Tutorial utorial Texts Series ies • • •
Matrix Methods for Optical Layout Layout , Gerhard Kloos, Vol. TT77 Fundamentals of Infrared Detector Materials , Michael A. Kinch, Vol. TT76 Practical Applications of Infrared Thermal Sensing and Imaging Equipment, Third Edition , Herbert Kaplan, Vol. TT75
• • • • •
Bioluminescence for Food and Environmental Environmental Microbiological Safety, Lubov Y. Brovko, Vol. TT74 Introduction to Image Stabilization Stabilization , Scott W. Teare, Sergio R. Restaino, Vol. TT73 Logic-based Nonlinear Image Processing , Stephen Marshall, Vol. TT72 The Physics and Engineering of Solid State Lasers , Yehoshua Kalisky, Vol. TT71 Thermal Infrared Characterization of Ground Targets and Backgrounds, Second Edition , Pieter A. Jacobs, Vol. TT70
• • • • • • • •
Introduction to Confocal Fluorescence Fluorescence Microscopy , Michiel Müller, Vol. TT69 Artificial Neural Networks An Introduction , Kevin L. Priddy and Paul E. Keller, Vol. TT68 Basics of Code Division Multiple Multiple Access (CDMA), Raghuveer Rao and Sohail Dianat , Vol. TT67 Optical Imaging in Projection Microlithography , Alfred Kwok-Kit Wong, Vol. TT66 Metrics for High-Quality Specular Specular Surfaces , Lionel R. Baker, Vol. TT65 Field Mathematics for Electromagnetics , Photonics, and Materials Science , Bernard Maxum, Vol. TT64 High-Fidelity Medical Imaging Displays , Aldo Badano, Michael J. Flynn, and Jerzy Kanicki, Vol. TT63 Diffractive Optics–Design, Fabrication , and Test , Donald C. O’Shea, Thomas J. Suleski, Alan D. Kathman, and Dennis W. Prather, Vol. TT62
• •
Fourier-Transform Spectroscopy Instrumentation Engineering , Vidi Saptari, Vol. TT61 The Power- and Energy-Handling Capability of Optical Materials , Components, and Systems, Roger M. Wood, Vol. TT60
• • • • • • • • • • • • • • • • • • • • • • • • •
Hands-on Morphological Image Image Processing , Edward R. Dougherty, Roberto A. Lotufo, Vol. TT59 Integrated Optomechanical Analysis Analysis , Keith B. Doyle, Victor L. Genberg, Gregory J. Michels, Vol. TT58 Thin-Film Design Modulated Thickness and Other Stopband Design Methods , Bruce Perilloux, Vol. TT57 Optische Grundlagen für Infrarotsysteme , Max J. Riedl, Vol. TT56 An Engineering Introduction to Biotechnology Biotechnology , J. Patrick Fitch, Vol. TT55 Image Performance in CRT CRT Displays , Kenneth Compton, Vol. TT54 Introduction to Laser Diode-Pumped Diode-Pumped Solid State Lasers , Richard Scheps, Vol. TT53 Modulation Transfer Function in Optical and Electro-Optical Systems Systems , Glenn D. Boreman, Vol. TT52 Uncooled Thermal Imaging Arrays , Systems, and Applications , Paul W. Kruse, Vol. TT51 Fundamentals of Antennas , Christos G. Christodoulou and Parveen Wahid, Vol. TT50 Basics of Spectroscopy, David W. Ball, Vol. TT49 Optical Design Fundamentals for Infrared Systems , Second Edition , Max J. Riedl, Vol. TT48 Resolution Enhancement Techniques in Optical Lithography , Alfred Kwok-Kit Wong, Vol. TT47 Copper Interconnect Technology , Christoph Steinbrüchel and Barry L. Chin, Vol. TT46 Optical Design for Visual Systems , Bruce H. Walker, Vol. TT45 Fundamentals of Contamination Control , Alan C. Tribble, Vol. TT44 Evolutionary Computation Principles and Practice for Signal Processing , David Fogel, Vol. TT43 Infrared Optics and Zoom Lenses, Allen Mann, Vol. TT42 Introduction to Adaptive Optics, Robert K. Tyson, Vol. TT41 Fractal and Wavelet Image Compression Techniques , Stephen Welstead, Vol. TT40 Analysis of Sampled Imaging Systems Systems , R. H. Vollmerhausen and R. G. Driggers, Vol. TT39 Tissue Optics Light Scattering Methods and Instruments for Medical Diagnosis , Valery Tuchin, Vol. TT38 Fundamentos de Electro-Óptica para Ingenieros , Glenn D. Boreman, translated by Javier Alda, Vol. TT37 Infrared Design Examples, William L. Wolfe, Vol. TT36 Sensor and Data Fusion Concepts and Applications , Second Edition , L. A. Klein, Vol. TT35
Tutorial Texts in Optical Engineering Volume TT77
Bellingham, Washington USA
Library of Congress Cataloging-in-Publication Data
Kloos, Gerhard. Matrix methods for optical layout / Gerhard Kloos. p. cm. -- (Tutorial texts series ; TT 77) ISBN 978-0-8194-6780-5 1. Optics--Mathematics. 2. Matrices. 3. Optical instruments--Design and construction. I. Title. QC355.3.K56 2007 681'.4--dc22 2007025587
Published by SPIE P.O. Box 10 Bellingham, Washington 98227-0010 USA Phone: +1 360 676 3290 Fax: +1 360 647 1445 Email:
[email protected] Web: spie.org
Copyright © 2007 Society for Photo-optical Instrumentation Engineers All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means without written permission of the publisher. The content of this book reflects the work and thought of the author(s). Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon. Printed in the United States of America.
Introduction to the Series Since its conception in 1989, the Tutorial Texts series has grown to more than 70 titles covering many diverse fields of science and engineering. When the series was started, the goal of the series was to provide a way to make the material presented in SPIE short courses available to those who could not attend, and to provide a reference text for those who could. Many of the texts in this series are generated from notes that were presented during these short courses. But as stand-alone documents, short course notes do not generally serve the student or reader well. Short course notes typically are developed on the assumption that supporting material will be presented verbally to complement the notes, which are generally written in summary form to highlight key technical topics and therefore are not intended as stand-alone documents. Additionally, the figures, tables, and other graphically formatted information accompanying the notes require the further explanation given during the instructor’s lecture. Thus, by adding the appropriate detail presented during the lecture, the course material can be read and used independently in a tutorial fashion. What separates the books in this series from other technical monographs and textbooks is the way in which the material is presented. To keep in line with the tutorial nature of the series, many of the topics presented in these texts are followed by detailed examples that further explain the concepts presented. Many pictures and illustrations are included with each text and, where appropriate, tabular reference data are also included. The topics within the series have grown from the initial areas of geometrical optics, optical detectors, and image processing to include the emerging fields of nanotechnology, biomedical optics, and micromachining. When a proposal for a text is received, each proposal is evaluated to determine the relevance of the proposed topic. This initial reviewing process has been very helpful to authors in identifying, early in the writing process, the need for additional material or other changes in approach that would serve to strengthen the text. Once a manuscript is completed, it is peer reviewed to ensure that chapters communicate accurately the essential ingredients of the processes and technologies under discussion. It is my goal to maintain the style and quality of books in the series, and to further expand the topic areas to include new emerging fields as they become of interest to our reading audience.
Arthur R. Weeks, Jr. University of Central Florida
Contents Preface
xi
1 An Introduction to Tools and Concepts 1.1 Matrix Method 1.2 Basic Elements 1.2.1 Propagation in a homogeneous medium 1.2.2 Refraction at the boundary of two media 1.2.3 Reflection at a surface 1.3 Comparison of Matrix Representations Used in the Literature 1.4 Building up a Lens 1.5 Cardinal Elements 1.6 Using Matrices for Optical-Layout Purposes 1.7 Lens Doublet 1.8 Decomposition of Matrices and System Synthesis 1.9 Central Theorem of First-Order Ray Tracing 1.10 Aperture Stop and Field Stop 1.11 Lagrange Invariant 1.11.1 Derivation using the matrix method 1.11.2 Application to optical design 1.12 Petzval Radius 1.13 Delano Diagram 1.14 Phase Space 1.15 An Alternative Paraxial Calculation Method 1.16 Gaussian Brackets
1 1 2 2 3 5 5 6 7 10 12 13 14 16 18 18 18 19 19 20 21 22
2 Optical Components 2.1 Components Based on Reflection 2.1.1 Plane mirror 2.1.2 Retroreflector 2.1.3 Phase-conjugate mirror 2.1.4 Cat’s-eye retroreflector 2.1.5 Roof mirror 2.2 Components Based on Refraction 2.2.1 Plane-parallel plate 2.2.2 Prisms
25 25 25 26 26 27 28 29 29 31
viii
2.2.3 Axicon devices 2.3 Components Based on Reflection and Refraction 2.3.1 Integrating rod 2.3.2 Triple mirror
47 49 49 52
3 Sensitivities and Tolerances 3.1 Cascading Misaligned Systems 3.2 Axial Misalignment 3.3 Beam Pointing Error
53 55 56 57
4 Anamorphic Optics 4.1 Two Alternative Matrix Representations 4.2 Orthogonal and Nonorthogonal Anamorphic Descriptions 4.3 Cascading 4.4 Rotation of an Anamorphic Component with Respect to the Optical Axis 4.4.1 Rotation of an “orthogonal” system 4.4.2 Rotation of a “nonorthogonal” system 4.5 Examples 4.5.1 Rotated anamorphic thin lens 4.5.2 Rotated thin cylindrical lens 4.5.3 Cascading two rotated thin cylindrical lenses 4.5.4 Cascading two rotated thin anamorphic lenses 4.5.5 “Quadrupole” lens 4.5.6 Telescope built by cylindrical lenses 4.5.7 Anamorphic collimation lens 4.6 Imaging Condition 4.7 Incorporating Sensitivities and Tolerances in the Analysis
59 59 60 60
5 Optical Systems 5.1 Single-Pass Optics 5.1.1 Triplet synthesis 5.1.2 Fourier transform objectives and 4 f arrangements 5.1.3 Telecentric lenses 5.1.4 Concatenated matrices for systems of n lenses 5.1.5 Dyson optics 5.1.6 Variable single-pass optics 5.2 Double-Pass Optics 5.2.1 Autocollimator 5.3 Multiple-Pass Optics 5.4 Systems with a Divided Optical Path 5.4.1 Fizeau interferometer 5.4.2 Michelson interferometer
61 62 65 65 65 66 67 68 69 72 72 73 75 77 77 77 79 80 81 82 84 92 92 95 99 99 101
ix
5.4.3 Dyson interferometer 5.5 Nested Ray Tracing
103 107
6 Outlook
111
Bibliography
113
Index
119
Preface This book is intended to familiarize the reader with the method of Gaussian matrices and some related tools of optical design. The matrix method provides a means to study an optical optical system in the paraxial approximation approximation.. In optical design, the method is used to find a solution to a given optical task, which can then be refined by optical-design software or analytical methods of aberration ration balancing balancing.. In some cases, the method method can be helpfu helpfull to demonstra demonstrate te that there is no solution possible under the given boundary conditions. Quite often it is of practical importance and theoretical interest to get an overview on the “solution space” of a problem. problem. The paraxial approach approach might then serve as a guideline during optimization in a similar way as a map does in an unknown landscape. Once a solution has been found, it can be analyzed under different points of view view using using the matrix matrix method. method. This This approa approach ch gives gives insigh insightt on how degree degreess of freedom couple in an optical device. The analysis of sensitivities and tolerances is common practice in optical engineering, because it serves to make optical devices or instruments more robust. The matrix method allows one to do this analysis in a first order order of approxima approximatio tion. n. With these results, results, it is then possible possible to plan and to interpret refined numerical simulations. In many cases, the matrix description gives useful classification schemes of optical phenomena or instruments. This can provide insight and might in addition be considered as a mnemonic aid. An aspect that should not be underestimated is that the matrix description represents a useful means of communicating among people designing optical instruments, because it gives a kind of shorthand description of main features of an optical instrument. The book contains an introductory first chapter and four more specialized chapters that are based on this first chapter. Sections 1.1–1.14 are intended to provide a self-contained self-contained introduction introduction into the method of Gaussian Gaussian transfer transfer matrices in paraxial optics. The remaining sections of the chapter contain additional material on how this approach compares to other paraxial methods. methods. The emphasis of Chapters 3 and 4 is on refining and expanding the method of analysis to additional degrees of freedom and to optical systems of lower symmetry. symmetry. The last part of Chapter 4 can be skipped at first reading. To my knowledge, the text contains new results such as theorems on the design of variable optics, on integrating rods, on the optical layout of prism devices, etc.
