source. This technique was employed by Michelson to measure the angular diameter of stars. Stars are so distant that imaging techniques are unable to resolve their diameters. If a star is as an extended, incoherent source with emanating from a continuous array of points extending across a diameter S ofthe star Figure 12-14b), then the spatial coherent width Is in (12-35) becomes I

s

< 1.22A ()

(12-36)

Here the factor 1.22 arises from the circular of the source, as it does in the Fraunhofer diffraction of a circular aperture. Since the angular diameter 6 a star is 262

Chap. 12

Coherence

I,

(a)

Ib!

Figure 12·14 Michelson stellar interferometer (a) used 10 determine a stellar diameter (b).

extremely small, Is will be correspondingly large. The movable slits were therefore 12- 14a, using mirrors that direct widely separated portions of arranged as in the radiation into a double-slit-telescope instrument. The of the visibility depend on the double slit separation a, whereas disappear when separation As Is is increased, the in is satisfied.

Example When Michelson used this technique on the star Betelgeuse in the constellation Orion, he found a first minimum in the at is 308 cm. Using an averdiameter of the star? age wavelength of 570 nm, what is the Solution Taking

as an equality, 1.22A

2.26

3.08

X

rad

Since Orion is known to be about 1 X 1015 mi away, the stellar diameter is s = rlJ 2.26 X lOS mi, or about 260 solar diameters.

period L

12-1. Determine the Fourier series for the square wave of

by

-L

2 < x <0

+L

o
Problems

263

1:z..2. A half-wave rectifier removes the half-cycles of a sinusoidal waveform, Eo cos wI. Find the Fourier series of the resulting wave. given by E 1:z..3. Find the Fourier transform of the Gaussian function by /(t) =

where h is the height and a the "width." You will also need the definite integral

Remember how to complete a

dx =

1:z..4.

12-5.

12-6.

1:z..7.

l:z..S.

t:z..9.

12-10.

l:z..n.

1:z..12. 12-13.

264

in your calculations.) Does the transform, as the treQw~n(:y spectrum, to the original "pulse" width? show the proper the Fourier transform. determine the power spectrum of a single square of amplitude A and duration To. Sketch the power spectrum. locating its zeros, and proportional to its durashow that the bandwidth for the pulse is tion. Two light filters are used to transmit yellow centered around a wavelength of 590 nm. One filter has a "broad" transmission width of 100 nm, whereas the other has a "narrow" pass band of 10 nm. Which filter would be better to use for an interference experiment? the coherence lengths of the from each. A continuous He-Ne laser beam (632.8 nm) is "chopped," a spinning aperture, into O.l-ns Compute the resultant line width lU, bandwidth At, and coherence The diameter of the sun viewed from the earth is approximately 0.5 degree. Determine the spatial coherence length for coherence, any variations in brightness across the surface. Let us consider. somewhat arbitrarily. that coherence will exist over an area that is 10% of the maximum area of coherence. Michelson found that the cadmium red line nm) is one of the most ideal monochromatic sources allowing fringes to be discerned up to a path difference of 30 cm in a beam-splitting interference such as with a Michelson interferometer. Calculate (a) the wavelength time of the souree. A narrow "<1"U-V"''''' filter transmits in the range 5000 ± 0.5 A. If this in front of a source of white light, what is the coherence length of the filter is transmitted Let a collimated beam of white light fall on one face of a and let the light emerging from the second face be focused by a lens onto a screen. Suppose that the linear dispersion at the screen is 20 By a narrow "exit slit" in the screen, one has a type of monochromator that provides a monochromatic beam of light. Sketch the setup. For an exit slit of 0.02 cm, what is the coherence time and coherence length of the light of mean wavelength 5000 A? A pinhole of diameter 0.5 mm is used in front of a sodium lamp (5890 A) as a source in a interference The distance from pinhole to slits is I m. What is the maximum slit space interference that are visible? Determine the linewidth in angstroms and hertz for laser light whose coherence length is 10 km. The mean wavelength is 6328 A. (a) A monochromator is used to obtain quasi-monochromatic light from a The linear dispersion of the instrument is 20 A/mm and an exit slit of 200 p,m is used. What is the coherence time and length of the light from the monochromator when set to give light of mean wavelength 500 nm? This light is used to form fringes in an interference experiment in which the light is first amplitude-split into two parts and then brought together again. If the optical path difference between the two paths is 0.400 mm, calculate the

Chap. 12

Coherence

tude of the normalized correlation function and the visibility of the resulting fringes. is 100 on an arbitrary (c) If the maximum irradiance produced by the what is the difference between maximum irradiance and background irradiance on this scale? 12-14. Determine the and base area of the cylindrical volume within which light received from the sun is coherent. For this purpose, let us assume "good" spatial coherence occurs within a length that is 25% of the maximum value given by Eq. (12-36). The sun subtends an angle of 0.5 0 at the earth's surface. The mean value of the visible the coherence volume also in terms of spectrum may be taken at 550 nm. number of wavelengths acrOSS cylindrical length and diameter. 12-15. (a) Show that the fringe visibility may be expressed

(b) What irradiance ratio of the

int'>rf~.,.in
V=I

and LlA is its linewidth. where A is the average wavelength of the 12-17. A filtered mercury lamp produces green light at 546.1 nm with a linewidth of 0.05 nm. The light illuminates a double slit of spacing 0.1 mm. Determine the visiof the on a screen 1 m away, in the vicinity of the m ::::: 20 order (See problem 12-16.) If the discharge lamp is replaced with a white light source and a filter of bandwidth 10 nm at 546 nm, how does the visibility change? 12-18. A Michelson interferometer forms fringes with cadmium red of 643.847 nm and Hnewidth of 0.0013 nm. What is the visibility of the fringes when one mirror is moved I cm from the position of zero path difference between arms? How does this when the distance moved is 5 cm? At what instance does the visibility go to zero? 12-19. (a) Repeat problem 12-18 when the light is the green mercury line of 546.1 nm with a Imewidth of 0.025 nm. (b) How far can the mirror be moved from zero path difference so that visibility is at least 0.85?

[I) Born, M., and E. Wolf. Principles of Optics. New York: Pergamon Press, 1959.

[2J Fowles. Grant R. Introduction to Modern Optics. New York: Holt, Rinehart and Winston, 1968. (3) Parrent, Mark J., and B. Jr. Theory of Panial Coherence. Englewood Cliffs, N.J.: Prentice-Hall, 1964. [4] Richard P., Robert B. Leighton, and Matthew Sands. The Feynman Lectures in Physics, vol. 1. Reading, Mass.: Addison-Wesley Publishing Company, 1975. Ch. 50. [5J Reynolds, 0., John B. DeVelis, George B. Parrent, and Brian J. Thompson. Bellingham, Wash.: SPIE Optical Physical Optics Notebook: Tutorials in Fourier Press, 1989. Ch. II, 18. [6] Jan. Coherence of Light. New York: Van Nostrand Reinhold Co., 1971.

Chap. 12

References

265

13

Holography

INTRODUCTION

Holography is one of the many flourishing fields that owes its success to the laser. Although the technique was invented in 1948 before the advent of laser light by the British scientist Dennis assurance of success was made at the University of ble by the laser. Emmett Leith and Juris first applied laser light to holography in and also introduced an ,rnll'V\,·t"",t axis technique illumination that we presently. The improvement in three-dimensional photography made possible so that by the has aroused unusual interest in nonscientific circles as applications of holography today also include its use in art and the advertising.

13-1 CONVENTIONAL VERSUS HOLOGRAPHIC PHOTOGRAPHY We are aware that a conventional is a two-dimensional of a three-dimensional scene, bringing into focus every part of the scene that falls within of the lens. As a result the photograph lacks the perception of the depth of with which we view a real-life scene. In contrast, the hologram depth or the provides a record of the scene that preserves these qualities. The hologram l:)U""""";;;U;> in effectively and preserving for later observation the intricate wavefront 266

of light that carries all the visual information of the scene. In viewing a hologram, this wavefront is reconstructed or released, and we view what we would have seen if ......c ..,<>"~ at the scene through the "window" by the The reconstructed wavefront provides perception and allowing us to look around of an object to see what is behind. It may be manipulated by a lens, for example, in the same way as the original wavefront. Thus a "hologram," as its etymology suggests, includes the "whole message." The real-life qualities of the provided by a hologram stem the relating to the phase of the wavefront in addition to its preservation of amplitude or Recording like ordinary photographic film and photomultipliers are sensitive only to the radiant energy received. In a developed photograph, for example, the density of the emulsion at each point is a function of optical energy received there due to the light-sensitive reaction that ..",iii",...,,., silver to its metallic form. When energy alone is the phase tionships of waves arriving from different directions and distances, and the visual lifelikeness the scene, are lost. To record these relationships as well, it is necessary to convert phase into amplitude information. The interferwaves provides the requisite means. Recall that when waves interfere to ence of a large amplitude, must be in and when amplitude is a minimum, the waves are out of phase, so that various contributions effectively cancel one another. If the wavefront of light a scene is made to interfere with a coherent reference then, the resultant interference includes repart of the with the reference wave garding the relationships of and, with every other part. The situation is sometimes described by referring to the reference wave as a carrier wave that is modulated by the signal wave from the scene. This provides a fruitful comparison with the techniques radio wave communication. All In conventional photography a lens is used to focus the scene onto a of the scene collected by the lens is fothe light originating from a single cused to a single conjugate point in the image. We can say that a one-to-one relationship exists between and image points. By contrast, a is made, as we see, without use of a lens or any other focusing The hologram is a complex interference pattern of spaced not an of the scene. Each of the hologram receives light from every point of the scene point iHuminates the entire hologram. There or, to put it another way, every is no one-to-one correspondence between object and points in the wavefront before reconstruction occurs. The hologram is a record of this "'..'''''T......n

13-2 HOLOGRAM OF A POINT SOURCE

To see how process is realized in practice, both making the and using the hologram to reconstruct the original scene, we begin with a very basic example, of a point source. In 13-1a, plane wavefronts of coherent, the In addition, monochromatic radiation illuminate a photographic wavefronts the after from object point O. The plate, when deinterference rings about X as a center. veloped, then shows a series of Point P falls on such a ring, for example, if the path difference OP - OX is an number of wavelengths, ensuring that the reference beam of plane wavefront light arrives at P in with the scattered subject beam of light. The develis called a Gabor zone zone lens-with circular transmitting is a varying function of radius. The zones, whose Sec. 13-2

Hologram of a Point Source

261

x

Plate

(al

(b)

I'igure 13-1 Hologram of a point source 0 is constructed in (a) and used in (b) to reconstruct the wavefront. Two images are formed in reconstruction.

zone plate is a "sinusoidal" circular because the optical density, and therefore the of the grating. varies as cos 2 r2 along the radius of the zone pattern. I sinusoidal plate is, in a hologram of the point O. The hologram itself is a series of circular interference that do not the object, 13-1 b, by the hnlnor",,,,,, but the object may be reconstructed, as in back into the reference beam without the of the object O. Just as light directed from 0 originally interferes with the reference beam to produce the zone rings, so the same reference beam is now reinforced in diffraction from the rings that diverge from the point 0'. The point 0' thus locates a virtual of the original point 0, seen on reconstruction by looking into the hologram. The condition for reinforcement must also be satisfied by a second point on the exit side of the a point I that is placed with 0' to the film. Clearly the set of from I to zones satisfies the same geometric as the distances from 0' to the zones. Thus the diffracted light also converges to point I, a real image of the original point 0 that can be onto a screen. in making of this object point 0 is moved farther away. the radius of Creases. For an off-axis object at infinity, zones are straight, fringes. The is then a grating hologram, formed by the of two plane wavefronts of light arriving at the plate along different directions. The grating hologram is further in 17. As explained there, the the angIe between these wavefronts, the finer the of the interference The fdmily of circular and straight, parallel we have been can be seen as special cases of two point-source interference, observed in planes perpendicular and parallel, respectively, to the axis points. (See the discussion relating to Figure 1O-5.) When object point 0 is by an extended or threedimensional scene, each point of the scene produces its own zone pattern on the film. The hologram is now a montage of zones in is coded all the information of the wavefront from the scene. On reconstruction, each set of zones produces own real and virtual and the original scene is reproduced. One usually views virtual by into the hologram. 13-1 b shows that when viewing the virtual in this way, undesirable forming the real image is also intercepted. Leith and Upatnieks introduced an off-axis technique. using one or more mirrors to bring in the reference beam from a different angle. so that the directions of the reconstructed real and virtual wavefronts are separated. I More the transmittance can he expressed as A constants. See problem \3-12.

268

Chap. 13

Holography

+

B cos2

where A, B, and a are

The two basic types of discussed in the are the Gabor zone plate and holographic corresponding to point at a finite distance and at an infinite distance from the plate, respectively. If the zone plate or Dr
T=

+

and higher-order images are eliminated. The compromise is that the transmittanee is now superimposed over a nonzero minimum transmittance To, and contrast is somewhat reduced. As we have just pointed out, the amplitude of the reference beam is made of the signal or object so that somewhat than the average the reference wave is modulated by the Even when the signal is zero, the referenee beam is of sufficient strength to the emulsion within its of linear response to radiant energy. The effect of variations in to variations in the contrast of the fringes, whereas variations in phase (or direction) of the waves produce variations in spadng of the fringes. Thus it is in the local variations of fringe contrast and spacing across the 110,loJ;!:nun that the corresponding variations in amplitude and phase of the object waves are encoded. High-resolution film is used to record this information faithfully. VHJUUL,",\J,

13-3 HOLOGRAM OF AN

KTI.:NlJtED OBJECT

One of many holographic techniques for an off-axis reference beam in conjunction with the beam of diffusely reflected from a three-dimensional scene is shown in 13-2a. A pinhole and lens is used to expand the beam from a laser. The expanded beam is then split by a semireflecting two coherent beams. One the reference beam ,is diplate BS to rected mirrors M I and M2 onto the photographic plate, as shown. The reflects diffusely from the and some of this beam, which we call the beam, also strikes the film, where it interferes with the reference beam and produces the hologram. Sec. 13-3

Hologram of an Extended Object

lal

y

13·2 (a) Off-axis hologrnphic system. (b) Orientation of film with reference beam in (a).

x (b)

We now make the previous qualitative explanation somewhat quantitative. Let the reference beam be represented by the field (13-0 at the of the film. The amplitude r = r y) of the reference beam can be assumed constant over the plane wavefront. The phase cp arises from the a between the film plane and the wavefront of the reference beam, as indicated in Figure 13-2b. If the top of the beam the film at x ~ 0, then cp is a linear function of distance x the film plane, since cp

(2;)a = (Z;) x

(13-2)

sin a

Thus the phase cp relates only to the tilt of the film relative to the ence beam and appears as an exponential factor in Eq. (13-1): reiwlei
Ir the reference beam were not present, the film would be illuminated ~~"I~'_' beam,

(13-3)

by the (13-4)

where s (x. y) is the amplitude of the reflected light at different points of the film and () = () (x, y) is a complicated function due to the variations in of the light

270

13

Holography

reaching the film from different parts of the subject. If the subject beam alone were the film would be in proportion to the irradiance of the subject for simplicity the constant factors between irradiance and square of amplitude, we write Is =

I

12 = El Es = [sex, y)J2

(13-5)

The function thus no phase of the subject beam. With the reference beam also however, the resultant amplitude at each point of the film-subject to the scalar approximation-is given by ER

+

Es

so that IF = 1

+ Es)(El +

12

MUltiplying the binomials, (13-6)

The right side of Eq. (13-6) is a function of x and y and so varies from point to point on the film The last two terms now the important function 6(x, y). I.JA~'U"'J'U

e- i(wt+81 (I This irradiance function describes the hologram. When the film is developed, its trdIlSmittance is determined by IF. To reconstruct the image of the scene, the hologram is situated in the r"t,p>r"nrp beam again, as in the formation of the Of course, the subthe reference the hologram, due to ject is now absent. When illuminated its transmittance modulates both the amplitude and the phase of the beam. As before, for constants, in terms

The resulting emergent beam can then be expressed, of the field

+

+

(13-9)

where we have together (13-7) and (13-8). We now interpret the three terms in (13-9) as the reconstruction of threc distinct beams from the hologram. Each beam is also illustrated in Figure 13-3. The first term,

= (r2 +

(r2

+ s2)re i(wl+
(13-10)

represents a beam modulated in amplitude but not in It therefore appears like the beam and passes through the hologram without deviation. it corresponds to zeroth-order diffraction. In analogy with the holographic The second term is Em

(13-11)

which describes the subject beam, amplitude-modulated by the factor r2. Thus the beam represents a reconstructed wavefront from the subject, the same a relative to the beam. Since this beam is essentlal.ly appears to come from the subject. Hence it Sec. 13-3

of an Extended

21'1

image

First-order

,..."r

be Figure 13-3 Reconstruction of hologram formed in Figure B-2a.

order

gram, as if coming from a virtual image behind the hologram. This virtual image is what we customarily view. The third term is given by (13-12) and represents the subject beam, modulated in both amplitude and phase. This beam reconstructs the subject beam of Eq. (13-4) but with phase reversal. Every delay in phase in Es now shows up as a phase advance. The image is turned inside out. Because of phase reversal, originally diverging rays become converging and focus as a real image on the viewing side of the hologram. The factor ei(2'1'), when compared with the phase term in Eq. (13-3), indicates an angular displacement of the image direction by 2a relative to the normal to the film plane. Notice that the off-axis system illustrated in Figure 13-2a produces a hologram in which the two first-order beams are separate in direction from each other and the zeroth-order beam. The virtual image can be observed clearly, without confusion from the other beams. The hologram made of an extended object shows the same essential features as the hologram of the point object. Photography by holography is a two-step process. Recall that in the making of a hologram, no lens is used, and the presence of the reference beam is essential. The light must have sufficient temporal coherence so that path differences between the two beams do not exceed the coherence length of the light; it must also possess sufficient spatial coherence so that the beam is coherent across that portion of the wavefront needed to encompass the scene. Of course, the holographic system must be vibration-free to within a fraction of the wavelength of the light during the exposure, a condition that is ea.<;ily satisfied when high-power laser pulses of very short duration are used to freeze undesirable motion. A three-dimensional view of the object from all sides can be produced on a holographic film that is wrapped around the object on a cylindrical form, as shown

Figure 13-4 Cylindrical film surrounding the subject records a 3600 hologram_

272

Chap. 13

Holography

in Figure 13-4. Light reaches the film both ""","',,<1,,, and with the help of a mirror at and scattered from the the end of the cylinder (the reference 360° hologram produces a view of the When viewed under the same conditions, fish from all sides.

13-4 HOLOGRAM PROPERTIES As stated earlier, the entire from each object point in the scene. As a result, any contains information of the whole each square is a hologram of the scene. If a hologram is cut up into small the resolution of the imwhole scene, although the reduction in through a small, square aperage. The situation is much the same as when in front of a window. The same scene is viewed, though with slightly ture parts of the window. Each perspective, as the opening is moved to view is complete, exhibiting both depth and parallax. Another interesting property of a hologram is that a contact print of the which interchanges the optically dense and transparent regions, has the same in use. The "negative" of a hologram alters neither fringe contrast nor and hence does not modify the stored information. Furthermore, the hologram may contain a number of separate exposures, each taken with the film at a different to the reference On each scene appears beam and with different wavelengths of in its own light when viewed along the of the scene, without mutual interference.

13-5 WllrlTI.:-UGH HOLOGRAMS

If the hologram of Figure 13-3 is viewed in a beam different color than of the fish will appear at that used in its construction, it can be shown that the a different The hologram, like the as a dispersthe continuously diselement. If the reference beam is white placed due to different spectral regions of the light overlap and produce a colored blur. By producing a hologram that restricts the possible angular views of the to one through a horiwntal slit, the confusion of is reduced. In in white light. The virtual reconstruction the hologram creates an improved image now appears colored. The particular color seen depends on. the direction a vertical line. Such along which the hologram is viewed as the head is moved is called a minbow hologram. Since the view is now restricted to what a slit placed in front of one would see by viewing the subject through a the rainbow hologram reproduces horizontal parallax but suffers from a loss of cal parallax. The hologram may be viewed in reflected light by the back side of the hologram with a thin layer of aluminum, which then serves as a mirror to redirect white light back through the hoi",..."...,... When the thickness of the emulsion is the spacing. may be considered a or hologram. The inthe the emulsion that behaves as terference fringes are now interference surfuces t'n.lct
White-Light Holograms

213

--Es

13..5 Formation of standing-wave by two planes in II volume plane waves oppositely

separation J80°, as in 13-5~ If the two monochromatic undistorted plane wavefronts. for the pattern anti nodal planes perpendicular to the beam directions and spaced A/2 apart, as planes produces, after film development, shown. The maximum irradiance in of excess free which function as partially reflecting planes. planes Of course the emulsion must itself possess high-resolution potential to record faithfully such detail. When illuminated from the reference beam direction with white for the developed hologram partially reflects light from each silver wavelength used in making the is reinforced by but only light of such multiple The physics of process is, of course, the same as equation, for diffraction from crystalline planes, governed by the mA = 2d sin () and illustrated in 13-6. Thus if a volume is illuminated at a given (), only the one wavelength that satisfies the Bragg loeally, where nar spacing is d, is reinforced and appears as a brightly reflected beam. The thicker the emulsion and the the number of contributing reflecting planes, the more selective the hologram will be in reinforcing the correct wavelength. If a volume hologram is made by multiple exposures of a scene in each of threc the reconstruction process with white-light illumination can produce a sional image in full color.

dsin (J

Figure 13..6 Constructive interference of reflected waves from of separation d is governed by lhe 8ragg rnA = 2d sin 6.

13-6 OTHER APPUCATIONS OF HOLOGRAPHY Holography offers a wide of the interference of light, has scribe a few. The itself a the wavebeen used as an alternate technique in interferometry, the science of length of light and interference to measure very small path lengths with prefish in Figure 13-3 and the fish are cision. Suppose that the hologram of and suppose that the same reference returned exactly to their 214

Chap. 13

Holography

beam illuminates the scene. In looking through the hologram, one now sees the virtual image superimposed over the object itself. Both are viewed with the same coherent light. If no change has occurred since recording the hologram, the view appears as if the subject or the hologram alone were in place. Suppose, however, that the model of the fish has undergone some small changes in shape, by thermal expansion, for example. Now the direct image of the object and the holographic image are slightly different, and the light forming the two images interferes, producing fringes that measure the extent of the change at specific locations, as in the case of Newton's rings. This techniqe is often applied to determine maximum stress points on the subject as pressure is applied, as in the case of an automobile tire, for example. The sensitivity of this technique has been dramatically demonstrated in holographic recordings of convection currents around a hot filament, compressional waves surrounding a speeding bullet, and the wings of a fruit fly in motion. Changes that occur over a period of time can be monitored in the same way, by returning the model to the holographic system. Another useful application of holography is in microscopy. When specimens of cells or microscopic particles are viewed conventionally under high magnification, the depth of field is correspondingly small. A photograph that freezes motion of the specimen captures in a focused image a very limited depth of field within the specimen. The disadvantages of this restriction can be overcome if the photograph is a hologram, which in a single snapshot contains potentially all the ordinary photographs that could be made after successive refocusings throughout the depth of the living specimen. The image provided by the hologram may be viewed by focusing at leisure on any depth of an unchanging field. In making a hologram with a microscope, the specimen is illuminated by laser light, part of which is first split off outside the microscope and routed independently to the photographic plate, where it rejoins the subject beam processed by the microscope optics. Furthermore it can be shown that, if the reconstructing light of wavelength AT is longer than the wavelength As used in "holographing" the subject, a magnification given by M -

(!) e:)

(13-13)

results, where p is the object distance (subject from film) and q is the corresponding image distance (image from hologram). Object and image distances are equal when the reference and reconstructing wavefronts are both plane waves. The content of Eq. (13-13) implies, for example, that if the hologram were made with laser Xradiation and viewed with visible light, magnifications as large as 106 could be achieved without deterioration in resolution. This prospect has contritiuted to the interest in developing X-ray lasers. X-ray holograms could provide strikingly detailed three-dimensional images of microscopic objects as small as viruses and DNA molecules. The ability to view a hologram with radiation of a different wavelength than that used in making the hologram offers other interesting possibilities, the use of an ultrasonic wave hologram to replace medical X-rays, for example, or the reading of a radar hologram with visible wavelengths. In fact, in his original work, Gabor proposed reconstruction of an electron wave hologram with optical wavelengths in an effort to improve the resolution of electron microscopes. The mention of ultrasonic holograms above implies that the waves producing a hologram need not be electromagnetic in nature. Indeed, the principles of holography do not depend on the transverse character of the radiation. Because of the ability of ultrasonic waves to penetrate objects opaque to visible light, hologram.. formed with such waves can be very useful. Opaque bodies that are promising candidates range from the human body to archeological tombs. Structures and cavities inside Sec. 13-6

Other Applications of Holography

275

can be revealed in three-dimensional images formed by ultrasonic holography. Figure 13-7 illustrates another application of ultrasonic holography to reveal objects under the surface of the ocean. G I and G 2 represent two phase-coupled generators radiating coherent ultrasonic waves. The wavefront from G 2 is deformed by an underwater object and interferes with the undeformed reference beam from G I. The deformations of the water surface represent an acoustic hologram. If this region is illuminated with monochromatic light, the light diffracted from the deformations can be photographed and converted into a visual image of the underwater object. The potential offered for submarine detection is an obvious military application.

Figure )3...7 Deformations in the surface of the water due to two coherent ultrasonic waves.

Holographic data storage also offers tremendous potential. Because data can be reduced by the holographic technique to dimensions of the order of the wavelength of light, volume holograms can be used to record vast quantities of information. As the hologram is rotated, new exposures can be made. Photosensitive crystals, such as potassium bromide crystals with color centers or the lithium niobate crystal, can be used in place of thick-layered photoemulsions. Because information can be reduced to such tiny dimensions and the crystal can be repeatedly exposed after small rotations that take the place of turning pages, it is said that all the information in the Library of Congress could theoretically be recorded on a crystal the size of a sugar cube! Information may, of course, be recorded in digital form and thus read by a computer, so that holographic storage offers a means of providing computer storage. In conjunction with the optical transport of computer information through optical fibers, information handling, storage, and retrieval can aU be done using light. A fuscinating aspect of holographic data storage lies in its reliability. Since every data unit is recorded throughout the volume of the hologram, in unique holograJ.f1ic fashion, damage to a portion of the hologram, while affecting the signal-to-noise level of the reconstructed image, does not affect its reliability. Information is not lost, as would be the case in other memory devices, where every bit of information has unique storage coordinates. In a reciprocal sense, computers are used to advantage in the science of holography by making possible the construction of synthetic holograms that faithfully represent three-dimensional objects. The object is first defined mathematically by specifying its coordinates and the intensity of all its points. The computer calculates the complex amplitude that is the sum of radiation due to the object and the reference wave and then directs the drawing of the hologram, which can be photographed and reduced to the appropriate fringe spacings required. For example, an ideal aspheric wavefront can be created synthetically to serve as a model against which a mirror may be shaped, using interference between the two surfaces as a guide to making appropriate corrections. Another area in which holograms may be very useful is in pattern recognition. Briefly. the procedure is as follows. A text is scanned, for example, for the presence of a particular letter or word. Light from the text to be searched is passed through a hologram of the letter or word to be identified in an appropriate optical system. The

276

Chap. 13

Holography

presence of the letter is indicated by the formation of a bright spot in a location that of the in the text. The hologram acts as a matched indicates the recoglnlZimg and that spectrum similar to the one recorded can be to holographic reading of microfilms, for examaptlUCatH)nS include the use of a memory bank of holograms of partieuconstructed from aerial photographs. Weapons could, by patselect proper It has also been suggested that robots could "1pnt.t·,, and be directed toward appropriate objects in the same way. Pattern recogin Chapter 25. nition is discussed In",..,,,,'" that redirect light may be used as inexpensive optical elein the place of lenses and mirrors. To cite one popular application, ments, laser readers of the universal product code on groceries use a spinning disc outfitted lenses. By continuously providing many angles of with a number of laser scanning, the product code can be identified even when the item is passed casually over the scanner.

13-1. Use Eq. (10-13) for the superposit.ion of two unequal beams to show that the irradiance pattern of a Gabor zone (the hologram of a is given approximately by 1

13-2.

13-3.

13-4.

13-5.

13-6.

13-7.

where A == 11 + 12 B and a = '1T/2sA. Here I. and h are the s is the distance of irradiances due to the reference and the object point from the and A is the For the approximation, assume the difference between the two beams is much smaller than s, so that we are looking at the inner zones of the hologram. (a) Show that if the local mtio of reference to beam irmdiances is a factor N at some of a then the of the resulting fringes is 2VN/(N + (b) What is the where the irradiance of the reference beam is three times that of beam? Show that the sepamtion d of in the formation of a holographic grating, as in Figure 13-5, is given by A/(2 sin 6), where 2f) is the angle between the coherent beams in the film and A is the of the beam<; in the film. Assume the beams are incident symmetrically on the film's surface. If the beams are laser beam'> of 488-nm wavelength and the between beam'> is 120°, how grooves per millimeter are formed in a plane emulsion oriented perpendicular to the fringes? Assume n == I. The angle between the signal and reference beams construction of a hologram is 20°. If the light is from a He-Ne laser at 633 nm, what is the Assume a refractive index of I for the emulsion. Suppose a hologram is to be made of object at a wavelength of 633 nm. What is the pel"mlSslltlle object does not move more than A/1O during the "A~""~'U", During the construction of a hologram, a beam splitter is selected that makes the amtimes that of the beam at the emulsion. plitude of the reference beam What is the maximum ratio of beam irnIdiances there? Let us suppose that as a theoretical limit, one bit of information can be stored in each A3 of hologram volume. At a of 492 nm and a refractive index of I determine the storage capacity of I of volume.

Chap. 13

Problems

'Z11

13-8. A volume hologram is made directed monochromatic beams of cocollimated laser light at 500 nm, as in 13-5. (a) Determine the spacing of the developed silver planes within the emulsion. (b) What is reinforced in reflected when white light is incident normally on the hologram? (c) Repeat (b) when the angle of incidence (relative to the normal) is 30°. Assume a film refractive index of I. 13-9. Two beams of planar wavefront, 633-nm coherent light, whose directions are 120" strike a emulsion. (a) Sketch the arrangement, showing the orientation of the planes of constructive in terference within the emulsion. (b) Determine the planar of the volume hologram. (c) At what angle of incidence relative to the silver planes is a wavelength of 450 nm reinforced? Assume n = I in the emulsion. 13-10. that the blue component of a white-light hologram is formed as in Figure 135, using light of 430-nm wavelength. If emulsion shrinkage is 15% during proccsswhat wavelength is reinforced by the fringes on reconstruction? How does this affect the holographic image under white-light viewing? 13-11. A hologram is constructed with ultraviolet laser of 337 nm and viewed in red laser at 633 nm. (a) If the original reference beam and the reconstructing beam are both collimated, what is the of the holographic compared with the SUbject? (b) What magnification would result if coherent X-radiation of I A. wavelength were available to construct the hologram? 13-12. (a) that the reconstructed wavefront from the of a point source produces both the real and virtual shown in l3-1 b. find the irradiance at the film due to the superposition of a plane and a spherical wave. Then, find the of the light transmitted by the developed film when irradiated by the reference beam. the terms as done in the discussion of a hologram of a three-dimensional subject. (b) Show that the phase delay of the at a point on the film a distance y from the is given by where d is the distance of the point source from the film. This result follows when y
[I J

Howard Michael. Principles of Holography. New York: John Wiley and 1975.

M. Holography. New York: Academic 1974. [21 [31 Stroke, W. An Introduction to Coherent Optics and Holography, 2d ed. New York: Academic Press, 1969. Vest, C. M. Holographic Interferometry. New York: John Wiley and 1979. [5] H. John, ed. Handbook Optical Holography. New York: Academic 1979. (6) Caulfield, H. John. "The Wonder of Holography." National Ge,?r!r.1nhic 165 (March 1984): 364. [7] Leith, Emmett N., and Juris Upatnieks. "Photography by Laser." Srjpm'itir American (June 24.

278

13

Holography

[8J Keith S. "Advances in American 1968): 40. [9] MethereJl, Alexander F. "Acoustical Holography." American (Oct. 1969): 36. [IOJ Emmett N. "White-Light Holograms." American (Oct. 1976): 80. [IIJ George 0., John B. DeVelis, George B. Parrent, Jr., and Brian J. Thompson. Ph\,dr,f]l Optics Notebook: Tutorials in Fourier Optics. Bellingham, Wash.: SPIE Optical Pn";",,.,,,.. ;,,o Press, 1989. Ch. 25-27.

Chap. 13

References

219

14 ",\ \

\

,, ,,

"\

\

Matrix Treatment of Polarization

INTRODUCTION r". ...r".<,,,.n,f<.fi,,n of a plane wave of radiation by a drawing such 8-7 is not applicable to ordinary wave, the electric field as vector always oscillates parallel to a fixed direction in space. Light of such character to be linearly polarized. The same can be said of the magnetic field vector, "1"'''''''''''1'' an orientation perpendicular to the field vector such that the direction of E )( B is everywhere the direction of wave propagation. Suth a wave be produced by a distant single-dipole oscillator or by a collection of dipole oscillators radiating in synchronization. Ordinary light, however, is produced by a ....." .. u .... of atomic sources whose radiation is not synchronized. Consider a beam of ordinary light, as that by a hot filament. The resultant E-field vector, generated from a collection of atoms, does not maintain a constant direction of oscillation, nor does it vary in a regular manner, proelliptically polarized or circularly Such ordinary light is simsaid to be unpolarized. Of course a beam of can consist of a mixture of polarized and unpolarized light. in which case it is said to be partially polarized. The possibility of polarizing light is related to its transverse character. If light were a longitudinal W.dve, the of polarized light in the ways to be described would simply not be possible. Thus the polarization of light constitutes proof of its transverse character.

280

In our mathematical of polarization, we shaH a matrix technique developed by R. Jones [I]. First we two-element column in various modes of Then we exmatrices or vectors to amine the physical elements that polarized and discover corresponding 2 X 2 matrices that function as mathematical operators on the Jones vectors. In the following chapter, we examine in more detail the physical processes that are responsible for pnXlulcUllg ...."1" ... 7<• .-1

14-1

REPRESENTATION UGHT: JONES VECTORS

POLARIZED

' ..AI""" ...."'. a ray of light directed perpendicularly out of the page, situated at the

of the axis system in 14-1. Let the E-field of the be by the and changes direcvector shown. Since the E-field varies continuously in tion every half-period, the shows the magnitude and direction of E at a parof E along the x-and y-axes be and Ey , reticular instant. Let the Then, in terms of the unit vectors i and j, (14-1)

E y

--------~------~x

o

Figure 14-1 of the instantaray in the neous E-vector of a +z-direction. Oscillations of the E-vector are equivalent to oscillations of the two orthogonal components, Ex and E}..

Introducing the space and time dependence of the component vibrations, Eox e i(kz-wt+'I'x)

(14-2)

and (14-3) for component waves in the +z-direction with tpx and tpy. Combining with (14-1),

C:UUVlI"UU",,"'.,L:.Ox

and EOy and

E= which may also be written

E=

+

(14-4)

The bracketed quantity,_separated into x-and y-components, is now recognized as polarization of the complex amplitude Eo for the polarized wave. Since the state the is completely determined by the relative amplitudes and of these we need concentrate on the complex amplitude, as a twoelement matrix, or Jones vector, (14-5)

Sec. 14-1

Mathematical

H.,nre..:,."nt~,ti..."n

of Polarized light: Jones Vectors

281

y

lei

(bl

(011

Figure 14·2 Representation of E-vectors of linearly polarized light with various special orientations. The direction of the light is along the z-axis.

Let us deterrnine the particular forrns for Jones vectors that describe linear. circular. elliptical polarization. In 14-2a. vertically travels in the +z-direction with its along the y-axis. a sinusoidally varying rnagnitude as it progresses, only the arnplitude of the electric fieid is ~1'rnbolized in both the y and negative y directions. [n this case we set and = A. say. In the absence of an cornponent, the t{>y rnay be to zero for convenience. by Eq. 04-5), the corresponding Jones vector is

o

when only the rnode of polarization is of interest, the amplitude A rnay be set equal to one. The Jones vector for vertically linearly is then sirnply [n This sirnplified forrn is the normalized form of the vector. In general, a vector [g] is expressed in normalized forrn when

lal 2 + Ibl 2 14-2b repreSel1lts h()riZ<)lltaUy polarized light, for

letting

On the other hand, Figure 14-2c linearly polarized light whose vltlratiorlS occur a line rnaking an a with to the x-axis. Both x- and ycornponents of E are simultaneously Evidently this is the case that reduces to the vertically polarized rnode when a = 90° and to the horiwntaUy polarized mode when a = 0°. Notice that to the resultant shown, the two perthat they rnust pass through toget her , their axes together. reach their rnaxirnurn values and then return together to continue the cycle. Figure 14-3a makes this sequence clear. Accordingly. since we require rnerely a relative of zero, we set t{>x t{>y O. For a resultant with arnplitude A, the perpendicular cOlmponcent are Eo" = A cos a and Eoy A sin a. The Jones vector takes the form

Eo =

[~::::::] = [~:::] = A[:~::]

(14-6)

For the norrnalized forrn of the vector, we set A = I, since a + a 1. Notice that forrn does indeed reduce to the Jones vectors found for the case a = 00 and a = 90°. For other for exarnple, a = 60°,

E,

Chap. 14

~

I [~I~ ~ [~I

[:::

Matrix Treatment of Polarization

y

(al

y

Figure 14-3 (a) Perpendicular vibrations in linearly polarized light with in the first and third quadrants. PerpendiclJlar vibrations 180" out of phase linearly light with E-vectors in the second and fourth quadrants.

fbI

where a and b are real numbers. the inclination of is given by

a

tan-I

(~:)

tan-I

(~)

(14-7)

Generalizing a bit, suppose a were a negative angle, as in Figure 14-3b. In of the vector is a negative number, since the sine is an odd this case, the function, whereas the element remains positive. The negative sign ensures that the two vibrations are 1800 out of phase, as needed to produce linearly polarized light with E-vectors lying in the second and fourth quadrants. Referring to Figure 14-3b means that if the x-vibration is increasing from the origin along its the must be increasing from the origin along its negative direction. The resultant vibration takes place along a line with negative slope. a Jones vector [g] with both a and b real numbers, not both zero, light at inclination angle a = tan-I (b/a). that, in determining the resultant vibration due we are in fact determining the appropriate Lissajous figure. If the difference between the vibrations is other than 00 or 1800 , the resultant E-vector traces out an ellipse rather than a straight line. Of course, the Sec. 14-1

Mathematical F/""nn:..,.. ntl'l,ti",n of Polarized light: Jones Vectors

283

straight line can considered a special case of the ellipse, as can the circle. Figure 14-4 summarizes the sequence of figures as a function relative phase =
-cf* * AI/> = 0°

AI/>

'-

= 45°

AI/>

-;

/

r-

L/

A4>

= 360"

AI/>

=

{ -45

4 "'" -, 4 "

r:- ~.

'-.

0

AI/> =

315"

./

AI/>

90"

= 135

0

Af!>

180"

~

./

{ -90"

AI/>

270"

=

{ -135" 225.

AI/>

:1:180"

....''''..'1''.>'' figures as a function of relative phase for orthogonal vibraamplitude. All allgle lead greater thall 180<' may also be reprelag of less thall 180<'. For all we have adopted the

Now suppose by 9(f. Then at the has reached its maximum displacement + A, is zero. A fourth of a period later, Ex is zero and + A, and so on. 14-5 shows a few samples in the process of the resultant vibration. For the cases illustrated where it is to make cpy > cpX' apparent the x-vibration leads the in the formulation of the E-field in contradiction results from our choice of (14-2) and (14-3), where the time-dependent term in the exponent is negative. To show this. let us observe the wave at z = 0 and choose q>x = 0 and cpy = E, so that cpy > CPX' Equations (14-2) and (14-3) then bec:OITle

ET. = Ey

=

The negative sign hefore E indicates a lag E in the y-vibration relative to the xvibration. To see that these equations the sequence in Figure 1 we take Eoy = A and E their real and set A cos wt

A cos ( wt

Recalling that w = verified. Also, since

-'11"2)

each of the cases in Figure 14-5 can

+ 14

A sin wt

Matrix Treatment of Polarization

wt

+

y

-fi y

p

E

---+--x

(=0

(=

I

I

8

Ey

=A sin 45°

Ex = A ccs 45°

Ibl

lal

leI

x

Figure 14·5 Resultant E-vibration due to orthogonal component vibrations of equal magnitude and difference of 90", shown at three different times. The points P represent the of the resu Itant. In (c) a sketch of the circular path traced E is also shown. Notice that the E -vector rotates counterclockwise in this case.

the tip of the vector traces out a circle of radius A. We now deduce the Jones vector for this case--where Eax = If'x 0, and 'PY = 'IT /2. Then

leads Ey-taking

- [~::~] [Ae:/2] A[!J

(14-8)

To determine the normalized form of the vector, notice that 12 + 1i 12 = + 1= 2, so that each element must be divided by V2 to unity. Thus the Jones circularly polarized light when E rotates counterclockvector (l/V2)[J] wise, viewed head-on. This mode is called left-circularly polarized light. leads Ex by 'IT /2, the result will be cin;uhlriv Similarly, if light with clockwise rotation, or right-circularly 1Jl),fllr.lze,(J (-'IT /2) in the normalized Jones vector,

Eo=~[l Notice that one of the elements in the Jones vector for polarized light is now pure and the magnitudes of the elements are the same. Due to the may not always be mathematical form of the vector, the actual character of the the Jones vector right-circularly immediately For polarized

the irradiThe prefactor of the Jones vector may affect the amplitude ance of the light but not the polarization mode. Prefactors such as 2 and 2i may therefore be ignored unless information regarding energy is required. Next suppose even though the phase difference between the component If A orthogonal vibrations is still 90°, the vibrations are of and EOy = say, is modified to give

[~J

counterclockwise rotation

and

clockwise rotation

These instances of are illustrated in Figure 14-4 for and Notice that a of 90" is equivalent to a lead of . The the x- or y-axis, as in Figure is oriented with its and In addition, either case may .,.."u1.",<> on the relative magnitudes of clockwise rotation of E around the ellipse (when Ey leads Ex) or COlJnterc:lo<:kVl(ise rotation (when E" leads Based on these observations, we conclude that a Jones one of which is pure reprevector with elements of Sec. 14-1

Mathematical Representation of Polarized light: Jones Vectors

285

<

Eox

Figure 14-6 Elliptically afj? 90".

POIlII1Zl~d

light for the

case

sents polarized light oriented along the xy-axes. The normalized forms of the Jones vectors now must a prefactor of IjVA 2 + B2. lt is also possible to produce elliptically polarized light with principal axes inclined to the xy-axes, as evident in Figure 14-4. This situation occurs when the other than m1T difference between component vibrations is some polarization} or {m + (circular or elliptical polarization oriented symmetrically Here m = 0, t, 2, .... For example, consider the case where about the Ex leads some angle e, that is, f{)y - f{)x = E. f{)" = 0, f{)' = e, 1::Ox = the Jones vector is

= [~:>~:;::J

- [b~re]

Using Euler's theorem we write be;'

The

= b(cos e + i sin e)

B

+

iC

vector for this general case is then

Eo =

[B : iC]

04-9)

where one of the elements is now a complex number r--;,---;:;:,----;: parts. The normalized form must be divided This form of the including all those discussed previously as special Jones vector is the most to show that the cases. With the help of analytical geometry. it is to whose Jones vector is by Eq. (14-9) is inclined at an angle a with the x-axis, as shown in Figure 14-7. The angle of inclination is determined from (14-10)

tan 2a

\

, \ \

\

\ \ \ \

\ \

\ \

Figure 14-7 Elliptically light orirelative to the x-axis. ented at an

Chap. 14

Matrix Treatment of Polarization

The ellipse is situated in a rectangle of sides 2Eox and 2Eoy. In terms of the parameters A, E, and C, the derivation of Eq. (14-9) makes dear that

Eox = A,

EOy =

VE 2 + C 2

,

and

E

tan-I

=

(~)

(14-11 )

Example Analyze the Jones vector given by

to show that it represents elliptically polarized light. Solution The light has relative phase between component vibrations of cpy - cpx = E = tan- 1 = 26.6°. Since Eo" = 3 and Eoy = V22 + 12 = V5, the inclination angle of the axis is given by

m

_ 1.

a-

2 tan

-I (2)(3)(Vs) cos (26.6°) - 35 go 9-5 -.

With this data the ellipse can be sketched as indicated in Figure 14-7. More precisely, the equation of the ellipse is given by

EX)2 (Ey)2 - 2 (Ex) (Ey) cos ( -Eox + -EOy EOx EOy

E

= sin2

E

04-12)

For this example, the equation of the ellipse is E2 1E2 + -; -

0.267 ExEy = 0.2

When Ex lags En the phase angle E becomes negative and leads to the Jones vector representing a clockwise rotation instead,

Table 14-1 provides a convenient summary of the most common Jones vectors in their normalized forms. It must be emphasized that the forms given in Table 14-1 are not unique. First, any Jones vector may be multiplied by a real constant, changing amplitude but not polarization mode. Vectors in Table 14-1 have all been multiplied by prefactors, when necessary, to put them in normalized form. Thus, for example, the vector = 2[1] and so represents linea!JY polarized light making an angle of 45° with the x-axis and with amplitude of 2V2. Second, each of the vectors in Table 14-1 can be multiplied by a factor of the form e icp , which has the effect of promoting the phase of each element by cp, that is, cpx ~ cpx + cp and cpy ~ cpy + cpo Since the phase difference is unchanged in this process, the new vector represents the same polarization mode. Recall that the vectors in Table 14-1 were formulated by choosing, somewhat arbitrarily, cpx = O. Thus, for example, multiplying the vector representing left-circularly polarized light by e i7f / 2 = i,

m

produces an alternate form of the vector. Clearly, given the second form, one could deduce the standard form in Table 14-1 by extracting the factor i. Sec. 14-1

Mathematical Representation of Polarized light: Jones Vectors

281

SUMMARY OF JONES VECTORS Eo

TABl.E 14-1 1.

Linear Polarization (&p

=

= tmr)

General:

Eo

Vertical: Eo

[°1]

WI

Ri~1

+ (ll.ct>

1)

-+ -+

III. Elliptical Polarization Left:

(ll.cf> = (m + !)-rr)

Eo

Lefl:

Righi:

288

Chap. 14

a]

sin a

Horizontal: Eo =

+ II. Circular Polarization

COS [

Matrix Treatment of Polarization

[~]

The usefulness of these Jones vectors will be demonstrated Jones matrices reJ)reserltll1lg polarizing elements are also developed. However, at this point it is alposslltlle to calculate the result the superposition of two or more polarized their Jones vectors. addition of left- and polar-

[~ +:J [~] or linearly polarized light of twice the amplitude. We conc1ude that linearly polarcan be regarded as made up of left- and polarized ized of vertilight in proportions. As another consider the cally and horizontaHy linearly in phase:

[~J + [~J [~J The result is linearly polarized light at inclination. Notice the addition of orthogonal of linearly is not unpolarized even though light is often symbolized such components. There is no Jones vector representing unpolarized or partially polarized light. I

14-2 MATHEMATICAL REJPRJ!:Sf:NJ:A JUNI:::i MATRICES

OF P04!.AlirlZf:RS:

can serve as optical elements that transmit light but the state physical mechanisms underlying their operation will be disc~ssed in the next chapter. Here it wiH be to categorize such nnll~rl:'7Plr~ in terms of their effects, which are basically three in number.

.of

~1,3rt:lvpIIV removes all or most of Linear Polarizer. The the E-vibrations in a given direction, while vibrations in the pelrpendicullaI direction to be transmitted. In most cases, the selectivity is not 100% efficient, so that transmitted Jight is partiaJJy polarized. Figure 14-8 illustrates the operation "" .................. light traveling in the +z-direction passes a plane whose preferential axis of or transmission axis (TA), is ver-

y

14-8 Operation of a linear

z

'A matrix that handles partially polarized Mueller matrices can be found in [2] and [3].

Sec. 14-2

n"I:''';7,M'

I x 4 Stokes vectors and 4 x 4

Mathematical Representation of Polarizers: Jones Matrices

289

tical. The unpolarized light is represented by two perpendicular (x and y) vibrations, since any direction of vibration present can be resolved into components along includes components only along the TA direction directions. The light and is therefore linearly polarized in the vertical, or y-direction. The horizontal the components of the original light have been removed by absorption. In the process is assumed to be 100%

Phase Retarder. The phase retarder does not remove either of the component orthogonal but introduces a phase difference between them. If light to each vibration travels with different speeds through such a retardation plate, there will be a cumulative phase acp between the two waves as they emerge. Symbolical1y, 14-9 shows the effect of a retardation plate on unpolarized light in a case where the vertical travels through the plate rnster the horizontal component. This is suggested by the schematic separation of the two components on the optical although of course both waves are simultaneously present at each point along the axis. The fast (FA) and slow axis (SA) directions of the plate are indicated. When the net difference acp = , the is called a quarter-wave when it is , it is called a Iullf retardation Mlve plate. y

SA __

z

Figure 14-9 Operatiol1 of a phase retarder.

Rotator. The rotator has the effect of rotating the direction ~f linearly poHght incident on it by some angle. Vertical linearly light is shown inciden t on a rotator in Figure 14- 10. The effect of the rotator elemen t is to transmit linearly polarized light whose direction of vibration has, in this case, rotated counterclockwise by an angle O. We desire now to create a set of matrices corresponding to these three of polarizers so that, just as the optical element alters the polarization mode of the actuallight beam, an element matrix operating on a Jones vector will produce the same result mathematically. We adopt a pragmatic point of view in formulating appropriate matrices. For example, consider a linear polarizer with a transmission axis the vertical, as in Figure 14-8. Let a 2 x 2 matrix the operate on vertically polarized light, let the elements of the matrix to be determined

290

Chap. 14

Matrix Treatment of Polarization

y

x

F'mguce 14-10 Onerrali.rm of a rotator.

z

be represented by letters a, b, c, and d. The resultant transmitted or product this case must again be vertically linearly light. ~y:rnooucaIIV [:

in

~] [~] = [~]

equation is equivalent to the a(O)

c(O)

~loph.",iC'

+ b(I) = + d(I) =

equations

0 1

from which we conclude b = 0 and d = L To determine elements a and c, let the same polarizer on horizontaHy light. In this case no light is trans" ....""", or

The corresponding algebraic equations are now

a(1) cO) from which a = 0 and c matrix is

+ b(O) + d(O)

0 0

O. We conclude here without further proof that the ap-

nrr.nri",tp

M =

[~ ~]

linear polarizer, TA vertical

04-

The TA horizontal, can be obtained in a similar manner and is included in ]4-2, near the end of this chapter. next that the matters as as linear polarizer has a TA inclined at 45° to the x-axis. To possible we allow light polarized in the same direction as-and perpendicular to-the TA to pass in tum through the Following the approach used

Sec. 14-2

Mathematical Representation of Polarizers: Jones Matrices

291

Equivalently,

a+b= c+d=l

or a b c

a-b

0

c

0

d

4. Thus the correct

d M

=

~[~

is

linear nnli!'lrl7£"r TA at 45°

;]

In the same way, a general matrix a can be determined. This is left as an exercise for M = [

nnll!'lri7pr

04-14) with TA at

()

sin () c~s ()]

(1

sin () cos ()

cases, with () = 90° and (1 includes Eqs. 04-13) and 04-14) as , respecti vely . Proceeding to the case of a phase retarder, we desire a matrix that will transform the elements into

+ ·xl

into

+ 'y)

and

Ins·J)e(~ti(m

is sufficient to show that

general form of a matrix

Thus

M

where

is acc:olTlplishe:d by the matrix operation

rpnrp,,~'n

retarder is

[ei€x.o] o

(14-16)

retarder

e"'y

and Ey represent the advance in of the Ex- and Ey-components of the light. Of course, Ex and Ey may be quantities. As a case, COlJlSIOler a quarter-wave plate (QWP) for which Ifl.E I = 71'/2. We c!istinguish the from the case for which Ex Ey case for which Ey - Ex = 71'/2 (SA horizontal). In the former case, let Ex -71' /4 and Ey = +71'/4. other choices-an infinite number of them-are possible, so that Jones matrices, like Jones vectors, are not This particular choice, however, to a form: common form of the matrix, due to its Ex

,n'-"""',nr

Similarly, when

Ex

>

QWP, SA

(14-17)

QWP, SA horizontal

04-18)

Ey ,

M =

eml{~

Corresponding matrices for half-wave plates (HWP), where I&

14

Matrix Treatment of Polarization

I

71', are given by

e-hT12 [OI ~J

M= M

[e~/2 e-~/2] = ehT/{01 ~]]

HWP. SA vertical

(14-19)

HWP, SA horizontal

(14-20)

elements of the matrices are identical in this case, since advancement of phase by '1T is physically equivalent to retardation by '1T. The only difference lies in the prefactors that modify the phases of all the elements of the Jones vector in the same way and hence do not affect interpretation of the results. for a rotator of angle f3 is that an E-vector oscillating at angle 6 be converted to one that oscillates linearly at angle (6 + f3). Thus the matrix elements must satisfy

b] [c~s 8] sm 6

(8 sin (6

COS

a [c d

[

+ f3)] + f3)

or

a cos 8 + b sin 6 = cos (8 + f3)

c cos 6 + d sin 6

sin (6

+ f3)

From the trigonometric identities for the sine and cosine of the sum of two angles, cos (6 sin (6

+ f3) + f3)

f3 6 cos f3 +

f3 6 sin f3

cos 6 cos

sin 6 sin

sin

cos

it follows that

a=cosf3 b = -sin

c=sinf3

f3

d=cosf3

so that the desired matrix is M = [cos f3 sin f3

an

-sin f3J cos f3

rotator through angle

+f3

(14-21)

The Jones matrices derived in this chapter are summarized in Table 14-2. As consider the production of circularly polarized by

TABLE 14-2 SUMMARY OF JONES MATRICES

I.

Linear polarizers TA horizontal

[

~ ~]

TA vertical

[~ ~]

TA at 45° 10 horizontal

I

!]

II. Phase retarders General

~] ~]

QWP, SA vertical HWP, SA vertical

[

~~x

QWP. SA horizontal HWP. SA hori7..ontal

e
0

-~]

III. Rotator Rotator

Sec. 14-2

(O~ 0

+ f3)

Mathematical Representation of Polarizers: Jones Matrices

293

y

x

Figure 14-11

Production of ";rrlll,,,!.,

ized light.

z

combining a linear with a QWP. Let the linear polarizer (LP) ",rl1<1'1'I£'1> VIll,ralJlflg at an , as in Figure 14-11, which is then the incident on the QWP is divided QWP. In this fast and slow axes. On a phase difference of 90° results circularly polarized Jight. With the Jones calculus, this process is equivalent to allowing the QWP to on the Jones vector for the

emi{~ ~J ~ [!] = (~)em/{ 1 polarized light with amplitude l/Yz polarized light. If the filst and slow axes of calculation shows that the result is IPTI'_C":rt'.

ngll[-c:trCll18l1} IInl'J'lrllv

mollitu(je of nnl"ri'7prl

in-

stead.

Example eu~.ml1l-WiaVe

result of allowing left-circularly plate.

to pass

an

Solution We first need a matrix that can represent the et!!ntlll-wiClve plate, a retarder that introduces a relative phase of 2?T /8 Ex = 0, lEX

e M = [ 0 This matrix is then polarized light:

294

14

0] [1 0]

ei.y =

0

e irr / 4

to operate on the Jones vector rpl",rp<;:pnf

Matrix Treatment of Polarization

the left-

The resultant Jones vector indicates that the light is elliptically and . Using Euler's equation to expand the components are out of phase by iYlr 4 e / , we obtain 1_

eiYlr/4

__

+ i(~)

and using our standard notation for this case, we have 1

- ~ ,r;;'

whereA = 1, B

v2

and C

1

~ ,r;;

v2 Comparing this matrix with the general form in Eq. (14-5), we determine that 1 and = l. Making use of . (14-10), we also determine that Of course, the Jones calculus can handle a case where polarized is transmitted by a series of polarizing elements, since the product of element matrices can represent an overall system IrUltrix. If light represented by Jones vector 'V passes sepolarizers represented by ~Jl h IDl 2 , IDl 3 • • • • , IDem. quentially through a series that is, (IDl m • •• IDl3IDl 2 IDl 1 )'V IDls'V matrix ~Jls

then the

=

~Jlm'"

IDl3 IDl2 IDl I •

14-1. Derive the Jones matrix, Eq. IS), representing a linear polarizer whose transmission axis is at an arbitrary (J with respect to the horizontal. 14-2. Write the normalized Jones vectors for each of the following waves, and describe completely the state of polarization of each. (a) E = ilio cos (kz !UI) cos (kz WI) (b) E

= ilio sin 27r

(c) E

sin

(± - it) + jEo sin - WI)

(d) E = ilio cos (kz - wt)

+

!UI

~)

- wI

+ ~)

sin (kZ -

+ jEo cos (kZ

ft )

14-3. Describe as completely as possible the state of polarization of each 'of the following waves, including amplitude and wave direction. Be careful to distinguish between the '\I=l. unit vector i and the complex number i (a) E 2Eoie;(kz-wt) (b) E EO(3i + 4j)e '(h-wl) (c) E = 5EoO ij)ei(h+WI) 14-4. Two linearly polarized beams are by E,

EO,(i - j) cos (kz

WI)

and

Ez

=

i

+ j) cos (kz

WI)

Determine the between their of polarization by (a) forming their Jones vectors and finding the vibration direction of each and (b) the dot product of their vector amplitudes. 14-5. Find the character of polarized light after in turn through (a) a half-wave plate with slow axis at 45"; (b) a linear polarizer with transmission axis at 45°; (c) a quar-

Chap. 14

Problems

ter-wave plate with slow axis horizontal. Assume the original light to be linearly polarized vertically. Use the matrix and the final Jones vector to describe the product light. (Hint: First find the effect of the HWP alone on the incident light.) 14-6. Write the equations for the electric fields of the waves in exponential form: (8) A linearly polarized wave traveling in the x-direction. The E-vector makes an anof 300 relative to the wave in the y-direction. The major axis of (b) A right elliptically the ellipse is in the z-direction and is twice the minor axis. A linearly polarized wave in the in a direction making an angle of 45° relative to the x-axis. The direction of polarization is the z-direction. 14-7. Determine the conditions on the elements A, and C of the general Jones vector 14-9), representing polarized light, that lead to the following special cases: (a) linearly polarized light; (b) elliptically with major axis aligned along a coordinate axis; (c) circularly In each case, from the meanings of A, 8, deduce the possible values of phase difference between component vibrations. 14-8. Write a l.'Omputer program that will determine of elliptically polarized light (14-12), with constants A, 8, and C, and from the equation for the ellipse, variable input parameter Ex. Plot the ellipse for the in the text,

14·9. Specify the polarization mode for each of the

fOlliow'lno

Jones vectors.

14-10. Linearly polarized light whose E is inclined at +300 relative to the x-axis is transmitted by a QWP with SA horizontal. Describe the mode of the product 14-11.

14-12.

14-13.

14-14.

14-15.

14-16.

the Jones calculus, show that the effect of a HWP on light linearly polarized at inclination angle ex is to rotate the plane of polarization through an angle of 2a. The HWP may be used in this way as a "laser-line rotator," the plane of polarization of a laser beam to be rotated without having to rotate the laser. An important application of the QWP is its use in an "isolator." For example, to prevent feedback from interferometers into lasers by back the beam is first allowed to pass through a combination of linear and QWP, with OA of the QWP at 45° to the TA of the Consider what to such light after reflection from a plane surface and transmission back through this optical device. linearly polarized with a horizontal transmission axis is sent another linear polarizer with TA at 45° and then through a QWP with SA horizontal. Use the Jones matrix technique to determine and describe the A beam passes consecutively through (I) a linear noll::lri7PY' wise from (2) a QWP with SA a linear with TA hori(4) a HWP with FA horizontal, (5) a linear with TA vertical. What is the nature of the product light? light passes through a linear polarizer with TA at 60" from the vertical, then through a QWP with SA horizontal, and finally through another linear polarizer with TA vertical. Determine using Jones matrices the character of the light after pass(a) the QWP and (b) the final linear nol:~Y'j7"Y' Determine the state of polarization of circularly polarized light after it is passed nor(a) a QWP; (b) an eighth-wave plate. Use the matrix method to support your answer_

14

Matrix Treatment of Polarization

14·17. Show that the matrix

[~i

i

]represents a

polarizer, converting any

incident polarized into right circularly-polarized light. What is the proper matrix to represent a left-circular ""I,~ .."·,,,.. ·, 14·18. Show that polarization t1ln be as a combination of circular and linear polarizations. 14·19. Derive the equation of the ellipse for light in (14-12). (Hint: Combine the Ex and equations for the general case of elliptical polarization, eliminating the space and time dependence between 14-2(1. (a) Identify the state of polarization to the Jones vector

and write it in the standard, normalized form of Table 14-1. (b) Lei this light be transmitted an element that rotates linearly polarized by +30". Find the new, normalized form and describe the result. 14-21. Determine the nature of the polarization that results from 04-12) when (a) EO = 1T /2; (b) = Eery = (c) both (a) and (b); (d) EO O. 14-22. A quarter-wave plate is placed between crossed polarizer and analyzer such that the angle between the TA and the QWP fast axis is 6. How does the emergent light vary as a function of 6?

REFERENCES [I] Jones, R. Clark. "A New Calculus for the Treatment of Optical SVstemls." Journal of the Optical 31 (1941): 488. [2] Walker, M. J. "Matrix Calculus and the Stokes Parameters of Polarized Radiation." 22 170. American Journal [3] Shurdiff, W. A. Polarized Production and Use. Cambridge, Mass.: Harvard University 1962. [4] Gerrard, and J. M. Burch. Introduction to Matrix Methods in Optics. New York: John Wiley and 1975.

Chap. 14

References

297

15

Production of Polarized Light

INTRODUCTION

Any interaction of light with matter whose optical are asymmetrical along light. directions transverse to the propagation vector provides a means of Indeed, if were longitudinal rather than transverse in its nature, transverse material along the propagation vector could not alter the sense of the oscillating E-vector, and the physical to be described here would have no polarizing or spatially selective effects on light beams. The experimental observation that light can be polarized is, therefore, dear evidence of its transverse nature. The processes that polarized are discussed in this chapter most under the following general areas: (I) dichroism, (2) reflection, (3) and is described as a mechanism that nwdifies polar(4) birefringence. Optical light. Finally, photoelasticity is briefly discussed as a useful application. 15-1 DICHROISM: POLARIZA nON BY SELECnVE ABSORPTION

A dichroic polarizer selectively absorbs light with E-vibrations along a unique direction characteristic of the dichroic material. polarizer easily transmits light with of absorption. E-vibrations a transverse direction orthogonal to the This preferred direction is called transmission axis (TA) of the polarizer. In the is linearly in the same as the ideal the transmitted

transmission axis. The state of polarization of the light can most be tested by a second dichroic polarizer, which then functions as an analyzer, shown in Figure 151. When the TA of the is at 90° relative to the TA of the nnl, .. ri'7pr the is effectively extinguished. As the analyzer is the light transrmtlte<1 by the pair increases, reaching a maximum when their TAs are aligned. If 10 represents the transmitted intensity, then Malus' law states that the irradiance for any angle () between the TAs is by 1 = 10

1)

() y

L. TA

-

Analyzer

Figure 15-1 Crossed dichroic functioning as a polarizer-analyzer No is transmilled the analyzer.

Malus' law is easily in conjunction with Figure 15-2. Notice that the amp1itude of the light from the is Eo cos (). The irradiance 1 (in W/m2) is then proportional to the square of this result. The impressive ability of dichroic materials to absorb light along one and to transmit light easily with E along a direction can best be understood by to a standard with mi15-3. Wavelengths of microwaves range roughly from crowaves, illustrated in I mm to 1 m. It is found that when a vertical wire grid. whose spacing is much smaller than the wavelength, intercepts microwaves with vertical linear po,larization,

TA

I

Polarizer

Analyzer

15-2 Illustration of Malus' law.

Sec. 15-'

Dichroism: Polarization by Selective

Ah-c:nlrntiin

299

y

Figuce 15·3 Action of a vertical wire on microwaves. Effective absorption of the vertical component of Ihe radiation occurs when A ~ grid spacing.

little or no radiation is transmitted. when the grid waves polarized in a direction perpendicular to the wires, there is efficient transmission of the waves. The explanation of this behavior involves a consideration of the interaction of electromagnetic radiation with the metal wires that operate as a dichroic pu•.auL.... Within the metal wires, the free electrons are set in oscillatory motion by the oscillations of the electric field of the radiation. We know that each electron so oscillating constitutes a dipole source that radiates electromagnetic energy in an directions, the direction of the oscillation itself. the superposition of an incident electromagnetic wave with vertical E-vibrations the radiation of thesc electron oscillators leads to cancellation in the forward direction. It turns out, in fact, that the of the electromagnetic wave originating with the oscillating plpt"tr,nn<:. is I out of step with that of the incident radiation, I so that no wave ean propagate in the forward direction. In addition, the oscillation of the electrons is not free. The friction due to with lattice ......."".·tA<'_ tions, for example, constitutes some dissipation of energy, which must attenuate the incident wave. The chief reason for the disappearance of the forward wave, howwaves. Horizonever, is destructive interference between the incident and tally linearly light incident on the vertical wire would suffer the same across the wire is inexcept that appreciable oscillatory motion of the hibited. As a result, the generated electromagnetic wave is reduced in strength and effective cancellation cannot occur. If the grid is rotated by 90", the vertical E",I'.r<>tuu,,, are transmitted and the horizontal The wire nnlilUI'7i"<:' microwaves as a dichroic absorber polarizes radiation. For optical wavelengths, the conduction paths analogous to the grid wires must be much closer together. The most common dichroic for light is Polaroid polyvinyl alH-sheet, invented in 1938 by Edwin H. Land. When a sheet of cohol is heated and its hydrocarbon molecules tend to align in the direction of stretching. The stretched material is then impregnated with iodine atoms, which become associated with the linear molecules and provide "conduction" electrons to the to the wire Some naturally occurring materi~uch as the mineral tourmaline, also possess dichroic properties to some degree. AU that is in principle is that the electrons be much freer to respond to an wave in one direction than in an orthogonal direction. In incident nonmetallic materialS, the electrons acting as dipole oscillators are not frce. In this with respect to the incident wave, case t.he wave they generate is not out of and complete cancellation of the forward wave does not occur. The energy of the as the wave advances the abdriving wave, however, is 'The theoretical basis for this statement is presented at the end of Section 27-1, applied to the free electrons of a metal.

300

15

Production of Polarized Light

sorber, so that the efficiency of the dichroic absorber is a tmlctlon of the thickness. The absorption follows the usual for att€muaU()n 1=

where 10 is the incident irradiance and 1 is the The constant a is the absorptivity, or absorption C()(;~'(14clel1t rial. In a good, practical dichroic a is of wavelength, that is, the material appears transparent and yet behaves as a linear polarizer for an optical wavelengths. This ideal condition is not in Polaroid Hsheet, which is less effective at the blue end the when a Polaroid H-sheet is crossed with another such sheet bination contributes a blue tint to the transmitted

15-2 POLARIZATION BY REFLECTION FROM DIELECTRIC SURFACES

Light that is from is at least partially polarized. This is most easily confirmed by looking through a piece of polarizing filter while rotating it about the direction of When the preferred E-direction of the is to the TA of the from which light is specularly reflected into the eye appear reduced in brightness. This is prethe preferred E-vibration cisely the working principle of Polaroid in light reflected from level into the eye turns out to be horizontal, the TA of is in the the Polaroids in a To appreciate the that underlies this phenomenon, consider Figure 154, which shows a narrow beam of incident at an arbitrary angle on a smooth, flat dielectric surfuce. The beam is conveniently represented by 15-4a) and one parallel two perpendicular E-vibrations, one (Figure 15-4b) to the plane of of the page, including the of incidence. Standard notation is to incident ray and the normal drawn to the refer to these components as and (parallel component). Alternatively, the mode is called the TE electric) mode, and the Ep mode is called the TM mode, since the B-component of the wave is transverse to the plane of when the E-component is parallel. Consider first the or The action of E. on the electrons in the surface the dielectric is to stimulate oscillations along the same direction, or perpendicular to the The radiation from all these electronic a of light in two directions only, the didipole oscillators adds up to

\ (al

(bl

lei

15-4 reflection of light at a dielectric surface. (a) TE mode. (b) TM mode. (c) Polarizalion at Brewl>ler's angle.

Sec. 15-2

Polarization by Reflection from Dielectric Surfaces

301

of the reflected and refracted both also linearly polarized perpendicuof incidence. The reflected and refracted rays are both in a direction COlTel,DQ,ndlinJ:t to maximum dipole radiation. perpendicular to the dipole axis. next the action of the ,or TM, component 15-4b). From the of the refracted beam (which may be calculated Sne11's law), we conclude that the E-field within the dielectric materials, and the axis of the dipole oscillations, is oriented perpendicular to the beam as shown. Notice that the dipole oscillations include a component the direction of the reflected beam. Recalling that a radiates only along directions small angles with the (J (X sin 2 0), we that only a fraction of the component of the light (compared with the component) in the reflected beam. both TE and TM modes together, it follows that the reflected light is partially polarized with a of the E. mode Since the energy of the incident beam is equally between Es and Ep components, it also follows that the refracted beam is polarized and component. richer in the This analysis should make it dear that when the dipole axes are in the same direction as the reflected ray. the component is entirely from the ,..,"11>",11>11 beam. and the reflected ray is linearly polarized in the In fact, if (lU',(lU;;U along the ray, the electromagnetic wave could only be a longitudinal wave! This orientation results when the reflected and refracted rays are to one another 15-4<:). The angle of incidence that a linearly polarized beam by reflection is 01" the polarizing angle, or Brewster's Combining Snell's Jaw 1'0....0 ...." ' "

with the trigonometric relation Or

90 01'

tan

01" we arrive at Brewster's law, I (::)

{15-2)

exist for both external reflection (n2 > nl) and internal travels and are clearly not the samc. For reflection when from air to with n = 1.50, for Of' 56.30 • For reflection when light travels in the opposite direction, 0" = . These angles are seen to be precisely complementary, as required by geometry and the definition of Brewster's While reflection at the polarizing from a dielectric surfuce can be used to produce linearly polarized light, the method is relatively inefficient. For reflection only 15% of the is from air to as in the example just found in the reflected beam. (The Fresnel which calculations of this kind, are treated in Chapter This can be remedied to a degree by stepwise of the beam as in a pile-ofp/ales polarizer (Figure 15-5). reflections by multiple of the dielectric at Brewster's angle both increas~<.j the intensity of the Es component in the integrated, reflected beam and, purifies the transmitted beam of this component. If plates are the transmitted beam a linearly polarized Pileof-plates polarizers are especially helpful in those regions of the infrared and ultraviolet spectrum where dichroic sheet polarizers and calcite prisms are ineffective. in the of inMultilayer, thin fi1m coatings that show Jittle terest in a similar manner and can be used as and transmitters. Another Int,~"""o".,'n application of polarization by reflection is the Brewster window. The 15-6) in the same way as a of the 302

Chap. 15

Production of Polarized

Figure 15-6 Brewster window. Brewster's law is satisfied for the TM mode at both surfaces.

TM linearly polarized light incident at Brewster's angle is at the first surface. The angle of Or, at the second surface also Brewster's law for internal reflection, so that the fully transmitted. The plate acts as a perfect window for TM nnl~ri'7f'rl The active medium of a gas laser is often bounded by two Brewster windows, at ends of the gas plasma tube. The light in the makes re}:lCaltOO passes the windows, on its way to and from mirrors beyond alternate ends of the gas tube. Upon each traversal, the TM mode is cOinpletely transmitted, whereas the TE mode is partially reflected (rejected), After many such traversals in the laser cavity, the beam is essentially free of TE and the is polarized in the TM mode. laser 75-3 POLARIZATION BY SeA ITERING

Before the polarization of light that occurs in we make a slight detour to discuss scattering in general, pointing out some familiar consequences of "r~ltf,p·rir.O" that are in themselves rather interesting. By the of light, we mean the removal of energy from an incident wave by a medium and the reemission of that energy in many We can think of the elemental or unit as an plp"f .."n1'" oscillator). The electron is set into forced oscillation the alternating electric field incident light and at the same frequency. The response of the electron to this force on the relationship between the wand the natural or resonant frequency of the oscillator woo In most resonant frequc~ncies lie predominantly in the ultraviolet to electronic oscillations) and in Sec. 15-3

Polarization by Scattering

303

the infrared (due to molecular vibrations) rather than in the visible. Because atomic masses are so much larger than the electron mass, amplitudes of induced molecular vibrations are small compared with electronic vibrations and so can be neglected in this discussion. Thus for light incident on most materials, w <:{ woo Calculations2 show that in this case, the induced dipole oscillations have an amplitude that is roughly independent of the frequency of the light. The oscillating dipoles, consisting of electrons accelerating in harmonic motion, are tiny radiators, antennas that reradiate or scatter energy in all directions except along the dipole axis itself. Such scattering is most effective when the scattering centers are particles whose dimensions are small compared with the wavelength of the radiation, in which case we speak of Rayleigh scattering. The scattering of sunlight from oxygen and nitrogen molecules in the atmosphere. for example. is Rayleigh scattering. whereas the scattering of light from dense scattering centers-like the droplets of water in clouds and fog-is not. In Rayleigh scattering. the well-separated scattering centers act independently (incoherently), so that, according to the conclusions of Section 9-3, their net irradiance is the sum of their individual irradiances. Now the radiated power can be shown to be inversely proportional to the fourth power of the wavelength of the incident radiation. Without deriving this Rayleigh scattering law, we can make the following hand-waving argument: The electric field of a dipole with a charge e accelerating back and forth along a line is proportional to the acceleration. If r = roe iwl , then the acceleration d 2 r/dt 2 = -w2roeiwl is proportional to the square of the frequency. Since the power P radiated is in turn proportional to the square of the electric field, it becomes proportional to the fourth power of the frequency. This is the Rayleigh scattering law. which is expressed by [9], p =

e 2 w 4 r2

0

I 27TEo c 3

Thus the oscillating dipoles radiate more energy in the shorter wavelength region of the visible spectrum than in the longer wavelength region. The radiated power for violet light of wavelength 400 nm is nearly 10 times as great as for red light of wavelength 700 nm. Rayleigh scattering explains why a clean atmosphere appears blue: Higher frequency blue light from the sun is scattered by the atmosphere down to the earth more than the lower frequency red light. On the other hand, when we are looking at the sunlight "head-on" at sunrise or sunset, after it has passed through a good deal of atmosphere, we see reddish or yellowish light, white light from which the blues have been preferentially removed by scattering. Scattering that occurs from larger particles3 such as those found in douds, fog, and powdered materials such as sugar appears as white light, in contrast to Rayleigh scattering. Here "larger particles" refers to the size of the scattering particle relative to the wavelength of light. In this case, the scattering centers (particles) are arranged-more or less-in an orderly fashion so that oscillators that are closer together than a wavelength of the incident light become coherent scatterers. The cooperative effect of many oscillators tends to cancel the radiation in all directions but the forward (refraction) direction and the backward (reflection) direction. In other words, the scattering due to these larger particles can be understood in terms of the usual laws ofreflection and refraction. However, the usual departures from perfectly ordered atomic arrangement lead to some scattering in other directions as well. The 2 Such

calculations, giving a quantitative treatment to these ideas, are given in Section 27-1.

3The more general theory of scattering. including larger scallering centers, is called Mie scattering after its creator. Mie scattering takes into account the size, shape, refractive index, and absorptivity of the scattering particles and includes Rayleigh scattering as a special case [4].

304

Chap. 15

Production of Polarized light

net amplitude of the coherent scattered radiation is now the sum of the individual or the power is to N 2 when there are N coherent oscillators. Although such scattering is much less effective, per oscillator, than Rayleigh the density of oscillators in this case leads to considerable scattering. It can be shown that the number N of such coherent oscillators responsible for is proporthe reflected radiation is proportional to ). 2 , so that the radiated tional to ).", the 1/). 4 dependence of isolated oscillators. Thus the scattered radiation is essentially wavelength independent, and and clouds appear white scattered light. particular interest in the context of this chapter, however, is the fuct that rather scattered radiation may also be polarized. This phenomenon can be is a and not a longitudinal, wave. As an example, simply, because consider a vessel of water to which is added one or more drops of milk. The milk molecules quickly diffuse throughout the water and serve as effective scattering centers for a beam of light transmitted across the medium. In pure water the light does not scatter sideways but only in the forward direction. The light scattered in various directions from the milk molecules, when examined with a polarizing filter, is found to be polarized, as shown in Figure 15-7. The perpendicular E components of the unpolarized light incident from the left sets the electronic oscillators of a scattering center into vibrations, reemitted in all Light scattered in any direction can include only those Evibrations executed by the oscillators, that along the y- and the z-directions. If scattered light is viewed from the -y-direction, it will be found to contain Evibrations the z-direction, but those along the y-direction are absent because would longitudinal E-vibrations in an electromagnetic wave. Similady, viewed along the z-direction, the z-vibrations are missing, and light is linearly polarized along the y-direction. Viewed along directions off the axes, the partially polarized. The beam shows the same as the inl'"iriP,nt light.

Figure IS-7

Polarization due to Unpolarized light incident from the lell: is scattered by a particle at the

In the same way, when the sun is not overhead so that its light crosses scattered down is found to be partially polarized. the atmosphere us, the seen by viewing the dear sky a polarizing filter while it The effect is is rotated. The polarization is not both because we see light that is multiply scattered into the eye and because not all electronic oscillators in molecules are free to oscillate in exactly the same direction as the incident E-vector of the weak and imperfect and so is polarization scattering is not used as a practical means of polarized In the area of nonlinear optics, however, the controlled scattering of from exby stimulated Raman, Rayleigh, and Brillou.in scattering, much research in modern optics. In such cases, the scattered light is modified by the "UJ.I-'UJJ\AJ

Sec. 15-3

Polarization by Scattering

resonant of the medium. An introduction to such nonlinear applications is given in Chapter 26. 754 BIREFRINGENCE: POLARIZATION WITH TWO REFRACTIVE INDICES Birefringent materials are so named they are able to cause double refraction, that the appearance of two refracted beams due to two indices of refraction for a material. We have already seen that anisotropy in the binding forces the electrons of a material can lead to anisotropy in the amplitudes of re..<;ponse to a electromagnetic wave and hence to of absorpto occur, however, the stimulating tion. Such a material displays dichroism. For optical must fall within the band of the material. Referring to Figure 15-8, we see that the dn/dw is less than zero-or "anomalous"over a certain interval. Such coincide with the existence of absorption bands in a given material. Typically. the absorption band lies in the ultraviolet, above optical frequencies, so that the material is transparent to visible light. In this case, even with anisotropy of electron-binding forces, there is Httle or no effect on optical absorption, and the material does not appear dichroic. the presence of anisotropic forces along the x- and y-directions leads to dispersion curves (like that of Figure 15-8) for refractive index nx to E,,vibrations and ny to The existence of both an n" and an ny for a given frequency is to be since different binding forces along these produce different interactions with the electromagnetic wave and, thus, different velocities of propagation v" and v}' through the The result is that such a while not appreciably still the property of birefringence. The critical physical here are the refractive index nand the extinction coefficient k (proportional to the absorption coefficient) for a given frequency of light. Each constitutes a part of the complex refractive

n = n + ik n

1..-_ _ _...1----"_ _ _ _

w

15-8 Response of refractive index as a function of frequency near an band. The band in which dn/d«1 < 0 is region of anomalous ili"'MP",inl'l

'*

Recapitulating, then, for an ideal dichroic nx = ny and whereas for an ideal k" = ky and n. ny • Both conditions require anis isotropic crystalline structures. The conditions are frequency dependent. birefringent in the spectrum, for example, and strongly dichroic in certain of the infrared Other common materials, birefringent in the visible mica, and even region, are quartz, The relationship of crystaHine asymmetry refractive of light in the medium may be understood a bit more clearly by 1'",,~c'lip ..i.,O' the case , which assumes a tetrahedral of calcite. The molecular unit of calcite is or pyramidal structure in the crystal. 15-9a shows one of these molecules, assumed to be surrounded identical structures that are similarly The

'*

306

15

Production of Polarized light

OA

ea

I I

t

OA Vi

E1

E1

(a)

tbl

t5-9 (a) Progress of light through a calcite crystal. Three atoms form the base of a telrahedron. The optic axis is parallel to the line the C and Ca aloms. (b) Rhombohedron of calcite. showing the optic axis, passes a blunt corner where the three face equal 102°,

carbon and oxygen atoms the base of the as shown, with carbon lying in the center the equilateral triangle of oxygen atoms. The calciwn atom is positioned at some distance above the carbon atom. at the apex of the pyramid. The unpoiarized light propagating the two different directions. First light entering from joining the carbon and atoms. All oscillations of the by the two transverse vectors shown. Since the molecule, and so also the is symmetric with respect to this direction, both E-vibrations interact with the in the same way when through the calcite. This of through the crystal is the optic axis (OA) ofthe For the entering from below, both E-components are perpendicular to the OA. next the light entering the crystal from the len. From this direction the two representative E-vibrations have dissimilar effects on the electrons in the base component parallel to the OA of the crystal causes electrons in the base to oscillate along a direction to the plane, whereas its counterpart E1. causes oscilthe plane. Oscillations within the the electrons tend to be confined due to the chemical bonding-take place more easily, that is, with than oscillations that are to the plane. Since E-osciHations in the oxygen plane (E 1.. OA) interact more strongly with the electrons. the ofthese component waves is reduced that V1. < vn. No interaction at all would make 0 = c. Since n c/o. we conclude that n1. > nil. The measured values for calcite are n1. = 1.658 and nn = 1.486 for A = 589.3 nm. As Table 15-1 indicates, the inequality may be in other materials. In materials that in the trigonal (like cakite), or systems. there is one unique direction through the crystal for which the atoms are arranged symmetrically. For example, the calcite molecule of 15-9a shows a threefold rotational symmetry about the optic axis. Such structures a single optic axis and are called uniaxial birefringent. when nR n1. > 0, the crystals are said to be uniaxial positive, and when this is uniaxial negative. Other systems. the triclinic, and orthorhombic, possess two Sec. 154

Birefringence: Polarization with Two Refractive Indices

307

TABLE 15-1 REFRACTIVE INDICES FOR SEVERAL MATERIALS MEASURED AT SODIUM WAVELENGTH OF 589.3 nm Sodium chloride Diamond Fluorite Positive: Ice Quartz (SiO!) Zircon (ZrSiO.) Rutile (Ti(h) Negative: Calcite (CaC03) Tourmaline Sodium Nitrate Beryl (Be3AIz(SiOJ)6)

Uniaxial (trigonal, tetragooal,

Biaxial (triclinic, monoclinic, orthorhombic)

1.544 2.417 1.392

Gypsum (CaSO.(2H 2 O» Feldspar Mica Topaz

nl'

nl.

1.313 1.5534 1.968 2.903

1.309 1.5443 1.923 2.616

1.4864 1.638 1.3369 1.590

1.6584 1.669 1.5854 1.598

nl

n2

nJ

1.520 1.522 1.552 1.619

1.523 1.526 1.582 1.620

1.530 1.530 1.588 1.627

such directions of symmetry or optic axes and are biaxial crystals. 4 Mica, which in monoclinic forms, is a Such materials then possess three indices of Of course there are cubic crystals such as salt (NaCl) or diamond (C) that are optically and possess one index of refraction. This is the case also for materials that have no large-scale crystalline such as glass or fluids, and thus these are also optically with a index of refraction. Naturally occurring calcite crystals are cleavable into rhombohedrons as a result of their crystallization into the trigonal lattice structures. The rhombohedron has only two corners where all 102°) are obtuse. These corners appear as the blunt corners of the The OA of calcite is through a blunt corner in such a way that it makes angles with the three faces there. A birefringent crystal can be cut and polished to polarizing elements in which the OA may have any desired orientation relative to the incident light. LonSllCler the cases represented in Figure 15-10. In both direc-OA-'-'

t

OA

t

t (bl

la)

(c)

15-10 Light entering a birefringent plate with its optic axis in various orientations. (a) Ught propagation along optic axis. (b) Ught propagation pe~pendiclLllar to axis. (c) Ughl propagation perpendicular to 0plic axis.

des'~nplllOn

of such crystalline systems, see, for example, Charles Kittel, Introduction to (New York: John Wiley & Sons, 1986).

308

Chap. 15

Production of Polarized Light

tions of the unpolarized light incident from the left are oriented perpendicular to the OA of the crystal. Both at the same speed through the crystal with of n-t. In (b) and however, [he OA is to one component and pel'pelo<11Cular to the other. In this case each component through the crystal with a different index refraction and speed. On the cumulative relative phase difference can be described in terms of difference between optical for the two If the thickness of the is d, the in optical paths is Il =

and

corresponding

In

l

-

nil I d

difference is

where '\'0 is the vacuum If the of the plate is such as to make 1l'P = iT" /2, it is a plate (QWP); if '1I', we have a half-wave plate (HWP), and so on. These are called zero-order (or sometimes first-order) Because such plates are thin, it is more to make thicker higher order, (2iT")m + '1I' /2, m 1, 2, 3, .. " A composite of two may also be joined, in which one plate compensates the retardance of all but desired 1l'P of the other. In this way we can optical elements that act as retarders. Mica and are commonly used as retardain the form of thin, fiat sandwiched between glass for added strength. the net phase retardation 1l'P is proportional to the thickness d, any device that allows a continuous change in makes a continuouslyadjustable plate. Such a device is called a £:or,nnen.'lOlJtlOr Figure 15 -11 working principle of a Soleil-Babine! compensator. talline quartz is used to form a fixed lower which is actually a optical contact with a quartz flat plate. Above is another quartz wedge, with r .. " . .nlP motion possible the inclined face. Notice the arrangement of the OA in this assembly. In (a) the position ofthe upper is such that light travels equal thicknesses of with the OAs aligned perpendicular to one another. retardation due to one thickness is then canceled by the other, yielding zero net retardaSliding the upper wedge to the left increases the thickness of the OA orientation relative to yielding a variable retardation up to a maximum, in (b), of two or 720°. by a micrometer screw allows small changes in to be made.

la)

Ib)

Figure 15-11 Soleil-Babinet compensator. The optic axes of the 5ented by dots and lines. The arrow shows the direction of light pensator. (a) Zero retardation. (b) Maximum retardation.

Sec. 15-4

nn.>ni',p·

are reprethe com-

Polarization with Two Refractive Indices

309

15~5

DOUBLE REFRACnON In the cases depicted in 15-IOb and c, the light propagating through the crystal may develop a net phase difference between E-components perpendicular and parallel to the crystal's but beam remains a single beam of light. If now the OA is situated so that it makes an arbitrary angle with to the beam direction, as in 15the light double refraction, that is, two refracted beams emerge, .Iabeled the ordinary and extraordinary rays. The extraordinary ray is so named because it does not exhibit ordinary Snell's law behavior on refraction at the crystal Thus if a calcite crystal is laid over a dark dot on a white piece of paper, or over an illuminated two images are seen while looking into the top surface. If the crystal is rotated about the incident ray direction, the PYI"rn,,>rtl nary is found to rotate around the image, which fixed in posItIon. the two beams emerge linearly in orthogonal orientaas shown. Notice that the ordinary ray is polarized perpendicular to the OA and so propagates with a refractive index of no = n.L clv1.. The extraordinary ray emerges in a direction perpendicular to the of the ordinary ray. Inside the crystal, the extraordinary ray can be described in terms of components polarized in directions both perpendicular and parallel to the axis. (This situation is discussed in the following paragraph.) The perpendicular component propagates with speed v 1. = , as for the ordinary ray. The other component, CIVil. The net effect of [he however, with a index ne = nil of both components is to cause the unusual bending of the extraordinary ray shown in Figure 15-12.

--'+--1-----'.1:--+-... Extraordinary ray _._ . . . . *-__-+~~:<>_'_--'I_e__------- Ordinary ray

OA

15-12 Double refraction.

The situation may be clarified somewhat by reference to Figure 15-13a, which shows one wavelet created by the extraordinary ray as it contacts the crystal surface at P. The incident E-vibration is shown resolved into components (aa) parallel to the OA and (bb) perpendicular to the ~A. The parallel propagates along the direction of VII. which must be perpendicular to 00, and the perpendicular component propagates along the direction of v j , which must be perpendicular to bb. Since each component travels with a speed determined the ,....trnt"';,j·p .. ,'U...."". til and n1.. the are unequal. For for Huygens' wavelet for the ray example, tiL > nil. so that t:.c < VII. is not spherical as in isotropic media but ellipsoidal as shown, with major axis proportional to VII and minor axis proportional to v1.. Figure 15-13b shows several such Huygens' ellipsoidal wavelets and the wavefront tangent to the wavelets. This plane which constitutes the new suriace of coru;tant phase, is perpendicular to the propagation vector k for the wave. The E of the wavefront is IntlF'rn"f>i1ilate between and . Notice that in this case of the extraordinary ray in an anDouble refracrion is a term used to describe a manifestation of birefringence in materials, alBirefringence indicates the of two refractive indices, it has literally the same whereas double refraction refers to the splitting of a ray of light into ordinary and extraordinary parts. S

310

Chap. 15

Production of Polarized light

OA,

, "", , E

", ,

"" a\ /

/

" /

b ' //

~ '-

~l

/ /

""

/ /

"" " OA "

/ / / /

Surface

(a)

fbI

Figure 15-13 (a) Creation of an elliptical wavelet by the extraordinary ray. The material in Ihis case is uniaxial lilce calcite. (b) Nonalignment of ray direction S and propagation vector II: for the extraordinary ray in hi ••>!·.i.,,,,,,,,,. material.

isotropic medium, E is not perpendicular to k. Since energy propagates in the direction of the Poynting vector, S = E X D, and since the ray direction is the V ultlE'.rnrlf'same as the direction of energy flow, the extraordinary ray with diate between V.l f)D shows the unusual refraction of Figure 15-12. The extraordinary ray is not to the rather, the ray direction is from the wavelet origin to the point of tangency of the elliptical wavelet with the wavefront. For the normal ray, on the other hand, due to the other E-component perpendicular to the OA, is normal; the wavelets are k .1 E. k II S, and the ray is to its wavefront. From 15-13a and the discussion it should be clear that the precise intermediate value of the velocity v of the ray on relrelative orientations of the incident ative contributions of ViI and v J , that is, on beam and the OA of the crystal. Thus both the velocity and index of of the extraordinary ray are continuous functions of direction. On the other refractive index of the ordinary ray is a constant, independent of Figure 15-14 is a plot of the refractive index versus wavelength for crystalline quartz. At any wavelength. the index for the ordinary ray is a constant, by the lower curve, whereas the index for the extraordinary ray fulls somewhere between the upper and lower curves, depending on the of the incident ray to the crystal axis. If the two refracted rays, linearly perpendicular to one another, can be physically then double refraction can be used to produce a linearly polarized beam of There are various devices that this. One of the most commonly is the Glan-air shown in 15-15. Two calcite prisms with apex (), as shown, are combined with their fdces opposed and separated by an air space. Their optic axes are parallel, with the orientation shown. At the point of refraction out of the first prism, the angle of incidence is equal to the Sec. 15-5

Double Refraction

311

1.59

<:

1.58

0

·u

i:! '@ 1.57

srowest extraordinary ray

....0

.. II>

.!:!

1.56

"0

.~

"iii Q. 1.55 '0 <:

~

n1

1.54

1.53 '--_-'-_--'-_---' _ _.L-_-'-_---'-_ _'------'

B

~

~

~

~

~

~

B

1~

Waverength (nm)

15-14 Refractive indices of crystalline quartz versus wavelength at IS"C. At II given wavelength the index for the extraordinary ray may fall between the two curves, whereas the index for the ordinary ray is fixed. (Adapted from Melles Griot. Optics Guide 3. 1985.)

15-15 Glan-Air prism.

apex 0 of the The critical angle for refraction into air is as usual by sin Be 1/n and so depends on the orientation of the E-vibration to the OA. For E II OA, n 1.4864 and Oe = , while for E 1. OA, n 1.6584 and Oe 37.1°. Thus by using prisms with apex angle intermediate between these values, the perpendicular component can be totally internally reflected while the leI component is transmitted. The prism serves to reorient the transmitted ray along the original beam direction. The entire device constitutes a linear polaris filled with some transparent material izer. When the space between such as glycerine, the apex must be modified. Several other for polarprisms constructed from positive uniaxial material (quartz) are illustrated 15-16. Notice that in these cases, the and extraordinary rays are separated without the agency of total internal reflection. In each case, the OAs of the two are perpendicular to one so that an in the first for instance, may become an EN-component in the second, with corresponding change in refractive Different relative indices for the two components result in different of refraction and separation into two beams. We see that materials are useful in fubricating devices that behave as linear polarizers as well as in retarders such as QWPs, considered earlier in this chapter.

(al Wollaston prism

Ibl Rochon prism

lei Sernarmont prism

Figul"e 15-16 Polarizing prisms. (a) Wollaston prism. (b) Rochon (c) Semamont prism.

312

Chap. 15

Production of Polarized light

'5-6 OPTICAl. ACTIVITY Certain materials possess a property called optical activity. When polarized light is on an optically active material. it emerges as linearly polarized light rotated from the the beam but with its direction of head-on, some materials produce a clockwise rotation (dextrorotatory) of the Efield, whereas others produce a counterclockwise rotation (levorotatory). Optically active materials include both solids example, and and liquids such as crystalline produce pentine and sugar in solution). Some either rotation, traceable to the existence of two forms of the crystalline structure that turn out to be mirror (enantiomorphs) of one another. Optically active of a beam of light and can be modify the state of rPr\..p~:pnfpif malhe:m~ltlc:ally by the Jones rotator given in Table 14-2. Notice that the rotator mechanism involved in rotating the direction of vibration of linearly polarized is distinct from the action of phase retarders, such as half-wave discussed in Sections 14-2 and 15-4, which may produce the same Optical activity is measured two linear polarizers originally set for extinction, that with their TAs crossed in perpendicular orientations 15When a certain thickness of optically active material is inserted between analyzer and polarizer, condition of extinction no exists because the of the is rotated optically medium. The exact of rotation can be measured by rotating analyzer extinction reoccurs, as shown. The rotation so measured depends on both the wavelength of the light and the thickness of medium. The rotation (in produced by a I-mm plate of active 15-2 gives the rotation p of material is called its specific rotation. quartz for a range of optical wavelengths. The amount of rotation caused by cally active liquids is much less by comparison. In the case of solutions, the rotation is defined as the rotation due to a to-cm thickness and concentration of I g of active solute per cubic centimeter of solution. The net of rotation f3 due to a path L through a solution of d grams of active solute per cubic centimeter is then (15-4) f3 = pLd where L is in decimeters and d is the concentration in grams per cubic centimeter. - 37°. The sign inFor 1 dm of turpentine rotates sodium light y

Polarizei'

Analyzei'

Figure 15·17 Measurement the optical activity is measured

Sec. 15-6

Optical Activity

activity. With the active material in the angle f3 to reestablish extinction.

313

TABU 15-2 SPECIFIC ROTATION OF QUARTZ

A (nm)

p (degrees/mm)

226.503 404.656 435.834 546.072 589.290 670.786

201.9 48.945 41.548 25.535 21.724 16.535

dicate.·, that turpentine is levorotatory in its optical activity. Measurement of the optical rotation of sugar solutions is often used to determine concentration, via (15-4).6 The dependence of specific rotation on wavelength means that if one views white light through an arrangement like that of Figure 15-17, each wavelength is roto a slightly different degree. This separation of colors is to as rotatory dispersion. Without giving a physical explanation of optical activity, we can, following Fresnel, a useful phenomenological description that enables us to relate specific rotation of an active substance to certain physical parameters. This description rests first on the demonstrated in the previous that linearly light can be assumed to consist of equal amounts of left- and right-circularly polarized light. Second, this description makes the assumption that the and right-circularly polarized components move through an optically active material with different velocities. V3£ and Vm, Since v = c/n, different n3£ and nm, may be defined for circularly light. Consider first the case of an inactive medium for which v~ = Vift. or, equivalently, n~, = n\1l and k3£ = km. Here k is the propagation vector whose magnitUde is related to wave speed by k = w/v. If the incident light is linearly polarized along the x-direction, as in Figure 15-17, it may be resolved into left- and right-circularly polarized 15-18 makes this clear the vector addition at three different times in an oscillation. The vector sum E executes oscillations along the x-axis as the fu.- and E3£-vectors rotate clockwise and counterclockwise, respectively, at equal rates. y

y

y

_x

15·18 Superposition of left- and right-circularly polarized light at different instants. The light is assumed to be emerging from the page.

60rdinary com syrup is often used in the optics lab to demonstrate optical activity.

314

Chap. 15

Production of Polarized light

cOI:J.s1l1er the consequences of assuming n!£ =I n'.¥!. Now the of the -C()ml'OTlents are not In general, their electric fields may be ex(15-5)

Em= where tor form

= (w/c)n!£ and

(15-6)

The complex amplitudes are given in vec-

(~O)(l, i)

and

t~ = (~o)(l,

-i)

(15-7)

COlrTelipo,nOJlng to Jones vectors for left- and right-circularly polarized modes, and of the two are ()!£

= k!£z - wi

(}'.¥!

k'!ltz - wI

(15-8) medium is one for which k!£ > k'.¥!. which also means that at some distance z into the medium, ()!£ > (}'!It for all t. at an arbitray instant in Figure 15-19a. The veclinearly polarized light but with an inclination angle to the The medium for which n!£ > 1llJi is therefore levorotatory. 15-19b the opposite case is also pictured, for which f3 is a negative angle and the is magnitude of f3 can be determined by nouemg the resultant E that determines the angle f3 is always the diagonal of an equal-sided parallelogram, so that the

or (15-9) y

y

left- and right-circularly polarized active medium. (a) Levorotatory:

Sec. 15-6

Optical Activity

315

Employing

(15-8),

f3 = Hk:e FinaUy, using k
kg; = ko1lr1l. and 71'Z

f3

Ao

=

271'/Ao, where Ao is the wavelength

(n:e - ng;)

(15-10)

Notice that the linearly light is rotated through an that is nrr,nnr_ tional to the thickness z of active medium, as verified experimentally. The action of the :£ and 9ft modes in producing the resultant light might be visualized in the folway. At incidence linearly polarized is resolved into :£ and circular modes, at z 0 and t = 0, begin with = fkA = O. If Vg; > V:e. the 9ft mode reaches some point along its path before the :£ mode. Until the mode E rotates at this according to the circular polarizaalone. As soon as the :£ mode however, the two tion of the 9ft mode modes superpose to fix the direction of vibration at an angle f3 in a linear mode. The relative phase between the two modes at this instant determines the angle f3, as ex......"."""'.£1 by Eq. (15-9). Since the frequencies of the two modes are identical, angle f3 constant thereafter. It should be emphasized that the of refraction involved in optical activity characterize circular birefringence rather than ordinary birefringence. The dices 1lr1l and n:e are much closer in value than n.L and nil, as can be seen in case of (Table I TABLE 15-3 REFRACTIVE INDICES FOR QUARTZ

A (fim)

396.8 762.0

1.56771 1.54811

1.55815 1.53917

1.55810 1.53914

1.55821 1.53920

Example Determine the specific rotation produced by a J-mm thick quartz plate at a wavelength of 396.8 nm.

Solution

Table

at A n:e

Using

396.8 nm, 1lr1l =

0.00011

(15-10),

f3

---'------'- (0.0001 J) = 0.8709 rad

in good agreement with Table 15-2 for 404.6 nm.

neighboring wavelength of

The above description does not explain why the velocities of the and 9ft circularly polarized modes should differ at all. We content ourselves for purposes of this discussion pointing out optically active possess or crystalline structures that have spiral with either left- handed or right-handed screw forms. Linearly polarized light transmitted through a collection of such molecules creates forced vibrations of electrons that, in response, move not only along a spiral but the spiral. Thus the effect of :£-circularly po316

Chap. 15

Production of Polarized light

larized light on a would be to be different from its effect on a right-handed spiral and lead to speeds through the medium. Even if individual spiral-shaped molecules confront the light in random orientations, as in a liquid, there will be a cumulative effect that does not cancel, as long as all or most of the molecules are of the same haJldedm$s. 15-7 PHOTOELASTICITY

Consider the following Two filters acting as polarizer and analyzer are set up with a source behind the pair. If the TAs of the filters are crossed, no light emerges from the If some birefringent material is inserted between them, is transmitted in beautiful colors. To understand this unusual consider where polarizer and analyzer TAs are crossed to the x-axis. Suppose that the birefringent at 45 0 and material in beam constitutes a half-wave plate with its fast axis (FA) vertical, as shown. Its action on the incident linearly polarized light is to convert it to linearly polarized light perpendicular to the original direction, or at -450 inclination with the x-axis. This can be understood by resolving the incident light 0 ortho~~onal ODmpOtlents along the FA and SA (slow axis) and with a 180 .-lifTP..,.,nl'p between them. As always, the effect of the HWP on linearly poJar20:. or. in this case, 900 • The same result follows light is to rotate it from use of the calculus:

[~ ~J

CJ [-!J

HWP FA vertical

LP at 45°

LP at -450 y

TA

x

FA TA SA

Polarizer

Retarder

15·20 transmitted by cross polarizers whel1 a birefril1gent material as a half-wave plate is placed betweel1 them.

from the HWP is now polarized along a direction that is fully transmitted If the retardation plate introduces phase than 1800 , is rendered elliptically polarized, and some portion of the light will still be tra!osrnitlted the analyzer. Only if the phase difference is 3600 or some multiple if the retardation plate functions as a will the charbe unmodified by the plate and Now recall that the phase difference Ilcp introduced dependent, such that

a

<>rrl<>t;,n"

plate is

wa1V'elt;~ngltn

Ao Ilcp Sec. 15-7

Photoelasticity

=

2Trd(n.c - nil)

(15-11)

311

where d is the thickness of the plate_ For is constant throughout the optical region nil) is 11 follows that .."t':l!.-d",r portional to the wavelength. Thus if the """,,,,'I·,tl"'," transmitted. light, in the arrangement of giving the whereas shorter visible wavelengths will transmitted light a predominantly reddish is now roSince the sum of tated by 90°, all components originally incident light, that the light transmitted under both transmission condiwhite light, it follows that the colors "'.... tions are complementary colors. Sections of quartz or calcite and strate the production of colors by birefringence, either under normal ('''t"hl,,,,,,,, crumpled piece of cellophane introduced 1"IP1U1~,pn cr(lsse:d ",."I", .. i7P,." shows a strikvariety of colors, enhanced by the through two or more thicknesses at certain points, so that fj..;p due to a change in thickness d. A similar effect is cellophane tape around a slide, allowing for Finally, may also vary point to point due to local nll_ Formed plastic pieces, such as a such tions due to localized birefringent safety goggles iilSerted between crossed rn<1,,",c,,,',, in those under in inmost blrefrmj;l,cl1l:e induced by mechanical to normally 1(:"f.... n'" i>Ul.J',,.UU\.1C stress or glass is the basis for the method of stress analysis called nnl?f(),PlllSl1lW that in such materials, OPrlctP"

la)

Ibl 15-21 PhOloelaslic stress patterns (a) lightly loaded at the center, (b) M. Francon, and J. C Thrierr, Springer-Verlag, 1962.)

318

Chap. 15

Production of Polarized

an axis is induced in the direction of the both in tension and in compreSSltOn. Since the degree of is proportional to the strain, prototypes of mechanical parts may be fabricated from plastic and subjected to stress Points of maximum strain are made visible by light transmitted through for when the stressed sample is positioned between the polarizers. light patterns for a beam under light and heavy stress is shown in Fig-

15·1. Initially unpolarized light passes in turn three linear with transmission axes at 0", 30", and 60°, respectively, relative to the horizontal. What is the irradiance of the product light, expressed as a percentage of the unpolarized light irradiance? 15-2. At what will light, externally and reflected from a diamond-air interhe cornpletely linearly polarized? For ..... u'u..,,, .... , 2.42. 15-3. Since a sheet of Polaroid is not an ideal not all the energy of the Evibrations to the TA are transmitted, nor are all E-vibrations perpendicular to the TA absorbed. Suppose an energy fraction a is transmitted in the first case and a fraction {3 is transmitted in the second. (3) Extend Malus' law by calculating the irradiance transmitted (j between their TAs. Assume larizers with radiance 10 • Show that Malus' law follows in the ideal case. 0.95 and {3 = 0.05 for a given sheet of Polaroid. the irradi(b) Let a ance with that of an ideal polarizer when light is through two , and 90". such sheet" having a relative angle between TAs of 0", 30", 15-4. How thick should a half·w.lVe plate of mica be in an where laser light of 632.8 nm is used? Appropriate refractive indices for mica are 1.599 and 1.594. 15-5. Describe what to unpolarized light incident on material when the OA is oriented as shown. You will want to comment on the considerations: or double refracted rays? Any phase retardation? polarization of refracted mys? (b)

(3)

OA I

OA (d)

(c)

OA

(e) Which would you use to make a quarter-wave 15-6. Consider a Soleil-Babinet compensator, as shown in 15-1 L the compensator is constructed of quartz and provides a maximum retardation of two of green mercury light (546.1 nm). Refractive indices of quartz at full 1.546. this are n (parallel) = 1.555 and n (perpendicular) (3) How does the total thickness compare with that of the flat in the tion of maximum retardation? (b) How do compare when the emergent light is circularly pollarizecl'! 15-7. A number of dichroic are available, each of which can be assumed

Chap. 15

Problems

319

that each passes 50% of the incident unpolarized light. Let the irradiance of the incident light on the first polarizer be 10 • (a) Using a show that if the polarizers have their transmission axes set at angle IJ the light transmitted by the pair is by 1

J5·8. 15·9. 15·10.

15·n.

= (~)

2

cos IJ

(b) What percentage or the incident light eneIgy is transmitted by the pair when their transmission axes are set at 0" and 90°, respectively? (c) Five additional polarizers of this type are placed between the two described with their transmission axes set at 15°, 30°,45°,60°, and 75°, in that order, with the 15° angle polarizer adjacent to the 0° polarizer, and so on. Now what of the incident energy is transmitted? What minimum thickness should a piece of quartz have to act as a quarter-wave plate for a of 5893 A in vaccum? Determine the of deviation between the two emerging beams of a Wollaston prism ('xmstructed of calcite and with wedge angle of 45°. Assume sodium light. A beam of is changed into circularly polarized light by passing it 0.003 cm thick. Caklllate the difference in the refractive indices for the two rays in the crystal, assuming this to be the minimum thickness SlJllJwm~ the effect for a wavelength of 600 nm. Sketch the arrangement, showing the OA of the and why it occurs. is incident on a water surface at such an angle that the reflected light is com-

(a) What is the of incidence? refracted into the water is intercepted by the top flat surface of a block with index of 1.50. The light reflected from the glass is completely linpolarJized. What is the angle between the glass and water surfaces? Sketch the arrangement, showing the polarization of the light at each stage. 15·12. In each of the cases, deduce the nature of the light that is consistent with the performed. Assume a 100% efficient polarizer. (a) When a is rotated in the path of the light, there is no intensity variation. With a QWP in front of (coming first) the rotating polarizer. one finds a variation in intensity but no angular position of the polarizer that gives zero intensity. (b) When a is rotated in the path of the light, there is some intensity variation but no position of the polarizer giving zero intensity. The polarizer is set to maximum intensity. A QWP is allowed to intercept the beam first with its OA parallel to the TA of the polarizer. Rotation of the polarizer now can produce

zero """'''''") from a source immersed in oil of refractive index 1.62 is incident on the p]ane fuce of a diamond (n also immersed in the oil. Determine (a) the of incidence at which maximum polarization occurs and (b) the angle of refraction into the diamond. 15·14. light in an optically active medium is found to be approxito the inverse square of the wavelength. (a) The rotation of is 20.5°. A glucose solution of unknown concentration is contained in a 12-cm-long tube and is found to rotate linearly polarized by 1.23° What is the concentration of the solution? (b) Upon through a I-mm-thick quartz red light is rotated about 15°. What rotation would you expect for violet 15·15. (a) What thickenss of quartz is to give an optical rotation of 100 for tight of 396.8 om? (b) What is the rotation of quartz for this wavelength'! The refractive indices

15·13.

320

15

Production of Polarized

IS-16.

IS-17.

15-18.

15-19.

IS-20.

1.5-21.

IS-22.

for quartz at this wavelength. for left- and right-circularly polarized light, are nL = 1.55821 and nR 1.55810, (a) A thin plate of calcite is cut with its OA parallel to the plane of the minimum thickness is to produce a quarter-wave path in~'r""""Q for sodium of 589 nm? (b) What color will be transmitted by a zircon 0.0182 mm when placed in a 45° orientation between crossed (a) Show that for internal and external reflection between the same two media must be complementary. (b) Show that if Brewster's is satisfied for a TM light beam a parallel plate (a Brewster window), it will also be satisfied for the beam as it leaves the plate on the opposite side. The indices of refraction for the fast and slow axes of quartz with 546 nm light are J.5462 and 1.5553, reSiJeCllvelV. (a) By what fraction of a wavelength is the e-ray retarded, relative to the o-ray, for every of travel in the (b) What is the thickness of a zeroth-order QWP? (c) If a multiple-order quartz 0.735 mm thick functions as a what is its order? (d) Two quartz plates are optically contacted so that they tions. Sketch the orientation of the OA of the two ference in thickness be such that function together like a zeroth-order QWP? When a triangle is viewed between crossed and with monochromatic light of 500 nm, a series of alternating transmission and extinction bands is observed. How much does (nJ. nl) vary between transmission bands to successive conditions for HWP retardation? The is It, in. thick. A plate of beryl is cut with the optic axis in the plane of the surfaces. Plane polarized light is incident on the plate such that the E-field vibrations are at 45" to the optic axis. Determine the smallest thickness of the plate such that the emergent light is (a) plane and (b) circularly nnl" .. i·"prl Find the at which a half-wave plate must be set to compensate for the rotation of a l.15-mm levorotatory quartz plate 546-nm wavelength In Chapter 20, the Fresnel equations are derived, showing that the percent irradiance R reflected from a dielectric plane surface for the TE polarization mode is by (20-23):

R= where (J is the angle of incidence and n is the ratio n2/n]. Calculate the reflectance for the TE mode when the light is incident from air onto glass of n 1.50 at the angle. (b) The reflectance calculated above is also valid for an internal reflection as leaves the into air. This being the case, calculate the net percentage of the TE mode transmitted a stack of 10 such relative to the incident irradiance 10 • Assume that the plates do not absorb light and that there are no multiple reflections within the plates. (c) Calculate the of polarization P of the transmitted given by P

11M

+

In,

where I stands for the irradiance of either polarization mode. 15-23. A half-wave plate is placed between crossed and such that the between the polarizer TA and the FA of the HWP is O. How does the emergent vary as a function of O?

Chap. 15

Problems

321

15-24. (3) Determine the rotation produced by the optical activity of a 3-mm quartz on a beam of light at 762 nm. (b) What is the rotation due to optical activity a half-wave of quartz using the same light beam?

[I] Shurciiff, W. A. Polarized

13J

[4J

[6]

[8] [9]

[10]

322

Production and Use. Cambridge, Mass.: Harvard University Press, 1962. and S. S. Ballard. Polarized Light. Princeton, N.l: D. Van Nostrand Shurdiff, W. Company, 1964. Jean and Harold E. Bennett. "Polarization." In Handbook of edited Walter G. Driscoll and William Vaughan. New York: McGraw-Hili Book Company, 1978. Meyer-Arendt, Jurgen R. Introduction to Classical and Modern 3d ed. Englewood Cliffs, N.J.: 1989. Ch. 4.2. and Spectroscopy. B~1on: Academic Press, Kliger, David S. Polarized Light in 1990. R. M. A., and N. M. Bashara. Ellipsometry and Polarized New York: North-Holland Publishing Company, 1977. Walermaln, Talbot H. "Polarized Light and Animal Navigation." Scientific American (July 1955): 88. Wehner, '''''~'''''fS'''' "Polarized Light ,.,,,,,,,,,,vu by Insect.,." Scientific American (July 1976): 106. Feynman, Richard P., Robert B. Leighton, and Matthew Sands. The Feynnum Lectures Mass.: Addison-Wesley Publishing Company, 1963. Ch. on Physics, vol. 1. 33. Weisskopf, Richard F. "How Light Interacts with Matter." In Lasers and Light. San Franl.'isco: W. H. Freeman and Company Publishers, 1969.

Chap. 15

Production of Polarized light

16

Fraunhofer Diffraction

INTRODUCTION The wave character of light has been invoked to explain a number of phenomena, classified as "interference effects" in preceding chapters. In case, two or more originating from a single source and separated by individual coherent beams of amplitude or wavefront division, were brought together to interfere. Fundamentally, the same effect is involved in the diffraction of light. In its simplest description, diffraction is any deviation from geometrical optics that results from the obstruction of a wavefront of light. For example, an opaque screen with a round hole .......... P<"'n·tc such an obstruction. On a viewing screen placed beyond the hole. the circle of light may show complex edge effects. This type of obstruction is typical in many optical instruments that utilize the portion of a wavefront passing through a round lens. Any obstruction, however, shows detailed structure in its own shadow that is quite unexpected on the basis of geometrical of the wave character light. if the Diffraction effects are a obstacle is not opaque but causes local variations in the amplitude or of the are observed. Tiny bubbles or imperwavefront of the transmitted light, such fections in a glass lens, for example, produce undesirable diffraction patterns when of optical are blurred by diffractransmitting laser light. Because the tion, the phenomenon to a fundamental limitation instrument resolution. More often, though. the sharpness of optical images is more seriously degraded by optical aberrations due to the imaging components themselves. Diffraction-limited optics is good optics indeed. 323

The double slit studied which light is blocked ance of the

constitutes an obstruction to a wavetront m at the two apertures. Recall that the irradiwas calculated treating the two openings as could be treated as points. A more complete analysis of this must into account the finite size of the slits. When this is done, the problem is treated as a diffraction problem. The results show that the interference earlier is modified in a way that accounts for the actual details of the observed Adequate agreement observations is possible through an application of the Huygens-Fresnel principle. According to Huygens, every point of a given wavefront of light can be considered a source of secondary spherical wavelets. To this, Fresnel added the that the actual field at any point beyond the wavefront is a superposition all these wavelets, taking into account both their amplitudes and Thus in calculating the diffraction pattern of the double slit at some point on a screen, one every point of the wavefront emerging from each slit as a source of wavelets whose superposition produces the resultant field. This then into account a continuous array of sources across both slits, rather than two isolated point sources, as in the interference calculation. Diffraction is often from interference on this basis: In diffraction phenomena, the from a continuous distribution of sources; in inbeams originate from a discrete number of sources. This is not, however, a fundamental physical distinction. A further of diffraction effects arises from the mathematical appf())dlmaltiolrls pcJSSIDIe when the resultant fields. If both the source of light and far enough from the diffraction ",,.,,,,,,.tn'r/> so that wavefronts at the and observation screen may be COlnSl,del'e
16-1 DIFFRACTION

"''''''''U",,,hl

A S/Il#GI..E

We first calculate the Fraunhofer diffraction pattern from a slit, a rectangular ('h~lrn,~te:lri7/~cl by a length much larger than its width. For Fraunhofer difthe source must be far enough away so thm the wavefronts light reaching 324

16

Fraunhofer Diffraction

the are essentially plane. Of course, this is easily accomplished in practice by the placing the source in the focal plane of a positive lens. Similarly, we another lens on the other side observation screen to be effectively at infinity by any point such as P on the slit, as shown in Figure 16- 1. Then the light from different portions of the wavefront at the screen is due to parallel rays of the slit (dashed According to the Huygens-Fresnel principle, we consider ......."'''.5 from each point of the wavefront as it reaches the plane at P by adding the waves according to the the slit and calculate the resultant principle of superposition. As shown in Figure 16-1, the waves do not arrive at Pin phase. A ray from the center of the for example, has an optical path that is an amount !l shorter than one from a point a distance s above the optical axis.

---------

~-----=::::::",l

p

Figure 16-1 Construction for del,~nnmtrI2 irradiance on a screen due to Fraunhofer diffraclion by a single-slil.

The plane portion of the wavefront at the slit opening a continuous array of Huygens' wavelet sources. We consider each interval of dimension tis as a source and calculate the result of all such sources by integrating over the entire slit wavelets at P the form width b. Each interval tis contributes I)

where r is the optical length from the interval tis to the P. The amplitude is divided by r because the spherical waves decrease in irradiance with dlS:taIlce in accordance with the inverse square law, that E2 ex l/r2 and E ex l/r. The amplitude at unit distance from the source point is then dEo. Let us set r = ro for the wave from the interval ds at s = O. Then any other wave originating at the val ds at height s, the difference in into account, the differential field at Pis

Sec. 16-1

Diffraction from a Single Slit

325

In the amplitude, dEo/(ro + .6.), the path difference .6. is unimportant, since .6. <:r; roo and .6. can be neglected there. The phase. on the other hand, is very sensitive to small differences. For intervals ds below the s is negative and the path difference is (ro corresponding to shorter optical paths to P. The amplitude of the radiation from each interval depends on the size of ds, so that when all such contributions are added by integration, we have the total effect at P. Accordingly. we write dEo

(16-3)

ds

per unit width of slit at unit distance away. For a point P where K is the at (J below the axis. relative to the lens center, the figure shows that .6. = s sin (J. With these modifications. the differential contribution to the field at P from an arbitrary interval ds is

Integrating over the width of the

we have

f

l

b2

ro

ds)ei(kYOWll

(I 6-4)

~~bI2

Since we are ultimately concerned with the irradiance, the square of the amplitude which we shall , we retain only the portion in parentheses and integrate: (16-5) Inserting the limits of EL ro

The phases of

---

(16-6)

ik sin ()

exponential terms

"UJ;U':,C;"t

f3 -

we

a convenient substitution,

sin ()

Then b To

where we we get

(16-8)

2if3

applied

to

exponential terms\ Simplifying,

sin f3 To

f3

(16-9)

The amplitude of resultant field at P, given by (16-9), includes the sinc function (sin f3)/f3, where f3 varies with () and thus the observation point P on the screen. We may give significance to f3 by interpreting it as a (16-7) indicates a terence. Since a phase difference is given in general by k.6., path difference associated with f3 of .6. (b/2) sin (). shown in Figure 16-1. Thus f3 represents the difference between waves from the center and either endpoint of the slit, where s b/2. The at P is proportional to square of the resultant amplitude there, or

326

16

Fraunhofer Diffraction

or 1

IO(Si~2/)

10)

where 10 includes all constant fuctors. (16-9) and (16-10) now permit us to distance from the axis at screen. plot the variation of irradiance with approaches 0: The sine function has the property that it approaches 1 as its

= its zeros occur when sin

f3

(16-11)

1

0, that is, when

f3 = Hkb sin 0) = rmr,

with m

I, ±2, ..

Equation 16-11 shows that the value m = 0 is excluded from this condition. The irradiance is as a function of f3 in Figure 16-2. Setting k = 271'/)., the condition for zeros of the sine function (and so of the irradiance) is rnA

b sin 0

(16-12)

= 0 or y

On screen, the irradiance is a maximum at 0 to zero at values y such that y s=

0 and drops

(16-13)

b

The approximation in (16-13) comes from setting sin 0 s= y / f, usually well justified, since 0 is a small The irradiance pattern is about y O. I

-sinc/3 - - - sinc2 fJ

/3

~

sinO

Figure 16-2 Sine function (solid line) plotted as a function of /3. The irradiance funclion (dashed line) for Fraunhofer diffraction is just the square of sinc /3. nonnalized 1010 at the center of the pattern.

The secondary maxima of the single-slit diffraction pattern do not quite full at the midpoints between zeros, even though this condition is more nearly approached as f3 increases. The maxima coincide with maxima of the sine function, points satisfying !--.._-'---c:---_'--

Sec. 16-1

Diffraction from a Single Slit

=0 327

Y

Y=P

gles

f3

______

16-3 Intersections of the curves y f3 and y = tan at which the sinc function is a maximum.

f3

~

____

~

____

~

~

____

~

____

~

~

~

_ _ _ _- L _ _ _ _

~

tan p

__

p

delennine the an-

or f3 = tan f3. An angle equals its tangent at intersections of the curves y = f3 and y = tan both plotted in 16-3. Intersections, excluding f3 = 0, occur at 1.431T than 1.51T), 2.467r (rather than 3.471T than 3.51T), so on, as can be verified with a hand calculator. The plot shows that intersecmidpoints more c10sely as f3 tion points aproach the vertical lines Thus, in the irradiance plot of 16-2, secondary maxima are skewed slightly away from the midpoints toward the central peak. Most of the energy the diffraction under the central maximum, which is much than the adjoining maximum on either side.

What is the ratio of irradiances at central secondary maxima?

maximum to the first of the

Solution The ratio to be calculated is

20.18 0.952

21.2

Thus the maximum irradiance of the nearest secondary peak is only 4.7% that the central

328

Chap. 16

Fraunhofer Diffraction

The central maximum represents the of the slit on a distant of the image are not sharp but reveal a series of screen. We observe that the maxima and minima that tail off into the shadow surroudning the image. These efdue to diffraction and will be seen in fects are typical of the blurring of other cases of diffraction to be considered. The angular width of the central maximum is defined as the angle M between the first minima on either side. Using Eq. (16-12) with m ± I and approximating sin (J by (J, we 2A b

14)

From (16-14) it follows that the central will as the slit width is narrowed. the length of the slit is very large compared to its width, the difpattern due to points of the wave front along the length of the has a very small width and is not prominent on the screen. Of course, the dimensions of the diffraction pattern also depend on the wavelength, as indicated in Eq. (16-14).

16-2 BEAM SPREADING

According to . (16-14), the angular spread of the central maximum in the fur field is independent of distance between aperture and screen. The linear dimensions of the diffraction thus increase with distance as shown in Figure 16-4, such that the width W of the central maximum is by

W=LM

i: ~ I

I

J

2LA b

(16-15)

w

: Jl --E::-----+ lib

I I I

11 I

1 - + - - - - L,----l.o-1l I

Figure 16-4 of the cenlral maximum in the fur-field diffraclion pattern of a singleslit.

We may describe the content of (16-15) as a linear spread of a beam of light, originally constricted to a width b. Indeed, the means by which the beam is nally narrowed is not relevant to the nature of the diffraction pattern that occurs. If one dispenses with the slit in 16-4 and merely assumes an original beam of width b, all our results follow in the same way. After collimation, a "parallel" beam of light spreads just as if it emerged from a single 0Pl~m!ng. Example UU'lJ",U'''' a parallel beam of 546-nm light of width b 0.5 mm across the laboratory, a distance of 10 m. Determine the final width of the beam due to diffraction spreading.

Sec. 16-2

Beam Spreading

329

Solution ~""':"':'----:---'-

= 21.8 mm

Thus even highly laser beams are to beam spreading as propagate, due to diffraction. It is a fundamental consequence of the wave nature of light that perfectly paranel beams of cannot exist. 14) is for rectangular aperThe beam spreading described we show in that a factor of 1.22 must tures. For circular accompany the wavelength. one must keep in mind that this treatment I The spreading described by Eq. assumes a plane wavefront of uniform (16-15) has been deduced on of or diffraction, which If L is taken small enough, for exmeans here that L must remain ample, the equation predicts a beam width less than b, contrary to assumption. Evithan some minumum L min , which gives a beam width dently L must be W = b, that

We may conclude that we are in the far field when L~

A more general approach leads to the tion in the form L

~

A

"""J,"U'V"';'

for fur-field diffrac-

stated

area of

----"'--A

(16-16)

16-3 RECTANGULAR AND CIRCULAR APERTURES

We have been describing diffraction from a slit a width b much smaller than its length, as illustrated in Figure 16-5a. When both dimensions of the slit are comas illustrated in Figure 16parable and small, each produces appreciable for the irradiance, as in 5b. For the aperture dimension a, we write, (16-10), where

a

(1)a

The two-dimensional pattern now gives zero irradiance for

mAl

y=b

or

(16-17)

sin 8 X,

Y saW,tJ.ed by

nAf

x=a

I A laser beam usually does not have constant irradiance across its diameter. In its fundamental that of (16-14) mode, the profile is a Gaussian funclion. Although its spread formula is with the beam diameter replacing h. the conslanl factor 1.22 is 1.27. Laser-beam ~n..,,,£I,,,o is treated in Chapter 21.

330

16

Fraunhofer Diffraction

Slit aperture (a)

(e)

lion image of a single square aperture, as in the rel=ir~;entatllon Francol1, aoo J. C. Thricrr,

Sec. 16-3

and Circular

(continued)

Id)

where both m and n screen turns out to be

The irradiance over the

rPr,rp,opn

18)

In calculating this the dimension of the slit is replaced by a double integration over both dimensions the aperture. Photographs of single aperture diffraction patterns for and square apertures are shown in 16-5c and d. When the is circular, the Int.>".."ti,," over the entire area of the since both vertical and horizontal dnnCllSI()I1S of the of Eq. (16-3), we ble. Thus, in sents the per unit area. The amplitude is then

We take as elemental area a rectangular 16-6a. the equation of a given by

x ds, shown in length x at s to be

x=

where R is the aperture radius. The ro

This

IRR

takes the form of a standard siR and y kR sin fJ;

tis UUAiJlI",l""

upon

dv

332

Chap. 16

Fraunhofer Diffraction

the substitu-

J,h')

,,

,,,_x------;O-; ,, ,,

(b)

(a)

Figure 16-6 (a) Geometry used in the integration over a circular aperlure. (b) The Bessel function J,(')I). The first zero of the function occurs at ')I = 3.832.

The integral has the value

I-, +

1

. e ryv

V1 -

2 7TJ,(y) v dv = - y -

where J1(y) is the first-order Bessel function infinite series,

J ( ) = 1 _ (y/2)3 , y 2 12 . 2

+

of the first kind, expressible by the

(y/2)5 }2 . 22 . 3

As can be verified from this series expansion, the ratio J1(y)/y has the limit ~ as y -7 O. Thus the circular aperture requires, instead of the sine function for the single slit, the Bessel function J" which oscillates somewhat like the sine function, as shown in the plot of Figure 16-6b. One difference is that the amplitude of the oscillation of the Bessel function decreases as its argument departs from zero. The irradiance for a circular aperture of diameter D can now be written as

1= 10(2J;Y)y.

where y == ~ kD sin ()

(16-19)

10 is the irradiance at y

-7 0, or at () = O. These equations should be compared with those of Eq. (16-17) to appreciate the analogous role played by the Bessel function. Like (sin x)/x, the function J1(x)/x approaches a maximum as x approaches zero, so that the irradiance is greatest at the center of the pattern «() = 0). The pattern is symmetrical about the optical axis through the center of the circular aperture and has its first zero when y = 3.832, as shown in Figure 16-6b. Thus the central maximum of the irradiance fulls to zero when

and

y =

(~) D sin () = 3.832

or when

D sin ()

= 1.22A

(16-20)

Equation (16-20) should be compared with the analogous equation for the narrow rectangular slit, mA = b sin 8. We see that m = 1 for the first minumum in the slit pattern is replaced by the number 1.22 in the case of the circular aperture. Successive minima are determined in a similar way from other zeros of the Bessel function. The irradiance pattern ofEq. (16-19) is ploued in Figure 16-7a. The pattern is simSec. 16-3

Rectangular and Circular Apertures

333

lal

fb)

Figure 16-7 (a) Irradiance pattern of amounl of lighl energy is diffracled of a eircular aperture. The circle order of diffraction and is known as and J. C. Thrierr, Atlas 1962.)

ilar 10 that of Figure 16-2 for a slit, has rotational symmetry about the 7b. The central maximum is cOlnsequ1enll} the circular aperture, and is dius of the Airy disc, according to Eg. (I

shown in diffracted that the far-field

of

ra-

(16-20

Chap. 16

Fraunhofer Diffraction

"16-4 RESOLUTION

In forming the Fraunhofer diffraction of a single slit, as in 16-1, we notice that the distance between slit and lens is not crucial to the details of the pata solid of light when the distance is tern. The lens If this distance is allowed to go to zero, aperture and lens as in the objective of a telescope. Thus the image formed by a telescope with a round objective is subject to the diffraction effects described by . (16-19) for a aperobject-a star, for exl'lmIDle-IS ture. The sharpness of the image of a distant then limited by diffraction. The image occupies essentially the of the Airy disc. An eyepiece the primary and providing further magnification of the diffraction formed by the lens. The limit of enlarges the is already set in the primary The inevitable blur that diffraction restricts the resolution of the instrument, that its ability to produces in the distinct for distinct object points, either physically close together (as in a microscope), or separated by a small angle at the lens (as in a Figure formed by a lens. The 16-8a illustrates the diffraction of two point (). If point objects and the centers of their Airy discs are both separated the angle is large enough, two distinct will be clearly seen, as shown in the are brought photograph of Figure 16-8b. Imagine now that the objects SI and patterns to substantiaIly, it becomes closer together. When their as distinct, that to resolve them as ~v,~",,..,.. ,,.... more difficult to discern the to distinct object points. A photograph of the two at the limit of resolution is shown in Figure 16-8c. Rayleigh's criterion for just-resolvable images-a somewhat " .. ,",itt-" .. " but useful criterion-requires that the centers of the image patterns be no of the Airy disc, as in 16-9. In this cOlrldl'tiOlrl, nearer than the angular the maximum of one falls directly over the as in Eq. 06-21), for limit of resolution, we (.M)min =

1.~2A

(16-22)

where D is now the of the lens. In accordance with this reSUlt, the minimum resolvable angular separation of two object points may be reduced (the resoluWe tion improved) by the lens diameter and the ,",v,.""",,,,, several applications of Eq. (16-22), with the following """ .....m .••

Example Suppose that each lens on a of binoculars has a diameter of 35 mm. How far apart must two stars be before they are resolvable by either of the lenses in the Solution According to

(16-22), X

10-5 rad

or about 4" of arc, using an average wavelength for light. If the stars are a distance d of around 30,000 light years, then near the center of our s is appro ximately their actual s = d M min

(30,000)( l. 92 x 10-5)

0.58 light years

some appreciation for this distance, consider that the planet Pluto at the of our solar system is only about 5.5 light distant. If the stars are Sec. 16-4

Resolution

335

s,------

(al

leI

(bl

Figure 16-8 (a) Diffraclion-limiled long as the Airy discs are well sep'aralcd (b) Separated of two incoherent the IWo are well resolved. (e) the limit of resolution (Photos from M. Alias of Oplical Phenomenon, Plate 16,

being detected by their Inl1""_,".,,,,,,I,sdl'H>th by dish antennas-the resolution If the lens is the objective of a problem of resolving nearby objects is ba!>ic~dlv mates, shall the fact that the wa vefli'onts ject points A and B are not minImum Xmin of two is then by 1

The ratio DI fis the numerical aperture, immersion Thus

Chap. 16

Fraunhofer Diffraction

incl:li'ICI'enl poinl sources al and 1. C. Thrierr, 1962,)

waves-the lenses being re. (16-22), be much less. in
value of 1.2 for a good oil-

I

I

I

I

L..- tJ.°mm-ll>-jI I I

16-9 dashed curve is

criterion for diffraction patterns. The observed sum of independent diffraction peaks.

Radius of Airy disk

Figure 16·10 Minimum angular resolution of a microscope.

The resolution of a microscope is roughly equal to the wavelength of light used, a fact that explains the of ultraviolet, X-ray, and electron microscopes in due to diffraction also affect the human eye, which The limits of may be approximated by a circular aperture (pupil), a lens, and a screen (retina), as in 16-11. Night vision, which takes place with large, adapted pupils of around 8 nun, is of higher resolution than daylight vision. Unfortunately there is not enough to take advantage of the situation! On a day the pupil Eq. (16-22) (~O)min diameter may be 2 mm. Under these 33.6 x 10-5 rad, for an average wavelength of 550 nm. Experimentally, one finds that a separation of 1 nun at a distance of about 2 m is just barely resolvable, giving (M)min 50 X 10-5, about 1.5 times the theoretical limit. One's own resolution (visual acuity) can easily be tested by two lines drawn 1 nun apart at indistances until they can no longer be seen as distinct. It is interesting to Sec. 16-4

Resolution

331

f'igure 16-11 Diffraclion by the eye with pupil as aperlure limits lhe resolution of objects sublending emin •

note that the resolution just determined for a 2-mm-diameter pupil is consistent with the value of t' of arc (29 X 10-5 rad) used by SneIJen to characterize normal visual Chapter 7). 16-5 DOUBLE-SUT DIFFRACTION

The diffraction pattern of a plane wavefront that is obstructed everywhere except at two narrow slits is calculated in the same manner as for the slit. The mathematical argument from that for the slit (16-4), where the Extracting limits of integration are now changed to those indicated in the amplitude ds

(16-23)

Integration and !>Ul"'U'U<:,"'" of the limits leads to EL ER = - - - ro ik sin 8

- e O/ 2 )ik(-a-b)sin6

+ e (I/2)ik(a+b)sin8

Reintroducing the !>"'l""tI''''<.'''''' of Eq. (16-7), involving the

f3

4kb sin 8

e O / 2)ilda-bhin8]

width (16-24)

and a similar one ..... ,,,,,,,11"''' the slit separation a, a ==

4ka sin 8

(16-25)

our equation is written more compactly as

Employing Euler's

c:<.iUI
ro

~f3 (2i sin (3)(2 cos a)

Figure 16-12 of slit width and separation for double-slit diffraction.

338

16

Fraunhofer Diffraction

Finally, sin 13 ER = - - - - cos a ro 13 The

(16-26)

is now

or

1

4/0 (

sin 13)2

T

2

(16-27)

cos a

where

10 =

2

(E:obY

as defined in (16-10) for the single slit. Since the maximum value of . (1627) is 4/0 , we see the double slit provides four times the maximum in the center as compared with the single slit. This is what should expected where the are in and add. On inspection of Eq. (16-27) we find that the irradiance is just a product of the irradiances found for double-slit interference and single-slit diffraction. The (16-10) for diffraction. The a factor, factor [(sin 13)/13]2 is that of a IS out a<; in (16-25), is cos 2

[ka (sin (J)] = 2

cos2

[1Ta (sin (J)] A

equivalent to the factor in . (10-19) for Young's slit. The sine and cosine are in Figure 16-13a for the case a = 6b or a = 613. factors of Eq. Because a > b, the cos2 a factor varies more rapidly than the (sin2 13)/13 2 factor. The product of the sine and cosine factors may be considered a modulation of the interference fringe by a single-slit diffraction as shown in 16I3b. The diffraction has a minimum 13 = nm. with m = ± 1, . . . , as shown. In terms of the spatial angle (), condition is diffra<~ticln

minima: mA

=

b sin (J

as in (16-12). these minima happen to maxima, fringe is missing from the pattern. a = p1T, withp = 0, 1, ... ,or when interference maxima: pA When the conditions expressed by point in the pattern (same (J), for missing orders.

ntp,rtp,'pnl'p

= a sin (J

(16-28) maxima occur for (16-29)

(16-28) and (16-29) are satisfied at the same one equation the other the condition

condition for missing orders: a

(~) b

(16-30)

or

Sec. 16-5

Double-Slit Diffraction

339

cos 2 a

, I

\

\

I

\

I

\

, 2"

"

12"

C<

2"

il

(a)

12" (b)

Figure 16-13 (a) Interference (solid line) and diffraction (dashed line) functions plotted for double-slit Fraunhofer diffraction when the slit separation is six times tbe width (a = 6b). (b) Irradiance for the double slit of (a). The curve represents the product of tbe and diffraction factors. (c) Diffraction pattern due to a single slit. (d) Diffraction pattern double-slit aperture, with each slit like the one that produced (c). (Both photos are from Francon, and J. C. Thrierr, Atlas of Optical Phenomenon, Plate 18, Berlin: Springer-Verlag,

(d)

Figure 16-13 (continued)

Thus when the slit separation is an multiple of slit width, the condition is met exactly. For example, when a = 2b, then p 2m = gives the missing orders of interference. For the case plotted in Figure 16-13a and b, a 6b, and the missing orders are those for which p = ± 12, and so on. Figure 16-13c and d photographs of a single-slit pattern and a double-slit with the same width. (What is the ratio of a/bin this Evidently, N is the first order at p ±N is far from the cenwhen a = Nb ter the pattern. To produce a simple interference pattern for two slits, one accordingly makes a ~ b so that N is A large number of fringes then fall under the central maximum of the diffraction As a trivial but satisfying case, observe that when a = b, Eq. (16-30) that all orders (except p 0) are missing. These dimensions cannot be satisfied, however, the two slits have merged into one and are unable to produce interference fringes. When a the resulting pattern is, of course, a single-slit diffraction pattern. 16-6 DIFFRACTION FROIIIIIIIIANY

For an aperture of multiple slits (a grating), the integrals of Eq. (16-23), together with Figure 16-12, are extended by integrating over N The individual slits are identified by the j in the following for the resultant ampliltud,e:

{f'-(Zi- 1 Il2

=_ ro

U1 i- l l(l+bl /2

)a+b

J=I

ds

1-(2j-lja-bJ/2

+

!

dS}

(16-31)

[(2j-l)a-bl/2

As j increases, of slits symmetrically placed below integral) and above (second integral) the origin are included in integration. When j 1, for example, . (16-31) to the double slit case, (16-23). Whenj 2, the next two slits are included, whose edges are located at ( - 3a and H- 3a + b) below the origin and H3a b) and H3a + b) above the origin. When j = all slits are accounted for. contained within the square brackets, Let us first concentrate on the which we shall to as J, temporarily. After integration and substitution of limits, we J

=

1 {lksi081(2]- I)a-bl/2 'k sm . 8 e

_

e-iksifl6((2J-I)a+bl/2}

I

+

1 ik sin

2 This is adapted 10 N even. For N one need not be concerned about the parity of N. For N small, however, N odd can be handled by taking the origin at the center of the central slit.

This

Sec. 16-6

is Jeft to the problems.

Diffraction from Many Slits

341

Using Eqs. (16-24) ad I

again for a and

f3.

b

With the help of Euler's equation. this can be written as sin f3){2 cos[(2j

J

l)a]}

or J

where we have the as the real part of the corresponding exponential. Returning to Eq. (16-31), we need next the sum S:

. f3

S = 2b sm

f3

N/2

Re

2: e

i(2j

II<>

j=1

Expanding the sum, this is S The series in brackets is a to find its sum, by

~""n'\,~f"'i£,

series whose first term a and ratio r can be used

=

Using Euler's equation, this can be recast into the form

+i

sin Na

- sin Na

~------~--------= ~------~------~

-2 sin a

2i sin a

whose real part is (sin Na)/(2 sin a). Then

S and sin Na sin a the square of this resultant I

gives the irraruance,

Jo(sinf3)2(si~Na)2 f3

(16-

sm a

10 includes all the constants. Although derived here for an even number N of slits, the result pyr.rp....pn . (16-32) is valid also for N odd problem 16-19). When N I (16-32) reduces to the results obtained for single- and .... respectively. By now we are with the factor in f3 relliresent envelope of resultant irradiance. Let us examine the fuctor (sin Na/sin a)2, which evidently interference between slits. When a 0 or some mUltiple of 'f1', the "'''''''P'''''''''''' that for such is a maximum. Vi....U ' "

. sin Na II m - sin a

Q-+tmr

342

Chap. 16

Fraunhofer Diffraction

lim N cos Na cos a

a-mrr

= ±N

"

'\

'\ '\

....

.... \.

I

....

\.

....

\.

....

'\

....

".....

lA,..

-A

I

cx;

lI\-

~

" , " "I

--1

21T

X

2X

3w w 3X

a

a

a

a

/3: sin 0:

1I\~ ---~

1T

a

J'

fal

"",---1 _..... 0

,,

"-

""""" "

\. \. \.

I

\. \.

.... \.

\.

sin 0:

\. \.

\.

a

....

\.

\.

"-

"

(bl

Figure 16-14 (a) Interference (solid line) and diffraction (dashed line) functions plotted for multiple-slit Fraunhofer diffraction when N = 8 and a = 3b. (b) Irradiance function for the multiple slit of (a). The irradiance is limited by the diffraction envelope (dashed line).

Sec. 16-6

Diffraction from Many Slits

343

so that the principal maxima of the interference pattern are proportional to N 2 in 8 and positive a is plotted together magnitude. interference function for N envelope in Figure 16-14a. The related irracliance is also plotted with the 2 small, in Figure 16-14b. Notice that the principal maxima are separated by N secondary maxima. Furthermore, as a Na equals an number of 'iT more often a, so that the numerator of interterence in ininstances result in N 1 minima bestances when denominator does not. described by the following equatween principal maxima. The situation is tions: with a

p'iT "8 Nora sm

p = 0, ±I,

N'

principal maxima occur for p

0, ±N, ±2N, ..

(16-33)

minima occur for p = all other values The practical device that makes use of multiple-slit diffraction is the diffraction grating. For N, its principal maxima are bright, distinct, and well sepAI"'£"£....L1I'n" to (16-33), with pi N m 0, ± I, ±2, .. , these maxima, the diffraction-grating equation is mA = a sin 8

(16-34)

where m is called the order of the diffraction. Some insight (16-34) is by examining 16-1 which by plane wavefronts of monochroshows representative slits of a grating matic light. Wavelets emerging from each slit arrive in phase at angular deviation f) from the axis if every difference like AB a sin 8) equalS an number m of wavelengths. When AB = mA, the (16-34) follows When all waves arrive in phase, the resulting phasor diagram is formed by adding N phasors all in the same "direction," giving a maximum resultant. At points the 16-14a are maxima result because principal maxima of a uniform phase between waves slits causes the minima, the gram to curl up with a smaller resultant. At each of so that cancellation is The phase difference between forms a closed 16-14a waves from adjoining slits and in the direction f) can be found from a represents half the difference between successive by recalling that the

Figure 16-15 Representative grating slits illuminated by collimated mOl1ochromalic light. Formation of lbe first-order diffraction maximum is shown.

16

Fraunhofer Diffraction

cipal maxima for the case N = 5. The diffraction ....""'......'....__u'r N very cussed further in some detail in the next chapter.

lan!t!--IS

dis-

at 546.1 nm is normally incident on a slit 16-1. A collimated beam of mercury green 0.015 cm wide. A lens of focal 60 em is placed behind the slit. A diffraction pattern is formed on a screen placed in the focal plane of the lens. Determine the distance between (a) the central maximum and first minimum and (b) the first and second minima. 16-2. Call the irradiance at the center of the centrd[ Fraunhofer diffraction maximum of a single slit 10 and the irradiance at some other point in the pattern I. Obtain the ratio 1/10 for a point on the screen that is ~ wavelength farther from one edge of the slit than the other. 16-3. The width of a slit is measured in the laboratory by means of its diffraction pattern at a distance of 2 m from the slit. When illuminated normally with a parallel beam of laser light (632.8 nm). the distance between third minima on either side of the maximum is measured. An average of several tries 5.625 cm. (8) Assuming fraunhofer diffraction, what is the slit width? (b) Is the assumption of fur-field diffraction justified in this case'! What is the ratio L/Lm;n? 16-4. In viewing the far-field diffraction pattern of a single slit illuminated by a discretespectrum source with the help of absorption filters, one finds that the fifth minimum of one component coincides exactly with the fourth minimum of the pattern due to a of 620 nm. What is the other wavelength? 16-5. Calculate the rectangular slit width that will produce a central maximum in its farfield diffraction pattern an angular breadth of 300, 45°. 90°, and 1800 • Assume a wavelength of 550 nm. 16-6. Consider the far-field diffraction pattern of a single slit of width 2.125 ILm, when illuminated normally by a collimated beam of 550 nm Determine (a) the angular radius of its central peak and (b) the ratio 1/10 at points making an angle of 8 = 5°,100, i5°, and 22S with the axis. 16-7. The Lick Observatory has one of the largebl refracting telescopes, with an aperture diameter of 36 in. and a focal length of 56 ft. Determine the radii of the first and second bright surrounding the disc in the diffraction pattern formed a star on the focal plane of the objective. The firbl two secondary maxima of the function [Jly)j-y]2 occur at -y 5.14 and -y = 8.42. 16-8. A telescope objective is 12 em in diameter and has a focal of 150 cm. of mean wavelength 550 nm from a distant star enters the scope as a nearly collimated beam. Compute the radius of the central disk of forming the of the star on the focal plane of the lens. 16-9. Suppose that a m gas laser emits a diffraction-limited beam at wavelength 10.6 ILm, power 2 kW. and diameter I mm. Assume that. by multimoding, the laser beam has essent.lalily uniform irradiance over its cross section. Approximately how large a spot would be produced on the surface of the moon, a distance of 376,000 km away from such a device, any scattering by the earth's atmosphere? What win be the irradiance at the lunar surface? 16-10. Assume a 2-mm diameter laser beam (632.8 nm) is diffraction limited and has a conblant irradiance over its cross section. On the basis of due to diffraction alone, how far must it travel to dOUble its diameter? 16-11. Two headlights on an automobile are 45 in. apart. How far away will the lights appear to be just resolvable to a person whose nocturnal pupils are just 5 mm in diameter? Assume an average of 550 nm. Chap. 16

Fraunhofer Diffraction

16-12. Assume the range of pupil variation during adaptation of a normal eye is from 2 to 7 mm. What is the range of distances over which it can detect the separation of I in. 16-13. A double-slit diffmction pattern is formed mercury green at 546.1 nm. Each slit has a width of 0.100 mm. The pattern reveals that the fourth-order interference maxima are from the pattern. (a) What is the slit separation? (b) What is the irradiance of the first three orders of interference fringes, relative to the zeroth-order maximum? 16·14. (a) Show that the number of fringes seen under the central diffraction in a Fraunhofer double-slit pattern is given by 2(alb) - I, where alb is the ratio of slit separdtion to slit width. (b) If 13 bright are seen in the central diffraction peak when the slit width is 0.30 mm, determine the slit separation. 16-15. (a) Show that in a double-slit Fraunhofer diffraction pattern, the ratio of widths of the central diffraction peak to the central interference is 2(alb), where alb is the ratio of slit separation to slit width. Notice the result is independent of wavelength. (b) Determine the peak-to-fringe ratio in particular when a = lOb. 16-16. Calculate by integrdtion the irradiance of the diffraction pattern produced by a threeslit aperture, where the slit separation a is three times the slit width b. Make a careful sketch of 1 versus sin 6 and describe of the pattern. Also show that your results are consistent with the generdl result for N slits, given by 06-32). 16·17. Make a rough sketch for the irradiance pattern from seven equally slits having a separation-to-width ratio of 4. Label points on the x-axis with values of a and {3. 16-18. A IO-slit aperture, with slit spacing five times the slit width of I x 10-4 em, is used to a Fraunhofer diffraction pattern with light of 435.8 nm. Determine the irradiance of the principal interference maxima of order I, 2, 3, 4, and 5, relative to the central of zeroth order. 16-19. Show that one can arrive al (16-32) by the origin of coordinates at the midpoint of the central slit in an array where N is odd. 16-20. A rectangular aperture of dimensions 0.100 mm the x-axis and 0.200 mm the y-axis is illuminated by coherent light of 546 nm. A l-m focal length lens intercepts the fight diffracted by the aperture and projects the diffraction pattern on a screen in its focal plane. (a) Whal is the distribution of irradiance on the screen near the pattern center, as a function of x and y (in mm) and 10 , the irradiance at the pattern center? (b) How far from the pattern center are the first minima along the x and y directions? (c) What fraction of the 10 irradiance occurs at I mm from pattern center the x and y directions? (d) What is the irradiance at the point (x 2, y = 3) mm? 16-21. What is the half width (from central maximum to first of a diffracted beam for a slit width of (a) A; (b) 5A; lOA? 16-22. A property of the Bessel function J1(x) is that, for large x, a closed form by sin x - cos x separation of diffraction minima far from the axis of a circular aperFind the ture. 16-23. We have shown that the secondary maxima in a slit diffraction pattern do not fall exactly half-way between minima, but are quite close. Assuming they are half-

way: Chap. 16

Problems

347

(8) Show that the irradiance of the mth secondary peak is

approximately by

(bl Calculate the percent error involved in this approximation for the first three secondary maxima. 16-24. Three antennas broadcal>l in at a wavelength of I km. The antennas are separated by a distance of i km and each antenna radiates in all horizontal direcinterference minima. tions. Because of interference, a broadcast "beam" is limited How many well-defined beams are broadcast and what are their angular half-widths? 16-25. A collimated light beam is incident normally on three very narrow, identical slits. At the center of the pattern on a screen, the irradiance is If the irradiance Ip at some P on the screen is zero, what is the phase difference between light arriving at P from neighboring slits? (b) If the phase difference between waves arriving at P from neighboring slits is "IT, determine the ratio I p/ I max. (c) What is Ip/lm .. at the first maximum? (d) If the average irradiance on the entire screen is I"", what is the ratio Ip/I_ at the central maximum? 16-26. Draw phasor diagrams the maxima and zero irradiance for a four-slit aperture.

[I] Ball, C. J. An Introduction to the Theory Diffraclion. New York: Jlf'fPHnnOn Press, 1971. [2] Max, and Emil Wolf. Principles 5th ed. New York: t'elgarnon Press, 1975. Ch. 8. [3] Longhurst, R. S. Geometrical and Ph\I"~ir.f11 2d ed. New York: John Wiley and 1967. Ch. 10, 11. W. Introduction 10 Fourier Oplics. New York: McGraw-Hill Book [4] Goodman. Company, 1968. Ch. 3.

348

Chap. 16

Fraunhofer Diffraction

17

The Diffraction Grating

INTRODUCTION The discussion of the

chapter continues here in the formal treatment of number of slits or apertures. The diffraction is first to handle light beams incident on the at an arbitrary angle. Performance parameters of practical interest are then developed in discussions of the range, dispersion, resolution, and blaze of a A brief discussion of interference gratings and several conventional types of follows.

17-1 THE GRATING EQUATION

A

multiple-slit device designed to take advantage of the sensitivity of its to the wavelength of the incident light is called a diffraction gratI>AjIHUj'VII developed in Chapter 16 may be for the case when the incident wavefronts of light make an with the of the as in 17 -I. The net path difference for waves then (17-1)

349

Figure 17-1 Neighboring slits illuminated by light incident at angle 8, with the grating normal. For light diffracted in the direction 8,., the net path differeoce from the two slits is

.11 + .12.

The two sine terms in the path difference may add or depending on the direction 8m of the diffracted light. To make Eq. (17-1) correct for all of diffraction, we need to adopt a sign convention for the angles. When the incident and diffracted rays are on the same side of the normal, as they are in Figure 17-1, considered positive. When the diffracted rays are on the side of the normal oPl0m.ite to that the incident rays, 8m is considered In the case, the net difference for waves successive slits is the difference .6. 1 - .6.2 , as evident in a modified sketch such as Figure 17-1. In either case, when .6. = mA, all diffracted waves are in pha<;e and the grating equation becomes a(sin 8;

+ sin

8m)

mA,

m = 0, ±l, ±2, ...

When it is not necessary to distinguish the subscript on the angle of diffrac8m , is often For each value of m, monochromatic of waveA is enhanced by the diffractive properties of the grating. By 07-2), the -8;. the direction of the incizeroth order of interference, m 0, occurs at 8m dent light, for all A. Thus of all appears in the central or zer'Otllorder of the diffraction pattern. orders-both plus and mllnm~-'DrOal spectr:al lines on either side of the zeroth For a fixed direction of incidence given by the direction of each principal maximum varies with wavedifferent wavelengths of length. For orders m 0, therefore, the grating light present in the incident beam, a feature that explains its usefulness in wavelength As a element, the is measurement and spectral to a in several wayS. 17-2a illustrates the formation of the spectrnl orders of diffraction for monochromatic Figure 17-2b shows the angular of the continuous spectrum of visible light for a particular grating. Note that second and third orders in this case partially overlap. Before wavelengths of spectral lines of overlap can be assigned, the actual order of the line must appearing in a (17-2). Unfirst be so that the appropriate value of m can be used in like the a grating greater deviation from the zeroth-order point for is not a one, the overlap ambilonger wavelengths. Thus, when the is often resolved experimentally by using a filter that removes, say, the shorter wavelengths from the incident light. In this way, the spectral range of the incident light is limited by filtering until overlap is removed and each line can he correctly range ac(:eplted identified. At other times it may be advisable to limit the by the by first an instrument of lower dispersion.

'*

350

Chap. 17

The Diffraction Grating

• em;2

_-+---------='""m; 1

-----t--~------"""'m;

G

-1

em ;-2

• (a)

Visible spectrum: 400-700 nm = 1: 9.2°_ 16.3°

m

m=2: 18.7"-34.1° m =3: 28.7°_57.1° m;+2 .:./~

m~~~~'

m;+l

~\ \

\

\~\ \ \\ \

\ \

\\ \

\' \\

! '"//

,

:

\

\,

'\

'\

\

\

\

II

I' /

:

/

/ I/ I

II

/

./

I

./

I // / /

\

\

/

/I

:

\

\

//

~/ / /

/

/

/'

(b)

Figure 17-2 (a) Formation of the orders of principal maxima for monochromatic light incident normally on grating G. The grating can replace the prism in a spectroscope. Focused images have the shape of the collimator slit (not shown). (b) Angular spread of the first three orders of the visible spectrum for a diffraction grating with 400 grooves/mm. Orders are shown at different distances from the lens for clarity. In each order, the red end of the spectrum is deviated most. Normal incidence is ass:umed.

17-2 FREE SPECTRAL RANGE OF A GRATING

The nonoverlapping wavelength range in a particular order is called the free spectral range, F. Overlapping occurs because in the grating equation, the product a sin (J may be equal to several possible combinations of rnA for the light actually incident and processed by the optical system. Thus at the position corresponding to A in the first order, we may also find a spectral line corresponding to 71./2 in the second order, 71./3 in the third order, and so on. The free spectral range in order m may be determined by the following argument. If AI is the shortest detectable wavelength in the incident light. then the longest nonoverlapping wavelength 71.2 in order m is coincident with the beginning of the spectrum again in the next higher order, or mAz = (m

Sec. 17-2

+

Free Spectral Range of a Grating

1)71.,

351

The free spectraJ range for order m is then given by At

07-3)

m

is smaller for higber orders.

Notice that this nonoverJapping Example

The shortest wavelength of n1""'"'''''' in a source is 400 nm. mine the free spectral range in the three orders of grating Solution

m Thus 400 t

400nm

F2 =

400 2

200 nm (from 400 to 600 nm in second order)

F3 =

3400

133 nm (from 400 to 533 nm in third order)

Fl

=-

400 to 800 nm in first order)

DISPERSION OF A GRATING As Figure 16- t 4b shows, higher diffraction orders grow less intense as they fall On the other hand, Figmore and more under the constraining ure 17-2b shows clearly that wavelengths are separated as their order increases. This property is described by the defined by

dBm

07-4)

dA

which gives the angular separation per unit range of waveljenll:tl1. The variation of 8", with A is described by the (from we may conclude

m

=---

(17-5)

If a photographic plate is used in the focal

it is convenient to describe the of on the plate in terms of a linnl."'",PYll1nn dy / dA, where y is measured along the Since dy f dB, the nj":nf'.",u"m is given by linear dispersion == dA

=f

dA

f9J

(17-6)

The plate factor is the reciprocal of the linear Example

of wavelength 500 nm is incident normally on a with 5000 grooves/em. Determine its angular and in first order when used with a lens of focal length 0.5 m. 352

Chap. 17

The Diffraction Grating

Solution The grating constant or groove a

5000 cm-

c~r'~_>"r,n

2

I

X

a is

10-4 cm

Clearly, for zeroth order, there is no For first (17-5) requires the diffraction angle 8 1, which ean be found from the grating equation (

(l)A a

500 X 10-7 2 X 104 = 0.25

Thus 8 1 = 14.5° and cos 8 1 0.968. The angular dispersion in the wavelength region around 500 nm can now be calculated:

m a cos 8m

qfj = - - -

cm)(0.968) = 5164 rad/cm

or 0

5.164

rad x 180 nm 7T rad

X 10- 4

The linear dispersion is then found from fqfj

(500 mm)(5.164 x 1O- 4 rad/nm)

and tbe plate factor is 1/0.258 range of almost 4 nm or 40 A.