xii
Preface
I tried to derive the results in a step-by-step way so that the reader might apply the methods methods presented presented here to her/his her/his design design proble problems ms with ease. I also also tried tried to organize the book in a way that might facilitate looking up results and the ways of how to obtain them. It would be a pleasure for me if the reader might find some of the material presented presented in this text useful for her/his her/his own engineering engineering work. Gerhard Kloos June 2007
Chapter 1
An Introduction to Tools and Concepts 1.1 Matrix Matrix M Metho ethod d Ray-transfer matrices is one of the possibile methods to describe optical systems in the paraxial approxima approximation. tion. It is widely widely used for first-orde first-orderr layout and for the purpose purpose of analyzing analyzing optical systems systems (Gerrard (Gerrard and Burch, 1975). The reason why the the para paraxi xial al appr approx oxim imat atio ion n is ofte often n used used in the the first first phas phasee of a desi design gn or of an opti optica call anal analys ysis is beco become mess obvi obviou ouss if we hav have a look look at the the law law of refr refrac acti tion on in vecto ectori rial al form form as follows: = n 2 b × N , (1.1) n1 a × N
. This where a is the vector of the ray incident on the interface with the normal N interface separates two homogeneous media with indices of refraction n1 and n2 . The refracted ray is described by the vector b. For optical-layout purposes, we need an explicit expression of this ray in terms of the other quantities because we have to trace the ray through the optical system. Using vector algebra, Eq. (1.1) can be rewritten in the following way: b =
n1 n2
a −
n1 n2
· · a − N
1 −
n1 n2
2
· a) 1 − (N a ) 2
. N
(1.2)
The form obtained like this is complicated and it is difficult to trace the ray without making use of a computer. Therefore, a linearized form of this law would be helpful for thinking about the optical system, and this is the motivation for starting with a paraxial layout. It would be a precious tool for analyzing optical instruments if the approximated description would also allow for cascading subsystems to describe a compound system. system. The method of ray-transfer ray-transfer matrices matrices provides provides this advantage advantage and cascading of subsystems is performed by matrix multiplication. Another aspect, which might be sometimes underestimated, is that paraxial descriptio descriptions, ns, and especiall especially y the matrix matrix method, method, provide provide a conven convenient ient shorthand shorthand notation to communicate and discuss ideas to other optical designers. In a way, this
2
Chapter 1
branch of optics is axiomatic like thermodynamics, for example. The framework of the underlying theory can be reduced to a limited number of basic rules and elements. But combining these rules and elements allows one to study a great variety of optical systems.
1.2 Basic Elements We will now look for linearized relations that describe three situations, namely, propagation of a ray, and its refraction and reflection. The matrices obtained in this way serve as building blocks of the matrix description.
1.2.1 Propagation in a homogeneous medium Let us first consider the propagation of a ray in a homogeneous medium. We assume that the ray propagates in the y –z plane and choose the z-axis as the optical axis. In any plane perpendicular to the optical axis, the ray can now be described by its distance from the optical axis, y , and by the angle β , which it has with a line parallel to the optical axis. As the ray propagates along the optical axis, these coordinates may change and take different values in different planes perpendicular to the optical axis. We now choose two reference planes separated by a distance t inside a homogeneous medium (Fig. 1.1) and determine the input–output relationship. The ray starts with the coordinates [ y (1) , β (1) ]. Due to the propagation along a rectilinear line, the angle remains unchanged, β (2) = β (1) .
(1.3)
The height in the second reference plane depends on the distance traveled and on the starting angle, y (2) = y (1) + t tan β (1) .
(1.4)
Figure 1.1 Propagation in a homogeneous medium. The two reference planes are at a distance t .
3
An Introduction to Tools and Concepts
Under the assumption that the paraxial approximation is valid, i.e., for small angles β (1) , we can linearize the trigonometric function in Eq. (1.4) as y (2) = y (1) + tβ (1) .
(1.5)
Equations (1.3) and (1.5) can now be combined and written as a matrix relation, y (2)
1
t
y (1)
0
1
β (1)
β (2)
=
.
(1.6)
The matrix depends on the distance of the two reference planes. We will later refer to it as the translation matrix T , defined as T =
1 0
t . 1
(1.7)
1.2.2 Refraction at the boundary of two media Now, we will try to obtain a linearized expression for the refraction of a ray at a spherical surface described by the radius R . This surface separates two homoge-
Figure 1.2 Refraction at a spherical surface. The spherical surface separates two media with refractive indices n 1 and n 2 .
4
Chapter 1
neous media of refractive indices n1 and n2 . Let us first draw a line representing the ray as it hits the spherical surface in a reference plane (Fig. 1.2). We consider how the input and output variables are changed in this single reference plane where the refraction takes place. The distance from the optical axis remains unchanged for the ray leaving the reference plane, i.e., y (2) = y (1) .
(1.8)
The change in angle is described by the law of refraction, n1 sin i (1) = n2 sin i (2) ,
(1.9)
where the angles i (1) and i (2) refer to the normal vector that is perpendicular to the surface. Assuming that the paraxial approximation is valid, Eq. (1.9) can be linearized as (1.10) n1 i (1) = n 2 i (2) . But we need expressions in terms of the angles β (1) and β (2) that are measured with respect to a line parallel to the optical axis. To obtain relations between these angles and the angles appearing in Eq. (1.9), we have a closer look at the triangles in Fig. 1.2. Applying the exterior angle theorem for triangles twice, we have i (1) = β (1) + α,
(1.11)
i (2) = β (2) + α.
(1.12)
Substituting these equations into Eq. (1.10), we find β (2) =
n1 n2
β (1) +
n1 − n2 n2
α.
(1.13)
Neglecting the small distance between the intersection of the spherical surface with the optical axis and the reference plane, we approximate the angle α appearing in Eq. (1.13) as y (1) ∼ tan α = . (1.14) R Linearizing the trigonometric function for small angles (tan α ∼ = α) , Eq. (1.14) is substituted into Eq. (1.13) and we have β (2) =
n1 n2
β (1) +
n1 − n2 n2 R
y (1) .
(1.15)
This is the linearized input–output relation we were looking for. In combination with Eq. (1.8), we can write it in matrix form as y (2)
1
0
n1 −n2 n2 R
n1 n2
β (2)
=
y (1)
β (1)
.
(1.16)
The corresponding matrix will be used later as the refraction matrix R , defined as R =
1
0
n1 −n2 n2 R
n1 n2
.
(1.17)
5
An Introduction to Tools and Concepts
Figure 1.3 The unfolding of a spherical mirror.
1.2.3 Reflection at a surface A geometrical consideration quite similar to the one that led to Eq. (1.17) can also be used to find the matrix for a spherical concave mirror. In this case, the output ray remains on the same side of the reference plane. It is interesting to note that we can formally obtain the matrix of an unfolded spherical concave mirror by setting n 1 = 1 and n 2 = −1 in Eq. (1.17), i.e., S =
1
0
− 2R
−1
.
(1.18)
Unfolding refers to the symmetry operation (or coordinate break) depicted in Fig. 1.3. This can be helpful in finding the matrix chain of a compound optical system. Please note that some signs might change in the system matrix with respect to the starting system because reference is made to an optical axis with a different direction after the coordinate break.
1.3 Comparison of Matrix Representations Used in the Literature In the literature, different notations used to write the ray-transfer matrices can be found. Many authors use coordinates that have nβ as the second coordinate, where n is the index of refraction (Guillemin and Sternberg, 1984). An advantage of this notation is that the determinant value of the ray-transfer matrices is always 1. This provides a useful check during calculations and can also simplify theoretical arguments based on the determinant. In the description used here, the determinant of the ray-transfer matrix A has the value det A =
n1 n2
,
(1.19)
with n1 as the refractive index of the medium at the entrance reference plane and n2 as the refractive index of the medium at the exit reference plane.
6
Chapter 1
The second coordinate nβ can also be introduced as a modified ray slope (Siegman, 1986) as dr(z) r (z) ∼ . (1.20) = n(z) dz The interpretation of this coordinate in terms of slope can be fruitful in some circumstances.
1.4 Building up a Lens With the prerequisite of Eqs. (1.7) and (1.17), we can determine the matrix of a spherical lens. The refraction at the first surface is expressed by the matrix R (a) . The ray is then propagated through the lens using the translation matrix T and finally refracted at the second surface of the lens. To describe this refraction, the matrix R (b) is used. The combined effect is calculated as the product of these matrices, (1.21) S = R (b) T R (a) . More explicitly, this equation reads as S =
1
0
n2 −n3 n3 R2
n2 n3
1
t
1
0
1
n1 −n2 n2 R1
n1 n2
0
,
(1.22)
where t is the thickness of the lens and R1 and R2 are the radii of the first and the second surfaces of the lens, respectively. Because the lens is in air, we can specialize the set of refractive indices as n1 = 1, n2 = n, and n3 = 1. Therefore, we have t 1 − nR−1 nt n 1 (1.23) S = . 1−n 1−n t n−1 n−1 1−n t − R − R + R R n 1 − R n
1
2
1
2
2
This might suggest the following abbreviations: P 1 = P 2 =
n − 1 R1 1 − n R2
,
(1.24)
.
(1.25)
With these abbreviations, Eq. (1.23) then takes the form S =
1 − P 1 nt
t n
−P 1 − P 2 + P 1 P 2 nt
1 − P 2 nt
.
(1.26)
The so-called thin lens is obtained by letting the lens thickness t tend to zero in Eq. (1.26), 1 0 (1.27) S = . −P 1 − P 2 1
7
An Introduction to Tools and Concepts
1.5 Cardinal Elements To identify the lower-left entry in the matrix of the thin lens, we first look at a lens described by a more general matrix of the form A =
a11
a12
a21
a22
.
(1.28)
Its focal plane can be found by letting a ray parallel to the optical axis pass through the lens and determine the distance b from the exit reference plane to the plane where it intersects the optical axis. Expressing this in matrix notation, we have 0
β out
or
0
β out
=
=
1 0
b 1
a11
a12
a21
a22
y in
0
,
a11 + ba 21
a12 + ba 22
y in
a21
a22
0
.
(1.29)
(1.30)
This implies that 0 = (a11 + ba 21 )y in .
(1.31)
This equation should hold for all values of y in . Therefore, it follows that a11 + ba 21 = 0 .
(1.32)
The position of the second focal plane of the lens described by the matrix A is therefore determined by a11 (1.33) b = − , a21 and we can identify b as the focal length f of the lens. Applying this result to the thin lens of Eq. (1.27), for which a 11 = 1 holds, we see that the lower-left entry represents the negative inverse of its focal length, i.e., the matrix of the thin lens is F =
1
0
− f 1
1
.
(1.34)
The second focal plane is one of the cardinal elements of a lens. The position of the first focal plane is calculated on the same footing, but by letting a parallel ray enter from the other side into the system or by finding the distance for which the light from a point source in front of the lens is collimated. In both ways, the following result is obtained for the position of the first focal plane: a=−
a22 a21
.
(1.35)
A straightforward way to obtain other cardinal elements is by direct comparison with the thin lens. We are interested in finding the positions of the planes with
8
Chapter 1
respect to which a lens given by the matrix A could be described similar to a thin lens. To this end, we take the following approach: 1
h2
a11
a12
0
1
a21
a22
1
h1
0
1
=
1
0
− 1f
1
,
(1.36)
where h1 and h2 are the distances that have to be determined. The corresponding planes are called principal planes and, together with the focal planes, they are the cardinal elements of a lens. After performing the matrix multiplication on the lefthand side, we have
a11 + a21 h1
a12 + a11 h1 + a 22 h2 + a 21 h1 h2
a21
a22 + a 12 h2
=
1
0
− 1f
1
.
(1.37)
The position of the first principal plane is therefore given by h1 =
1 − a 11 a21
,
(1.38)
measured with respect to the first reference plane of the lens. The position of the second principal plane is at 1 − a 22 (1.39) h2 = , a21 measured with respect to the second reference plane of the lens. A beautiful illustration of the principal planes concept is given by Lipson et al. (1997). We can trace typical rays through the lens and draw this on a piece of paper. If we now fold this paper along the lines that represent the principal planes, we can hold it in such a way that the part between the principal planes is perpendicular to the other parts. These other parts are combined to represent a simplified arrangement (Fig. 1.4), which corresponds to a thin lens. This is in complete analogy to Eq. (1.36). The results on the cardinal elements are collected in Fig. 1.5. With these prerequisites, we can state the cardinal elements of the thick lens given by Eq. (1.23). The equation for the focal length f of the lens is 1 f
=
n − 1 R1
+
1 − n R2
−
(n − 1)(1 − n)t nR1 R2
.
(1.40)
Its principal planes are at h1 = −f h2 = −f
n − 1 t R1 n
1 − n t R2 n
,
(1.41)
.
(1.42)
An Introduction to Tools and Concepts
9
Figure 1.4 Principal planes visualized by folding. The optical system is described by the matrix A . It has the focal points F 1 and F 2 and its principal planes are at h 1 and h 2 , respectively.
Figure 1.5 Cardinal elements. The focal points F 1 and F 2 and the positions h1 and h2 of the principal planes serve to characterize the optical system given by the matrix A .
10
Chapter 1
1.6 Using Matrices for Optical-Layout Purposes In the derivation of the position of the second focal plane, we considered the optical arrangement formed by a lens, which was given by the matrix A , and a translation matrix T , i.e., (1.43) S = T A. On this combined optical arrangement, the condition s 12 = 0 was then imposed to ensure that the ray height in the output plane was independent of the ray angle in the input plane. We used this condition because in the paraxial approximation, it characterizes a point in the second focal plane. This way of reasoning can also be applied to other situations. Its application to the first focal plane is convenient; to this end, we consider the combined arrangement given by the matrix product, S = AT .
(1.44)
We then impose a condition on the combined matrix S that expresses (in the linear approximation) that a bundle of rays at a given ray height y in but with different angles β in in the entrance plane of S will be transformed into a parallel beam, i.e., a bundle of rays with the same angle, at the exit plane. The general input–output relation is y out s11 s12 y in . (1.45) = β out s21 s22 β in
To ensure that β out has a single value for a given ray height y in , it has to be independent of β in . A look at the input–output relation suggests that this condition is met if we choose s22 = 0 . (1.46) This choice determines the distance contained in the translation matrix and thereby the position of the first focal plane, which corresponds to the matrix A . At this point, we have conditions for the first focal plane (s 22 = 0) and for the second focal plane (s11 = 0), and we might ask: what is the characteristic feature of a ray-transfer matrix S that describes imaging? The rays leaving at a point at y in in the object plane with different angles β in intersect in a point at y out in the image plane. If the matrix A describes a lens, we have to add two spacings on both sides to model imaging, so we have S = BAG, (1.47) with B = 10 1b and G = 10 1g . Considering the input–output relation again, we find that y out is independent of β in if
s12 = 0 .
This is the characteristic feature of a matrix S that represents imaging.
(1.48)
11
An Introduction to Tools and Concepts
We can apply this condition immediately to find the imaging relation for a thin lens. The corresponding matrix chain is S =
1 0
b 1
1
0
− 1f
1
1 0
g = 1
b f
1 −
g + b −
− 1f
1 −
bg f
g f
.
(1.49)
Using s 12 = 0, we have the well-known imaging condition 1
+
g
1 b
1
=
f
,
(1.50)
which expresses that b varies in a hyperbolic way as a function of g and vice versa. The signs of the distances are positive in Eq. (1.50) because the direction of the distances is chosen as the direction of the optical axis. To find a relation for the first focal plane, we asked under which conditions parallel rays leaving the system might be independent of the input angle. Alternatively, we can consider the situation where the rays leaving the system are independent of the ray height in the entrance plane. This is the case if a collimated input beam is transformed into a collimated output beam. Making reference to the input–output relation for S , we see that setting s21 = 0
(1.51)
ensures that β out does not depend on the ray height y in in the input reference plane. Because collimated rays are considered, no additional translation matrices have to be introduced here and therefore S = A. Earlier, we related the matrix entry a12 to the negative inverse of the focal length of an optical system via Eq. (1.33). This matrix entry takes the value of zero now, which corresponds to the case of an afocal system. Typical examples for such systems are telescopes. In the paraxial approximation, we might model a telescopic arrangement using thin lenses. We choose two lenses with focal lengths f 1 and f 2 , separated by a distance d . Concatenation of the corresponding matrices gives us the system matrix S =
1
0
− f 12
1
1 0
d 1
1
0
− f 11
1
=
1 −
− f 11
−
1 f 2
d f 1
+
d d f 1 f 2
1 −
d f 2
. (1.52)
Now, we impose the condition that s 21 = 0 should hold. This implies that
−
1 f 1
−
1 f 2
+
d f 1 f 2
= 0 .
(1.53)
(1.54)
The setting of d = f 1 + f 2 solves this equation and we have S =
− f f 21
f 1 + f 2
0
f 1 f 2
12
Chapter 1
Figure 1.6 The significance of zero-matrix entries.
for the system matrix of the telescopic arrangement. It represents a Newtonian telescope if both focal lengths are positive. If the focal length of the first lens is negative, the matrix describes a Galilean telescope, which is composed of a concave and a convex lens. Optical arrangements of this type also serve as transmissive beam expanders (Das, 1991) and intracavity telescopes (Siegman, 1986). The results on the significance of special matrix entries are summarized in Fig. 1.6.
1.7 Lens Doublet We encountered telescopic arrangements as the first examples of a lens doublet and we now have a closer look at optical systems composed of two lenses. The matrix that describes two lenses separated by a distance d forms the starting point of our discussion: d 1 − f d 1 S = . (1.55) 1 1 d d − f − f + f f 1 − f
1
2
1 2
2
13
An Introduction to Tools and Concepts
The term s 21 is related to the focal length of the doublet (measured with respect to its second principal plane). 1 f
=
1 f 1
+
1 f 2
−
d f 1 f 2
.
(1.56)
[As shown before, this principal plane is at a distance z = ( 1 − s11 )/(s21 ) from the second reference plane of the system.] To facilitate the discussion, it is convenient to reference the intermediate distance to the second focal plane of the first lens and to the first focal plane of the second lens by setting d = f 1 + E + f 2 .
(1.57)
With this setting, the equation for the focal length of the doublet reduces to f = −
f 1 f 2 E
.
(1.58)
It can now be discussed in terms of the signs of the three parameters that intervene. Depending on whether f 1 < 0 or f 1 > 0, f 2 < 0 or f 2 > 0, or E < 0, E = 0, or E > 0, twelve cases can be distinguished. The case where f 1 > 0 and f 2 > 0 and E = 0, for example, represents the Galilean telescope. At this point, it is near at hand to make a distinction between divergent (f < 0 ) and convergent (f > 0) doublets in terms of their three parameters. A compound microscope represented as a doublet is characterized by f 1 > 0 and f 2 > 0 and E > 0, and it is interesting to note that it is an example of a divergent system (Pérez, 1996)
1.8 Decomposition of Matrices and System Synthesis In the layout of a new optical system, it is advantageous to know how the raytransfer matrix of a given optical system can be factorized. Let us consider the design of an optical device with given properties and that some of these features can be expressed in terms of a system matrix. To realize the device, it is now of interest to systematically explore in which ways a device with the given features can be realized. To this end, it is useful to divide the device into subsystems, the combination of which would create the desired functionality. In the matrix description, this is equivalent to considering matrix products of the target matrix, and this is where factorizing the system matrix comes into play. The problem of a synthesis of optical systems using this approach has been studied in depth by Casperson (1981). In what follows, we will consider optical systems that have both their object and image planes in air. Therefore, n1 = 1 and n 2 = 1 and the determinant of the system matrix S can be written as det S =
n1 n2
= 1 .
(1.59)
14
Chapter 1
Therefore, the condition s11 s22 − s 12 s21 = 1
(1.60)
is contained implicitly in Eqs. (1.61) and (1.62). A generalization is possible and can, for example, be found in the work of Casperson (1981). The appropriate factorization depends on the matrix entries. If we consider a nonimaging problem, we can assume s 12 = 0 for the system matrix. Such a matrix can be factorized as S =
s11
s12
s21
s22
=
1
0
s22 −1 s12
1
1
s12
1
0
0
1
s11 −1 s12
1
.
(1.61)
If the lower-left entry of the system matrix can be assumed to be nonzero (s21 = 0 ), i.e., if we do not look for an afocal system, the following matrix decomposition is appropriate: S =
s11
s12
s21
s22
=
1
s11 −1 s21
0
1
1
0
1
s22 −1 s21
s21
1
0
1
.
(1.62)
What is left are the cases in which both s12 = 0 and s21 = 0. These cases correspond to optical systems that are imaging and afocal devices. In the above-cited work, four possibilities for a decomposition of this diagonal matrix are given. The system matrix is either decomposed in a product of matrices A and B with a 21 = 0 and b 21 = 0 as S = S =
s11
0
0 s11
s22 0
0
s22
s11
−ts22
0
s22
= =
1 0
t 1
s11
−t s11
0
s22
1 0
,
(1.63)
t , 1
(1.64)
,
(1.65)
.
(1.66)
or a product of matrices with a 12 = 0 and b 12 = 0 as S = S =
s11
0
0 s11
s22 0
0
s22
= =
1
0
− 1f s11
1 0
s22 f
s22
s11
0
s11 f
s22 1 0
− f 1
1
Depending on the application, the matrices appearing in the product can then be further decomposed by applying the same set of rules.
1.9 Central Theorem of First-Order Ray Tracing We will now turn to a theorem that is of prime importance to ray tracing using the matrix method. It can be applied to different sets of rays. Its main content is that the number of rays necessary to characterize an optical system in the linear approximation is rather small.
15
An Introduction to Tools and Concepts
Let us consider two rays labeled a and b that are traced through the optical system described by the matrix A . Each ray vector entering the system is mapped onto an output ray vector as follows: ya
yaout
→
βa yb
βaout ybout
→
βb
βbout
,
(1.67)
.
(1.68)
The mapping is given by the system matrix A . Therefore, we have yaout
ya
= A
βaout ybout
βa yb
= A
βbout
βb
,
(1.69)
.
(1.70)
We assume that we can completely determine the four ray coordinates and that we want to use this information to determine the system matrix A. Its entries are therefore the unknown variables of the problem, and we can state it by rewriting the above equations as the following system of linear equations:
ya
βa
0
0
yb
βb
0
0
0
0
ya
βa
0
0
yb
βb
a11
ya
βa
yb
βb
a12 a21 a22
=
yaout ybout βaout βbout
.
(1.71)
Because the matrix is partitioned, two sets of linear equations can be solved independently. If the determinant D = det
= 0 ,
(1.72)
the problem has a unique solution, namely,
det a11 =
det a12 =
det a21 =
det a12 =
yaout βa ybout
βb
D ya yaout yb ybout
D βaout βa βbout βb
D ya βaout yb βbout
D
,
(1.73)
,
(1.74)
,
(1.75)
.
(1.76)
16
Chapter 1
D = 0 is equivalent to the condition that the input ray vectors are linearly independent. We can therefore conclude that the ray-transfer matrix is completely determined if we know a set of two linearly independent input ray vectors and the corresponding output ray vectors. In the linear approximation, the passage of any other third ray through the system is then known because we can trace it through the system using the matrix A . Putting it in different words, the theorem states that in the approximation used, the input–output relation is completely characterized once the input and output data of two linearly independent rays are known. This gives the theoretical basis of why an optical system can be characterized to such an extent by just tracing the principal ray and the axial ray.
1.10 Aperture Stop and Field Stop The aperture stop is defined as the opening of an optical system that limits the input angle at zero height in the object plane. A ray with these coordinates can be transported through the system. If the input angle of a ray is slightly greater than this critical angle, the ray is blocked. We might have several candidates in the system to cause this blockage, and which of them forms the aperture stop can be determined in the following way using the matrix method. We label the free diameters of the candidates as y (k) . To every candidate now corresponds a matrix P (k) that maps the start ray into the reference plane at z (k) , y (k)
β
(k)
= P (k)
0
β
This implies that β
(k)
=
y (k)
(k)
.
(1.77)
(k) p12
.
(1.78)
The aperture stop is at the position z (k) for which β (k) takes the minimum value of all the candidates. It has the height y as = y (k) if (k ) is the label for that minimum. The axial ray is the ray that starts at zero height in the object plane and that passes through the aperture stop at the maximum possible height. If we suppose that the matrix P describes the mapping of the ray from the object plane to the aperture plane, we can trace this ray to that plane using P
0
yas
β
=
β
.
(1.79)
Its start coordinates in the object plane are y in
0
β in
=
yas p12
,
and this ray can now be traced through the complete optical system. We describe the second part of the system, i.e., the part between the aperture plane and the image
17
An Introduction to Tools and Concepts
plane, by Q . Therefore, the system matrix is S = QP ,
(1.80)
and the coordinates of the axial ray as its leaves the system are y out
β out
yas
=
s12
p12
s22
.