0.258 mm/nm

3.87 nm/mm. One mm of film tben spans a

At normal incidence, the grating equation can be incorporated with the angular dispersion relation to qfj _

m _(a sinA 8) (a cosI 8)

a cos 8 or

qfj

= tan 8 A

(

Thus tbe dispersion is actually independent of the constant at a angle of diffraction and increases rapidly with 6. At a given of diffraction, the effect of increasing the grating constant is to increase tbe order of the diffraction there, as Eq. (17-5) clearly sbows. 17-4 RESOLUTION OF A GRATING

Increased dispersion or spread of wavelengths does not by itself make neighboring wavelengths appear more distinctly, unless tbe peaks are tbemselves sharp enough. By tbe resoluTbe latter property describes tbe resolution of tbe recorded tion of a grating, we mean its ability to produce distinct peaks for closely spaced wavelengths in a particular order. Recall tbat tbe resolving power is defined in by A

Sec. 17-4

Resolution of a Grating

(17-8) 353

where (8A)min is the minimum wavelength interval of two spectral components that are just resolvable by Rayleigh's criterion (Figure 16-9). For normally light of wavelength A + dA, and principal maximum of order m, we have the grating (17-2),

a sin 0 To Rayleigh's criterion this mum of the neighboring wa1"elf~nQlth

m(A

+ dA)

0) with the must coincide peak in the same order, or

a sin 0

(17-9) mini-

(17-10)

as can be deduced from both 16-14a and Eq. (16-33). Equating the members of (17-9) and (17-10), AidA = mN. Since dA here is the ."i"'nl"rn resolvable wavelength the resolving power of the Eq. (

mN

07-11)

For a of N grooves, the power is proportional to the order of the diffraction. In a order of diffraction, the resolving power nrr,.,,
(a Si~ Om) :

m= mN or

m=

Wsin A

(17-12)

According to (17 -12), the resolution a grating at diffracting angle Om depends on the width of the grating rather than on the number of its grooves. For a fixed ratio of fJm)/A, however, the grating equation also fixes the ratio mla. Thus

354

Chap. 17

The Diffraction Grating

a grating with fewer grooves and a larger grating constant requires that we work at a higher order m, where there is increased complication due to overlapping orders. Such confusion in high orders is sometimes alleviated by a second dispersing instrument that the first again but in a direction orthogonal to the first. One such instrument is described later in this chapter.

17-5 TYPES OF GRATINGS Up to this point we have been imagining the diffraction grating to be an opaque in closely spaced slits have been introduced. Fraunhofer's nrl,mn,,,1 graltm~~s were, in fact, wires wound between closely spaced of two parallel screws or parallel lines ruled on smoked glass. Strong used ruled metal coatings on glass blanks. Today the typical grating master is made by diamond point ruling of grooves a low-expansion base or into a of aluminum or that has been vacuum-evaporated onto glass base. The or blank. itself must first be polished to closer than AI 10 for green light. The development of ruling machines capable of ruling up to 3600 sculptured grooves per millimeter over a width of 10 in. or more, with suitably uniform depth, shape, and spacing, has been an impressive and technological achievement. involving interferometric and electronic servo-control have been used to enhance the of most modern engines. High-quality grating masters ruled over widths as as 46 cm or more have become feasible. A grating may be designed to operate either as a transmission grating or a the In a transmission grating, light is periodically transmitted clear sections of a blank, into which grooves as centers have been or the light is transmitted by the entire ruled area but periodically retarded in phase due to the varying optical thickness of the grooves. In the first case, the is a transmission amplitude grating, functioning like the slotted, opaque in the second case, the grating is a transmission phase grating. In the reflection the groove are made highly reflecting, and the periodic of waves from reflection of the incident light operates like the periodic a transmission grating. Research-quality gratings are usually of the reflection type. A section of a plane reflection grating is shown in Figure 17-3. The path difference

Figure 17-3 reflection grooves illuminated by light incident at 8j with lhe normal. For light diffracted in the direclion 8m the net path difference of the two waves is.6., - .6.2 _

Sec. 17-5

of Gratings

355

between equivalent reflected rays of light from successive groove the difference

a sin 01

rPM'''',.. •• '''"

is

a sin Om

where both rays are assumed to have the direction after diffraction "iJ'.""'U''"''' by the rnA an interference principal maximum results, so that the angle Om. When t::. grating equation is the same as for a transmission rnA =

0;

+ sin Om)

The same sign convention also applies to the Om and 0;: When Om is on the opposite side of the normal relative to 0;, as in Figure 17-3, it is considered negative. The zeroth order of interference occurs when mOor Om = -0;, that is, in the direction of from grating, as a for all wavelengths. The metallic coating of the reflection grating should be as highly reflective as possible. In the ultraviolet range 110 to 160 nm, coatings of magnesium flouride or lithium fluoride over aluminum are generally used to enhance reflectivity, and below 100 nm, gold and platinum are often used. In the infrared regions, silver and gold coatings are both effective. The light diffracted from a plane grating must be focused by means of a lens or concave mirror. When the absorption of radiation by the focusing elements is severe, as in the vacuum ultraviolet (about I to 200 nm), the focusing and diffraction may both be accomplished by using a concave grating, is, a concave mirror that has been ruled to form grooves onto its reflecting surface.

17-6 BLAZED GRATINGS

The absolute efficiency of a grating in a given wavelength region and order is the ratio of the diffracted light energy to the incident light energy in the same wavelength on a grating, for example, increases the region. Increasing the number of light energy throughput. The zeroth-order diffraction principal maximum, for which a wdSte of energy, grating "nl,..,,'''''''''' there is no The zeroth it will be recalled, contains the most intense interference maximum because it coincides with the maximum of the single-slit diffraction envelope. individual grooves so that the diffraction envelope maxiThe technique of mum shifts into another order is called blazing the grating. To understand the effect of blazing, consider Figure 17-4 for a transmission and 17-5 for a reflection grating. For simplicity, light shown transmitted or reflected from a groove, even though diffraction the cooperative contribution from many grooves. In each figure, (a) illustrates the situation for an unblazed grating and (b) shows the result of shaping the grooves to shift the 0) from the zeroth-order (m = 0) interference diffraction envelope maximum (f3 or principal maximum. Recall that the diffraction envelope maximum occurs where f3 = 0, that is, where the far-field path difference for light rays from the center and of any groove is zero. A zero path difference for these rays implies the conthe optics: For transmitted light, the diffraction peak dition of is in the direction of the incident beam; for reflected light, Figure it is in the direction of the specularly reflected beam. By introducing prismatic grooves in ure 17-4 or inclined mirror faces in 17-5, the corresponding zero path difference is shifted into the directions of the refracted beam and the new reflected which now to the case f3 = O. While the diffraction envelope is thus shifted by the shaping of the individual grooves, the interference maxima remain fixed in position. Their positions are determined by the grating equation, "' . . . . . . .'l'>

356

Chap. 17

The Diffraction Grating

~a)

17·4 In an unblazed transmission grat(a), Ihe diffraclion envelope maximum at f3 0 coincides with the zerolh-order inlerference at m = O. In the blazed grating (b), they are separated.

(bl

la)

(bl

17·5 In an unblazed reflection (a), the diffraction mum at f3 = 0 coincides with the zerolh-order interference at m O. In grating (b), lhey are separated.

maxiblazed

in which are measured relative to the plane of the grating. Neither this plane from (a) to (b) in Figure nor the groove have been altered in 17-4 or 17-5. The result is that the diffraction maximum now favors a principal maximum of a higher order (I m 1> 0), and the redirects the bulk of the light energy where it is most useful. It remains to determine the proper blaze angle of a Consider the of Figure 17-6, where a beam is incident on a groove face at angle reflection 81 and is at arbitrary 8, both measured relative to the normal N to the grating plane. The normal N ' to the groove face makes an angle 8" relative to N. This angle is the blaze angle of the grating. Now let us require that the diffracted reflection from the groove face and the beam both the condition of condition for a principal maximum in the mth order (88m). The first condition is Sec. 17-6

Blazed Gratings

351

~=---L-----L-'---------N

Figure 17-6 Relation of blaze angle incident and diffracted beams.

(Jb

to the

satisfied by making the angle of incidence equal to the angle of reflection relative to N': 8; - (Jb = 8m + 8b , or

8b

- 8j

8m

-

-

07-13)

2

The second condition is that the angle 8m satisfy the gmting equation, mA = a(sin OJ

+ sin 8m )

(l7-14)

Equation 07-13) shows that the blaze angle depends on the angle of incidence, so that various geometries requiring different blaze angles are possible. In the geneml case, the equation that must be satisfied by the blaze angle is found by combining Eqs. (17-13) and (17-14). Taking into account the associated sign convention, the gmting equation becomes rnA = alsin 8i

\.

I

+

sin (28b

-

(Ji)l

(17-15)

We consider two special cases of Eq. (17-15). In the Littrow mount, the light is brought in along or close to the groove IDce normal N', so that 8b = 8 and 8m = -8 as is clear from Figure 17-6 and Eq. (17-13). For this special case, Eq. (17-15) gives j

j ,

Littrow:

rnA = 2a sin 8b

or

8b =

sin~1 (~)

(17-16)

Since the quantity a sin 8b corresponds to the steep-face height of the groove (Figure 17 -6), we see that a gmting correctly blazed for wavelength A and order m in a Littrow mount must have a groove step of an integml number m of half-wavelengths. Commerical gmtings are usually specified by their blaze angles and the corresponding first-order Littrow wavelengths. In another configumtion, the light is introduced instead along the normal N to the gmting itself. Then (J; = 0 and (Jb = -8m !2. Equation (17-15) now gives normal incidence:

Ob =

~ sin- {

:A)

(17-17)

Example (a) Consider a 1200 groove/mm grating to be blazed for a wavelength of 600 nm in first order. Determine the proper blaze angle. 358

Chap. 17

The Diffraction Grating

(b) An echelle grating is a pitched grating designed to achieve resolution by in high orders. Consider the operation in order m 30 of a commercially available echeHe grating with 79 grooves/mm, blazed at an angle of 63°26', and ruled over an area of 406 x 610 mm. Determine its resolution when used in a Littrow mount. Solution

(a) In a Littrow mount,

16), the blaze angle must be

On the other hand, if the grating is used in a mount with light incident normal to the grating, then from (17 -17), (Jb

= 4sin- ' [(l)(600 x 10 6)(1200)] =

(b) In a Littrow mount, the grating returns, along the incidence direction, light wavelength

m

755 nm

The total number of lines on the grating is N (79)(610) = 48,190 so that = mN (30)(48,190) = 1,445,700 at the blaze the resolving power is wavelength of 755 om. The minimum resolvable wavelength interval in this region is then dA min A/Wi, or 0.0005 nm. Actual resolutions be somewhat less than the theoretical due to grating imperfections. The high resolution is gained at the expense of a contracted spectral range of only A/m = 755/30 = 25 om.

17-7 GRADNG REPLICAS The expense and difficulty of manufacturing gratings prohibit the routine use of grating masters in spectroscopic instruments. Until the technique of making ....,."Iir-#u relatively inexpensive copies of the masters-was developed, few research scientists owned a good To make a replica the master is first coated with a layer of nonadherent material, which can be off the master at a later overcoat of aluminum. A layer of resin is This is followed by a then spread over the combination, and a substrate for the future replica is placed on top. After the resin has hardened, the replica grating can be from the master. The good replica usually serves as a submaster for the routine production of other replicas. Thin replicas made from a submaster are mounted on a glass or fused silica blank and a highly reflective overcoat of aluminum is added. This is the usual form in which the gratings are made commercially available. Replica gratings can be purchased that are as as or better than the ma"ters, both in performance and useful life. The efficiency of deep-groove replicas may be better than that of the master because the replication process transfers the smooth parts of the groove faces from bottom to top, improving performance. 1~8INTERFERENCEGRADNGS

The availability of intense and highly coherent beams of light have made possible the production of gratings apart from the rulings produced by grating engines. As early as 1927, Michelson the possibility of photographing straight interferSec. 17-8

Interference

359

c

p

M

(a)

M

(bl

Figure 17-7 (a) Michelson system for producing interference gratings, including collimator C, mirror M, and photographic plate P. (b) Holographic system for producing interference fringes including collimator C, bearnsplitrer BS. mirrors M, and light-sensitive plate P. (c) Production of interference fringes in the region of superposition of two collimated and coherent beams intersecting at angle 26.

(e)

ence fringes using an optical system such as that shown in Figure 17-7a. Two coherent, monochromatic beams are made to interfere, producing standing waves in the region between the collimating lens and a plane mirror. The resulting straight-line interference maxima are intercepted by a light-sensitive film, inclined at an angle. When developed, straight-line fringes appear. Interference gratings produced by such optical techniques are also' called holographic gratings, since a grating of uniformly spaced, parallel grooves can be considered as a hologram of a point source at infinity. Other interferometric systems. such as that shown in Figure 17-7b, are essentially those used to produce holograms. Today the interfering wave fronts are photographed on a grainless film of photoresist whose solubility to the etchant is proportional to the irradiance of exposure. The photoresist is spread evenly over the surface of the glass blank to a thickness of 1 J-Lm or less by rapidly spinning the blank. When etched, the interference pattern is preserved in the form of transmission grating grooves whose transmittance varies gradually across the groove in a sine-squared profile. A reflective metallic coating is usually added to the grating by vacuum evaporation. The fringe spacing d, as shown in Figure 17-7c, is determined by the wavelength of the light and by the angle 20 between the two interfering beams, according to the relation d = A/(2 sin 0). In addition to freedom from the expensive and laborious process of machine ruling, the predominant advantage of the interference grating is the absence of peri360

Chap. 17

The Diffraction Grating

odic or random errors in groove positions that produce ghosts and grass, respectively. Thus interference gratings impressive spectral purity and provide a On the other hand, control over groove profile, high of the is not easily which affects the blazing and thus the achieved. The groove profiles of normal interference gratings are sine-squared in form and so symmetrical, rather than sawtooth-shaped. as are the usual blazed ings. Under normal incidence, a groove profile results in an equal disin the positive and orders of diffraction. When used under tribution of nonnormal incidence, however, it is possible to disperse light into only one diffracted order (other than the zeroth order), and it has been shown that in this case the distribution light does not depend to a extent on groove Efficiencies in this configuration can be comparable to those blazed gratings. Nevertheless, various efforts are in progress to produce groove shapes more like those of ordinary blazed by exposing the photoresist to two wavelengths of radiais more saw-toothed in shape, for example, or subsetion whose Fourier quent modification of the symmetrical grooves by argon-ion etching or in a variety of other ways. Interference are not practical in the production of coarse, echellelike gratings . ., 7-9 GRATING INSTRUMENTS

An instrument that uses a as a spectra1 element is around a particular application. An inexpensive transmission the type of grating selected grating may be mounted in place of the prism in a spectroscope, where the spectrum is viewed with the eye, by means a telescope focused for infinity. The light inciis rendered parallel by a primary slit and collimating lens. Redent on the These may be search grade however, make use of reflection spectrographs, which record a portion of the spectrum on a photographic plate, photodiode array, or other image detector, or spectrometers, where a narrow portion the spectrum is allowed to pass through an exit slit onto a photomultiplier or other may be scanned by rotating the light flux detector. In the latter case, the possible; we describe briefly a few of the There are a number of 17-8 shows the basic Littrow mount, where a fomore common ones. cusing element is used both to collimate the light incident on the plane grating and, in the reverse direction, to focus the light onto the photographic plate placed near light is incident along the normal to the slit. Recall that in the Littrow the groove faces. The Littrow condition is also used in the echelle spectrograph (Figure 17-9), which is designed to take advantage of the high dispersion and reso-

lens

17-8 Littrow-moonted graling. Photographic plate and entrance slit are separated along a direction transverse to the plane of the drawing.

Figure 17·9 Side view of the echelle spectrograph. The echelle is positioned directly over the slit-to-mirror path, but the plate is offset in a horizontal direction.

Slit

Sec. 17-9

Grating Instruments

361

lution with As discussed previously, the useful order is small, so that a second concave is used to disperse the overlapping orders in a direction perpendicular to the dispersion of the echelle grating. In the figure, a incident on the echelle, located near the slit and concave mirror collimates the diffracted by the echeHe is dispersed oriented with grooves horizontal. The again by the concave grating, oriented with grooves verticaL The second grating also focuses the two-dimensional spectrum onto the photographic plate. Figure 1710 shows a Czerny-Turner system in a spectrometer. from an entrance slit is directed by a mirror to a first concave mirror, which collimates the light incident on the grating. The diffracted light is incident on a second concave mirror, which then focuses the spectrum at the exit slit. As the grating is rotated, the disspectrum moves across the slit. When the instrument is used specifically to select individual wavelengths from a discrete spectral source or to allow a narrow wavelength range of spectrum through the exit slit, it is called a mOIWchromator. Entrance slit

Mirror

Exit slit

....--=-----_. .- - - - -.. . . . --....:::=-1 Figure 17·1(1 Czerny-Turner spectrometer.

Other instruments dispense with secondary focusing lenses or mirrors and rely on concave gratings both to focus and to the light. The grooves on a concave grating are equally relative to a plane projection of the surface, not relative to the concave surface it.'ielf. In this way, spherical aberration and coma are eliminated. instruments are used for wavelengths in the soft (l to 25 nm) and ultraviolet regions, extending into the visible. The Paschen-Runge 17 -11, makes use of the Rowland circle. If the surface is tangent at its center with a circle having a diameter equal to the radius of curvature of the concave grating, it can be shown that a slit source placed anywhere on the circle well-focused spectral lines that also full on the circle. If the source and slit, grating, and plate holder are placed in a dark room at three stable 'positions de-

Grating

Figure 17·n Paschen-Runge mounting for a concave grating. Diffrdcted slit formed at the Rowland circle. For a of 1200 grooves/mm and 6, = 30°, the first-order spectrum for wavelengths between 200 and 1200 nm fall between the - '}20 and 56°, 56°

362

17

The Diffraction

termined by the Rowland circle the equation, the basic requirements of the are met. Since typical radii of curvature for the grating may be around 6 m, the space occupied by this spectrograph can be large. The first three orders of diffraction are most commonly used. Typical angles of vary within the range 30° to 45°, and angles of diffraction may vary between on the of the grating normal to 85° on the same side of the normal as the slit. Thus much the Rowland circle is useful for recording of the In Figure 17-11, the first-order spectrum spread the Rowland circle is shown for (J; = 38° and a grating of Jines in this way may suffer rather severely astigmatism. The Wadsworth spectrograph (Figure 17-12) eliminates astigmatism by adding a primary mirror to collimate the light incident on the grating. In so doing, the with the Rowland circle. Spectra are observed over a to the normal, perhaps lO° to either side. To range making small record different of the spectrum, the grating can be rotated and higher orspectrograph is capable of more compact ders can be used. This version of a construction than is the J..'ac ..h'.... Mirror

Slit

Plate

17·12 Wadsworth mount for a concave

17·1. What is the in second order between of waveLengths 400 nm of 5000 1!1lIUV"':SIIL.:III and 600 nm when diffracted by a 17-2. Describe the in the red wavelength °lnm and in nmfmm) for a transmission 6 em wide, containing 3500 grooves/em, when it is focused in the third-order spectrum on a screen by a lens of focal length 150 cm. (b) Find !llso the resolving power of the under these conditi(lns. 17-3. (a) What is the between the second-order principal maximum and the neighboring minimum on either side for the Fraunhofer pattern of a 24-groove grating having a groove separation of 10- 3 em and illuminated by light of 600 nm? (b) What slightly longer (or slightly shorter) wavelength would have it.'> second-order maximum on top of the minimum adjacent to the second-order maximum of 600 nm light? (c) From your results in parts (a) and (b), calculate the power in second order. Compare this with the resolving power obtained from the theoretical resolving power fonnula, Eq. II). 17-4. How many Lines must be ruled on a transmission so that it is of resolving the sodium doublet (589.592 nm and 588.995 nm) in the first- and secondorder spectra? 17-5. (a) A grating spectrograph is to be used in first order. If crown bringing the light to the entrance slit, what is the first wavetenl1t11 Chap. 17

Problems

363

that may contain second-order lines? If the optics is quartz, how does this change? Asswne that the absorption cutoff is 350 nm for crown and 180 nm for quartz. (b) At what of diffraction does the of occur in each case for a of 1200 grooves/mm? (c) What is the free range for first and second orders in each case? 17-6. A transmission having 16,000 lines/in. is 2.5 in. wide. in the green at about 550 nm, what is the resolving power in the third order? Calculate the minimum resolvable difference in the second order. 17-7. The two sodium D lines at 5893 A are 6 A apart. If a with only 400 is available (a) what is the lowest order in which the D lines are re!iOI'/e
Chap. 17

The Diffraction

17·16. A

is needed that, working in first is able to resolve the red doublet {Jroduced by an electrical discharge in a mixture of hydrogen and deuteriwn: 1.8 A at 6563 The can be produced with a standard blaze at 6300 A for use in a Lit(b) the number of grooves trow mount. Find (a) the total nwnber of grooves per millimeter on the grating with a b1a7-c angle of , (c) the minimwn width of the 17-17. An echelle is ruled over 12 em of width with 8 grooves/mm and is blazed at 63°. Determine for a Littrow configuration (a) the range of orders in which the visible spectrum (400 to 700 nm) appears; (b) the total nwnber of grooves; (c) the resolving power and minimum resolvable wavelength intenral at 550 nm; (d) the dispersion at 550 nm; the free spectral range, the shortest wavelength present is 350 nm.

[I] Davis, Sumner P. Slon, 1970. [2] Hutley, M. C.

JittJrnrJ'inn

Grating Spectrographs. New York: Holt, Rinehart and Win-

Gratings. New York: Academic [3] James, J. F., and R. S. Sternberg. The Design of Optical ,"n"rU'nm,plpy< London: Chapman and Hall Ltd., 1969. Ch. 5, 6. r,nt>lnt''"' " Sciemijic American (June [4] Ingalls, Albert G. 45. [5] Feynman, Richard Robert B. Leighton, and Matthew Sands. The ,",p"n""nn in Physics, vol. l. Ke;aOJlng, Mass.: Addison-Wesley Pul)iishin,g

Chap. 17

References

365

18

iSM

1.0

----,--=_........Ijt:--O::::::::=..----'---_

_C(v)

-1.0

fresnel Diffraction

INTRODUCTION In the last two chapters we have dealt with Fraunhofer diffraction, situations in which the wavefront at the diffracting may be considered planar without appreciable error. We tum now to cases where this constitutes a bad aplJroJ{uoatlon cases in which either or both source and observing screen are close enough to the aperture that wavefront curvature must be taken into account. Collimating lenses are not required, therefore, for the observation of Fresnel, or near-field, diffraction patand in this sense, their study is simpler. The mathematical treatment, however, is more complex and is almost always handled by approximation techniques, as we will see. Fresnel diffraction patterns form a continuity between the patterns characterizing geometrical optics at one extreme and Fraunhofer diffraction at the other. In geometrical optics, where light waves can be treated as rays propagating along straight we to see a sharp of the aperture. In practice, such images are formed when the observing screen is quite close to the In cases of Fraunhofer diffraction, where the screen is actually or, through the use of a lens, effectively far from the aperture, the diffraction pattern is a fringed image that bears little resemblance to the aperture. Recall the Fraunhofer double-slit pattern, for example. In the intermediate case of Fresnel diffraction, the image is essentially an image of the aperture, but the edges are

366

18-1 FRESNEL-KIRCHHOFF DIFFRACTION INTEGRAL A arrangement is shown in 18-1. wavefronts emerge from a point source and encounter an At the aperture, the wavefront is still substantially spherical, because the aperture is not far from the source. Diffraction effects in the near field on the other side of the aperture are then of the Fresnel type. The distance from the source S to a representative 0 on the wavefront at the aperture is r I, and the distance from the point 0 to a representative point P in the field is r. Compared to Fraunhofer this case requires special treatment in several ways. Since approaching waves are not plane, the distance ,.' enters into the calculations. the distances,. and,.' are no longer so much than the size the aperture that Fraunhofer diffraction As a result, the variation rand ,.', both with different points 0 and with field points must be taken into account. Finally, because the direction from various aperture points 0 to a given point P may no longer be considered approximately constant, the dependence of amplitude on direction of the wavelets at the ture must be This correction is handled by the obliquity factor to be cussed presently.

s

Figure 18·1 Schematic defining the paramePIers for a typical Fresnel diffraction.

Employing the Huygens-Fresnel principle, as in Fraunhofer diffraction, we seek to find the resultant amplitude of the electric field at P due to a superposition of all the Huygens' wavelets from the wavefront at the aperture, each emanating a "point" on the wavefront, an elemental area 00. Let the contribution to the resultant field at P due to such an elemental area be by the wave, dEp

(d~O)eikr

(18- 1)

The wave amplitude dEo at the aperture is proportional to the elemental area, so we can write dEo IX El. da (18-2) The amplitude source, or

at point 0 is the amplitude of the

"nh",rIL''.l

wave originating at the (I

for a proportionality constant then, we have, combining these eQ1Jalilons, da Sec. 18-1

Fresnel-Kirchhoff Diffraction Integral

(18-4) 367

The field at P due to the se<:ondal:Y wavelets integral

the entire

is the

(18-5) it does not take into account the Equation (18-5) is incomplete in two ways. to their direction, obliquity which attenuates the diffracted waves as described earlier. For the present, we call this factor F(8), a function of the 8 between the of the radiation incident and at the aperture point O. Second, it does not take into account a curious requirement, a 90° shift of the diffracted waves relative to the incident wave. We will return to each of these (XIints in the following The corrected integral is the Fresnel-Kirch/wff diffraction formula,

-ikE 27T

= - -s where the factor -i

II

eik(r+r')

F(8)-- da rr'

e- i7f / 2 ..".'"""0"'''' the required F(8)

=

(18-6) shift, and

l+cos8 2

WdS developed Fresnel and limits the This integral on a more rigorous theoretical basis by Kirchhoff. The ad hoc assumptions by Fresnel were shown by Kirchhoff to follow naturally arguing from integral theorem whose functions are scalar function solutions to the electromagnetic wave equation. I The result by Eq. (18-6), however, still involves mations, that the source and screen distances remain relative to the l'IN'rtl:lrp dimensions and that the aperture dimensions themselves remain relative to the wavelength of the optical disturbance. The integration specified by Eq. (18-6) is over a closed surface including the aperture but is assumed to a contribution only over the aperture itself. In arriving at this result. Kirchhoff assumed as boundary conditions that wave function and its gradient are zero directly bewithin the itself, they have hind the opaque parts of the aperture, and the same value as they would in the absence of the aperture. These assumptions make the derivation of Eq. (18-6) possible but are not entirely justified. Furthermore, in the theory described here, the E-field wave function is a scalar function whose absolute square yields the irradiance. We know that, in the close vicinity of the aperture, more rigorous methods must be that take into account the vector properties the electromagnetic field, including polarization effects. Nevertheless, the Kirchhoff theory suffices to yield accurate results for most practical diffraction situations. In the limiting case of Fraunhofer diffraction, (18-6) is simplified by asthat (I) the obliquity factor is roughly constant over the due to the small spread in the diffracted light and (2) the variation of distances r and r f remains small relative to that of the exponential function. When all constant (or approximately constant) terms are taken out of the integral and included in an is simply constant C, Eq. (

I This derivation is found in many places. for example. [I] and [21. but ability that is beyond the Slated level of this textbook.

368

18

Fresnel Diffraction

r~ires

mathematical

which is a statement of the Huygen.;;-Fresnel principle and corresponds to the inteused in Chapter 16 to calculate Fraunhofer patterns there. For situation.;; in which the assumptions of Fraunhofer diffraction fail, we are left with (18-6). This integration in not easy to carry out for a given Fresnel offered satisfactory methods for simplifying this task, or avoiding We apply these methods in the simpler cases to be considered here.

f 8-2 CRITERION FOR FRJE'SI1'1EL DIFFRACTION

Before with these cases, we wish to establish a practical criterion that determine..;; when we should use Fresnel than the simpler Fraunhofer treatment already presented. It will suffice to consider the case when both S and P are located on the central axis through the as in Figure 18-2. Notice that the dimension indicated by b. is zero when the wavefront is plane. The mp,rnlVI<: of Fraunhofer diffraction suffice, however, as as b. is less than the wavelength of the light. From Figure 18-2a we may express this quantity as (18-7) or, equivdlently,

r'(l - 2~:2)

p where we have approximated the nll'
using the first two terms of . Since p == r', the condi-

>A

(18-9)

and similarly, for the diffracted Wdve curvature in

>A Combining Eqs. (18-9) and (18-10), the may be expressed by near

(bl

Sec. 18-2

Criterion for Fresnel Diffraction

(18-8)

18-2b, 10)

of Fresnel, or near-field, dittrai;!iclO (18-11)

Figure 18-2 view of 18-1. The curvature of (a) incident and (b) diffracted wavefronts is small when tl is small.

369

Of course, this condition also applies to the other dimension (transverse to h) of the When h is taken as the maximum extent of the aperture, not shown in aperture in either direction or as the radius of a circular aperture, (18-9) or 10) may also be expre.;;sed approximately by the condition near field:

d <

A

A

(18-12)

where d represents either p or q and A is the area of the aperture.

18-3 THE OBUQUITY FACTOR The effect of the obliquity factor on the secondary wavelets originating at points of the wavefront was introduced by Fresnel. Recall that according to Huygens, a point source of secondary wavelets could radiate with equal effectiveness without regard to direction. This peculiarity would produce new wavefronts in both forward and reverse directions of a propagating wavefront, although the reverse wave does not exIf point 0 in Figure 18-3 is the of secondary wavelets that arrive at an arbitrary point P in the field, then the correct modification of amplitude as a function of the 8 is by

a

(i)o +

cos 8)

(18-13)

where ao is evidently the amplitude in the forward direction, 8 = O. Notice that justification for relation can also be found in Kirchhoff's derivation.

a = 0 in the reverse direction. The

Figure 18-3 ractor.

Illustration or the obliquity

18-4 FRESNEL DIFFRACTION FROM CIRCULAR APERTURES Suppose the aperture in Figure 18-1 is circular. Fresnel a clever tec:hnlqlJle for analyzing this special case without having to do the explicit integration of (18-6). He devised a method for dealing with the contribution from various parts of the wavefront by dividing the into zones with circular symmetry about the axis SOP. The configuration is sketched in which shows an eITleQ~mg spherical wavefront at S. The zones are defined by circles on the wavefront, spaced in such a way that each zone on the average, A/2 farther from the field point P than the preceding zone. In Figure 18-4a, then, rl = ro + A/2, r2 = ro + A, ... , rN = ro + NA/2. This means that each successive zone's contribution is exactly out of phase with that of the preceding one. Of course, each of these half-period, or Fresnel, zones could be subdivided further into smaller parts, 310

18

Fresnel Diffraction

(a)

(b)

Figure 18·4 (a) Fresnel circular half-period zones on a wavefront emerging from an aperture. (b) Phasor diagram for circular Fresnel zones. Each indicate the avhalf-period zone is subdivided into 15 subzones. Individual erage phase angle of the subzones and are progressively by 5% to simulate the effect of the factor. The amplitude 01 represents the first half-period zone, and A represents all the zones shown, about

for which the phase varies from one end to the other by 'IT. One can show that the re..<;ultant contribution these subzones has an intermediate between the phases at the zone and end, such that cessive half-period zones are 'IT, or 180°, apart. This is also clear each half-period zone is a phasor diagram in Each of the small phasors represents the contribution from one subzone. The first half-period zone is completed after a number of such phasors produce a phasor oppoin direction to first. The amplitude Q] (dashed line) the resultant of the subzones from zone. Notice that it makes an angle of 90° relative to the reference or axial so that the phasor from the first For a large number of the phasor diagram bezone has a phase of comes circular and the of Ql is the diameter of The obliquity 18-4b by each succeeding phasor slightly shorter than one. Thus the do not close but spiral inward. The summation of the waves at P from each half-period zone can be expressed as " " " " , " L t .. ",

+ ... or (18-14) The successive zonal au,v..:..",,,,,, are affected by three different considerations: (1) a gradual increase with N due to zonal areas, a gradual decrease with N due to the inverse square law effect as P increase, and (3) a to the first of these, gradual decrease with N due to the obliquity fuctor. With Sec. 18-4

fresnel Diffraction from Circular

371

it can be shown that the surface area

of the Nth Fresnel zone is given by

+ (2N

(18-15)

The quantity (Alro) is very smaU in most cases of interest. If the second term in the brackets is accordingly neglected compared to the first, Eq. (18-15) describes zones with equal areas (independent of N), given by (18-16) The existence of the second term, however smaU, indicates increases in zonal areas with N and corresponding small increases in the successive term'> of Eq. (18-14). Now one can show that these increases are canceled by the decreases that arise from the second consideration, the effect of the inverse square law. This leaves only the obliquity factor, which is responsible for systematic decreases in the amplitudes as N increases. A phasor diagram for the amplitude terms of Eq. (18-14) is shown in as each zone contribution is added. The resultant phasors are shown in Figure 18-5b. The individual phasors in Figure 18-5a are separated vertically for clarity. Each phasor is out of phase with its predecessor by 180" and is also shorter, due to the obliquity factor. The resultant in 18-5b at the start of the phasor al and terminate at the end of the phasor aN for any number N of contributing zones. Notice the large changes in the resultant phasor AN, for sman N, as the contribution from each new zone is added. For N large, the diagram shows clearly that the resultant amplitUde AN approaches a value of AI? ad2, or half of that of the first contributing zone. The resultant amplitude is seen to oscillate between magnitudes that are and smaller than the limiting value of a1/2, de1J""IUUJIj>:. on whether it an even or an odd number of contributing zones. A A, __ _______ ..... _____ ~

~

~

A 2 +--

AN

_

:

AN + ,--.~ . - - - - 1 1 - -.... I

I I

---------..1----

.

I

I I

la)

..

Ib)

Figure 18-5 Phasor diagram for Fresnel half-period zones. Individual are shown in (a) lind the resultant phasors at each step in (b).

372

Chap. 18

Fresnel Diffraction

A,

= 81

A:t

8, - 8 2

A3

a,-a2+ a3

careful study of may be expressed

where N is even, the resultant <>n'''Irn,,,,y"¥u,jr,,lu

(18-17)

N

and where N is odd by

+

N

We may use either conclusions:

J. If N is small so that al

(18-18) 18) to make the following

aN,

tially al , that of the first zone zero. 2. If N is large, as in the case either N even or odd, the zone, or 01/2. These conclusions produce some taBy. For example, suppose an aperture coincides with the first to admit the second zone as well, tude at P! Now remove the opaque structed wavefront contribute. The the first-zone aperture alone. QU'P"""'''\.-, the unobstructed "'n'~rh"rA alone. Such results are ",,,,nriH;:1rlHH experience; they ne;cessar

amplitude is essenresultant amplitude is near approaches zero, and for half that of the first contribut-

can be verified experimenmeasured at P when a circular v .....,. u.u"'" the wider almost zero ampJiso that all zones of an unobbec:olJoes aj2, or half that due to proportional to the square of ~ that due to the first-zone ~aC~I'Llse not apparent in ordinary is understood.

Diffraction pa[(ern due 10 an disc, the Poisson spot (From M, Cagnet, M, Francon, AI/as of Optical PhenumBerlin: 1962,)

Sec. 18-4

Fresnel Diffraction

313

Another "'Ul''''"l1>I'LIII that is of some historic interest follows from a consideralion of the effect at P when a round obstacle or disc just covering the first zone is substituted for the aperture. The light reaching P is now due to all zones except the first. The first contributing zone is therefore the second zone, and by the same arguments as those used above, we conclude that light of amplitude ad2 occurs at P. Thus the irradiance at the center of the shadow of the obstacle should be almost the same as with no disc present! When Fresnel's paper on diffraction WdS presented to Poisson argued that this prediction was patently absurd and so undermined its theoretical basis. However, Fresnel and Arago showed experimentally that the spot, now known somewhat ironically as Poisson's spot, did occur as predicted. The diffraction pattern of an opaque circular disc, including the celebrated Poisson spot, is shown in Figure 18-6. As often happens in such cases, conclusive on hand, a century before the experimental evidence was argument. Here is an illustration of the need to fit experimental results into a successful conceptual framework if they are to make an impact on the scientific world.

18-5 PHASE SHIFT OF THE DIFFRACTED LIGHT

The first in 18-5 is drawn, rather arbitrarily, in a horizontal direction, seen, however, the first and the other phasors are then related to it. As we behind that of phasor. due to the first Fresnel zone, has an effective phase of the light reaching P along the axis. The directly propagated light could therefore be re['r~;entea by a phasor in the vertical direction, making an of "1T /2 with al. Now the resultant phasor of N zones is also in the direction of al. We are forced by these observations to conclude that phase of the light at P, deduced from the Fresnel zone scheme, is at variance by 9(t relative to the phase of the light reaching P directly along the axis. To remove this discrepancy and to make the result'> agree with the phase of the wave without diffraction, Fresnel was forced to assume that secondary wavelets on diffraction leave with a gain in phase of "1T /2 relative to the incident wavefront. The factor of i introduced in Eq. (18-6) for this purpose appears naturally in the Kirchhoff derivation of the same equation.

18-6 THE FRESNEL ZONE PLATE

Examination of Eq. (18-14) that if either the negative or the positive terms are eliminated from the sum, the resultant amplitude and irradiance could be quite large. Practically, this means that every other Fresnel zone in the wdvefront should be blocked. Figure 18-7 shows a drawing of 16 Fresnel zones in which alternate in reduced zones are shaded. If such a picture is photographed and a a Fresnel zone plate is produced. Let the incident on such a size is wavefronts. Then the zone radii required to make the zone plate consist of zones half-period zones relative to a fixed Held point P can be calculated. From Figure 18-8, radius RN of the Nth zone must R~

which can be written as

374

Chap. 18

Fresnel Diffraction

(18-19)

p

18·7

18-8 Schematic for the calculation of Fresnel zone plate radii.

Fresnel zone plate.

Since we restrict our discussion to cases where A/ro ~ 1, the second term in square brackets is compared with the first. For example, taking A = 600 nm and ro = 30 cm, A/ro 2 X . The zone plate radii are thus given approximately RN

(18-20)

YNroA

Evidently the radii of successive zones in 18-7 increase in proportion to The radius of the first (N = 1) zone determines the magnitude of ro, or point P on axis for which the configuration functions as a zone plate. If the first zone has radius R 1 , then successive zones have radii of 1.41R 1 , 1 , 2R., and so on. Example If light of wavelength 632.8 nm illuminates a zone plate, what is the first zone radius relative to a point 30 cm from the zone plate on the central axis? How many half-period zones are contained in an aperture with a radius 100 times

Solution

(18-20),

Using Rl

=

= 0.0436 cm

Since R 0:: VN, N increases by a of 104 when R increases by a factor of lQ2. Thus a radius of 4.36 cm encompasses 10,000 Fresnel zones. If the third, fifth, ... zones are transmitting, as in Figure (18-14) becomes Au;

al

then

+ a3 + as + a7 + a9 + all + al3 + alS

with 8 zones contributing. When these few zones are reproduced on a smaller scale, the obliquity is not very important, and we may approximate A16 8al. By comparison, this amplitude at P is 16 times the (ad2) of a wholly unobstructed wdvefront. The irradiance at Pis, (16)2, or 256, times as even for an aperture encompassing only these 16 zones. If Pis 30 cm aWdy, as in the previous example, the radius of this aperture, by . (18-20), is only 1.74 mm for 632.8 run light. This concentration of light at an axial point shows that the zone plate as a lens with P as a focal Rearranging (18-20), we call the distance ro the first focal point (N 1) with focal length /1 given by (18-21) Sec. 18-6

The Fresnel Zone Plate

375

There are other points as well. As the field P approaches the zone along the axis, the same zonal area of radius RI encompasses more half-period zones. In ]8-20), when is fixed, N mcrea..<;es as Yo Thus as P is iil2 for the same zonal radius R I • At moved toward the plate, N = 2 when ro this point of the zones covers two half-period zones and all zones cancel. When ro =:: ii/3, N = 3, and three zones contribute from the zone of radius . Of two cancel, but one is left to contribute. The next three are opaque, and so on, as in Eq. {

+

A

(18-22)

Of first six zones, one zone contributes rather than three, so that the ir~ that at roo argument may, of course, be extended to radiance at ro/3 is when the original zone of radius RJ includes five zones, and the irradiance is:is that at ro, and so on. Thus other maximum intensity points the axis are to be found at

NA'

Nodd

(18-23)

Example What are the focal

for the zone plate described in the (18-23) to£ether with Eq. (18-20),

Solution

Ri NA so that so on.

/1

30/1

30 em,

/1

= 30/3

NA

N IO em,

/5

30/5 = 6 em, and

18-7 FRESNEL DIFFRACTION FROM APERTURES WITH RECTANGULAR SYMMETRY Diffraction by straight rectangular apertures, and wires are all conveniently Eq. handled another approximation to the Fresnel-Kirchhoff diffraction (18-6). Forthis let the source S in Figures] 8-1 and] 8-2 a slit, so that the wavcfronts emerging from S are cylindricaL Recall that cylindrical waves can be expressed mathematically in the same form as spherical waves, except that the amplitude decreases as l/Vr so that the irradiance decrease.-, as l/r. pursuing Fresnel's quantitative treatment of cases, consider qualitatively what we might by using the concept of Fresnel half-period zones. This time the zones are rectangular strips along the wavefront, as in Figure 18-9. We wish to show that the sum of all phasors now gives the end points of a curve called the Cornu spiral. As the average phase at P of the light from each zone advances by a half-period, or 11'. In Figure I the rectangular strip zones are shown above and below the axis SP. Unlike the Fresnel circular zones, the areas of the zones fall off markedly with N so that su(:ceSSJ'{e phasor amplitudes of the zonal contributions are distinctly shorter. A phasor diagram for the complex amplitudes from the zones above axis might look like Figure 18-10. If the first zone is subdivided into smaller which advance by equal phases, the corresponding phasors can be by 01, 02, . . . • as shown. 376

Chap_ 18

Fresnel Diffraction

~ )

S

ro+N(~)

N

--

2 0

P

1

A

TO

+ '2

2

N

-(b)

(al

Figure 18-9 Fresnel (a) edge view and (b) front view.

strip zones on a cylindrical WlIVefront in an

T

I I

I I

I I I /

I I Al I A2 I I I I

I

I

I I

/

I

I

I

/

I

I

/

I

I

I

I

I I

/

I

I

I

I

1/

I

I

1/ II 1/

III

Figure 18·10 Phasor for the first two half-period Fresnel zone strips, each subdivided into smaller zones of equal prnse incre-

1/

t

It

o

ment.

When first half-period zone has been included, the last phasor is advanced by 7T relative to the first and ends at T. The sum of an these contributions is then the 18-4b, the resultant phasor has a phasor AI. In the case of circular zones, angle of 7T /2 and the point T falls on the vertical axis. Because of the rapid magnitudes in this case, the angle is less than 7T /2. After decrease in of the second half-period zone, the phase changes through the by another 1800 and the last phasor ends at B. The resultant phasor, which includes two full half-period zones, is By continuing this process, one sees that the Sec. 18-7

Fresnel Diffraction from

with Rectangular Symmetry

377

phasors a smooth curve, which in to a limit point the eye of the A AR from 0 to E then the contributions of half the unimpeded wavefront, above the axis SP in Figure 18-9a. A similar argument for the zones below the axis would lead to a twin spiral, represented in the third quadrant of this Cornu spiral and connecting at the origin O. If the coordinates of all are known, the amplitudes due to contributions from any number of zones can be determined from such a drawing and the relative irradiances compared. The quantitative treatment that allows us to make such calculations follows.

18-8 THE CORNU SPIRAL If we neglect the of obliquity factor and the variation of the product rr' in the denominator of Eq. (18-6), the Fresnel-Kirchhoff integral may be approximated by

cfI

(18-24)

da

Ap

all constants are into C. We assume that the surface closed including the aperture is zero everywhere over the self, so that we need the only over the aperture in the of Figure 18-11a. A view, which shows the curvature of the cylindrical wavefront, is shown in 18-1 I b. The distance r + r I may be determined approximately from this For h ~ p and h ~ q,

r'

= (p2 + h2)1/2 =

p(] + ;:y/2

Thus

and similarly, r

It follows that

r + r'

==

(p

+ q) +

(~ + !) 2 1 P

1

1 q

-+-

D=p+q and L

(18-25)

we have

Then

378

Chap. 18

r

+ r' ==

=

c

D

h2

+-

(18-26)

2L

(18-24) becomes

Fresnel Diffraction

II

eik(o+filIZLl

da

)(

lal

Figure Is-n (a) Cylindrical wavefroms from source slit S are diffracted by a rectangular aperture. (b) view of (a).

Ibl

If the elemental area da is taken to be the shaded strip in da = W dz, h = z, and

I8-lla,

(18-27) The exponent kz2/2L = 7TZ2I LA. Making a change of variable, we let

z=

v

or

v

=

(18-28)

whereby the magnitude of Ep is

Using Euler's theorem on the integrand and recombining constants again, we may write (18-29)

Sec. 18-8

The Cornu Spiral

379

The two integrals in this form are the Fresnel integrals, which we name C(v)

==

S(v)

==

f f 0

cos

(1TV2) 2 dv

(18-30)

sin

dv

(18-31)

Table 18-1 provides numerical values of these definite integrals for various values of v. As we shall see in several applications, choice of the upper limit v in the Fresnel TABLE 18-1

380

FRESNEL INTEGRALS

v

C(v)

S(v)

v

C(v)

S(v)

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40

0.0000 0.1000 0.1999 0.2994 0.3975 0.4923 0.5811 0.6597 0.7230 0.7648 0.7799 0.7638 0.7154 0.6386 0.5431 0.4453 0.3655 0.3238 0.3336 0.3944 0.4882 0.5815 0.6363 0.6266 0.5550 0.4574 0.3890 0.3925 0.4675 0.5624 0.6058 0.5616 0.4664 0.4058 0.4385 0.5326 0.5880 0.5420 0.4481 0.4223 0.4984 0.5738 0.5418 0.4494 0.4383

0.0000 0.0005 0.0042 0.0141 0.0334 0.0647 0.1105 0.1721 0.2493 0.3398 0.4383 0.5365 0.6234 0.6863 0.7135 0.6975 0.6389 0.5492 0.4508 0.3734 0.3434 0.3743 0.4557 0.5531 0.6197 0.6192 0.5500 0.4529 0.3915 0.4101 0.4963 0.5818 0.5933 0.5192 0.4296 0.4152 0.4923 0.5750 0.5656 0.4752 0.4204 0.4758 0.5633 0.5540 0.4622

4.50 4.60 4.70 4.80 4.90 5.00 5.05 5.10 5.15 5.20 5.25 5.30 5.35 5.40 5.45 5.50 5.55 5.60 5.65 5.70 5.75 5.80 5.85 5.90 5.95 6.00 6.05 6.10 6.15 6.20 6.25 6.30 6.35 6.40 6.45 6.50 6.55 6.60 6.65 6.70 6.75 6.80 6.85 6.90 6.95

0.5261 0.5673 0.4914 0.4338 0.5002 0.5637 0.5450 0.4998 0.4553 0.4389 0.4610 0.5078 0.5490 0.5573 0.5269 0.4784 0.4456 0.4517 0.4926 0.5385 0.5551 0.5298 0.4819 0.4486 0.4566 0.4995 0.5424 0.5495 0.5146 0.4676 0.4493 0.4760 0.5240 0.5496 0.5292 0.4816 0.4520 0.4690 0.5161 0.5467 0.5302 0.4831 0.4539 0.4732 0.5207

0.4342 0.5162 0.5672 0.4968 0.4350 0.4992 0.5442 0.5624 0.5427 0.4969 0.4536 0.4405 0.4662 0.5140 0.5519 0.5537 0.5181 0.4700 0.4441 0.4595 0.5049 0.5461 0.5513 0.5163 0.4688 0.4470 0.4689 0.5165 0.5496 0.5398 0.4954 0.4555 0.4560 0.4965 0.5398 0.5454 0.5078 0.4631 0.4549 0.4915 0.5362 0.5436 0.5060 0.4624 0.4591

Chap. 18

Fresnel Diffraction

integrals is determined by the vertical dimension of the diffraction aperture, measured in terms of the aperture height z. Now the irradiancc at P, since Jp ()( 1Ep 12, is given by (18-32) If the values of the Fresnel integrals are plotted against the variable v, as real and imaginary coordinates on the complex plane, the resulting graph is the Cornu spiral (Figure 18-12). According to Eg. (18-32), a straight line drawn between any two points of the spiral must be proportional to an amplitude of the electric field at P, since we have, by the Pythagorean relation,

where C and S are distances along the axes of a rectangular coordinate system. The origin v = 0 corresponds to z = 0 and therefore to the y-axis through the aperture of Figure 18-11a. The top part of the spiral (z > 0 and v > 0) represents contributions from strips of the aperture above the y-axis, and the twin spiral below (z < 0 and v < 0) represents similar contributions from below the y-axis. The limit points or eyes of the spiral at E and E' represent linear zones at z = ±oo. Furthermore, we may show that the variable v represents the length along the Cornu spiral itself. The incremental length dl along a curve in the xy-plane is given in general by

is(v) I

1.5 0.7 0.6 -

----

/

...........

.......-: ~

0.4

!I t- ~ ~ \ 1\ \ )); }

0.3

2.

0.5

'"~V 1 I

0.1 0.8

0.7

o.~~

y-x

:--

I

I

\

,

,

I

"~

0.1 0

0.1

0.2

J-~ 0j3

0.4

0.5

I

,1.0

~

0.6

I

~

0.7

C(v)

°i

S

I

0.2 0.3

2.0

{ if(£, \~

I 0.2

0.1

.0.5.

Ix-- +--....

1.0

0.3

\

~

0.2 0.4

t'\

"><;

~

"'-- f--'"

,\ J

f

/

0.4 0.5 0.6 0.7

1.5

Figure 18-12 able v.

Sec. 18-8

The Cornu spiral. Intervals along the spiral measure the vari-

The Cornu Spiral

381

Here x and yare the Fresnel dP = [cos

C and S, .."'''''''''''''tt

2(W;2) +

(

W;2) ]dv

2

or simply, dJ

(18-33)

dv

18-9 APPLICATIONS OF THE CORNU SPIRAL Approximate evaluations of the Kirchhoff-Fresnel of the Cornu We examine a few cases.

are possible with the help

Unobstructed Wavefront. The vertical dimension of the aperture ranges from -00 to + 00. The resultant amplitude Ep on the Cornu spiral in this case is a phasor drawn from E' to E, as shown in Figure 18-13. These have -0.5) and (0.5, 0.5), as is by evaluating coordinates

Joo cos (W;2) dv

1 00

=

o

sin

dv

0.5

0

with the help of the definite integral formula .fu cos (x 2 ) dx = and y-components of Ep are both unity and = v'2, or, Ip

=

Then the xEq. (18-32),

= lu

(18-34)

Other irradiances may convenientJy be compared to this value of lu obstructed wavefront.

2/0 for the un-

is(v)

1.5

0.5

--1.0

---.-__=:::=--4f'--=--:-'=------1.... C(v)

-1.0

---

-0.5

18-13 The representing lhe unobstructed wavefront has a 00 the

-1.5

Cornu spiral of

vi

Straight Edge. Fresnel diffraction by a straight is pictured in 18-143. At the field P on the the edge of the geometric shadow for which v = the upper half of the zones and Cornu spiral are effective. The resulting z amplitude, shown as OE in Figure 18-14b, a magnitude of 1/v2 and, consequently, Ip =

382

Chap. 18

Fresnel Diffraction

(18-35)

p'

p

p"

la) iSlv'

Vertical screen position Ie)

(b)

(d,

(a) Straight-edge ditfraClioo, (b) of the Cornu spiral in analyzstraight-edge diffraction. (c) Ir"uli:••""P 10 straight-edge Fresnel (From M, M, (d) Diffraction fringes from a Francon, and J, C Thrierr, Alias of /'henomenon, Plate 32, Berlin: Springer-Verlag, 1962)

Sec, 18-9

Applications of the Cornu

383

The irradiance is plotted as point P in Figure 18-14c. For a lower point P" on the screen, we now consider the zones relative to the axis O"P", drawn from P" to the wavefront at the aperture. For P", the point 0" marks the center of the wavefront, just as the point 0 marks the center of the wavefront relative to point P. Thus above the axis O"P", the new "upper half of the wavefront," some of the first zones do not contribute to the irradiance at point P" on the screen, due to the obstruction. Of course, all the bottom half of the wavefront is similarly blocked off. The contributing zones, beginning at a finite positive value of z and continuing to 00, are represented by the amplitude BE on the Cornu spiral. As the observation point P" moves from P to lower points on the screen, the representative phasor point B slides along the Cornu spiral from 0, with its other end fixed at E. One sees that the amplitude, and so the irradiance, must decrease monotonically, as shown in Figure 18-14c. The edge of the shadow is clearly not sharp. On the other hand, for a point P' above P, we conclude that relative to its axis 0' P', all the upper zones plus some of the first lower zones contribute. The corresponding amplitude in Figure 18-14b is like DE. As P' moves up the screen, D moves down along the spiral. In this case, as D winds around the turns of the spiral, the length of DE oscillates with various maxima and minima points, as shown in Figure 18-14c.

Example Calculate the irradiance at the first maximum above the edge of the shadow.

Solution

At the first maximum point, the tail of the phasor is at the extreme point G relative to E. From Figure 18-12, we read from the curve the value v = 1.2 at this point. Then from Table 18-1, C(1.2)

=

-0.7154

whereas at E, C(oo) = S(oo)

= 0.5.

Ep = [(-0.7154 - 0.5)2

and

S(1.2)

=

-0.6234

The magnitude of the phasor GE is then

+

(-0.6234 - 0.5)2]1/2 = 1.655

and

Ip = (1.655)2/0 = 2.74/0 = 1.37/u The irradiance at the first maximum is 1.37 times that of the unobstructed irradiance. Similarly, calculating the magnitude of the first minimum amplitude HE, one finds that 1p = 0.781u, and so on. The irradiance curve approaches the value 1u due to the unobstructed wavefront. A photograph of the pattern is given in Figure 18-14d. Notice that, from Figure 18-14a, it is possible to relate points at elevation yon the screen to corresponding points at elevation z on the wavefront, such that

y =

(p ; q)z

The value of z determines the length v on the Cornu spiral. permitting quantitative calculations of screen irradiance to be made.

Single Slit. If the diffracting aperture is a single slit of width w, as in Figure 18-15a, then z = w, and by Eq. (18-28),

av = 'J-U [2 az = [2 'J-U 384

Chap. 18

Fresnel Diffraction

W

(18-36)

istvl

1.5

1.0

--

p'

p

-1.0

-1.5

tbl

tal

Figllre 18-15 Fresnel diffraction from a single slit (a) and ils amplitude represen(b). tation on the Cornu

Once L is calculated from (18-25), the interval avon the Cornu Note that v plays the roJe of a universal, dimensionless spiral can be allowing one Cornu spiral to serve for various combinations of p, q, and A. For example, if p q = 20 cm and A = 500 nm, av = for a slit width of 0.01 cm. To calculate the irradiance at P, a length of t; 0.632 along the Cornu as shown, determines the amplitude spiral, symmetrically placed about the Ep • For a point ]ike P I above P, the contributing zones form a group in the lower half of the unobstructed wavefront. Their center point z' below the axis 0 I P' determines the center v' of the length av along the spiral, by Eq. (18-28). As P' moves av slides toward the lower eye of the as farther above the axis, the shown. Although av remains fixed in length, its placement on different portions of the spiral chord and a When P' is below is placed the upper In this way, the irradiance of the entire the axis, pattern can be calculated. From this approach, one can reason that the diffraction pattern of the slit is symmetrical about its center and that the while oscilis never zero. Example

Let the wavelength of the light be 500 nm and the slit of Figure 18- t Sa be I mm in width. Light emerges from a source slit, as shown, that is 20 cm from .1"Tr.-," ......... slit. The diffraction pattern is observed 30 cm from the slit. irradiance at a of 1 mm above the axis at the screen? Solution The parameter L pq/(p + q) (20)(30)/(20 + 30) = 12 cm. We are looking for the irradiance at a point like P' in Figure IS-ISa. Cont ...;." .. ;; ....... zones are like those included in the chord FG of Figure IS-ISb, but with the spiral length FG moved somewhat toward the lower end. Fewer zones to P I than make a contribution from the upper half of the wavefront I mm. they do relative to P. At the screen, point P I corresponds to y corresponding point on the wavefront is

z Sec. 18-9

(p ~ q)Y ~(l)

Applications of the Cornu Spiral

0.4 mm 385

Then the corresponding point on spiral length given by

Cornu spiral

G) is determined by

x 10-3)

2.3094

To find the lower point Vz (like F), note that the length dv is independent of the point of observation and can be found from

12 dz = VtA

slit. Thus dv = 5.

where w = 1 mm is the width of V2

= VI

-

dv

w

= 2.3094

and

- 5.7735

The contributing amplitude is the chord from VI to V2 on the Cornu spiral. The coordinates of these are found by interpolation in Table 18-1, giving For VI: For

V2:

=

C(-3.464I)

0.6199

and

-0.4988

S(2.3094) = 0.5594

and

S(-3.4641)

-0.4204

The amplitude is calculated from

= [0.6199

(-0.4988)]2

+ [0.5594

(-0.4204)]2 =

1

Thus the irradiance at the point is Ip = 2.21110 = 1.11 The irradiance at the screen. I mm above axis, is 1.11 times the irradiance an unobstructed wavefront there. Wire. Suppose now the narrow slit of 18-15a is replaced a long, but thin, opaque obstacle such as a wire (Figure I If the width w of slit and wire are equal, then there is an exact reversal of the transmitting and blocking zones of wavefront. Now all parts of the Cornu are to be used in calculating the """'''''''''''J<, amplitude except that designated by in 18-16b. This iSM

I I

I

P'

1---------+

f

jl w

-f I

I I I

-1.5

tal

(bl

Figure 18-16 Fresnel diffraction from a wire (a) and its amplitude representation on the Cornu spiral (b).

386

18

Fresnel Diffraction

situation clearly yields two which must be added together. when field points The interval av omitted slides along the like P I are considered. The in the actual photograph of Figure 18-17. An example of a more Ovv1ml"uw.,at'lkiU Fresnel pattern than those considered above is also in Figure 18-18.

Figure 18·17 Fresnel diffraction M. Francon, and J. C. Thrierr, Atlas of 1962.)

fine wire. (From M. Cagnet, PhenomenOll, Plate 32, Berlin:

18·18

Sec. 18-9

Applications of the Cornu

Fresnel shadow of a screw. M. Francon. and 1 C. Phenomenon, Plate 36, 1962.)

387

18-10 BABINET"S PRINCIPLE J8-15 and 18-16, in which dear and opaque regions complementary If one of the apertures,

like those of

say B, are put into and the amplitude at some point of the screen is determined for each, the sum of these amplitudes must equal the unobstructed there. This is the content of Babinet's principle, which we express as (18-37)

with A and B any two apertures. For for the slit and wire apertures we have considered, notice that in 18-16b we may express the phasor addition at P on the screen by E'F+FG+GE

E'E

where E + GE represents the amplitude due to the wire, FG the amdue to the slit, and E represents the unobstructed ampJitude. principle is a point where Eu O. An case of Then, by Eq. = -EB and lA lB at the point. In practice, Fresnel dif= 0 without an aperture. Fraunhofer diffraction does not produce amplitudes fraction however. as in the case of the pattern formed by a point source and a 0, essentiaHy. LOmr)lernelllens. For the region outside the smaJJ Airy tary introduced into such systems outside central image, of the same identical diffraction Thus positive and pattern produce the same diffraction pattern. This discussion is appropriately reminiscent of the fact that positive and negative holograms produce the same graphic

IS-I. A I-mm diameter hole is illuminated by plane waves of 546-nm light. According to which (near-field or may be applied to the difthe usual fraction problem when the detector is at 50 cm, I m, and 5 m from the 18-2. A 3-mm diameter circular hole in an opaque screen is illuminated normally by plane waves or wavelength 550 nm. A small photocell is moved the central density of the diffracted beam. Determine the locations of the recording the first three and minima as the photocell approaches the screen. IS-3. A distant source of sodium (589.3 nm) illuminates a circular hole. As the hole increases in diameter. the irradiance at an axial point 1.5 m from the hole passes alternately through maxima and minima. What are the diameters of the holes that produce (a) the first two maxima and (b) the first two minima? 18-4. Plane waves of monochromatic (600 nm) are incident on an aperture. A detector is situated on axis at a distance of 20 cm from the aperture (a) What is the value of R l , the radius of the first Fresnel zone, relative to the detector? (b) If the aperture is a circle of radius I cm, centered on axis, how many zones does it contain? (c) If the aperture is a zone plate with every other zone blocked out and with the radius of the first zone equal 10 Rl in (a)], determine the first three focal lengths of the zone plate. 18-5. The zone radii given by (18-20) were derived for the case of plane waves incident on the aperture. If instead the incident waves are spherical, from an axial 388

18

Fresnel Diffraction

source at distance p from the aperture, show that the necessary modification vields

18-6. 18-7.

18-8.

18-9.

where q is the distance from aperture to the axial point of detection and L is defined II p + I/q. by IlL (a) and (b) of problem 18-4 when the source is a source to cm from the aperture. Take into account the results of problem 18-5. A point source of monochromatic light (500 nm) is 50 cm from an aperture is located 50 cm on the other side of the aperture The detection (a) The portion of the aperture plane is an annular ring of inner radius 0.500 mm and outer radius 0.935 mm. What is the irradiance at the detector, relative to the irradiance there for an unobstructed wavefront? The results of problem 18-5 will be helpful. (b) Answer the same question if the outer radius i" 1.00 mm. (c) How many zones are included in the annular ring in each case? By what perecntage does the area of the 25th Fresnel half-period zone differ from that of the first, for the ease when source and detector are both 50 cm from the aperture and the source at 500 nm? A zooe plate is to be having a fucallength of 2 m for He-Ne laser light of 632.8 nm. An ink of 20 zones is made with alternate zones shaded in, and a reduced transparency is made of the drawing. (a) If the radius of the first zone is 11.25 cm in the what reduction factor is

(b) What is the radius of the last zone in the drawing? 18-10. A zone plate has its center half zone opaque. Find the diameters of the first threc focuses light of wavelength 550 nm at 25 cm clear zone.." such that the from the plate. 18-11. For an incident plane wavefront, show that the areas of the Fresnel half-period zones relative to an observation at distance x from the wavefront are approximately constant and equal to ?TAx. Assume that Alx is much smaller than l. 18-12. Light of wdvelength 485 nm is incident normally on a screen. How is a circular openJn~ in an otherwise opaque screen if it transmits four Fresnel zones to a point 2 m away? What, approximately, is the irradiance at the point? 18-13. A slit of width! millimeter is illuminated by a collimated beam of of wavelength 540 nm. At what observation point on the axis is L\v = 2.5? 18-14. A source slit at one end of an bench is illuminated by monochromatic mercury light of 435.8 nm. The beam diverging from the source slit encounters a second slit 0.5 mm wide at a distance of 30 cm. The diffracted light is observed on a sereen at 15 cm farther along the bench. Determine the irradiance terms of the unobstructed irradiance) at the sereen (a) on axis and (b) at one of the geometrical shadow of the diffracting slit. 18-15. A slit illuminated with sodium is placed 60 cm from a straight and the diffraction is observed a photoelectric cell, 120 cm beyond the straight Determine the irradiance at (a) 2 mm inside and (b) I mm outside the edge of the gcometrical shadow. 18-16. Filtered green mercury light (546.1 nm) emerges from a slit 30 cm from a rod 1.5 mm thick. The diffraction pattern formed by the rod is examined in a plane at 60 cm beyond the rod. Calculate the irradiance of the pattern at (a) the center of the geometrical shadow of the rod and (b) the edge of the shadow. 18-17. For the near-field diffraction pattern of a straight calculate the irradiance of the second maximum and minimum, using the Cornu spiral and the table of Fresnel intevalues given. 18-18. Fresnel diffraction is observed behind a wire 0.37 mm which is placed 2 m from the light source and 3 m from the screen. If light of wavelength 630 nm is

Chap. 18

Problems

compute, the Cornu spiral, the irradiance of the diffraction pattern on the axis at the scrcen. Express the answer as some number times the unobstructed irradiance there. 18-19. Calculate the relative irradianee (compared to the unobstructed irradiance) on the optic axis due to a double-slit aperture that is both 10 cm from a point source of monochromatic light (546 nm) and 10 cm from the observation screen. The slits are 0.04 mm in width and to center) by 0.25 mm. 18-20. diffraction is produced using a monochromatic light source (435.8 nm) at 25 cm from the slit. The slit is 0.75 mm wide. A detector is placed on the 25 cm from the slit. (a) Ensure that far-field diffraction is invalid in this case. (b) determine the distance above the axis at which single-slit Fraunhofer diffraction the first zero in irradiance. (c) Then calculate the irradianee at the same point, using Fresnel diffraction and the Cornu the result in terms of the unobstructed irradiance. with uniform opaque particles. When a distant point source of through the plate, a diffuse halo is seen whose angular width is about 2°. Estimate the size of the particles. (Hint: Use Babinet's principle.)

(I]

390

and Emil Wolf. Principles of Optics, 5th ed. New York: Pergamon Press, 1975. Ch. 8. Robert. Modem Optics. New York: John Wiley and Sons, 1990. Ch. 9. John B. B. Parrent, Jr., and Brian J. Thompson. Notebook: Tutorials in Fourier Optics. Bellingham, Wash.: SPIE Optical Press. 1989. Ch. 9. Longhurst, R. S. Geometrical and Physical Optics. 2d ed. New York: John Wiley and 1967. Ch. 13.

18

Fresnel Diffraction

19 111 14

12

0,1

02

03

0.4

05

OJ!

Theory of Multilayer films

INTRODUCTION of mf,'rlf'rl',nrp in single-layer dielectric films has been treated. in its esThe scntials, in 10. Many useful and interesting of thin films, oowever. make use of stacks of films. It is to multiple layers while ""',If,.,..... control over both refractive index (choice of material) and of flexibility in dcthickness. Such techniques provide a , ... f'>rt{'rp.~rp with almost any frequency-dependent yo",.,,,,,,,,,..,,...,. or transmittance characteristics. Useful applications such coatings indude antireflccting multilayers for use on the lenses of optical instruments and disavailable from near play windows; mUltipurpose broad and narrow band-pass ultraviolet to ncar infrared wavelengths; thermal reflectors and cold mirrors. which respectively. and are used in dichroic mirreflect and transmit filters deposited on the faces of beam splitters rors consisting to divide light into green, and blue channels in color cameras; and and in interferhighly reflecting mirrors for use in gas ometers. rather detailed calculations inComputer technique.... have made routine volved in the analysis of multilayer mm performance. The stack that will meet prespecified characteristics, remains a formidable task. In this we develop a lransfer matrix to the film and characterize its The approach differs from that used in treating 391

multiple reflections from a thin film in Chapter II. There we added the amplitudes of all the individual reflected or transmitted beams to find the resultant retlectarlce or transmittance. It will be more efficient, in the general treatment that to consider all transmitted or reflected beams as already summed in corresponding electric fields that the general boundary conditions required Maxwell's The relationships we require from electromagnetic theory, presented in Chapter 8, are summarized here. The energy of a plane, electromagnetic wave propagates in the direction of the Poynting given by

s=

xB

The magnitudes of electric and .....,F........ E

fields in the wave are related

=

vB

where the wave speed can also be expressed n

(19-1)

(

the refractive

c t;

(19-3)

The wave speed in vacuum is a constant, equal to

c where

= ---:==

(19-4)

and JLo are the permittivity and permeability, respectively, of free space. Eqs. (19-3), and (19-4), the magnitudes of the and electric fields can also be related by Eo

B

E

v

19-1 TRANSFER MATRIX . Our analysis is carried out in terms of the defined in 19-1. An incident beam is with E chosen for the moment in a direction perpendicular to the plane of incidence. in mind, that for normal incidence and Ell are equivalent since a unique plane of incidence cannot be specified.) The beam un,t",.·""".: external reflection at the plane interface (a) separating the external medium of index no from the nonmagnetic (J-L = JLo) film of index nl. The transmitted portion of the beam undergoes an internal reflection and transmission at the plane interface (b) separating the film from the substrate of index ns. each beam the Efield is shown--by the usual dot notation-to be pointing out of the page (-z-direction), and the B-field is shown in a direction consistent with (19-1). Notice that the y-component of B must reverse on The define a terminology for the magnitudes of the electric fields at the boundaries (a) and (b). For example, the sum of all the multiply reflected beams at interface (a) in the process of emerging from the film, represents the sum of aU the mUltiple beams at interface (b) and directed toward the substrate, and so on. In this way, we account for multiple beams in the interference. We assume that the film is both homogeneous and isotropic. We assume further that the film thickness is of the order of the wavelength of so that the difference between multiply and tmnsmitted beams remains small compared with the coherence length of the monochromatic light. This ensures that the 392

Chap. 19

Theory of Multilayer Films

Substrate

Film

B

E

E

B

E

B

v

(a)

(b)

Figure 19-1 Reflection of a beam from a The diagram defines boundary conditions to write Eqs. (19-6)-(19-9). Note tities used in to the plane of incidence. bold dot is used to denote directions

\ ,

a

beams are essentially coherent. The width of the incident beam, finally, is assumed to be compared with its lateral displacement due to the many reflections that contribute significantly to the resultant reflected and transmitted beams. Boundary conditions for the electric and fields of plane waves incident on the interlaces and (b) are simply stated: The tangential of the resultant E- and B-fields are continuous acrOSs the that their magnitudes on either side are equal. For the case considered in Figure 19-1, E is everywhere tangent to the planes at (a) and (b), whereas B consists of both a tangential component (y-direction) and a component (x-direction), Thus the boundary conditions for the electric field at the two interlaces become

Eo = Eo

+

=

E,l

+

(19-6) (19-7)

Corresponding equations for the

magnc~tic

field are (19-8) (19-9)

(19-8) and (19-9) in terms of electric fields with the help of

(19-5), (19-10) (19-11)

where we have written (19-12)

Sec. 19-1

Transfer Matrix

cos 6,1

(19-13)

cos 612

(19-14) 393

difference 5 that develops due to Now from Ell only because of a one traversal of the film. Using half the phase ifielren(;e calculated in (10-33) for two of the film, we have

5

(~:)nlt cos 8

= koll =

(19-15)

11

Thus (19-16) In the same way. (19-17) "hn~l ..."t"

(19-16) and (19-17) we may conditions at (b), expressed by

+

(19-18) (19-19)

)t~r'f>mllrt1trIO

for the moment the rightmost for Etl and Eil in terms of

~P'~I1IPf'~

these equations may be solved yielding (19-20)

E'I

(19-21) (19-20) and (19-21) into the equasubstituting the expressions from tions (19-6) and (19-10) for boundary (a), the result is I ( i s:nl

+

Ea

Eb cos 5

Ba

Eb (iYI sin 5)

+ Bb

5)

cos 5

where we have used the identities

(19-22) and (19-23) relate the net fields at one boundary with those at the other. They may be written in matrix form as

EO] [Ba

[COS 5 iYI sin IJ

-

i

S~

[Eb]

cos 5

Bb

The 2 x 2 matrix is called the IIU"-'''''. matrix of the film,

IDl

(19-24) f'''nr''''''''nt~'rl

[:~: ~::]

in

by (19-25)

If boundary (b) is the of another film layer, rather than the substrate (19-24) is still valid. The fields and Bb are then related to the fields the back boundary of the second film layer by a second transfer matrix. then for a multilayer of number N of layers,

[!:J 394

Chap. 19

IDl I £JJl 2 £J.n 3

Theory of Multilayer Films

•••

ffi'N

[!:J

An overall transfer matrix, representing the entire stack is the product of the individual transfer matrices, in the order in which the light encounters (19-26) We return now to Eqs. (19-7), (19-10), and (19-11) to make use of those members previously ignored in first finding the transfer matrix. Those remaining are (19-27)

= E12

(I 9-28)

= 'Yo(Eo - Erd Bb =

(19-30)

For the fields as represented by Eqs. (19-27) to (19-30), the transfer matrix, (19-24) and (19-25), may be written as

] E + [ 1'o(~o ErJ )

(19-31)

Equation (19-31) is equivalent to the two equations, I

+

r.,ti~~ •• "",

r

+

mill

r) = m2d

'Yo(i where we have used the

r

(l9-32)

mJl'Ys!

+

(19-33)

m22'Y.t

and transmission coefficients defined by

==

and

t

(19-34)

Eo

and reflection

Equations (19-32) and (19-33) can be solved for the coefficients in terms of the transfer-matrix elements to t = ----------~~---------

(19-35)

r

(19-36)

'}'omll

+

'YO'Y.mJ2

+

m21

+ 'Ys m 22

Equations (19-35) and (19-36), together with the transfer-mattix as defined by Eq. (19-24), now enable one to evaluate the reflective and transmissive properties of the single or multilayer film represented by the transfer matrix. Before continuing with applications of these equations, we must take into account the necessary modification of the theory that results when the incident electric field of 19-1 has the other polarization. that in the of incidence. ""~JIJU'>'" that E is chosen in the original direction of Band B is rotated accordingly to maintain the same wave direction. If the equations are developed along the same one finds that only a minor alteration of the transfer matrix becomes necessary: In the expression for 'YI, Eq. (19-13), the cosine factor now appears in the denominator rather than in the numerator. Summarizing, E .1 plane of incidence: E Sec. 19-1

'YI = nl

cos 611

(l9-37)

II plane of incidence:

Transfer Matrix

395

Notice that for normal incidence, where and are indistinguishable, we have cos OIl I, and the expressions are equivalent. For oblique incidence, however, results must be calculated for each polarization. An average can be taken for unpolarized light. For example. the reflectance becomes (19-38)

19-2 REFLECTANCE AT NORMAL INCIDENCE We apply the theory now for the case of normally incident light, the case most commonly found in Results apply quite well also to cases of near-normal dence. The beam remains normal at all so that all angles are zero. In Eqs. (19-12) to (19-14), the cosine factors in the y-terms are all unity. The matrix elements from ( appropriately modified to become

mil

cos

i sin I)

l}

mn

cos I}

(19-39)

are substituted into (19-36). After cancellation of the constant V EoILo and some simplification, we find (19-40) The reflectance R. which measures the reflected ",,,ilh>rll'P is defined by (19-41) To calculate first notice that the reflection coefficient r is complex and that it the general form

r=

A C

+ iB + iD

so that

1r 12 = rr * = A

C

+ iB + iD

A - iB = --:::-_---:C - iD

By inspection then, we may write normal incidence (19-42)

Example A 400- A. thick film of (n = 2.10) is deposited on glass (n = 1.50). Determine the normal reflectance sodium light. Solution The phase difference is given by 21T

A"(n1t)

Chap. 19

21T

= 589.3 (2.1)(40)

Theory of Multilayer Films

= 0.8956 rad

so that cos l3 (19-42),

0.6250 and sin l3 = 0.7806. Then, substituting into Eq.

R ThusR A plot of reflectance versus optical path difference fl. is shown in where the is calibrated in ratios of fl./ A. Each curve cor'res:polnds ent film index, but the glass substrate index has been chosen n. 1.52 in aU cases. The magnitude of the film index nl evidently determines whether the reflectance is enhanced (for nl > ns) or reduced (for nl < ns) from that for uncoated The lead either to curves show that quarter-wave thicknesses, or odd mUltiples optimum enhancement (high-reflectance coating) or to maximum reduction (antireflection These minima or maxima points in R can be made to occur at fl. of the film thickness. Notice various or any even multiple of a that for fl. nl < n" never just that from the uncoated glass. An single coat, reflects more than the uncoated glass at any wavelength. The periodic variation in R with fl., which is proportional to the film provides a way of monitoring film thickness in the course of a film deposition. The case of quarter-wave film thickness, t

A

Ao

4

4nl

20 18 16 14 12

~

Il::

10 8 6 4 2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Path difference/wavelength

Figure 19-2 Reflectance from a film layer versus normali7Mi path difference. The dashed line represents the uncoated glass substrate of index n, 1.52.

Sec. 19-2

Reflectance at Normal Incidence

1.0

makes the phase difference, Eq. (19-15), j) 21Tnlt/Ao = 1T/2, so that cos j) = 0 and sin j) = I. In this case, Eq. (19-42) reduces to normal incidence R quarter-wave thickness

(19-43)

From (19-43), it follows that a perfectly antireflecting film can be fubricated index nl ~. If the substrate is with a coating of A/4 thickness and glass, with ns 1 the ideal index for a nomeflecting is nl = 1 assuming an ambient with no 1. A compromise choice among available coating materials is a film of MgF2, with nl 1.38. For this film, (19-43) predicts a reflectance of 1.3% in the visible where the uncoated (set nl = no) would reflect about 4.3%. This difference a saving of light energy in an optical system where multiple surfaces occur. For example, after only six or three optical components in series, 93% of the incident light sursuch vives in the case of MgF2 compared with 77% in the case of uncoated glass.