(1.81)
While the axial ray starts at zero height in the object plane and passes through the aperture stop at its margin, the principal ray starts at the marginal height of the object (if this corresponds to the field stop) and passes through the aperture stop at zero height. In the matrix description, we can express this relation by using the matrix P , which describes the mapping from the object plane to the plane of the aperture stop, as 0 y field P . (1.82) = β β
To be able to trace the principal ray through a complete system, we need its input angle, which we can calculate from the following equation: β=−
p11 p12
y field .
(1.83)
Therefore, the input coordinates of the principal ray are given by y in
=
β in
y field y field − pp11 12
,
(1.84)
and the output coordinates after passage through the whole system are y out
β out
= y field S
1
− pp11 12
.
(1.85)
It is interesting to note the following symmetry that exists between the axial ray and the principal ray: 0
yas
P
β
y field β
=
for the axial ray,
β
= P −1
0
β
for the principal ray,
(1.86)
(1.87)
where the corresponding inverse matrix has been used. Das (1991) expressed this symmetry relation by writing “ . . . the field stop is nothing but the new aperture stop, when the object is placed at the center of the actual aperture stop.” (Please note that in writing the symmetry relation it was assumed that the extension of the object can be identified with the extension of the field stop. This is quite often the case, but more intricate situations are possible.)
18
Chapter 1
1.11 Lagrange Invariant 1.11.1 Derivation using the matrix method We know that det A =
n1
(1.88)
n2
holds for a ray-transfer matrix. During the derivation of the central theorem of firstorder ray tracing, the following result was obtained for two linearly independent rays that pass through the system described by this matrix: A =
a11
a12
a21
a22
1
yaout βb − ybout βa
ya ybout − yb yaout
D
βaout βb − βbout βa
ya βbout − yb βaout
=
.
(1.89)
If we use this result in Eq. (1.88), we have (yaout βb − ybout βa )(ya βbout − y b βaout ) − (βaout βb − β bout βa )(ya ybout − yb yaout ) (ya βb − yb βa )2
=
n1 n2
.
(1.90) Performing the multiplications, we find yaout βbout − βaout ybout ya βb − βa yb
=
n1 n2
.
(1.91)
This equation can now be rearranged slightly, to separate input and output quantities, as n1 (ya βb − βa yb ) = n 2 yaout βbout − β aout ybout . (1.92)
We can therefore conclude that the following quantity is conserved during the passage through the system: L = −n(ya βb − β a yb ).
(1.93)
This is the Lagrange invariant.
1.11.2 Application to optical design We now turn to an application of the Lagrange invariant that is useful when designing imaging systems. Let us consider the invariance condition given by Eq. (1.92) and specialize it for the case where ray a is the axial ray and ray b is the principal ray. From the discussion earlier, we know that these are two linearly independent rays and are therefore suitable to characterize the system. These rays pass through an imaging system. As reference planes, the object plane and the image plane are natural choices, so we have n(yar βpr − βar ypr ) = n (yar − βar βpr ypr ).
(1.94)
19
An Introduction to Tools and Concepts
In both these planes, the height of the axial ray is equal to zero. Therefore, the above equation reduces to n(βar ypr ) = n (βar ypr ).
(1.95)
The quantities βar and βar can be identified with the aperture of the system in the object plane and the image plane, respectively, and yar and yar are the heights of the object and the image. In this way, the Lagrange invariant allows one to establish a direct relation between these quantities. This relation is, for example, of use to study the magnification and angular magnification of an optical instrument.
1.12 Petzval Radius The Petzval radius is the reciprocal of the curvature of the image field. This is a concept of aberration theory, and it is surprising that we can determine its value from paraxial quantities. The quantities to be considered are quite similar to those that we encountered building up a lens, namely, P 1 =
n − 1 R1
P 2 =
,
1 − n R2
.
We state the relation for the Petzval radius R p of a system made up of N refractive surfaces, without proof, as 1 ∼ = Rp
N
k =1
Dk nk
,
(1.96)
where Dk = nk /Rk is the refractive power of the k th surface, nk is the difference in refractive indices, and Rk is the radius of curvature of the k th surface. A proof can be found in textbooks on optics (Born, 1933; Born and Wolf, 1980) The Petzval radius should tend to infinity to have a flat image field. This corresponds to finding a combination for the right-hand side of the equation that brings its value close to zero. An example for an optical device that minimizes this quantity is Dyson’s system (Dyson, 1959). This optical arrangement will be considered in more detail in Section 5.1.
1.13 Delano Diagram The Delano diagram or y –y¯ diagram is a visual tool of paraxial analysis (Delano, 1963; Shack, 1973; Besenmatter, 1980). It is created by tracing a principal ray ( y) ¯ and an axial ray (y) through an optical system and by drawing the corresponding ray heights (y¯ , y ) in a single diagram. The position on the optical axis does not appear explicitly in this diagram and this representation is therefore somewhat abstract compared to a usual ray trace. But in many cases it can give an overview that is like a kind of shorthand notation for the system. Figure 1.7 shows four Delano diagrams that represent the different cases of matrices with zero entries.
20
Chapter 1
Figure 1.7 Delano diagram, four cases.
1.14 Phase Space Looking at the set of coordinates used in matrix optics, it is near at hand to try a representation in a plane using the ray height as one coordinate, and the ray angle as the other coordinate. A set of rays in a given reference plane can then be represented as a surface in this abstract plane. To familiarize ourselves with this concept, let us choose a set of rays whose coordinates are represented by a rectangle in this so-called phase space. We would like to see what might be the effect of the translation matrix T on this set of rays.
0 → β0
0 β0
+
tβ0 0
,
(1.97)
y0 → β0
y0 β0
+
tβ0 0
,
(1.98)
0 → 0
0 , 0
y0 → 0
y0 0
.
(1.99)
(1.100)
Graphically, this can be expressed as in Fig. 1.8. The corresponding mapping for the refraction matrix is shown in Fig. 1.9.
An Introduction to Tools and Concepts
21
Figure 1.8 Effect of the translation matrix in phase space. The spatial coordinate is represented along the y -axis and the angular coordinate is represented along the β -axis.
Figure 1.9 Effect of the refraction matrix in phase space. As in Fig. 1.8, the spatial coordinate is represented along the y -axis and the angular coordinate is represented along the β -axis.
It is an important feature of this phase space that the “volume” (or “surface” in the two-dimensional case considered here) is conserved if we consider mappings between input and output planes where the refractive indices are 1. This conservation is a consequence of Eq. (1.19). From the theoretical point of view, it is more convenient to use coordinates (y, nβ). The corresponding “volume” (or “surface”) is then always conserved. The phase-space approach is common in laser technology (Hodgson and Weber, 1997).
1.15 An Alternative Paraxial Calculation Method An alternative paraxial method (Berek, 1930) uses distances s k measured along the optical axis and ray heights h k measured perpendicular to it as a set of coordinates for the description of a ray. For readers who use this method, it might be interesting
22
Chapter 1
to see how both methods are connected. They will be familiar with the so-called transition equations. These equations state how a ray is transformed that leaves a lens surface labeled with k and and that reaches another surface (labeled with k + 1) situated at a distance e k as follows: sk+1 = s k − ek , h k +1 sk+1 = , uk+1 hk sk = , uk
(1.101)
(1.102)
(1.103)
where u k+1 and u k are angles. We can combine these equations to obtain h k +1
uk
=
hk
uk
− ek .
(1.104)
Setting u k+1 = u k , we can write the linear relation, hk+1
uk+1
=
1
−ek
0
1
hk
uk
,
(1.105)
which is quite similar to the relation for the translation matrix T . The other method also makes use of the relation nk
1 Rk
−
1 sk
= n k
1 Rk
−
1 sk
(1.106)
for the two sides of a refracting surface with radius Rk . Using uk = hk /sk for the paraxial angle, we can write this equation as follows: uk =
hk Rk
1 −
nk
nk
+
nk nk
uk .
(1.107)
Assuming h k = hk in addition, we have an augmented linear relation of the form hk
uk
=
1
0
nk −nk Rk nk
nk nk
hk uk
.
(1.108)
This relation corresponds to the refraction matrix R of the matrix description.
1.16 Gaussian Brackets While cascading matrices in order to determine a system matrix, the recursive character of the problem became clear. But at this point, we were unable to state the underlying recursion law that would allow us to express the ray that finally leaves
23
An Introduction to Tools and Concepts
the system, and that is described by the product matrix, as a function of the input ray without performing the matrix multiplication. The use of the mathematical concept of Gaussian brackets makes it possible to state this recursion formula. An introduction to the algebra of Gaussian brackets and the recursion law for cascaded linearized optical systems was given by Herzberger (1943). In this text, the Gaussian brackets are defined by the following recursion formula:
[a1 , . . . , ak ] = a 1 [a2 , . . . , ak ] + [ a3 , . . . , ak ],
(1.109)
with [ ] = 1. This implies
[a1 ] = a 1 ,
(1.110)
[a1 , a2 ] = a1 a2 + 1,
(1.111)
[a1 , a2 , a3 ] = a 1 a2 a3 + a1 + a3 ,
(1.112)
[a1 , a2 , a3 , a4 ] = a 1 a2 a3 a4 + a1 a2 + a1 a4 + a3 a4 + 1. (1.113) From Herzberger’s article, we take a description of a lens using his symbols, but arrange the linear transformation as a matrix as follows:
x2 ξ 2
=
d 12 φ n12 1
1 −
φ1 + φ2 −
− nd 1122
d 12 φ φ n12 1 2
1 −
x1
d 12 φ n12 2
ξ 01
.
(1.114)
where x is the distance between intersections of the optical axis and the ray with the reference plane and ξ is the inclination angle of the ray. The subscripts indicate the corresponding surfaces. φk is the refractive power of the k th surface, d k,k +1 is the distance between the k th and the (k + 1)st surface, and nk,k +1 is the refractive index between the k th and the (k + 1 )st surface. Using Eqs. (1.110)–(1.112), this can be recast in the following way:
x2
=
ξ 2
φ1 , − nd 12
− nd 1122
x1
d 12 , φ2 n12
ξ 01
12
φ1 , −
d 12 , φ2 n12
−
.
(1.115)
For the general case, the recursion law reads as follows (Herzberger, 1943):
x = ξ
d
φ1 , − nd 12 , . . . , − nk−1,k
12
φ1 , . . . , −
k −1,k
d k −1,k , φk nk −1,k
d
− φd 1122 , φ2 , . . . , − nkk−−11,k,k −
d 12 , . . . , φk n12
x1
ξ 01
. (1.116)
This equation establishes a link between the input coordinates and the output coordinates of an optical system. The optical properties of the system are described by the matrix entries, which are Gaussian brackets. In order to state these entries explicitly in terms of refractive powers, distances, and refractive indices, the recursive relations have to be evaluated using Eq. (1.109) in a kind of backtracking procedure. The approach of the matrix method is advantageous, because it allows us to obtain the system matrix by concatenating matrices, i.e., by matrix multiplication.
24
Chapter 1
In cases, where it is useful to state input–output relations in a recursive way, the method of Gaussian brackets provides an alternative approach. In the framework of the matrix method presented in this text, I found no way to state the recursion law with a similar conciseness. Therefore, I would like to draw the reader’s attention to the Gaussian-brackets method that allows such a formulation. It is beyond the scope of this book to derive the method here, and interested readers readers are referred referred to Herzber Herzberger’ ger’ss work. work. In optical design, Gaussian Gaussian brackets brackets were applied to the layout of zoom systems (Pegis and Peck, 1962; Tanaka, 1979, 1982).
Chapter 2
Optical Components 2.1 Componen Components ts Based on Reflection Reflection Mirrors are a key component of many optical devices. The matrix for the reflection at a spheri spherical cal mirror mirror was was consid considere ered d in the introduc introductor tory y chapte chapterr. We will will now now discuss reflectors more generally.
2.1.1 Plane Plane mirror mirror The matrix describing the reflection at a plane mirror can be obtained by taking the matrix for reflection at a spherical reflector and letting the radius of the spherical mirror tend to infinity. In this way, the unity matrix is obtained as A =
1 0
0 1
.
(2.1)
The signs that appear in this matrix are surprising at first, and it is instructive to derive derive the matrix also in an alternati alternative ve way. way. In Fig. 2.1, the reflection reflection of a ray at a plane mirror mirror, which which is perpendic perpendicular ular to the optical optical axis, is depicted. depicted. In the matrix matrix representation used here, we unfold the ray using the reference plane of the mirror as the plane of the coordinate break. Figure 2.2 shows the result of this “unfolding.” It is this coordinate break that causes the positive signs in the matrix of the plane reflector.
Figure 2.1 Plane 2.1 Plane mirror.
26
Chapter 2
Figure 2.2 Unfolded 2.2 Unfolded plane mirror.
2.1.2 Retroreflector Retroreflector Unlike a plane mirror, a retroreflector redirects the beam back into the same direction from where it came. In the matrix description, this is expressed by a negative sign of the matrix entry a22 . Reflection at a retroreflector is combined with a change in the height height of the beam. beam. The height height of the incide incident nt beam is change changed d from from y in to y out = −y in . This corresponds to a parallel shift and is expressed by a negative sign of the matrix entry a 11 . The complete matrix of the retroreflector reads as A =
−1 0
0 −1
.
(2.2)
(2.3)
More generally, a retroreflector has the following matrix: A =
−1 0
a12 −1
.
2.1.3 Phase-conjugate Phase-conjugate mirror An element that redirects an incident beam into itself without any shift in ray height would be described by the following matrix (Lam and Brown, 1980): A =
1 0
0 −1
.
(2.4)
Ther Theree are are devi device cess that that expl exploi oitt nonl nonlin inea earr opti optica call effe effect ctss and and that that are are able able to oper operat atee in such a way on an incident laser beam. To describe these so-called phase-conjugate
Optical Components
27
mirrors in the paraxial approximation, the matrix stated above can be used. The nonlinear effect itself is far beyond the scope of this approach. Retroreflectors, on the other hand, can be advantageously modeled using the matrix method.
2.1.4 Cat’s-eye retroreflector A prominent example of this type of optics is the cat’s-eye retroreflector. It basically consists of a lens and a plane mirror (Fig. 2.3). The distance between the lens and the mirror is chosen as the focal length of the lens (with respect to the backward principal plane of the lens). The lens will be approximated as a thin lens. To describe the plane mirror, we can use the matrix given earlier [Eq. (2.1)]. The distances involved are first designated by g and b. Following the ray through the unfolded arrangement, we find the following matrix chain: S =
1 0
×
1 − f 1
g 1
1 0
b 1
0 1
1 − f 1
1 0
0 1
1 0
b 1
1 0
g 1
0 1
.
(2.5)
After performing the matrix multiplications, we have S =
1 + 2 f b f g 2 b f f
− 1
− 1
g − 2 f
2g + 2b − 4 bg f g b 2 f f − 1
g2 + 2 f
b f
− 1
+ 1 − 2 f b
.
(2.6)
We know that letting b = f makes the arrangement work as a cat’s-eye retroreflector. It is instructive to see what happens if we choose the entrance and exit planes of the system either at the position of the lens or at a distance f in front of it. The first alternative corresponds to setting g = 0 and the second one to letting g = f . In this way, we find S(g = 0 , b = f ) =
−1 0
2(f − g) −1
Figure 2.3 Cat’s-eye arrangement.
(2.7)
28
Chapter 2
and S(g = f, b = f ) =
−1 0
0 −1
.
(2.8)
The form of the second matrix is equal to the matrix of a retroreflector stated in Eq. (2.2).
2.1.5 Roof mirror A roof mirror is formed by two plane mirrors meeting at a right angle. This optical arrangement is also being designated as a double mirror (DeWeerd and Hill, 2004). We will consider the plane that is perpendicular to the roof edge. In this plane, the mirror can be unfolded as shown in Fig. 2.4 using an x –y plane as the plane of symmetry. The coordinates of the rays that leave the mirror after reflection can be found by tracing a rectilinear line through the unfolded arrangement. If we first ignore the coordinate break caused by the unfolding operation, the figure shows the passage of a ray through a distance 2 t . This propagation can be described by the following matrix: 1 2t T = . (2.9) 0 1
Additionally, it has to be taken into account that the orientation of the reference axis changes due to the unfolding operation. The corresponding sign change of the coordinates in the new coordinate system can be seen in Fig. 2.4 and expressed by the following matrix: −1 0 (2.10) . 0 −1
Combining, this gives the component matrix, S =
−1 0
0 −1
1 0
2t 1
=
−1 0
−2t −1
.
(2.11)
Therefore, a roof reflector acts like a retroreflector [Eq. (2.3)] in the plane that is perpendicular to the roof edge.
Figure 2.4 Unfolding the roof mirror.
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29
Figure 2.5 Arrangement with roof mirror and plane mirror.
It is interesting to see how a combination of this roof mirror and a plane mirror as depicted in Fig. 2.5 would act on a ray. To this end, it is advantageous to unfold the optical arrangement with respect to the reference plane of the plane mirror. This leads to the following matrix chain: S =
−1 0
−2t −1
1 0
1 0
b 1
0 1
1 0
b 1
−1 0
−2t −1
. (2.12)
The evaluation of this matrix product gives the system matrix of the arrangement of Fig. 2.5 as follows: 1 4t + 2b (2.13) S = . 0 1
This equation has some similarity to the equation of a plane mirror, but is of the following more general form: A =
1 0
a12 1
.
(2.14)
2.2 Components Based on Refraction Lenses are of course very prominent examples of optical components based on refraction. Because their description using the matrix method are treated in other chapters, other components based on refraction are considered here.
2.2.1 Plane-parallel plate The plane-parallel plate is a component that is often encountered in optical setups. It can be a simple glass plate used for path-length compensation or a component that is used to influence the polarization and has the form of a plate. Its system
30
Chapter 2
matrix can be concluded as a special case of the system matrix of a thick lens [Eq. (1.23)] in air by letting both radii tend to infinity. In this way, we find S =
1
t n
0
1
,
(2.15)
where t is the thickness of the plane-parallel plate and n is its refractive index. To recall this equation during design work, we may also apply the linearized law of refraction twice. In matrix notation, this reads as y (1) β (1)
=
1 0
0 1 n
y in β in
(2.16)
y (2) β (2)
(2.17)
for the rays entering the plate and y out β out
1 0
=
0 n
for those leaving it. In between, they pass through a homogeneous medium of refractive index n . Putting this together, we have S =
1 0
0 n
1 0
t 1
1 0
0 1 n
=
1
t n
0
1
.
(2.18)
This matrix can be used to determine the shift z depicted in Fig. 2.6, which occurs if a plane-parallel plate is introduced into a convergent beam. For comparison, we first describe the situation without a plane-parallel plate in the following way:
y out,1 β out,1
=
1 0
z1 1
y in β in
1 0
t 1
=
1 0
t + z1 1
y in β in
. (2.19)
At the intersection with the optical axis, we have y out,1 = 0. This implies z1 = −
y in β in
− t .
Figure 2.6 Shift caused by a plane-parallel plate.
(2.20)
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31
The situation with a plane-parallel plate can be described by
out,2
y β out,2
=
1 0
z2 1
1
t n
0
1
in
y β in
=
1
t n
0
y in β in
+ z2
1
. (2.21)
Using y out,2 = 0, we find z2 = −
y in β in
−
t n
.
(2.22)
The difference of Eqs. (2.22) and (2.20) gives the shift z we are looking for in the paraxial approximation,
z = z2 − z1 = −t
1
n
− 1 .
(2.23)
Because the term in brackets is generally negative, the shift z takes a positive value as is expected from Fig. 2.6.
2.2.2 Prisms Prisms have a vast range of applications in several fields of optics. In optical spectroscopy, for example, an important branch of science, prisms serve both as dispersive means (Demtröder, 1999) and as components for beam shaping. In optical data storage technology, prisms are also used for beam shaping purposes, namely, to circularize the asymmetric beam emitted by a semiconductor laser (Marchant, 1990). 2.2.2.1 Two types of prisms
It is useful to divide prisms into two distinct groups that have different properties with respect to dispersion. A convenient graphical tool to make this distinction is the so-called “tunnel diagram” (Yoder, 1985). To draw this diagram, the prism has to be optically unfolded as if it has a surface from which the beam of light is reflected back into the prism. Figures 2.7–2.10 show two examples that are representative for the two groups. The tunnel diagram in Fig. 2.8 shows that the Dove prism (Fig. 2.7) is equivalent to a plane-parallel plate. There is a variety of prisms that can be understood in terms of this basic component. The second prism (Fig. 2.9), on the other hand, can be reduced to a prism with an apex angle using the tunnel diagram (Fig. 2.10).
Figure 2.7 Dove prism.
32
Chapter 2
Figure 2.8 Tunnel diagram of the Dove prism.
Figure 2.9 Prism with one internal reflection.
Figure 2.10 Tunnel diagram of the prism with one internal reflection.
We can conclude from these examples that there are the following: 1. Prisms that are reducible to a plane-parallel plate. 2. Prisms that are not reducible to a plane-parallel plate. Another way of stating this important difference is talking of nondispersive (Wolfe, 1995) and dispersive prisms (Zissis, 1995). The first case can be treated with the matrix of a plane-parallel plate, which has already been derived. We will therefore have a closer look at the other type of prisms. Thin-prism approximation for dispersive prisms If both the prism angle α and the incidence angle measured against the normal of the first face of the prism are small,
Optical Components
33
Snell’s law can be linearized and the approximated expression for the deviation angle δ is as follows (Heavens and Ditchburn, 1991): δ = (n − 1)α.
(2.24)
It is interesting to observe that the angle of incidence does not appear in this approximated expression. In terms of the matrix method, the thin prism can be written in the following way: y y 1 0 0 . + (2.25) = 0 1 β β −(n − 1)α
We will apply this later to get an approximated expression for an axicon. Trigonometric description of dispersive prisms In many cases, the linear approximation is not sufficient and it is necessary to perform the trigonometric calculations to trace a ray through a prism. Using beam shaping, especially laser beam circularization, is a representative example of how prism arrangements can be analyzed with trigonometric transfer functions. Semiconductor lasers emit with a high beam divergence perpendicular to the junction and with a low beam divergence parallel to it. In several applications, it is of importance to transform this elliptical Gaussian beam into a circular Gaussian beam or, more generally speaking, to adapt the elliptical Gaussian beam to a given optical system by expanding or compressing it along an axis. There are also applications in which the intensity profile that results in a plane perpendicular to the optical axis differs from a circular one after the Gaussian beam is shaped. Only beam shaping of collimated light is considered here, i.e., prisms that can be used to perform the expansion or compression after the beam emitted by the laser diode has passed a collimating lens are described. Of course, prism arrangements are not the only way to realize beam shaping in an optical system. The reader who would like to know more about alternative techniques is referred to Dickey and Holswade (2000). 2.2.2.2 Brewster condition
To minimize losses, the so-called Brewster condition important. Reflective losses at a surface are minimal if a polarized beam is parallel to the plane of incidence, i.e., a p-polarized beam, is incident at Brewster’s angle (Young, 1997; Marchant, 1990). This angle depends on the refractive index in the following way: φ0 = arctan(n).
(2.26)
The Brewster condition is also often expressed by saying that the ray reflected by the surface and the ray refracted into the medium are perpendicular with respect to each other, i.e., φ0 = 90 deg − φ1 , (2.27)
34
Chapter 2
where φ 1 is the angle that the refracted ray forms with the surface normal pointing into the medium. This implies that the following equation is an alternative expression of the Brewster condition in terms of the angle φ 1 : φ1 = arccot(n).
(2.28)
2.2.2.3 Refracting prism Relation of incident and exit angle in the case of a single prism To derive a relation between the angle of incidence φ 0 and the exit angle φ 3 of the refracting prism characterized by its apex angle α and the refractive index n, the law of refraction has to be applied twice. This leads to
φ3 = arcsin(n sin φ2 ), 1 sin φ0 , φ1 = arcsin n
(2.29)
(2.30)
where φ1 and φ2 are the corresponding angles inside of the prism as depicted in Fig. 2.11. In a prism, these angles are related by
φ1 + φ2 = α.
(2.31)
If these equations are put together, the following relation is obtained:
φ3 = arcsin n sin α − arcsin
1 n
sin φ0
.
(2.32)
In an analogous way, the corresponding equation for the inverse relation can be derived as 1 sin φ3 (2.33) φ0 = arcsin n sin α − arcsin . n
Figure 2.11 Refracting prism.
Optical Components
35
From Eq. (2.32), a relation can be derived that links the incidence angle and the apex angle for the case where the beam exiting the prism is perpendicular to the second surface of the prism, i.e., φ3 = 0. Using the trigonometric identity arcsin x = 90 deg − arccos x , Eq. (2.32) can be reformulated as follows:
φ3 = arcsin n sin(α − 90 deg + φ0 ) .
For φ 3 = 0, this equation implies that α − 90 deg + φ0 = 0 .
(2.34)
(2.35)
This relation is used later. The transfer function for the beamwidth altered by a single prism To study the beam-shaping effect of a prism on a collimated beam, it is appropriate to consider first how a single refracting surface changes an incident beam of width w0 . The angle of incidence of this beam, with respect to the surface normal is, designated by φ0 . The angle φ1 , with respect to this normal inside a medium of refractive index n is, determined by Snell’s law as
φ1 = arcsin
1 n
sin φ0 .