19-3 TWO-LAYER ANTIREFLECTING FILMS Durable coating materials with arbitrary refractive indices are, of course, not immediately available. Practically speaking then, single films with zero reflectances cannot be fabricated. By using a double layer of quarter-wave-thickness films, however, it is possible to achieve essentially zero reflectance at one wavelength with available coating materials. At normal the transfer matrix of a film of quarter-wave thickness is anI

.!..-] 1'1

= [ .0

0

11'1

is found, according to Eq. (19-26),

transfer matrix Wi for two such the product

-l'dl'z, and

Matrix components are mil = -'Y2/I'I, m22 these values in Eq. (19-36), the result is

mil

= mZI =

(19-44)

r

Incorporating the refractive indices through the use of then squaring to get the reflectance, normal incidence l"rl"pr_W""J£' thickness Zero reflectance is predicted by

Chap. 19

(19-45) when

of Multilayer Films

(19-12) to (19-14) and

09-45)

R

~ 398

O. Using

noni

, or (19-46)

n. = 1.52

Glass

19-3 Antireflecting double layer, A/4-A/4 thickness films.

For a substrate (ns = l.52) and incidence (no I), the ideal ratio for the two films is I'Idnl = 1.23. The requirement is met quite well using zirconium dioxide (n2 = 2.1) and cerium trifluoride (I'll I both good coating materials. The ratio of refractive indices for Ceh and Zr02 of 1.27 produces a reflectance of only 0.1% according to Eq. (19-45). The is shown in Figure 19-3 and as curve (a) in Figure 19-4. zero reflectance at some wavemay not satisfy the very common need to reduce reflectance over a broad reof the visible spectrum. Curve (a) is rather on both sides of its minimum at 550 nm. Broader regions of low reflectance result for >../4->../4 coatings when the substrate index is larger than that of the adjacent film layer, that ns > n2. In such cases the index is "stepped down" substrate to ambient. Indices ""~.".. c, .. to satisfy this condition are in infrared aP1pli(;atllon values of ns are available, as in the case of with n. refractive indices is given in Table 19-1. Broader of low reflectance also be-

,,

10

I

9

,,

8

I

7

I I

I

6

I

I

~ 5

I

I:t

4

3

I

I

\

, I

2

I

I

I

!!JIIIII"'-__ '

.-....... ... ~ ...

450

350

Uncoated glass

l-

550

-- ..

.. ~. 850

Wavelength Inm)

19-4 110

"I "2 Sec. 19-3

J and ",

1.65, 1.85.

"2

Reflectance from a double-layer film versus wavelength. In all cases 1.52. Thicknesses are determined at A 550 nm. (a) A/4-A/4; 2.1. (b) A/4-A/2: ", = 1.38, 1.6. (c) A/4-A/2: ", 1.38,

"2

Antireflecting Films

399

TABI..E 19-1

REfRACl1VE INDICES fOR SEVERAL COATING MATERIALS

Material

Visible (- 550 nrn)

Cryolite MgF2 Si
1.30-L33 1.38 1.46 1.55-2.0 1.60 1.65 1.8 2.0 2.1 2.35 2.35 2.4

Near infrared (-

1.35 1.44 1.5-L85 1.55 1.59 1.75 1.95 2.0 2.2 2.2

3.3 4.0

come [X)Ssible in the visible region of the spectrum, once the restriction of equal A/4 coatings is relaxed. For example, curves (b) and (c) of Figure 19-4 show two such solutions to the problem, where the inner is A/2 thick, as illustrated in Figure At the wavelength of 550 nm, for which the A/4 and A/2 thicknesses are determined, the A/2 has no effect on the reflectance, and the double layer behaves like a single A/4 layer. At other wavelengths, however, the layer helps to keep R below values attained by a single A/4 layer alone. For n = 1.85 [curve (c)l, two minima near R = 0 appear. Although reflectance at nm is 1.2Mb, greater than for the A/4-A/4 coating of curve (a), it remains at values less than this over the broad range of wavelengths about 420 to 800 nm. For nz = 1.6 [curve (b)l, the response of the double layer, while more reflective, is flatter over the visible spectrum. Still other practical solutions for douare alble-layer anti reflecting films become possible if the thicknesses of the lowed to have values other than mUltiples of A/4. The curves of 19-4 have been calculated the theory presented in this chapter. The overall transfer-matrix elements are first determined by forming the product of the transfer matrices of the individual layers. In these elements, the phase difference I) is as a function of A, and the film thickness is determined by the A/4 or A/2 at a single wavelength. These matrix elements are then used in Eq. (19-36) for the coefficient. When. the reflectance as a function of wavelength is determined. Although the ca1culations can be tedious, they are easily done using a programmable calculator or computer.

Figure 19-5 Antireflecting double layer using A/4-A/2 thickness films. Reflectance curves are shown in Figure 19-4.

400

Chap. 19

Theory of Multilayer Films

19-4 THREE-LA YER ANDREFLECTING FILMS

The was used to the spectral of threelayer The use of three or more layer coatings makes possible a broader, low-reflectance region in which the response is flatter. If each of the three one can show that a zero occurs when the relayers is of 11./4 fractive indices (19-47) One such solution is shown in Figure 19-00 and plotted as curve ure 19-7. improvement when the middle is of 11./2 .""•.,.",,,"'" in Figure 19-6b and curve (b) of Figure 19-7.

-A

"4 A 2 ~

____________

n. : 1.52

n, = 1.52

lal

(hI

9 8 1

6

A ~-L_4

Figure 19..(1 Antireflecting (a) wavelength (b) Quarter-half-quarler wavelength layers. Reflectance curves are shown in Figure 19-7.

,, , ,, ,, , \

\

\

,

Uncoated

t-~~~~~"-~~~~"-

4

3

,, ,

\ \

\

2

\ \

\ 350

\

'...' ...

450

650

550

750

850

Wavelength (nm)

Figure 19-7 Refleclance from films versus In all cases = I and n. 1.52. Thicknesses are determined al A = 550 run. (a) A/4-1I./4A/4: n, = 1.38, n2 2.02, n3 = 1.8. (b) 1I./4-A/2-A/4: n, = 1.38. nz 2.2, n3 = 1.7. no

Sec. 19-4

Films

401

19-5 HIGH-REFLECTANCE LAYERS of the in a A/4-A/4 double-layer optimized for antireflection so that the order is air - high low - substrate, aU three rPf"I","tpn beams are in phase on emerging from the structure, and the reflectance is enhanced rather than A series of such double layers increases the reflectance and the structure is a high-refiectnnce or dielectric mirror. We derive now an expression for the reflectance of this type of structure, shown schematically in Figure 19-8, where High and Low high- and lovv-rt!fractive indices, respectively. transfer matrix for one double layer of A/4-thick coatings at normal incidence is the product of the individual film just as in the case of the double-layer antireflecting films: is

l~ ~H

!))'HL =

or IDlHL

-I 'YH

0

IDcH !))k

~L [~:L _ 0

] [ 0

0

0

(19-48)

]

]

. l'YL

-'YH 'YL

..,....,......;.,.,.--------........ot

High ...... Low High Low

19·8 High-R~fiec:tanlce stack of double layers with alternating high- and low-refractive indices. Reflectance curves are shown in Figure 19-9.

n.

in series,

For N similar double

IDl

(ID{HI!)){LI)

(IDlH2 !l)(l.2) • •. (IDlHN IDlLN)

l -;H

(IDlH 9JlL)N

Substituting the double-layer matrix, Eq. (19-48),

9)l

o]N

'YH

[( -'YLr

: (-;Hr

o

For normal 'YL

402

19

=

Theory of Multilayer Films

0]

and

'YH

(IDlHL)N

(19-49)

so that (19-50)

The matrix elements of the transfer matrix representing N high-low double layers of A/4 thick in are thus (19-51) these 36), we

"'."'........'" in the expression for the ..""f1'Pr't1r\n coefficient, Eq. (19at (19-52)

When numerator and denominator of Eq. (-ndnH)N/ns and the result is squared to

R

= max

by the factor we have

[
+

1]2 I

(19-53)

stack like that of (n = 1.46) and ZnS (n

What is the

for

19-8 incorporates six double layers on a (n = 1.48) substrate. of 550 nm at normal incidence?

=

Solution Substituting directly into

we get

R= or R

99.1%.

(19-53) predicts 100% when either N approaches infinity or when (nz/nH) approaches zero. Some data these tendencies are given in sees that the reflectance quickly 100% for several douTable 19-2. ble Since the smallest ratio of ndnH yields best reflectances, high-reflectance TABLE 19-2 REFLECTANCE OF A HIGH-lOW QUARTER-WAVE STACK Reflectance for N = 3 n,,'lh-ln,w layers versus nunH

1.0 0.91 0.83 0.77 0.71 0.67 0.625 0.59 0.56 0.53 0.50

Sec. 19-5

Reflectance versus N when ndnH = 0.587 for alternating double layers of MgF2 and ZnS

R (%)

N

R (%)

4.26 21.01 40.82 57.77

I 2

39.71 73.08 89.77 96.35 98.72 99.56 99.85 99.95

70.44

79.35 85.48 89.67 92.55 94.56 95.97

High-Reflectance

3 4 5 6 7 8

403

100

. I I

00

•

---------

(bl-,

"", .. ,

.II ,,,'

80

,-----Ie) -~-"""""" "

"I

,\ ,

\

... •

'~,

,

Ih

00

Q:

",

I

70

~

'\

,

',' \

I

'/

\

\

50

...

\

40

....

.1\

'\

\

I \

30

i

20

i

'\.Ii

10

\ \

\.. .' \,

. ----------...~\ ~...s;

Uncoated glass

,

~" ~

I

350

450

550

650

750

850

Wavelength (nm)

Hgure 19-9 Spectral reflectance of a high-low index Slack for (a) N = 2 and 6 double layers. Curve (c) represents an N = 2 stack with an additional (b) N high-index to the substrate. Layers are A/4 thick at A = 550 nm. In all cases, nH = 2.35, nL 1.38, n. = 1.52, and no 1.00.

stacks may be fabricated from alternating layers of MgF2 (n2 1.38) and ZnS (nH = 2.35) or Ti0 2 (llH = 2.40). The reflectance given in Eq. (19-53) represents the maximum reflectance at the wavelength Ao, for which the layers have optical of Ao/4. For other form, containing the wavelengths the transfer matrix must be used in its wavelength-dependent phase differences. curves for N 2 and N = 6 double-layer stacks have been calculated and plotted in Figure 19-9. Curve (c) shows the improvement in the maximum reflectance that results when an extra high-index layer is between the substrate and last low-indc;;x layer. The width of the high-reflectance in these curves is nearly independent of the ratio number of double layers used but increases when the ratio nz./nH increases. 19-9, representing alternating MgF2 and ZnS layers on Outis 1.70 in side the central stopband, the reflectance oscillates between a series of maxima and minima. The center of the stopband can be shifted by depositing layers whose thickness is A/4 at another A. Except for light energy lost by absorption and ""';'11t,, ...., during passage through the dielectric the transmission of the structure is given by T (%) 100 - R (%). Thus such structures can be as band-pass filters whose spectral transmittance is essentially the inverse of the spectral reflectance. Narrow band-pass filters that behave like Fabry-Perot eta Ions can be fabricated by separating two dielectric-mirror, multilayer structures with a spacer of, say, MgF2 film. Narrow wavelength regions that satisfy constructive can be produced far in wavelength so that all but one such is easily filtered out by a conventional absorption color The result is a filter with a pass bandwidth of perhaps 15 A and 40% transmittance.

404

Chap. 19

of Multilayer Films

19-1. Show that when the incident E-field is parallel to the of incidence, '}II has the form given in (19-37). 19-2. A transparent film is on glass of refractive index 1.50. (a) Determine values of film thickness and refractive index that will produce a film for normally incident of 500 nm. (b) What reflectance does the structure have for incident of 550 nm? 19-3. Show from Eq. (19-42) that the normal reflectance of a single half-wave thick deposited on a substrate is the same as the reflectance from the uncoated substrate.

of (n = 1.46) is deposited to a thickness of 137 nm on a 19-4. A single substrate (n I Determine the normal reflectance for light of (a) 800 nm; (b) 600 nm; 400 nm. Verify the reasonableness of your results by comparison with 19-2. 19-5. A 596-A-thick of ZnS (n = 2.35) is on (n = I Calculate the normal reflectance of 560 nm light. 19-6. Determine the theoretical refractive index and thickness of a (n = 4.0) such that normal reflectance is zero at a wavelenl~th posited on of 2 p.. m. What actual material could be used? 19-7. A double layer of quarter-wave layers of Ab (n = 1.60) and cryolite (n 1.30) are in turn on a glass substrate (n 1 (a) Determine the thickness of the layers and the normal reflectance for light of 550 nm. (b) What is the reflectance if the are reversed? 19-8. Quarter-wave thin films of ZnS (n = 2.2) and (n = 1.35) are in turn on a substrate of silicon (n = 3.3) to minimum reflectance at 2 p..m. (a) Determine the actual thickness of the (b) what difference does the ratio of the film indices differ from the ideal? (c) What is the normal reflectance .... rrvl"'''pt1l? 19-9. By with the appropriate transfer show that a quarter-wave/half-wave double as in Figure 19-5, produces the same reflectance as the quarter-wave layer alone. 19-10. Write a computer program that will calculate and/or plot reflectance values for a double under normal incidence. Let include thickness and indices of the layers and the index of the substrate. Check results againsfFigure 19-4. 19-11. Prove the condition given by Eq. (19-47) for zerO reflectance of 'nr,f''''_I~v,'r rlll>'rt"'r_(1I,,~,rr"'f'_W"V'" films when used with normal incidence. Do this by .. ~.~---:-; the composite transfer matrix for the three quarter layers and the matrix elements in the calculation of the reflection coefficient in Eq. (19-36). 19-12. the materials given in Table 19-1, a three-layer multifilm of wave thicknesses on a substrate of germanium that will give nearly zero reflectance for normal incidence of 2 p..m radiation. 19-13. Determine the maximum reflectance in the center of the visible for a formed using nL 1.38 and reflectance stack of high-low index double nH = 2.6 on a substrate of index 1.52. The are of equal optical thi,ckrless res]porldulg to II quarter-wavelength for of average 550 nm. The as in index material is encountered first the incident normal incidence and stacks of (a) 2; (b) 4; (c) 8 double layers. 19-14. A stack of alternating index layers is to at 2 p..m in the near infrared. A stack of four double layers is made of of germa-

Chap. 19

Problems

405

nium (n = 4.0) and MgF2 (n = 1.35), each of 0.5-pm optical thickness. Assume a substrate index of 1.50 and normal incidence. What reflectance is produced at 2 pm? 19-15. What theoretical ratio of high-to-Iow refractive indices is needed to at least 90% reflectance in a high-reflectance stack of two double of quarter-wave layers at normal incidence? Assume a substrate of index 1.52. 19-16. Show that Rmax in Eq. (19-53) approaches 1 when either N approaches infinity or when the ratio ndnH approaches zero.

[I] Dobrowolski, 1. A. and Filters." In Handbook of Optics, edited by Walter G. Driscoll and William Vaughan. New York: McGraw-Hili Book Company, 1978.

[2] Knitll, Z. Optics of Thin Films. an Optical Multilayer Theory. New York: John Wiley and Sons, 1976. Heavens, O. S. Thin Film Physics. New York: Barnes and Noble, 1970. [4] Macleod, H. A. Thin Film Optical Filters. New York: American Elsevier Publishing Company. 1969. Chopra, Kasturi L. Thin Film Phenomena. New York: McGraw-Hili Book Company, 1969. Scientific Ameri[6] Baumeister, Philip, and Gerald Pincus. "Optical Interference can (Dec. 1970): 58. NUssbaum, Allen, and Richard A. Phillip..<;. Contemporary Opticsfor Scientists and neer~·. Englewood Cliffs, N.J.: Prentice-Hall, 1976. Ch. 8.

406

Chap. 19

Theory of Multilayer films

20 {J

Incident light

Circularly polarized light

fresnel Equations

INTRODUCTION The basic laws of reflection and refraction in geometrical optics were earlier on the of or Fennat's principles. In this chapter we regard and rPfr<'l("f1"',n as an electromagnetic wave and show that the laws of can also be deduced from this of view. More importantly, this approach also leads to the Fresnel equations, which describe the fraction of incident energy transmitted or reflected at a plane surfuce. These quantities will be seen to depend not only on the change in refractive index and the angle of incidence at the but also on the polarization of the Finally, the important between internal and external are clarified.

20-1 THE FRESNEL EQUATIONS Consider 20-1, which shows a ray of light at point P on a plane interface-the xy-plane-and the resulting reflected and refracted rays. The plane of incidence is the xz-plane. Let us assume the incident light consists of plane harmonic waves, expressed by (20-1)

401

y

Interface - -

Flgure:ro.l Defining diagram for incident. reflected, and transmined rays at an XY-plane interface when the electric field is perpendiclular to the plane of incidence, the TE mode.

where the of coordinates is taken to be point O. The wave vector E of the incident wave is chosen in the so that the wave is linearly The direction of the corresponding magnetic field vector B is then determined to ensure that the direction of E x B is the direction of wave propagation k. This mode of polarization, in which the E-field is perpendicular to the plane of incidence and the B-field lies in the plane of incidence, is called the transverse electric mode. If instead B is transverse to the plane of incidence, a case to be considered the mode is a transverse magnetic (TM) mode. An arbitrary polarization direction represents some linear combination of these two special cases. The reflected and transmitted waves in Figure 20-1 can be expressed, respectively, in rorms like that of the incident wave of (20-1): (20-2) (20-3) In the boundary plane xy, where all three waves exit simultaneously, there must be a fixed relationship between the three wave amplitudes (and thus their irradiances) that has yet to be determined. Since such a relationship cannot depend on the arbitrary choice of a boundary point r nor a time t, it follows that the of the three waves, which depend on rand t, must themselves be equal: . r - WI) = (14 . r - wrt) = (kl

.

r

Wit)

(20-4)

In particular, at the boundary point r = 0 of Figure 20- I, or W

=

£liT

= w,

so that an frequencies are equaL On the other hand, at t plane, (20-4) yields: k· r =

408

Chap. 20

Fresnel Equations

.r

k,· r

(20-5)

0 within the boundary

(20-6)

cOlrlcllllSlCJnS can be drawn from the relations of Eq. (20-6), First notice that by subtracting any two these are equivalent to

(k

(k - kt) . r

k,)' r

= (k,. -

k/) , r

=0

(20-7)

determined by the Equation requires that the vectors kr and kt lie in the vectors k and r. Thus aU propagation vectors are coplanar in the and waves in the plane of incidence. we conclude that the reflected and Next, consider the first two members of (20-6), which govern the relationship between the and waves. In terms of the angles in ure 20-1, they are equivalent to kr sin 0 = krr sin Or

both waves travel in the same ......"',fin"'" their wavelengths are identical and so k = kr • Therefore, we have the law of reflection: the last two members of

0 = Or

(20-6) are

(20-8) to

krr sin Or = k,r sin 0/ Writing kr = w/vr

(20-9)

nrw/c and k, = n,w/c, Eq. (20-9) becomes law of ..",tro"t1r.n·

(20-10)

We now to further the at the boundary with the help of boundary conditions arising out of Maxwell's equations, and treated in texts on plprtrlrll' .... and magnetism. We employ them here without proof. These boundary conditions that the components of both the and fields parallel to the boundary plane be continuous as the boundary is crossed. In terms of choices made for the direction of E in Figure 20-1, the requirement for the electric field is

E+

(20-1 I)

where we have described the total field at the side of the boundary as a superposition of and reflected waves and at the other side as the transmitted field alone. Notice that all three fields are parallel to the boundary plane and in the +y-direction. In the case of the corresponding magnetic the compoby nents the boundary condition are B cos 0 -

cos 0 = B, cos 0,

(20-12)

where we have made use of Eq. (20-8). The negative indicates that the Br-component is along the -x-direction in the reflected beam. Equations (20-11) and (2012) are correct for the E and B vectors as chosen in 20-1. If a different by the E vector incident wave (and choice is for l) and (20-12) appear with a also B to keep the wave the Eqs. change of signs. However, the physical import of these equations is the same when they are interpreted in terms of their original figures. (20-11) and (20-12) for the TE Before pursuing the significance of we parallel their development ror the TM pictured in 20-2. Analogous to Eqs. (20-11) and (20-12), we now have B

-EcosO+

Sec. 20-1

The Fresnel Equations

+ Br = cosO=

(20-13) cos 0,

(20-14)

y

Interface

Plane of incidence

z Figure 20-2 Defining diagram for incident, reflected. and transmitted rays at an XY-plane interfuce when the magnelic field is perpendicular 10 Ihe plane or incidence, the 1M mode.

(20-11) through 14) are valid for the instantaneous values of the at the boundary. Because the equality of their phases. they are also valid for the amplitudes of the fields. The magnetic fields of Eqs. (20-12) and (20-13) can be expressed in terms of electric fields through the relation (20-15) Writing the index of refraction for incident media as nl and n2, respectively, (20-11) through (20-14) can be recast as follows:

E+ TE:{

= E,

(20-16) cos 6,

nlE cos 6

(20-17)

(20-18)

-E cos 6 + E, cos 6 Next. eliminating

coefficient r

-E, cos 6,

of equations and

each

(20-19) for the reflection

= E,/ E, TE:

r

TM:

r

cos 6 - n cos cos 6 + n cos 6,

Er E E

=

n cos 6 n cos 6

+

cos cos 6,

(20-20) (20-21)

where we have introduced a relative refractive index n ndnl' Finally, since nand 6, are to 6 through Snell's law, sin 6 = n sin 6" 6, may be eliminated using

n cos 6, = 410

Chap. 20

Fresnel Equations

(20-22)

The results are then TE:

cos (; r = E = -c-os-(;-+-Vrn::;::2=-=s=in::::::2=(;

TM:

r

n 2 cos (; 2

E

n cos (;

Yn 2

sin 2

-

(;

(20-24)

+

Returning to EqS. (20-16) through if Er is instead of lead to the following equations describing the transmission coefficient TE: TM:

t

E

=

2 cos (;

cos (;

(20-25)

+

t =-

(20-26)

E

(20-25) and (20-26) can also be found more quickly by using Eqs. (20-16) and (20-18) written in the form TE: t = r + I TM:

nt = r

+

1

into which the result,> expressed by (20-23) and can be conveniently substituted. Equations (20-23) through are the Fresnel equations, reflection and transmission coefficient'>, the ratio of both and transmitted E-field amplitudes to the incident E-field amplitude. In reality, measured reflection and transmission coefficients depend on losses from a nonplanar surface.

Calculate the reflectance and transmittance for both TE and TM modes of light incident at 30° on glass of index 1.60. Solution Using Eqs. (20-23) and (20-24), rTE

rIM

=

cos 30 - Y1.6 2 cos 30

-

sin 2 30

+

-0.2740

= 0.1866

=

Thus = 7.51% RTM

and

TTE

= 3.48%

= 1

92.5%

=1-

96.5%

20-2 EXTERNAL AND INTERNAL REFLECTIONS In dealing with the interpretation of these equations, it is necessary to between two physically situations:

Sec. 20-2

external reflection:

n! < nz or n =

internal reflection:

nl

> nz

External and Internal Reflections

or n

nl

nl

~'U"",b"'U'

> 1 < 1 411

External reflection 0.6

0.4

0.2

-0.2

1

- - - -____

r-------_

-0.4

0.6 -0.8 _ -1.0 '-_.....L_ _ 10 20 ~

__'__ _...l__..........J.....L_

30

40

50

_L..._ _.l........._

60

70

__'__

80

_._:III

90

Angle of incidence

Figure 20-3 Reflection (r) and transmission (t) coefficients for the case of external reflection, with n = n2/n, 1.50.

20-3 is a plot of (20-23) through (20-26) the case of external 1.50. Notice that at both normal and grazing incidence-angles reflection with n of 0° and 90°, and TM modes result in the same magnitudes for tOOr the reflection or transmission Negative values of r for both the TE and TM modes indicate a of the E- or B-field vectors on reflection and will be discussed The fraction of power P in the incident wave that is reflected or transmitted, called the reflectance and the transmittance, respectively, depends on the ratio of the squares of the amplitudes. reflectance = R =

Pi

= r2 =

(Er)2 E Ot)

. transmittance = T = PI = n (cos - - t2 Pi cos 0/

(20-27) (20-28)

These are justified later in this chapter. Reflectance is plotted as a function of the of incidence in 20-4. The curve for the case of external of reflection, TM mode, indicates that no wave energy is reflected when the incidence is near 60°. More precisely, RTM = 0 when 0 = tan- J n, or Brewster's angie. symbolized by Op as the polarizing angle. This condition is also evident in the 20-3 and of the numerator in Eq. (20-24). RTf: does not go vanishing of rTM in to zero under this condition, so that reflected light contains only the TE mode and is linearly polarized, with RTf!: 15%. For the case n 1.50 used in 20-3 and 20-4,Op = 56.31°. At normal incidence (0 = 0°), for both TE and TM modes, Eqs. (20-23) and (20-24) simplify to give R

(20-29)

or a reflectance of 4% from glass of refractive index n 1.5. Keep in mind, however, that n is a function of wavelength. As the angle of incidence increases to graz412

Chap. 20

Fresnel Equations

100

I

90 -

I I I

!

80 -

II

70

I

60

~ Q;:

•

50

11

40-

'I

:1

Ii

Internal reflection

20

TE

.......

10

I

/

,

10 -

.... ,,"

,I,:II

I

II

!

TM I

,

20

,

30 I

40:

I

I

I

e'p

External reflection

Ir I ,;

30

33.69°

(Ie

60

50 41.81°

(lp

70

80

90

56.31"

Angle of incidence

Figure 20-4 Reflectance for both external and internal reflection when n, = I and 112 = 1.50.

1.0

0.8

•

III

,,I1 ,:

0.6

II

,1

r 0.4

,I

,

I

, I

I I

0.2

,I

I

: I

I

I

0

I I

:, I I

41.8° (I'

-0.2

-0.4

p

10

20

30

40

50

60

70

80

90

Angle of incidence

coefficient for the case of internal reflection with

Sec. 20-2

External and Internal Reflections

413

incidence (0 = 90°) both RTE and become unity, although RTM remains quite small until Brewster's angle has been exceeded. The reflection coefficient for the case of internal reflection is shown in Figure 20-5 with n = 1/1.5, as when light emerges from glass to air. Evidence of phase changes and of a polarizing, or angle may also be seen here. However, 20-5 shows that in this case both rTf, and rro reach of unity before the angle of incidence reaches 90°. This is the phenomenon of total internal reflection, which occurs at the critical angle Oe sin-I n. For the example of glass (n = 1.5) 20-5, 33.7" and (Je = 41.8°. When sin (Jc > n, the used in is and both rTE and rTM are complex. Their magnitudes, however, are easily shown to be unity in this range, giving total reflection for (J > (Je. Reflectance internal reflection is also graphed in 20-4.

0;

20-3 PHASE CHANGES ON REFLECTION The values of the reflection coefficient in 20-3 and 20-5 indicate that r IE in certain situations. Evidently the electric field vector may reverse direction on reflection. Equivalently, in such cases there is a 1T-phase shift of Eon retlecuoln. as the following mathematical demonstrate.<;:

Er

-!r!E =

Thus in the case of external reflection, Figure 20-3, a 1T-phase shift of E occurs at any of incidence for the TE mode and for (J > Op for the 1M mode. When reflection is internal, we conclude that a 1T-phase shift occurs for the TM mode for (J < (J;. However, the situation in the region (J > where r is complex, require.'> further inve.'>tigation. When (J > (Jc = s1n- 1 n, the radical in Eqs. (20-23) and (20-24) becomes imaginary, and the equations may be written in the form

TE:

r

1M:

r

(20-30)

n 2 cos

(J

+ rYsin2

(J -

n2

(20-31)

In either equation, the reflection coefficient takes the r (a - ib)/(a + ib). Since the and imaginary parts of the numerator and denominator are the same, except for a sign, the magnitudes of the numerator and denominator are equal, and r has unit amplitude. The phase of r may be investigated by expressing Eq. (20-30) in complex polar form, as r

where, for the TE mode, tan a = (J). If we write r 2a, and expressions enabling the calculation of 4> are: phase of r, 4>

414

Chap. 20

Fresnel

(~)

TE:

tan

1M:

tan (1:.2)

e- i4>, the

= --c-os-(J--

(20-32)

-~-­ n cos (J

(20-33)

=

2

180

-----

--- .......I I

I I I I I I I

160 140

I

120 100

it -5 lil 1'0

.c:

80

"-

60 40

20

20

10

30 Angle of incidence

Figure 2O..(i

n

= nt/nl

Phase shift of electric field for infernally reflected rays, with

I/L5.

Clearly, the phase difference introduced on reflection may take on values other than and depending on the angle of The phase 4>, as determined (20-32) and is plotted in 20-6. plotted is the relative from phase shift «PrM Notice relative phase is at an angle of near 50°. Two such consecutive internal reflections thus produce a 90° difference beresulting in circularly polartween the perpendicular components of the Summarizized This technique is utilized in the Fresnel rhomb (Figure

(f

I ncident light

Figure 20-7 The Fresnel rhomb. With the incident light at 45" to the plane of incidence, two reflections produce equal amlPIit1Jde TE and TM amplirudes with a 90° or circularly light. For relative n J .50, the angle 6 . The device is effective over a wide range of wavelengths.

Circularly polarized light

Sec. 20-3

Phase Changes on Reflection

415

these results for the case of internal reflection,

1800, 0°

(20-34)

{ 2 :rctan 00 { 2 arctan Phase shifts for both TM and TE modes and are summarized in Figure

o

both internal and external reflection

01----"'" 30

60

90

Angle of incidence

Angle of incidence

(al TE mode, external reflection

(b) TE mode, internal reflection

!

'11"1-

I I I I I I

I

6p

I I I I I

01-------11/

I

30

90

I

60

(J'

p

o 90

Angle of incidence (c) TM mode, external reflection

Angle of incidence (d) TM mode, internal reflection

Figure 20-8 Phase changes between incident and reflected cidence. Discontinuities occur at 0,. = 41.8", Op 56.3°, fractive indices of TI, 1 and Tl2 = 1.50.

versus angle of in33.7° for re-

0;

Example What is the shift of the TM and TE rays reflected both externally and internally in the preceding example? Solution For this reflection, 416

Chap. 20

Fresnel

Oc = sin-I

C~6)

=

38.T

Op = tan-I (1.6) = 58.0°

0;

l

= tan-- (/6) = 32.0°

Since the angle of incidence of 30° is less than either 0; or Oc, Eqs. (20-34) and (20-35) or Figure 20-8 require that for internal reflection, CPTM = 1800 and CPTE = 0°, while Figure 20-8 shows that for external reflection, CPTM = 0° and CPTE = 180°. Thus a 7T-phase shift occurs between the externally and internally reflected rays for both the TE and TM modes. A general conclusion can be drawn from the phase changes for the TE and TM modes under internal and external reflection: For angles of incidence not far from normal incidence, the phase change."; that occur in internal reflection (0 for TE, 7T for TM) are the reverse of those at the same angles for external reflection (7T for TE, 0 for TM). For both TE and TM modes. the two reflected beams experience a relative 7T-phase shift. For a thin film in air, we are interested in the relative phase shift between rays reflected from the first surface (external) and the second surface (internal). Inspection of Figure 20-8 shows that a relative phase shift of 7T occurs in the TE mode for internal angles of incidence less than 0, and in the TM mode for internal angles of incidence less than 0; . The corresponding external angles of incidence at the first surface are 90° (TE mode) and Op(TM mode). Thus in the TE mode a relative phase shift of 7T occurs for all external angles of incidence, but in the TM mode this is true only for external angles less than Op.

20-4 CONSERVATION OF ENERGY From the point of view of energy, it must be true that the rate of energy input or incident power must show up as the sum of reflected power and transmitted power at the boundary, or

P; = Pr

+

P,

(20-36)

If we represent the reflectance R as the ratio of reflected to incident power and the transmittance as the ratio of transmitted to incident power,

R = Pc and Pi

T = P, Pi

(20-37)

Eq. (20-36) takes the form

I=R+T

(20-38)

The irradiance I is the power density (W/m2), so that we may write, in place of Eq. (20-36), I,A;

= IrAr + I,A,

(20-39)

The cross-sectional areas of the three beams (see Figure 20-9) that appear in Eq. (20-39) are all related to the area A intercepted by the beams in the boundary plane through the cosines of the angles of incidence, reflection, and refraction. We may then write

Sec. 20-4

Conservation of Energy

417

Figure 20-9 of cross sections of incident. reflected, and transmitted beams.

Of course, (}; (}r by the law of reflection. Also using the relation between ance and electric field <>,.."nlitl1rlp

and the facts that Vi = V r , and we arrive at the equation

Ei

Er ,

correspond to the same

since

(20-40)

is just a complicated way quantity index n, which we can show as follows:

expressing

Vi V,

ViE;

In

at this

=n

relative

(20-41)

we have ILl

JLo

ILl

for nonmagnetic materials and the relation p.E

the velocity

a plane electromagnetic wave. Incorporating

(20-40 in Eq.

(20-40), (20-42)

Dividing the equation by the left me!mber it becomes COS (}I) 2 +n ( - t COS

418

Chap. 20

Fresnel

OJ

(20-43)

where the reflection transmission coefficients rand t have been introduced. Now the quantity r2 is just the reflectance R: R =

~

t = (!:Y

(20-38), it follows that the transmittance Tis ex-

Comparing (20-43) with n ..p·","~.rI by the relation

COS (J/) 2. n -- t (

T

(20-44)

cos (Jj

that T is not t 2 since it must take into account a different and direction in a new medium. The change speed modifies rate of energy propagation and thus the power of the beam; the change in direction modifies the cross section and thus the power density the beam. However, for normal incidence, reduce..<>; to T and becomes normal nCIIGefllce: 1 =

+ nt 2

Graphs of T versus of incidence are most with (20-38).

(20-45) using Figure

to-

20-5 EVANESCENT WAVES In discussing the propagation a light wave by total internal reflection (TIR) we shall mention (Chapter 24) the phenomenon of· :Tu.':.,·-UJ.tI''.. through an optic coupling of wave energy into another medium when it is brought enough to the reflecting wave. This loss of energy is described as frustrated total reflection. The theory presented in this chapter puts us in a good position to describe this phenomenon quantitatively. The transmitted wave at a refraction is given in Eq. (20-3) as

where, according to the coordinates chosen in Figure 20-1, k, . r

= k, (sin (J.,

cos

k, . r = k. (x sin 6/

0) . (x, y, 0)

+ Y cos 6,)

We can express cos 6. as cos 6.

VI

- s1n2 6/

where we have used law in the last step. At the critical angle, sin 6 = n and cos 6 = O. For angles such that sin 8 > n, when TIR occurs, cos 8. becomes pure imaginary and we can write cos 6, = Thus the exponential factor sin 6 k, . r = k,x - -

n

Sec. 20-5

EvanescentVVaves

6 + ik/ Y ~sin2 -2

n

419

If we define the real, positive number,

a == k, then the transmitted wave may be

PVIr'lrp""",/i

as

E,

The factor describes an exponential decrease in the amplitude of the wave as it enters the rare medium the y-direction. The other exponential include the number i, them harmonic with unit When the wave penetrates into the rare medium by an amount

y=

1 a

(20-46)

the amplitude is decreased by a factor The energy of this evanescent wave medium unless a second medium is introduced into its Although detrimental in the case of cross-talk in closely bound fibers . - -... .,'0 ".,rM.~,,,,,r thickness protective cladding, the frustration of the total internal reflection is put to use in devices such as variable output made of two right-angle whose separation along their diagonal faces can be carefully adjusted to vary the amount of evanescent wave coupled from one prism into the near to the surface other. Another application involve.·;; a prism face of an so that the evanescent wave from the prism can be coupled into the waveguide at a angle (mode) of propagation.

Example Calculate the penetration depth of an evanescent wave unc:ler:goi.ng TIR at a glass (n 1.50)-to-air such that the is attenuated to 1/e its Assume of wavelength nm is incident at the interwith an angle of 60°. Solution Since Be depth is given by

(1.5) (20-46):

= 41.8°, TIR occurs at 60°. The pe[letrati()f1

y = --;====== = 0.0% pm

20-6 COMPLEX REFRACTIVE INDEX We wish now to show that when the reflecting surface is the Fre.'ine1 equations we have derived continue to be valid, with one modification: index refraction becomes a number, an imaginary part that is a is treated at length in measure of the absorption of the wave. This Chapter 27, where we examine the frequency dependence of the real and imaginary parts of the reff"dctive index. When the reflecting surface is that of a homogeneous dielectric-the case we have been discussing in this conductivity u of the material is zero. The conductivity is the proportionality constant in Ohm's law,

j = uE 420

Chap. 20

Fresnel Equations

where j is the current density (A/m2) produced by the field E. In such cases, both the E- and B-fields a differential wave equation of the form

V2E

(:2)

(20-47)

as described in Chapter 8. We have written harmonic waves satisfying in the form

E=

. (20-47) (20-48)

Now if the material is metallic or has an appreciable conductivity. the tunloalmel!ltaJ Maxwell equations of electricity and magnetism lead to a modification of 47) and (20-48). The differential wave equation to be satisfied by the

_(~)iPE +

-

c2

(20-49)

iJt 2

Note that, compared with (20-47), the new wave Eq. (20-49) includes an additional term involving the conductivity and the derivative of E. As a result, when a harmonic wave in the form of Eq. (20-48) is substituted into Eq. (20-49), we find that the vector k must have the complex magnitude

k Since the refractive the complex number

=; [1 + iC:w)]'/2

n is related to k by n

(c/w)k, the refractive index is now (20-51)

or we write, in general, where Re (ii) = nR and In! (ii) nl. Combing I) and (20-52) and their real and imaginary parts, the optical constants nR and nl can terms of the conductivity by the equations

nj- nl = 1

if the

character of k in the form

is introduced into the harmonic wave, Eq. (20-48), the result is E

(20-55)

We conclude from Eq. that the wave propagates in the material at a wave and is absorbed such that the amplitude decreases at a rate governed by the exponential factor Thus Re (ii) = nR must as the ordinary reindex, and 1m (ii) nl, called the extinction coefficient, determines the rate of absorption of the wave in the conductive medium. This absorption, due to the energy contributed to the of conduction in the is l1"'~'('''lIn'''l1 by the decrease in power I with distanoe (20-56) Sec. 20-6

Complex Refractive Index

421

By comparison with the power density as determined from 1 IX

where

EI\

1

1=

(20-57)

ah.·~nrJrJtilm r,rlPftiri.ont

and so we see that the

a is related to the extinction coefficient

nl

2wn/

a =-c-=

A

(20-58)

20-7 REFLECTION FROM METALS Replacing n by ji in the Fresnel eQu:atlC)flS,

Introducing

TE: TM:

ER

ji

as

nR

TE:

ER E

cos (I

ER E

ji2

TM:

+

cos (I

cos (I ji2

cos cos

(20-23) and (20-24), we have for

(20-59)

+ (I (I

+

(20-60)

(20-59) and (20-60), these equations take the

(20-61)

E

ER E (20-62) R = 1ER/ E 12 , the cOI1npJt~x quantity ER/ E can first be ...Ull" ....."A numbers in the (a + ib)/(c + id), so that

In \..ru\wW,ru.Ul~ the ..pH.,."t..

In the process, we must take the square root of a COI1nplt~x number, which is done by putting it into polar form. For example. if z A + then, in

and the square root

hi>r.nn'l'P'(:

(20-63) The complex expression in . (20-63) can then returned to the general COI1npl(~X C + iD using Euler's equation. These mathematical steps are easily performed 20-10, the results of such with a programmable calculator or a computer. In calculations are shown two metal surfaces, solid sodium and single reflectance in the is characteristic of metallic as shown by the curves for sodium at a wavelength of 589.3 nm. Strong crimination between the TE and TM modes in the incident radiation is exhibited by the curves for single crystal gallium surfaces.

422

Chap. 20

Fresnel Equations

TE

100

TM 90 80

10 60

~

50

Q;:

40 -

Single crystal gallium: 3.1, n l = 5.4

30

nR

- - - Solid sodium: n R =0.04, nl= 2.4

20

10

90

30

20

10

Angle of incidence

Figure 20-10 Reflectance from metal surfaces using Fresnel's equations. The values of nR and f4 are given for sodium light of A 589.3 nm.

20-1. Show that the vanishing of the reflection coefficient in the TM mode, Eq. leads to Brewster's law. 20-2. The critical angle for a certain oil is found to be 33°33', What are its Brewster's anfor both external and internal reflections? 20-3. Determine the critical angle and polarizing angles for (a) external and (b) internal rel1ections from dense flint of index n = 1.84. 20-4. For what refractive index are the critical and (external) Brewster angle when the first medium is air? 20-5. Show that the Fresnel Eqs. to (20-26), may also be by TE:

r

TM:

r

sin sin (6

= ---'-_"""":"

+ 6,)

tan (6

6,)

tan (6

+ 6.)

t ::;;:

2 cos 6 sin 6, sin (6 + 6,) cos (6 - 6,)

20-6. Confirm (20-3) and by computer programs to calculate and/or using (20-23) through (20-26). the value of n to produce plot the for the case of external and internal reflection from diamond (n = 20-7. Write a computer program to calculate and/or plot the reflectance curves of Figure 20-4. Also plot the transmittance. 20-8. Write a colnpllter program to calculate a plot of the phase shifts as a function of angle of as in Figure for 6 > ()c.

Chap. 20

Problems

423

20-9. A film of fluoride is onto a glass substrate with optical thickness equal to one-fourth the wavelength of the light to be reflected from it. Refractive indices for the film and substrate are 1.38 and 1.52. respectively. Assume that the film is nonabsorbing. For monochromatic incident normally on the determine (a) reflectance from the air-film (b) reflectance from the sursurface without the film; (d) net reflectance (c) reflectance from an from the combination [see Eq. 20-10. Calculate the reflectance of water (n 1.33) for both (a) TE and (b) TM polarizations when the of incidence are 0", 10",45", and 90". 20-n. is incident upon an air-diamond interface. If the index of diamond is 2.42, calculate the Brewster and critical for both (a) external and (b) internal reflections. In each case distinguish between polarization modes. 20-12. Calculate the percent reflectance and transmittance for both (a) TE and (b) TM modes of light incident at 50" on a glass surface of index 1.60. 20-13. Derive and (20-26) for the transmission coefficient both (a) eliminating E, from (20-16) to (20-19), and (b) using the corresponding equations for the reflection with the relationships between reflection and transmission coefficients implied by (20-16) and (20-18). 20-14. is reflected from a surface of fused silica of index 1.458. (a) Determine the critical and polarizing (b) Determine the reflectance and transmittance for the TE mode at normal incidence and at 45" (c) Repeat (b) for the TM mode. (d) Calculate the difference between TM and TE modes for reflected rays at of incidence of 0°, 20°, 40", 50", 70", and 90°. 20-IS. A Fresnel rhomb is constructed of transparent material of index 1.65. (a) What should be the apex angle 8, as in 20-?? (b) What is the difference between the TE and TM modes after both reflections, when the angle is 5% below and above the correct value? 20-16. Determine the reflectance for metallic reflection of sodium light (589.3 om) from steel, for which nR 2.485 and nl = 1.381. Calculate reflectance for (a) TE and (b) of incidence of 0°, 30°, 50°, 70", and 90°. TM modes at of incidence of 0°, 30°, and 60°. Do this 20-17. Determine the reflectance from tin at Real and parts of the for the (a) TE and (b) TM modes of complex refractive index are 1.5 and for light of 589.3 nm. 20-18. (a) What is coefficient for tin, with an imaginary part of the refractive index equal to 5.3 for 589.3 nm light? is 99% of normally incident sodium light absorbed in tin? (b) At what 20-19. (a) From the power conservation requirement, as expressed by Eq. show that for an external reflection the transmission coefficient I must be less than I, but for an internal reflection t may be greater than I. (b) Show fUrther, the Fresnel Eqs. and (20-26), that as the of incidence the critical angle, t' must a value of 2 in the TE in the TM mode. mode and (c) Plot the transmis..'lion coefficient I' for an interface between glass (n = I and air. 20-20. A narrow beam of 546 nm) is rotated 90° by TIR from the potenuse face of a prism made of glass with n 1.60. (a) What is the depth at which the amplitude of the evanescent wave is reduced to of its value at the surface? (b) What is the ratio of irradiance of the evanescent wave at 1 JLI11 beyond the surface to that at the surface? I

424

20

Fresnel ....... ''' ....1..."

REfERENCES [ I] Dichtburn, R. W. Light, vol. 2. New York: Academic 1976. Ch. 14. Longhurst, R. S. Geometrical and Physcial Optics, 2d ed. New York: John Wiley and Sons, 1967. Ch. 21. [3] Bruno. Optics. Reading, Mass.: Addison-Wesley Publishing Company, 1957. Ch.8.

Chap. 20

References

425

21

Laser Basics

INTRODUCTION The laser is the most important optical device to be in the past 50 years. Since in the 1960s, rather quiet and unheralded outside the scientific community, it has provided the stimulus to make one of the most rapidly growing fields in science and technology today. an optical amplifier. The word The laser is is an acronym that by the stimulated emission of stands for key words here are amplification and stimulated emission. The theoretical of laser an optical amplifier was made by action as the basis early as 1916, when he first predicted the existence of a new stimulated emission. His theoretical work, however, remained until 1954, when C. H. Townes and co-workers developed a microwave .. UIIJ. .,...... based on stimulated of radiation. It was called a maser. Shortly th",...,."n",.. in 1958, A. Schawlow and C. H. Townes adapted the principle of masers to light in the visible region, and in 1960, T. H. Maiman built the first laser Maiman's laser incorporated a for the laser amplifying medium and a t<~I'rv_,",f'-rnr optical cavity as the resonator. Within months of the arrival of ruby laser, which emitted deep red light at a wavelength of 694.3 nm, A. Javan and associates developed the first gas the helium-neon laser, which emitted light in both the infrared (at 1.15 j.Lm) and visible (at 632.8 nm) spectral reg,IOIlS.

426

Following the birth of the ruby and helium-neon (He-Ne) lasers, other laser devices followed in rapid succession, each with a different laser medium and a different wavelength emission. For the part of the 1960s, the laser was viewed by the world of industry and technology as a curiosity. It was referred to " In the and 1970s, all in as "a solution in search of a that changed. The laser came into its own as a unique source of intense, coherent light. Today, fueled by many sources, new laser applications are discovered almost with the fiber-optic cable and semiconductor weekly. and the optics industry. the laser has revolutionized In this chapter we review the essence of Einstein's prediction of the existence of stimulated emission, examine the essential elements that make up a laser, dethe of the laser in simple terms, and list the characteristics of laser Finally, by way of a summary, a table is provided that that it so along with their important operatlists some of the popular lasers in existence ing parameters.

21-1 EINSTEIN"S QUANTUM THEORY OF RADIAnON In 1916, while the fundamental processes involved in the interaction of electromagnetic with matter, showed that the existence of equilibrium between matter and radiation required a previously undiscovered radiation process called stimulated emission. According to Einstein, the interaction of tion with matter could be explained in terms of three basic processes: stimulated abspontaneous and emission. The three are illustrated in Figure 21-1. Stimulated absorption, or simply absorption, Figure 21-1a, occurs whenever is incident on matter with radiation containing photons of energy hv = EI

Before

After

El Incident hI'

Radiation

Eo

Eo

III

Atom

(a) Stimulated absorption

hI'

Eo---(bl Spontaneous emission

Incident 2hv

hI'

Radiation

Eo---(cl Stimulated emission

Sec. 21-1

21-1 Three basic processes that affect the passage of radiation through matter. Note that hll = El Eo.

Einstein's Quantum Theory of Radiation

427

ground-state Eo and arbitrary excited state energy E 1 • The resonant photon energy hv is the atom from energy state to E 1 • In the process, the absorbed. Spontaneous emission, figure 21-1b, takes place whenever atoms are in an excited state. No external radiation is required to initiate the emission. In this process, when an atom in an excited state spontaneously up its energy and faUs to a photon of energy hv = is released. The photon is in a ranthe photon is dom direction. Even if external radiation is off in a direction that is completely uncorrelated with the direction of the pxt"pnUI radiation. Quite contrast, stimulated requires the presence of external radiation. When an photon of resonant energy hv E1 passes by an atom in excited state , it "stimulates" the atom to drop to the lower state, Eo. In the process, the atom releases a photon of the same energy, direction, phase, and polarization as that of the photon by. The net effect, is two of one, or an increase in the intensity of the incident identical photons in the "beam." It is precisely this process of stimulated emission that makes possible the amplification of light in lasers. Einstein A and B Coefficients. the existence of stimulated grew out of his desire to understand the mechanisms involved in the interaction between electromaguetic radiation and matter. A review of his study is both interesting and informative. As a model for this study, we shall assume that matter collection of IS In equilibrium with a blacklJO(jly radiation field. The atoms and the resonant radiation are contained in an 21-2 shows a enclosure at some temperature T and interact with one another. simplified of two-level atoms and radiation (photons) bound inside of an arbitrary unit volume. If thermodynamic equilibrium exists, the number of atoms N2 at energy level the number of atoms NI at energy level , and the number of phowill all remain constant. and proctons in the radiation field occur at a constant esses that add and remove photons the leaving the total photon number unchanged. At the same time, every N2 atom moving to during an emission process, there will be an NI atom This condition of during an absorption so that M and N2 will not balance is depicted in 21-3, in terms of atoms from level to and from toE,.

Spontaneous

Enclosure

FJgUre 21-2 A blackbody at temperature T emits radiation that interacts with the atoms in the blackbody.

428

Chap. 21

Laser Basics

Stimulated

21-3 Radiative processes that affect the number of atoms at energy £, and £2. The two emission processes remove atoms from level £2 and add them to level £,. The absorption process involves transitions from £, 10 £2.

Of considerable importance in the quantum theory of radiation and the operaof the Einstein A21 , B 21 , and B 12. Their tion of the laser is the significance is best appreciated in the context of Figure 21-3, where each is related to a radiative process: Spontaneous Emission (All)' Atoms at spontaneously EI to the radiation (photon to level Et, adding photons of energy hv decreases. The rate of population). At the same time, the population N2 of level decrease is proportional to the population at any time, that is,

If spontaneous emission alone takes place, the solution to

equation

N2 (t)

The N2 population decreases with a time constant T 11 A zl , depleting the number N2 at level at a rale N2IT, or All , and increasing the number NJ at level at the same rate. The constant T is referred to as the spontaneous radiative lifetime level measured in units the coefficient A21 is referred to as the radiative rate, S l , The coefficient All is a constant, characteristic of the atom. Note carefully that (dNz/dt)spon, makes no reference to the prior presence or absence of a radiation field. Stimulated Emis....ion (B 21 ). In this process the rate at which the N2 atoms are from level to level EI is proporstimulated by the radiation field (photons) to tional both to the number of atoms present and to the density of the radiation field, or

( dN2) dt ,,.,

-B21 N2P (v)

where the photon density is expressed as a function of frequency by the factor p(v). Absorption (B d. Absorption is also a stimulated process, since it depends of the photon field. In stimulated absorption and stimulated on the from energy emission are inverse processes. The rate at which NI atoms are level to E2 is given by (v)

The

BI2 is a constant characteristic of the atom. It turns out that and are related; are equal only under conditions of nOlnaj~1!f:'ne,raey of the quantum states that to energy levels and levels are those in which two or more states share the same With the three basic processes of absorption, spontaneous emission, and stimulated emission related quantitatively to A and B coefficients, we focus on several of Einstein's assumptions and their implications:

B21

1. Thermodynamic equilibrium at arbitrary T exists between the radiation field and the atoms. 2. The radiation field p (v) has the spectral distribution characteristic of a blackbody at temperature T.

Sec. 21-1

Einstein's Quantum

of Radiation

429

3, The atom population densities NI and N2 at energy levels EI and are distributed according to the Boltzmann distribution at that temperalure, Nt and N2 are constant in time. 4. Population From Figure 21-3 and of atoms in level is

(1) and (4), it follows that the rate of

(21-1) assumpWJns (2) and (3), we write for the spectral energy density p (v) of t

p(v) = 81Thv

3

-~-

(21-2)

and for the Boltzrnarm distribution of atoms between the two N2

Nt

(21-2) and (21 v is the frequency of radiation, such that hv the blackbody and k is the constant. It should has been written for the case of energy also that Eq. the algebra and yet not materially affect levels. This small concession will the conclusion we shall reach. (21-1) for p(v) and substituting for Nt/N2 from (21-3), we obtain In

T

p(v)

Equating this

.,.v.",.,.'O<)

(21-4) for p(v) to that

---:-~--

=

in Eq. (21-2), 81Thv 3

--:--::-=--

Rearranging to isolate mUltipliers of the term e hv / kT , 81Thv 3

A21 _ (

B2t

81T~V3)

=

0

(21-6)

C

We now have one equation the three Einstein coefficients: According to the assumptions made, (21-6) must be true for temperature T. can he so only if the term that mUltiplies ehv/kT and the remaining term in pal'enthe:ses are each identically zero. Then it follows at once that (21-7) and (21-8)

(21-2) for p(v), spectral energy density with dimensions for M A, spectral radiant exitance for a blackbody in h~"'A~J,r ~r~g_ (2-12) from A and AA 10 v and l:I.v to get M,dA = multiplies M.dA by 4/c to get the expression above for

430

Chap. 21

laser Basics

follows from One first changes van· 1)-'. Then one 8im-

The importance of Eqs. (21-7) and (21-8) cannot be overestimated. Taken together, they tell us the following. 1. The fundamental Einstein coefficients , B21. and are all interrelated. If one is known, by measurement or calculation, all are known. 2. The stimulated emission coefficient and the (stimulated) absorption coefficient BI2 are at least for the case of non degenerate energy states. This equality certainly the observation, made earlier, that stimuand absorption are inlated the new process discovered by verse processes insofar as rate of occurrence goes. Note carefully, however, (v) differ, dependthat the rates dNddt = N2B2IP(V) and dM/dt = Nl on the population densities N2 and M. If Nz is greater than NI and a radiation field interacts with the atoms, stimulated exceeds and If, however, is greater than photons will be added to lion exceeds stimulated emission and photons will be removed from the field. The first case (Nz > M) leads to an increase in p(v), an amplification. The second case (M > N z) leads to a decrease in p(v), an attenuation. For the than N 1 • This is the condition laser to operate, it is necessary that Nz be population inversion. Without a popUlation inversion-a condition that runs rnlntr,u" to the equilibrium popUlation densities predicted by the Boltzmann distribution-laser action cannot occur. 3. Since is proportional to the reciprocal of the cube of the frequency v, the higher the frequency (the shorter the wavelength), smaller B21 becomes in comparison with A21. Since is related to stimulated emission (which is related to spontaneous emission leads to photon amplification) and if any, to photon amplification), it would seem that (which contributes short wavelength radiation (ultraviolet or for example) would be more difficult to build and Such has been the case, even though lasers of shorter wavelengths have been developed rather extensively. 4. Although the relations between AZI , B21 , and were derived based on the condition of thermodynamic equilibrium, they are valid and hold under any condition. The while operating, is hardly an enclosure in thermodyare namic equilibrium. Yet the A and B coefficient relationships, because characteristic of the atom, are equally valid whether the atom is in the intense radiation field of a laser cavity or in a hot furnace that can be treated as a blackbody in thermodynamic equilibrium. Two important the successful of a laser emerge from a review of Einstein's study of the interaction of electromagnetic radiation with matter. The first is that there is a process, stimulated emission, that leads to light amplification. The is that a population inversion of atoms in energy levels must be achieved if the stimulated emission process producing coherent photons is to outrival the absorption process removing We use these ideas in describing how as a device and consider the indithe laser operates, but first let us examine the vidual parts essential to its operation. 21-2 ESSENTIAL ELEMENTS OF A LASER

The laser device is an optical oscillator that emits an intense, highly collima.zeor:r!ea'm of coherent radiation. device consists of three elements: energy source or pump, an amplifying medium, and an optical cavity 0 Sec. 21-2

Essential Elements of a laser

431

lal laser

I

a Pump

(b) Pump

M{

Resonator

L

Ie) Resonator

r ·1

I

I@ Cd@@\ :@@§@ II 1@r=J (:;l! \

i

I

\

laser medium (dl Medium

Figure 21·4 Basic elements of a laser. (a) laser device with output laser beam. (b) External energy source, or pump. The pump creates a population inversion in the laser medium. The can be an optical, electrical, chemical, or thermal source. The pictured are only symbolic. (c) Empty optical or resonator, bounded by two mirrors. (d) Active cavity, or laser vjJu"""m inversion and stimulated emission work in the laser amlplifi,catilDn of light.

These three elements are shown schematicaJiy in Figure 21-4: as a unit in Figure 21-4a and in 21 c, and d. The pump is an external a population inversion in the laser medium. As explained section, amplification of a light wave or field will occur in a laser medium that exhibits a population two energy levels. Otherwise the light wave passing through the laser medium will be
Chap. 21

laser Basics

Although there are numerous other pumps or excitation processes, we cite one more process which has some historical significance. The first laser, developed by T. Maiman at the Hughes Research Laboratories in 1960, was a pulsed ruby at the visible red wavelength of 694.3 nm. 21-5 shows a which impurity ions in the ruby rod, drawing of the ruby laser device. To excite the Maiman used a helical flashlamp filled with xenon gas. This particular method of exciting the laser medium is known as optical pumping. It is the only practical method that can be used to pump liquid or solid media. Shield Ruby rod Output mirror End mirror

Beam direction

Power supply

Figure 21·5 Components of a ruby laser ~)'stem. The shield helps to reflect light from the flashlamp back into the ruby rod.

The Laser Medium. The amplifying medium or laser medium is an imof the laser device. Many lasers are named the type laser portant for helium-neon (He-Ne), carbon dioxide (C02), and medium neodymium:yttrium aluminum The laser which may be a gas, liquid, or solid, determines the wavelength of the laser radiation. Because of the selection of laser media, the range of available laser wavelengths extends from the ultraviolet well into the infrared sometimes to wavelengths that are a sizable fraction of a millimeter. Laser action has been observed in over half of the known with more than a thousand transitions in gases alone. Two of the most widely used in gases are the 632.8-nm radiation molecule. Other commonly neon and the 1O.6-p.m infrared radiation from the 21-2 at the used laser media and their radiations are listed in end of this In some the amplifying medium consists of two parts, the laser host medium and the laser atoms. For example, the host of the Nd:YAG laser is a of yttrium aluminum (commonly called YAG), whereas the laser atoms are the trivalent neodymium ions. In gas lasers consisting of mixtures of gases, the disum;ucln between host and laser atoms is generally not made. of the medium is its ability to The most important support a population inversion between two energy levels of the laser atoms. This is accomplished by exciting (or pumping) more atoms into the energy level in the absence of pumping, there will exist in the lower level. As mentioned be no popUlation inversion between any two levels of a laser medium. Ac, where AE - E 1 , the cording to the Boltzmann distribution, N 2/ M . Pumping, level E2 will always be less populated than the lower level sometimes vigorous pumping, is required to produce the "unnatural" condition of a of population inversion. As it turns out though, due to the different Sec. 21-2

Essential Elements of a laser

433

aV8lUaitHe atomic energy levels, only certain pairs of energy levels with appropriate spontaneous lifetimes can be "inverted," even with vigorous pumping. The Resonator. Given a suitable pump and a laser medium that can be inverted, the third basic element is a reasonator, an "feedback that directs photons back and forth through the laser (amplifying) medium. The resonator, plane or optical cavity, in its most basic form consists of a pair of carefuHy system, as shown in or curved mirrors centered along the optical axis of the 21-4. One of the mirrors is chosen with a as close to 100% as possible. The other is with a reflectivity somewhat less than 100% to allow part of the internally reflecting beam to escape and become the useful laser output beam. The geometry the mirrors and their separation determine the structure of the field within the laser cavity. The exact distribution of the electric field pattern across the wavefront of the laser beam, and thus the on the construction of the resonator cavtransverse irradiance of the beam, and mirror surfaces. Many transverse irradiance p'dtterns, called TEM modes, are usually in the output laser beam. By suppressing the gain of the higher-order modes-those with intense electric fields near the of the beam-the laser can be made to in a fundamental mode, the TEMoo mode. The transverse variation in the irradiance this mode is Gaussian in with peak irradiance at the center and decreasing irradiance toward the edges. It is to compare a laser resonator and its geometry to a Fabry-Perot resonator bounded by plane, parallel mirrors. back to 11-8, where we show a two-parallel-plate cavity for the Fabry-Perot the condition for resonance is given as 2t cos 0, = mA. If we made up of two parallel plane by a distance t where Or laser photons that reflect axially back and forth between the mirrors, the Fabry-Perot condition for resonance becomes, simply, mAI2 L. This is often stated as "an integral number of half-wavelengths fitting between the end mirrors," a requirement that reminds us of standing waves in a string vibrating between two fixed ends. to think of a laser resonator as a Fabry-Perot resonator, with Thus it is several modifications. In laser resonators, the cavity is generally bounded by curved rather than plane mirrors, and the empty cavity characteristic of a Fabry-Perot resonator is filled (or partially filled) with a gain medium in the laser resonator. Neverthe resonance condition for the axial (or longitudinal) modes is the same for the two resonators.

21-3 SIMPUFIED DESCRIPTION

LASER OPERATION

We have described briefly the basic elements that comprise the laser How do these medium, and resonator-work? We know basically that photons of a certain resonant energy must be created in the laser cavity, must interact with atoms, and must be amplified via stimulated all while bouncing back and between the mirrors of the reSO'1ator. We can gain a reasonably accurate, though qualitative, understanding of laser operation by studying -6 and 21-7. Figure 21-6a shows, in four steps, what happens to a typical atom in the laser medium the creation of a laser photon. Figure 21-6b shows the actual energy level diagram for a helium-neon laser, with the four steps described in Figure 21-6a identified. Figure 21-7 then shows same process while focusing on the behavior of the atoms in the laser medium and the photon population in the laser cavity. Let us now examine these figures in turn. 434

21

laser Basics

Pump levels

N2

•• . . . . . . . . .

Resonant photon

E n e

--0f\r-

g

hI' = E2

y

0

Upper laser level

--0f\r-

Light

amplification

E,

N,

El

0 Eo

£.2

lower laser level

Decay to ground level

Ground level

(a) 21.1

i

CD

GJ

;r

21 5

Collision energy transfer 19.8

~

23 5

i i

I I I I i i

,, I

:>

i i

,

.!II. >~

Ql

c:: w

I

I I

18.6

CD

c.f;)

E ....

::>Ql

o..C W

,,, , i

I I I I

,, I I I i i

,

,, , ,, ,, ,, I

17.4

i

, I

,,, , i

!}1S

,, i

(b)

Figure :11-6 Four-step energy cycle associated with a lasing process, for both (a) four-level laser and (b) a particular laser. the helium-neon laser. (a) Fouffor a laser alom involved in the creation of laser (b) Energy level for the helium-neon laser, the production of the O.6328-Mm laser line in terms of the four steps (circled numbers) outlined in (a).

Sec. 21-3

Simplified

of laser Operation

435

In 1 of Figure -6a, energy from an appropriate pump is coupled into the laser medium. The energy is sufficiently high to excite a number of atoms from the ground state Eo to several excited states, collectively labeled E 3 • Once at these the atoms spontaneously decay, through various chains, back to the ground state Eo. Many, however, preferentially start the back by a very fast (usually radiationless) decay from pump levels E3 to a very level, This decay process is shown in step 2. Level is labeled as the "upper laser level." It is special in the sense that it has a long lifetime. Whereas most excited levels in an atom might decay in times of the order of 10-8 s, level is metastable, with a typical lifetime of the order of 10- 3 s, hundreds of thousands of times longer than other levels. Thus as atoms funnel rapidly from pump levels to they begin to pile up at the metastable level, which functions as a bottleneck. In the process, Nl grows to a large value. When level does decay, say by spontaneous it does so to level , labeled the "lower laser level." Level is an ordinary level that decays to ground state quite rapidly, so that the population N. cannot build to a large value. The net effect is the population inversion (N2 > N 1) required for light amplification stimulated emission. Once the population inversion has been established and a photon of resonant energy hv E2 EJ passes by anyone of the N z atoms in the upper laser level (step 3), stimulated emission can occur. When it laser amplification begins. Note carefully that a photon of resonant energy EI can also stimulate absorption from level El to level thereby losing itself in the process. Since N z is greater than N., however, and as shown earlier, the rate for stimulated ernllSSlon Nzp(v), exceeds that for stimulated B1ZN1P(v). Then light amplification occurs. In that event there is a steady increase in the incident resonant photon population and lasing continues. This is shown schematically in step 3, where the incident resonant photon approaching from the "left" leaves the vicinity of an atom in duplicate. In step one of the inverted Nz atoms, which dropped to level EI during the stimulated emission process, now decays rapidly to ground state Eo. If the pump is still operating, this atom is ready to the cycle, thereby ensuring a population inversion and a constant laser beam output. In Figure 21-6b, the pump energy 1) is supplied by an electrical discharge in the low-pressure gas mixture, thereby elevating ground state helium atoms to higher energy states, one of which is represented by the 21 S level. Then by resonant collisional transfer-made possible because the 21 S level of helium is nearly equal to the level of neon-step 2 is achieved as excited helium atoms pass their energy over to ground state neon atoms, raising them to the neon level. This process produces the population inversion required for effective· amplification via stimulated emission of radiation. The stimulated emission process 3) occurs between the neon levels 3s z and 2P4, the transition with the highest probabilityZ from to any of the ten states. This transition rise to photons of wavelength 0.6328 Ikm, photons that are amplified via emission and the common red beam characteristic of helium-neon lasers. FinaUy, in step the neon atom in energy state 2P4 decays by spontaneous emission to the Is ground level. Once back in the ground state, it is again available to undergo collision with an excited helium atom and to repeat the While Figure 21-6b relates the four steps to the emission of the He-Ne 632.8nm laser line, other transitions from the 3s to the 2s and levels have also been made to lase. One such transition, to the 1.1523 pm line, is indicated in the figure. Now let us look at Figure 21-7. It shows essentially the same action but does

436

Chap. 21

Laser Basics

Mirror 1

Laser medium

I

(al

Mirror 2

I I (bl

Ie)

(dl

lei

Ifl

21-7 development of laser oscillation in a typical laser cavity. (a) Quiescent laser. (b) Laser (c) Spontaneous and stimuJated emission. (d) Light amplification begins. (e) Light amplification continues. (f) Established laser operation.

in terms of the behavior of the atoms in the laser medium and the photon populabetween the mirrors of tion in the cavity. In (a) laser medium is shown the optical resonator. Mirror I is essentially 100% reflecting, while mirror 2 is partially reflecting and partially transmitting. Most of the atoms in the laser medium the black dots. In (b), external energy are in the ground state. This is shown from a fJashlamp or from an electrical discharge) is pumped into the medium, Excited states are raising most atoms to the excited levels (EJ in Figure 21 shown circles. During the pumping process, the popUlation inversion is established. The light amplification process is initiated in (c), when excited atoms (some in 21-6) spontaneously decay to level . Since this is the atoms at spontaneous emission, the photons given off in process radiate out randomly in and are all directions. Many, leave the sides of the laser be several photons-let us call them "seed" lost. Nevertheless, there will photons-directed along the optical axis of the laser. These are the horizontal arrows shown in (c) of Figure 21-7, directed perpendicularly to the mirrors. With the seed photons of correct (resonant) energy accurately directed between the mirrors the stage for stimulated is and many Nz atoms still in the inverted state set. As the seed photons pass by the inverted N2 atoms, stimulated emission adds identical photons in the same direction, providing an ever-increasing population of coherent photons that bounce back and forth between the mirrors. This buildup process, shown in (d) and (e), continues as as there are inverted aloms and resonant energy photons in the cavity. Since mirror 2 is partially transparent, a of photons incident on the mirror out through the mirror. These photons constitute the external laser beam, as shown in (0. Those that do not leave SO

2 A readable, comprehensive discussion of the helium-neon laser, with energy level diagrams and transition probabilities, is given in [4].

Sec. 21-3

Sirnnlliti~~rI (1"":>l'ri.,ttr.n

of Laser Operation

431

through the output mirror are reflected, recycling back and forth through the cavity medium. In summary then, the laser process on the following: 1. A populotion inversion between two appropriate energy levels in the laser medium. This is achieved by the pumping process and the existence of a metastable state. 2. Seed photons of proper energy and direction. coming from the ever-present spontaneous emission process between the two laser energy levels. These initiate the stimuloted emission process. 3. An optical cavity that confines and directs the number of resonant energy photons back and forth through the laser medium, continually exploiting the population inversion to create more and more stimulated emissions, thereby creating more and more photons directed back and forth between the mirrors, and so on. wave (the cavity photon popula4. Coupling a certain fraction of the laser tion) out of the cavity through the output coupler mirror to form the external laser beam.

Comparing Fabry-Perot and laser Resonators. In closing out this section, it is instructive to compare the operation of the Fabry-Perot cavity onator) with that of a laser oscillator (resonator) and laser amplifier. Figure 21-8 shows the comparison with three separate sketches. One can see readily that a laser oscillator (c) combines the characteristics of a Fabry-Perot cavity and laser amplifier [(a) and (b)]. 21-8a shows a cavity. resonant to a wavelength Ao, such that the resonance condition, mA o/2 = is satisfied. As the drawing indicates, with two plane parallel mirrors of equal reflectivities (say, 95%), separated by a distance L, the output power is equal to the input power, with, of course, a substantiaJly circulatory power confined between the mirrors. (There is no no gain medium between the two mirrors.) The next sketch (Figure 21-8b) shows the operation of a laser amplifier-a gain medium with popUlation inversion and stimulated emission-but with no cavity (that no mirrors). In this instance, an input beam (generally a laser beam of wavelength A21 ) enters the gain medium, which contains an inverted population be£2 and EJ, for which EI = he/A'}.l. input tween two energy laser beam then energy via the process of stimulated emission as it passes through the medium, finally with significantly amplified power. The last sketch in Figure 21-8c shows how a laser oscillator combines the elements a Fabry-Perot cavity and a bare gain medium and gives rise to a laser device and a laser beam. In this geometry the medium, pumped by an external = hc/A2l . In addition, source, contains inverted levels and , such that the cavity is resonant to the wavelength A2l, since the condition mA2l /2 = L is satisfied for this particular wavelength. Thus, with a resonant and a gain medium, the essential requirements for laser oscillation are met. Now that we have some understanding of laser operation and the origin of the laser beam, let us next look at the characteristics of light.

438

Chap. 21

laser Basics

Input

Output

2222222??l&2?V$

lll7ffffiwzzzz;$

110

110 k - - - - L - - - -.......

lal Fabry-Perot cavity

Pump energy

(bl laser amplifier

Gain medium

Laser 22%W~ output (A:!1)

leI laser oscillator

Figure 21-8 Comparing the essential characteristics of a Fabry-Perot laser amplifier, and laser oscillator. (a) Fabry-Perot cavity with identical mirrors MI and M 2 , spaced a distance L apart. The cavity is resonant to a wavelength.\o. Oulput power equals input power. (b) A bare medium (no end mirrors), pumped by an external energy source, has inverted population levels E2 and EI such that E, - EI he/A21' A coherent input beam of wavelength ALI is amplified as it passes the gain medium. (c) A laser oscillator combines elements of a Fabry-Perot resonant cavity and a gain medium 10 produce a coherent laser beam.

Sec. 21-3

Simplified Description of laser OnArfltin,n

439

21-4 CHARACTERISTICS OF LASER LIGHT Monochromaticity. The light emitted by a laser is almost pure in color, almost of a single wavelength or frequency. Although we know that no light can be truly monochromatic, with unlimited sharpness in wavelength definition, laser light comes far closer than any other available source in meeting this ideal limit. The monochromaticity of light is determined by the fundamental emission process wherein atoms in excited states decay to lower energy states and emit light. In blackbody radiation, the emission process involves billions of atoms and many sets of energy-level pairs within each atom. The resultant radiation is hardly monochromatic, as we know. If we could select an identical set of atoms from this blackbody and isolate the emission determined by a single pair of energy levels, the resultant radiation. although weaker. would be decidedly more monochromatic. When such radiation is produced by nonthermal excitation, the radiation is often calledfiuorescence. Figure 21-9 depicts such an emission process. The fluorescence comes from the radiative decay of atoms between two well-defined energy levels E2 and E •. The nature of the fluorescence, analyzed by a spectrophotometer. is shown in the lineshape plot. a graph of spectral radiant exitance versus wavelength. Note carefully that the emitted light has a wavelength spread 1lA about a center wavelength Ao. where Ao = c/vo and Vo = (E2 - E1)/h. While most of the light may be emitted at a wavelength Ao , it is an experimental fact that some light is also emitted at wavelengths above and below Ao. with different relative exitance, as shown by the lineshape plot. Thus the emission is not monochromatic; it has a wavelength spread given by Ao ± tJ.A/2, where tJ.A is often referred to as the linewidth. When the linewidth is measured at the half maximum level of the lineshape plot, it is called the FWHM linewidth, that is, "full width at half maximum."

(a)

Wavelength

(bl

Figure 21-9 fluorescence and its spectral content for a radiative decay process between two energy levels in an atom. (a) Spontaneous decay process between well-defined energy levels. (b) Spectral content bf fluorescence in (a), showing lineshape and linewidth.

In the laser process. the linewidth tJ.A shown in Figure 21-9 is narrowed considerably, leading to light of a much higher degree of monochromaticity. Basically this occurs because the process of stimulated emission effectively narrows the band of wavelengths emitted during spontaneous emission. This narrowing of the linewidth is shown qualitatively in Figure 21-10. To gain a quantitative appreciation for the monochromaticity of laser light, consider the data in Table 21-1, in which the linewidth of a high-quality He-Ne laser is compared to the linewidth of the spectral output of a typical sodium discharge lamp and to the linewidth of the very narrow cadmium red line found in the spectral emission of a low-pressure lamp. The conversion from 1lA to tJ.v is made by using the approximate relationship, tJ.v = ctJ.A/A~. The data of Table 21-1 show that the He-Ne laser is 10 million times more monochromatic than the ordinary discharge lamp and about 100,000 times more so 440

Chap. 21

laser Basics

..,'"c;; .t: '"x

100

-

...c;; 0

'iii 'f.,:'!

'">

50

[.611] Laser

FlgIIre 21·10 Qualitative of linewidths for laser emission spontaneous emission involving the same pair of energy levels in an atom. The broad peak is the lineshalpe of spontaneously emitted light between levels £2 and £, before lasing The sharp peak of laser between levels £2 is the and £1 begins.

.~

0;

Spomaneous emission

a:

0 Wavelength in arbitrary units

TABLE 21-1

COMPARISON OF LlNEWIDTHS

Light source

Helium-neon laser

Center wavelength Ac (A)

5896 6438 6328

FWHM linewidth fj,.A (A)

I

0.013 10-1

FWHM linewidth tJ./J (Hz)

9 X 1010 9.4 X 108 7.5 X 103

than the cadmium red line. No ordinary light source, without "'1'."""""""" filtering, in the output of typical can approach the degree of monochromaticity lasers.

Coherence. The optical property of light that most distinguishes the from other light sources is coherence. The laser is regarded, quite correctly, as the first truly coherent light source. Other light sources, such as the sun or a gas charge are at best only coherent The subject of coherence can treated quite rigorously and mathematically, a statistical interpretation. Some of that was done in Chapter 12, which was devoted entirely to coherence. At this point we bypass mathematical analysis and sense, only for a qualitatively useful underdescribe coherence in a standing of laser coherence. Coherence, simply stated, is a measure of the degree of phase correlation that exists in the radiation field of a light source at different locations and times. It is often in terms of a temporal coherence, which is a of the degree of monochromaticity of the light, and a spatial coherence, which is a measure of the uniformity of across the wavefront. To obtain a qualitative understanding of temporal and spatial coherence, consider the analogy of water WdVes created at the center of a quiet pond by a regular, disturbance. The source of disturbance might be a cork bobbing up and down fushion, creala regular of moving crests and troughs, as in Figure 21 t I. Such a water wavefield can to have temporal and spatial coherence. The temporal coherence is perfect because there is but a single wavelength; the crest-to-crest distance remains constant. As long as the cork keeps bobbing regularly, the wavelength will remain fixed, and one can predict with accuracy the on the surface. The spatia) coherence of the location of all crests and wavefield is also because the is a small source, generating waves, circular crests, and troughs of ideal regularity. Along each wave then, the spatial variation of the phase of the water motion is zero, that is, the surface of the Sec. 21-1

Characteristics of laser light

441

c

-___

Figure 21-n Portion of a perfectly coherent waleI' wavefield created by a regularly bobbing cod; at S. The wavefield contains perfectly OTdered wavefronts. C (crests) and T (troughs), representing water waves of a wavelength.

water all along a crest or trough is in step or in phase. Again, one can predict with great accuracy, on the the vertical displacement of the water The water wavefield

above can be rendered temporally and spatially process of replacing the single cork with a hundred corks and causing each cork to bob up and down with a different and randomly varying periodic motion. There would then be little correlation between the behavior of the water surface at one position and another. The wavefronts would be highly irregular geometrical curves, changing shape haphazardly as the collection of corks continued their jumbled, disconnected motions. It does not require much imagination to mOve conceptually from a collection of corks that give to water waves to a collection of atoms that rise to light. Disconnected, uncorrelated creation of water waves results in an incoherent water wavefield. Disconnected, uncorrelated creation of light waves results, similarly, in an incoherent radiation field. To emit light of high coherence then, the radiating region of a source must be sman in extent (in the limit, of course, a single atom) and light of a narrow bandwidth (in the limit, with A>. equal to zero). For real light sources, neither of these conditions is attainable. Real light sources, with the exception of laser, emit light via the uncorrelated action of many atoms, involving many different wavelengths. The result is the generation of incoherent light. To achieve some measure of coherence with a nonlaser source, two modifications to the emitted light can be made. First, a pinhole can be used with the light source to limit the extent of the source. Second, a narrow-band filter can be used to decrease significantly the the coherence of the light linewidth all. of the light. Each modification given off by the source, but only at the expense of a drastic loss of light In contrast, a laser source, by the very nature of its production of amplified light via emission, ensures both a output and a high of correlation. Recall that in the process of stimulated emission, each photon added to the stimulating radiation has a phase, polarization, energy, and direction identical to that of the amplified light wave in the laser cavity. The laser light thus created and emitted is both temporaHy and spatially coherent. ]n fact, one can describe or model a real laser device as a very powerful, fictitious "point source," looff monochromatic in a narrow cone angle. cated at a distance, 21-12 summarizes the basic ideas of coherence for and sources. For typical both the coherence and temporal coherence of laser light are far superior to that for light from other sources. The transverse spatial coherence of a single mode laser beam extends across the full width of the beam, whatever that might be. The temporal coherence, also called "longitudinal spatial coherence," is many orders of magnitude above that of any ordinary light source. The incoherent by the

442

Chap. 21

laser Basics

lal

(bl

(d)

21·12 A tungsten lamp requires a pinhole and filter to partially coherent The lighl from a laser is naturally coherent. (a) lamp. The tungsten is an extended source that emits many wmvelf:nglthS. The emission and change lacks both and spatial coherence. The wavefronts are in a manner. (b) Thngsten lamp with An ideal pinhole limits the extent of the tungsten source and improves the coherence of the However, the light still lacks temporal coherence since all wa"elelnpIl1~ present. Power in the beam has been decreased. (c) and filter. Adding a good narrow-band filter further the power bol improves the temporal coherence. Now the light is "coherent," bot the available power is far below that initially radiated by the lamp. (d) Laser. from the laser has a degree of spatial and temporal coherence. In addition, the output power can be very high.

coherence time te of a laser is a measure of the average can continue to predict the correct phase of the laser The length is related to the coherence time ctc , where c is the of light. Thus the coherence length of beam along which the phase of the wave remains For the He-Ne laser in Table 21-1, the coherence time is of the ordet.: of milliseconds (c(]lmr!an~d with about 10- 11 s for light from a sodium lamp), and the co(compared with fracherence for the same laser is thousands of tions of a centimeter for the sodium lamp).