(2.36)
If φ0 is not zero, the beamwidth is increased after transition from a medium of lower refractive index to a medium of higher refractive index. Figure 2.12 shows a cut through a surface that forms the boundary between air and the prism material. The beamwidth after refraction is called w1 . From the figure, the following relations for the cosines of the two angles are obvious: w0 , cos φ0 = (2.37) h w1 . cos φ1 = (2.38) h
Figure 2.12 Change of beamwidth at a refracting surface.
36
Chapter 2
If they are combined, a relation follows that is an expression for the beamwidths as a function of incident and exiting angle, i.e., w1 w0
=
cos φ1 cos φ0
.
(2.39)
Together with Snell’s law [Eq. (2.36)], this leads to the beamwidth transfer function for a refracting surface, w1 w0
=
cos{arcsin[(1/n) sin φ0 ]} cos φ0
.
(2.40)
The corresponding equation for the other refracting surface of the prism, i.e., the transition from a medium with refractive index n to air, is w3 w1
=
cos[arcsin(n sin φ2 )] cos φ2
.
(2.41)
In a prism of apex angle α , the link between the known angle φ1 and the angle φ2 is given by α = φ 1 + φ2 . In this way, one finds w3 w1
=
cos[arcsin(n · sin{α − arcsin[(1/n) sin φ0 ]})] cos{α − arcsin[(1/n) sin φ0 ]}
.
(2.42)
Substituting for w1 from Eq. (2.40), the beamwidth transfer function for a prism with the apex angle α is obtained as w3 w0
=
cos[arcsin(n · sin{α − arcsin[(1/n) sin φ0 ]})] cos{arcsin[(1/n) sin φ0 ]} cos{α − arcsin[(1/n) sin φ0 ]}
cos φ0
.
(2.43) This equation shows that the transfer functions of the refracting surfaces can be multiplied to obtain the transfer function of the prism. This is a general property of the beamwidth transfer functions. A prism for expansion along one axis It is of interest to consider the situation where the light beam is incident on the prism in accordance with the Brewster condition [Eq. (2.26)], i.e., sin φ0 (2.44) = n. cos φ0
To avoid reshaping at the other surface of the prism, i.e., compression in the direction of the axis that has been expanded before, it is convenient to have the beam exiting the prism perpendicular to its rear surface (φ 3 = 0 ). Combining this condition [Eq. (2.35)] with the Brewster condition leads to the equation, α = 90 deg + arctan(n).
(2.45)
Optical Components
37
Figure 2.13 Expanding prism.
This implies that the following relation between the apex angle of the prism and its refractive index should hold: (2.46) α = arccot (n). In this situation of practical importance (Fig. 2.13), the relation for the beamwidths is especially simple, i.e., w3 (2.47) = n. w0 This formula can be concluded from Eq. (2.33) using the Brewster condition and the additional condition expressed in Eq. (2.46). To this end, it is helpful to consider the argument α − arcsin(sin φ0 /n) first. Using Snell’s law and the trigonometric identity arcsin x = 90 ◦ − arccos x , one has α − arcsin
sin φ0 n
= α − 90 deg + φ0 .
(2.48)
Combining this equation with Eq. (2.46) and the Brewster condition, it follows that α − arcsin
sin φ0 n
= 0 .
(2.49)
This implies that the first factor in Eq. (2.43) is equal to 1. Using Snell’s law and the trigonometric identity stated before, the second factor can be expressed as cos[arcsin(sin φ0 /n) ] cos φ0
=
sin φ0 cos φ0
.
(2.50)
If this is combined with the Brewster condition (tan φ0 = n), one finds that the second factor is equal to the refractive index n .
38
Chapter 2
It seems worthwhile to present a shortcut to derive Eq. (2.47) for the case where it can already be assumed that the second surface of the prism leaves the beamwidth unaltered, i.e., w3 (2.51) φ3 = 0 ⇒ = 1 . w1 In this case, it is sufficient to consider w1
cos φ1
=
w0
cos φ0
.
(2.52)
Using the Brewster condition ( 1/ cos φ0 = n/ sin φ0 ), it is concluded that w1
=
w0
cos φ1 sin φ0
n.
(2.53)
The sine term can now be expressed by Snell’s law ( sin φ0 = n sin φ1 ) as w1 w0
=
cos φ1 sin φ1
= cot φ1 .
(2.54)
At this point, it is convenient to make use of the alternative expression of the Brewster condition ( cos φ1 = n) in order to obtain Eq. (2.47) and we have w3 w0
=
w3 w1
= n.
w1 w0
The type of prism considered here has numerous applications in optical devices. In the literature, the abbreviation M = w1 /w0 for magnification can sometimes be encountered in conjunction with the following equation (Hanna et al., 1975): M =
1 n
n2 − sin2 φ0
1 − sin2 φ0
.
(2.55)
Its equivalence with Eq. (2.52) can directly be seen using Snell’s law (sin φ0 = n sin φ1 ). A prism for compression along one axis An analogous relation holds for a prism that reduces the beamwidth along an axis instead of expanding it, namely,
w3 w0
=
1 n
.
(2.56)
Such a prism can be realized by letting a beam pass undeviated through the first surface, so that the beam shaping occurs at the exit surface (Fig. 2.14). In contrast to the beam-shaping effect described earlier, in this case a transition from a medium with a higher refractive index to a medium with a lower index of refraction is of importance (n sin φ2 = sin φ3 ).
Optical Components
39
Figure 2.14 Compressive prism.
Equation (2.56) can be derived in a way very similar to the derivation of Eq. (2.47) if it is assumed that w 2 /w0 = 1 holds, i.e., that perpendicular incidence on the first surface is realized. Additionally, it has to be assumed that tan φ3 = n
⇔
cot φ2 = n.
(2.57)
(2.58)
Combined with Snell’s law, the two equivalent equations lead to w3 w2
=
cos φ3 cos φ2
=
1 cot φ2
=
1 n
,
and combined with the assumption of perpendicular incidence finally gives w3 w0
=
w2 w0
·
w3 w2
=
1 n
.
(2.59)
Tolerancing For the purpose of tolerancing, it is advantageous to have explicit formulas for the quantities of interest as functions of physical quantities that are controlled by adjustment (the incidence angle φ 0 ) or by manufacturing (the refractive index n, the apex angle α ). Equations (2.32) and (2.43) are appropriate starting points for such a sensitivity analysis. An important point is the question of how the exiting angle changes if the input angle φ 0 is not well adjusted. In Fig. 2.15, the output angle is plotted as a function opt opt of the deviation φ = φ0 − φ0 from the optimum angle φ0 = arctan(n) for an expanding prism (Fig. 2.15) made of the standard glass BK 7. The wavelength considered is λ = 405 nm. This determines the refractive index used in Eq. (2.32), which is n = 1.53024. In Fig. 2.16, the change of beamwidth calculated from Eq. (2.43) is shown as a function of the deviation φ for the same prism.
40
Chapter 2
Figure 2.15 Exit angle versus adjustment error for the prism of Fig. 2.13.
Figure 2.16 Beamwidth versus adjustment error for the prism of Fig. 2.13.
Figure 2.17 gives a representation of the change of the exiting angle with disadjustment for a compressive prism (Fig. 2.14). To allow for comparison with Fig. 2.9, the same material and wavelength as before are chosen in this example. Figure 2.18 shows the dependence of the beamwidth on a deviation from the optimum input angle. Being a function of wavelength and temperature, the refractive index that has to be considered in the analysis might change in practice, depending on the conditions
Optical Components
41
Figure 2.17 Exit angle versus adjustment error for the prism of Fig. 2.14.
Figure 2.18 Beamwidth versus adjustment error for the prism of Fig. 2.14.
under which the laser is operated. Figure 2.19 shows how the exiting angle changes in the case of an expanding prism (Fig. 2.13) if there is a deviation n = n − nopt from the optimum refractive index. The apex angle α is considered to be a fixed value. The same holds for the angle of incidence. Again, Eq. (2.32) can be used for a simple analysis. Figure 2.20 shows the same dependence for a compressive prism (Fig. 2.14).
42
Chapter 2
Figure 2.19 Exit angle versus a change in refractive index for the prism of Fig. 2.13.
Figure 2.20 Exit angle versus a change in refractive index for the prism of Fig. 2.14.
2.2.2.4 Two-prism arrangements
Figures 2.21 and 2.22 show two ways of arranging two prisms. The ways that the dispersion of light is influenced by these arrangements is quite different. While the dispersion of the prisms adds if the two refracting edges are on the same side, it subtracts if the prisms are arranged in a way that the apex angles are on different sides (Barr, 1984). The first case corresponds to typical prism arrangements applied
Optical Components
43
Figure 2.21 Arrangement of two prisms to increase dispersion.
Figure 2.22 Arrangement of two prisms to decrease dispersion.
in optical spectroscopy to increase dispersion, while the second case is of importance here. Arrangements of two prisms in which the second case is realized are common in devices for optical recording (Okuda et al., 1995) and also have been used in dye laser systems (Niefer and Atkinson, 1988). Relation of incident and exit angle in the case of a two-prism arrangement Having a look at the result for a single prism, the following equations can be written immediately:
φ7 = arcsin nII sin α II − arcsin φ3 = arcsin nI sin α I − arcsin
sin φ4 nII
sin φ0 nI
,
,
(2.60) (2.61)
where the angles are designated as depicted in Fig. 2.22 and α II is the apex angle of the second prism and nII is its index of refraction, while the variables α I and n I describe the first prism. In the equation for φ4 , the relative orientation of the two
44
Chapter 2
prisms intervenes. It is expressed by the variable β = φ 3 − φ4 as
φ4 = arcsin nI sin α I − arcsin
sin φ0 nI
− β .
(2.62)
If this expression is substituted into the equation for φ7 , the output angle of the two-prism arrangement is found as a function of the input angle, i.e.,
φ7 = arcsin nII sin α II − arcsin − arcsin
sin φ0 nI
− β
1
nII
sin arcsin nI sin α I
.
(2.63)
The transfer function for the beamwidth altered by two prisms Exploiting the fact stated earlier that the beamwidth transfer function can be composed as the product of the transfer function of the refracting surfaces that intervene, the transfer function for the two prisms can now be written directly as
w7 w0
=
cos(arcsin{nII · sin[α II − arcsin(sin φ4 /n II )]}) cos[arcsin(sin φ4 /n II )] cos[α II − arcsin( sinnIIφ4 )] ×
cos φ4
cos(arcsin{nI · sin[α I − arcsin(sin φ0 /n I )]}) cos[arcsin(sin φ0 /n I )] cos[α I − arcsin(sin φ0 /n)]
cos φ0
.
(2.64)
The link between the two prisms is again established via the following equation:
φ4 = arcsin nI sin α I − arcsin
sin φ0 nI
− β .
(2.65)
Expansion or compression If two expanding prisms of the type shown in Fig. 2.13 are combined, the following set of equations is valid in the ideal case, i.e., when no adjustment errors and no manufacturing errors are present:
First prism,
φ0 = arctan nI ,
α I = arccot nI , φ4 = −β,
Second prism,
(2.66)
(2.67)
−β = arctan nII ,
α II = arccot nII .
Under these conditions, the beam leaves the last prism perpendicular to its second surface, i.e., φ 7 = 0. The beamwidth ratio after passing the two prisms adjusted in this arrangement is w7 (2.68) = nI nII . w0 In practice, two prisms of equal refractive index are often used.
Optical Components
45
Figure 2.23 Exit angle versus adjustment error.
Of course, it is also possible to arrange two prisms in such a way that the incident beam is perpendicular to each of the first surfaces of the prisms to realize compressive beam shaping. If α I = arccot(nI ) and α II = arccot(nII ) hold, one finds 1 w7 (2.69) = I II . w0 nn Tolerancing The formulas presented here [Eqs. (2.63)–(2.65)] allow for the investigation of a variety of dependencies. As an example, the sensitivity of the exit angle on the input angle for a pair of expanding prisms of BK 7 at a wavelength of 405 nm is shown in Fig. 2.23. 2.2.2.5 Prism with one internal reflection
A modified prism is sometimes used in data-storage applications in order to realize a 90-deg deflection. A description of the component can be found in a U.S. patent filed by Marchant (1988). The front surface of the prism can also serve as a beamsplitter to divert a laser-monitoring signal from the main optical path to a power detector (Latta et al., 1992). Figure 2.9 shows the working principle of this prism. The tunnel diagram (Fig. 2.10) shows how this component corresponds to a prism with the apex angle α . Using α = 90 deg − β , the equations derived for a single prism can directly be applied as
φ3 = arcsin n cos β − arcsin φ0 = arcsin n cos β − arcsin
1 n
1 n
sin φ0 sin φ3
,
(2.70)
,
(2.71)
46
Chapter 2
w3 w0
=
cos[arcsin(n cos{β + arcsin[(1/n) sin φ0 ]})] cos{arcsin[(1/n) sin φ0 ]} sin{β + arcsin[(1/n) sin φ0 ]}
cos φ0
.