Directionality. When one sees the thin, pencil-like beam of a He-Ne for the time, one is struck by the high of beam direcNo other source, with or without the of ates a beam of such precise definition and The astonishing degree of directionality of a laser beam is due to the geometriof the laser cavity and to the monochromatic and coherent nature of light 0",""" .."."", in the cavity. Figure 21-13 shows a and an external 4>. The mirrors laser beam with an angular spread shown are shaped with surfuces concave toward the cavity, thereby "focusing" the reflecting light back into the cavity and forming a beam waist at one position in the Sec. 21-4

Characteristics of Laser Ught

443

Beam divergence

angle

External laser beam

Figure 11·13 External and internal laser beam for a beam spread, measured by the heam fjJ. an effective aperture of diameter D, located at the beam

cavity. The nature of the beam the laser and its characteristics outside the cavity are detennined by solving the rather complicated problem of electromagnetic waves in an open cavity. Although the details of this analysis are beyond the scope of this discussion, several results are worth It turns out that the beam-spread ~ is given by the relationship ~ = 1.27A D

(21-9)

where A is the wavelength of the laser beam and D is the diameter of the laser beam at its beam waist. One cannot help but observe that is quite similar to that obtained when calculating the angular spread in light by the diffraction of plane waves passing through a circular aperture (Chapter 16). The pattern consists of a central, bright circular spot, the disk, surrounded by a series of bright rings. The essence of this phenomenon is shown in Figure 21-14. The diffraction angle (J, tracking the Airy disk, is given by (J

= 2.44;\ D

(21-10)

where A is the wavelength of the collimated, monochromatic light and D is the diameter of the circular aperture. Both (21-9) and (21-10) depend on the ratio of a to a diameter; they differ only by a constant coefficient. It is .""uu,."",,-, then, to think of the angular ~ inheren. in laser beams and in -9) in terms of diffraction. If we treat the beam waist as an effective circular aperture located inside the laser cavity. then by controlling the sire of the beam waist we control the diffraction or beam spread of the laser. The beam waist, in

Jo'igure 11·14 Fraunhofer diffraction of plane waves through a circular Beam diverof the Airy gence angle e is set by the disk.

Plane waves of wavelength )..

Chap. 21

laser Basics

practice, is determined by the design of the laser cavity and depends on the radii of curvature of the two mirrors and the distance between the mirrors. Therefore, one ought to be able to build lasers with a given beam waist and, consequently, a given beam divergence or beam spread in the jar field, that is, at sufficiently great distance L from the diffracting aperture that L ~ area aperture/A. Such is indeed the case. Several comments may be helpful in conjunction with Figures 21-13 and 2114. Plane waves of uniform irradiance pass through the circular aperture in Figure 21-14; that is, the strength of the electric field is the same at all points along the wavefront. In Figure 21-13 the wavefronts that pass "through" the effective aperture or beam waist are also plane waves, but the irradiance of the laser light is not uniform across the plane. For the lowest-order transverse mode, the TEMoo mode or Gaussian beam, the irradiance of the beam decreases eXP9nentially toward the edges of the beam in accordance with the Gaussian form e Sy'l/D2, where y measures the transverse beam direction and D is the beam width at a given position along the beam (see Figure 21-15). The circular aperture in Figure 21-14 is a true physical aperture; the beam waist in Figure 21-13 is not. It is interpreted as an effective aperture when beam spread from a laser is compared with light diffraction through a real aperture. With the help of Eq. (21-9), one can now develop a feel for the low beam spread, or high degree of directionality, of laser beams.

lIe 2 irradiance

divergence

profi Ie curves

Figure 21-15 A Gaussian TEMoo beam. The diameter (transverse widlh) of the beam is measured between the IIe 2 profile curves. These curves are the locus of points where the irradiance of the beam, on either side of the optical axis, has decreased 10 II e 2 of ils value al the center of the beam. The Gaussian Irradiance variation is shown al two positions along the beam, where the beam diameters are D, and D 2 , respectively. The diameter D, is greater than D, because the beam is spreading according to the beam divergence angle cP = 1.27lt.ID.

Example (a) He-Ne lasers (632.8 nm) have an internal beam waist of diameter near 0.5 mm. Determine the beam divergence. (b) Since we can control the beam waist D by laser cavity design and "select" the wavelength by choosing different laser media, what lower limit might we expect for the beam divergence? How directional can lasers be? Suppose we design a laser with a beam waist of 0.5-cm diameter and a wavelength of 200 nm. By what factor is the beam divergence improved? Solution (a) ~ = I.27A = (1.27)(632.8 x 10- m) = 1 x 10- 3 rad 'I' D 5 X 10- 4 m .6 9

This is a typical laser-beam divergence, indicating that the beam width increases about 1.6 cm every 1000 cm. Sec. 21-4

Characteristics of laser light

(b) A.. 'I"

=

1.27A

D

=

(1.27)(200 x 10

5xlO-3m

9

m)

=5

.1

X 10- 5

rad

Thus the beam divergence angle becomes about 5 x 10-5 rad, roughly a 30fold decrease in beam spread over the He-Ne laser described in part (a). This beam would spread about 1.6 em every 320 m. Clearly, if beam waist size is at our command and lasers can be built with wavelengths below the ultraviolet, there is no limit to how parallel and directional the laser beam can be made. The high degree of directionality of the laser, or any other light source. depends on the monochromaticity and coherence of the light generated. Ordinary sources are neither monochromatic nor coherent. Lasers, on the other hand, are superior on both counts, and as a consequence generate highly directional, quasi-collimated light beams.

laser Source Intensity. It has been said that a I-mW He-Ne laser is hundreds of times "brighter" than the sun. As difficult as this may be to imagine, calculations for luminance or visual brightness of a typical laser, compared to the sun, substantiate these claims. To develop an appreciation for the enormous difference between the radiance of lasers and thermal sources we consider a comparison of their photon output rates (photons per second). Example (a) Small gas lasers typically have power outputs P of I mW. Neodymium-glass lasers, such as those under development for the production of laser-induced fusion, boast of power outputs near 10 14 W! Using these two extremes and an average energy of 10- 19 J per visible photon (E = hv). determine the approximate range of photon output from lasers. (b) For comparison, consider a broadband thermal source with a radiating surface equal to that of the beam waist of a 1- mW He-Ne laser with diameter of 0.5 mm (or area of A = 2 X 10-7 m2 ). Let the surface emit radiation at a wavelength of 633 nm with a linewidth of LU = 100 nm and temperature T = 1000 K. Determine the photon output rate of this thermal source. Solution (a) The rate of photon output is given by P/hv. Thus

10- 3 J/s 10- 19 J/photon

P

hv P

hv

=

10 14 J/s

10- 19 J/photon

10 16 photons Is

=

33

10 photons/s

(b) The photon output rate for the broadband source can be calculated from the equation3 thermal photons Is

I

== A2 --:--:-:::::--1 LlA LW

(21-11)

'One can obtain Eq. (21-11) from Eq. (21-2) by changing from p(v) in units of energy per unit volume per frequency range [E/(vol-~jI)l to units that give thermal photons per second [E/(hv-tll)]. In this process, one must change geometry accordingly, from an isotropic radiating volume element ~V (Eq. 21-2) to thai of a radiating surfuce element ~A (Eq. 21-11).

446

Chap. 21

Laser Basics

The frequency v ciA 4.74 x 10 14 Hz. The bandwidth tJ.v 13 (c/ A2) I:M X 10 Hz. The argument the exponent is

hv kT

~----"""'::"''!'''''''-------'-

Substituting all values into oodis

= 22.77

thermal photon output

sec-

(2 x

(633 x :0-9)2 m2)

We find a rate of only about 109 photons/sf This value is seven orders of magnitude smaller than the photon output rate of a low-power l-mW and 24 orders of magnitude smaller than a powerful neodymium-glass The comparison is summarized in Figure 2 t - t 6.

1016 photons/s (per 2 X 10-6 sri

Beam waist area 12 X 10-3 cm2 ) lal

Source area (2 X 10-3 cm2 ) (bl

Figure 21-16 Comparison of photon output rates between a low-power He-Ne gas laser and a hot thermal source of lhe same radiating surface area. (a) l-mW HeNe laser (A 633 nm). (b) Broadband thermal source (Lambertian), with male values of T 1000 K, aA = 2 X 10- 3 cm2 , Ao = 633 nm, and 100 nm. Note that the laser emits aU the photons in a small solid (- 2 x 10-6 compared with the 2w solid angle of the thennal source.

We see also from Figure 21-16 that the laser emits 1016 photons/s into a 6 acting as very small solid angle of about 2 x 10. sr, whereas the thermal a Lambertian source, radiates 109 photons/s into a forward, hemispherical solid angle of 211" sr. If we were to ask how many thermal photons/second are by the thermal source into a solid angle equal to that of the laser, we would the answer to be a mere 320 phcltools/s: =

320 photons/s

The contrast between 10 16 photons/s for the laser source and 320 ptlc)torls/s for the thermal source is now even more dramatic. Carrying our comparison of source intensity between laser and nonlaser sources one step further, we can determine how the radiance [W/(cm2-sr)] of a highpower, nonlaser source compares with that of a low-power laser.

Sec. 21-4

Characteristics of laser

447

We choose for the nonlaser source a super-high-pressure lamp, capable of a source radiance to 250 W/(cm2 -sr). lamps were about the best high-radiance sources available before the advent of the laser. For the source, we choose a 4-mW He-Ne laser with diameter of 0.5 mm and beam of 1.6 mrad, at a wavelength of 633 nm. The geometry is shown in Figure 21-17. It be clear from the figure that the laser can be considered a radiating that of the beam waist. The power of the surface is and the surface into the solid dictat(:d the laser. Mirror

I/> = 1.6 X 10- 3 rad

_h

r

AreaB-

@

D = 0.5 mm Ll.A

20

2 X 10-3 cm 2

21-17 Radiance of a Hc-Ne laser. The radiance of the laser (beam waist) "seen" by the detector is about 1(1' W/cm2 -sr.

Solution the definition for L" in terms of laser power <1>", source or beam-waist area and solid angle dn, dAdn

the radiance of the laser is found to be 106 W/cm2-sr, as follows: area BjR 2 , where Figure 21-17, the solid angle an area B Then substituting into

(21-12) Wpl"PrTlno

to

= 1T[R tan

(21

the radiance is

4<1>"

1Tl/J2dA Again, the comparison between laser source and high-intensity mercury lamp is dramatic. We conclude that where brightness and radiance are important in the of light sources, the laser stands alone. Focusing Ught to a spot is a lenge. It is in geometrical to show a positive lens focusing a beam of perfectly collimated light to a "point Figure 21-18a). We know that a point unattainable even in the limit of geometrical (A ~ 0), aberration-free lenses do not exist. Nevertheless, the of focusing light to a diffraction-limited point has long been a goal. Now the laser, with its coherent, nearly collimated beam, has made that ideal attainable. ure 21-18b shows the difficulty involved in focusing ordinary light to a tiny spot. First, the light emitted is incoherent. Second, the source of light cannot be too small, a point source, for then either the light generated be

Chap. 21

laser Basics

Ideal beam

I:A -1> 01

lal Ideal source

Focused image of thermal source

Ibl Ordinary source

Focused laser beam

laser

lei laser source

Figure 21-18 Focused beams from various sources. (a) Ideal, collimated beam is in accordance with geometrical (b) Incoherent focused to a fictitious image of size hi ~ A. radiation from a thermal source is focused to a (c) Coherent laser beam is focused to a diffraction-limited spot of diameter d A.

insufficient or the source would melt. combination of a nonpoint source and infixed more or less by the laws of coherent light leads to large magnification in geometrical The on the other hand, radiates intense, coherent lighi that appears to come from a distant "point source." By its unique properties then, it overcomes the precise limitations that frustrate one's attempts to focus thermal radiation to a tiny spot. 21 18c shows a laser beam focused by a lens to a diffractionlimited of incredibly small equal to the wavelength of the focused It can be shown that a laser beam, with beam tjJ, incident on a lens of length f, whose diameter is several times larger than the width of incident beam, is focused to a diffraction-limited of diameter apprcixirnately equal to (21-13) as shown in Figure 21-19. The beam divergence angle tjJ is equal to t.27A/D, as given previously in (21-9). Note carefully that Din (21-9) refers to the diameter of the beam waist in the laser that "determined" the beam divergence, and d

Sec. 214

Characteristics of Laser Light

Figure 21-19 TEMoo laser beam of beamspread angle cP.

in

(21-13) refers to the diameter of the laser beam focused location of the focused laser is essentially at the (s' I), although a careful shows that this in even if a good one. With the help of 13), we can make several the spot size of focused laser beams. With a lens of focal inClldelr1t laser light of beam tP = 10-3 to 10-4 rad, spot of the order 5 4 of 10- m to 10- m (or 100 pm to 10 pm) in diameter can be obtained easily. If we compare these diameters with the wavelength of the carbon dioxide (A = 10.6 we see at once that laser light-indeed all laser be focused to sizes of the order of a Wa'/elt::mH Equation (21-13) indicates that focusing laser light down to small can be "",-,111<;;1/<;;.., by lenses with short focal and laser beams with divergences. As long as aberration-free lenses of high quality are the focal length can be chosen as short as is The beam of a usually determined at the time the laser is can still be reduced with the additional 21-20, a collimated laser beam of width found in beam expanders. In W; and divergence tP. is focused the first lens of the beam (focal lengthII ) to a of diameter d IltPIJ in accordance with Eq. (21 13). The second a distance.li from the with Ji > ji. collects the light expandfrom the focused spot and essentially recollimates it. The beam divergence of the recollimated beam is to (21-14) where fd Ii = "'ifW; is the beam eXf)'OrMilOn ratio. The validity of to show. The incident by the first has a dl ji tPi. By the principle of of1ight, if the expanded

1-(4) is not diameter were to be

Expanded beam Beam expander

21-20 laser beam.

450

Chap. 21

Beam expansion as a method of reducing beam divergence of a

Laser Basics

redirected to the left and focused spot at the same location. so that

the second lens, it would form the identical hcf>J' Since d l necessarily,

II cf>; = hcf>f cf>f

h cf>;

If the beam expansion ratio is hIII = the beam divergence of the expanded laser beam is ~ that of the incident beam. Expanding the beam width by a factor of 5 achieves a reduction in beam by the same rnctor. What has been gained? Suppose a laser beam is expanded 10-fold an appropriate beam expander. The outgoing beam then has had its beam divergence decrease:o by a factor of 10. If the expanded beam is then focused to a tiny spot with a of arbitrary length I. the diameter of the spot will be -k that achievable with the unexpanded beam and the same lens. This reduction in diameter leads to a loo-fold reduction in size area and thus, for any given laser beam power, to a lOO-fold increase in focused spot Laser energy focused onto small areas makes it to drill tiny holes in hard. dense make tiny cuts or welds, make high-density recordings, and generally carry out industrial or medical procedures in target areas a wavelength or two in size. In ophthalmology, for example, where Nd: YAG lasers are used in ocular surgery, target of 109 to 10 12 W/cm2 are required. Such irradiance are readily with the help of beam expanders and suitable focusing optics. as was discussed in Section 7-5 problem 7-10).

LASER

TVI:t£:'C!'

AND PARAMETERS To this point we have examined the basic assumptions that led Einstein to the existence of stimulated identified the essential parts that make up a laser, described in a general way how a laser and studied the characteristics that by way of summary, we tum our make lasers such a unique source of attention to the identification of some of the common lasers in existence today and to parameters that distinguish them from one another. Lasers are classified in many ways. Sometimes they are grouped to the laser medium: gas, liquid, or solid. Sometimes the state of matter to how they are pumped: flashlamp, electrical disare classified ""'Uj:.\'~' chemical actions, and so on. Still other classifications divide them according to the nature of their output [pulsed or continuous wave (cw)] and according to their of emission (infrared, visible, or ultraviolet). spectral No particular classification scheme has been chosen for the lasers in of the 30 or 40 common Table 21-2. Those identified are, in away, a cross lasers on the market today. A careful examination of Table 21-2 serves as an introinclude data duction to the state of laser technology. For each laser listed, the on emission output power (or in some cases, energy per pulse), nature of output, beam beam and Table 21-2 of gas lasers CO2 , nitrogen); solid state (ruby, cludes Nd:YAG, Nd-glass); liquid or lasers; semiconductor lasers (gallium arsenide); the excimer gas lasers (argon fluoride); chemical lasers (hydrogen fluoride); and ion lasers (argon ion). Both pulsed and continuously operating (cw) lasers are represented. Taken as a whole, Table 21-2 includes lasers whose wavelengths vary from 193 nm (deep to 10.6 f-tm infrared); whose cw power outputs vary Sec. 21-5

Laser

and Parameters

451

~

TABLE 21-2

LASER PARAMETERS FOR SEVERAL COMMON LASERS

of output 632.8 run 694.3 nm Carbon dioxide

1O.61Lm

337 nm Nd:YAG (solid)

1. 064 lL m l. 06 lLm

Argon ion (gas)

488 run or 514.5 nm 400-900 run (tunable)

Dye

Argon fluoride (excimer) I"Iv!l1r02'en fluoride (chemical) Gallium arsenide (semiconductor diode)

0.1-50 mW 0.03-100 J

20-800 mW

193 nm 2.6-3 ILm

U.Ul-l:5U W cw

780-900 nm

or 2-600 mJ per 1-40 mW cw or average

per

1 The efficiency here is the overall over wall-plug electrical power in. The quantum energy to the pump energy

Beam divergence

Efficiencyl

cw

0.5-2 rnrn 1.5 mm-2.5 cm

0.5-1.7 mrad 0.2-10 mrad

<0.1% <0.5%

cw

3-4mm 2 X 3-6

1-2 mrad 1-3 x 7 mrad

5-15% <0.1%

1-300 mW

5

Beam diameter

X

30 rnrn

cw

0.75-6 mm 3 mm-2.5 cm

2-18 mrad 3-10 mrad

0.1-2% 1-5%

cw

0.7-2 mm

o.4-l.S mrad

<0.1%

0.4-0.6 rnrn

1-2 mrad

10-20%

6 x 23-20 X 32 mm

2-6 mrad

<0.5%

2 mm-4 cm

1-15 mrad

0.1-1%

200 x 600 mrad (oval in

1-20%

too

often referred to as a "wall-plug" efficiency, to denote the ratio of la~er-beam power out always than the wall-plug efficiency, is defined as the ratio of the laser's transition

from 0.1 mW to 600 W; whose beam divergences vary from 0.2 mrad (circular cross section) to 200 x 600 mrad (oval cross section); and whose overall efficiencies (laser energy out divided by pump energy in) vary from less than 0.1 % to 20%.

21-1. Beginning with the expression for rate of spontaneous decay of atom density N2 at excited energy level £2. dN2) ( dt spoIll

_ A21

Nz

show that an initial popuJation density Nw decreases to a value in a time T equal to I/A 21 • 21-2. Assume that an atom has two energy levels by an energy corresponding to a wavelength of 632.8 nm, as in the He-Ne laser. If we suppose that all the atoms are located in one or the other of these two stares, what fraction of atoms is in the upper state at room T = 300 to the Boltzmann distribution? energy density I~n~r~'vf{ 21-3. Show that for the blackbody p(v) =

flmhv 3

follows from the form given in Eq. (2-12) for the blackbody [power/(area- AA)].

"V",,-u,,u

radiant exitance

by variables from A, dA to v, dv and then mUltiplying the result by 4/c. (See footnote I.) 21-4. Treating the sun as a blackbody radiator, determine its spectral energy p(v) in the visible near A = 550 nm. Assume the SUrfdce temperature of the sun to be about 6000 K. 21-5. Why should one lasing at ultraviolet to be more difficult to attain than lasing at wavelengths? your answer based on the ratio Azd B21 and the meaning of the A zI , B21 coefficients. 21-6. Calculate the ratio of stimulated to spontaneous transitions for green light at 0.5 J,Lm within a dense plasma at a temperature of 5000 K. That is, determine the value of the ratio (dN/dt)se (dN /dt),p

-BzINzp(v) -A 21 N2

What does the numerical value of this ratio imply? 21-7. (0) Given the center Ao and linewidth AA for the three entries in Table 21-1 (ordinary discharge lamp, Cd 10\1/-pl'essure lamp, and He-Ne is as given in the table. verify that the FWHM linewidth AI' is a He-Ne laser line for example, than a sodium (b) How much line from a sodium lamp? 21...ft Suppose that the coherence time of a light beam is equal to the reciprocal of its frequency linewidth (FWHM). What then is the coherence time and coherence length of the He-Ne laser in Table 21-1? 21-9. A He-Ne laser has a beam waist (diameter) equal to about I mrn. What is its beamspread angle in the far field?

21

Problems

21-10. Consider a broadband thermal source with a circular surface of diameter 0.5 mm (roughly the size of beam waists in He-Ne Let the surface at a temperature of 1000 K emit at 633 nm with a Iinewidth (FWHM) of 100 nm. Use Eq. (21-11) to show that the thermal photon output rate is about 5 x Id' photons/so 21-11. Consider a I-mW He-Ne laser emitting at 632.8 nm with a FWHM frequency linewidth of t.w = let Hz. Assume that the full-angle beam divergence tfJ in the fur field is tfJ = 1.27A/D, where D is the diameter of the aperture (beam waist). See Figure 21-17. (a) Show that the spectral radiance fJ.L./ t.w = tfJ./(fJ.A fJ.O tfJ. is the laser power. fJ.A is the area of a radiating surfuce, fJ.O is the solid angle the laser radiates into, and t.w is the spectral bandwidth (FWHM) of the laser emission-is independent of the value of the diameter D of the diffra,:ting ...""rturf' (b) Obtain a numerical value for the spectral radiance described above. 21-12. With reference to the beam shown in Figure 21-20, = 10 and the beam divergence of the incident beam is I mrad. (a) What is the beam of the expanded beam? (b) If the expanded beam is focused a subsequent lens of power 10 diopters, what is the beam waist at the focal (e) If the power in the incident is 1 mW, what is the irradiance (W/m2) at the focal spot? 21-13. Refer to Figure 21-8a. Assume that 1 W of incident laser power of resonant wavelength Ao is incident on the cavity. (a) What is the output (b) How much power circulating within the cavity? (e) Describe what must be at mirror MI to make your answers to (a) and (b) valid. 21-14. For a Nd:YAG laser, there are four pump levels located at 1.53 1.653 eV, state energy level. 2.119 eV, and 2.361 eV above the (8) What is the associated with the photon energy to populate each of the pump levels? (b) Knowing that a Nd:YAG laser emits photons of wavelength 1.064 /Lm, determine the quantum efficiency associated with each of the four pump levels. 21-15. To operate a Nd:YAG laser, 2500 W of "wall-plug" power are for a power supply that drives the arc lamps. The arc lamps provide pump energy to create the population inversion. The overall laser system, from power in (to the power supply) to power out (laser output is characterized by the component efficiencies: 80%-power supply VjJ'i;IiIlIVII 30%-arc lamps for 70%-optical reflectors pump light on laser rod 15%-for spectral match of pump light to Nd:YAG pump levels 50%-due to internal losses (8) the efficiencies into account sequentialJy as they "occur," how much of the initial 2500 W is available for power in the output beam? (b) What is the overall operational (wall-plug efficiency) for this laser?

[I] [2J

B. A. Lasers, 2d ed. New York: John Wiley and Sons, 1971. A. E. An Introduction 10 Lasers and Masers. New Vorl<: McGraw-Hill Book 1971. A. E. Lasers. Mill Valley, Calif.: University Science Books, 1986.

Chap. 21

laser Basics

[4] Thompson, G. H. B. Physics of Semiconductor Laser Devices. New York: WileyInterscience, 1980. [5] Verdeyen, J. T. Laser Electronics. N.J.: Prentice-Hall, 1981. [6] Unger, H. G. Introduction to Quantum Electronics. New York: Pergamon Press, 1970. [7] O'Shea, D. c., W. R. Callen, and W. T. Rhodes. Introduction to Lasers and Their plications. Reading Mass.: Addison-Wesley 1978. [8] Klein, M. V. Optics. New York: John and Sons, 1970. [9] Saleh, B. E. A., and M. C. Teich. Fundamentals New York: John Wiley and Sons, 1991. [101 Schaw]ow, Arthur L. "Optical Masers." :lcientificAmerican 1961): 52. [II] Schawlow, Arthur L. "Advances in Ma.'iCrs." Scientific American (July 1963): 34. American (Apr. 1966): 32. [121 Pimentel, George C. "Chemical Lasers." [13] Lempicki, Alexander, and Harold Samelson. Lasers." Scientific American (June 1967): 80. C. K. N. "High-Power Carbon Dioxide Lasers." Scientific American (Aug. 1968): 22. Peter. "Organic Lasers." Scientific American 1969): 30.

[16] Panish, Morton B., and Izuo Hayashi. "A New Class of Diode Lasers." Scientific American (July 1971): 32. American (Feb. 1973): 88. William T. "Metal-Vapor Lasers." W. T. "The C3 Laser." SCientific American (Nov. 1984): 148.

21

References

22 Mirror surface

Mirror-lens combination

I

: III I

,,

:......- - I

Characteristics of Laser Beams

INTRODUCTION

We now turn our attention to the beam generated by the laser and the propagation characteristics of this beam. We shall see that a laser beam has characteristics of both plane and spherical waves, while retaining an identity all its own. In its simmode-the fundamental laser beam takes the form of spherical wavefronts, with the a transverse irradiance ' localized near the axis. In other, more forms, referred to as higher-order modes or Hermite-Gaussinns, the field takes on transverse irradiance distributions that depart from the simple variation and exhibit an ordered pattern of "hot " In many cases, the output laser beam consists of a mixture of modes: the fundamental and several modes. In this chapter we examine the nature of the laser beam and describe its general characteristics. We shall study its properties at positions near the laser (near field) and far from the laser (far field), We shall specify a method for deterits characteristics-involving wavefront curvatures and transverse irradiance I While the words inlensity and irradiance are often used interchangeably in laser literature, we shall use irradiance when we mean to deseribe power per unit area (e.g .• W/m2) at some location on a (see Table 2-1). plane along the beam

456

distributions-while it propagates through arbitrary optical systems. After a rather thorough study of the fundamental mode of the laser beam, the so-called TEMoo Gaussian beam mode, we examine the higher-order transverse modes and their transverse irradiance distributions. 22-1 THREE-DIMENSIONAL WAVE EQUATION AND ELECTROMAGNETIC WAVES In Chapter 8, Wave Equations, we the propagation of transverse waves in a homogeneous medium. We found that the one-dimensional differential wave equation describing such motion is written as

o

1)

Here y describes the transverse wave displacement, z denotes the coordinate along t stands for time, and v represents the wave speed in the wave propagation the homogeneous propagation medium. ) is generalized to three dimenWhen the one-dimensional wave equation sions for electric field displacements E{x, y, z, t) in homogeneous media devoid of free charges or currents, the wave equation takes the form

=0 is the Laplacian operator, n is the refractive index of the propagating in vacuum. medium, and c is the wave

Plane Wave Solutions to the Wave Equation. The so-called plane wave solutions to the three·dimensional wave equation may be expressed in the comform E(r, t) = 3) where the propagation and position vectors are given by k

= ikx + jky + kkz

and

r

= IX + jy + kz

and w = Ik Ie/ n. Plane wave solutions are useful as base functions in the Fourier of ele:ctrorrlagnetlc analysis of arbitrary wave forms and as fields reaching a detector from a distant point source. They are referred to as plane waves because the surfaces of constant phase (k . r) are geometric planes perpendicular to the wave vector k. The essential features of propagating plane waves have been discussed in Chapter 8 and are shown in Figure 22-1. 2

Spherical Wave Solutions to the Wave Equation.

The spherical

solutions that satisfy the wave equation can be written as i l A,. ) E- (r,u''I',t

Eo(e, cf» r

e

i(krwtl

(22-4)

The appropriate geometry and fundamental are shown in Figure 22-2. acThe spherical wave solutions are useful in the formulation of diffraction

vector

needs to distinguish the unit vector i from the im~:"in"rv number i from the propagation constant k.

Sec. 22-1

Three-Dimensional Wave Equation and

v=T and the unit

1<>"'trrllm'''U'1n,,ti...

Waves

451

Plane waves: E lx, y,z,tl = Eo e ~I... -

o~

wll

______.-____~____-+______+-____~______~______.-~P~ro~p=a~g=at=io=n~~ I!

(I! . rl is a eonslant for r from point plane

Iv!

axis

.£ n

22-1 The surfaces of constant phase k ... for waves are OP{.rru'trJc,,1 planes oriented in a direction to the wave veclor k and speed v c:/n the homogeneous medium of refractive index n. Transverse plane Z

lx, yl

o

x Three-dimensional geometry

Two-dimensional representation of spherical waves

22-2 The surfaces of constant for spherical waves are geometrical radiating outwardly from a point source O.

and Kirchhoff. They represent the mathematical forms used for the secondary wavelets that radiate out from the diffraction aperthey a valid of ",Ip"tr,nrn",u_ ture (see Section 3-1). In netic energy propagating away from a point source. When the distance from the point source becomes very the waves can be treated as plane waves, with (22-3) replacing Eq. (22-4) as a mathematical description.

Chap. 22

Characteristics of Laser Beams

22-2 PHASE VARIATION OF SPHERICAL WAVES ALONG A TRANSVERSE PLANE To help us understand the nature of the laser beam, it will be useful at this point to determine the phase distribution of the electric field E associated with spherical waves along a plane perpendicular to a given direction of propagation. In Figure 222, this plane is shown as the x-y plane, and the direction of propagation is along the z-direction. At any point (x, y) on the transverse plane z = R, the spatial part of the electric field is given by E(x, Y)Z=R = (constant)eil
(22-5)

where r = (R2 + x 2 + y2)1/2 and R is the perpendicular distance from the source to the transverse plane. It is clear from Figure 22-2 that R is also the radius of curvature of the spherical wave that is tangent to the plane at (x, y) = (0, 0). Rewriting r as (22-6) and restricting the (x, y) points in the transverse plane to positions near the axis of propagation, such that (x 2 + y2) ~ R2, we can approximate3 Eq. (22-6) as r

==

R

x2 + y2 + ----"2R

(22-7)

In turn, then, Eq. (22-5) for the spatial part of the electric field E at point (x, y) on the transverse plane z = R takes on the explicit form (22-8) We shall use Eq. (22-8) later as a condition for identifying a spherical wave that is crossing a transverse plane, as long as we restrict the examination of the electric field E to points near the axis of propagation.

22-3 BASIS FOR DEFINING LASER-BEAM MODE STRUCTURES Before we proceed further with the description of laser beams, it is necessary to say a few words about the variation of the electric field in a plane transverse to the beam, specifically, how this variation is used to characterize the mode .structure of the beam. In this regard, it may be helpful to draw an analogy between the displacement of the skin on a musical drum and the value of the electric field across a transverse plane of the laser beam. For a drum, the fundamental mode occurs when the skin fastened around the rim moves up and down as a whole, with greatest motion occurring at the center, producing a single-humped profile. Analogously, the fundamental mode for a laser beam exists when the electric field has its maximum value at the center of the beam and drops off in accordance with a Gaussian profile. This profile is a "hump" that decreases symmetrically in aU directions from the center toward the edges of the 3Defining (x 2 + Y')/R 2 as u', and expanding (I + U')·/2 in accordance with a binomial series for which u 2 «: I, Eq. (22-1) follows from Eq. (22-6) when smaller, higher-order terms are neglected.

Sec. 22-3

Basis for Defining laser-Beam Mode Structures

459

Drum skin displacement

I

t

Laser beam

I

i

(b)

(e)

Comn,an!<;on of the mechanical dlsiplal:errlenl of a drum skin and the of a laser beam. (a) Instantaneous drum skin in its fundamental mode. Vertical shown is any olher diameter. (b) Instanof the electric field variation diameter in a transverse to the fundamental mode, the electric the same any other diameter.

beam.4 Figure 22-3 compares the displacernc)[lt instant. of the skin on the fie1d vector across beam. drum and of the The notation used to describe the beam is mode of the 5 TEM refers to TEMoo, a case of the "Transverse Electric Magnetic. The denote [he number of times the electric field goes to zero, rpC"""t·t",,,, xy-coordinate axes superimposed on the beam's transverse TEMoo denotes a "single-humped profile" with no zeros along either x or y (or transverse beam diamennnr,')£];"h.,,~ zero. ter) except, of course, at the edges where the cOIrrf:spondulJ!: burn ~"..~ .• ~~ In this then, electric field variations for TEM oo , , and TEM II appear 22-4. For examn

y

y

y

y

x

x

y

y

y

y

m=O n 1

-+---...,..---H-x

Figure 22-4 Electric-field variations across trall!i~f!rse corresponding burn pauerns such beams would TEMmn modes.

a target-for different

4 The edges of a VI!!!nIlIllV drum head are well-defined, of a laser beam are not. Still, the analogy, in is USerll!. 'The lransverse modes denoted here by the notation TEMmn generated, of course, in a mirror-bounded, open·sidewall optical resonator, the laser 6 So named because of the pattern produced by a nU!fHln:te:nslltv beam when it "bums" into a target

460

22

Characteristics of laser Beams

the TEMOI mo.de sho.ws no. E-field zeros alo.ng the x-directio.n ,,,,u,,,,,,,,,,",-, profile) and o.ne E-field zero. alo.ng y-directio.n with zero at the center po.int). Such a variatio.n produces a burn (o.n target) that co.ntains two. ho.rizontal, dark areas centered above and a "no.-burn" regio.n. A the o.ther cases piccareful examinatio.n o.f the E-fieJd variatio.n and burn tured-and. in fact, fo.r all TEMmn modes-discloses similar correlatio.ns between E-field variatio.n and "burn" areas. Fo.r the sake o.f simplicity in this treatment, we shall with the beam, who.se burn is essentially a dark circle bum intensity at the center (o.n axis).

22-4 GAUSSIAN BEAM SOLUTION FOR LASERS If we were to. examine the character o.f a laser beam, we wo.uld find that its spherical with lo.ng radii o.f curvature that increase as the beam alo.ng the axis. The co.mwavelTo.lilt and irradiance variatio.n o.f a TEMoo laser beam passing thro.ugh a ('."ntH'Ta.na Jens appear as sho.wn in Figure 22-5.

lens

Figure 22-5 An external laser beam, confined essentially to within the Ije 2 -irradiance guidelines. is focused a converging lens. The incoming beam is highly collimated so thai its wavefronls are very nearly planar. The converging lens reshapes the wavefronts and focuses the beam-forming a beam waist-IO lhe of the lens. The beam as il propagates on past the beam waist.

The so.lid guidelines, above and belo.w the z-axis, the of points fo.r which the beam's electric field irradiance in a transverse directio.n· is to. lle 2 o.f its value o.n axis. Thus these lines are used to. define a continuo.usly changing beam width. The dashed arcs transverse to. the z-axis indicate the wavefronts o.f the commo.n phase fronts for a collimated beam. Since the geometrical Jaser beam are "mo.re-or-less" we can choose a trial so.lutio.n for the laser beam's electric field E(r, t) o.f the mo.tivated by a one-dimensio.nal wave, that a trial so.lutio.n7 o.f the form E(r, t)

u (x, y.

term U(x, y, z), when ('\pl'pnnIfIP('\ irradiance and phase variatio.ns o.f l:(j~lall(m (22-9) is casl in the sumed 10 be essentially along Iric field E(x, y, z. I) in the sealar form of

Sec. 22-4

z)ei(kz-Wl)

nr(\vI/1p"

(22-9)

the details that ac(;ur.:ltellv

aPlJlro~jm.lIion. That is, a wave is asthe z-direction, thereby permilling us to express the elec(22-9).

Gaussian Beam Solution for lasers

461

exponential term, in Eq. (22-9), reflects the "more-or-Iess" plane wave nain determining the function U (x. y. z). we ture of the solution. If we are would then have a useful solution describing the actual laser beam. Determination of the U(x, y, z) Function. Since our trial solution in (22-9) must satisfy the wave we substitute it into and obtain a defining equation for the yet function U(x, y,

ei(kz-wtl[CPU + iPU + cPU + 2ik iJU iJx z iJy2 iJz

(k2 -

2 W C2

)U]

0

10)

the second-order variation of U with z since the monatonic attenuation of y, z, t) with axial propagation distance is small. However, we retain the firstorder variation of U with z, as well as the variation of U with x and y. The last term (22-10) vanishes outright, w kc. Thus Eq. (22-10) can be simplified, us with the following equation that U(x, y, z) must obey, We

iPU

1)

-+ 2 -+ iix

iiy2

. (22-11) is a nontrivial partial differential equation with complex terms. To it, we make an "educated" at a solution, motivated in part by the cylindrical symmetry which we the laser beam's electric field magnitude to have in the transverse direction, and in part by the complex nature which the solution U(x, y, z) must exhibit. Thus we "guess" the form U(x, y, z)

(22-12)

where p(z) and q(z) are general as yet undetermined and subject to constraints imposed by Eq. (22-11), the defining equation for U(x, y, z). After substituting Eq. (22-12) into (22-11), we obtain

2ikU q

~ _ (x 2 + y2)U - ZkU

~

+

k2

_

(x 2 + y2)U

~

=

0

which, when

[(~k _2k ~) If of

(x 2 +

+

o

+

(22-

is to be a solution for all x and y, then each term multiplying a power "coefficients" of (x 2 .+ yl)O and

yZ) must equal zero separately. Thus yZ)I, when set equal to zero, yield iJz

(22-14)

q

and iiq = 1

(22-15)

iJz Let us now explore the meaning of the (22-15) on the functions q(z) and p(z).

eXDlreSS4~d

by

(22-14) and

8A derivation of the Gaussian beam equation is given in Reference [I], Chapter 16. Those who wish to become familiar with the details of this derivation are encouraged to examine man's scholarly treatment.

462

Chap. 22

Characteristics of laser Beams

Definition of the Complex Radius of Curvature. We see from Eq. We also note, by the (22-14) that, in general, p and q may be (22with the second term in (22-8), that second term in the exponent of q(z) ought to resemble, in the radius of curvature of the wavefront, if indeed the for the laser beam are to be spherical in nature. Having little to then, we make an extremely "shrewd guess" and write q as a complex term calling it the complex radius of curvature and writing it in the

.! = ij

I R

+ i~

(22-16)

1TW2

of curvature of the wavefront and w is the transverse dimension where R is the the beam. 16) is an educated guess, dictated more by hindsight Quite candidly, Eq. dividends. We see that l/ij, as expressed than foresight. Nevertheless, it bears in (22-16), is indeed complex, that its real part I/R is related to the real curvature R of the wavefront, and that its imaginary part (A/1TW 2) is related to the transverse dimension w of the beam. As a matter of fact, we shall see that w-called (unfortunately) the "spot size" and so confused with a diameter-is a measure of the w is measured from the on-axis position beam's half-width (radius). in a transverse plane to the position where the beam has de(22-16) into (22creased to 1/e 2 of its on-axis value. Thus, if we introduce 12), we get

Vex, y, z} =

17)

for p whose identification we defer until later in this chapter, we now have an explicit expression for V (x, y, z), and thus, together with Eq. (22-9), a workable form for the space and time variation of the electric field in a laser beam. 10 Spherical Gaussian problem for the mathematical back into our original guess,

rf"l1,rp",pnt>lh(~n

At point, we have "solved" the of a laser beam. Inserting (22-17)

E(x, y, z, t) = V(x, y, z,

we

t)ei(kz-M)

the Gaussian beam representation for the E-field, y, z, t) =

18)

The first exponential term in (22-18), identical with the second exponenearHer for use at this the getial term of ometryof wavefront for the laser beam as a spherical surface. second exponential term indicates that the electric field in any transverse plane (at a fixed z) falls value on axis. The last exponential term conoff as a Gaussian function, II with tains phase information dependent on z. At this point, as far as p(z) is concerned, we note that it must satisfy the condition (Jp/(Jz i/q. 9 Some texts give (22-16) with a negalive sign between the two terms on the right (for example, see [I], [3J, and [6]). If one writes (22-9) with the term elk' as we have, Eq. (22-16) follows. If one writes Eq. (22-9) with the term , (22-16), with a negative sign, follows. In either case, in the conlext of a consistent set of equations, the characteristics of a Gaussian laser beam are found to be the same. lOWe note that this is but one very useful solution to Eq. (22-11). Discussion of other solutions is deferred to Section 22-7. of a "Gaussian" variation of Ihe form II It is assumed that the reader is familiar with the + y2. for a cylindrically beam where y2 exp

Sec. 22-4

Gaussian Beam Solution for lasers

463

Thus we have in finding a beamlike solution to the wave equation. It has explicitly the form in (22-18) and is referred to as a spherical Gaussian beam. Along the beam propagation axis (z-direction), the electric field is harmonic in nature, but with a wavelength approximately by ).. = 211 /k, due to the presence of the p(z) term in the last exponent. In any transverse plane the electric drops off in a Gaussian manner with distance the conclusions are illustrated in Figure 22-6. Elx, Y• .zo)

E(z)

(a) Axial variation

22-6 Approximate axial variation and transverse variation of tbe electric field tude E(x, y, z) in the laser beam. Here w is the beam width at Eo/e.

(b) Transverse variation

22-5 SPOT SIZE AND RADIUS OF CURVATURE OF A GAUSSIAN BEAM Suppose that at some transverse plane, z 0, the Gaussian beam has a planar wave0) - 00. Then, from Eq. we have front, that is, R(z i)..

ijo

or ijo

.1IW~

-1--

)..

where ijo and Wo are defined as ij(z = 0) and w(z 0), The plane z = 0 locates the beam waist. The Wo is, therefore, the spot size at the 22-7. beam waist, as iI1ustrated in For any other transverse plane z 4= 0, it fonows from (22-15) that

= ij(O) + z Irradiance variation in transverse plane

)(0,

Yo

Figure 22-7 Gaussian spherical beam propagating in !be z-directioo. The spot size w(z) at the beam waist (planar wavefront) is defined as woo The half-angle beam divergence () A/(7TWo) is valid only in the far field. Note the in transverse irradiance as tbe beam propagates to tbe right.

Chap. 22

Characteristics of laser Beams

(22-20)

where z is the axial distance measured from the beam waist. This is the with propagation for the laser beam. If we combine obtain ~ q(z) =

Writing nator,

Z -

.1rW~

1-

A

(22-21) in reciprocal form, we obtain, after rationalizing I q(z)

(22-21) denomi-

z -;-----;-:+

in (22-22) with their counterparts in Equating the real and imaginary (22-16), the equation l/q(z), we obtain expressions for R and w(z), in terms of the spot size Wo at the beam waist and the distance z the beam waist to the transverse plane in question. That is, (22-23) and (22-24) Equation specifies the radius of curvature of the laser-beam wavefront as a function of distance z from the reference plane, that the beam at z = O. Note that, if the spherical wavefronts were concentric about z = 0, then equality R(z) z would be true. indicates that is not the case; the all wavefronts. However, center for wavefronts is not fixed at z = 0 when the distance z ~ 1rwUA, then R(z) ;;;; z, and we do have nearly spherical wavefronts concentric about z O. The specified by

is known as the far .field. 13 It is in this that one can treat the diffraction from as Frnunhofer diffraction. In the limit of Eq. the beam waist by' and (22-24) take on the far-field forms

R(z)

z

(22-26)

A --z 1rWo

(22-27)

and

w(z)

12 For our wave is signified by a positive R. wbereas a converging wave, toward some point, is by a negative R. J] In practice, the "strength" of the inequality is understood to be 20 10 50 times the value TrwUA. In reality, there is always a "gray area" between the near field (where Fresnel diffraction theory for wavefront calculations is required) and (where Fraunhofer diffraction is acceptable). When in doubt, one can always perform the accurate Fresnel calculations and compare with the approximate Fraunhofer results.

Sec, 22-5

Spot Size and Radius of Curvature of a Gaussian Beam

465

Thus the or detector is said to be in the fur field relative to the laser, when the distance from beam waist to target is much greater than TrwUA. In the fur field, it is clear from Eq. (22-27) that w(z), the size, grows linearly with z. Since the halfangle spread 8, specified in Figure 22-7, is small in most applications (a tan 8 8. Thus the (FF), half-angle l4 beam diverradian or less), gence for the laser is defined by (22-28) 17'Wo

In Figure the implications of Eqs. (22-26) and (22-27) are evident. At large distances the the has a spherical shape the larger the beam center of curvature at the waist. Also, the smaller the waist Wo, divergence, or spread, for the beam propagating away from the beam waist. precisely as dictated by Eq.

, I

,

Z

T

I I I I

Figure 22-8 Two laser beams wilh a beam waist at:z = O. The smaller the beam waist Wo. tlie the beam 8.

I I

Z=

0

With

(22-23) and (22-24), we have expressions for the radius curvaw(z), in terms of Wo, the size at the beam transverse plane. where R and z, the distance from the beam waist to and w(z) are to be found. Conversely, given w(z) and R(z) for a particular transverse plane, we can locate the beam waist and its spot Wo as follows. Consider Figure 22-9, where WI and RJ in the transverse plane at Zl are both known. ExpressR(z), and the beam spot

Qo=-

22-9 TIie basic law of propagation for IIie complex radius of curvature ii. Knowing w" RI al z" one can determine ZI and woo Or, conversely, knowing Wo and the beam waist location, one can calculate all olher ii. 4 f Note carefully Ihat in 21 we denoted the/ull-angle beam cI,vI'rppnN' with the letter c/J. Here we denote the halfangle beam divergence with the leller 6.

Chap. 22

Characteristics of Laser Beams

ing

ql

in tenns of RI and we can write

WI,

with the aid of

. (22-16), and making use of Eq.

(22~21).

=

ZI

.1TW5 I--

Then, if we rationalize left side appropriate1y and nary parts in Eq. (22-29), we obtain ZI

1

(22~29)

A

real parts and imagi ~

ARI)2 + (-

(22-30)

1TW~

and Wo

Therefore, with WI and RI known, the location of the waist and its detennined from Eqs. and (22-31). In addition, with Eqs. one can determine R(z) and w(z} for any other transverse plane ,"Pi:llIf"!P beam waist. Example Let us illustrate the use of several equations we have derived. 4-mW, TEMoo helium-neon (He-Ne) laser (A = 632.8 nm) with mensions given below. Jeft mirror (R I = 2 m) is 100% right mirror (R2 ~ (0) is a partially plane mirror through which the 4-mW output beam passes. beam waist in the laser cavity (L = I occurs at the where we have chosen the reference plane Z = O. dashed profiles the in the For stable laser oscillation, the curvatures the wavefronts match the curvatures of the mirrors at the mirror surfaces.

(a) (b) (c)

Wo at beam waist. the the spot size W on the rear laser mirror. the cornph!x radius of curvature q(z) at z = -I m and

z = O. (d) Detennine the location (z = 0). (e) What is the field?

Sec. 22-5

Ztl'

of the far field from the beam waist vergellce On for this laser in the fur

Spot Size and Radius of Curvature of a Gaussian Beam

467

Solution (a) Use Eq. (22-23), with R(z) = -2 m, z = 632.8 nm, and solve for woo Substituting directly,

1 m, and A =

-2= 1[1 + Solving for Wo gives 4.49 x 10-4 m (22-24) with Wo (b) Use 632.8 nm:

W(Z)2 = (4.49

0.45 mm. 4.49 X 10- 4 m, z

x 1O-4) { 1 +

=

1 m, and A

=

632.8 X 10-9 x 14 x (4.49 x

giving w(z) = 6.35 X 10-4 m == 0.64 mm. Thus the spot size increases from a radius w = 0.45 mm at the waist to w = 0.64 mm at the rear mirror. (c) Using Eq. (22-21), Atz

-1 m:

ij(z)

z

0:

ij(z)

0

Atz

i7TW~

= -i

(d) Use with 4.49 x 10-4 m and A = 632.8 Z/'f

~

-1 - i

-1-

A

Wo

X

and A known: 10-9 m. Then

3.14 x (4.49 x 10- 4)2 632.8 X 10-9

or

Zf<1o'

ZtF

~ TrwUA, where

~ 1.0 m

Applying the that ~ means 20 to 50 times the far field for this laser is 20 to 50 m from the Fraunhofer diffraction calculations are valid. (e) Use (22-28):

A

-:----:--:------;:- =

4.49

X

Wo

10-4 fad

we conclude that

== 0.45 mrad

22-6 LASER PROPAGATION THROUGH ARBITRARY OPTICAL SYSTEMS With the basic Jaw of propagation the complex radius of curvature ij in Eq. (2220) and the defining equations for the real radius of curvature R(z) and beam width w(z) in Eqs. (22-23) and (22-24), we are able to characterize the beam parameters for laser propagation in any homogeneous medium of refractive index n. We now wish to the question of how the beam when it is modified by an arbitrary one that contains mirrors, prisms, and so forth. It is helpful to note the similarity between the behavior of ordinary spherical waves encountered in geometrical optics and Gaussian spherical waves encountered here. In Figure 22-10, the basic law of propagation for each type of wave and the effect of a lens on reshaping the propagating wavefront is illustrated. One is struck immediately with the correspondence between R(z) for ordinary waves and ij(z) for Gaussian spherical waves in the equations. For example, the law of propagation of ordinary spherical waves is by Rz

Chap. 22

= R, + (Z2

Characteristics of laser Beams

Zl)

Ordinary spherical waves

Gaussian spherical waves

Simple propagation

22-10 Corre!;poilldence between ordinary spherical waves and Gaussian SiJh,encal waves. If one knows the law for spherical waves, one can infer a similar law for Gaussian spherical waves by simply rev'''''';'''/!, R(z) with ij(z) as sbown above.

Simple lens effect

as we all know. Then, in accordance with a result already derived in Eq. (22-20), the basic law of propagation the laser beam is given by

= ql + (Z2 -

(22-33)

an equation to except that R been by q. U"","''''',", the initiative, we extend this correspondence I!> to other basic laws, for '-""""1..0'''', simple lens law. For ordinary spherical waves, a common form of the law is R2

RI

1 /

(22-34)

where RI and R2 are the radii of curvature at incidence and refraction, respectively, and/is the focal length Figure 22-10). If q is to replace R for the laser beam, we obtain at once 1 1 I q2

This relationship predicts refraction by a simple thin

I,,,

=

the

ql

/

"'''','UP''UF,

of the incident laser beam after

General Lew of Leser Propagation. With the correspondence between R(z) and q(z) established, we can, with the help of the matrix methods outlined in Chapter a powerful, for laser-beam propagation through an arbitrary optical 22-11. An incident ray with (YI, al) is incident on the entrance of an arbitrary optical system described by the overall system matrix,

Upon from the the same ray has (Y2, a2). radius of curvature R of an ordinary spherical wave can be related to its appropriate paraxIS The

Sec. 22-6

correspondence we refer to here can be shown to be rigorous, not just analogous.

laser Propagation through Arbitrary Optical

SV'Q"''''rn",

469

Figure 22-11 Propagation of ordinary cal waves an arbitrary oplical system via matrix formulation.

ial ray parameters y and a simply by (22-36)

R

We know from optics that ray I is changed into ray 2 by an optical system, the change can be reJ:)re~;enlle<1 by the ABCD system matrix as foHows: (22-37) Then it must be true that Y2

=:

+

and

a2

= CYI

+ Da,

(22-38)

Dividing the first equation by the second. and using Eq. (22-36), we obtain

+B

R2 = CR + D I

(22-39)

the basic result in Eq. (22-39) to a Gaussian spherical wave, by simply with ij(z), we obtain at once

+B +D

ABCD propagation law

(22-40)

The relationship between ij2 and ijl in Eq. (22-40) enables one to determine the of the laser beam after it passes through an optical system. One need only know the beam (ijl) as it enters the and the the optical system. Eq. (22-40) is a overall 2 x 2 matrix useful and powerful result. It is often referred to as the ABCD propagation law for example, Figure 22-12 illustrates a typical laser spherical Gaussian laser beams. to which the ABCD propagation law can be applied. Mirror surface

Mirror-lens combination

Figure 22-12 Geometry for a He-Ne laser system. With the parameters for the laser system one can use the ABeD propagation law 10 delermine the location f and size of the beam waist wo( f) outside of the laser.

470

Chap. 22

Characteristics of Laser Beams

The problem depicted in 22-12 is to determine the location l of the external beam waist and its size Wo. Clearly, Eq. can be used if one knows a value for iit somewhere in cavity, say at the mirror (plane '6), and the ABCD matrix for the optical system that must, consequently, extend from the left mirror though the mirror-lens combination and on to the transverse plane containing the external beam waist. The value of ijz derived with the aid of 40) then pinpoints both l and Wo for the externally focused beam waist. The outline of the solution is given in the that follows.

Example For the He-Ne laser geometry given in Figure 22-12, use the ABCD propagation law to determine the size Wo of the external beam waist and its lance from the outer surface of the mirror-lens combination. Determine the complex radius of curvature iit at the plane mirror in the cavity. Then develop an ABCD matrix for to the external beam waist. Determine the cOlmplex radius of curvature at external beam waist by the ABCD propagation law:

Cijl

+B +D

Then, from determine the spot size Wo and location of the beam waist. The steps that follow are in outline form. Details wiJI be assigned as problems. (8) Determine an expression for ijl. (b) Determine the ABCD matrix for the optical from the plane mirror to external beam waist. (c) Use the ABCD propagation law to relate ij2(l) to ijl. Solution

(a)

plane mirror, ----j>

iA or q- I 1Twi

--

A

Relate the wavefront curvature at Rz--where the wavefront curvature matches the mirror curvature-to the spot size WI at the plane mirror and the distance z from plane to curved mirror R2 • Eq. (22-23) to write

where A 0.633 x 10-6 m, Z2 = 0.7 m, and R2 = 2.0 m. (Notice that sign conventions for laser cavity mirrors a',sign R positive for mirrors concave toward the cavity, R negative otherwise.) Solve for WI to get WI 4.38 X 10- 4 m. ijl = -i1Twi/A -0.952i. (b) From the plane mirror to the external beam waist, the system matrix is

16 As already alluded to and used in the example, it can be shown that the wavefront curvatures of the beam at the mirrors match the curvature of the mirror surfuces.

Sec. 22-6

laser Propagation through Arbitrary Optical

S"."t",rnc>

471

Carrying out the indicated matrix multiplication, one gets:

A B] [C D

=

[I - 0.53e 0.7 + 0.63e] -0.53 0.63

(c) Substitute into the propagation law given above, using the results of parts (a) and (b): A

I - 0.53e

ql and Eq.

B = 0.7

+ 0.63f

C

= -0.53

D = 0.63

-0.952i

16),

e

Equating real and imaginary parts, the solution gives = 0.06 m and W2(€) = 0.54 mm. Thus the external beam waist has a size W2(€) of 0.54 mm, located 6 cm from the R3 surface of the mirror-lens combination. Collimation of a Gaussian Beam. A laser beam is a Gaussian beam with a long waist, as shown in Figure 22-13. The collimated beam length is arbitrarily defined as the distance between two symmetrical, transverse planes on either side of the beam waist, the two planes being those in which the spot size W (z) has increased by a factor of V2 over the spot size Wo at the waist.

Figure 22-13

A collimated laser beam and the

cor:res~lon{ling

Rayleigh range ZR.

In Figure 22-13, the beam leaving the laser passes through a Galilean-type beam expander and forms a beam waist Wo some distance to the right of the convergis defined as the distance from the beam waist Wo to ing lens. The Rayleigh mnge the transverse plane where w(z) has the value V2wo, or, equivalently, where the w{z) = beam area has doubled. The Rayleigh range can be obtained by V2wo in and solving for Z thus w{z) = V2wo

AZ )2]111 = Wo [ I + ( 1T:ij

(22-41)

from which it follows that 1TW~ A

(Rayleigh range)

Thus the collimated region extends over two Rayleigh ranges, one on either side of beam waist. As shown earlier in Figure 22-8, the smaller the spot size Wo of the beam at the waist, more rapidly the beam diverges as it leaves the waist and so the smaller the Rayleigh range, or distance over which the beam remains with a nearly constant diameter and planar wavefront. 412

Chap. 22

Characteristics of Laser Beams

Selecting Beam-Shaping Optics for Optimum Beam Propagation.

Gaussian beams are often passed through apertures such as mirrors, lenses, beam expanders, and telescopes. If such "apertures" are not to restrict the we need to develop a criterion for size amount of laser energy passing of the aperture relative to the spot of the beam. To find fraction incident power that passes through a circular of a, we return to the TEMoo Gaussian beam represendiaphragm, tation, (22-18), for the electric field across a plane transverse to the beam: E(x, y, t, t) =

(22-18)

Since the beam irradiance is proportional to the of the field tude, the total power fIl lol in the beam is obtained by evaluating the integral

fil,,,,

II

1E(x, y, z, t) 12 dA

II

EMz)

_-21X"+v")/w"lzl

dA

A

A

where the integration is over the entire cross-sectional area of the beam. When this is done, one obtains for the total beam power 7r[w(z)j2

2 Here with

is the electric field amplitude at beam center for a transverse plane at z size w. Eo(z) =

w(z)

Making use of the expression in Eq. (22-18) and calculating the fractional power fIl rrac = fIl(r = a)/fIlIOI passing through a circular aperture of radius a, one sets up the integration as follows:

y, z, t) 12 dA

and obtains (22-43) Figure 22-14 shows the fractional power fIl(r = a)/fIl,ol that. passes through a circular aperture radius a, versus the ratio a/ w, where w is the size of the beam at the aperture location. Notice that 86% of the beam gets through (14% blocked) when the radius equals the size, just under 99% gets through the radius is increased to 1.5 times the spot Thus, if each beam-shaping element that passes laser beam in a optisystem has a diameter equal to 3 times the beam size (d 3w), nearly 99% of the beam through. Even though nearly all the power gets at d we should note that diffraction effects caused circular apertures produce effects on irradiance patterns in the near field and a reduction of on-axis irradiance in the fur field of about 17% [1 J. To the diffraction one enlarges the aperture so that d == 4.5w. Beam-shaping optics with diameters that satthe d == 4.5w criterion transmit essentially 100% of the beam power without additional diffraction effects on the beam. With Eq. for the Rayleigh range and the d = 4.5w (99% transmission), one can calculate the Rayleigh range for typical Sec. 22-6

Laser Propagation through Arbitrary Optical Systems

473

~

____?w,

I

I

f2-t

I

I I

I I I

,

Aperture geometry

8

I

t

I I I I

I I

50% 8=

1.5 w

Normalized aperture radius

alw

Figure 22-14 Fractional power transmission of a Gaussian beam of spot size w through a circular aperture of radius a.

lasers as a function of aperture diameter. Results are shown in Figure 22-1S for a He-Ne, HF (hydrogen fluoride), and CO2 laser on a'log-log plot. For if the aperture diameter 4.Sw = 1 cm, the collimated beam length 2ZR is equaJ to 24.S m for He-Ne light at A 632.8 nm and 1.46 m for CO2 light at 10.6 #Lm. For an aperture diameter 4.Sw 2 cm, the collimated beam length becomes 98 m for the HeNe laser and S. 8 m for the CO2 laser, as the square of the aperture diameter. HF(3pml

200

100m 80

~ £:! OJ

50

01 C

f!

1:1

OJ

m

,§ '0 u

20

10m 8 5

0.2 0.1 em

0.5

0.8 lem

B 10em Aperture diameter Cd)

2

5

20

50

Collimated range 2ZR versus aperture diameter d, for a laser beam a waist woo assuming that the aperture diameter d 4.Sw, where

414

Chap. 22

Characteristics of laser Beams

Focusing a Gaussian Beam. Let us now address the problem of focusa Gaussian laser beam with a lens of focal length f. We take a rather general case, that of a beam with waist WOI located a distance ZI to the left of the lens, incident upon a positive thin lens and to a with waist W02 located a distance Z2 to the right of the lens. The pertinent is shown in Figure 22-16. Plane at

Plane at Z2

Zl

Figure 22-16 A Gaussian beam of waist Wm and half-angle beam divergence (JI is focused by a thin, lens of focal length f. The beam is focused to a beam waist Wo2 a distance 22 away.

The problem before us is as follows: Given the beam waist %1 and distance Z, on the incident side of the determine the beam waist Win and the distance Z2 on the image side of the lens. We shall use the ABeD propagation law, Eq. (22-40), which involves the ABeD matrix from transverse to transverse plane matrix is

f

ZI

+ Z2

I

(22-44)

f Then the ABeD propagation law becomes

Making use of Eq.

which gives us atZI

and

ib =

and substituting into (22-45), we obtain a rather complicated equation that contains the desired unknowns, Zz and Win, After real and imaginary parts, we sort out the unknowns W02 and Z2 in the form:

-+(1 ~ f

ZI)2

+l

WOI

We note from case, "" f. Apparently the Gaussian beam is not necessarily to a in the focal plane of the lens, that is, at a distance fto the right of the lens. We can, make some practical (22-46) and considerably. First, consider If WOI ~ wm-l:nat a strong positive lens is used in the focusing the first term in 46) is small in comparison with the second term and the focused beam Sec. 22-6

Laser

rUJlJi:llQ(;IlLlOI

through Arbitrary Optical Systems

415

given approximately by (22-48)

=

for the half-angle beam diver(h == A/7T Wm is the defining gence Oa. (22-47), if we have a physical system for which Next, in f)2-not an uncommon situation l7-then it follows that (227TW~I/ A > (ZI 47) reduces to (22-49) =f (22-47), in any case, reduces to (22-49) if the beam Wm is located in the focal of the lens = f). If we assume that WOI is approximately equal to the of the lens, then by multiplying the numerator and of (22-48) by 2 and using the f / d for the fnumber of the lens, where d is the lens diameter, we definitionf # obtain (22-50)

W02

In general, then, the smaller the f # of the ,,-,,-,,u""lolens, the smaller the beam waist at the focused 22-7 HIGHER-ORDER GAUSSIAN BEAMS

solution for spherical Gaussian laser beam, derived earlier as Eq. (22-18), represents the lowest order-that is, fundamental-transverse electromagnetic mode that exists in the open-sidewaH laser cavity. modes-exist that do not have a pure profile for their irradiance variation in the transverse plane. Let us return to our earlier development in this chapter and generalize (22-12), which we first guessed to have the form, [; (x, y, z) Eo e;(P(z)+lk(x2 +y2)1I2ij(zlJ 12) as (22-51) functions g(x/w} and h(y/w) admit a more complicated variation of the beam irradiance with x and y. Substitution of (22-51) into (22-11) .. the defining equation for leads to w

2

+

2ikx I W

-

lOW) g , W

-

iJz

h' h

1 h"

-+--+ 2 w

h

2

o 3

4

11 In many cases, laser beams are first in cross section before being focused to a small spot, as in the case of laser welding. In these instances, the beam waist Wm may be of the order of a centimeter or more, ensuring thai the inequality 7TW~,/" ~ (z, - 1)2 is satisfied. (See, for example, 9 at the end of the chapter.)

416

Chap. 22

Characteristics of Laser Beams

denote differentiation with respect to the of g and h. 11S[lectlon of that for a given z, that is, for a transfirst bracketed series of tenns, the g-expression, is a function of x the is a function of y alone; third is mdep(:nd,ent of both x and y; and is identically zero, since ij + z is the propagation law that we hold to be valid. Then Eq. (22-52) can be satisfied arbitrary z only if (I) the a s a y , (2) a constant, say thereby (3) leaving the third tenn equal to Al + il 2 • ,I?-f;XDlreS!SlOll, we write _1 g"

+ 2ikX(~ _ ~ dW)!L

g

q

W

W

dZ

= -AI

(22-53)

g

We can show, with the help of Eqs. (22-16) and by I iA

ses is

dW

ij

tain

term in

lUl""" . .'L";;.)'

W

ilz

'/TW

2

and a change in variable, ~ = the Hermite differential equation,

V2x/ W in

(22-53), we ob-

2

'H: dq A,w ~-+--q

il~

This

2

o

is known to have a solution only if

AI 2

= 2m, wherem = 0, 1,2, ...

The solutions to Eq. (22-55) are the well-known Hermite

Hm(~)

(22-57)

can be obtained for each appropriate m from the generating

Hm(~} Solution of

(22-56)

nnJ:vn,C)ml!nt.~

Hm(~) = Hm(V;X)

g({;) = where the function

(22-55)

=

(_l}me~2 ~:I

(22-58)

(22-58) for m = 0, I, 2, ... gives

m m

=

0:

Ho(~)

I:

HI(~) = 2~

I (22-59) W

2 the second condition and for the manner identical to that for the g-expression, we Vzy/w, able TJ 2 d h 2 ilh ilTJ2

TJ dTJ

+

2 "''',,''''U," in Eq. (22-52), in a

using the change of van-

o

(22-60)

for which the solutions are again the Hermite polynomials: where n = 0, I, Sec. 22-7

Higher-Order Gaussian Beams

(22-61) 411

obtained for all n from the generating expression, Eq. Before effects that the Hermite hu"-hn,,, .. g(t) and h(l1) have consequences of on the transverse nature of the beam irradiance. let us the third condition on Eq. (22~52), iJp

ij

AI

2/(- =

ilz

+ A~•

(22-62)

0, which is necessarily where AI = 4m/w 2 and true when m n 0 then g(t) = h(l1) 1 and we are back to (22~62) reduces to the pure Gaussian spherical wave solution. In this case, = i/ij, the equation for p obtained the general are not equal to zero, case, for which AI and must remain the defining equation for the function p(z). Then we must solve A

ilz to ohtain an ..v ....."',""••," ij (z) and then suIting equation to ohtain

Substitution of Eq. (22-16) into (22-63) for we can int'egrate the refor R (z) and

i p(z) = -In 2 With the help of

(22-63)

ij(z)

- (m

+

n

+

(22~64)

I) arctan

we can form the term

as

e -ilm tn+ I )arc,."(Az/m"~)

(22-65)

w(z)

where we have used Eq.

for w(z) to simplify

"rhe Hermite-Gaussian Beam Solutions. for p. ij. g(t). and h(l1), and putting them aU together in the the electric field E(x, y. z, t) as initially introduced in E(x, y,

Z,

of the eX[lression. our results expression for we obtain finally,

t) phase

(22-66)

Tbis expression for y, z, t) includes the higher~order modes as well as the Gaussian spherical wave (when m n 0) derived earlier. first of terms describes the amplitude variation of the electric field at any transverse plane z. The and Hn Hermite polynomial terms permit a finer-tuned variation in the transverse electric field than that described by the pure Gaussian term

The variations in transverse deviate more and more from the pure Gaussian with higher integers, m and n. These are, of course, of the modes. The second set of terms, a colTIPllicated phase, describes the nature of the wavefront as a function of m and n. we see that for m = n 0, the phase term reduces to the form characteristic the pure Gaussian spherical wave derived earlier. It can be shown that the associated with an arbitrary beam mode is given by

478

Chap. 22

Characteristics of Laser Beams

= [q

jmnq

+

m+n+l( ~

arctan

ZR

arctan

ZI)] 2L C

ZR

where q is the axial mode number (not the complex radius of curvature); m and n are nTP'OP"" associated with the Hermite polynomials Hm and Hl1 ; ZI and Zz are the distances from the beam waist inside the laser to mirrors M, and M2 , respecL is length between and is the Rayleigh range. J',"'1I'0."" expression for E(x, y, z, t), (22-66), is called the Hermiteto the wave for a laser cavity. Any arbitrary beam can be as a linear combination of Hermite-Gaussian beams, each of which has the same propagation law, ij + z. and Irradiance Patterns for Hermite-Gaussian Beams. With the help of Eq. (22-66) we can sketch the transverse electric field and irradiance variation for several of the lower-order modes-small m and n-and predict the nature of the "burn" pattern. Eq. (22-66) by setting w(z) for Without loss of generality, we can some transverse plane z = Zo to V2 units of This makes the arguments for ....... '·11111' .. polynomials simply x and y. In addition, recognizing that the (22-66) do not contribute to the we can write rather nrr:,n'-}Irtii'1rnnl expressions for E(x, y, and 1 y, 20) 12 as follows: E(x, y,

20)

(22-68)

ex: Hm(x)H II (

IE(x, y, 20) 12 ex: [Hm(x)]2 [Hn(

(22-69)

of the patterns IE(x, y, Zo) 12 for the 22- shows the x-variation in the transverse plane at Z zoo From symmetry, it is clear that identical would be obtained for the for two cases, m = 0 and m l, 22-17 shows Hm(x) exp (-x 2 /2), X-variation of electric field

Hermite polynomial

,

,

H(x)

m 0

Burn

X-variation of irradiance

I

,,, , --.-,,, , ,

Ho(x) = 1

I

I

,, ,,

-"'----t---"'---+- x :

TEMoo

H(x)

m

---r-.,

,

1

H 1 (x) =2x

:

---+--..... x:, ,, ,

,

Figure 22-17 Laser-beam electric field and irradiam;e variations in the x-direction for two values of the Hermite integer m. Corresponding burn patterns for m = 0, n = 0 and m I, n I are shown.

Sec. 22-7

Higher-Order Gaussian Beams

419

finally, [H",(x) exp (-x 2 /2)J2, the sketch tern to be 18 After studying along the the zeros at Thus, for m = I, there is a single zero x type of variation is to be expected for the direction. Putting the x- and (22-69) and its corresponding beam irradiance in the transverse plane. ter, these are commonly referred to have the Gaussian spherical beam with When m 0 and n 0, we have the patterns for low values of the Hermite uUpo;pr;;: 69) and 22-17-are reproduced

'*

'*

evidence of the burn trraOIHU1lce variation has m zeros of the Gaussian envelope. on either side. The same exp (-y2/2W in the ym, n pair, we see that determine the laserthe beginning of this chapWhen m n 0, we variation in irradiance. Some of the irradiance n--pn~atc:talJ!le from (22-

22-18 different orders of Hermite·Gaussian lographed in the output beam of a laser UMill!tlor

Beginning with the £''l
22-1.

_ ...... -'_.J

burn pattern for several as they might be

for the geometry shown in the propagation where ~TW'''mw;*"

r

(b) Then use E(x, ybR above to obtain

IS To ensure understanding of the sketch problem 22 -17, to compute the next row. for In

22

Characteristics of Laser Beams

22-17, the reader is invited, in

22-2. Show that substitution of E(r, t) = U(r)ei(h-wf) in the wave leads to Eq. (22-10). 22-3. A TEMw He-Ne laser (A = 632.8 nm) has a beam waist Wo (al z = 0) of 0.5 mm and a half-angle beam divergence of (J 0.4 mrad. At a distance z 50 m from the beam waist, detennine (8) the value of the complex radius of curvature i/o at the beam waist; (b) a numerical expression for the complex radius of curvature q, in meters, from each of the following expressions:

I q

I +.

--;; = -R

A

l--------r-() 11"W Z

an

dq

(Hint: Is the transverse plane. 50 m from the beam the far field? If so, what doe.:; this say about R and z?)

22-4. (a) In problem 3, use

(22-23) and

far enough away to be in

to determine R(z) and

W

at

z = 50 m.

(b) Is it true thai R(z) == z in the far field? Is it true that tan (J == w(z)/z, where (J is the beam divergence, can be used as I:l good approximation to determine at z = 50 m? 22-5. A TEMou He-Ne laser (A = 0.6328 /Lm) has a that is 0.34 m long, a fully reflecting mirror of radius R = 10 m (concave inward), and an output mirror of radius R = 10 m (also concave inward). and the boundary condition that (a) From the symmetry of mirror wavefront and mirror curvatures match at the mirrors. detennine the location of the beam waist in the Set z 0 at this location to be the reference plane. (b) Determine the beam waist woo (c) Determine the beam spot size w(z) at the left and right cavity mirrors. (d) Determine the beam divergence (J for this laser. (e) Where is the far field for this laser if you use the criterion Zn ~ (f) If the laser emits a constant beam of power 5 mW, what is the average where ZFF = 50(11"wUA)? irradiance at the 22-6. Relative to Figure 22set up the matrix for the output element of the laser, that the mirror-lens combination with thickness 0.004 m, mirror surface curvature of R21 2 m, lens surface curvature of IR31 = 0.64 m, and lens of 1.50. refmctive (a) Using the definitions given for the refraction and translation matrices in ...."'a."'v. 4, set up the ABeD matrix for this element as follows:

Pay particular attention to the changing of nand n for the two refractions and to the conventions for R2 and R, in the matrix formulations. Within rounding approximations, you should find I

A [C

BJ D

[1.0()07 =

-0.5318

0.0027J 0.9979

(b) Since L = 0.004 m is a very small dimension with IR21 2 m or IR31 ::;; 0.64 m, the ABCD calculation, '''IJ'.d\,'''l; the translation matrix with the unit matrix

What then is the result of the ABeD matrix for this "thin lens"? Chap. 22

Problems

481

22·7. (8) Since the output element described in problem 6 is essentially a thin lens. compare the ABCD matrix obtained in problem 6(b) with the thin-lens

and deduce the focal length of the output element. (b) Use the expression for the focal length of a thin lens, I

1 with careful attention to thin-lens and obtain the focal length of the thin-lens output element. How do the results for parts (a) and (b) compare? designed around Figure (a) determine an cxJlfe!.siolfl 22-8. Relative to the for ib at the plane mirror; (b) solve 22-23 for the spot size value WI; (c) obtain a numerical value for 'b; (d) multiply ib by the ABCD matrix to obtain ih; (e) use Eq. (22-16) and from part (d) to obtain € and w(€). 22·9. (8) Relative to the calculations in the example (refer to 22-12) in which the external beam waist is focused at € == 0.06 m with a waist size w(€) = 0.54 mm, use (22-46) and (22-47) with 22-16 to obtain values for W02 and %2. How do these results compare with those for wo(€) and t obtained in the example? one cannot use the approximations (b) Explain

18

and

Z2

== f

in this instance. 22·10. Refer to the externally focused laser beam shown in 22-12, with beam waist wo(€) = 0.54 mm, located at 0.06 m from the output element. (a) Calculate the far-field distance for the focused beam waist. (b) Calculate the far-field beam divergence for the laser beam that emerges from the focused beam waist. (c) insert a lOx beam expander in the beam at a distance z = 30 m past the focused beam waist. Calculate the beam spot size w at the entrance and exit faces of the heam expander. (d) Now place a thin lens of focal length 10 cm and appropriate diameter at a dislance of 20 cm from the output face of the beam With reference to Figure 22-16 and Eqs. (22-46) and (22-47), calculate %2 and Woo fOr the newly foIA/rrwOl and cused beam. Could you have used the approximate formulas "'b2 Z2 f in this instance? Why? How do the calculations for the exact formulas and approximate formulas compare? 22·n. Given the PYrw""~i,,,n 4>(r == a)

= _1 4>'0'

III

y, z, t) 12 dA

Aper ute

and (22-18) for y, %, t), carry out the indicated integration over a circular aperture of radius r a to show that the fractional power of a laser beam through the aperture is by

Chap. 22

Characteristics of Laser Beams

22·12. Carry out the integration for

II

dA

Aperture

over the entire cross section of a TEMoo beam and show thai the total power the beam is equal 10

in

4>'01 == 22-13. 22-14.

22-15. 22-16. 22·17.

[I]

how you can use an adjustable circular aperlure (iris) and a power meter to determine the spot size W of a TEMoo laser beam at any position the beam. Determine col.limated beam al!. where ZR is the Rayleigh range, for a TEMoo Nd:YAG laser beam (A = 1.064 #km) focused by lenses of aperture diameters d = I cm, 2 em, 3 em, and 5 em, respectively. Assume that the lens diameter d is related d 4.5(Yzwo). Refer to Figure 22-15 to the focused beam waist WI) by the for geometry and similar calculations made for He-Ne, HF, and COz TEMoo lasers. Given the generating function, (22-58), for Hermite polynomials Hm(~), where ~ = V2x/w, verify the parlicular cases for m = 0, 1,2, . . . in Eq. Fill in the steps to show how (22-65) follows from Refer to Figure 22-17. Extend the "table" to include the case m 2, n = O. Thus, in a third row, sketch in curves for column I, Hm(x), column 2 for the x-variation curves of the electric column 3 for the x-variation curves of the irradiance, and column 4 for the bum pattern.

SIPpml'ln

Anthony E. Lasers. Mill Valley, Calif.: University Science Books. 1986. Ch.

16, 17. B. E. and M. C. Teich. Fundamentals of Photonics. New York: John and Sons, 1991. Ch. 3,9. Guenther, Robert. Modern Optics. New York: John Wiley and Sons, 1990. Ch. 9. [4] Moller, K. D. Optics. Mill Calif.: Science 1988. Ch. 15. Gerrard, A., and J. M. Burch. Introductioo to Matrix Methods in Optics. New York: John Wiley and Sons, 1975. [6] J. T. Laser Electronics. Cliffs, N.J.: 1981.

[2]

Chap. 22

References

483

23 laser

Laser Applications

When one endeavors to chronicle current in a rapidly ev()lvlnl! as laser technology, one accepts the fuet that such outdated almost as rapidly as it is written. For presented in vv...... lU\./v"'''. obsolescence is distressing. I it seems worthwhile to describe applications of as exist today. Such applications some of the both illustrate the of the remarkable properties of laser light and serve as a benchmark of laser technology in the mid-1980s and early 1990.'1.

INTRODUCTION

The variety of ways in which lasers can be seems to be Hmited only by human creativity and ingenuity. In a few short oeca
484

It of course, the unique properties of the laser-monochromaticity, directionality, coherency, and brightness-that account for its wide acceptance and usefulness. In each application, one or more these properties is exploited to achieve a 0) goaL Most applications can be organized into two general lasers and interactions and (2) lasers and information, outlined in 23-1. In the former, lasers interact with matter and cause desirable changes, either permanent or detect, store, and process informatemporary. In the latter, lasers are used to tion. TABLE 23-1 A CLASSIFICATION OF LASER APPLICATION S

Lasers and interactions

Lasers and infonnation

Materials proces!.inJ!.

Communication Information nrnc"~'''n'' Optical sensing Oplical Optical storage Printers and copiers Metrology Holography Alignment Poinl-of-sale scanners Entertainment and display

Welding Drilling Heat treating Medical Marking and Microfabrication Laser spectroscopy Laser-driven energy

The broad class of that involve the interaction of lasers with matter includes the industrial machining and of therapeutic and surgical uses in medicine, laser-drivcn energy sources, scribing, and microfubrication in semiconductor and computer technologies. The broad class that inincludes information optical and lasers and ranging, optical communication, entertainment, and copiers, metrology, and alignment facilitators in construction and agriculture. In this chapter we examine several major areas of laser applications in each of the two main divisions.

23-'1 LASERS AND INTERACTION

The laser beam with its density, excellent directionality, monochromaof coherence interacts with different media in many ticity, and ferent ways. The ability of laser light to cause changes in the matter it strike.. makes it ideal for many laser applications, especially those in the fields of material proce..smedicine, and laser-beam fusion. In the following we some of the impressive achievements made with lasers in these areas.

Material Processing. In the industrial processing of materials, lasers are used to cut, weld, drill, perforate, and heat-treat a of substance... laser is an ideal source for these tasks because it delivers appropriate thermal manipulated, the laser energy in a tightly focused beam. Flexible and can be optimally shaped to direct intense laser power, accurately and repeatedly, on almost any target or work surface. When we consider that a 500-kW laser of I-mrad beam divergence can be focused to power densitie<>; of 10 12 W/cm 2 on one appreciates immediately the attractiveness of lasers in material proc-

Sec. 23-1

lasers and Interaction

485

The gaseous carbon dioxide (C0 2 ) emitting at 10.6 pm, and the solidstate neodymium (Nd:YAG) laser, emitting at 1.06 pm, are two lasers used to alter materials and surfaces. The three most popular applications of lasers in material are usually involved with metals in cutting, and heat treating. though, lasers are used to process and T other nonmetal materials. are compatible with automated and computer-aided manufacturing processes, because they can be programmed to move over the surface of a workpiece and perform computer-controlled cutting and operations. The flexible nature of lasers makes them especially suitable for prototype fabrication. An automobile manufacturer, for can a one-of-akind plastic dashboard on a computer and later, with the help of the computer. direct a laser accurately to cut out the desired Thousands of with output powers extending into the kilowatt levels, are in materials-processing applications. Lasers are used to perform .....&."",."... used welding on a wide range of small components, such as hermetically sealed plprtrnn'lr components. are used to weld automobile transmission parts to increase productivity. are used to weld structural assemblies to improve control. They are used in job shops to cut sheet metal accurately and They are rubber, cloth, and wood. also used to make precise, rapid cuts in In the automotive and aerospace industries, noncontact hole drilling by lasers is a critical process. Accurate and holes for of air and liquid flow in hydraulic and pneumatic components are p'tcrltcrl~II\I precisely controlled laser beams. Localized of metal exposed to wear and fatigue is accomplished by laser hardening and cladding processes. Laser is used successfully to machine brittle materials such as fired ("f'r'~rrliN: and silicon wafers. Precise adjustment of microcircuits and networks, a rnaindustrial task, is handled easily with lasers. Low-power lasers are used to of applications. Lasers also provide a convemeasure, and detect flaws in a nient means of marking thousand" of parts for identification and The process of laser cutting is typical of laser machining in industry. Figure 23-1 shows a line drawing of a simple fixed-beam laser system with for a laser cutting application. The assist gas is used to "blow away" the material in the cut while at the same time enhancing the burning and melting proces-". The actual laser system bears little resemblance to the schematic depicted in Figure I. A laser machining center that can cut, drill. and weld parts of any ge-

Laser

J.'igure 23·1 Laser fixed-beam system for cutting operation. (Adapted with permission from Lasers and Applications.)

486

Chap. 23

laser Applications

23-2 A heavy-duly laser machming thai and weldof aimosl any geometry. (Reproduced [JeflmiSl!ion from Lasers and Applications _)

OtT.",,-,! under adverse conditions laser are used to 23-3 shows a typical industrial laser. The plot shows the of the laser during the used to generate the data in through aluminum, at any given the laser in question slices

The housings that accomto operate the laser. performance of a high-power cut as a function of the linear to note that the laser

.."'".....'prl

lIl1'."'''I,.""

n"rrorrn"l1c" data for a industrial used in metal-cut(Reproduced permission 6, No.6, November 1984, a publlc.1ilion of Ihe Laser Inslitute of America.)

1l111i1--DOWl!f

o

50

100

150

Cuttmg speed (mm/sl

Whether lasers are used to perform countless other special in material processing, it is aDlJaf-ent have become inextricably linked with industry and the The of lasers with robots represents one of the innovative of the laser that is now being pioneered. Currently, the dUlY systems that team robot units. In a major automobile plant, CO 2 laser works with an dashboard repeatedly and lUI -.-Il'"'' robot to trim with high quality ....o.d,,'!')1 aplllICait,ons will multiply many times lasers and robots continue lC('I~~'l'lulliV

cut, human tisslle. In to lasers and Interaction

tbat tbe laser beam-a scalpel of coagulate, open, and beal some detail bow lasers are used additions to hospital

487

Lasers in the hands of doctors the medical world. (lIIuslralion by

as and therapeutic typical wavelengths, the five laser types the

::IOlnn.::"..

cal a visible beam aligned. The tunable dye a beam that contains ,,/,n",,"u·.. (400 nm) to the near infrared He-Ne lasers-are used The CO 2 laser, one of the emits (ir) energy at 10.6 I-Lm body cells (issues are 70% to or, if focused more sharply, can and seals as lasers, to is their Slon of the beam from the laser to articulated Unlike the CO2 laser, the laser beam at 488 nm mate well ferred readily from the laser Using flexible that

Chap. 23

laser Applications

more power, emits low in end of the visible region spotter beamsand medical practitioners, absorbed by water. Since make a clean mCISI{Jn

1.06 ",m and the argon-ion laser energy is thus transwithout significant power of laser beam'> and

TABLE 23-2

LASER TYPES IN MEDICINE

Uses

Laser type

Wavelength

Power levels

Excimer Argon fluoride Krypton fluoride

193 nm (ultraviolet; invisible) 248 nm (ultraviolet; invisible>

Pulsed, == 1 1/cm2 with pulsewidlh of about 10 ns (== lOR W /cm2 )

Cutting, incising, ablative photodecomposition (ophthalmology, cardiology. and artruoscopy)

Argon-ion

488 nm (blue-green; visible)

Continuous; up to 10 W

Photocoagulation, welding, vaporization (dermatology, ophthalmology, general surgery)

Tunable dye (pumped with an argon-ion laser)

631 nm (Ted; visible)

Continuous; 3-4 W

Pholoaclivation (treatment of tumOTS)

Nd:YAG

1.06 JLm (infrared; invisible>

Continuous; up 10 maximum of

Photocoagulation, vaporization, perforation (ophthalmology, gastroenterology, dermatology, urology, and tumors)

60-100 W

Carbon dioxide (C02 )

10.6 JLm (infrared; invisible)

Cominuous; up to 80 W

Tissue vaporization, incision, excision (dermatology. gynecology, gastroenterology, and neurosurgery)

fiberoptic scopes (endoscopes), surgeons are able to perform major abdominal surgery, for example, through an incision as small as ] inch. The blue-green argon-ion laser beam is unique in that it is selectively absorbed by red and brown substances, such as red blood cells and melanin pigment spots. It can therefore stop retinal bleeding (photocoagulation) and weld detached retinas. It can fude port-wine birthmarks and tatoos, penetrating the nonpigmented upper layer of skin without harm. The Nd: YAG laser, with fiberoptic scope, has also been used effectively in photocoagulative applications-controlling or stopping the bleeding of ulcers and large tumors or blood vessels deep in the body. The Nd: YAG laser has been used with some success to seal tiny bleeding vessels of the retina, thereby ameliorating a disease (retinopathy) that often leads to blindness. To accomplish this, the Nd:YAG laser is optically focused to a diameter of 1/1000 inch. In one of its more recent dramatic applications, aNd: YAG laser beam has been focused to a 30-ILm spot within the eye. By concentrating powers of over I billion W/cm 2 , pla.'ima breakdown occurs in the vitreous humor. Subsequent acoustic shock waves rupture unwanted, opacified membranes along the vision axis of the eye. This unique example of noninva.'iive pain-free surgery wa.'i treated in more depth in Chapter 7. Quite recently, both the Nd:YAG and the excimer lasers have been tested in an important assault on arteriosclerosis (hardening of the arteries) and critically

Sec. 23-1

lasers and Interaction

blocked blood vessels of the heart. "'''''.,..",,''' fiber-optic scopes, threaded slowly through unwanted fatty deposits and blasted away. This ,,,,'nn,,,, promising, remains in the exway of a summary, Table 23-3 list" medical fields wherein lasers and where they hold the future. A quick glance remany of the human body that come under the purview of these fields mdllC~ttes clearly the extent to which lasers have been or will be involved in the treatment of human ailments, It is apparent that almost every major part of the of laser therapy. body, or small, stands to benefit from some TABLE 23-3 MEDICAL FIELDS INVOLVED WITH LASERS Current fields of Ophthalmology Gynecology Dermatology Cardiology

eyes female replrodlJcti've organs skin heart and blood vessels intestines tumors/cells

cr~noenterology

Oncology Future fields of Otolaryngology Podiatry Urology Dentistry

nerves ears and throat feel urinary tract teeth

laser-Induced Fusion. It has long been the of scientists to harness an source of energy. With the of thermonuclear phenomena and fusion that came in the early seemed a little closer to realization. In the well-known D-T nuclear reaction that fuses two isotopes of hydroI'erl--deute:rmm and tritium-together and releases 14 MeV of kinetic energy per of producreaction in the process, scientists sought to emulate the sun's energy. The fusion reaction, however, high pressures and Major technologies have been to meet the stringent reIlT£',mf',nf!: of producing a confined, high-density at temperatures approachis laser-induced futhose on the surfuce of the sun. One of these is rather simple. In laser fusion, a pellet of fusion fuel, usually a and tritium, is irradiated uniformly over its surface with bighmixture of energy laser circumferentially spaced around the as shown in Figure off (an ablation) 23-5. The of the sunace of the pellet causes a of the outer material of the pellet. The rapid of the surface is accompanied, in rum, by an inward-propagating compressional wave (an implosion) that creates the conditions for fusion, high density and In the interior of the densities reach 10,000 times that of rise to 100,OOO,OOO"c. These conditions, if maintained for about 1 ps, lead to the fusion reaction and the release of enormous amounts of energy per reaction. the laser (10.6 /Lm) and the Nd:YAG/Nd-glass lasers (1.06 /Lm), in combination, are being used to irradiate the D-T pellets. The CO2 laser fusion is under development at the Los Alamos National Laboratory in New technology is at the Lawrence Livermore in California. Laser power output with the NOVA Nd490

23

Laser Applications

L

'~

9 A

A

A

~' A L

A

A

,(f

A

A

L

Figure 23-5 Symmetrically arranged lasers (L) irradiate a tiny deuterium-tritium pellet. Instantaneous heating of the surfuce causes ablation (A). an outward expansion of the surfuce, The reaction, an inward-directed cOl1[1nrl~c;sil:mal wave (C), creates the high density and temperature needed for the fusion reaction.

i ...""r.. <1;lhl .. level of 1 trillion W, making NOVA most powerful in ",n.",'..,t'",n Nevertheless, both experiment and theory show that shorter than I p.m offer significant advantages in the production the wave. recent laser fusion experiments have been performed at of 0.53 p.m and 0,35 p.m. These experiments, in agreement with theoretical predictions, corroborate the efficacy of shorter wavelengths in the ignition and thermonuclear burn process. It is anticipated that short-wavelength lasers-the excimer lasers or the free electron lasers-will play an imrole in laser fusion. If and when laser fusion becomes ec(mclmicall:y J(~S:IOlC;, we will have realized one of our most cherished dreams, a energy.

23-2 LASERS AND INFORMATION

We live in an age of exploding information, Computers provide instant access to worldwide sources information, available from a wealth of 'data banks. In the that scan, midst it lasers are playing a leading role. They are used in and store information. Laser printers and checkout scanners sense, are already on the market. Laser sensors are used to detect pollution, wind "IA""""", and weather patterns. Lasers and fiberoptic cables are chl:mging COlmnlurnCllLasers and hologranls are pointing the way an efficient ".nr"u'f' and of burgeoning amounts of survey of laser applications in information Pf()ce:SSUlg Laser Communications. The capacity of any communication channel is .....""",,,,h,,.... ,,,1 to the width III of its frequency band. Even with monochromatic is very the bandwidth be enormous when 10 is very

a communication system that uses light waves, where high center tre.qut~nciesfo near 10 14 Hz are available, should in principle carry many times the information now carried by radio and microwave at much Sec. 23-2

lasers and Information

491

lower center frequencies. If, in addition, the monochromatic light can be made coall the necessary ingredients communication by light are at hand. When the laser on the scene in the 1960s, a solution to the requirements monochromaticity, coherency, and wide bandwidth became available. Some 30 years later, with laser communication technology firmly in place, the Df<)mISe is well on the way to realization. volumes of information are transmitted over long distances by communication that involve satellites, coaxial cables, and waveguides. Each of these systems involves waves of longer wavelength and lower frequency than does laser light. For example, the frequency in center of the visible spectrum is about 100,000 times than the frequency of 6-cm waves used in microwave-radio the theoretical information capacity of a typical light wave is about )00,000 times greater than that of a microwave. Let us see how this comes about. Long··di~.ta!lce communication systems on principle of multiplexing, the transmission of many different message.o;; (information) over the same The voice requires a frequency band from 200 to 4000 a band 3800 Hz wide. A telephone can, therefore, can be transmitted on any band that is 3800 Hz wide. It can be carried. for example, by coaxial cable in the megahertz frequency band between 1,000,200 Hz and I occupying about 0.4% of the available coaxial carrier frequency. By contrast, it can be carried also by a He-Ne laser beam (632.8 nm, in the frequency band between 473,800,000,000,200 Hz and 4.738 x 10 14 473,800,000,004,000 Hz, where it uses less than one billionth of J% of the available laser-beam For microwaves, considerably shorter than the waves carried by cables, the contrast is not so the carrying capacity of the laser exceeds that of the microwave a factor of 100,000. ~UJ)POise we are able to develop a laser cornmUnl.catlon quency 4 x 10 14 Hz and operating bandwidth of 10 x Hz tern, on a beam, could carry about 2.5 million conversations, or 2000 .:>UllU"C<111'vV"" TV programs. When the first telephone system was built, each carried only one conversation. Today each line carries hundreds of calls and cable transmits up to 1000 an hour, but lines are still smaller in size than a can be fitted with 144 fibers. pair of fibers can carry 672 simultaneous telephone messages on laser light pulses. For the J44-fiber cab1e, considerably smaller and bulky than the telecoaxial cables, this translates into 50,000 phone messages. information-carrying systems use what is time-division mUltiplexing. Such a system divides the voice signal into many small bits and fills the normal, short in talking patterns with other the carrying For a telephone conversation, process works as follows. Sound waves are converted into waves in the mouthA sampler device then measures the of each elecpiece of the trical wave 8000 per second and assigns to each a binary number, the same numbering used in languages. In this way each sound is coded electronically. is all at the rate of millions of bits per second. The biare transmitted through the optical and converted into sound waves by a reverse process at the opposite end of the telephone conversation. He-Ne and diode (the gallium arsenide semiconductor family) are currently mated with in today's optical communication (see Chapter 492

Chap. 23

laser Applications

Fiber Optics). The and low-dispersion of silica-based mode optical fibers transmission of billions of of information per second over distances in excess of 100 miles. Long-wavelength light (1.3-1.5 J.Lm), coupled with dominate optical communications applications, both on land and under water. Current laser used in space transmit as high as 1 billion bits per second, a rate that could handle the entire information content of the Encyclopaedia Brittanica or the simultaneous program content of 14 color TV stations, all in about 1 s. The essential that make up a atmospheric laser communications system are shown in 23-6. Video, and data information are processed by multiplexing electronics and fed to the laser. A modulated laser beam is then transmitted through the atmosphere and collected at the detection site. photo-detector signal, with reverse multiplexing, is reconverted into the voice, and data

De-multiplexing electronics

Multiplexing electronics

Figure 23-6 FUIlctional arrangement of an laser communication system. with permission from Lasers and Applications.}

Information Processing. Not only are lasers making a on communications technology. which deals with the transmission and of information-underwater, over land, and through space-but they are also driving tec:hnOl()~lC~S in optical data optical data readout or and laser Lasers are ideal in such because of their .",.",......... and focusability. made with for example, media for optical storage and were discussed in Chapter 13. lasers most often used in information processing are the argon, and diode lasers. Information in our is an ongoing and challenge. For example, NASA alone receives 1000 trillion bits of data from satellites and space probes each year, data that must be interpreted, and stored. With the seemingly exponential growth in data generated and collected, it is clear that fast, reliable for information represent one of our critical needs. Sec. 23-2

lasers and Information

493

stora~:e

,1"v",I,,,'·" the use of lasers to store digitized information on of permanent or erasable imprints. U'I'.""""""' information can be divided, the binary digit or bit, .....,1'>"'."'''''"' as a 1 or a O. bits make a byte, and a such a.'> the letter or or two the numbers 27 or The byte 01 00000 1 might I'p.nl''''''I"n 01000010 might B, and so on. Currently, high-density optical systems accommodate in the of 4 x 109 bytes (4 Gbytes) on one side of a 14-m. A of 4 Gbytes translates into about -'....~/.V"V sized documents! The average access time is of the order of 0.] s to In optical data a laser source is modulated appropriately input of the data described above. The laser writes bits of data on a nh,,,t,,,...,,_ active in the shape of a 30-cm or 12-cm disk. Here it causes either a permanent or change in the disk's optical behavior at precisely located, micron-size pits. The pits result from surface absorption of the laser energy with subsequent surface ablation. For playback or readout, a lower-power laser is rI ..·"",tArl onto the and its light is reflected, or in some cases transmitted, to the exact pattern of detail written on the surface. The reflected (or transmitted) laser beam is then detected by appropriately positioned photodiodes that translate the nals received back into the original electronic data pattern. Permanent are made for storage of archival materials, by the of Congress, for Popular compact discs (CDs) for recording or playback of audio or video information are also of this kind. Erasable recordings, while not yet availare under It is only a matter of time erasable recordbecome of modern data storage technology. printers, capable of producing some of the fastest, clearest printing availre(J,lacing conventional printers, both and dot-matrix. Traditional create an image by forcing an ink ribbon onto paper mechanieither with a set of small pins, as in dot-matrix or with a fully formed as in daisy-wheeL Dot-matrix print or graphics but suffer from coarse resolution; daisy-wheel printers offer text quality but cannot do graphics. Both, being mechanical printers, are and slow by running at of several pages a minute. By creates with a laser beam that scans rapidly across a ph;otOICOIIU!lICU selectively discharging certain areas on the drum (see Figure with of opposite polarity, adheres to the photoconductive "''',>1<,<0 to paper with both pressure and heat. In the process, a spots (coherence and focusability pins and creates text of much higher quality. In the laser printing process runs smoothly, quietly. and quickly. Printing rates may vary from several pages per minute to two pages per second. depending on the cost of the printer. Remote Sensing. Recent successes of space combined with adtechnology, have set the stage for remote Simply SC[ISl[lg involves the detection of laser-scattered light from targets are relative to active laser The laser systems-sources and detectors-are frequently on space platforms. whereas the remote fur below, may be the or the crust. Ground-based systems, also prominent but not u~""'.".,c:u used to detect poUution and to monitor global technique used, often referred to as laser or UDAR (light detection and depends on the presence of a laser source, a suitable in494

23

laser Applications

Multisided rotating mirror

k--------i--+t-H-l++H---------""'iIlMirror

Laser

Figure 23-7 with

Schematic of a optical system for a laser from Lasers and Applications.)

Mirror

(Adapted

terroJl:ati4Jn, and detectors. For any application, an ideal laser must have appropriate wavelength, tunability, output power, pulse repetition rate, and amplitude and frequency stability. It should, in addition, have high operating efficiency, be mechanically rigid, and possess a operating un'Ul.U". Current laser sources include continuous wave carbon dioxide lasers, pulsed Nd:YAG and lasers, Nd:YAG-pumped lasers, excimer AIGaAs-pumped Nd:YAG and an entirely new class of continuously tunable, solid-state lasers, such as the cobalt fluoride (CoMgF) lasers. Taken together, the family of special lasers provides wavelength coverage applications. from 0.66 p.m to 10.6 p.m, sufficient for most with the interrogating laser The whatever their nature, must light and a mea<;urable change that can be detected. Such changes, returned to the detection system by the back-scattered laser radiation, include nU4JreSCf:noe, Raman-shifted wavelengths, and Doppler each of which' provides important information on identity, or movement. LIDAR such as differential absorption Raman LIDAR, and Doppler LIDAR, are designed to detect specific information content locked in the backscattered radiation. The state of the art in remote and the ahead is immediately evident from a review of two current applications involving the measurement of wind and earth crust shifts. The detailed movement and speed of tropospheric winds is of considerable interest in the study of weather patterns. To ferret out such information, a space-based C(h laser system would operate from an orbiting ei ther an operational or the space shuttle, at distances from 250 km to 800 km above the From such altitudes, high initial laser power is needed, or the backscattered radiation is too weak to detect. The incident laser radiation is backscattered by dust particles carried along by the wind. Calculations of wind and direction are based on the Doppler frequency shifts in the reflected Sec. 23-2

lasers and Information

495

radiation. A laser, to measure wind speeds with an accuracy of I mIs, requires to-I every second or so, with a frequency stability of several hundred kilohertz or better. power requirements of several kilowatts are needed to operate such laser The challenge is the availability of high power, stored and accessible during the space mission. The use of a space-borne LIDAR to detect movement of the earth's crust is of value in determining strains near zones, swelt prior to volcanic and large-area land Based on time-of-flight measurements of very short (nanosecond) laser space to ground retrofiectors and back, and use of simple methods of distances to parts of the earth's crust are determined to within I or 2 cm! By repeating the measurements at apprq:lriate time intervals, significant earth movement is detected and monitored. Systems under consideration for earth crustal studies include a Nd:YAG laser with pulse widths less than 0.5 ns and rates of the order of 10 per second. Backscattered detection on board the can be with existing 6-in. diameter collecting telescopes and photomultiplier eleClr,OnJlCS. LIDAR techniques, coupled with laser "",,,tp,,,,,, the detection of pollution, identification of concentrations, or global measurements atmospheric gases and aerosols. Optical-sensing technology has been used successfully to monitor smokestack emissions from industrial plants or to detect gas leaks in pipelines. On a broader scale, atmospheric concentrations of ozone, water vapor, and sulfuric acid can be measured with notable sion. In particular, laser absorption relying on of differential and resonance have already demonstrated their usefulness. Global measurements to determine atmospheric aerosol concentrations are of much interest and are within the capability of Such determinations illuminate the role that aerosols in the warming or of our planet as well as the formation haze layers in the t ..nnnc:nh,p.rp The recorded successes of such space as Voyager, the Infrared Astronomical Satellite, and several Landsat orbiters, coupled with advances in technology, that spaceborne remote of the earth's atmosphere and crust is a rapidly technology. 23-3 Mll'RE RECENT DEVELOPMENTS z

During the 19908 and laser applications continued to spawn strong activity in the marketplace. technology moved ahead surely and steadily 'in well-established areas of bar-code scanning, optical storage and memory, communications, and therapeutic in medicine, and the entertainment/display industry. In the early 1990s, the development of new laser materials and devices continued at a rapid pace, guaranteeing the user a wide choice of lasers nnp.."Jinn tinuum of from 0.3 Mm to I Mm, with available powers watts to tens of watts. Promising new with names such as tiumi,um-SG!DD'hi,·e. cobalt-doped magnesium fluoride, alexandrite. Nd: YLF (neodymium yttrium lithium fluoride), and NYAB (neodymium yttrium borate), joined the workhorse lasers of the C02, Nd:YAG, argon, and gallium arsenide. 2The contents of this section are intended to present a flavor-certainly not a cornprlehelnsi\le coverage-of laser and applications in the early 19908.

Chap. 23

laser Applications

Diode lasers now compete with the old standby, the He-Ne laser. Thrp"tpnin to replace the lamp-pumped lasers, diode-pumping offers an almost ideal match between pump and absorber, higher operating efficiencies, better reliabilities, and less cooling requirements. In a slightly different arena, self-frequency-doubling such as NYAB and MLNN (neodymium and magnesium-doped LiNbOJ ) are used to directly, green light near 531 nm. In the world of optical communications, erbium-doped fiber optic amplifiers, pumped with diode are under development to replace costly repeaters in transoceanic, long-haul communications systems. We shall survey in a bit more detail some of the ongoing progress in several fields: defense-related laser interests, and laser research in the microworld.

Medical Applications. In laser remains extremely both as a and as a surgical tool. In rooms, for example, and Nd: YAG lasers are demonstrating new successes in gall bladder operations (laparoscopic laser cholecystectomy). While gall bladder operations with lasers are on the increase, laser angioplasty (the use of a hot-tip to plaque in blocked arteries) is on the decrease. Instead, laser-assisted balloon angioplasty (inflating a small balloon to widen the occluded in the artery) seems to be the currently technique. Back surgery, too, is sometimes possible, using the selective heating of a laser beam to remove ruptured disk material between vertebrae of the spine. In ophthalmology, excimer lasers output) are still being considered as Two techniques are currently in use, both into correct volving a reconfiguration of the corneal surface. One is called radial keratotomy (making carefuHy controlled grooves in the corneal surface); the other is corneal In diagnostics, resculpting removal of corneal search programs are under way to develop a lead-salt tunable diode laser technology for clinical The goal of this research is to use nonradioactive isotopes as tracers to help detect physiological dysfunctions. Instrumentation involving highresolution infrared spectroscopy is being designed to perform isotopic analysis of exhaled human breath (a noninvasive procedure). This procedure promises to be an important future tool for clinical of diseases such as diabetes. Military-Related Applications. techniques account for the pinpoint accuracy now possible in the delivery of weapons. In the hardware and systems of current interest, lasers continue to be at heart of ImIJrOVements for range target satellite communication svs;lelllS. radar-based navigational and remote sensing. The laser ~V~rf':nl~ developmental activity in these fields include the high-power CO2 lasers (diode-pumped and frequency doubled), erbium-doped YAG lasers, diode lasers, to a extent, free-electron (FEL) and (HF, OF) lasers. include 532 nm, run, 830 nm, 1.06 p.m, 1.54 p.m, The wavelengths of 3 p.m, and 10.6 p.m, with pulse from miUijoules to .... uuJ ..JU ..~" lasers and the Microworld. Extraordinary new laser devices are currently showing up at the frontiers of subatomic research. The unique properties of laser enable to examine protein molecules attached to the inner membrane of a red blood cell. Physicists are able to levitate millions of sodium atoms in stainless-steel containers and release them simultaneously in fountains of

Sec. 23-3

More Recent Ue'Vel'ODlllents

491

hydrogen atom an atom away Chemists can observe a from a CO2 molecule. Femtosecond lasers of lO-15 second) can freeze the motion of atoms and molecules, it possible to take snapshots of chemical reactions. Beams of laser light, in laser traps, are used to corral of atoms and to move them around at will. Techniques referred to as "optical molasses" use laser to create electromagnetic forces that can slow down atoms. The recontrolled atoms, observed in both free fall and motions, enable measurement of gravitational with greatly increased accuracy and the development of new generations atomic clocks. Further, single laser beams, as "optical tweezers," hold in place or nudge around microscopic as DNA molecules, the study of their internal and external structures. Lasers are an essential part an atomic force microscope (AFM), a remarkable that offers a quick and accurate method for YnP'''''''',I',",o the atomic details of a surfuce. The AFM glides over microscopic surcarefully sensing and the outlines of individual atoms. Such microscopes offer motion control down to the angstrom level, laser displacement forces as low as 10- 10 newtons, and scan sizes on the order of 100 #Lm2. Underthe atomic surface detail of insulating and conducting is essential, for example, in the fabrication of reliable fiber

Other Applications. In we have to survey some of the more important applications of the laser in modern technology. The survey was OI]~alllIU~ around two and information and interactions with This to review a number of laser applications in processing, laser-initiated fusion, communication, information processing, and remote sensing. As important as are, they by no means exhaust the different ways in which the laser is being applied. Lasers also find and construction straight line direct applications in civil that is not there); in agriculture as systems for laser the land; in the semiconductor industry, lasers are the microtools needed for precision trimming and cutting in device-processing operations; and in where combine flashy colors and music in exotic, dynamic n<>tt"r,1c

23-1. Consider a pulsed infrared-type laser. with a peak power of 10 MW and beam divergence of I mrad, focused on a work area (target) by a lens of focal length 20 cm. What is the power density on target? waist), it has a certain "depth of focus" 23-2. When a laser is focused down to a spot d which the beam waist does not increase appreciably. This is by the rei ation

beam at its own the to which "focused." (a) Consider a CO2 , ter of 8 mm and

498

Chap. 23

of the D is the diameter of the incident laser waist, A is the and p is a tolerance factor that sets the focused beam is allowed to expand and still be considered 1O.6-JLm laser from it" own beam-waist diameimpinging on a germanium lells of focal length 60 mm. Deter-

laser Applications

23-3.

23-4.

23-5.

23-6.

23-7.

mine the depth of focus at the focal spot if the beam diameter is to increase no more than 10% from its minimum value at the waist (p = 1.10). (b) The beam waist D' at the focal plane of the germanium lens is given by D I = /cfJ. where cfJ = 1.27>./D is the beam divergence of the incident laser beam. Calculate D' and the diameter at the end point of the depth-of-focus region d for the tolerance factor in (a). Consider the cutting performance of the laser, illustrated graphically in Figure 23-3. Explain why the same laser penetrates less deeply in aluminum (2.7 g/cm3) than it does in steel (7.8 g/cm 3) at a transverse cutting speed such as 25 mm/s. In formulating your answer, consider the importance of reflection and absorption coefficients, as well as thermal conductivity. A pulsed Nd:YAG laser beam used in eye surgery emits a pulse of energy of about 1 mJ in 1 ns. The laser beam is focused to a tiny spot of 30-lLm diameter in the interior of the eye. (a) What is the irradiance at the focal spot? (b) Assuming the eye to have an index of refraction of 1.33 and a permittivity of about 78 EQ, estimate the electric field strength at the focal spot. (Hint: Use Eq. (8-42), appropriately modified.) In the field of communication theory, it is well known that the information capacity C of a signal of average power S in the presence of additive white-noise power N in a channel of bandwidth B is given by C = B log (l + S/ N). (Note that channel capacity C is directly proportional to bandwidth B, justifying the great interest in lasers for communications purposes.) Suppose the signal-to-noise ratio, S/ N, is 9 and the available bandwidth in a laser communication system is about 4000 MHz (only 0.001 % of the carrier frequency, which is around 4 x 10 14 Hz). (a) What is the information capacity C? (b) How many telephone conversations of bandwidth 4000 Hz could be carried by such a laser system? A laser system, fixed on a geosynchronous satellite, uses aNd: YAG laser that emits 1.06 ILm radiation in a highly collimated beam of 5-lLrad beam divergence. (a) If the laser is 36,205 km above the earth. what is the minimum diameter of the laser-beam "footprint" on the surface of the earth? (b) If the laser emits pulses of 200-MW power, what is the average electric field per pulse in the laser beam at the earth's surface? (Hint: See Eq. (8-42).) Personnel who work with high-power lasers or with unshielded laser beams must take measures to protect themselves from laser damage to the skin and eyes. Towards that end, nominal hazard zones (NHZ) for given lasers can be defined. A NHZ outlines the conical space within which the level of direct, reflected, or scattered radiationduring normal operation-exceeds the assigned levels of maximum permissible exposure (MPE). Exposure levels outside the conical region are below ~e designated MPE level. The conical region, with tip emanating from the laser, extends to a range R NHZ with sides spreading at the beam divergence angle cfJ. The range for a nonfocused laser beam is given by RNHZ =

1[(

4>

4P 7T

MPE

)1/2

- d

]

where P is the laser power, cfJ is the beam divergence. MPE is the maximum pennissible exposure in power/area, and d is the aperture diameter of the laser exit port. (a) Draw a sketch of the NHZ (conical shape), labeling the extent of the cone, RNHZ , and the full-angle beam divergence cfJ. (b) Determine RNHZ for a continuous wave Nd:YAG laser (A = 1.06 ILm) of 50 W power, 3.0 mrad beam divergence, MPE of 5.1 x 10- 3 W/cm\ a 10 s exposure, and an exit aperture diameter of 3.0 mm.

Chap. 23

Problems

499

23-8. The range RNHZ to problem immediately onto a lens is given by

for a laser that directs its

laser beam

where RNHZ defines the extent of the cone emanating from the point of beam focus, f is the focal length of the lens, b is the diameter of the beam exiting the laser, P is the power of the laser, and MPE is the maximum exposure. (a) Draw a sketch of the NHZ (conical space) for this focused laser beam. (b) Determine RNHZ for a continuous wave Nd:YAG laser of 50 W power, exit beam diameter of 5.0 mm, and capped by a lens of focal 7.5 em, if the MPE of this laser for an 8-hr exposure is 1.6 x )0- J W/cm 2 •

REfERENCES .. Scientific American (June 1991): 84. Goldman, Leon. Applications of the Laser. Malabar, Fla.: Robert E. Publishing Company, 1982. Ready, J. F. Industrial Applications New York: Academic 1978. D. H., and H. L. Wolbarsht. Safety with Lasers and Other Optical Sources: A Comprehensive Handbook. New York: Plenum 1980. Lasers and Optronics. Morris Plains, NJ.: Elsevier Communications. Trade journal, published Optics and Photonics. Washington, D.C.: Optical of America. Published monthly. Photonics Spectra. Pittsfield, Mass.: Laurin Publishing Co. Trade journal, published monthly. Vali, Victor. "Measuring Earth Strains by Laser." Scientific American 1969): 88. Berns, Michael H., and Donald E. Rounds. "Cell Surgery Laser." Scientific American (Feb. 1970): 98. Faller, James E., and E. joseph Wampler. "The Lunar Laser Reflector." Scientific American (Mar. 1970): 38. Moshe and Arthur P. Fraas. "Fusion by Lasers." Scientific American (June

[II Berns, Michael W. "Laser

[21 [3]

[41 (5)

[61 [7J

[8] [91 [10] [11]

21.

[12] Feld, M. and V. S. Letokhov. "Laser Spectroscopy." Scientific American (Dec. 1973): 69. [13] EmmeU, John L., John Nuckolls, and Lowell Wood. "Fusion Power by Laser Implosion." Scientific American (June 24. rI4] Zare, Richard N. "Laser Separation oflsotopes." Scientific American (Feb. 86. (l51 Ronn, M. "Laser .. Scientific American 1979): 114. [16] Tsipis, Kosta. "Laser Weapons." Scientific American (Dec. 1981): 51. [l7J LaRocca, Aldo V. "Laser Applications in Manufacturing." Scientific American (Mar. 1982): 94. iYTI)SCIJpeS." Scientific American (Apr. 1986): 94. [18] ,...mJ"'.'...... , Dana Z. [19} Jewell, Jack L, James P. Harbison, and Axel Scherer. "Microlasers." Scientific American (Nov. 1991): 86.

500

23

laser Applications

24 A

Fiber Optics

INTRODUCTION

The channeling of light through a conduit has taken on great importance in recent times, as we have seen in our review of laser applications. This is especiaHy true because of its applications in communications and laser medicine. As long as a transparent solid cylinder, as a has a refractive index than that of surrounding medium, much of the light launched into one end will emerge from the other even though attenuated, due to a large number of total internal reflections. A comprehensive treatment fiber optics requires a wave approach in which Maxwell's equations are solved in a dielectric medium, subject to boundary conditions at the fiber walls. In this chapter we adopt a simpler and more intuitive approach, describing the propagating wavefronts by their rays, although we appeal in some contexts to wave such as phase and ence. 24-1 APPUCATIONS

The simplest use of optical fibers, either or in bundles, is as light pipes. For example, a flexible bundle of fibers might be used to transport light from inside a vacuum system to the outside, where it can be more easily measured. I The bundle the rods and cones of the human eye have been shown to function as light pipes along their lengths, as in fibers.

501

into two or more sections at some point to act as a aPII)ic:atiC)ns, the fibers can be randomly distributed ......"'"./S is required, however, the fiber ends at the input are coordiat the output. To maintain this coordination, fibers at both ends are The fiberscope consists of a bundle of such to exequipped lens and eyepiece. It is routinely used by amine regions of the lungs, and duodenum. Some of the fibers function as Jight pipes, from an external source to i]]uminate inaccessible areas return the internally. Fibers can be bound rigidly by fusing their outer coating or cladding. In this way, fiber-optic are made for use as windows in ray tubes. Furbundles are tapered by and stn!tclun:g, ther, when such can be magnified or in depending on the relative areas of output mces. The power of imaging fibers depends on the accuracy of fiber be on the individual fiber d. A r-n,"''''· ....l<,'H1P ment and, as estimate ll] of fiber resolving power RP is given by RP (lines/mm) =

d:)

can produce a high resolution about 100 Thus a lines/mm. The most ""'--"''''<><'''''' applications of fiber optics lie in the area of communicafollowing section. tions, the 24-2 COMMUNICATIONS

~V~TjlC'_

OVERVIEW

No application has impetus to the rapid than ha'i voice or video communication and data transmission. The optic conduits or waveguides over conventional TU""_Ul1rp crowave waveguide systems are impressive. The replacement of dio waves by light waves is especially attractive, since the int;orrnal:iOll-can'vin carrier wave increases directly with the width of pacity of available. of copper coaxial cable by """'r_,C)","'" greater communications capacity in a lighter cable that less space. Additionally, in contrast with metallic conduction techniques, communication by light offers of electrical isolation, immunity to interference, and the Iree04Jm from The latter is especially important of inas in computer networks that handle .." •• h£lpnt 24-1, we give an overview of the ron',..,.,'"........ I''' in a fiber-optic communications the information to be message At the input end of the transmitted is converted by some type of transducer from an electrical signal to an it is reconverted from an optical one. After transmission the to an The fiber serves as an waveguide to propagate the inover a distance that formation with as little distortion and loss of power as can from meters to thousands of kilometers. electrical signal from The message source might be audio. mlcfC>pl1one; it might be visual, providing an a video camera; be digitally encoded information. like "'V'''IJ'~'''''' data in the form of a Analog and digital formats are into one another. so that 502

Chap. 24

Fiber Optics

Sound: microphone Visual: video camera Data: computer

Amplification filtering demodulation

PIN

APD

Figure 24·1 Overview of a hh,>r.r....hr<: com· munication system.

choice of format for transmission through the fiber is always av'llla:DU~. of the original nature of signal. purpose of the modulator is to perform this conversion when desired and to this onto the carrier wave generated by the carrier source. The carrier wave can be to contain the information in various ways, usually amplitude modulation (AM), frequency modulation (FM), or modulation. 2 See Figure 24·2. In fiber-optics the carrier source is either a lightemitting diode (LEO) or a laser diode (LD). Section 2-5.) ideal carrier wave is a single-frequency wave of adequate power to propagate long distances through the fiber. In fact, no light source can produce radiation of a single LD LED are by a band with central wavequency. length A spectral width dA. The LD is preferable because it approaches the ideal Signal

Carrier

AM modulated

Signal

Carrier

FM modulated

Digital pulse modulated

2

Some of the techniques of

Sec. 24-2

Figure 24-2 Three forms of modulation in which a carrier wave is modified to carry a sig· nal. Amplitude modulation; center: fre· quency modulation; bottom: modulation in which a pulse is either present ("on") or missing ("off").

modulation are discussed in

Communications System Overview

26.

more closely than the LED; however, other considerations make ac(:epltable in some systems, as we shall see. 24-1, the carrier source output into the optic fiber is represented by a square pulse. As this pulse propagates through the fiber, it suffers both attenuation and (change in shape) due to several mechaThe may typically. a glass or plastic filament 50 J,Lm nisms to be in diameter. If the fiber is very long. it may be necessary to intercept the signal at with a repeater that amplifies and restores the remote end of the fiber, the light signal is coupled into a optical signal back into an electrical signal. This service is a semiconductor most commonly a PIN diode, an avalanche Dnolo'm£Il11JO{I~!r (Section Of course, the response of the detector ",,,,,I,,I-,,,L'i to the frequency of the signal received. The detector handled by a processor, whose function is to recapture the origfrom the a process that involves filtering and ama conversion. The message output may lUll .... ' ' ' ' ' ' ' ' by (audio), by cathode-ray tube (video), or by computer input (digital). tTp,,,...'nrv

24-3 BANDWIDTH AND DATA RATE

The more the greater is the range of freuu,~ ..... , .... " required to The output of a stereo system is more fuithful to the original is the output of a telephone receiver because a greater frequency range is devoted to the process of reproduction. The range of frequencies required to modulate a for a telephone channel is only 4 kHz, whereas the bandwidth of an FM station is 200 kHz. A commercial TV """u"",cu.,-, both sound and video uses a mtnr'Tll'IItlnln.C'l'Irl"VllnC potential of a light beam becomes evident when we calculate the ratio of frequency to bandwidth, a measure of the number of that can be on the carrier. For a TV station using a 300-MHz this is MHz/6 or 50; for an optical fiber using a carrier of 1 (3 x lOll MHz) to carry the same information, the ratio is (3 x MHz)/6 MHz or More information can be sent by optical fiber when distinct pulses can be transmitted in more frequencies or, in the case of digital latter case, suppose that i bits (on or oJlpulses) arc required to of an analog signal. According to the sampling theorem, 4 an must be sampled at a rate at least twice as high as its to be faithfully In the case of the TV channel of 6 this means that 2 x 6 MHz or 12 x 10" Since each is described using 8 bits. the data rate is % Mbps per second). Data rates are limited by the as well as by fiber distortions that prevent distinct sec in discussions to follow. 3Tbeorclical slUdies and applications are well under way that use lightly erbium-doped fibers, from the tran'imitter end so that the fiber itself serves as a distributed amplifier. As such. Ihe libel' is ahle 10 overcome ils losses and reduce or eliminate the need for repeater stations. See [Ill and lUll. 4The sampling theorem is also discussed in Section 25-2. in connection with Fourier transform spectroscopy.

504

Chap. 24

Fiber Optics

24-4 OPTICS OF PROPAGAnON

We consider now the manner in which through an optical fiber. The conditions for successful propagation are developed here mainly from the point of view of geometrical optics. In addition, we consider the meridional rays, which intersect with the fiber's central axis. 5 Consider a short section of a pictured in 24-3a. fiber itself has refractive nl, the medium cladding) has index n2, and the end are exposed to a medium of index no. Ray A entering the left face and transmitted to point C on fiber where it of the fiber is refracted is partially refracted out of the fiber and partiaHy reflected internally. internal ray continues, diminished in amplitude, to D, then to and so on. After mUltiple reflections the ray wi]] have lost a part of its energy. Ray A does not meet C, conditions for total internal reflection, that is, it strikes the fiber surface at D, such that its angle of incidence 'P is less than critical 'Pc, or 'P

< 'Pc

=

Ray E, on the other hand, which enters at a smaller Om with to it is refracted parallel to the fiber strikes the fiber surface at F in such a way sur:fuce. Other rays, as in 24-3b, incident at angles 0 < Om, experience total internal reflection at fiber surface. rays are along the fiber by a

B

A

0'

()'

(b)

Figure 24-3 (a) Propagation of light rays through an optical fiber. Ray B defines the maximum input cone of rays satisfying total internal reflection at Ihe walls of the fiber . (b) Propagation of a light ray through an optical fiber.

5 Other rays, the skew rays, do not lie in a plane a piecewise path through the fiber.

Sec. 24-4

of Propagation

the central fiber axis. These rays take

505

succession of such reflections, without loss of energy due to refraction out of cylinder. However, depending upon the of of the material to the light, some attenuation occurs by absorption. B thus represents an extreme ray, defining the slant face of a cone rays, the condition for total internal reflection within the fiber. The all of which maximum half-angle Om of this cone is evidently related to the critical angle of reflection cpc. At the input face, 110 sin 0", :::::: nl sin 0:'" and at a point .

Ib

sm CPc: = -

n,

Using the cp,. +

fact, (J;' 90° - l/>c, and the iden thy, I, these relations combine to give the numerical aperture, N. A_

==

no sin Om

= nl cos cpc

(24-4)

=

If no = I, the numerical aperture is simply the sine of the cone of meridional rays , rays coplanar with the fiber through the fiber by a series of total internal reflections. numerical aperture clearly cannot be than unity, unless no > I. A numerical aperture of 0.6, for example. corresponds to an acceptance cone of 74°. The light-gathering ability of an optical fiber increases with its numerical aperture. Also from Figure the skip distance L. between two successive propagating in the fiber is given reflections of a ray of

d cot 0'

(24-5)

where d is the fiber diameter. Relating 0' to the entrance angle 0 by Snell's law,

=d

(24-6)

J00 , and d 50 IJ-m, (24-6) For example, in the case no = I, nl = 1.60, (J gives L. 152 IJ-m. Thus in one meter of fiber, there are approximately IlL. or 6580 reflections. Table 24-1 lists various core and cladding possibilities, for which the critical numerical and skip distances have been calculated. With so many reflections occurring, the condition for total internal reflection must be acfiber. Surface scratches or irregularities, as curately met over the length of dust, moisture, or grease, become sources of loss that rapiC1ly diminwell as ish light energy. If only 0.1% of the light is at cach reflection, over a length of I m, this attenuation would reduce the energy by a of about 720. Therefore. it is essential that it be coated with a layer to the quality of the of or called cladding. Cladding material need not be highly transTABLE 24-1 Core/cladding

CHARACTERIZATION OF SEVERAL OPTICAL FIBERS 110

Glass/air Plastic/plastic Glass/plastic Glass/glass

nl

n2

'P,

emu

1.50

1.0 1.39

4Ur

68_9"

91lO" 32S 24S

1.49 1.46 1.48

1.40

1.46

NOTE: The reciprocal of lhe sic ip distance ( 1/L, or ter 100 p,ID and al 0 0""•.

506

24

Fiber Optics

73.5° 80.6°

14.0"

N.A.

IlL.

I 0.54 0.41 0.24

8944 3866 2962 1657

per meier) is calculated for a fiber of diarne-

parent, but must be compatible with the fiber core in terms of expansion coefficients, for example. The index of refraction n2 of the cladding. where nz < nl, influences the critical angle and numerical aperture of the fiber. The cladding around the fiber cores has another important function, which is to prevent what is called frustrated total internal reflection from occurring. When the process of total internal reflection is treated as the interaction of a wave disturbance with the electron oscillators comprising the medium, it becomes apparent that there is some short-range penetration of the wave beyond the boundary. Although the wave amplitude decreases rapidly beyond the boundary, a second medium introduced into this region can couple into the wave and provide a means of carrying away energy that otherwise returns into the first medium. Thus if bare optic fibers are packed closely together in a bundle, there is some leakage between fibers, a phenomenon called cross talk in communications applications. The presence of cladding of sufficient thickness prevents leakage, or, to put it more obliquely, negates the frustration of total internal reflection. 6 The optic fiber cores discussed above are assumed to be homogeneous in composition, characterized by a single index of refraction nl. Light is propagated through them by multiple total internal reflections. Such fibers are called step-index fibers because the refractive index changes discontinuously between core and cladding. They are multimode fibers if they permit a discrete number of modes (or ray directions) to propagate. When the fiber is thin enough so that only one mode (a ray in the axial direction) satisfies this condition, the fiber is said to be single-nwde. Restrictions on possible modes will be described later. Another type of fiber, the graded-index fiber, is produced with an index of refraction that decreases continuously from the core axis as a function of radius. All these types are discussed in the sections that follow.

24-5 ALLOWED MODES

Not every ray that enters an optical fiber within its acceptance cone can propagate successfully through the fiber. Only certain ray directions or modes are allowed. To see why, we consider the simpler case of a symmetric planar or slab wave guide, shown in Figure 24-4. The waveguide core of index nl ha<; a rectangular (rather than cylindrical) shape and is bounded symmetrically above and below by cladding of index n2. A sample ray is shown Wldergoing two total internal reflections from the core-cladding interface at points A and B. Recalling that the ray represents plane waves moving up and down in the waveguide, it is evident that such waves overlap and interfere with one another. Only those waves are sustained that satisfy a reso-

, ,, d

:

I

tp ,/

:--"

------L--------------ll----A'

Figure 24-4 Section of a slab waveguide showing a successfully propagating ray or one of the possible modes. The geometry is used to determine the condition for constructive interference.

6This topic was treated quantitatively in Section 20-5.

Sec. 24-5

Allowed Modes

507

nance condition. Notice that points A and C lie on a common wavefront of such waves. If the net change that develops between points A and C is some multiple of then the wavefronts constructive and corresponding ray directions are allowed. The net phase change is made up of two parts, the optical path difference A and the change 2cf;r that occurs due to the two total reflections at points A and B. Thus the self-sustaining waves must satisfy the condition

where m is an integer. geometrical extensions denoted by the dashed lines in Figure 24-4, identifying triangle ACA " make it evident that

A

AB

+

=

A'B

+

BC

2n,d cos tp

(24-7)

so that the possible modes are given by

m

+

A

7T

since < 7T. the second term is one at most and is negligible compared with waveguide has an inthe first term. Thus each successful mode of propagation in teger mode number m, related to a direction tpm and given (24-8)

m

For our present purposes, the precise number of allowable modes is not as important as the qualitative dependence the mode order m on the characteristics. Notice that low-order modes~m sman~correspond to tp == 90°, or ray directions that are axial, and high-order modes-m to rays that propagate with tp near tpc, or at ray The number of propagating modes is at tbe critical angle. the value of m when cos tpm has its maximum tpm tpc. from (24-4), nl cos tpc = N. we can write

2d m = A N. A.

+ I

=

2d

A

+1

(24-9)

We have added I to the total number of modes to account for the "straight through" mode (m = 0) at tp = Finally, we should point out that, because two independent polarizations are possible for the propagating wave, 7 total number of modes is twice that by (24-9) above. This analysis for the slab waveguide has served to elucidate the physical reasons for mode restriction. The analysis giving the possible modes in a cylindrical fiber is based on same physical principles, but is more complicated and is not devel,om:o here. It is shown 3] that, in this case,

mma. =

~(~ N. AJ

10)

Notice that, as for the slab waveguide, the number of possible modes increases with the ratio d/ A. Thus diameter fibers are multimode fibers. If d/'A is small enough to make m < 2, the fiber allows only the axial mode to propagate. This is 7The two orthogonal polarization directions are described in Section 15-2.

508

Chap. 24

Fiber Optics

the monomode (or single mode)fiber. The required diameter for single-mode performance is found by imposing the condition mmax < 2 on Eq. (24-10), giving

d A

2 'IT (N. A.)

- <---A more careful analysis [2] indicates that single-mode performance results even when d 2.4 - <---(24-10 A 'IT(N. A.)

Example Suppose an optical fiber (core index of 1.465, cladding index of 1.460) is being used at A = 1.25 jJ.m. Determine the diameter for monomode performance and the number of propagating modes when d = 50 jJ.m. Solution The N. A. is then V(I.465 2 - 1.462 ) = 0.12l, and the required diameter for monomode performance is, using Eq. (24-11) 2.4 ( ) d < 'IT (0.121) 1.25 jJ.m

or d < 7.9 jJ.m

On the other hand, if d = 50 jJ.m, the fiber is multimode with mmax

=

4['IT 1~~5(0.121)

r

=

115

giving the number of propagating modes according to Eq. (24.10).

24-6 ATTENUATION The intensity of light propagating through a fiber invariably attenuates due to a variety of mechanisms that can be classified as extrinsic and intrinsic losses. Among the extrinsic losses are inhomogeneities and geometric effects. Inhomogeneities whose dimensions are much greater than the optical wavelength can result, for example, from inadequate mixing of the fiber material before solidification and from an imperfect interface between core and cladding. Irregularities of a geometric nature include sharp bends in the fiber as well as microbends, both of which cause radiation loss because the condition for total internal reflection is no longer satisfied (see Figure 24-5). Other extrinsic losses occur as light is coupled into and out of the fiber.

(a)

Loss

(b)

Sec. 24-6

Attenuation

Figure 24-5 Radiation loss from an optical fiber because of (a) a sharp bend and (b) micro-defects at the fiber surface. Loss occurs where the condition for total internal reflection fails. Notice that in (b) the defect is also responsible for mode coupling. in this case a conversion from a lower to a higher mode.

509

are due to the restrictions of numerical aperture, as due to inevitable reflections at the interface, the so-called Fresnel and size of the light source may also be ill-adapted to Of course such losses also occur at the output fiber is fed to a detector. Still other losses become imwherever connectors, couplers. or splices are necessary. j" ...,,,,tr'h of coupled fiber ends. involving core diameter and lat'""""'''''"h,nn and numerical aperture incompatibility are posslb,le and can lead to losses when not properly corrected. Intrinsic losses are due to absorption, both by the core material and by residRayleigh scattering from microscopic inhomogeneities, diual and wavelength. The core material-silica, in the smaller than the fibers-absorbs in the region of its electronic and molecular transition 24-6. Strong absorption in the ultmviolet occurs due to electronic in the infrared is due to molecular vibmtional and bands. bands. Both uv- and ir-absorption decrease as wavelengths approach the visible region. Figure 24-6 shows a minimum of absorption at around 1.3 ILm. Residual impurities, such as the transitional metal ions (Fe, Cu, Co, Ni, Mn, Cr, V) ticular, the hydroxyl (OH) ion, also contribute to absorption, the last I and 1.73 ILm. Rayleigh scattering. with its significant absorption at 8 occurs from localized variations in the density or chamcteristic 1/A4 For an optical fiber transmitting at refractive index of the core

At

100r-------------------~1---------------------.

I I

Total fiber loss

Glass absorption in infrared

I I

10

,,

,

I"~"~

Glass absorption in ultraviolet

" "\ ,

""

\

0.1

"" ""

Scattering loss II

,

'I I' I " I I

0.5

0.6

0.7

1.2 1.5

2

3

5 10

Wavelength (urn)

Figure 24-6 Conlributions to the net attenuation of a germamUlm-tlopetl glass fiber. [From H. Osanai, T. Shioda, T. S. Araki, M. HnlrHYI'lCll. Izawa. and H. Takara, "Effects of Dopants on Transmission Loss of Low-OH-Content Optical Fibers," Electronics Letters 12, No. 21 (October 14, 1976): 550. Adapted with permission.]

• Rayleigh scauering was discussed in greater detail in Section 15-3.

510

Chap. 24

Fiber Optics

1.3 p,m rather than, say, 800 nm, a seven-fold reduction in Rayleigh scattering losses. Absorption losses over a length L of fiber can be described by the usual exponential law for light irradiance I, I

where a is an attenuation or absorption coefficient for the fiber, a function of wavefor the absorption in length. 9 For optical fibers, the decibels (db) is given by (24-13)

to 10gIO

where PI and refer to power levels of the at cross sections (I) and (2), illustrated in Figure 24-7. The distance z is usually chosen to be a standard 1 km. For a loss £l
E -"" :a

3.0

:s!. c

.2 'lil ::! C
~

0 800

900

1000 1100 1200 1300 1400 1500 1600 Wavelength (mm) la)

5000

2000

E

:a:s!. c;; 0

'';:;

1000 500

'"::!c;; !B

(1)

«

(2)

200 100 50 400

Light

500

600

700

800

Wavelength Inm)

(bl

24-7 Schematic used coefficient for a fiber.

10

define the absorption

Figure 24·8 (a) attenuation for all-glass multimode fibers. (Courtesy Glass Works.) (b) allenualion for aU-plastic fiber cable. Mitsubishi Rayon America, Inc.)

rays that strike the fiber wall at smaller angles of incidence travel a through the same axial length L of the absorbing medium, a is also a function of the

Sec. 24-6

Attenuation

distance

511

advances have been made in reducing the absorption of fused silica so that 10fibers rated at 0.2 db/km (operating at 1.55 /Lm) are available. Plastic are less but not nearly as Their overall attenuation is at Glass are therefore preferleast an order of magnitude higher than able in long-distance applications. Figure 24-8 compares spectral absorption in silica and fibers.

24-7 DISTORTION Light transmitted by a fiber may not only

power by the mechanisms just menit may also lose information through pulse broadening. When input light is modulated to convey information, the signal waveform becomes distorted due to several mechanisms to be discussed. The major causes of distortion include modal distortion. material dispersion, and waveguide dispersion, in order of rlp"rp'.~nla severity.

Modal Distortion. Figure 24-9 indicates schematically the input of a into a The output pulse at the other end suffers, square wave (a digital in from both attenuation and distortion. Modal distortion occurs because at the output. propagating rays (fiber modes) travel different distances in broadening the square wave, as Consequently, these rays arrive at different ray; the disshown. The shortest distance L from A to B is taken by the by the propagating ray that reflects at the crittance L I from A to B is !PC' The Land L' are related, as suggested by the geometry in ical Figure 24-9, by L L'

I
Thus the time interval M between the two rays is given

M T~

_

Tmm

=

~' _ ~ = ~(::

I)

where v is the of light in the fiber core. Since v c/n. this result is conveexpressed as a temporal spread per unit length. in the form 14)

Example Suppose the has a core index of 1.46 and a cladding index of 1.45. Determine the modal distortion for this fiber. 512

Chap. 24

Fiber Optics

Solution

Using Eq. (24-14),

8(:!.) L

=

3

X

1.46 (1.46 - 1.45) = 10- 4 km/ns 1.45

34 ns/km

The pulse broadens by 34 ns in each km of fiber. 10 Clearly, this broadening effect limits the possible frequency of distinct pulses. Modal distortion can be lessened by reducing the number of propagating modes. Consequently, the best solution is to use a monomode fiber, with only one propagating mode. The next best solution is to use a graded index (GRIN) fiber, which is described next. The Graded Index (GRIN) Fiber. A GRIN fiber is produced with a refractive index that decreases gradually from the core axis as a function of radius. Figure 24-10 shows the GRIN fiber profile, together with the profile of the ordinary step-index fiber, for comparison. In the GRIN fiber, a process of continuous refraction bends rays of light, as shown. Notice that at every point of the path, Snell's law is obeyed on a microscopic scale. Ray containment now occurs by a process of continuous refraction, rather than by total reflection. Refraction may not suffice to contain rays making steeper angles wth the axis, so GRIN fibers are also characterized by an acceptance cone. When the index profile is suitably adjusted, the rays shown in Figure 24-lOc form isochronous loops, an aspect of graded-index fiber that is responsible for reducing modal distortion. Like ordinary fibers, GRIN fibers are also dadded for protection. The variation of refractive index with fiber radius is given [4], in general, by

n(r) = nl

~1 - 2 (~r ~,

0

:S

r

:S

a

(24-15)

where ~ == (nl - n2)/nl and nl = [n (r)]max. The parameter a is chosen to minimize modal distortion. For a = I, the profile has a triangular shape; for a = 2, it is parabolic; for higher values of a, the profile gradually approaches its limiting case, the step-index profile, as a -;. 00. Minimizing 87 for all modes requires a value of a = 2. Thus the parabolic profile shown in Figure 24-10 is optimum. It can be shown [5] that for this case, pulse broadening is given approximately by modal distortion: (GRIN fiber, a = 2)

(24-16)

Comparing with modal distortion in the step-index fiber, Eq. (24-14), we can write

8(itRIN

=

~ (:.1 ~) = ~ 8(itl

The factor ~/2 thus represents the improvement offered by a GRIN fiber. For the example used previously, where nl = 1.46 and n2 = 1.45, we have ~/2 = 1/292. The GRIN fiber reduces the pulse broadening effect of modal distortion in this case by a factor of 292.

10 Actual values are somewhat better than predicted by Eq. (24-14) due to mode coupling or mixing (rays may switch modes in transit due to scattering mechanisms that, on the average, shift power from higher and lower modes to intermediate ones) and due to preferential attenuation (higher modes taking longer paths suffer greater attenuation and so contribute less 10 overall pulse spreading). For longer distances, this leads to a modified dependence of the form, M ex Vi.

Sec. 24-7

Distortion

513

2a

r-------------~--~n

(al

r-----~----~-n

' - - - - - - - = n o " - - ' -- ----------

Ib)

no (e)

24-10 (a) Profile of a £rade
Material Dispersion. Even if modal distortion is absent, some pulse broadening stm occurs because the refractive index is a function of wavelength. Dispersion for a silica fiber is shown in Figure II. Since no light source can be precisely the light in the fiber is characterized by a spread of determined by the source. wavelength has a different refractive index and therefore a speed through the Pulse broadening occurs because each component arrives at a slightly different time. The more monochromatic the light, the less the distortion due to material To be detected as a pulse, the output must not spread to the extent of significant with neighboring this requirement a limita-

514

Chap. 24

Fiber

Dispersion in fused quartz Refractive index versus wavelength

1.415

1.410

1.465

x

Ql

"C

.5 Ql >

'ts ....

'" 'Iii a::

1.460

1.455

III ..........

1.450

III .......

III .......

III ........ 111 ........ 111

1.445

......::±----i

1.440 0.4

0.6

0.8

1.2

1.4

1.6

Wavelength 111m)

24-11

Dispersion in fused quartz.

lion on the frequency of input pulses or the rate at which bits of information may be sent. 24-12 illustrates "".tpr". ."...."'n.:.nn by showing the progress of two in a fiber at wavelengths AI and Az. If the corresquare pulses (initially sponding refractive are nl and n2, the figure implies that nl > nz. These wavelengths are two in a continuum by the spectral width liA of the source, as the width of the source's output at ",,,'-IO"",,, as shown. we describe the Because the optical fiber is of propagation of a pulse by its group Vg (':seCtlClfl time T required a signal of anthe fiber is therefore gular frequency w to travel a distance L by dw

T(W) If the pressed by

Light source

bl n

dk

bandwidth is

"'1

times per unit distance is ex-

n Jl

1 - - -_ _A_2_ _

lti2-~t4\

~ ---.....1A"':"""-' Sec. 24-7

Distortion

24-12 Symbolic representation of material A square wave input arrives at the fiber end at different times, depending on wav'elellgth The spectral output of the light source is characterized both by a central wavcA and a width aA.

515

Now the first derivative dkldw can be calculated from k = 27T/A = nwlc, where n is a function of w. This gives

! (n -

dk dw

where we have used the proportion the second derivative, we write

wIdw

(24-17)

A

-AIdA in the last step. Progressing to

d (dk) LU

dA dw

(24-17), giving

and substitute

!

(dn c dA

or simply, material dispersion:

1>(£)

-MLU

(24-18)

(Alc) where M is a property of the core material defined by the From 18), we (d 2nldA2), involving the second derivative of the of a temporal pulse spread per unit of width see that M has the per unit of fiber length. Values of M (in units of ps/nm-km) for pure silica are in 24-13.

Material dispersion for pure silica

200 175 150

E "'E" c:

125

S

100

~

I

~

\

0



\.i+ '+"

c:

.~

\

75

r,+,

c.
:0

50


!!l '" ::2

. 1'+,

'i:

25

"'+-... t .... 0

~-,

-25 I

-50

0.7

O.B

0.9

1.1

1.2

1.3

!

r-+-i-_. I

1.4

I

1.5

'i-~--i1 1.6

Wavelength (,urn)

Figure 24-13 Material dispersion in pure silica. Tbe quantity M. r ..nr.."".nlh,o the pulse broadening (ps) per unit of spectral width (nm) per unit of fiber length against the Pulse broadening becomes zero al 1.27 ILm (km), is and is negative as wavelength increases further.

516

Chap. 24

Fiber Optics

1.1

Example 24- 13, calculate the pulse spread due to material dispersion in for both a LED and a LD light source. Consider the source waveto be 0.82 /-Lm, with a spectral width of 20 nm for the LED 1 nm for the more monochromatic LD. Solution

24-13

At 0.82 /-Lm, then LED: LD:

a value of near 110

;)

(110 ps/nm-km)(20 nm) = 2.2 ns/km

;)(T/L)

(llO ps/nm-km)(1 nm)

At 0.82 /-Lm, the LD is 20 times better than the ""r'IPr';nr monochromaticity.

0.11 ns/km as a direct result

its

also that pulse dispersion is much than that due to modal distortion. Material becomes only when modal distortion is reduced, in both monomode and GRIN in the presence of modal distortion, the advantage of of a LD over a LED is lost. In applications where fiber lenJ;(ttls are short plastic fibers and LED sources may well the best compromise between performance and cost. Finally, notice from 24-13 that M actually passes through zero at around 1 /-Lm, so that material dispersion can be reduced by finding light sources that operate in this "IJ""""" We shall extend the previous numerical example to determine the bandwidth limitation due to pulse spreading. Pulse distortion limits transmission frequency and information rate in a way that we can roughly estimate. Let us use as a reasonable for successful between neighboring pulses that their separation &r be no less than half their T:

T

1

l57 > 2 or fJT > 2v where v is

frequency. It follows that the maximum U""I.I""""'" Vmax

For the lows

=

0.5 ;)7

examples

LED: LD:

V max

or

Vma.

L = ;)

we calculate an

L

0.5 2.2 ns/km 0.5

v""", L = 0 II k . nsl m

0.5

(24-19)

UAIUIi:llt<;;

bandwidth, as fol-

GHz-km

k

= 4.5 GHz- m

Waveguide Dispersion. The last pulse-broadening effect to be discussed is called dispersion, a effect that oelperlOS parameters. Compared with modal distortion and material waveguide "This value corresponds UAllmm:IY to lhe so-called 3-db bandwidth, the modulation frequency at which the signal power is reduced one half due to signal distortion.

Sec. 24-7

Distortion

511

is a small that becomes important only when the other DUllse-OfOall.lening effects have been essentially eliminated. its presence is important in determining the wavelength at which net fiber dispersion is zero, as we shaH see. The of the index with wavelength leads to disperAn effective refractive index netT for the wave is sion, as previously defined neff = c / V g , where Vg is the group velocity. Waveguide leads to a variation of netT with A for a fixed-diameter even in the absence of material It can be shown [6] that neff = nl cpo Since cp between 90° and cpc, and sin cpc = , it follows that netT varies between nl cp = WO) and n2 (at cp = CPJ. Thus neff for an axial my depends only on the core index; for a my at the critical it depends only on the cladding index. The variation of netT is quite in quite small. 24-14 waveguide small because nl - n2 between the my dispersion, in the my representation. For a given mode, the and the fiber axis with A. Thus the my paths and times for two different wavealso vary with A, to pulse broadening. IPn ......' "

11

Figure 24-14 Symbolic rejJIresentallion U}~"..., •• ir! .. dispersion. A square wave arrives al Ihe fiber end al different limes, deon even in a <1i"......sionllp""-" medium. fur anyone mode, the is a function of the wa'll'elelU!:th

/
The variation of neff with A simulates material dispersion and can be handled (24-18) by simply replacing n by netT: quantitatively by material dispersion: waveguide dispersion:

5(i)

=

~(-L7)

-- - ~c d n tT

u

-M/U 2

e

== -M'llA

(24-18) (24-20)

We can appreciate the relative contribution of material and by comparing values of M and M'. In Figure 24-13, M mnges from about 165 to - 30 ps/nm-km over the spectmI mnge of 0.7 to 1.7 p,m. Values of M' for fused quartz over the same mnge have values of only about 1 to 4.5 ps/nm-km [6]. For example, the calculation carried out for material using a LED source at 0.82 p.m, with M 110 ps/nm-km, gave a pulse broadening of 2200 For the same wavelength, M = 2 of 2/110 times as or only 40 ps/km. Figure 13 shows that M for material dispersion becomes zero at around wavelengths. Waveguide dispersion, 1.27 p,m and then becomes negative for positive. Combination of the two thus shifts the waveon the other hand, is length of zero net toward a of about 1.31 p.m in a ical fiber. Sources operating at or near this are thus in reducing broadening and increasing transmission rates. In discussing attenuation earlier, we pointed out that minimum absorption in silica fibers occurs at around 1.55 p.m. The closeness of the for minimum absorption and minimum dispersion has motivated attempts to both conditions by shifting the curve tozero at 1.55 p.m instead of wards so that it passes 1.31 p.m. Means of modifying dispersion curve include the use of multiple cladI

518

Chap. 24

fiber Optics

the and variation ding layers, control of the core/cladding index profile a in GRIN fibers. way of summary, we have discussed three principal ways of pulse t)roaa~:nmIJl: in fibers: (I) usc a monomode fiber to eliminate modal distortion, (2) usc a light source of small width to reduce material and (3) use a light source in a spectral region where both attenuation and dispersion are as small as Clearly. the length of and component costs play major roles in determining the of the best for a Spe:CUlIC apPI1C
24-1. The bandwidth of a single telephone channel is 4 kHz. In a particular !>)'Stern, the transmission rate is 44.7 Mbps. In an actual system, some channels are devoted to functions such as synchronization. In this system, 26 channels are so devoted. How many telephone channels can the system accommodate? 24-2. Determine the theoretical limit to the number of TV station channels that could transmit on a optical beam of I #Lm wavelength. 24-3. (a) Show that, for a ray traveling at the angle relative to the fiber the skip di!>tance Ls can be expressed

(b) How many reflections occ.."Ur per meter for such a ray in a step-index fiber with nl = 1.460, n2 = 1.457, and d = 50 #Lm?

24-4. Refractive indices for a fiber are 1.52 for the core and 1.41 for the cladDetermine (a) the critical angle; (b) the numerical aperture; (c) the maximum incidence angle 0", for light that is totally internally reflected. 2+5. A step-index fiber 0.0025 in. in diameter has a core of index 1.53 and a cladding of index 1.39. Determine (a) the numerical aperture of the (b) the acceptance angle (or maximum entrance cone (c) the number of reflections in 3 ft of fiber for a ray at the maximum entrance angle, and for one at half this angle. 24-6. (8) Show that the actual distance x. a ray travels during one skip distance is by Hid

Xs

sin 8

where 8 is the entrance and the fiber is used in air. (b) Show that the actual total \,,.,,,..,.,,,,,,x, a ray with entrance angle 0 travels over a total length L of fiber is given by

(c) Determine XX, L" and x, for a lO-m-Iong fiber of diameter 50 #Lm, core index of 1.50, and a ray entrance of 10°. 24-7. How many modes can in a fiber with nl 1.461 and n2 1.456 at 850 nm? The core radius is 20 #Lm. 24-8. Determine the maximum core radius of a fiber so that it only one mode at 1.25 #Lm for which nl 1.460 and n2 = 1.457. 24-9. Consider a slab waveguide of AIGaAs, for which HI 3.60 and n2 = 3.55. How many modes can propagate in this if d = SA and d 50 A?

24

Problems

519

24-:10. A flux of 5 p,W exists just inside the entrance of a fiber 100 m long. The flux just inside the fiber exit is only 1 p,w. What is the coefficient of the fiber in db/km? 24-11. An fiber cable 3 km long is made up of three I-km lengths, spliced together. has a 5-db loss and each splice contributes a I-db loss. If the input power is 4 what is the output power? 24-12. The attenuation of a I-km length of RG-191U coaxial cable is about 12 db at 50 MHz. the input power to the cable is 10 mW and the receiver sensitivity is I p,W. How can the coaxiru cable be under these conditions? If optical fiber is used with a loss rated at 4 db/km, how long can the transmission line be? lJe"()OIPea silica fiber has an attenuation loss of 1.2 db/km due to Rayleigh scatterwhen of 0.90 p,m is used. Determine the attenuation loss at 1.55 p,m. 24-14. (a) Show that the attenuation db/km is given by a = 10 logw (l

24-15.

24-16.

24-17. 24-18.

24-19. 24-20.

24-21.

24-22.

24-23.

f)

the overall fractional power loss from to output over a I-km-long fiber. (b) Determine the attenuation in db/km for fibers having an overall fractional power 90%, and 99%. loss of Determine (a) the length and (b) transit time for the and shortest trajectories in a fiber of I km, having a COre index of 1.46 and a index of 1.45. Evaluate modal distortion in a fiber by calculating the difference in transit time by an axial ray and a ray at the maximum enthrough a I-km fiber trance of 35°. Assume a fused silica core index of 1.446. What is the maximum that produce nonoverlapping on due to this frequency of case of modal dispersioll'! Calculate the time between an axial ray and one that enters a fiber at an angle of IS". The core index is 1.48. Calculate the group between the fastest and slowest modes in a I-km-Iong stepindex fiber with 111 1.46 and a relative index differellce a (111 - 112)/112 = 0.003, a source at wavelength 0.9 p,m. Plot the refractive index profile for a GRIN fiber of radius 50 p,m and with 111 = 1.5 and a = 0.02. Do this when the profile parameter a "= 2 and repeat for a = to. Calculate the delay due to modal dispersion in a I-km GRIN fiber with a 2. The maximum core index is 1.46 and the cladding index is 1.44. what factor is this fiber an improvement over a fiber with 111 = 1.46 and 112 1.44? Equation allows calculation of bandwidth for distances less than the equilibrium length of fiber footnote Assume an equilibrium of 1 km and the 3-db bandwidth of a multimode fiber determine for this fiber whose pulse 20 ns/km. Determine the material in a I-km length of fused silica fiber when the source is (a) a LED centered at 820 nm with a spectral width of 40 nm and (b) a LD centered at 820 nm with a width of 4 nm. The total delay time aT due to both modru distortion and material is given by

Determine the total time in a I-km fiber for which III = I A = 820 nm, and IlA 40 nm. 24-24. Waveguide dispersion is measured in a silica fiber at various wavelengths width of 2 nm. The results are: diode sources with a 520

Chap. 24

Fiber

a

1%, laser

!l(T I L) (ps/km)

0.70 0.90

1.88 5.02 7.08 8.40 8.80

LlO 1.40 1.70

M ' versus A in the range 0.70 to 1.70 #Lm. dispersion in ps/km at A = 1.27 and 1.55 #Lm for a source with a spectral width of I nm. 24-25. Compare pulse broadening for a silica fiber due to the three principal causes-modal distortion, material and waveguide dispersion-in a step-index fiber. The core index is 1.470 and the cladding index is 1.455 at A 1 #Lm. Assume a LED source with a width of 25 nm. The values ofthe parameters M and M' are 43 pSlnm-km and 3 ps/nm-km, respectively. (8) Determine each by calculating AT for a I-km of fiber. (b) Determine an overall broadening AT for a I-km length of using """,n",'t"r

(b) Determine the

(AT)2

(ATm
+

+

(ATwg)2

[1] Siegmund, Walter P. "Fiber Optics." In Handbook

[3]

[4]

[6] [7] [8] [9] [10] [11] [12] [13] [14]

[15] [16] [17] [ 18]

Optics, edited by Walter G. Driscoll and William Vaughan. New York: McGraw-Hili Book Company, 1978. Yariv, Amnon. Optical Electronics, 3d ed. New York: Holt, Rinehart and Winston, 1985. Ch. 3. Cheo, Peter K. Fiber Optics Devices and Englewood Cliffs, N.J.: PrenticeHall, 1985. Ch. 4. Gloge, D., and E. A. J. Marcatili. "Multimode Theory of Graded-Core Fibers." Bell Syst. Tech. J. 52 (Nov. 1563. Miller, Stewart E., A. J. Marcatili, and Li Tingye. "Research toward OpticalFiber Transmission Systems." Proc. IEEE 61, no. 12 (Dec. 1973): 1703. Palais, Joseph C. Fiber Optic Communications. Englewood N.J.: Prentice-Hall, 1988. Guenther, Robert. Modern New York: John Wiley and Sons, 1990. Ch. 5. Karim, Mohammad A. Devices and Systems. Boston: PWS-Kent Publishing Company, 1990. Ch. 9. Nerou, Jean Pierre. Introduction to Fiber Optics. Sainte-Foy, Quebec: Les Editions Le Griffon D' Argile, 1988. Waldo T. Fiber Optics Communications, Experiments and Indianapolis: Howard W. Sams and Co., 1982. J. Whitley. "Long Span Fiber Amplifiers." Applied OpUrquhart, Paul, and tics 29, no. 24 (Aug. 1990): 3503. Kapany, Narinder S. "Fiber Optics." Scientific American (Nov. 1960): 72. Busignies, Henri. "Communication Channels." American (Sept. 1972): 98. J. S. "Communication by Optical Fiber." Scientific American (Nov. 1973): 28. Yariv, Amnon. "Guided Wave Optics." American (Jan. 1979): 64. Mandoli, Dina F., and Winslow R. in Plants." Scientific American (Aug. 1984): 90. Katzir, Abraham. "Optical Fibers in Medicine." American (May 1989): 120. Emmanuel. "Lightwave Communications: The Fifth Generation." Scientific American (Jan. 1992): 114.