(2.72) The condition of perpendicularity of the exiting beam with respect to the last surface gives φ3 = 0 , (2.73) −β + φ0 = 0 . The Brewster condition [ φ0 = arctan(n)] corresponds to β = arctan(n).
(2.74)
Therefore, this type of prism can be reduced to the type of prism discussed earlier, making use of the tunnel diagram. 2.2.2.6 Prism with two internal reflections
At first glance, it seems difficult if not impossible to realize the parallelism of incident and exiting beams with a single refracting prism oriented in such a way that the incident beam is perpendicular to its first surface. Fantone (1986, 1991) proposed a type of prism that fulfills these requirements. His design (Fig. 2.24) is based on two reflections inside of the prism and the following equation: cos(3β) = n cos β,
(2.75)
where n is the refractive index of the prism material and β is the angle depicted in Fig. 2.24. It is related to the apex angle α in the following way: α = 90 deg − β . The design equation stated by Fantone (1986, 1991) can be derived by taking a look at Fig. 2.24 and referencing the corresponding tunnel diagram. From the drawing, it can be seen that δ must complement the angle ϕ B to 90 deg in order to realize the parallelism condition, i.e., ϕB = 90 deg − β.
(2.76)
The tunnel diagram that represents the single prism and takes the two reflections into account is given in Fig. 2.25. There are two equations that follow directly from this diagram, namely, α t = 90 deg − β t , β t = 3 β.
Figure 2.24 Prism with two internal reflections.
(2.77) (2.78)
Optical Components
47
Figure 2.25 Tunnel 2.25 Tunnel diagram of the prism with two internal reflections.
Using α t = φ 1t + φ2t , it can be concluded that α t = φ2t .
(2.79)
The combination of these three equations gives φ2t = 90 deg deg − 3β.
(2.80)
Using ϕ A = φ2t , Eqs. (2.76) and (2.80) are combined with Snell’s law as n sin ϕ A = sin ϕ B ,
(2.81)
and the design design equation (2.75) follows follows directly. directly. These These considerations considerations on prisms were restricted to beams of collimated light. If uncollimated light is used, astigmatism becomes an important feature (Naumann and Schröder, 1992). In the technical literature, a multitude of prism-based arrangements are described scribed that are beyond beyond the scope of this introductory introductory text. A sophistica sophisticated ted device for optical data storage described by Saito et al. (2000) is based on a birefringent crystal prism, for example. Aspects of achromatic prism designs are treated for optical recording applications (Kay and Gage, 1997) and for applications in dye laser systems (Duarte and Piper, 1982, 1983; Duarte, 1985a, 1985b). Additional resources on this subject can also be found in the patent literature (Duarte and Piper, 1983; Luecke, 1997; Grove and Shuman, 1992).
2.2.3 Axicon Axicon devices devices Axicons (McLeod, 1954, 1954, 1960) are components components featuring rotational symmetry symmetry with respect respect to the optical optical axis. Unlike Unlike a lens whose surface surface can be thought thought of as being created by letting a part of a circle rotate around the optical axes, axicons often resemble a cone, i.e., their surface can be thought of as being created by letting a straight line rotate around around the optical axis. Figure Figure 2.26 shows an axicon that is used for linear focusing (Rioux et al., 1978), i.e., being illuminated by collimated
48
Chapter 2
Figure 2.26 Axicon 2.26 Axicon used for linear focusing.
light, it creates a focal line extended from the intersection of its exit surface with the optical axis to the point designated by L. Making use of the matrix method, it is easy to derive an expression for the length of the focal line in terms of the axicon parameter parameters. s. We assume that the angle α is small. Under Under this conditi condition, on, we can use the equation derived for a thin prism [Eq. (2.25)] to model the passage of a ray through the axicon as y (1) β (1)
y in β in
=
1 0
0 1
+
0 −δ
, with δ = (n ( n − 1)α.
(2.82)
This ray then propagates through air, y (2) β (2)
y (1) β (1)
y in β in
=
1 0
t 1
=
1 0
t 1
−t δ −δ
+
.
(2.83)
The condition that the incident light is collimated (β in = 0 ) implies that y (2) = y in − t δ.
(2.84)
The length of the focal line is now determined as the t value for which y (2) = 0 holds and y is at its maximum height, i.e., L =
y in,max δ
=
y in,max (n − 1)α
.
(2.85)
Annular and radial focusing are other applications of axicons to laser machining (Rioux et al., 1978). Of special interest are adaptive axicon devices as annular beam expanders (Rioux et al., 1978; Ichie, Ichie, 1994). Here, Here, an arrangeme arrangement nt will be considered considered that decreases the diameter of an incoming collimated beam (Fig. 2.27). The difference in height H , which is obtainable with the axicon arrangement, can be found using the following formula: H = d
tan(β − α) α ) α )/ tan γ ] 1 − [tan(β − α)/
, with β = arcsin(n sin α) and α = 90 deg deg − γ .
(2.86)
Optical Components
49
Figure 2.27 Adaptive 2.27 Adaptive axicon. The adaptive axicon is characterized by the angle γ . The two components that form the adaptive axicon are separated at a distance d .
It can can be deri derive ved d from from a cons consid ider erat atio ion n of tria triang ngle less that that give givess tan tan (β − α) = H/(a + d ) and tan γ = H /a . To get a better understandi understanding ng of this expression, expression, we approximate it using tan(β − α) α ) tan γ as follows: H ∼ α ). tan(β − α). = d tan
(2.87)
Further approximated and using the law of refraction, this reads as H ∼ = d arcsin(n sin α) − α .
(2.88)
Now, we use the fact that α = Θ , where Θ is the angle of the axicon depicted in Fig. 2.27, and linearize the equation as H (d ; n,Θ) ∼ )Θ . = d(nΘ − Θ ) = d (n − 1)Θ.
(2.89)
This is the approximated expression that links the difference in beam diameter to the state of the compound axicon and to its parameters n and Θ = 90 deg deg − γ .
2.3 Components Components Based on Reflection Reflection and Refraction Refraction 2.3.1 Integrati Integrating ng rod The integra integratin ting g rod is treate treated d as an examp example le of a light light guide. guide. This This optica opticall comcomponent is often used to homogenize light distributions (Chang and Shieh, 2000; Magarill, 2002).
50
Chapter 2
Figure 2.28 Integrating rod.
The component seems to be quite simple at first glance, but it provides some interesting features of unexpected complexity. Figure 2.28 shows a ray propagating through a light guide. As before, we are interested in having an explicit formula that gives the ray leaving the integrating rod as a function of the input ray and the parameters of the rod, namely, its length T , the height d describing the entrance surface, and its refractive index n. The entrance surface is assumed to be quadratic. The ray entering the light guide undergoes refraction and the matrix describing this reads as 1 0 (2.90) A = . 0 n1
Depending on its entrance height and entrance angle, the ray propagates through the light guide along a straight line before it hits one of the surfaces that are perpendicular to the entrance surface for the first time. From the corresponding triangle, we see that this propagation can be described using the matrix B=
1 t(y,β) 0 1
, where t(y, β) =
d − y
tan β
.
(2.91)
The ray is then reflected back in the direction of the optical axis. At this point, we distinguish two cases as shown in Fig. 2.29. Both cases have in common that they contain a part of the passage of the ray through the rod, during which it is reflected in such a way that its ray coordinates are the same after the reflections. It is therefore possible to conceptually separate N “unit cells” with this property from the rod and to describe them by C=
N
1 0
0 1
.
(2.92)
Proceeding further, either case 1 or case 2 can occur, and the corresponding matrices are −1 0 D −+ = for case 1, (2.93) 0 1 D +− =
1 0
0 −1
for case 2.
(2.94)
To obtain an explicit formula later, we combine them into one matrix. To this end, the integer exponent M is introduced, which is a function of the start ray coordinates and the rod parameters, D=
(−1)M 0
0 (−1)M +1
.
(2.95)
Optical Components
51
Figure 2.29 Integrating rod, two cases.
From Fig. 2.29, we see that this matrix is then followed by a matrix describing a propagation of the ray along a rectilinear line, E=
1 0
t rest 1
.
(2.96)
And finally, the ray is refracted at the exit surface of the light guide, F =
1 0
0 n
.
(2.97)
The exponent M is determined using the div operation, which gives the integer part of a division. For example, 7 .8 div 2 evaluates to 3. M corresponds to the number of reflections minus 1. From a geometrical consideration, this number is calculated as 2d (2.98) M = T − t(y, β) div s where s = . tan ϕ
The overall length of the rod is
T = t(y, β) + M s + t rest .
(2.99)
From this relation, we can determine the distance appearing in matrix E . This can now be put together to obtain the system matrix of the integrating rod as S = F EDCBA = ( −1)M
1
1 [t(y,β) − t rest (y,β)] n
0
−1
,
(2.100)
52
Chapter 2
with t rest (y,β) = T − t(y, β) − M s , t(y, β) = (d − y )/ tan β , and M and s as in Eq. (2.98). This system matrix is quite different from the matrices encountered earlier insofar as it contains the coordinates of the ray. It provides an explicit formula to calculate the ray leaving the light guide as a function of the parameters of the light guide and the input ray. The matrix S reflects two important properties of a mixing rod. The intricate dependence of the entry s 21 on the input coordinates indicates the mixing property with respect to the y coordinates. The entries s21 = 0 and s22 = −(−1)M , on the other hand, indicate that the angle is conserved with the exception of a change in sign.
2.3.2 Triple mirror Triple mirrors are used as retroreflectors in interferometers and spectrometers, as well as in measurement devices for civil engineering. Unfolding the triple mirror is different from the geometrical operations that we used until now. Drawing the tunnel diagram for a prism, we used a reflection with respect to a plane that allowed us to unfold the device and to make it more accessible to an analytical treatment. Also, in the case of the light guides studied before, we used a symmetry operation that consisted of reflection with respect to a plane. To unfold a triple mirror, it is advantageous to use a reflection with respect to a point. This operation is called an inversion.
Chapter 3
Sensitivities and Tolerances The analysis of the sensitivity of an optical system with respect to misadjustment is an important part of the optical design. The matrix method provides useful tools to study tilt, decentering, defocusing, and beam pointing error in a systematic way. We will start with a simple example and consider the surface in Fig. 3.1 that separates two media with different indices of refraction. The surface is slightly tilted at an angle γ . From Eq. (1.17), we know how the situation without tilt is described by a matrix. The question is now how this relation has to be modified to incorporate the effect of tilt. Starting with the linearized form of the law of refraction, we find from Fig. 3.1 that n1 β (1) − γ = n2 β (2) − γ .
Solving this equation for β (2) and taking into account that the distance of the ray is not changed in the reference plane, this can be expressed as follows: y (2) β (2)
=
1 0
0 n1 n2
y (1) β (1)
+ γ
0 1 − nn1 2
.
(3.1)
Figure 3.1 Tilted surface. The surface that separates the two media with refractive indices n1 and n 2 is tilted by an angle γ .
Chapter 3
54
Figure 3.2 Tilted element. The optical element is tilted with respect to the optical axis and also decentered by y1 .
An additional additive term appears in this equation. The matrix in Eq. (3.1) is the same that is used to describe the untilted surface. It is a typical feature of considerations with respect to tilt and decentering that they can be described by a so-called error vector that represents the effect of misalignment. We will now have a look at how an error vector is transformed if the misaligned element cannot be described by a single reference plane, but if two reference planes separated by a distance L are needed. An approach involving a coordinate transformation to the coordinate system of the misaligned element is presented in the literature on laser technology (Siegman, 1986). Figure 3.2 shows an optical element that is decentered and tilted. Decentering and tilt are described by the error vector 1 = y1 (3.2) β1
with respect to the optical axis. In the coordinate system of the misaligned element, the vector R1 is transformed by the element matrix A to the vector R2 . The transformation law for the error vector is different and is described by a matrix that propagates the error vector along a distance L. Both transformations are then combined for the transformation of the main coordinate system. Figure 3.3 gives an overview of the transformations. To analyze tilt and/or decentering, we have a choice between two alternative methods. On the one hand, we can use 2 × 2 matrices and two-dimensional vectors to study the influence of error vectors on rays propagating through the system. On the other hand, we can use 3 × 3 matrices to describe the same situation in homogeneous coordinates. In these coordinates, an equation of the form y (2) β (2)
=
a11 a21
a12 a22
y (1) β (1)
+
y β
(3.3)
(3.4)
is expressed as y (2) β (2) 1
=
a11 a21 0
a12 a22 0
y β 1
y (1) β (1) 1
.