24

References

521

25 Aperture Image

s

L1

L2

L3

Fourier Optics

INTRODUCTION

Two rather extensive areas in which the Fourier transform is central to applications in optics are treated in this chapter, although, necessarily, somewhat cursorily. The first is included under the general heading of optical data imaging and processing and the second, Fourier-transform spectroscopy. Both are included within a branch of referred to generally as Fourier optics, in which the Fourier transform, convolution, and correlation are central concepts of mathematical analysis. Optical data processing takes advantage of the fact that the lens constitutes a Fourier-transform computer, capable of transforming a complex two-dimenresolution and at the sional pattern into a two-dimensional transfonn at very speed of light. The pattern of a spatial object by the lens is shown pattern to be a two-dimensional Fourier transform, or spectrum, of the input. may be manipulated in turn, masks or filters to modify the final image produced by a second lens in a process called spatial filtering. Since vdrious details of the image can be modified by appropriate filtering, this technique is exploited in such areaS as contrast enhancement and restoration. If the image is compared directly with a second the two may be optically correlated. Such correlation is for example, in the problem of pattern recognition. By such optical means, two-dimensional pictures or text are processed at once, without the necessity of sequential of the Optical data processing a fruitful convergence of the fields of optics, infonnation science, and holography. As in 522

many other fields, the availability of the laser as a coherent source has insured rapid growth. Fourier-transform spectroscopy capitalizes on the fact that the spatial or temporal variations of an irradiance due to polychromatic radiation can be Fourier-transformed into a spectral decomposition of the This tecJhmque makes possible another application of interferometry with distinct advantages for spectroscopy. Fourier-transform spectroscopy is the subject of the second part of the pre:seru chapter.

25-1 OPTICAL DATA IMAGING AND PROCESSING Fraunhofer Diffraction and the Fourier Transform. We wish to show that the Fraunhofer diffraction is, within certain approximations, the Fourier transform (Section 12-1) of the E-field amplitude distribution in the plane. the one-dimensional transforms presented in Chapter 12:

2~ i~OO g (k)e-

j(x) g(k)

=

i~oo

ila

dk

1)

dx

(25-2)

Equation 1) states that an arbitrary, nonperiodk function j(x) can be synthesized summing a continuous distribution of plane waves with amplitude distribution g given by (25-2), The functionsj(x) and g(k) are said to be a Fouriertransform so that they are Fourier transforms of one another. Symbolically, g(k)

=

j(x) The

j(x)}

(25-3)

l{g(k)}

(25-4)

of the inverse transform of a >"".LA'L'" returns the function, that is, gp-I{g(k)}

j(x)

j(x)

(25-5)

dk,

(25-6)

In two dimensions, the transforms take the form +",

j(x, y) =

(2~)2 JJg(kx, +'"

g(kx, ky )

=

JJj(x,

dxdy

(25-7)

The function of two variables j(x, y) can thus be synthesized a distribution of plane waves, each with amplitude g(kx, ky ) and constant phase, such that

+ yky

constant

(25-8)

The quantities k" and k. are the spatial jrequency (2Tl'/ A) components needed in the expansion to represent the desired function j(x, y). The individual plane waves in the continuous distribution intersect the xy-plane along the straight lines defined by Eq. (25-8). As kJ( and ky vary, the slopes of these lines vary. Thus the synthesis involves plane waves that vary in direction. Consider the Fraunhofer diffraction due to an arbitrary aperture situI. Plane monochromatic waves ated in an as shown in Figure ,·hY'f .. " , · .

Sec. 25-1

Optical Data Imaging and Processing

y

Figure 2S-1 in the

Fraunhofer diffraction in the spectrum XY-plane due 10 an aperture

from the aperture (xy) plane. The diffraction pattern is observed in the XV-plane, which we shall call the spectrum plane, a distance Z the axis. The contribution dEp at an arbitrary point P due to the light amplitude an elemental area da UU'"U':lI;'; point 0 in the is given (25-9) from point 0 to point P. the obliquity for a spherical wave whose amplitude decreases r. The quantity is the source strength, or amplitude per unit area of l'lnprhllrf' in the neighborhood of point O. The combination Esda is then the ampli1) from 0 due to the elemental area da. If the tude at unit distance (r aperture is not uniformly illuminated or is not uniformly transparent, then = Es(x, y) and is called the aperture function. In wand k refer to the properties of the incident and radiation. The point P in the spectrum plane is a distance ro from the of the xy-coordinate system in the plane. The r may be referred to the distance ro as follows. From the geometry apI, in Figure where r is the

0, Eq. (25-9)

+ (Y

- y)2

= Xl +

+

= (X - X)2

+

(Z - 0)2

and

so that

r2 = r~

2xX

2yY

+ (x 2 +

10)

Although the dimensions X and Y in the spectrum may be appreciable, the dimensions x and y are typically negligible in '"'''''''V.''''I<'VII with ro for far-field diffraction. Accordingly, the terms Xl and y2 are and Eq. (25-10) is rewritten as

r= r{ I 524

25

Fourier

2 ..!.--~.--::.

1/2

11)

In this form. Eq. (25-]]) is immediately ad~IDtaible to approximation by the binomial expansion (1 + uy = 1 + Wu + . . . . only the first two terms.

roll

r

(25-12)

In Eq. (25-9), the distance r appears in both the amplitude and the phase. In the amplitude it can be safely approximated by the Z between planes, but in the phase we use the approximate Then (25-13) so that, upon integration over the area

the

we have

dxdy If we are interested in the """"I'tr'" ..... plane, we may set Z

of

rp"U"'fP

Ep =

(25-]4)

electric field in the

= 1 and write

ff

E.(x. y)eik(xX+YY)/ro dx dy

(25-15)

introducing the angular spatial frequencies,

k" ==

kX ro

and

to each point (X, Y) in the

k y == kY ro

16) 15) may be ex-

"''''''I'trlltn

pressed as

Ep(kx , k y) =

ff

E.(x,

dx

(25-17)

In this form, Eq. (25-17) may be compared We see that and E. are related Iml'el1'ie transform, as in Eq. (25-6), is

E.(x, y) =

(2~y

ff

Ep(kx ,

(25-18)

Within the approximations made, we have shown that the tern k y ) is just the two-dimensional function described by E.(x, y). The continuous of constituent recuonal plane wave." is responsible for the "bending" of the light into the of the two-dimensional diffraction pattern. Optical Spectrum Analysis. The Fraunhofer diffraction pattern of a is most conveniently displayed using a positive lens, as in 25from a monochromatic (temporally coherent) point source \"..,"U
Sec. 25-1

Optical Data Imaging and Processing

525

y y

x

m; -2

•

Figure 25-2 Fraunhofer diffraction of a Ronchi

by any photographic negative. For simplicity we shall the function to vary like a square wave, such as would be produced by a Ronchi ruling, a of parallel straight lines with large space, whose opaque and transparent regions are of equal width. Since the Fourier tr.msform is an amplitude (not an ance) transform, we describe the square wave in 25-3 by amplitude of the "".nittprl light. We refer to the ratio of tm.nsmitted to amplitudes as caHed the transmission, in contrast with the ratio i.rrddiances It!10 , which we the transmittance. Transmittance is then the square of the transmission. The aperture function, involving amplitudes, may also be called the transmission lion. Lens L 2 acts a..<; a Fourier-transform lens. With a transmL<;sion function in its first plane, the Fraunhofer diffraction pattern, which is its Fourier tr<>ncf".....r,m produced in the second focal plane, the spectrum, or output, The ....v .......,' .. ing acts as a coarse grating, producing a series of bright that correspond to the various orders of diffraction. Since the Ronchi rulings are aligned parallel to the xaxis in the plane, the spectrum of bright in the output plane occurs along the Y-direction, as shown. Now, according to the grating equation, d sin 6

where d is the tical axis by

d

f

period of the ruling. Spots appear at distances

19)

the op-

We wish to show now that this series of bright spots in the spectrum of frerequired in a Fourier the aperture or transmission function

... u •..,....... ,..,'"

25-3 Transmission function of d due to a Ronchi ruling, in which opaque and Iransmitting widths are

I - - - - t - d --00-

526

Chap. 25

Fourier Optics

spatial in the Fourier 16). In the Y-direction these are given by

....1;.... '...

were

kY

f Since the transmission function for the Ronchi is a periodic square function, it is represented by a set of \.AI'-""'''\'''''''' in a Fourier series rather than by a of in a integral Let us introduce a wave continuous number or "normalized" form of the spatial in the Fourier by

Then, substituting for from Eq. (25-21) and for Y from Eq. (25-20), we the spectrum of spatial frequencies displayed in the diffraction

for

m d The central spot with m 0 thus corresponds to a frequency Vy = 0, the DC component, in analogy with electrical frequencies. The first-order (m = 1) spots above and below the central represent the frequency VYI = lid. order (m > I) higher harmonics by mvYl. We see when the frequency of the square wave is larger (more spaced rulings with smaller d), the fundamental in the Fourier is also larger, and the separabe familiar from our of tion Y1 = AfI d is increased~a fact that should the diffraction ","'UHi!';. A Fourier of the square function calculated in Chapter 12 for a square wave as an even gives the Fourier

f(Y) = "21

2 ( cos kY + 3"I cos 3kY + 5I cos 5kY + . . .) + 7T

Here we find a constant (k 0) term of 4corresponding to the DC component or central spot of the diffraction pattern; a term with fundamental (spatial) t;,p"l1pn£'v kl 27Tld; and terms higher odd , .... The absence of the even harmonics at first be puzzling, on the of Eq. (25-23), since it woud lead us to all the higher harmonics in the representation. The even harmonics, however, are those corre.<;ponding to the missing o~ders in the grating These orders are expected when the ratio of slit ""'.""..,,"-'" the width of the slit opening, precisely the case in the Ronchi squares of the coefficients in the Fourier series are proportional to the Irnloullloes the corresponding diffraction spots. Suppose now that the transmission function is not a square wave but a sine ruling have gradually such that the wave. [f the lines of the transmitted we have the sinusoidal grating. the Fourier series to this kind of aperture function, it clear that orders in the spectrum higher than m = ] do not appear. only one is required to represent a sine wave. Why then does the """"rotrJ1n> also show a central the DC component with m = O? A little thought make clear that an aperture function cannot be produced with both 25-4a. A photopositive and negative portions, like the pure sine wave of negative, at points of ideal opacity, may produce an amplitude E = 0 but

Sec. 25-1

Optical Data

1......... I'.. r'n

and Processing

521

E

E

I - - - - - - - - - - - -__ wt

25-4 Sinusoidal amplitude or transmission functions including negative displacements as in (a), and with all displacement~ positive, as in (b). AClual aperlure functions do 001 have (a) E = Eo sin wt

(b) E = Eve

+ Eo sin wt

cannot provide negative values. Thus the sinusoidal a transmission function like that of Figure 25-4b, in which the sine wave is by a DC bias. It is the component Eoc in the figure that accounts for the zeroth-order difof the Fourier sinusoidal grating is the diffracproduced by two slits. As shown when considering the interference of two point sources, the resultant diffraction pattern is a series of fringes whose pattern across the fringes is or sinusoidal in form. A film or other type of detector records the irradiance pattern in the form shown in 25-4b.

Optical Filtering. We have seen that the back focal plane of the transform plane in which a Fourier transform of the aperture or transmission is located. If this spectrum plane now serves in turn as a new aperture function for a second lens L 3, a focal length away the back focal plane the second lens recieves the Fourier transform of the new aperture function. This second Fourier transform is thus the transform of the of the original WI\,""'II and so return..<; the original an image of the is formed there. This conclusion also follows a simple apthe ray included in plication of the laws of geometrical optics, evident Figure Each diffraction in the spectrum plane, with (X, Y), repref"r="'"'~"'f""<' ....."."".,t in the aperture as we have pointed out. spots now helps to the of t~e aperture in How is this image affected if the light from one or more of these Aperture

Image

Spectrum I I

s

L1

L2

Figure 25-5 Optical filter.

528

Chap. 25

Fourier Optics

L3

diffraction spots is blocked, so that its contribution to the image is subtracted out? From our knowledge of Fourier series, we conclude that the finer features of the image disappear when spots corresponding to the higher spatial frequencies are blocked. If all spots are blocked except the DC component, or undeviated diffraction beam-say by an iris diaphragm centered on the centrdl spot-the image plane is illuminated but no image details appear. As the circular opening of the diaphragm is gradually widened, higher spatial frequencies are admitted and the image gradually sharpens. The physical operation of opening the diaphragm is thus analogous mathematically to the systematic inclusion of higher and higher frequency terms in the Fourier series representing the aperture function. Optical filtering is the process of intentionally blocking certain portions-that is, certain spatial frequencies-present in the diffraction pattern, to manipUlate the image. Suppose, for example, that the aperture function is the superposition of two sine waves that are produced by back-to-back sinusoidal gratings with parallel rulings but different line spacings or spatial frequencies. The diffraction pattern consists, in addition to the direct beam, of two pairs of light spots, each pair due to one of the spatial frequencies present. If one of these pairs is blocked, that frequency is eliminated, or filtered from the illumination. The image is a sinusoidal pattern of the other frequency. This example shows how optical filtering is applied to the extraction of desired periodic signals from background noise or, on the other hand, to the elimination of periodic noise from a desirable signal. As another example, suppose the aperture function is a television picture in which horizontal raster lines are visible. The diffraction pattern due to this function may be quite complicated, but the raster lines, like a Ronchi ruling, produce a series of diffraction spots along the vertical direction in the spectrum plane. If a rectangular-shaped, opaque shield is used to block the contribution of these spots, the raster line frequencies are filtered out and the final image is a reproduction of the TV picture but without the raster lines present. This technique was used to remove video scan lines from the video micrograph of a diatom frustule, as shown in Figure 25-6. From the point of view of optical filtering, then, it should be clear that a diaphragm, which blocks all but those frequencies near the direct beam, functions as a low-pass optical filter; a diaphragm, which blocks only those frequencies near the direct beam, functions as a high-pass optical filter; and an annular ring, which blocks the lowest and the highest frequencies, functions as a band-pass filter. A case in point is the suppresion of low spatial frequencies, or high-pass optical filtering, to enhance the contrast in a photograph. (Recall the importance of the high-frequency components in a Fourier series when synthesizing the fine features" of a function, like the corners of a square wave; Section 12-1.) More complex filtering has also been used in image restoration, for example, in the deblurring of lunar photographs.

Optical Correlation. As we have seen, an image of the two-dimensional object situated in the aperture plane is formed in the image plane of the optical filter (Figure 25-5). Suppose now that in the position of the image plane we insert a second object, that is, another photographic transparency containing an image, so that the image of the original object is superimposed over that of the second. Then the amount of light passed by the second object at any point depends both on the amount of light available in the image and the transparency of the second object. Let the light so transmitted be intercepted by an additional lens and, in its second focal plane, be monitored by a film or light detector, as shown in Figure 25-7. We have, in effect, added an optical spectrum analyzer to the optical filter of Figure 25-5. In the output plane where the detector is placed, we expect to measure the spectrum or Sec. 25-1

Optical Data Imaging and Processing

529

!b)

(a)

25·6 (a) Video micrograph of (shown vertically). (b) Video mu'rn<,rn,Yi video scan lines. (Photos by Gordon

scan lines filtered 10 remove 1he lJulvf'rsily of Pennsylvania.) rt+"'~ien[e(]

COlinCIOC

by the

transmiued means of compardeter-

vU.",UIJU< occurs, a ca'le to the other, no'we'ver with the regions of

s

L1

Spectrum analyzer

Optical filter

Figure 25·7 a spectrum

Chap. 25

Fourier

Oplical correla!or formed

cm"hi"':rtrinn

of an

filter and

throughput and correlation are reduced. If the first obis a photographic of the block letter A and the secis of a similar a high degree of correlation should be obtained when the are properly positioned; on the other hand, if the second object is a photographic image of the letter B, the maximum light be significantly reduced. This technique of pattern to the and counting of small particase of blood cells, or to the search for characph(lto~~~)bs, medical X-rays, and fingerprint files. more precisely in mathematical terms. Let the first be illuminated uniformly by light of unit amplitude, and be by E,(X, y). The transmitted light, ampli....."l".""" at the position of the second object in the image plane, rer:lr~;cnted - y). The change to negative coordinates is required relative to the object. If now the second transmission function is y), the transmitted is the product function E , ( - x, -y) E2(X, y). The Fourier transform or spectrum of this composite transmission function is formed in the that the diffraction pattern there is described by 00

II

(25-25)

To concentrate only on the spatial kx ky

or DC component, in the pattern, we set the to zero, so that

II

(25-26)

Both transmission functions y) been referred to xy-coordinate system origins that differ tralnsi:!ltlC)ll along the Z-, or optical, axis. If the by an arbitrary translation given by comfirst object is shifted in the ",,,,,,rlmrp ponents (qx, qy), for trrurlSnnSSlon function must reOect a trrurlslation of origin within the pYr.rp~:~prl more generally by 00

II The lion,

in

- x, qy -

is an example of

y) dxdy

(25-27)

two-dimensional convolution func-

x, qy

(25-28)

y)f2(X, y) dx dy

If~

~

~

= fl(X, y)

then the

uv~.au "v

nte:gnm(l of may be written as positive the correlotion function, (25-29)

II eo

4lu(qx, qy)

Sec. 25-1

fl(X + qx, y + qy)

Optical Data Imaging and

Prnlr:"'~:!O:inln

y) dx dy

Further. when!1 andJi are merely shifted versions of the same function. we instead of the autocorrelation function, y) dx

(25-30)

Transmission functions with inversion symmetry are imaged in such a way that the actual image inversion due to the lens is not Let us briefly examine the autocorrelation integral of is a product of two functions and is nonzero only at those y) both functions have nonzero values. With (qx. q,.) fixed, the integral is the area under a curve representing the product of the two functions. This area, which we call dearly depends on the If (q" q,.) are so that there is no overlap of the funcchoice of the area and correlation are zero. q. and q,. are both zero, the functions '""-'",...'.....,_, yielding a product curve with the maximum area and correlation. As an eX~lml)le, Figure 25-8. we have chosen as a function the top half of a circle. As one curve is translated along the x-axis relative to the other, their autocorrelation varies as a function of the parameter Bx. the displacement of their y-axes. The example illustrates a one-dimensional correlation.

, I

I j

\ \ \ \

I

I j

~ 8=-4

I<'igure 25·8 One-dimensional aUlocorrelation «1>(B) of a semicircle with rddius 3 as a function of the displacement parameter B. Several relation curve are referred to the specific translalions thai

We see from Eq (25-29) that the correlation is the DC component. or spectral point of the Fourier transform, or Thus a detector, on axis at the output plane in the optical of Figure 25-7, it is only to the measures correlation. More precisely. measures a quantity proportional to the square of correlation. As the object in the aperture plane is translated along its the light energy in the direct beam producing the correlation function ¢l 12 (qx). A given function can be simultaneously correlated with other functions separating the reference functions as at the position of the second object. The DC "_".rn/'r

532

25

Fourier Optics

ing to channel are kept separated in the output by a cylindrical lens as the lens. The method of pattern recognition just described is perhaps the simplest to unbut many techniques with various developed. we can only mention briefly one other that makes use The technique was introduced by Vander Lugt in UV"V~,l"l11 is made of a particular pattern to be "recognized." Let us .-iich·il"".;,"," by the function f. The plate is situated 1:'01L1rller-tralnSionn plane. The resulting is called a or a Vander Lugt filter, its originator. This holographic filter is subsequently used in the Fourier together with various test patterns distributions 8.,82, ... , in the object plane. It can be shown [1, 2] that, in three angularly beams result, so that three distinct appear in the plane. One of these is centered on the optical axis, and the other two, otr-aJ!llS. respectively, the convolution and the correlation of the f and 8 When test pattern 8 matches the desired patternf, the rnr'rpl'"tirm with a bright, central spot, and pattern recognition is achieved. '-JU'''''''' correlation techniques have been developed that allow recognition of a of its size or orientation. Matched using have been devised to reduce the background in coherent light systems. Another Model of Imaging: Convolution. In the preceding sections, we have as (I) a result of diffraction or Fourier analysis, produca and their recombination, or Fourier to form the image. We wish now to introduce some of the mathematical formalism and terminology commonly used to discuss Fourier transformations in approach to imaging. plane) Consl,der a two-dimensional aperture by some intervening optical system. We assume the two sets of axes are similarly oriented, as in Figure 25-1. In the case of a optical system, there is established a one-to-one correspondence between object and image of I. Let the irradiFor simplicity, we shall assume a lateral ance of such a (hypothetical) perfect be 10 (X, Y) == 10 (x, y). In reality, each object point is spread out over conjugate image point, due to diffraction and aberration. In this model, the resultant image is considered to be of such "blurred" image points. In a linear system, these elementary ....".""""'" are simply additive. Let the actual irradiance over the image from 10 (X, Y) to Ii (X, Y) clearly by Ji (X, Y). The ....h~lrl'l.~tl'lr17.'" the optical system and is accomplished by a third function. called the rU1i!Ctlon. G(x. y, X, Y). For example, in the case of an aberration-free function describing the 16-3). G is if we assume the point to be space-invariant (independent of object coordinates), it can on the relative displacement of

x, Y

G(x, y, X, Y)

if the light from the object plane is

Ilf the

Sec. 25-1

incoh(~reJflt

y) irradiances add,· and we

is coherellt, the sum is a vector sum of complex electric field amplitudes.

Optical Data Imaging and

I'Jrr.....,""

533

point (X, Y) due to all object points (x, y):

can write for the irradiance at the

!:.!!;!]

II~

P(X -

objecl irradiance

~, Y - y~ dx dy

(25-31)

point spread function

The integral in (25-31) is called the convoiuJion 2 of the functions 10 and G, usually abbreviated by

Ii == 10 ® G

(25-32)

Suppose that we calculate the Fourier transform of each of these functions, represented by 9'(10 ), 9'(11), and 'J'(G). The convolution theorem problems) states that the Fourier transform of the convolution two functions is equal to the product of their individual transforms. Symbolically.

® G) = 9'(10) X 'J'(G)

(25-33)

The content of Eqs. and (25-33) can be summarized by stating that convolution in real space corresponds to multiplication in Fourier space. Combining this result with our understanding of the equivalence of Fourier transform and spatial frequency spectrum (or Fraunhofer function) of an aperture function, we can read as follows: The spatial frequency """,·('f.-urn of image is equal to product of the frequency <:'I1r'ctrnm ject irradiance times the spatial frequency of the point The last of these, is caned the transfer junction because it trairtsf4'!rs or the object spectrum into the image ",.",·.. f.-lIm Thus the OTF is used to the performance of an optical 1>V,,"".,. .... As an example of the thcorem. recall the results given in (1632) for the Fraunhofer diffraction of a grating. There we found that the two product functions could be interpreted separately as diffraction from a slit and intertprpnc'p. from multiple (negligible width) slits. these functions are the Fourier transforms of their we can say that (1) in Fourier space, the grating diffraction pattern is given product of the I-inllrlt'r transform of the single-aperture function and the of the array sources the grating; or (2) in real space, the aperture function is a function. convolution of the slit-array aperture function with the single-slit second formulation is suggested by 25-9. Practically speaking, if one knows the Fourier transform of simple aperture functions, one can more easily cal-

Aperture funclion for an array of slits

Aperture function for a single slit

Aperture function for a grating

Figure 25-9 Synl00lic represenlalion of the convolulion lheorem for a grating. 2 This has other imponant applicalions in physics. II requires the multiplicalion of one function at each point by the v.i:lole of another function and then the summation of the results. Hence it is also called or superposition

534

Chap. 25

Fourier Optics

culate the Fraunhofer that results from more complicated aperture by using the convolution thcorem. System Characterization of the imaging capacity of an by simply citing its resolving power does not give an adequate assessment of the system's performance. The .....".1'"....",-1 criterion of is the transfer function (OTF). To test an system properly, both high and low spatial are .."",,, ..,>.-1 As usual, low spatial frequencies are sufficient to image the gross details of an object, while are required to reproduce the finer One technique of an optical system is to use a series of test ...,f'f"n"" with sinusoidally each at a different spatial frequency K. When illuminated, incoherent imaging of the test pattern takes place. Let us assume a system magnification of I. produced by a linear optical system is also same spatial but with a modification of amplitude soidal at as shown in 25-lOa. The OTF encompasses both modifications when it is form: written in

OTF

(MTF) e i{PTF)

where MTF is the modulus and PTF is the phase. When either is known as a function of MTF is the modulation transfer function and PTF is the phase tran.ifer function. corresponding to object and are described by by their contrast modulation 1', and 1'1 =

1'0

....:;;;;::.:.--=

MTF ----------Omax

1.0

0.5

(8)

(bl

25-10 (a) Irradiance sinusoids of object and image, both of spatial frequency K. The optical system has unit magnification. (b) Modulation transfer function (MTF) for three optical systems plotted against

Then the MTF and OTF are given, simply, by MTF = 1'/11'0 and

PTF

(25-36)

MTF at various spatial curves shown in Figperformance than resoluure 25-lOb, allows a more complete evaluation of a MTF of 1 as the tion alone. Three systems are shown characterized. AU spatial approaches zero, but indicate different resolution limits as MTF becomes zero. A clearly shows the B has a lower frequency limit than system C, but better performance at lower tfClCjUe:nclles. Sec. 25-1

Optical Data Imaging and Processing

535

25-2 FOURIER-TRANSFORM SPECTROSCOPY Fourier-transform spectroscopy represents an elegant alternative to traditional methods of spectrum analysis. The special advantages of this technique have led to widespread applications in research and industry. Employing as a spectrometer an instrument such as the Michelson these advantages derive both from the at signal and from the presence of entire use of a large at signal output. The large energy throughput that results from the use of a large of aperture is called the Jacquinot advantage, whereas the simultaneous the entire range a single scan of the is referred to as the Fellgett, or multiplex, advantage. Thus the is not limited, as arc prism and grating spectrometers. by the presence of narrow slits that restrict both the wavelength interval irradiance available at anyone time. In addition. the technique is of high limited in principle only the and the region analysis. sample width of the input The aperture and integrated throughput of the Michelson interferometer make it useful as a Fourier-transform spectrometer. It will be shown in the following treatment that spectral distribution, or spectrogram (irradiance versus wave of the incident on a Michelson is the Fourier transform of the irradiance distribution. or interferogram (irradiance versus path difference), of its two-beam interference as a function of mirror movement. 25-11 schematically shows the Michelson interferometer, which uses a beam spJ itter SP to equal-amplitude portions of a spectral input beam from source Sand reooite them again reflection from mirrors M I and M 2. The beams are collected at detector D. Let the electric fields of the interfering beams for a particular wave number k (= 21T /11) component in the light source, on arrival at the tector, be by =

Eo cos (hI - wI)

(25-37)

cos (h2 - Wi)

(25-38)

and the two beams have a physical path difference of x = X2 Xl betwccn separation and recombination. The time-averaged irradiance for the k component at the detector is then It = + £2)2) which

as also calculated in Chapter 10,

lk

210( 1

+ cos

Figure 25-11

Elements of a Michelson interferometer used as a Fourier -trallSform spectrometer.

536

Chap. 25

Fourier

where 10 represents the time-averaged of one beam. there will be a spread of k values in the source, h ean be interpreted as irradiance I (k) per unit k interval at k, an integrated irradiance over all wavelengths of I =

f'" I (k) dk = f'" 2lo(k) dk + ftc 2fo(k) cos (kx) dk

(25-40)

0 0 0

The first term in the result behaves as a term, representing the constant inteirradiance due to all components in the two noninterfering beams betwcen the two beams added together. The second term represents and can be considered as a positive or negative deviation from the constant term, dependent upon the path difference x. Irradiance fluctuations about the constant comprise the spectral distribution given

I (x)

L" I (k) cos (Ia) dk

which is the Fourier transform of the

I(k) =

(25-41)

Cnf'l't'rnc"",,.m

(~:) L" I (x) cos (kx) dx

(25-42)

detection output I (x), as a function of path difference x, at a point on optical axis the system enables one to the spectral irradiance distribution I (k) as a function of wavenumber by the Fourier-transform inte25- I 2 three experimental sample intergration indicated in (25-42). In ferograms are shown, produced by a Michelson interferometer using various spectral inputs. Such interferograms are approximated for the purposes of Fourier-transform calculations When the I (x) is a discrete set of sample points, continuous Fourier transform is allowed to go over into sums and is referred to as a discrete Fourier transform. The use of finite sampling intervals across a finite total sample width or window leads to limitations both in the resolvthat is unambiguously ing power of the instrument and in the minimum handled by the transform calculation. It can be shown that the restriction of data to a finite window Xw limits the resolution of the spectral distribution so that the minimum resolvable wavelength interval is given (25-43) yielding a

res01vm~

power of

One sees that the resolution is by using sample widths. For example, cm, a total path difference or of 1 cm, a mirror movement of results in a resolving power at 500 nm of 20,000 and a resolution 0.025 nm. Spectrometers have been built with mirror displacements of a meter or more, yielding resolving powers of 1(f or However, another important limmust be into account. Because true interferogram is only approximated at a specific sampling interval (nm/reading), a well-known phenomenon in sampling theory called aliasing plaees a limit on the smallest wavelength that can be unambiguously processed by this method. Wavelengths present in the input radiation, which are smaller than a particular show up as wavelengths in the transformed Such overlapping of wavelengths can be avoided observmust be sampled at a rate at the Nyquist criterion of sampling theory: The Sec. 25-2

Fourier-Transform Spectroscopy

537

(al

(b)

lei Path difference

Figure 2S-1l produced by a Michelson interferometer ferent light sources. (a) He-Ne laser. (b) Hg source. violet filter. (c) unfiltered.

least twice as high as its highest-frequency component. It is this criterion is also used in the production of modern where an audio signal sampling rate of 50 kHz ensures accurate of 20 kHz. Expressed in terms maximum audio parameters, the criterion states equivalently that, to avoid aml:s1[lg lerval must be less than the wavelength nr"""p,,t minimum wavelength is by

difsource.

to note that

giving N - I sampling intervals. where N is the total number of now that a large X w , which is beneficial in producing good resolution, may range of the spectrometer, unless N is also suitdetrimental in limiting the ably large. The maximum number of data points, however, is limited by I""nlmnnt,,"r data-storage requirements and by computer time in handling the calculations. number of operations performed by a computer in calculating the spectral this setion I (k) is roughly equal to N 2 • Use of the Cooley-Tukey algorithm for fies of calculations reduces the number of calculations to about N N and is 538

Chap. 25

Fourier Optics

known as the fast Fourier transform. For a transform using 1000 data to around 10,000, a considerpoints would be reduced from 1,000,000 able of computer time and expense. In the just discussed, if the input raOl1atJIOn includes wavelengths in the visible and near ultraviolet, then N could not be less than without jeopardizing the correct analysis of wavelengths as small as 300 nm.

25·1. (8) Calculate the distances from the axis of the first three Ronchi ruling with transmitting slits of width 0.25 mm, as in sume laser irradiation of 632.8 nm and a 50-cm focal lens. (b) What is the wavelength corresponding to the fundamental (c) the three lowest angular in a Fourier repres;ent:attcm (d) What are the ratios of irradiance of the first three relative to the irradiance of the "fundamental"? 25--2. (8) When two transmission functions are put by physically placing two "<:n,,,rpnri,p<: back to back in the how must the combined transmission function relate to the individual transmission functions? Consider an function formed by two perpendicularly crossed Ronchi rulWhat would you expect to see in the 25-3. The of film is defined as the common of its opacity. The opacity, in tum, is just the reciprocal of the transmittance T. (8) show that optical density is equal to T. (b) Show that the total optical density of several film is just the sum of their individual optical densities. (c) What is the transmittance of five layers of film, each with an opacity of 1.25? What is the net optical density of the combined 25·4. The sinusoidal transmission of a grating varies as 5 sin (ay), in units. (a) To produce faithfully the sinusoidal variation in the transmittance of the grating, what bias is required in the transmission function, 100% maximum transmission? (b) Sketch the aperture function with and without the bias term. (c) What is the irradiance function at the detector for unit irradiance incident at the 25-5. Prove the convolution theorem, that is, prove that if h(x)

=:=

f(x) ® g(x)

then 2it[h(x)] = 2it[f(x)]2it[g(x)]

of unit height 25-6. Plot the convolution in one dimension of two identical square and of 6 units for the sinusoidal 25-7. Determine the one-dimensional autocorrelation function A sin (WI + a). function y 25-8. (a) The output of a Michelson spectrometer is fed to a ph{)w.:letc::ctcDr The input is of 5 mm/s, mercury green light of 546.] nm. If one mirror translates at a what is the frequency of modulation of the l'l"JtU'~"'ll is the yellow light (b) What is the beat: of the photocllrre:nt (II of at 5889.95 A and 5895.92 A? 25-9. The mirror translation in a Michelson spectrometer is 5 cm. What is the minimum reat (a) 632.8 nm and at (b) I fLm? solvable

Chap. 25

Problems

539

25-10.

from a mercury falls on the heam spliner of a student Michelson spectrometer. shorter than 360 nm are fillered from the light. The mirror translation rate is 71.5 nm/s. The rate at which spectrogram data is sampled is 1.28 is fed to the computer for Fourier-transform A total of 256 data ",".,'''1>.''. Find the window width Aw; (b) minimum resolvable wavelength interval at 400 nm; (c) minimum that is not subject 10 aliasing; (d) minimum sam9;rr,nrtiino to the Nyquist criterion. difference executed by a Fourier-transform spectrometer operating in is 2.78 mm. lis range is from 4400 to 400 cm- l . (a) What is its resolution in wave number? (b) How many data points must be taken over the scan to avoid aliasing within this (c) What is the scan rate if one run is completed in 30 s?

E. G. Fourier

[I]

An Introduction. 2d ed. New York: Halsted Press, 1987.

Ch. 4. 5. W. Introduction to Fourier Optics. New York: McGraw-Hili Book [3] Almeida. Silverio P .• and Guy Indebetouw. "Pattern Recognition via Complex Spatial Filtering." Applications of Optical Fourier Transforms. edited by Henry Stark. New York: Academic Press, 1982. [4] Duffieux, P. M. The Fourier and Its Applications to Optics, 2d ed. New York: John Wiley and Sons. 1983. [5] Fran~on, M. Optit'al Formation and New York: Academic Press, 1979. [6] Lee, S. H. ed. Optical IHfnrl1"",i"" Prl'lrl',~,~il~"_ Fundamentals. New York: SpringerVerlag, 1981. Robert John. Introductory Fourier Tr{m.~&lrJm .'\nl'l·trfl_~r,'mv New York: Academic [7] 1972. [8] Griffiths, Peter, and James A. de Hascth. Fourier Transform Infrared Spectrometry. New York: John and Sons. 1986. Ch. 1-4. [9] D. and P. L. Pedrotti. "fourier Transforms and the Use of a Microcom.. American Journal of Physics 50 puter in the Advanced (1982): 990. 1101 Classical Optics. San Francisco: W. H: Freeman and ComE in Optics. New York: John Wiley and Sons, 1965.

LI3]

[141 [15] [16] [17]

540

Williams, Charles S., and Orville A. Becklund. Introduction to the Optical Transfer Function. New York: John Wiley and 1989. Smith, E Dow. "How Images Are Formed." Lasers and Readingsfrom Scientific American, p. 59. Sun Francisco: W. H. Freeman and Publishers, 1969. Campbell, W., and Lamberto Maffei. "Contrast and Spatial .. Scientific American (Nov. 1974): 106. Bracewell. Ronald N. "The Fourier Transform." American (June 1989): 86. James,1. E, and R. S. Sternberg. The London: Chapman and Hall Ltd., 1969. Ch. 3. Reynolds, George 0., John B. DeVelis, B. Parrent. Jr.. and Brian J. Thompson. Wash.: SPIE Physical Optics Notebook: Tutorials in Fourier Engineering 1989.

Chap. 25

Fourier Optics

26 =-=-==+--?--7«---.,3>-
Modulated beam

Piezoelectric crystal

Nonlinear Optics and the Modulation of Light

INTRODUCTION The treated in most of this text. including the processes of tT""","""''',,,,r,,, reflection. refraction, and full in the ......,tQ...,~ .. " caned linear optics. When we speak of linear we assume that an optical disturbance propagating through an optical medium can be described by a linear wave equation. As a consequence of this assumption, two harmonic waves in the medium of superposition. without due to the medium obey the interference of the waves, regardless of the itself or as a result of the sHy of the light. Only the wavelength and velocity of a light beam in a transparent material are required to describe its behavior. We now know that when the light becomes great linear optics is not adequate to describe the situation. With the advent of the more intense and we find that the properties of the coherent light made available by the such as refractive become a function of the intensity of the When two or more light waves interfere within the medium, the principle of superlight waves interact with one another and with the position no longer holds. an extension of the linear theory that medium. These rwnlinear phenomena to the radiation. allows for a nonlinear response of optical we define more the area of optics, describe In this some of the new properties that are found, and discuss some of their practical applications in the modulation of light beams. 541

26-1 THE NONUNEAR MEDIUM Nonlinear phenomena are due ultimately to the inability of the dipoles in the optical medium to respond in a linear fushion to the E-field associated with a light beam. Atomic are too massive and inner-core electrons too tightly bound to respond to the alternating at the frequency of (~10 14_1O15 Thus the outer electrons of the atoms in a material are primarily responsible for the polarthe beam's E-field. 1 When the oscillations of these ization of the optical medium electrons in response to the field are the to the Efield, as described later in Section 27-1. However, as the the intensity of the beam, strict proportionality begins to fail, as the harmonic oscillations of a simple spring become increasingly anharmonic as the amplitude of the oscillations Another means of nonlinear behavior without beam is to choose the optical frequency near a resonant frequency of the oscillating dipoles. a technique widely utilized in nonlinear spectroscopy and known as resonance enhancement [I]. The polarization of a linear medium by an field E is usually written in the form (26-1) P = EoXE where X is the susceptibility and Eo is the vacuum permittivity. When from are small, it is possible to represent the modification of the suscepltl bility in a nonlinear medium by a power series in the form

+ ... When substituted into Eq.

P

1), the polarization takes the form

Eo(XIE

P

or

=

(26-2)

PI ~

linear

+ X3 E3 + - . -)

+ X2 +

,(P2

+

P3

(26-3)

+ - . .~

small nonli'near terms

where the subscripts on X match the powers of E and reflect the decreasing magnisusceptibility coefficients tude of the higher-order term.,,_ The linear and characterize the optical properties of the medium, and this relation between P and E completely characterizes the response of the optical medium to the field. linear optics in which the polarizaThe term PI in Eq. (26-3) tion of the medium is simply proportional to the E-field. the E-field is very the coefficients of the higher-power terms are too small to an ow higher-power terms to influence the polarization Only with the availability of intense, coherent light have these higher-order terms become important. The coherence of laser allows the beam to be focused onto small spot sizes of the order of a wavelength. producing E-fields of around J010 VIm, on the order of the fields binding electrons to nuclei in the medium. Peter Franken and his associates are " ..~.rl,;,~..'1 with the first nonlinear coherent optics experiment [2], conducted at the InnJPr.,.n! of Michigan in 1961. The team focused the coherent 694.3-nm from a pulsed ruby laser onto a quartz and detected second harmonic generation, the presence in the output of a weak ultraviolet coherent radiation component at 347.15 nm, twice the frequency or half the wavelength of the exciting light. This nonlinear phenomenon is discussed in the next section. t We speak here of the electric polarization, rather than the wave polarization that of Chapters 14 and 15. 2 See the discussion on focusing laser beams in Section 21-4.

542

Chap. 26

Nonlinear Optics and the Modulation of Light

VIlaS

the subject

26-2 SECOND HARMONIC GENERA nON Second harmonic "".,,, ...~tu'n results from the contribution of the second-order term in Eq. EoX2 E 2

P2

in which the polarization term P2 of the optical medium is proportional Figure 26-1 shows the polarization as a function of the to the square of the E-field for the linear case and the deviation from linearity due to this second-order term. p

Figure 26-1

---+---*"-----'----;,.....- E sponse of polarization P to an field E. For equal and the response of the metrical in the case the nonlinear (curved line) response. In this case, the negative field Eo produces a greater than a positive field of the same

It can be shown that the second-order term makes no contribution to pomaterial, or one a center of symmetry. A larization in an a center of symmetry is characterized by an inversion center, such atomic arrangement rethat if the radial coordinate r is changed to - r, the mains unchanged and so the crystal responds in the same way to a physical should for a influence. In such a crystal, reversing the in any physical Tt1l.J." we should have both

Pz

=

EoX2( + E)2

and

- P2

Because the E-fieId is squared, P2 = -Pz. which can O. The as do not possess inused by Franken, and many other in adversion symmetry. They can, therefore, manifest second harmonic dition to other second-order phenomena to be described nrp
Eo cos wt

substitution into Eq. (26-4) gives I + cos 2wt)]

wt

we have substituted the P1

=

UUUUJ''''-,'"

+

identity for cos 2 wt. Then cos 2wt

(26-5)

The second-order polarization consists of a term of twice the frequency of the apthat represents optical plied optical field as wen as a constant or DC rectification. Thus the dipole oscillations constituting the time dependence of the polarization generate electromagnetic radiation of frequency 2w, which is nrl"
Second Harmonic Generation

543

'''F\fv°if

Applied optical field at III

'''f\f\f\f t3)PJ\;¥ (5)

Nonsymmetrical polarization

Fundamental at (J)

t=============

Figtlre 26-2

E-field (I) and its effects in a nonlinear medium: a nonsymmetrical polarization and its radiative Fourier components (3. 4, 5).

Constant negative bias

the fundamental frequency w. Figure 26-2 illustrates the Fourier of the nn"""'",rr."t'ri("~1 polarization that produces them. the light generated at frequency 2w travels Because of at a different in the optical material than the light at frequency w. The two inOne terfering waves are periodically in and out of step as they traverse the can show [3 j that the output is proportional to

.

'}

SlOe-

where L is the distance into the crystal and k is the wave propagation constant, Ilk k2A> - 2kw, When Ilk = 0, the crystal shows no to nwlc. is present dispersion and intensity factor above is a maximum. Because in Ilk 0 and the factor describes the resultant beof the sinc tween the fundamental and second harmonic. Reference to the a (see Figure 16-2) shows that the change in argument of the function minimum to a maximum is 7T 12. The corresponding change in L is called the coherence length. 3 Thus and

*'

The {'"In"",,,'"-''' length can also be expressed by =-=--

c

(26-6)

3This coherence Chapter 12.

544

Chap. 26

is not to be confused with the quantity of the same name presented in

Nonlinear Optics and the Modulation of

where we have used nw k = -

c

and

llk

c

c

2w - lln

c

Here Ao is the vacuum wavelength of the fundamental and lln the fractive index for fundamental and second harmonic.

(IJttefl~n(:e

in re-

Example the fundamental A 0.8 ILm and the indices in KDP (potassium dihydrogen phosphate) are 1.4802 for the second harmonic and 1.5019 for the fundamental. What is the maximum crystal thickness useful in generating second harmonic light. Solution Substitution into Eq. (26-6) 1.4802)

9.21Lm

This calculation shows that the maximum crystal thickness useful in generating second harmonic is typically in this case around 10 times the wavelength of the fundamental. Crystals with thicknesses equal to their coherence length are impractically small. of nonlinear crysA that can circumvent the small coherence tals involves making use of their As described in Section 15-5, the refractive index (and so the velocity) of the extraordinary (E) ray varies with direction through the crystal. If a direction through the is chosen such that n2w for the I1w for the the fundamental and second harmonic waves remam In and the crystal can be a centimeter or so thick. This technique is called index matching or phase matching and is clarified by 26-3, which shows how the ellipsoids the velocity versus crystal direction for the E- and O-rays intersect along the direction of matching. Second harmonic is not the nonlinear phenomenon that results from the of the polarization on the elcctric field. Table 26-1 lists others, as well as several that depend on the next higher order of approximation, where EoX3E J • For example. notice that for third-order nonlinear proc Optic axis

E-ray velocity surface

O-ray velocity surface

Sec. 26-2

Second Harmonic Generation

Figure 26-3 Velocity ellipsoids for orthogonally polarized beams in a birefringent medium. The is spherical and intersecls the E-ray along a direction (shown relative 10 lhe optic axis) for which both rays have the same

545

TABLE 26-1

LINEAR AND NONLINEAR PROCESSES Nonlinear third order: P3 = EoX3 E 3

Nonlinear second order:

Linear firS! order: p, ::: fiuX,E

P2

fioXzE2

Materials lacking inversion symmetry: Second harmonic generation Three-wave mixing Optical rectification Parametric amplification Pockels effect

Classical optics: Superposition Reflection Refraction Absorption

Third harmonic gellieraition Four-wave mixing Kerr effect Raman scattering Brillouin scattering phase conjugation

on,>I'_""""" nonlinear phe-

esses, third harmonic generation can occur. of the nomena are discussed in the paragraphs that follow.

26-3 FREQUENCY WI'XING

When two or more incident beams with different frequencies are allowed to interfere within a nonlinear dielectric material,Jrequency mixing can occur. Let two jnT"'r..,>r .... '" incident waves of WI and W2 be in the form £ = £01 cos

WJt

or, in the equivalent exponential £ = ~

(e""I'

+ e- iwl ') +

+

Second-order polarization P2 = requires the square of the incident field. Thus, by observation, we can see that the square of £ produces harmonics in 2w 1, 2w2 , WI - W2, and WI + W2. Similarly, third-order polarization produces frequency in all of two or more incident beams. is the process known as parametric amA case of frequency plification. Instead of 2w1 -. W3, as in second harmonic generation, it is possible to achieve a situation in which W3 - . WI + W2, with power flowing from a pump wave at W3 into signal and idler waves at WI and W2. (Notice that in second harmonic generation the power flow is reversed, from the wave at WI to a wave of double the fremore suppose a sman wave at quency at W3 _) To visualize this Ws and a pump wave at wp interact within a nonlinear medium. A difference or idler frequency Wi Wp Ws is produced as a beal frequency. This idler frequency can in turn beat with the pump frequency to enhance the signal frequency, Ws = Wp Wi- In this process, both idler and signal waves are amplified, and idler frequencies rnlrrp.,,,-nrmt1 drawing power from the pump wave. When to resonant frequencies in the nonlinear acting as a tuned Fabry-Perot the parametric oscillator is a tunable source of radiation. Tuning the cavity is accomplished by varying the refractive index of the cavity through control of temperature or an applied DC field. Some of the nonlinear processes itemized in Table 26-1 are discussed in the together with several of their device applications in the probeams. By light modulation, we refer to any means of duction of modulated modifYing the amplitude (AM), frequency (FM), phase. polarization, or direction of a light wave. The purpose of modulation is to render the wave capable of carrying information, as described in Chapter for example, where light beams carry information through optical Light and shutters can accomplish some 546

Chap. 26

Nonlinear Optics and the Modulation of light

modulation mechanically. We are interested here in the modulation that is accomplished the refractive index of a material through the use of elecor means. The basic equation describing nonlinear behavior was as Eq. (26-3), a relation between the polarization of the medium and the applied E-field. In dealing with crystalline which represent most of the useful materials, it is customary to nonlinearity of the index n an equation [4] analogous to for the susceptibility: 1 I 2= +rE+ n

(26-7)

4 respectively, where rand R are the linear and quadratic electro-optic is no other effect present (like strain) that can modand We assume that ify n. The refractive in the absence of an applied is no. In general, the refractive depends on the propagation direction and wave polarization relative to the crystal axes. Since E is a vector field, the coefficients rand R are tensors that reflect the crystal Depending on the of many tensor components may vanish or become equal to others. reducing the total number of independent elements' to represent a particular crystalline material [5].

26-4

POCKEtS EFFECT The Pockels effect results from the linear term in Eq. (26-7), where E is an applied DC field. This effect can be a special case of two-wave where one of the waves is the wave and the other a field of zero frequency. enough to produce The optical E-field can be since the DC field is itself nonlinear behavior. In the DC field redistributes electrons in such a way that birefringence is induced in an otherwise isotropic material, or new axes appear in naturally Since the Pockels effect is a second-order effect materials having relative to the 26-3), it is not found in inversion symmetry. All crystalline materials exhibiting a Pockels are also piezoelectric, that is, they show induced birefringence due to mechanical strain. it Since the effect was discovered in 1893, long before the discovery of the was well known even before intense optical fields became available. In one configuration of the Pockels cen, the natural optical of the crystal is aligned parallel to the Fast and slow axes are induced in a normal to the applied field, as shown in Figure 26-4. If the Pockels cell crystal is rotated until the FA and SA are at to the x- and y-axes, a vertically polarized light the field direction has equal amplitude compowave Eo incident on the nents on FA and SA. experience different refractive and different speeds through the The crystal therefore behaves as a retarder (Section 14-2), and the component waves emerge with a phase dltlen~nce. 41t is important to distinguish the "order" of an electro-optic process as determined from (26-3) for the polarization and as determined from (26-7) for lin' or tw. For example, the Pockets effect is a second-order effect (involving by the first criterion and a first-order effect (involving r) the second. Both criteria are used. 5 The linear electro-optic tensor p,] is defined by the relation ,o.O/n'), = ~JP'JEj. with i 1,2, 3,. ..6 andj = x, y, z. For in crystals oftriclinic symmetry, aU the 18 tensorelemenlS are required; in zinc blende (GaAs), one is in crystals. all elements are zero. (> Sillgle crystals can be divided inlo 32 symmetry classes; of these, 20 show Ihe Pockels effect.

Sec. 26-4

The Pockels Effect

547

y

,,

,, " 'x

SA

Figure 26-4 Pockels cell schematic. The retardation action due to an applied voltage V is suggested by a separation on axis of the polarized components transmitted by the fast and slow axes of the crystal.

z

One component advances in phase by !:1(j) and the other lags in phase by !:1(j) while traversing the crystal of length L, so that their relative phase on emerging is given by cP = 2!:1(j). Now !:1(j) = (27T / Ao)L!:1n. where Ao is the vacuum wavelength and L!:1n represents the optical-path difference. We find !:1n from Eq. (26-7), which, for small changes, can be approximated by dO/n2) = rEo The E2 term is considered negligible in the Pockels effects. Thus dn n

-2-3 = rE l!:1n I

==

r

-n~E

2

Substituting into the phase equations above, we find

cP

27T

3

= ~rnoEL = 1\0

27T 3 -rno V Ao

where V = EL is the voltage applied across the length of the cell. Notice that the phase difference cP is independent of the crystal length. If the Pockels cell is to behave as a half-wave plate, we need to make cP = 7T. The half-wave voltage required is then Ao VHW = - 23 rno

(26-8)

Example Suppose the cell is made from a KD*P crystal of I-cm thickness and the optical wave has a wavelength of 1.06 /-.Lm. What half-wave voltage is required?

548

Chap. 26

Nonlinear Optics and the Modulation of Light

TABLE 26-2 LINEAR ELECTRO-OPTIC COEFFICIENTS FOR REPRESENTATIVE MATERIALS

Material (wavelength if not 633 run) KH 2 P04 (KDP) KD2 P04 (KD*P) (NH.)H1 P04 (ADP) A = O.546nm LiNbO, (lithium niobate) LiTa03 (lithium tantalate) GaAs (gallium arsenide) A = 1O.61Lm ZnS (zinc sulfide) A = O.61Lm Quartz

Linear electrooptic coefficient'

Refractive index

r (pm/V)

no

II 24.1 8.56

L51 1.48

30.9 30.5 1.51

2.18 3.3

2.1

2.36

1.4

L54

1.51

2.29

"Depending on crystalline symmetry, materials have more than one dec"n.. n""~ coefficient. Only one has been listed here for use in a Pockels cell. These and others may be found in [3].

we find r = 24. I x

Solution From Table index of 1.51. Then

v = __l_.06 . . . . . . ._x---:_ _-..,. 2(24.1 x

.51)3

Thus an applied voltage of 6.4 kV transforms the

=

m/V and a

r"i",..,,.,,tn,',,

6390 V into a half-wave

Recall (Section 15-7) that in this case the of polarization of the emergent is rotated by 90° relative to the plane of polarization of the light. If a linear polarizer with TA along the intercept<; the light, as in Figure 26-5, the tmnsmittance of the is zero when V 0 and maximum when V = VHW • Variations in V therefore modify the polarization state of the emergent rendering it elliptical, in general, with an x-component that can be transy

TA

x

linear polarizer

a half26-S The effect of wave voltage to a Pockels cell when the system and analyzer. The includes a crossed beam is transmitvertically ted as a horizontally polarized beam. For other values of applied voltage, the beam incident on the analyzer is elliptically and is only transmiued.

Sec. 26-4

The Pockels Effect

549

mitted by the In effect. the polarizer-analyzer pair transforms phase modulation into amplitude modulation. Thus we see that the transmittance of the can be modulated by in the voltage. Variations of a voltage ",,.,,,,r'lrn'I"l,,,,,,.r\ on V are transformed into variations in light intensity in such a device, modulaJor. known as a Pockels The cell can be used also with the field oriented orthogonally to the that simplifies placement of the electrodes. In the the electrodes are usually end-rings that allow tt....,.me.h and still provide a reasonably uniform field in the crystal. The transmittance of the beam can be expressed [6] by the relation: 1=

V)

• 2 (1T sm ---

2 VHW

(26-9)

in advantage of the more linear region of the a is often inserted between the initial polarizer and the Pockels This has the effect of producing 50% transmittance when V 0 so that the operating point is located at P in the figure, rather than at the origin. variations in modulating voltage, if not too large, then occur with a system response that is Two other closely related applications of the Pockels cell are illustrated in Figthe cell is used as a Q-switch that allows sudden ures 26-7 and 26-8. In dumping of the energy stored in a laser. Without an applied voltage, the Pockels crystal transmits the beam from a laser cavity without changing its to pass vertically polarized light, restate of polarization. A Glan-laser

100

Vow

Applied voltage V

1r!2

Phase difference

Modulating voltage

Chap. 26

Figure 26·6 Transmittance curve for the Pockels cell modulator. Without a quarter-wave plate, the transmittance is zero when the apvoltage is zero. Using a quarter-wave between and modulator, the 50% at operating point P when is zero, Under these conditions, the modulator more linearly to an

Nonlinear Optics and the Modulation of

Horizontally polarized beam rejected

Horizontally polarized beam HR mirror prism Pockels crystal (al

Horizontally polarized beam

}'igure 26·7 Light-controlling action of a Pockets cell. used as a Q-switch. The conliguration in (a) producc... low lran..mission at zero-cell and in (b) high transmission at half-wave vollage. In (b). the incident and reflected beams are separated for darity. Repealed re-entries of the beam into the laser cavity initiates stimulated emission that produces Ihe laser pulse.

(bl

Output beam

Other pulseshaping r:ml'l""ne,nIS

l'igure 26-S Light-controlling action of a Pockels cell. used as a cavity When half-wave is applied to the Pockels cell. it rotates the linear polarization of the laser beam so that il can be dumped by the prism.

HR mirror

jects the beam. With half-wave applied, the polarized beam from the laser cavity is rotated by 90° and is accordingly by the Glanlaser prism, the beam to be back-reflected from a high-reflectance mirror. The configuration now permif,s rapid traversal of the beam back and forth through pulse of laser the laser cavity and stimulated emission occurs, producing an radiation. In Figure the Pockels cell is used to initiate dumping. When half-wave voltage is applied, the polarization state of the la<;er radiation is rotated 90°, so that it can be extracted with the help of the polarizing 26-5 THE KERR EFFECT When the optical medium is isotropic, as in the case of liquids and the Pockels effect is absent and the polarization is modified by the third-order7 electro11n Eq. (26·7), the Kerr effect arises from the £2 term involving the coefficient,

Sec. 26-5

The Kerr Effect

UI"ltmlllt:

electro-optic

551

optic effect, better known as the Kerr effect. This effect, like all third-order effects, occurs whether or not a material possesses inversion symmetry. Actually, the Kerr effect was the first electro-optic effect to be discovered (1875), despite its thirdorder dependence. Kerr cells usually contain nitrobenzene or carbon disulfide in the space between two electrodes across which a voltage is applied, as indicated in Figure 26-9. The applied electric field induces birefringence with an optic axis parallel to the applied field. Light traversing the cell thus encounters two refractive indices for polarizations parallel and perpendicular to the optic axis, and phase retardation results. In this case, Eq. (26-7) becomes

Illn I = R-n6£2 2

(26-10)

Figure 26-9 Kerr cell. The applied voltage creates a field that is perpendicular to the beam direction. As with the Pockds cell, a modulating voltage produces phase modulation that is converted to amplitude modulation by the polarizer-analyzer pair.

Modulating voltage

Experimentally, the difference between nE and no is found to obey a relation of the form, (26-11) where K is the Kerr constant. Equating Eqs. (26-10) and (26-11), we find the relationship between K and R to be

K

= Rn6 2A

As explained for the Pockels cell, the relative phase retardation for the ordinary and extraordinary components is

~ = 211' Liln A

Introducing the Kerr constant through Eq. (26-11), ~

= 211'KV2L d2

where we have set V = Ed and d is the interelectrode distance. To function as a half-wave plate, ~ = 11', and we find that the required voltage VHW is given by d VHW =-== V2KL

552

Chap. 26

Nonlinear Optics and the Modulation of light

(26-12)

TABLE 26-3 KERR CONSTANT FOR SELECTED MATERIALS

K (pm/V')

Material 589 nm, Rf)

Glass (typical) Carbon disulfide (CS,) Water (H 2 0)

4 X 10-6 0.001 0.036 0.052

Nitrotoluene (Cs H1 N02) Nitrobenzene (C;HsNO,)

1.4 2.4

(STP)

Example Consider a nitrobenzene Kerr cell for which K 2.4 x 10- 12 m/V2 26-3) at room temperature and A = 589.3 om. If dIem and L what half-wave voltage is needed? Solution

3 cm,

Substituting into the equation

0.01

26.4 kV

Thus the Kerr cell behaves as a half-wave plate at a voltage of around considerably higher than for a typical Pockels cell.

kV,

Kerr cells can be used as modulators in the manner described for Pockels cells. Because of the voltages required, and because of the toxic and explosive naPockels cells are usually preferred. Nevertheless, Kerr cells apIPIU;a(llOn as high-speed shutters and as a substitute for mechanical choppers, capable of response to frequencies in the range of 1010 Hz. can often found operating as in pulsed lasers.

26-6 THE FARADA Y EFFECT In contrast to the electro-optic discussed to this point, the Faraday is a first-order • !:In C( B) magneto-optic interaction. s When a material is light is passed through it along the placed in a magnetic field and linearly direction of the field, the light is found to linearly polarized, but with a net rotation f3 of the of polarization that is proportional both to the thickness d of the sample and the strength of the field B, to the relation,

f3 = VBd

(26-13)

Here V is the Verdet constant for the usually expressed in minutes of angle per Gauss-cm The Verdet constant is both temperature and wavelength dependent. Its on the wavelength is explained in what follows. An aspect of the Faraday rotation is that the sense of rotation re1a..,.,o ......,>.i.... field direction for a given independent of the prop8The magnetic field analogues of the Kerr effect (where /:..n £2) are the Voigt effect (in gases) and the Cotton-Mouton effect (in liquids), in which a constant magnetic field is applied normal to the light-beam direction. Both are very small ellects and will not be discussed further here. C(

Sec. 26-6

The Faraday Effect

553

agation direction of the light. Thus, repeated forward and backward traversals of the on the of rotation (3. This bematerial by a light beam has a cumulative havior contrasts with that exhibited in the closely related phenomenon of optical activity, discussed in Section 15-6. The optical rotation of the polarized light can be understood as circular birefringence, the existence of different indices of refraction for left-circularly and rightcircularly polarized components (Section Recall that linearly polarized light is equivalent to a combination of right- and left-circularly polarized components. Each component is affected differently by the applied field and traverses the sample with a different speed, since the refractive index is different for the two The end result consists of left- and right-circular components that are out of phase and whose superposition, upon emerging from the Faraday rotator, is linearly polarized light with its plane of polarization rotated relative to its original ori.en!tatjlon A classical [7] of the angle of rotation (3 predicts a relation of the form e ~dn)Bd 2m c dA

(3=

(26-14)

with e and m the electronic charge and mass, c the speed of light, A the wavelength, (26-13), theory preand dn/dA the rotatory dispersion. By comparison with dicts a dependence of the empirical Verdet constant V given by V = _e_ A dn 2mc dA

(26-15)

When the constants are evaluated and V is eXlpresse:d in the standard units of mini (G-cm), Eq. (26-15) becomes V

=

dn

1.0083A dA

We note that the Verdet constant is proportional to both the wavelength of the light and the induced rotatory dispersion in the medium. Measured values for Vat 589.3 nm in Table are modulation, "',,,'v''',''', The Faraday practically field at very high lre.:juenclles 26-10 shows schematically the rotator, a or liquid cell whose axis of symmetry is aligned with a magnetic field. The figure shows a field B. established by TABLE 26-4 VERDET CONSTANT FOR SELECTED MATERIALS V (minta-em)

Material

H2 O Crown Flint CS2 Co. NaCI KCI Quartz ZI1S

554

Chap. 26

A

589 nm 0.0131 0.0161 0.0317 0.0423 0.0160 0.0359 0.02858 0.0166 0.225

Nonlinear Optics and the Modulation of light

Plane of polarization rotates inside material Exit polarization (vertical)

Incident polarization (horizontal!

8~~

CD Head-on view

Head-on view

Figure 26-10 Faraday effect producillg rotatioll of the plane of polarization.

current windings and indicates a rotation of the polarization in the same sense as the current the field. This particular geometry defines a positive Verdet con26-11 illustrates the principal application of the Faraday rotator as an stant. rotator situated between a polaroptical isolator_The isolator consist" of a izer-analyzer pair. As shown, the incident vertically polarized Ught is rotated counterclockwise by the Faraday rotator and in this orientation is fully transmitted by the analyzer. Optical elements (not shown) farther down the line are responsible of this radiation along the optical axis. undesirable back reflections In traversing the Faraday rotator a second time, the polarization vector of the in the same rotational sense, so that it reflected light is rotated an additional emerges horizontally polarized and encounters the polarizer at an angle of 90° with the polarization direction of the original beam. In this state, it is rejected by the popreventing it from continuing back into the optical system, in highpower laser systems, it can damage optical component". Thus the optical isolator effectively isolates the optical from stray

Figure 26-11 Faraday rotator used betweell a polarizer-analyzer pair 10 produce optical isolation uf the optical system providing the incident light.

Example Let us calculate the required length of SF58 flint glass, a Verdet constant of 0.112 min/G-cm for 543.5-nm light, if it is to produce the 45° rotation of the polarization vector required in an optical isolator when the magnetic field has a value of 9 kG. (26-13), we have

Solution d

Sec. 2&6

Ji. VB

The Faraday Effect

555

26-7 THE ACOUSTO-OPTIC EFFECT

Photoelasticity (Section 15-7) is the change in refractive index of a crystal due to mechanical stress. This phenomenon makes possible the AO or acousto-optic ef feet-the interaction of optical and acoustic waves-in which a longitudinal acoustic wave launched by means of a piezoelectric transducer produces a periodic mechanical stress in the crystal. The acoustic wave (see Figure 26-12) consists of a series of compressions and rarefactions (longitudinal vibrations) in atomic density and so a periodic-although small-variation of the refractive index about its normal value. Light incident on this structure is scattered to a greater extent from regions of higher refractive index. The scattering is called Brillouin scattering and is another thirdorder effect that does not require a medium possessing inversion symmetry. Train of compressions and rarefractions

High

low High

low High

low High

low High Low High

---.--t--;.,,----------=-.~ Undeflected light beam

low h -+--'------' Index of refraction

Moving sound waves at speed v

Piezoelectric transducer

Figure 26-12 Variations in the refractive index of a medium due to the passage of a harmonic acoustic wave (left) and the scattering of an incident optical beam by the induced "planes" (right). The inset shows the relationship of wave vectors required by momentum conservation when the acoustic wave has the direction indicated.

If the crystal is thin enough, variations in refractive index along its length lead to corresponding variations in the speed of light so that the crystal behaves as a transmission phase grating (Section 17-5). Some of the incident light beam is diffracted into various orders, according to the diffraction grating equation, rnA = d sin Om, where the grating constant d = As, the acoustic wavelength, and Om is the angle of diffraction in rn th order. This is the so-called Rarnan-Nath regime [8]. If the crystal is thicker, regions of higher refractive index represent planes normal to the direction of the acoustic wave, as suggested in Figure 26-12. In this case, the light wave is diffracted in a way similar to that of X-rays from crystalline planes in Bragg scattering (the Bragg regime). The light wave traverses the crystal with a speed that is around five orders of magnitude greater than the speed of the acoustic wave. This means that the induced grating appears essentially stationary to the light wave. A diffracted light beam appears in any direction in which portions of the wavefront reflected (1) from different parts of a given plane and (2) from successive planes obey the usual condition for constructive interference, that is, the path difference must be an integral number of wavelengths. The first requirement is satisfied by

556

Chap. 26

Nonlinear Optics and the Modulation of Light

of scatter Ihal is equal to the of incidence. This condition is shown 26-12. The second is the condition. The path Inf~rp.11Cp. between the incident and waves is made up of the two segments e. This leads to indicated in the figure. By geometry, each has a magnitude of As the _"I'~""V" mA :::::

sin

.

e=

¢Iv

sm2

(26-16)

a worked out in this context by Brillouin in 1921 and identical to the Bragg equation for X-ray diffraction. 9 In 16), the angles and optical waveIenl;th are those measured within the medium. (However, see problem 26The condition for diffraction maxima can be found by an alternative arnature of waves I). incident makes use of the energy light beam with wave vector k can be considered a flux of photons, each hw and momentum hk, where !U is the frequency and h is the Planck constant divided by 211'. In the same way, the wave with wave vector ks can be COlISl(lenXl a flux of quantized called phofUJns, each of energy hws and momentum fik... The acousto-optic interaction then consists of the or collision of these particles in which both energy and momentum are conserved. two wave vectors k and k" as well as the wave vector k' of the diffracted are waves are shown down in also shown in Figure 26-12. The figure. Conservation of momentum in a collision between a photon and a phonon requires that k' k k•• as indicated in the inset vector triangle. On the other hand, if the wave is reversed in the corresponding vector triangle is satisfied by k' "" k + In

k'

k

k,

(26-17)

With the we interpret this to mean that an incident photon combines with a phonon to the diffracted photon; with the negative sign, the incident photon is considered to yield an additional phonon to the acoustic wave, as well as a photon to the diffracted beam. Thus 17) describes both the and the absorption of a phonon by the crystal We now show that Eq. (26-17) is to the Bragg diffraction condition. Since light are of the order of 10 14 while are generally than 10lO Hz, both !U and !u' are much greater than !Us, that

w' tude,

=W

k'

and

=k

(Figure 26-13) for the wave vectors then shows that, in 9, or, in terms of """V'-'J..."'".

A=

w'

18)

sin 9

equation, with m = 1. of energy of the

m",O'nl_

requires that !U ±!Us

(26-19)

9 An important difference between X-ray diffraction and light diffraction in the Bragg is occurs in a continuous manner from II thick, sinusoidal grating. rather than from disthai light crete planes. As a consequence, only the first-order diffraction, m I, occurs in the effect. In this connection, see Section 13-2
Sec. 26-7

The Acousto-Optic Effect

551

ksin8

26-13 Wave vector triangle in the approltirn!llion r k 'I = Ik I k. The angle II is the Bragg angle. The goornetrical relationship of sides is to the diffraction coodition.

This result shows that the diffracted photon differs in frequency-however litdefrom the photon by the amount (I)., or on the direction of the wave. turns out to be another way of at the Doppler effect for light. When the incident light encounters the acoustic wave approaching, the scattered is greater. and when departing in Figure 26-12), the scattered frequency is less. Example To an idea of the magnitude of the diffraction angle in first order. us consider a typical case in the incident light has a of 550 nm of 200 MHz and a of 3000 m/s. and the acoustic wave has a Solution Then

1.5 x I Using

m

(6),

sin 0 =

A

2As

550 X 10 9 (2)(1.5 x 10-:5)

0.0183

so that

o=

1.050

The acousto-optic (AO) effect can be applied to the modulation of a by controlling its (AM), frequency (PM), or its direction. 26-14 illustrates one means of achieving AM modulation a thin AO material in the Raman-Nath regime. The fraction of light removed from the zero-order fracted beam depends on the magnitude of the induced stress and so on the amplitude of the modulating RF signal. The slit allows only the modulated zero-order beam to be transmitted. Other applications make use of the beam deflection capabilback to 26it should be evident that both ities of the AO effect. wave cause a change a change in frequency and a change in direction of the in the direction of the diffracted beam. If the frequency-sensitive position of the output beam is detected by a photodetector array, the AO device can be used as a spectrum analyzer. because the frequency of the diffracted beam is shifted by an amount equal to the acoustic the beam can be frequency modulated. In this case, the aims to minimize the angular of the light: when used as a spectrum analyzer, the aims instead to maximize it. Another application of the device as a beam deflector to initiate laser-cavity 26-15. When no acoustic wave is applied, the beam dumping is illustrated in (I) bounces back and forth in the laser building up energy to a maximum 558

Chap. 26

Nonlinear Optics and the Modulation of light

First-order diffraction

Incident

light beam

=-=-:::c:.=+-----3J-E-~E-_+--- Modulated beam

26-14 Modulation of a light beam an acousto-optic grating in the Raman-Nath The modulated signal driving the pie,~lectI'ic crystal is transferred to the output beam in zero order. Only the zero- and firstorder diffracted beams are shown.

Piezoelectric crystal

beam

Laser cavity

26·15

HR mirror

Cavily dumping of a laser us-

an acousto-optic beam deflector. Turning on

HR mirror

the acoustic wave deflects the beam outside the laser cavity and initiates cavity dumping.

value. Turning on the acoustic wave causes a deflection of the beam (2) out of the cavity, thereby dumping the energy stored in the

26-8 NONLINEAR OPTICAL PHASE CONJUGATION Optical phase conjugation (OPC) another third-order, nonlinear pheIt was observed relatively recently. the nomenon with some fascinating first publications oecurring in 1971-1972 by researchers in the Soviet Union_ It is so named because the nonlinear interaction a beam that, mathematically, is the spatial complex conjugate of the wave. This means that a new wave is produced that exactly reverses the direction overall factor of the primary retraces the path of the original beam. Thus the phase conjugate wave of the original wavefront. beam and, at each position, the exact We shall, for the moment, view the process as a unique type of reflection and refer to the nonlinear medium that creates the phase conjugate wave as a phase conmirror (PCM). To appreciate the uniqueness of the process, consider Figure wave is which reflections from ordinary mirrors. In (a), a reflected from an ideal (infinite) plane mirror. The reflected wave is also a plane wave. To express mathematically the reversal in direction of the incident wave, the of the kz term is changed from minus to plus. Notice then except for the of the wI term, the reflected wave is the complex the wave and has the properties of a phase conjugate wave as described above: It retraces the 26shows the path of the incident beam and is its phase-reversed replica. the same process for an incident spherical wave. All that is needed to reversed replica on reflection is a concave spherical mirror whose curvature exactly as matches that of the wavefront at incidence. For an Sec. 26-8

Nonlinear Optical Phase Conjugation

I

I I I

""'"iI

M

I

I

.......II I I

I I I

(a) Plane wave

(b) Spherical wave

M

Ie) "Nearly plane" wave

26-16 Three examples of a phasereversed replica of an incidenl wavefroot produced by an ordinary mirror. In each case, the rays corresponding to the wavefront are everywhere normal 10 the mirror surface 00 reflection. A mirror handles all cases and also responds to instantaneous changes in the incident wavefront.

shown in (c). the amplitUde (r) is and includes the amplitUde and phase such a wave as a factors that describe its deviation from a plane wave. We plane wave that has been shaped by passing through a distorting medium, by diffraction, or modulation. phase reversed replica is expressed by laking the complex conjugate of both 'l'(r) and ikz, in other words, the spatial part of the wave equation. To produce this wave with ordinary mirrors, we would need to construct a mirror that matched the wavefront of the incident wave at the instant of reflection. The uniqueness of the peM is that the phase conjugate replica is proof the incident wavefront, as long as the PCM has an duced regardless of the aperture large enough to receive the entire wavefront. Unlike an ordinary mirror, the peM is able to respond immediately to varying spatial and temporal features of an incident wave, such that a wave is continually Notice that the for the phase waves of Figure. 16 imply waves that are equivalent physically to those that result by leaving the spatial part unchanged and reversing the sign of t. Because of this, the phase conjugated Wdve is often referred to as the "time reversed replica" of the incident wave. If one could make a video of the incident wave and then play it backwards, one would see an exact reproduction of the behavior of the wave itself. 26-17a further contrasts the mirror with a peM. With an ormirror, a diverging continues its reflection, whereas with a PCM, the reflected wavefront converges so that it exactly matches the curvature of the incident wave at every point and returns to its point of origin. To elaborate this consider what one would see looking into an ordinary mirror and a peM, as in 26-17b. In the first case, the of the from point, "Pti .."tUHY tip of the nose can be seen by means of light from the mirror, and entering the eye. With a peM, such a diverging wavefront would be returned as a converging wavefront that collapsed at the original point on 560

Chap. 26

l\Ionlinear Optics and the Modulation of Light

Ordinary mirror

Phase conjugate mirror

Phase conjugate mirror

26-17 Differenceg between the action of an ordinary and a wavefront and (b) formalion of an conjugate mirror in (a) reflection of a

the nose. Thus the eye looking into a PCM does not see an the face but views only the light that scatters from the pupil. eye sees a uniformly lit mirror whose brightness depends on the intensity of the light scattered toward the PCM from the cornea covering the pupil. way of OPC is to see it as real time holography. recall with the help of 26-18 the procedure for making a conventional hologram (Section A photographic emulsion is exposed (1) simultaneously to a reference beam and an object which interfere to produce a complex to tern within the emulsion. In the next step (2), the film is ."..E",·,,,.,.,,·,, pattern and the In the third (3), the beam as a phase-reversed of the again illuminating the (3) Reconstruction Reference beam

Reference beam

(1) Formation

, ,, , , ,, ' ",'" , I,," 0"/

(2) Development

-<'

(virtual)

<::"-. . Hologram: three-dimensional ')J interference pattern

(real)

(4) Reconstruction

Reference beam

26-18 Conventional holography. The three steps are sequentiaL

Sec. 26-8

(3) and (4) are alternative tech-

Nonlinear Optical Phase Conjugation

561

1-----

Pump (1) ---11-_»_-1

beam

/

/

Input beam

Pump (2)

beam

OUtput(4) PC beam

Figure 26-19 Real-time hologrn(ily or phase conjugation. The four-wave that produces the output beam occurs simultaneoosiy. The output beam (4) corresponds to the real image beam in conventional hnl"oY",rim

o

(1) + (3) : Real time hologram (2) + )(1) + (3)] : Bragg reflection; PC beam (4)

hologram with the beam. When the reference has the original tion, both real and virtual images of the original object appear, with the virtual image located at position of the original object. If the reference beam is reversed as shown. In OPe, (4), real image appears in the the processes (1), (2), and (4) occur simultaneously, or in "real time." Figure 26-19 shows the PCM exposed to counter-propagating beams (l) and (2), which take the place of the reference beams in Figure 26-18. Similarly. the input and output beams beams. We can consider beams (I) (3) and (4) take the place of the object and and (3) as interfering to produce a real-time hologram, complex regions of alternating refractive index induced by alternating regions of high and low light intensity. As pump beam (2) encounters this structure, Bragg occurs, producing the ""Y, .. ,~,,,t,,, beam (4), a replica of the beam (3). Notice that the roles of beams (1) and could just as well be interchanged in this £1"'''''.....'tion, so that Bragg occurs when beam (I) encounters the interference patterns produced by beams (2) and (3). This technique of producing an ope wave is called four wave mixinf!" involvas it does four interfering waves in a nonlinear medium. 10 to 26-20. Two of the beams are the pump or reference beams that are incident on the from opposite directions. these beams are plane waves of the same wavelength as that of the third input beam, the signal beam whose OPC is desired. The signal beam can have any direction, but its direction determines that of its phase ~~".l~,!::\-'-' The amplitudes of the pump AI and A2 , are much than those of the and conjugate or A4 , so that they can be considered constant within the volume of the nonlinear medium. Analysis of this interac[9] shows thaI the PC output beam has an amplitude proportional to the amplitudes of the other beams: (26-20) A, Pump beam

~

....

Nonlinear medium

PC output

A2 Pump beam

Figure 26-20 Conventional geometry for conjugation by four-wave mixing. beams AI and Az are directed antiparallel and are much stronger than the signal beam A3 • The phase conjugate output of the signal beam is the beam shown as A 4 •

IOMedia used for nonlinear optical phase conjugation upon the wavelength region of interest but are wide including liquids, vapors, and solids. In a small sampling of these materials, we list alcohol, carbon disulfide, sodium and iodine vapors, liquid crystals, ruby, lithium niobate. and semiconductors such as Ge, Si. CdTe. and HgCdTe.

562

Chap. 26

Nonlinear Optics and the Modulation of Light

This

that the phase conjugate beam can be modulated by apmg;-spllllfilly or temporally-one or more of the input beams. nr,\,....,rti,,,,,, of the OPC beam, implied in the preceding, have interesting applications. include aberration correction, and tracking. These properties follow the basic nature of the PC wave, as described above. consider a transparent, distorting medium like frosted which is placed in the path of a light beam on its way to a PCM. The wave is modified in a nonuniform conjugate exactly retraces its and reverses its manner. but its modifications, so that return passage through the medium undoes or "heals" the original distortion. This means that a beam by passing through severe can be recovered "reflection" from a an optical PCM that sends it back the same system. It means that, if the light beam originates from a point source, divergence and effects are reversed is returned through the system, so that the PC beam converges to the when the original point. Furthermore, if the source point moves, the returning beam adjusts so that it continues to point to the source, the pointing and tracking property. I of the usefulness of these will now be given. OthA few 26-21 ers can be found in the references provided at the end of the to concentrate illustrates the basic action of self-targeting, using a PCM. It is Target scatters radiation

Lens

Laser .-.m,,,lifi,pr (introduces disltorlion,sl

PCM

Figure 26-21 Pointing and properties of a PC beam are illustrated in this scheme. A highly directed and beam is returned to the

II Although the aPr~lcsIII01~S described here are theoretically plernenled commercially.

Sec. 26-8

Nonlinear

Phase Conjugation

have not all been im-

563

a high-energy laser beam onto a target, precisely the goal of laser fusion. for is a A laser amplifier that can deliver example, where the directivity high energy invariably introduces distortions in the beam that impair of the beam and the ability to focus it accurately onto a small area. problem can laser that has directivity at the expense of be circumvented by using a power. Some the light from the laser that scatters from the particle enters the laser amplifier, emission and amplification. Distortions result is a produced in this process are removed by using a PCM as a reflector. on the high-power directed laser beam that retraces route to converge original scatterer. In this process, it is fairly assumed that the light traverses the amso quickly that time-varying inhomogeneities in the medium are not beyond the capadetrimental to the healing process. To increase the power on laser amplifier, it is to arrange a number of laser amplifiers bility of a to pick up some of the scattered, pulsed radiation and to synchronize their return pulses to the target. in serves to indicate A slight lions this basic scheme. the seed laser is aboard a laser amplifier is situated at a ground station. The satellite can initiate a return PC beam from the ground station that heals the distortions introduced both by atmospheric turbulence and by the amplifier. Not only does this make possible satellite tracking, but it also for communication between and If one of the pump beams is pulsed, the PC beam will turn on and off accordingly; if one of the pump beams is information-modulated, this modulation is transferred to the return PC beam, as (26-20) implies. area of fiber-optic transfer of Another interesting application of OPC lies in information. Recall 24-3} that a light sent an optic fiber is gradually degraded both modal and spectral dispersion. Figure 26-22 shows a arrowoblength of fiber (fiber l) that is used to transmit an image, in this case, ject. If light reflected downward by the beam splitter is focused, the distortions intr ...l1I1('p£I by the fiber result in a degraded image. To correct this situation, Jight transmitted the beam is from a PCM and directed as shown to a BS

Restored Image Fiber 2

Fiber 1

PCM

Distorted image

Figure 26-22 Restoration of image quality to transmitted by an optical fiber. The healing of the PC beam is accomplished by returning the beam through a second, identical fiber.

564

26

Nonlinear Optics and the Modulation of

length of fiber (fiber 2), which we assume to be identical to fiber l. This arrangement allows the second fiber to compensate for the distortions by the first, so that it is possible to a high-quality image at a remote location.

26-1. Write out the third-order terms of the polarization for a single beam described a plane wave with amplitude and frequency w. Whal frequencies appear in the polarization wave? 26-2. Write out the third-order terms of the for two-beam interaction, where waves having and and WI and W2, the beams are respectively. What frequencies are radiated by the n"ll",r'7""'An 26-3. Write out the second-order terms of the polarization where the beams are plane waves having amplitudes Eo., Eu2. and E0.1 , and frequencies WI, W2, and W,' What are radiated by the polarization wave? 26·4. show that the linear electro-optic effect is found only in crystals inversion symmetry. 26·5. (3) Determine the coherence for second harmonic generation in KDP when laser light at An = 694 nm. Appropriate refractive insubmitted to pulsed dices are n (694 nm) 1.505 and n (347 nm) 1.534. (b) The measured cohereoce length of barium tilanate at Au:= 1.06 p.,m is 5.8 p.,m. Calculate the change in refractive index at A = 0.53 p.,m. 26-6. Determine the half-wave voltage for a longitudinal Pockels cell made of ADP (ammonium dihydrogen phosphate) at A 546 nm. What is its 26-7. A longitudinal Pockels cell is made from lithium niobale. Determine the in refractive index and the difference produced by an applied of 426 V when the light beam is from a He-Ne laser at 632.8 nm. The length of the crystal is I cm. 26-8. Using Eq. show that the transmittance of a Pockels cell can also be written as I = 1m.. sin 2 (3) At what values of V and IP than 7.ero) is the transmittance zero? (b) If the Pockels cell is by an ordinary half-wave plate, what is the irradiance when V = 0 and when V = VHW ? 26-9. in what kinds of media are both longitudinal Pockels and Kerr effects To get some idea of their relative strengths, compare them by calculating the ratio of retardations an appropriately 10 kV. Derive an for this ratio. Then do a numerical calculation by assuming a hypothetical medium with "typical" values of r = 10 pmIV, K I pm/V2, L = 2 cm. d = I cm, and no = 2. Take A 550 nm. 26·10. Calculate the length of a Kerr cell using carbon disulfide required to produce halfwave retardation for an applied voltage of 30 kV. The electrodes of the cell have a separation of 1.5 cm. Is this cell practical'! 26-11. 19) is equivalent to the Doppler effect for light. Use the fact that freQUI~IlC:y shift !J.v for light reflected from a object is twice that prn"m,'T1Y,O from a moving or I1.v = where v is the frev its velocity in the medium, and Up is the componelll of the object velocity to the light wave's propagation direction. Use the geometry of 26-12 and the Bragg condition. 26-12. The speed of sound in glass is 3 km/s. For a sound wave having a width of 1 cm, calculate the advance of the sound wave while it is traversed by a light wave. Take II = 1.50 for the glass. What is the of this result?

Chap. 26

Problems

26·13. (a) Show that a small f:l8 around the direction of the diffmcted beam can be expressed
where A is the wavelength in the medium. tw exceeds the beam divergence is a practically useful number N called "number of resolvable spots." This serves as a of glVmg the number of resolvable that can be addressed the beam deflector. If the beam divergence is by the diffroction A/D, with D the beam diameter, show that

(c) The foctor by which

M = 'filv. Of)

N = -

26-14.

26-15.

26-16.

26-11.

26-18.

26-19.

26.21.

where T is the time for the sound to crOss the optical beam diameter. (d) As a numerical example, consider modulation of the sound in the mnge 80-120 MHz in fused quartz, where u, = 5.95 x 105 cm/s. If the beam diameter is 1 cm, determine the number of resolvable spots. What acoustic frequency is of a acoustic wave, launched in an acoustoso that a He-Ne laser beam is deflceted by 1°? The of sound in the is 2500 mls and its refractive index at 632.8 nm is ] .6. In (26-18), the of the light and the angle are those measif the medium is isotropic and its sides are paralured within the medium. Show lel to the direction of a plane acoustic wave, the equation also holds for the waveof diffraction measured outside the medium. and Determine the difference in deflection for a He-Ne laser beam that is Braggscallered by an acoustic plane wave when the frequencies are 50 MHz and 80 MHz. 1.76 and a sound of 11.00 km/s. The acoustic crystal is sapphire, with n 26-11, that uses ZnS as the active medium. an optical isolator, as in Let the field be produced by a solenoid directly onlO the ZnS crystal at a turn density of 60 turns/em. Assume A = 589 nm. A of SF57 glass with sides and 2.73 em in is placed between the poles of an A small, central is drilled the pieees 10 allow of a linearly polarized He-Ne laser beam the sample and parallel to field direction. The field is set at 5.098 kG. (8) When red He-Ne laser light (632.8 nm) is used. the measured rotalion is 900 min. Determine the Verdet constant for the (b) When green He-Ne laser light nm) is used, the measured rotation is 1330 min. Determine the Verdet constant for the glass. A 5-cm liquid cell is situated in a magnctic field of 4 kG. The cell is filled with carbon disulfide and linearly sodium is transmilled the cell. and the circuthe B..field direction. Determine both the nel rotation of the rh •• .--c""\" of at Ihis waVelenjgl11 of a nonsymmetrical pulse before and after reflection from an ordinary mirror and before and after reflection from a PCM. In the laller case, a.<;sume that the PCM is "turned on" by the pump beams at the inslant the entire pulse has moved inside the PC medium. Show how this effect might be used to correct broadening in an optical fiber. necessary, consult (12J.) Sketch an arrangement using a PCM 10 a sharp. high-intensity of a lenses. This mask onto the photo-resist layer on a semiconducting chip without without placing a mask in direct contact provides a means of doing with the necessary, consult

Chap. 26

Nonlinear Optics and the Modulation of light

[3] [4] [5]

[6]

[8] [9]

[10] [II]

[12] [13] [14]

[15] [16]

[17] [18] [19]

Butcher, P. N., and D. Cotter. The Elements Nonlinear Optics. New York: Cam1990. Ch. 6. bridge University and G. Weireich. "Generation of Optical Franken, P. A., A. E. Hill, C. W. Harmonics." Rev. Letters 7 (1961): 118. Ammon. Optical Electronics, 3d ed. New York: Holt, Rinehart and Winston. 1985. Ch. 8. Ivan P. An Introduction to Devices. New York: Academic Press, 1974. Ch. 3. Yariv, Ammon. Optical Electronics, 3d ed. New York: Holt, Rinehart and Winston, 1985. Ch. 9. Wilson, and 1. F. B. Hawkes. Optoelectronics: An Introduction. London: PrenticeHall International, 1983. Ch. 3. Pedrotti, Frank and Peter Bandellini. ROiation in the Advanced Laboratory. Am. 1. Phys. 58 1990}: 542. Guenther, Robert. Modern Optics. New York: John Wiley and 1990. Ch. 14. Yariv. Ammon. Optical Electronics, 3d ed. New York: HolI, Rinehart and Winston, 1985. Ch. 16. Ammon, and Robert A. Fisher. In Optical Phase Conjugation, edited Robert A. Fisher, Ch. L New York: Academic Press. 1983. Giuliano. Concello R. "Applications of Optical Phase Conjugation." 34, no. 4 (April 1981): 27. Shkunov, Vladimir V., and Boris Ya. Zerdovich. "Optical Phase Conjugation." Scientific Americcm (Dee. 1985): 54. David M. "Applications of Optical Phase Conjugation." Scientific American 1986): 74. Karim. Mohammad A. Electro-optical Devices and Systems. Boston: PWS-Kent Publishing Company, 1990. Ch 7. Boyle, W. S. Wave Communications." Scientific American 1971): 40. Giordmaine, 1. A. "The Interaction of Light with American (Apr. I 38. Miller, Stewart E. "Communication by Laser." American (Jan. 1966): 19. Nelson. Donald F. "The Modulation of Laser " Scientific American (June 1968): 17. Pepper. David M., Jack Feinberg, and Nk-olai V. Kukhtarev. "The Photorefractive Effee!." American (Oct. 1990): 62.

Chap. 26

References

567

27

I

nR

J\t

1.0

If

,, I

I

I

I nl J I //

I

,,

0.01

\

\

\

10 16

Frequency {rad/sl

Optical Properties of Materials

INTRODUCTION

waves that encounter materials create a of interactions of the medium. Forces are on charges by the because of the motions of the also by the electric field of the waves waves. In responding to these fields, the charges magnetic field of waves. Thus themselves oscillate and act as radiators of secondary in determining the net at some point, the fields of both source waves and the charged oscillators must be laken into account. In the case of ordinary fields, assumed smaller than those now attainable with high-energy lasers, the net to be a linear of the constituent fields. of aH the microscopic conlributions to the resultant field can, for certain be described by m~lcn[)Sc:o[)ic the optical constants of the material. In this chapter we show in particular how the refractive index and coefficient for isotropic conducting (metals) and nonconducting or materials can be understood. In order to do this we make use of Maxwell's and the mathematical techniques of vector calculus. FIE'rt.eOt1MH'np·tH'

27-1 POLARIZATION OF A DIELECTRIC MEDIUM

We take as our model a simple dielectric, that is, a nonconducting material whose properties are isolropic. nonconducting we mean that the meOUJlm, metal, contains no free charges are associated with the __ ~':L•.. __ • 568

; r!nlr",,;/" we with the electrons bound to such nuclei. physical properties we consider are 1n(]lepen(Jen so that we may treat the physical constants as scalar Apto such a medium causes charge displacement, in which plication of an distribution bound to the nuclei shifts in a opposite to the the The shift may occur in a polar molecule, , because the that the centers of its positive molecule has a permanent electric distributions do not coincide. In this case application of the field and produces some of the molecules so that, on the average, the positive toward end of the dipole is in the direction of the field. The counteracted motions of the molecules. The shift in tion may also occur in nonpolar molecules, such as 01. in and negative charge distributions normally have the same effective center. Application of the field results in a shift of the electron cloud relative to an induced dipole. In case, the dipole moment p due to each atom or molecule is of the by product of the displaced charge q and the and charge in the atomic dipole. or

=

p

as In tive toward the material electric field.

-qr

(27-1)

27-]a. The direction of the dipole moment is from the negacharge. The magnitude of the dipole moment for a given the influence of a given on how easily charge is displaced P of the medium is then said to be the collective by the sum of dipole moments

'AI~,JL"f'"

P = -Ner where N is the number elementary dipoles per unit the

(27-2) yv.uu ...

and e is the magnitude of

-m,"(v E

q

v

p

-eE

(a)

(b)

Figure 27·1 The electric dipole. {al Alignment with the field. (b) Forces acting on a dipole when the electric field has the direction indicated.

Electrons behave as though the forces binding them to the nuclei are elastic forces Hooke's law, where the restoring force is proportional to the disopl~;ltely directed. The more massive nuclei can be considered stah£\,n
Polarization of a Dielectric Medium

569

In K is the force constant of the effective spring, m is the electronic mass, and '}' is a frictional constant with dimensions of reciprocal time. Notice that the force (ev x on the electron due to the of the radiation is omitted; it in compared with the due to the electric field. When the applied E-field is static, there is no oscillation of the dipoles, so that both velocity and acceleration of the electron vanishes. In this special case, (27-3) reduces to -Kr

eE

or, eliminating r with the help of Eq. (27-2), the static

nnlnr"7n

is given by

p=-K ~UI)f)Olse

(27-4)

now that E is the field of a hannonic wave with a time dependence and that the oscillations respond with a similar dependence, the corresponding derivatives dr/dt -iwr and d 2r/dt 2 = -eE -mw - imw'}'

r=-------2

which, when substituted in given by

(27-5)

+K

. (27-2), now yields a 2

p = (

-mw

2

Ne -

imw'}'

+

K

)E

(27-6)

Note that Eq. agrees (27-4) in the case of a static specified by the conditions w = 0 and'}' O. In all other cases, polarization is a function of the radiation wand, because the coefficient of E is complex, the polarization may possess a phase relative to as we will see. The in the interior of field E in Eq. (27-6) should the actual field at the the medium. This local is a superposition of the applied and the field that results from all the other dipoles aligned in a polarized medium. It is shown in standard texts on and magnetism that the latter contribution is given by P/3Eo. where Eo is the pennittivity of free space. Thus (27-7) Retaining the symbol E for the applied field and substituting (27-6),

into

p P now appears twice in the equation, and we can solve for it explicitly. in Eq. (27-8) equal to F for the moment, we conclude

......"'••""tnr

p The multiplier of E can now be

E to be

-

510

Chap. 27

Optical

Prr,n",rii.,,,,,

of Materials

lW'}'

the

Defining

wii as the quantity in parentheses. that is. K

(27-10)

m (27-9) becomes

P the

""'/':.un'........,

.

IWy

(27-11)

E

of this complex ""'.",",,"""'" for P.

Clearly, P can increase dramatically as W -+ Wo, so that Wo represents a resonance frequency for the dipoles of the medium. Equation (27 -II) has the same form as the "":I'''''''''vu of motion of a driven harmonic oscillator with damping. As the frequency approaches the resonance Wo of the oscillator, the amplitude of the vibrations becomes very large subsides as the increases beyond Wo. In the case of a dielectric medium, the increase of dipole moments at resonance results in a maximum polarization. Equation 11) also illustrates a lency-oelperlOent phase shift between the applied field and the polarization. Far from resonance, we may set 'Y = 0, corresponding to negligible damping. Then w ~ Wo, P and E have the same sign and the are oscillating in phase with the field. Beyond resonance. however, when W ~ Wo, P and E have opposite signs, indicating a phase difference of 180°. Free electrons respond in this manner. When w == Wo, near resonance, the vibrations are The term in the denomiequivalent to mUltiplicanator in this case is not negligible, and the division by shift between E and P. tion by i, indicates a 90° The dependence of P on E, as given in Eq. (27-11), can now be used to discover conditions which plane waves are able to in a dielectric. The fundamental wave equation for waves in the dielectric is a consequence of the Maxwell equations.

27-2 PROPAGATION OF UGHT WAVES IN A DIELECTRIC The four Maxwell equations may be written in the V·E =

12) Eo

ilB

VxE V· B xB

form

(27-

ilt

=0 ilE

at

14)

+J

(27-15)

Eo

In these equations P is the charge density, which in includes both free and bound as indicated by P = Pb + PI' In a dielectric, however, PI = O. It is magnetism to show that the bound-charge standard in a course in electricity density is related to the polarization Pb Sec. 27-2

Propagation of

= -V, P

Waves in a Dielectric

(27-16)

511

reJ:)r~;ents

the current density and can arise from both free J = Jb + Jfo In a dielectric where Pf 0, Jf 0 that

also.

Jb = With these

ilP

at

(27-17)

the four Maxwell equations for a dielectric can be written

('o.,,,f .. ,,,,

V. E = _-_V_"_p

(27-18)

Eo

aB

VxE

V·B xB

(27-19)

at o

(27-20)

aE + I ap

at

Eo

(27-21)

at

Now we take the curl of both sides of

vx

x

(V x E)

where we have •.,t" ..... I.", ... ,,,,,>,-1 time in the last identity Vx

member

B)

(27-22)

diflfen::nti:aticln with respect to space and can be reexpressed by the

x

(27-23)

In a homogeneous is to produce a net surface charge density, while density Ph = 0 unchanged. The internal charge density is zero in any internal closed surface, every bit of charge that moves into the enclosed volume in response to a polarizing field is balbit of that moves out. The surface-charge density appears anced by an because balancing is not there. Thus by (27 -16) and (27-18) we 0 and substitute the of Eq. (27-23) into Eq. (27conclude that V . E 22), giving (27-24) For the right member we may make use

Maxwell's equation (27-21) and write I

+-

a2 p

&l

(27-25)

The last term is expressible in terms

[I + -meo---=----::---For a harmonic wave as E (PEtal]. = -w 2 E, Eq. (27-26)

, in which case V2 E becomes

(27-26)

= -k 2 E and (27-27)

We conclude that the analysis of tric requires in general that the 512

Chap. 27

Optical

to""'" of Materials

Prn ... "" ...

prc)paJgatJing in a homogeneous dielecconstant k be a complex number.

lphnulU

the real and imaginary parts of k by (27-28)

k=kll+ikl

this form into the expression for a harmonic wave, we have

E

(27-29)

The factor in kl represents a depth-dependent absorption of an otherwise harmonic wave, and kl measures the attenuation of the wave. By taking square of the of both sides of Eq. the result describes instead the energy flux I =

where a: stant is cornpllex,

nh.'t,orJrJil£IH

2Tr A

k

If we

irl"'....i!'"

of the mediun. If the propagation conindex, for we can write

r."pr.,r'PH'

the real and imaginary

(27-30) of the complex refractive index by (27-31)

index and nl is caUed the extinction coefficient, it foland (27-30) that

where nil is lows from

+

ik, =

(~) (nil

+

in,)

the relations (27-32) and (27-33) Writing

as

and relating this

to (27-34) can be found by member (27-35)

The right member can also be written as the sum of a real and imaginary part. The complex term is first numerator and denominator by the the denominator. The simplification, is

= I +

+ i(-,-(,-:.,zo,------",wZ)...!...' Yz-----=-z--:::2) W +w'Y LU

(27-36) Sec. 27-2

UI..l,tlY
of

Waves in a Dielectric

573

Now by comparing the right members of

n/i - nl

and (27-36), (27-37)

1+

and (27-38) The can be solved simultaneously nJ and nR. The appearance of the mass m in the denominator of these equations shows that electronic oscillations are more important than ionic oscillations in the index of refraction. may be significant in the resonance, however, where bracketed term balances the small prefactor the mass. 27-2 shows both nR and nJ calculated from (27-37) and (27-38) as a function of driving frequency w. The by the extinction coefficient is seen to peak at the resonant Wo. real refractive index exn"r"I>n,"I>c a sharp rise and fall as w increases toward and passes through resonance, which it increases again. approaching the value nR I at frequencies. The where nR decreases with frequency is to the usual dispersion media and is called the of anomalous dispersion. A resonance as Wo for the dielectric means photons of frequency Wo, there is a high probability of absorption. Absorption of such a photon correto a transition of Eo = Iiwo = lifo in the structure of the material. As W is there will be a series of resonance characteristic of material. If such a resonance occurs in the visible range frequencies, for exthe material absorbs a portion of the spectrum and appears colored, while t""nc,.... ,tt,nn the remainder. Transparent materials like glass have resonance frequenIn terms of our simand ultraviolet regions but not in the of a dielectric, we interpret the existence a number of resonance frequencies to mean that electrons experience different of freedom in re(27-34) is usually sponse to the applied field. To take this into account formally, to include a number of terms summed over resonant frequencies Wj, I

Ne 2

+-

mEo

L -Wj- : - - = -i')'jw --

(27-39)

j

1

~,

I1R

- 1.0

I

c

\r-

'1;;

,i

- 0.1

c

o

':ce

I

';(

I \\

TIl

I

I

I

-8'" :~

W

0,01

\

\

\

I //

Angular frequern~y dlepeilldeoce uf the refrdctive index nil and the extioction coefficient nl for a dielectric. Assumed values are We = I X 10"" 8-', 'Y "" 1014 1;-', and N = I X 1()28 m- l • Figure 21·2

\

to 16 Frequency Irad!s)

574

Chap. 27

""."ri,.,." of Materials

P ....

where fj, called the oscillator strength for the resonance Wh represents the fraction of dipoles having this resonant frequency. The rigorous treatment of this problem requires the application of quantum theory. The Dispersion Equation. The variation in refractive index with frequency, described by Eq. (27-39), is what we mean by dispersion. We wish to show that the Cauchy dispersion equation, introduced in Eq. (6-17) as an empirical formula, can be deduced from Eq. (27-39) under certain simplifying assumptions. We shall assume a single resonant frequency Wo in the ultraviolet, such that frequencies in the visible obey the inequality, W ~ Wo. We shall also assume that damping is negligible by setting 'Y = o. Since all dipoles have the same resonant frequency, f = I and Eq. (27-39) takes the form:

Notice that, for W ~ Wo, as for a gas, the refractive index is nearly constant. As W approaches Wo from below, the refractive index increases slightly, as shown in Figure 27-2. The slowly increasing index with frequency (decreasing with wavelength) is characteristic of normal dispersion. To derive the Cauchy dispersion equation, let us first expand the frequency factor in a binomial series:

so that

Writing express

W

=

27TC/A and gathering constants A', B', and C ' appropriately, we can n 2 = A'

B'

C'

+ -A2 + -A4 + ...

We may take the square root of both sides and, since each higher-order term in the expression is less than A " use the binomial expansion again on the right member. After re-collecting constants, we get

B

C

n=A+-+-+'" A2 A4 which is the Cauchy relation introduced earlier to describe normal dispersion.

27-3 CONDUCTION CURRENT IN A METAL In metals. the existence of "free" electrons, not bound to particular nuclei, modifies the treatment outlined above for dielectrics. Although there are also bound electrons, the response of the free electrons dominates the electrical and optical properties of the medium. So, in Eq. (27-3), we set K = 0, and the equation of motion becomes

dv

mdi + myv Sec. 27-3

Conduction Current in a Metal

= -eE

(27-40) 575

eQl.liatl()O may be conveniently "'vr,...,.",.,rI in terms defined by

= -Nev

J where J has (Sf) units J rather than v,

the conduction current den(27-40

amperes per square meter. Writing Eq. (27-40) in terms of

dJ

(27-42)

+ yJ

dl

In the case where the applied field is the harmonic wave E current to vary at the same rate and J takes the (-iw

In the

• we expect the . Equation (27-42) then

+ y)J

(27-43)

or DC. ca'ie specified by ru

0,

J= The static

C01taUC1l11tlV

(27-44)

u, defined by

law,

J = uE then takes the

(27-45)

form

Ne 2

u=my

(27-46)

Since conductivities are usually measured, we rewrite Eq.

J = (

in terms of u,

)E

U

I - iru/'Y

27-4 PROPAGATION OF LIGHT WAVES IN A METAL An

wave propagating in the conducting medium satisfies Maxwell's (27-15). Although free exists in the metal, the inPI is zero. The free is so mobile that it trele-Cllan!e volume in response to an applied field, the of local quickly charge densities. The appropriate Maxwell equations are then ""'"U~~U

(27-48)

V' E = 0

aD

VxE=--

ot

V·D

=0 ilE

c2V x D = -

ot

J

+£0

As before, V x (V x -VzE because V . E = 0 in the identity of 23). Calculating same quantity by taking the curl of Eq. (27-49), we have

:::::: V x (-

~t = -~(V x u

516

Chap. 27

PrnnertlA!::

at

of Materials

D) =

_.l2 o2E c all

__£oc1_2 (oj) at

where we have used (27 -47), we conclude

Representing J with the help of

(27-51) in the last

I (ifE) + _1_ £oc

aE

at

2

i1h wt ),

For plane, h",,·....,rm waves given by E Eoe the aD[,rO['f1 (27-52) can be calculated to derivatives required

space and time

(27-53) where we have also made use of the fact that II£oJ.Lo, with J.Lo the permeability of vacuum. Again, we find that the propagation constant must be a complex number to properly describe the propagation of the wave in a metal.

27-5 SKIN DEPTH

Before with the general case described by Eq. (27-53), we pause to consider the special case in which the w of the radiation is small enough to allow as a approximation to (27-53), k2

Expressing i as

=

iwuJ.Lo

and taking the square root of both sides, (I

k

k as the complex number coefficients by

k=

+ i)(u~wr2 kR

+

as before, we can identify the real

(27-55) and the real and

refractive indices by

c kR

(27-56)

W

and

( Zweo

~)1/2

The tion,

character of k, when introduced into the plane. harmonic wave equaas in Eq. (27-29) to

E= The real exponential factor describes absorption. When the radiation has penea depth of z llki' therefore, the amplitude has to lIe of surface value. This distance is the skin depth, l5, where l5=.!..= k/

(27-58)

u. For 3-cm and is evidently smaller for better conductors with example, the skin in copper, with conductivity of 5.8 x 107/!l-m, is only about 6.6 x 10-5 cm. Sec. 27-5

Skin Depth

511

27-6 PLASMA FREQUENCY Returning to the case of Eq. (27-53) and introducing there the complex rewe write

n2

=

(E-w k)2 = I

After multiplying the complex term by

w(I -

ry I ry > this becomes

n2 = 1 _ The numerator in the identified as the square

+ ---:---'--'--;--

ILoUc2 y w 2 + iwy

(27-59)

term must have the same dimensions as w 2 and is a plasma frequency given by

= ~c2yu

= ILOC2y(Ne2) = Ne my

2 (27-60)

m€o

where we have use of both Eq. (27-46) and relation l/€o~. The plasma frequency is a resonant frequency for the free oscillations of the electrons about their equilibrium positions. Inserting it into Eq.

n2 = 1 -

w2 w2

P

+ iwy

(27-61)

the plasma turns out to be a critical whose determines whether the refractive index is real or imaginary. This can be seen by neglecting the y-term, valid (w ~ y), in which case (27-61) is simply (27-62) of the metal is Equation now complex and radiation is whereas for w > W p • the is real and the metal is transparent to the radiation. Returning to . (27-61), we find, as before, two equations from which the real and imaginary refractive index can be calcu1ated. real imaginary parts in (27-63) we find

These equations, solved simultaneously, permit calculation of curves such at.; those in Figure 27-3. The curves cross at w (w; - y2)112, as is evident from Since typically wp ~ y. the crossover occurs at w == w p • dividing the and the opaque (and highly The plasma frequency for in the visible to near-ultraviolet so that they are opaque to visible and transparent to ultraviolet radiation at sufficiently high frequency.

578

Chap. 27

Optical Properties of Materials

1.0

__-----.....,1.0

"'.... 1: (l)

~ '1;; 0.10 8 c

o

.~

.'w""

0.01

0,01

Figure rI-] frequency dependence of the refractive index HI? and the extinction coefficient HI for Values assumed are

1.63 x 10 16

andy=4.1 x lO13 s- l • of the curves coincides

Frequency (rad/5)

..,.,,,,.1;<1,1,, to the good insulator and good we have treated sepaare materials. like semiconductors, for which of these extreme cases ""th"."" to the properties. Such materials appreciable contributions properties from both free and bound charges and accordingly must be for both types of behavior.

27-1. In

the dielectric constant, is related to tbe refrac-

the "electrica1 constant"

tive index by K

n2

(a) Show that if Kf/ and KI are the real and then

parts of the dielectric constant,

and

Calculate nf/ and nl for a in terms of K1 , at frequencies high enough such that KI = Kf/. 27·2. Show that in a nearly medium, the absorption coefficient is related to the conductivity and refractive index a

3770'

27·3. Write a computer program to calculate and lor plot real and imaginary parts of the refractive index for a dielectric the frictional parameter 'Y, the resonant fre4~uelncy wo, and the dipole density N. Check your program Figure 27-2. 27-4. Assume that aluminum has one free electron per atom and a static conductivity by 3.54 x ]07/0-m. Determine (a) the frictional constant 'Y; (b) the plasma frequency; (c) the real and imaginary parts of the refractive index at 550 nm. Chap. 27

Problems

579

27-5. Show that Eq. (27-58) for the skin depth at low is an adequate approximation when w ~ l' and w ~ 27-6. Calculate the skin in copper for radiation of (a) 60 Hz and (b) 3 m. First ensure that the of 27-5 are satisfied. (Handbook data for copper: u 5.76 x IO'/O-m.) 27-7. Compare the skin depth of (a) aluminum, with conductivity of 3.54 x 10'/0- m and (b) seawater, with conductivity of 4.3/0-m. for radio waves of 60 kHz. 27-8. Calculate the skin depth of a solid silver waveguide component for 1O-cm microwaves. Silver has a conductivity of 3 x 107/O-m. Explain why a more economical silver-plated brass will work as well. 27-9. The energy density of red of wavelength 660 nm is reduced to one-quarter of its original value by passage 342 cm of seawater. (a) What is the absorption coefficient of seawater for red light of this wavelength? (b) At what depth is red reduced to 1% of its original energy density? 27-10. Write a program to calculate and/or plot the real and parts of the refractive index for a metal, the frictional parameter l' and the plasma frequency. Check your results Figure 27-3. 27-11. Determine the theoretical content of the constants A, B, and C used to express the dispersion equation.

Richard P., Robert B. and Matthew Sands. The !"p'Imnwn vol. L Mass.: Publishing ('nlmnl.nv ~nITlmlPt"f,,,.lrl Arnold. Optics. New York: Academic Press, 1964. Ch. 3. Javan, Ali. "The Optical Properties of Materials." Scientific American 1967): 238. [4] R. S. Geometrical and Physical Optics, 2d ed. New York: John Wiley and 1967. Ch. 20. John. Concepts oj Classical San Francisco: W. H. Freeman and Company 1958. Ch. 4. Bruno. Optics. Reading, Mass.: Addison-Wesley [6] 1957. Ch. 8.

580

27

Properties of Materials

Suggestions for further Reading

ARTICLES ON OPTICS FROM SCIENTIFIC AMERICAN (chronolofJicalorder} Kirkpatrick, Paul, "X·rny Microscope." (Mar. 1949): 44. Evans. Ralph M. "Seeing Light and C",lor." (Aug. (949): 52. Wald, George. "E;ye and Cmnera:' (Aug. 1950): 32. WiISOll. Alberl G. '"The Big SciJmidl.·· (Dec. (950): 34. Kclrer. Alber!. "Revival by UghI. ... (May (951): 22. Muller. Erwin W. "A New Microscope." (May 1952): 58. Ingalls. Alberl G. ":Ruling Engines." (June (952): 45. James, T. H. "Photographic Developmen!." (Nov. 1952): 30. LaMer. Victor K .• and Millon Kerker. "Light SCllIteroo by Particles." (Feh. 1953): 69. Waterman. Talbot H. "Polarized Light and Animal Navigation." (July 1955): 88. Rush. 1. H. '"The Speed of Ughl." (Aug. 1955): 62. Sperry. R. W. "The Eye and tile 8mio." (May 1956): 48, Milne, L. J" and M. 1. Milne. "EJecuical Evems in Vision," (Dec, 1956); 113. Gordon, James P. "Tile Maser." (Dec, 19581: 42, Land. Edwin H. "Experiments in Color Vision." (May 1959): 84. ",",Id. George, "Ufe and Lighl." (OCI. 1959): 92, McClam, Edward. Jr. "The 6QO.Foo! Radio Telesc.ope," (Jan. 1960): 45,

581

Bloom, Arnold L "Optical Pumping." (OeL 1%0): 72. Kapany, Narinder S. "Fiber Optics." (Nov. 1960); 72. Amon, DanieL "The Role of ughl in Photosynthe>;is. (Nov. 1960): 104. Butler, W. L, and R. ). DowllS. "ughl and Plant Development." (Dec. 1960): 56. Hess, Eckhard H. "Shadows and Depth Perception." (Mar. 1961); 138. SCh_low, Arthor L "Optical Masers." (June 1961): 52. Pritchard, Roy M. "Stabilized Images on the Relina." (June 19(1): 72. Evans. Ralph M. "MlIxwell', Color Photograph." (Nov. 1961): 118. Wallach. Hans. '~11le Perception of Neutral Colors." (Jan. 196.1): 107. Scbawlow, Arthor "Advances in Optical MaorefS." (July 19(3): 34. Giardm.ine, J. A. "The Interaction of Light with ughl." (Apr. 19(4): 38. Leith, Emmet! N., and Juris Upalnieks. "Photography by Laser." (June 19(5); 24. Miller, Stewart E. "Communication by Laser." (jan. 19(6): 19. Pimentel, George C. "Chemical Lasers." (Apr. 1966): 32. Rock, Irvin, and Charles Harris. "Vision and Toucl1." (May 1961): 96. Morelutd, Fred F .. Jr. ··ught.Emitting Semiconductors." (May 1967): 108. Hubbard, RUlh. and Allen Kropf. "Mo!erular Isomers in Vision." (June 19(7): 64. Lernpicki, Alexander, and Harold Samet""n. 'Uquid Laser,." (June 1967): 80. Javan, Ali. "The Optical Properties of Materials." (Sepl. 19(7): 238. Penningtoo, Keith S. "Advances in Holography." (feb. 1968): 40. HlIflrromn, Svcn R. "Photon Echoes." (Apr. 1968): 32. Nelson, Donald F. "TIle Modulation of La"" Light." (lune 1968): 17. Patel, C. K. N. "High·Power Carbon Dioxide 1""""." (Aug. 1968): 22. Feibcr)l, Gerald. "Lighl." (Sept. 1968); 50. Wei"'kopl, Victor f. "How Ugh! Interacts with Mal!er." (Sept. 1968): 60. Connes, Pierre. "How Light is Analyzed." (Sept. 1968); 12. Smith, F. Dow. "How Images Are Formed." (Sept. 1968): 96. Jones, R. Clark. "How Images Are Detected." (Sept. 1968): 110. Schawlow. Arthur L "Laser ught." (Sept. 1968); 120. Herriol!, Donald R. "Applications of Laser Light." (Sept. 1968): 140. Oster, Gerald. "The Chemical Effects of Ught." (Sept. 1968): 158. Hendricks, SterlinF B. "How Ught Interacts with Living MllIter." (Sept. 19(8): 174. Fitc!J, James Marston. "The Control of the Luminous Envirornnent." (Sept. 19(8): 190. Neiss
Suggestions for Further Reading

Panish, Mor1on B., aoo I'JJO Hayashi."A New CIas.,ofDiode Users." (July 1971): 32. Ingham, M. F. "The Spectrum of the Airglow." (lan. 1972): 78, Ashkin, Arthur, "The Pressure of Laser Ugh!." (Feb. 1972): 62, Kellermann. K, I. "lntercont;ocnlaJ Radio Asuunomy." (Feb, 1972): 72, Busignies, Henri, "Communication Channels," (Sepl, 1972): 9K Silrvast, William T, "Melal-Vapor Lasers," (Feb. 1973): 88. Cook. J, S, "Communi""li"n hy Opli<:aJ Finer," (Nov, 1973): 28, l"'ld. M, S" aoo V, S, Lelokhov. "Laser Speetroscopy," (Dec, 1913): 69, Tien. p, K, "Int
'Wurtrnann, Richand j, "The Effects ofLigln on the Human BOOy," (July 1975): 68. Beck, JucOO, "The Perception of Surfuce Color." (Aug, 1975): 62, Roo;, John, 'The Resources of Binocular Perceplion," (Mar, 1976): 80, 'Wunner, Rudiger. "Polarized Light Navigation by Insects," (luly 1976): 106, Price, William H, 'The Photographic Lens," (Aug, 1976): Leilh, Emmet! N, "Wbite-Light Holograms." (Oct. 1916): /lO, Zare, Richar~ N. "La,er Separation of ISOlopes," (Feb, 1977): 86, NU&Senzvelg, H, Moyses, '"The Theory of the Rainbow." IApr. 1977): 116, Horridge, G, Adrian. "The Compound Eye of Insecls," (July 1977): 108, Boyle, W, S, "Ught-Wave Communications," (Aug. 1(77): 40. Land, Edwin H. "The Retinex Theory of Color Vision," (De<;. 1977): lOR Dickinson, Dale F. "Cosmic Masers," (lu,", 1978): 90. Eberhan:l. aoo Ralph Feder, '"The Opli('1; of Long-Wavelength X Rays," (Nov, 70, Land, Michael E "Animal Eyes with Mirror Optics." (Dec. 1978): 88, 'nIriv, Amooo. "Guided Wave Optics," (lan. 1(79): 64, Roon. Avigdor M. "Laser Chemistry," (May 1(79): 114, Quate. ('"Ivin, "ACOOSIic Microscope," (Oel, 1919): 62. Nassau, Kurt. "The Causes of ('..oIor," (Oct. 1980): 124, Thomas, D. E. "Mirror Images," (Dec. 1980): 206, Yelloll. Joho L If. "Binocular Depth Inver,ion," (July 1(81): 148. T,ipis, Kosta, "Laser Weapons," (Dec. 1(81): 51. LaRocca, Aklo v, "Laser Appicatiom; in Manufacturing," (!\fur. 1982): 94, Ballcall, J, N.. and L Spitzer, lr. "The Spare Telescope," (July 1982): 40, Abraham, Eitan, et ai, "The Optical C.omputer." (Feb, 1983): 85, Poggio, Tomaso. "Vision by Man and Machioc." (Apr, 1(84): 106, Maooeli, Dina F.. and Winslow R. Briggs, "Fiber Optics in Plants," (Aug, 1984): 90, Habing, Harm J., and Gerry Neugebauer, 'The Infrared Sky," (Nov, 1984): 48. Tsang, W. T. "The C' Laser," (Nov, 1(84): 148. Brookner, Ell. "Phased-Array Radars," 11-00, 19851: 94, Vladimir v.. and Boris Ya, Zel'dovieh, "Oplical Phase Conjugation," (Dec, 1985): Pepper, David M. "Applications of Optical Pbsse Conjugation," (Jan, 1986): 74,

n,

Suggestions for Furthe, Reading

Hosk'n. Michael. "William Herscbel aIId the Making of Modern Astronomy." (Feb. 1986): 106. AmJerson. Dana Z. ''Optical Gyroscopes." (Apr. 1986): 94. S., and Stuarl M. Anslis. "The Perceplion or ApPareJIl Motion."

Thom". R. Sciascia, Lynette Linden, and Jerome Y. LMIIitn. "The Colors or 19l11il: 84 M. "PholOnjc MaI.rial •. " (0<:1. 1986): 146. Abo·Mustafa. Yaser S., and Demclri Psaltis. ''Oplic.al Neoral Computers." (Mar. 1987): 88. Julie L, and Denis A. Baylor. "How Pholore",plor Ceils Respond 10 Ugh!." (Apr.

40. Wimck, Herman. "Synchrotron Radialion." (Nov. 1987): 88. Livmgstone, Marga",t S. "Art, Illusion and lhe Visual System." (Jan. 198M): 78. Slusher, Richard E .. and Bernard Yurte. "Squee7.ed Light." (May 19881: SO. Lnretz, Jan<: F.. and George H. Handelman. "How the Human Eye Focuse.s." (July 1988): 92. "Delecling Individual AlOIn> and Molecules wilh Lasers." (Sepl.

Nalha"", Jeremy. "The Gooes ror Color VISion." (Fch 1989): 42. Katz", Abraham. ''Optical Fibers in Medicine." (May 1989): 120. Brncewell, Rooakl N. "The Fourier Transform." (lu"" 1989): 86. Romer, Grant 8., and Jeannelle Delamoir. ''TIle Flrsl Color Photographs," (De<;. 1989): 88. Davld M., Jack Feinberg, and Nicolai V. Kukhlarev. "The Photorefractive Effect." 1990): 62. Howells. M.kolm R.. lanos Kin, aoo William Sayre "X-ray Micr
Suggesllons for I'u rther Reading

Answers to Se ted Problems

Chapter 1

Chapter 2

Chapter 3

(a) 6.6 x 10- 34 m (b) 3.9 A 3.6 x 1O l7 W 3.10 and l.77eV 0.024 A; 2.7 x 10- 22 kg-m/s 1-9. (a) 1.49 x 10- 18 kg-m/s (b) 4.45 x 1.10. 3.75 x 10 17 I-I. 1-2. 1-3. 1-4.

1O~16

m (c) 4.22 x 10- 16 m

2-1. 3.9 - 7.9 x 1014 Hz 2-2. 2050lm (b) 39.8 W/sr~ 163 cd (c) lOS W/m2; 4.1 x lOS (d) 9.95 W/m2; 40.8 Ix (e) 0.0195 W; 0.0803 1m 2·3. (a) He-Cd appears about L3X brighter (b) about 2.4 mW 2-4. (a) 900 cd (b) 85.4 Im/m2 or Ix 2-5. 1.055: I 2-6. 320 Ix 2-7. (a) 1.7 x Hfcd/m 2 (b) 21TL 2-8. 0.97 1m 2-11. 5800 K 2-12. (a) 0.4830 J-tm (b) 0.0756 W 2-13. 6266 462.5 nm 2-14. 6105 K 3-1. t 3-2. I

=

n/x;)/c

+

y2)

+ 70(x 2 +

y2)t/ 2 - 135x

+ 800

0

585

3-3. 4.00 mm 3-4. 3 with top edge of mirror at a height m ... UI>IV between the person's eye level and the top of the head 3-5. The ray emerges from the bottom at 45°. 1.60 3-6. Reflection from the bottom 3-7. 1.55 3-8. 1.153 em 3-9. 8 em 3-10. Light from the bubble is refracted through the surface, both directly and after reflection from the spherical mirror; 3.33 em and 10 em. 3-11. 12.5 em; 75 em 3-12. 10 em behind the near surface; 3 x 3-13. (a) / lhR/(nz - nl) (b) R > 0 (convex) and R < 0 respeetively 3-14. (a) center, ~ aetual size 6.4 em behind the ~ aetual size 3-15. Virtual, inverted, 15 em from the window, twice the object size 3-U•• 13.0 cm 3-17. +20 em or -20 cm 3-18. 22.5 em behind the lens; L50 times the actual size 3-19. (a) -6.7 em (b) - JO em or -60 em 3·20. -50 em 3-21. 3.33 mm in front or the objective; ereet and magnified 3·22. Final image between lens and mirror at 21134/from virtual, inverted, and Tr the original size 3-23. (a) 33.3 em, 2X (b) 86.67 em, 2x (e) 7.37 em, -0.316x 3·24. 1.63 3-25. 150 em and 600 em; inverted 3-26. (a) 10,5, -2.5 diopters; 12.5 (b) 8.33 m- I , 4.17 24 em 3-30. = n/(n - 1)

3·3t. At

5

5'

= 2/

3-32. (b) }';j(tl tan 81) 3·33. Incident on plane side: 8 em beyond lens; on eurved side: 5.33 em 3·34. +40 cm, +30 em; - 30 em, -40 em 3-35. 25,000 ft

Chapter 4

4·1. /1 -62.05 em;J2 ;;;::: 46.66 em; r = 2.91 em; 5 -0.98 em 4·2. (a) /1 = 14.06 em r 1.17 em; 5 -0.73 em (b) 8.92 em left of lens center (e) 9.78 em left of lens center, with 9.6% error 4-3. virtual image at 6.67 em left of second vertex, 0.556 in. high 4·4. Ji 11.51 em;f2 15.31 em; r 0.400 em; s == 1.16 em; v = 4.20 em; w 2.64 em; SI = 18.9 em from H2; hi = -l.18 cm 4-5. (a)!1 = -20.15 em r = 10 em = -5; v = 10 em = -w (b) Image is 61.38 cm from center and m = -2.05x 4-6. (a) y = 1 em; 0: -5.73° (b) A = 1 x/IO; B = 10/3 + 2x/3; C = -1/10; D = 2/3 (e) x to cm 4-8. p = -4.17 em. q = +2.17 cm, r -0.83 cm, 5 = -2.17 em,/I -3.33 em, /z 4.33 cm 4-9. /1 -20cm,/z = +20em,p -30 em, q = +IOem, r -lOem, S =: -Wem +23.3 em, q = + 18.7 em, p = -18.3 em, r = -1.67 em, 4-10'/1 = -16.7cm,/z S = -4.61 em 4-11. (a) A L B = 0, C = - -k, D -2 (b) Input and output full at eOflJU!;ate object and image IJV~~"'../ll". A is identical with the linear magnification. 4-12. (a) p -2, q = /1 = +6, r = 4, S = -4 in. (b) 2 in. beyond ball

586

lens

Answers to Selected Problems

*,

4-13. (a) Elements of system matrix: A = B ~, C = , D ~ (b) P = 140, q 160, , = s 10,[, 150, .12 150, all in em 4-14. (a) A = B 0.9676, C 0.009182, D = 1.033 (b) /1 108.9 em, h. -108.9 em, p = 112.5 em, q 106.3 em, , = 3.62 em, s 2.57 cm 100cm 4-15. (a) A j-s'/6;B=¥+2- ss'/6+s';C LD -s/6+ I (b) s' (4s + 12)/(s m = ~ - s' /6 (c) s' = 6~ em; m = -0.429 (d) s' 4 em to second local plane; s 6 cm to first focal plane 4-16. A :::: V . 7 J 7 J J . B = C -0.009284; D = 0.8448;, = v 16.72 mm; s w = -6.53 mm; p -90.99 mm; q = 101.18 mm;/, = 107.71 mm; the film is a distance q = 101.2 mm behind the last lens surface. 4-17. A

C

=

n

nL

R J

= -=--n'R 2

+ n'R I n 3.180 cm and a'

RJR2

D=!!..+ n'

4-21. For a = 0", s' -23.51°; for a == s' and a' = 6.081° 4-22. s' = -49.525 em; a' == 3.371"; Q = 2.912 4-23. For h = I mm, s' 98.20 mm, a' = -0.567°; for h = 5 mm, s' = 102.45 mm, a' = -2.723°

Chapter 5

Chapter 6

(d) s' = 0, +H, + 14.4 em a = +0.0954 mm; by = + 1.22 mm; b, = +3.91 mm a = 0.015 mm; by == 0.49 mm; b, = 3.9 mm (a) 0.0296 mm (b) 0.021 mm (a) 0.60 mm (b) 1.2 mm b, = 1.48 mm; by = 0.15 mm bz = 0.974 mm at h = 1 em, 3.84 mm at h 2 em, 8.44 mm al h 3 em, 14.53 mm at h 4 em, 21.81 mm at h = 5 em 5-10. b, = 1.82 em; by 0.970 mm 5-12.ForO' 0.7,'1 I7.65and'2=-I00em;forO' 3,'1 7.50 and '2 = 15.0 em 5-13. " = 18.62 em, '2 -33.75 em 5-14. optimum 0' closer to + I than to 5·15. (a) +0.714 (b) n = 17.5 em;'2 105 em (e) -0.714, reverse the lens 5-16. (a) 0.8 (b)'1 16.7 em;'2 = -150 em (c) reverse the lens 5-17. +20 and -20 em 5·18. answers the same 5-19. 17.7 em 5-20. (a) R = 15.7 em (b) .12 = -3.476 em 5-21. '11 = 8.5168 em, '22 -434.89 = 20.0000 em, = 20.0096 em, 20.0096 em 5-22. (a) '11 = 3.4535 em, '22 12.6576 em (b) = 5.0000 em,1c = 5.0026 em, = 5.0026 em (e) 0.3695 em-I, Pw -0.1695 em-', 0.01802, Il.m 0.03928 (d) yes 5-23. (a)," -5.2415 em. '22 = 53.1840 em (b) /tD -4.5770 em, 12D = 8.4399 em (e) /D -10.0000 em, Ic = 10.0050 em, /F = -10.0050 em 5·2. 5-3. 5-5. 5·6. 5-7. 5-8. 5-9.

6-1. Entranee pupil is the stop; exit pupil is 3.33 em in front of the lens, with an aperture of 3.33 em; image is 10 em behind the inverted and 2 em long. 6·2. Exit pupil is the stop; entrance pupil is 4.29 em behind the lens, with an aperture of 3.43 em; image is 10.5 em behind the inverted, and 3 em long. Answers to Selected Problems

587

6-3. Entrance pupil is the stop; exit pupil is 12 cm in front of the lens, with an aperture of 6 cm; is 10.5 cm behind the lens, and 1.5 cm long. 6-4. (b) 20 cm of L2 (c) both at LI (d) 8.57 cm beyond L2 and 1cm in diameter (e) field stop at A, entrance window in object plane with I cm diameter, exit window in image with I cm diameter (f) 2.86° 6-6. 53' 6-7. (a) crown: A 1.511, B = 4240 nm 2 , nD = 1.523: flint: A = t B= 13,190 nm2, no 1.715 (b) crown: -4.146 x 10- 5 nm I; flint: 1.290 X 10- 4 nm- t (c) crown: 3110,1.9 A; flint: 0.61 A 6-8. (a) 50.0" (b) 1/55.5 (c) A = I B 6073.7 nm2 ; 4.297 x 10-5 nm- I (d) 1 12 m 6-9. 0.01909 6-10. 2.16' 6-11. 4.82", 4.37" 6-12. (a) M, = 1 x W/m2 , I, 3.98 W/sr. 1.59 x 103 W/m2-sr (b) 7.81 x 10'5 W (c) 18 W/m2 6-14. 5.7 cm 6-15. (a) f 53.3 cm, 13.33 em, 1.86 em (b) r = -100 em; s = -40 cm 6-16. 5.3 to 7.0 n 6-17. 1.3 x 107 Ix 6-18. (a) 0.90 em 5.45 em, 3x 6-19. (a) 27.8 mm (b) f /3.1. f /5.4, f /9.4 (c) 16.0, 9.26, 5.35 mm (d) 0.03,0.09, 0.27 s 6-21. (a) 2.8 em (b) lOx 6-22. (a) 320x (b) 0.516 em 6-23. (a) 46.7x (b) 8.68 em 6-24. 5 em 6-25. 14.9 cm 6-26. (a) 7x (b) 2 em (c) 5 mm (d) 2.3 em (e) 337 ft 6-27. (b) 7.50x; 8.70x 6-28. 1.05 cm 6-29. (a) 8 em, 3X (b) 7.38 em, 2.6x 6-30. 1.25 em farther from the objective 6-31. 12.5x (b) 15x (c) 0.13 em, 3 mm (d) 3.8" 6-33. -2.5 ft; 180x

Chapter 7

588

7-1. +41.6 D 7-2. (a) 8.33 mm; + 120 D (b) 42.86 mm; +23.33 D 43.65 mm, measured from its second principal or 42.38 mm from its second surface; +22.9 D 7-3. (a) 22.34 mm from cornea (b) 21.60 mm from cornea 7-4. (a) A = 0.75846. B = 5. C = -0.05011, D = 0.65180 (b) Focal are 13.01 mm in front and 22.34 mm behind the cornea; principal points are 1.96 mm behind and 2.38 mm behind the cornea. 7-5. Block-letter sizes are 1.309 in. for 0.436 in. for 20/100; 0.262 in. for 20/60; 0.087 in. for 0.065 in. for 20/15. Leuer details are kblock letter size in each case. 7-6, (a) +3.2 D (b) yes 7-7. -2.0 D; 21.4 em 7-8. (a) argon doubled YAG (b) CO 2 (e) Nd:YAG; krypton red 7-9. (a) myopia; astigmatism (b) myopia (e) hyperopia (d) hyperopia; astl.gmatism 7-10. (a) Glass absorbs 10.6 p,m light. (b) 4.4 x 10- 4 rad (c) 14.5 p,m (d) 3 x 1(1' W/cm 2 7-n. (a) H1' W (b) 5 p,m (c) 5.1 x 10 12 W/cm2 7-12. (a) 1.04 x 10- 2 (b) 783 W/em 2 (c) 75.000 7-13. (a) 0.21 1Icm2 (b) 4.6

Answers to Selected Problems

ChapterS

8-1. 3.18 x IOtO W/cm 2 8-2 (a) y = •

(x

4

+ 2.51)2 + 2

8-3. (a) (I) and (2) qualify because they satisfy the wave equation; more simply, if 2 2 W = Z + vI, they are functions of w: y = A sin (4'7Tw) and y = Aw • (b) (i) v = I m/s in -z-direction; (ii) v = I m1s in +x-direction. 8-4. 10 m!s in + x-direction 8-5. (a) '" = 2 sin 2'7T(zI5 + t13) (b) '" = 2 sin (2'7T/5)(z + ~t) (c) '" = 2 exp [(2'7Ti(zI5 + (13)] 8-6. (a) y = 5 sin ('7TxI25) (b) y = 5 sin ('7T 125)(x + 8) 8-7. (a) 0.01 cm (b) 1000 Hz (c) 628.3 cm- I (d) 6283 s·t (e) I ms (f) 10 cm/s (g) 10 cm 8-8. (a) + I in y-direction (b) -CIB in x-direction (c) C in z-direction 8-9. y = 15 sin (kx + '7T 13) 8-10. (b) 90°, 60°, 0°, -90°, 108° (c) Subtract 90° from each. 8-U. (a) A sin (2'7T/A)(z - VI) (b) A sin (2'7T/A)(V2x ± vt) (c) A sin (2'7T/A)[(V3/3)(x + V + z) ± vl1 8-15. E = 1028 VIm; B = 3.43 x IO-~ T 8-16. (a) 5 x 10- 7 T (b) 19.88 W/m2 8-17. (a) 1.01 x I(}' VIm, 3.37 x 10 6 T (b) 4.76 x Hr ' /m 2 -s (c) E = 1010 sin 2'7T(1.43 x lQ6r + 4.28 x IO l4 t), r in m, t in s 8-18. (a) 8.75 x 10- 3 W/m 2 , 2.57 VIm (b) 2 x IOu W/m2, 1.23 x 108 Vim, 0.410 T 8-21. v = 0.168c 8-22. v = -0.9I7c 8-23. 2 III = 0.12 A

Chapterg

9-1. (a) The waves move in opposite directions along the x-axis, E t to the right, E2 to the left, with equal speeds of ~ m/s. (b) t = ~ s (c) x = 1 m 9-2. (b) E" = 8.53 sin (wt + 0.2'7T) 9-3. E" = 6.08 sin (2m + 1.13) 9-4. Y = 11.6 sin (wr + 0.402'7T) 9-5. E = 1.71 sin (m + 4.57) 9-6. (a) 2 Vim (b) 0.2 Vim 9-7. "'(1) = 2.48 sin (201 + 0.8'7T) 9-8. (a) v.., = vp [I - (wln)(dnldw)] (b) v" < Vp

9-9. cl1.56 9-10. vp = e1L5; Vg = el1.73 9-12. V R = A = constanl 9-14. 2(vle)vo 9-15. 14 cm; 1.57 cm; 0.785 cm; 0 cm/s; T seconds 9-16. (a) 1.5 cm; 25 Hz; 20 em; 5 m/s; opposite directions (b) 10 em o em/s; 7.40 x 104 cm/52

Chapter 10

10-1. 10-2. 10-3. 10-4. 10-5.

(a) 11,945 and 21,235 W/m2

(b) 18,723 W/m 2

(e) 51,903 W/m2

(e) -3 cm;

(d) 0.96

0.86, 0 0.8; 3.73/1 1.78, 2.55,4.00, 13.9

Lloyd's mirror interference fringes are produced, aligned parallel to the slit, and separated by 0.273 mm. The irradiance of the pattern is given by I = 410 sin1 (tt5 y). with y measured in cm from the mirror surface. 10-6. 509 nm 10-7. 514.5 nm 10-8. To acquire coherent beams; 560 nm

Answers to Selected Problems

589

10-9. 10-10. to-H. JO-I2. 10-13. 10-14. 10-15. 10·16. 10-17. 10·18. 10-19. 10-20. 10·21. 10-23. 10-25. 10-26.

Chapter 11

Chapter 12

(a) 83.3 em (b) 83.3 fringes (c) 150 nm 556 nm, 455 nm 20.3' 6'5/1 35'40" 9.09 x 10- 5 em; orders 4 and 3, respectively 498 rim 1.33; 103 nm (8) 2.78% (b) 89.3 nm (c) 1% Soap film becomes wedge-shaped under 15 1.16 x 10- 3 em 1.09 mm; 184

the angle of the

is 1'14".

3m 603.5 nm; 2.39 rum; 2.87 x 10-4 cm 928 nm

11·1. 436 nm 11·2. One mirror makes a of 0.0172" with the of the other. reflected through the beam splitter. Fizeau fringes result. 11·3. 23.75 p,m 11-4. (a) 80,000; (b) 79.994 11-5. (a) 11 I + NA/2L (b) 153 11-6. (a) 11.2" (b) 45.9° 11·7. 79.1 nm or A/8 r' 0.28; II' 0.9216 (d) 274.253, 19.8 VIm; 11-8. (a) 980 VIm (b) 30° 7.8%. 0.041% (e) 903, 70.8 VIm; 85%, 0.52% (f) 258 nm H·IO. (8) (b) 0.01013 cm 11-11. (a) 3.996 x 106 3.16 X 106 (e) 0.318 mm (d) 6.29 A (e) 0.002 A 11-12. (a) 329.670 (b) 361 (c) 9.8 x 1011 11-13. 2.18 cm 11-14. 16 11-15. 0.161 rum 11-16. (a) 360" (b) 1800 (c) 2 11-17. I; 0.47 12-1. /(x)

= (4/11")(sin kx + i sin 3kx + ~ sin 5kx + ... )

12·2. /(/)

= -

12-3. g(w)

12-4.

12.5. 12-6. 12-7. 12-8.

12-9. 12-10. 12-H. 12·12.

Eo 11"

Eo

2Eo cos 2w1 311"

+ 2 cos wi + -

1511"

cos 41Tt

+ ...

uh

If the width of the first is u, the width of the second is IIu. Thus the spectrum broadens as the Gaussian narrows, and vice versa. 1g(w) 12 = (A 2-d/411"2)(sin U/U}2, where u = wTo/2 The narrow-band filter has a coherence length better by one order of magnitude: 3A8 x 10- 5 m 0.013 nm; lOW 3 em 0.0243 mm (a) 0.00138 nm (b) 1 ns 2.5 mm 0.0625 cm; 2.08 x 10- 12 s 0.144 em 4 x 10-1 A; 3 x 10" Hz

Answers to Selected Problems

12-13. 12-14. 12-15. 12-17. 12-18. 12-19.

Chapter 13

13-2. 13-4. 13-5. 13-7. 13-8. 13-9.

(a) 2.08 x 10- 12 s, 0.0625 cm (b) 0.36,0.36 1.01 x 10-4 cm, 2.90 x 10- 6 cm2 ; 1.8, 35

(c) 53

(b) 2.55 0.998,0.63 0.937,0.686, 15.95 cm (a) 0,0, 0.596 cm (b) 0.895 mm (b) 0.866 1.82 p.m

63.3 mls 1.8 x 1010 bits (a) 250 nm (b) 500 nm (c) 433 nm (b) 365 nm (c) 38° ]3·10. 365 nm; blue components shift into ultraviolet and are 13-11. (a) 1.88x (b) 6330x

Chapter 14

14-2. (a) (b)

[

~ t}linearlY

[! J: _1_ [

linearly

1

-1-0 Y2

(d)

at -45 at +45

0

0

] : right-elliptically

",,1"1'"'7,,11

at +450

i)

~ [! lleft-cirCUlarlY polarized

14-3. (a) linearly polarized x-direction, traveling in +z-direction with amplitude of 2Eo (b) polarized at 53.1" relative to the x-axis, traveling in the +z-direction with amplitude of 5Eo (c) right-circularly polarized, traveling in with amplitude of 5Eo

14-5. right-circularly polarized 14-6. (a) E = Eo(\13 j + k)e i4kx - w t) (b) E = Eo(2k (c) E = kEo exp {i[(x + y)klY2 cut]} 14-7. (a) C = 0, 1mT (b) B = 0, (m + i)1T (c) B 0, A 14-9. (a) linearly polarized. 0: = 18.4", A = ViO (b)

14-10. 14-13. 14-14. 14-15. 14-16. 14-17.

±C, (m

+ 4)11"

polarized. A = t (c) right-elliptically polarized; semi major axis = 5 along y-axis, semiminor horizontal. A 5 (e) left-circularly axis 4 along x-axis (d) linearly 0: = 56.3°, A = (g) left-elliptically polarized, A = 2 (f) 53.1°, 0: 10 polarized, E right-elliptically symmetrical with x- and y-axes, Eo,/ Eoy \13 right-circularly polarized light no light emerges (a) right-ellipticallY polarized, axis along x-axis (b) vertically linearly polarized (a) linearly al ±45° (b) elliptically nm."rJ"'p,

[!

~iJ

14·20. (a) Elliptical

"",I,.,r",<>.i,,,, with inclination

0:

= -25.09r:

[~+ :3:'] Answers to Selected Problems

591

(b) Elliptical polarization with inclination

--===[ 14-21. (a) (b) (c) (d) 14-22. I =

2.6519 ] -0.6651 + i(5.1%2)

elliptically with semi-axes Eo. and Eoy with coordinate axes elliptically with principal axes at 45° to coordinate axes circularly polarized, centered at origin. radius of Eo linearly polarized with EOy/ Eo., 10(2 sin} 8 8)

Chapter 75

IS-I. 28.1% 15-2. 67 S; 22S sin 2 8} 15-3. (a) lo{O.5(a 2 + p2) (b) 0.4525/0 versus versus 0.37510 : both 0.0475/0 versus 0 154. 0.0633 mm retardation, any polarization possible (b) single 15-5. (a) single refraction, no phase unpolarized (c) same as (a) (d) double refraction, no phase retardation in each separated beam, each beam lincarly polarized (e) cases (a) and (c) 15-6. (a) difference of 0.121 mm (b) 0.015 mm 15-7. (b) 0% (c) 33% 15-8. 0.0162 mm 15-9. 20° 15-10. 0.005 IS-U. (a) 53.12° (b) lIS 15-12. (a) mixture of and polarized light (b) elliptically polarized light 15-13. (a) 56.2" (b) 33.8° 15-14. (a) 0.05 glml (b) about 46° 15-15. (a) 0.200 mm (b) 50" 15-16. (a) 8.57 x 10- 5 em (b) green 15-18. (a) OJ)091 (b) IS JIm (c) 12 (d) IS JIm 15-19. 3.15 x 10- 4 J5-20. (a) 36.8 JIm (b) 18.4 JIm 15-21. 14.7° 15-22. (a) 14.8% (b) 2.03% of 10 (c) 0.92 15-23. 10 sin2 (28) 15-24. (a) 42.5" (b) 0.600

Chapter 76

16-1. 16-2. 16-3. 16-4. 16-5. 16-6. 16-7. 16-8. 16-9. 16-10. 16-11. 16-12. 16-13. 16-14. 16-15. 16-18.

(a) 0.218 em (b) 0.218 em 0.090 (a) 0.135 mm (b) 139 496 nm 2. 1.44,0.778, and 0.55 JIm (a) 150 (b) 0.678, 0.166.0, 0.0461 1.68 x IO- J em; 2.75 x IO- J cm 8.4 x 10 4 em 9725 km in diameter; 2.69 x 10- 11 W/m 2 5.2 m 5.3 miles 75.7 to 265 m 0.400 mm (b) 0.8106,0.4053, 0.09006 (b) 2.10 mm (b) 20 0.255,0.0547,0

Answers to Selected Problems

16-20. (a) 1 16-21. 16-22. 16·23. 16-24. 16-25.

Chapter 17

17-1. 17·2. 17-3. 17-4. 17-5.

17-6. 17-7. 17-8. 17-9. 17-10. 17·11. 17-12. 17·13. 17-14. 17-15. 17-16. 17·17.

Chapter 18

LnBD'Cer 19

18-1. 18-2. 18-3. 18-4. 18-6. 18-7. 18-8. 18-9. 18-10. 18-12. 18-13. 18-14. 18-15. 18-16. 18-17. 18-18. 18-19. 18-20. 18-21.

19-2. 19-4. 19-5. 19-6. 19-7. 19-8.

(b) 5.46 mm

x; 2.73 mm along y

(e) 0.895 along x; 0.629 (d) 0.005 (a) 90" (b) liS (e) 5.7 0 IliJ == (ltID)(lfeos 6) (b) 4.7%, 1.8%, 0.84% [or In = I, 2, 3. respectively m = 0; 6 1/2 = 30" (a) 120° (b) Ip = (Wm., (c) Ip == 1m ., (d) Ip = 3/., 13°18' (a) 0.0823"/nm; 0.464 nm/mm

(b) 63,000 (a) 8.66' (b) 612.5 nm (or 587.5 nm) (e) 48; 48 987; 494 (a) 700 nm, 360 nm (b) 57.1°,25.6" (c) 350 nm and 175 nm for crown 180 nm and 90 nm for quartz 120,000; 0.069 A (a) third order (b) any width smaller than light beam 21.8 em, in each case (b) 9, in each case (c) 21.8 cm, 4.37 em, and 0.0029 em, respectively (a) 8750 grooves/em (b) t8.H9" (e) 37.77° (d) 7.88 nm/deg (a) 7000 (b) 0.018 mm (a) -5.7° to + 1I.so (b) 100,000 (c) 10 A/mm (d) I m about 5000 grooves/em (a) 1.16 ",.m (b) 18.4 (a) liS (b) 11.8" 3550 grooves/mm; reduces it (a) 3647 (b) 1200 grooves/mm (c) 3.04 mm (a) 557 to 318 (b) 960 (c) 388,800; 0.014 A (d) 0.41 0 /nm (e) 5.5 A

near. near, fur maxima: 409, 136, 81.8 em; minima: 204.5, 102, 68 em (a) 1.88 and 3.26 mm (b) 2.66 and 3.76 mm (a) 0.0346 em (b) 833 (c) 20 em, 6.67 em, 4 em (a) 0.02 em (b) 2500 (a) 4X (b) very zero (c) 5; 6 0.0012% (a) 1/100 (b) 50.31 em 1.05, 1.48. 1.82 mm 1.97 mm zero 14.8 em (a) 0.066/"

(b) 1 (b) I (a) 0.018/" (b) 0.0538/" 1.19/.: 0.86/" (a) 0.0119/"

0.55/" 21% (b) 0.145 mm (c) 0.645/. 19 I'm (a) 102 nm, 1.22 (b) 0.084% (a) 2.81% (b) 3.17% (e) 4.26% 32.3% 2; 0.25 J.tm; (a) 859 A of aluminum oxide, 1058 A of cryolite; 0.0003% (a) 227 om and 370 nm (b) 10% (e) 1.2%

Answers to Selected Problems

(b) 15.6%

593

19·12. For

1.35), SiD (n = 1.5),

(n

from surface to substrate:

(n = 19·13. (a) 81.1%

Zns

(b) 98.4% (c) 99.99%

19·14. 99.96% 19-15. 2.24

Chapter 20

Chapter 21

20-2. 61°4'; 28"56' 20·3. Oe = 32.9°, Op = 61S, 0; 28S 204. 1.272 20·9. (a) (b) 0.233% (c) 4.26% 1.26% 20-10. (a) 2.01%,2.10%, 100% (b) 2.01%, 1.91%,0.274%, 100% 20·11. (a) TM: Op = 67"33', no Oe; TE: no Bp. no Be (b) TM: Bp 22"27', Be = 24"24'; TE: no Op, Oe 24°24' 20·12. (a) R = T = 86.15% (b) R = 0.62%, T = 99.38% 20-14. (a) 0< 43.3", Op = 55.6", 0; = 34.4" (b) R 3.47%, T R = T = 91.79% (e) R 3.47%, T R T = 99.33% (d) 180", 180",0",41.0", 27.9", 0° 20-15. (a) 59"51' (b) 97.0° and 82.3" 20-16. (a) 29.3%, 45.4%, 100% (b) 29.3%, 14.9%, 5.4%, 100% 20-17. (a) 84.7%,90.9% (b) 82.5%,80. 69.5% 20·18. (a) 0.113 nm-I (b) 41 nm 20-20. (a) 0.164 ftm (b) 5.1 x 10- 6

= 1.2

2)·2. Nzl NI 214. 1.30 x

IO- IS

X

10

33

J/mJ-Hz

21·6. 0.00318; spontaneous emission is about 314 times larger than stimulated emission. This is to be expecte
Chapter 22

22-3. (a) -i{1.24 m) (b) for each, approximately, 50 m -i(L24 m) 22-4. (a) Rso = 50.03 m; Wso = 20.15 X 10- 3 m 22-5. (a) center of cavity (b) 0.51 mm (c) 0.51 mm (d) 0.4 mrad (e) (f) 245 ftW/em 2

2:

64.6 m

~]

22-6. (b) [-0.;3125 22·7. (a) 1.88 m (b) 1.88 m 22-8. (a) (b) 0.438 mm + BD + + (d) (e)

22-9.

~'-'-!...:-----::-;;-:::--:::;:--:--;~---.::.:.

t W02

22-10. (a)

ZFF

6 em; W:2(t) 0.54 mm = 0.543 mm; Z1 = 0.0663 m ~ 1.46 m (b) 0.371 mrad (c)

== 0.1 m;

== 1.81

10- 1 m 22·14. 14.6 m; 58.3 m; 131.2 m; 364.5 m (d)

594

Z2

W02

Answers to Selected Problems

X

+

Wln

.

A

l1TWi(t')

== 1.113 em;

Wool

= ILl3 em

Chapter 23

23-J. 3.18 x 1010 W/cm2 23·2. (a) 0.7 mm (b) D' = 101 #Lm; diameter at = HI #Lm 23-4. 1.4 x lOIS W/m2 (b) 1.35 x lOS Vim 23-5. (a) 4 x 109 Hz, or the channel bandwidth (b) 106 23-6. (a) 181 m (b) 2.42 x 1(1' VIm 23-7. (b) = 371 m 23·8. (b) RNHZ = 29.9 m

Chapter 24

24-1. 24-2. 24-3. 24·4. 24-5. 24-6. 24·7. 24-8. 24-9. 24-10. 24-B. 24-12. 24-13. 24-14. 24-15. 24-16. 24-17. 24-18. 24-20. 24·21. 24·22. 24-23. 24-24. 24-25.

Chapter 25

25-1. (a) 0.633, 1.898,3.164 mm 0.50 mm (c) 12.57, and 62.83 cycles/mm (d) I : ~ : 25-2. (a) product 25-3. 0.48 25-4. (a) 5 units of amplitude (c) 25[1 + sin (ay)]2 25·7. ('1TA 2/W) cos (WT) 25-8. (a) 18.3 kHz (b) Hz 25-9. (a) 0'()4 A (b) 0.1 25·10. (a) 2.86 x 10-3 cm (b) 5.59 nm (c) 224 nm (d) 0.80 readingfs 25-11. (a) 3.6 (b) 2450 (c) 0.093 mm/s

Chapter 26

26-1. 26-2. 26-3. 26-5. 26-6. 26-7. 26-8.

672 50 million (b) 1284 (a) 68.1° (b) 0.567 (c) 34S (a) 0.64 (b) 79S 3281 (c) 432 #Lm; 429 #Lm; 10.07 m 159 10.2 #Lm 12 and 120, counting both polarizations -70 db/km 0.080 mW 3.33 km; 10 km 0.136 db/km (b) - 1.25 db, -6.02 -10 db, -20 db (a) 1.0069 km; 1 km (b) 4.900 #Ls; 4.867 p,s 431 ns; 2.32 MHz 77.2 ns 14.6 ns/km 457 ps; 1/146 25 MHz (a) 4 ns (b) 0.4 ns 48.9 ns (b) 3.9 ps/km; 4.3 ps/km (a) 50.5 ns; 1.075 ns; 0.075 ns (b) 50.5 ns

w, lw lw l , lwz, 2w1 + Wz. 2w1 - Wz, + WI, 2w z 0, 2w 2w z , WI ± Wz, WI ± W). Wz ± w) " (a) 5983 nm (b) 0.046 9.84 kV; VHW is independent of the 7.9 x 10- 6 ; '1T /2 (a)

(b) At V := 0, I l at V = VHW , I 26-9.
Answers to Selected Problems

WI, WI, Wz

=0

595

26·12. The sound wave advances 150 nm, which is wave thus appears practically to the 26-13. (d) 67 26-14. 221 MHz 26-16. 2.97' 26-17. For a 5-cm the current is 31.8 A. 26-18. (a) 0.0647 min/G-cm (b) 0.0956 min/G-cm 26-19. 14.1°; 0.0712 #-tm-I Chapter 27

596

27·1. 274. 27·6. 27·7. 27-8. 27·9. 27·11.

(b) 11/ = 0.455 v'iG; I1R = 1.099 (a) 4.80 x 10 13 8- 1 (b) 1.38 x 10 16 S l (c) (a) 0.856 cm (b) 6.63 #-tm (a) 0.35 mm (b) I m 1.7 #-tm (a) 0.405 m- I (b) 11.4 m A (I + 0')1/2; B = 2(7Tc/wo)2(a + J) 1/2 C = 2(7TC/WO)4 O' (Ja + 4)(a + 1)-3/2 where a

Answers to Selected Problems

for A = 500 nm. The ullrasonic beam.

I1R

"1 = 3.92

Index

597

Cataract, 163, 1M Cauchy dispersIOn formula, 119, 122.

515 Ca\"ity dumper, 551 Qu~rgc-coupl(if device. 26---21 ChicfnlY. 112 whit'. 116 Chmmatic lateral,

Claddmg. L"O? laser. 433. 49&-97 Cnddmgton s~pe 95, 97 Cocfficiencc of fim~~c, Coherence: correlation function, 255 degree or, 256--58 l.a~er. 441---43 Icnglh and lime, 253, 443 panml, 241, 25459 relation ... , .'>umman-, 25~ :.patial, 241. 259--63. rempci1:l1. 247. 25J.--54, Coherel1l"e Icnglh. noolincar cr}">UI:I, 54445

Deulenum arc l.amp, 21 n Dextn,rotatory. 31 J Dichroism. 298--- 301 DIt:"lo.'lrk. l>implc, 568-69 cooManl, 198,579 film, 211-21 391-104

~'Iral range, 351-52 hnlognlphic. 360 ins.lrurncn(}.. 361-~63 interference, 359-61 irradiancc. ).112-43 re."olving power, 353-55

577-79

35556

bJcJordillllr}' '''Y. 31"-12, 545.552 E}e:

('....ones. eye.

Coojugalc points. 41 C...oostmctive mter!creoce, 200. 203, 205---7 214 C'nnvolutlon. 522, 533-35 G:Joke triple« Icn:.. ll'5. 130 Cornea, 152. 154. 162---66 Cornu ~pmu, 376-86 C()rre!aliun. 522, 529-33 runction, coherence, 255 413-14 5ll!i

515-76

[)-&VlR..on-ClCfmCr

expenment, 3 de Broglie wao.:elcngth, 3 Degree of whcrcfK."e, 256-58, 262 JJepth of field, 126, 12829 JJevc prism, 124 OcsIructive interference, 200, 2'03, 20.5-1, 214

Index

2,

Fibers, optic: grarnode (rnooomode)~ 501. 509 s1c~indcx. 507, 513-14 field of liew. 113-16 Field stop. I 15 Film:

324 374---76

258, 262. 265.

[MK.llrooous rays, 41

Index

Inde)(

uf w<"lucncy, 119, 306, 571J-79

{obie.

574~75,

3U~

Resolutloo:

Radtal keratotomy. Radiarn.oe. JO~12

"""",me'e"

164~

Index

537

801

Xenon arc lamp. 21

602

Index