Sensitivities and Tolerances
55
Figure 3.3 Transformation used to describe a misaligned element. This figure gives a scheme for how the coordinates in the reference system and those in the coordinate system of the misaligned element are interrelated.
The advantage of the use of homogeneous coordinates stems from the fact that every transition can now be described by cascading matrices and no additional vectors appear.
3.1 Cascading Misaligned Systems To perform the multiplication of the 3 × 3 matrices, it is convenient to organize them as partitioned matrices. Equation (3.4) then takes the form (2)
r
1
=
A T 0
b 1
r (1) 1
,
(3.5)
T is the transposed vector of the two-dimensional zero vector. where 0 The product of two matrices of this kind can now be calculated as follows:
A(2) T 0
b (2) 1
A(1) T 0
b (1) 1
=
A(2) A(1) T 0
A(2) b (1) + b (2) 1
.
(3.6)
We observe that the error vector b of a system with N subsystems is formed according to the rule b = A(N) · · · · · A(2) b1 + A(N) · · · · · A(3) b (2) + · · · + A(N) b (N −1) + b (N) . (3.7)
Chapter 3
56
This implies that, for the general case of N 3 × 3 matrices, the product matrix can be expressed by the following formula (Kloos, 2007): N −1 k +1
1
A T 0
b 1
=
A(k)
k =N
(
A(l) )b (k) + b (N)
k =1 l =N
T 0
1
,
(3.8)
1
with k=N A (k) = A(N) · · · · · A(1) . Please note that the symbol is used here to designate a matrix multiplication (and not a multiplication of scalar quantities).
3.2 Axial Misalignment Up to now, we have been looking at decentering and tilt. Now, we will also take an axial misalignment as defocusing into consideration. To analyze the tolerance of a given subsystem described by the matrix A, we can proceed in this way: first, the deviation z from the undisturbed position is expressed by a matrix as follows:
1 0
z 1
.
(3.9)
Depending on the application, there are two possibilities to multiply the matrices and it might even be more appropriate to use two axial-misalignment matrices to model the subsystem. We take a tolerance of the entrance reference plane of the subsystem as an example. This leads to A
1 0
z 1
=A+
0 0
a11 a21
z.
(3.10)
As before, we are able to separate the matrix of the subsystem and a term describing the effect of the misalignment. In the case of axial misalignment, the corresponding term is an error matrix and not an error vector. This description provides a means by which to study the effect of the deviation z on an optical device with the system matrix S = A(N) · · · · · A(1) . Assuming that the matrix A corresponds to the k th subsystem of the optical device, we find for the error matrix, S = A(N) · · · · · A(k+1)
0 0
(k) a11 z (k) a21 z
A(k −1) · · · · · A(1) .
(3.11)
As an illustrative example, let us consider the cat’s-eye retroreflector formed by a lens and a plane mirror. If we model the lens as a thin lens and assume that the
Sensitivities and Tolerances
57
mirror is well positioned at a distance f with the respect to the lens, the system matrix is −1 2f S = (3.12) 0 −1
if the entrance and exit reference plane is coincident with the thin lens. If we now assume that the mirror is moved with respect to the lens by a distance z , we find S = 2z
− f 1
1
1 f 2
− f 1
.
(3.13)
The factor 2 in the error matrix stems from the fact that the ray passes two times through the spacing that is changed by the deviation. Alternatively, we might refer to an entrance and exit reference plane at a distance f in front of the lens. In this case, the results for the system matrix and the error matrix of the cat’s-eye retroreflector appear as follows: S =
−1 0
0 −1
,
S = 2z
0 1 f 2
0 0
.
(3.14)
3.3 Beam Pointing Error The matrix method is also a convenient tool with which to study the beam pointing error. For example, we might consider a laser beam that should enter a given instrument parallel to the optical axis. We are interested in seeing how a deviation from this alignment will affect the performance of an optical instrument. In this case, we can start with y in , βin
where β in describes the deviation from the target alignment. Tracing this ray through a system with the system matrix S leads to an error vector of the form yout s12 in (3.15) = β . βout s22
Chapter 4
Anamorphic Optics Most of the examples treated until now featured symmetry with respect to the optical axis and it was sufficient to consider a projection onto the y –z plane to find the system matrix. But there are many optical systems that are not symmetric with respect to the optical axis. In those cases, it is necessary to use an extended form of the matrix method.
4.1 Two Alternative Matrix Representations
×
The generalization is straightforward and leads to a 4 4 matrix description. There are two ways to state the corresponding matrices, namely,
or
x out α out y out β out x out y out α out β out
= =
m11 m21 s11 s21
m12 m22 s12 s22
a11 a21 c11 c21
a12 a22 c12 c22
r11 r21 n11 n21 b11 b21 d 11 d 21
x in α in y in β in
x in α in y in β in
r12 r22 n12 n22 b12 b22 d 12 d 22
x in y in α in β in
.
(4.1)
(4.2)
The second form has the advantage that algebraic commutation relations are fulfilled between the matrices A, B , C , and D and that it allows for a generalization of the determinant relation known from 2 2 ray matrices with determinant 1 (Siegman, 1986, p. 618). The first form reduces to the following partitioned matrix if there is no coupling between the two orthogonal planes:
×
x out α out y out β out
=
m11 m21 0 0
m12 m22 0 0
0 0 n11 n21
0 0 n12 n22
.
(4.3)
In the case of rotation symmetry, M and N would be equal. In the case of some illumination devices such as headlamps, for example, they can be rather different.
Chapter 4
60
One would like to have a collimated beam in the y –z plane and a high divergence in the x –z plane to have a broad illumination in the plane of the road. In such a case, n22 0 in the paraxial approximation and m22 takes a value different from zero in accordance with the divergence angle that has to be reached. We will refer to the first form as expressed by Eq. (4.1) as representation A and the second form given by Eq. (4.2) will be designated as representation B.
=
4.2 Orthogonal and Nonorthogonal Anamorphic Descriptions The matrix given by Eq. (4.3), in which the block matrices R and S vanish, describes what might be called an orthogonal anamorphic system. In the more general case of a nonorthogonal anamorphic system, there is coupling between the orthogonal planes, and the block matrices R and S have nonzero entries that reflect this coupling. Examples of an orthogonal anamorphic system are encountered describing an anamorphic lens or a prism in a coordinate system oriented in accordance with the main axes of these components. In these cases, the anamorphic symmetry is inherent to the optical component under study. The rotational symmetry can also be broken if a component with rotational symmetry is tilted with respect to the common optical axis of a system. Components with a three-fold symmetry can be analyzed using a nonorthogonal anamorphic description. Another example for a component with nonvanishing block matrices R and S is an anamorphic lens with its main transversal axes oriented in such a way that they are rotated with respect to the two axes of the coordinate system used.
4.3 Cascading The matrices of representation A or B can be cascaded by multiplying the corresponding block matrices. We consider the passage through two optical systems I and II. In representation A, the overall matrix then takes the form
M II S II
R II N II
M I S I
RI N I
=
M II M I S II M I
II
I
II
I
+ R S + N S
M II R I S II R I
II
I
II
I
+ R N + N N
. (4.4)
If subsystem II can be described as orthogonal anamorphic arrangement, we have
x out α out y out β out
=
M II M I N II S I
M II R I N II N I
x in α in y in β in
.
(4.5)
Anamorphic Optics
61
For the case where an orthogonal anamorphic description is appropriate for subsystem I, it follows that
x out α out y out β out
x out α out y out β out
=
M II M I S II M I
x in α in y in β in
x in α in y in β in
II
I
II
II
I
R II N I N II N I
.
(4.6)
In the case where both subsystems can be described as orthogonal anamorphic in the same coordinate system, the product matrix for the compound system takes the simpler form of an orthogonal anamorphic arrangement,
=
M II M I 0
0 N II N I
.
(4.7)
The matrix of a compound system described by representation B also follows by multiplication of block matrices,
II
A C II
II
B D II
I
I
A CI
B DI
=
II
I
A A C II AI
+B C +D C
I
A B C II B I
II
I
II
I
+B D +D D
. (4.8)
Due to the different arrangement of spatial and angular coordinates in representation B, the symmetry expressed by Eq. (4.7) is less apparent in this matrix representation. But this representation has other advantages.
4.4 Rotation of an Anamorphic Component with Respect to the Optical Axis Although cascading subsystems by concatenating the corresponding matrices is relatively simple, describing the rotation of an anamorphic component or system with respect to the optical axis is somewhat more involved. Making use of matrix representation B, we can state the coordinate transformation as follows (Hodgson and Weber, 1997):
x y α β
x α y β
= −
= −
cos ϕ sin ϕ 0 0
sin ϕ cos ϕ 0 0
0 0 cos ϕ sin ϕ
−
0 0 sin ϕ cos ϕ
x y α β
,
(4.9)
where ϕ is the angle of rotation. The rotation matrices appear as block matrices in Eq. (4.9). This is not the case if we turn to matrix representation A, i.e.,
cos ϕ 0 sin ϕ 0
0 sin ϕ cos ϕ 0 0 cos ϕ sin ϕ 0
−
0 sin ϕ 0 cos ϕ
x α y β
.
(4.10)
Chapter 4
62
In order to be able to continue to calculate with block matrices, we might introduce the following matrices:
= =
Cϕ
cos ϕ 0
S ϕ
sin ϕ 0
0 cos ϕ 0 sin ϕ
,
(4.11)
.
(4.12)
With these abbreviations, Eq. (4.10) reads as
x α y β
= −
Cϕ S ϕ
S ϕ Cϕ
x α y β
.
(4.13)
4.4.1 Rotation of an “orthogonal” system Representation A
We are interested in the matrix of the anamorphic component after rotation. We will start with the simpler case of a component to be rotated that has an orthogonal anamorphic description,
x out α out y out β out
=
0 Y
X 0
x in α in y in β in
,
(4.14)
where X and Y are quadratic submatrices. They form the system matrix E as
=
E
0 Y
X 0
.
(4.15)
This matrix is transformed according to the following transformation law into the new, angle-dependent system matrix E (ϕ) : E (ϕ)
= T − E T , 1
ϕ
ϕ
(4.16)
(4.17)
where T ϕ−1 is the inverse matrix of T ϕ , or E (ϕ)
T ϕ E T ϕ ,
= T
where T ϕT is the transposed matrix of T ϕ . The transposed matrix can be formed easily in the block-matrix notation and we find E (ϕ)
=
Cϕ S ϕ
−S
ϕ
Cϕ
X 0
0 Y
Cϕ S ϕ
−
S ϕ Cϕ
.
(4.18)
Anamorphic Optics
63
After multiplication, we have E (ϕ)
=
Cϕ XC ϕ S ϕ XC ϕ
+ S Y S − C Y S ϕ
ϕ
ϕ
ϕ
Cϕ XS ϕ S ϕ XS ϕ
− S Y C + C YC ϕ
ϕ
ϕ
ϕ
.
(4.19)
By definition, Cϕ and S ϕ are diagonal matrices. The above equation can therefore be simplied as E (ϕ)
=
Cϕ2 X S ϕ2 Y S ϕ Cϕ (X Y )
+
−
S ϕ Cϕ (X Y ) S ϕ2 X Cϕ2 Y
−
+
.
(4.20)
×
The upper-right and lower-left 2 2 block matrices are equal. The matrix difference that appears in these block matrices expresses the transversal anisotropy of the anamorphic component or system. (The case X Y would correspond to a situation of transversal isotropy and could be treated with the simpler description of the former chapters.) To use this result in systems design or analysis, we have to make reference to its matrix elements. The block matrices appearing in the above equation are therefore given in expanded form to facilitate their use as follows:
=
Cϕ2 X
+ S Y
cos2 ϕx 11 cos2 ϕx 21
S ϕ2 X
Cϕ2 Y
sin2 ϕx 11 sin2 ϕx 21
2 ϕ
+
S ϕ Cϕ (X
= =
2
ϕy11 2 ϕy21
cos2 ϕx 12 cos2 ϕx 22
2
sin2 ϕx 12 sin2 ϕx 22
+ sin + sin + cos + cos
− Y ) = sin ϕ cos ϕ
ϕy11 2 ϕy21
x11 x21
−y −y
11 21
x12 x22
−y −y
2
+ sin + sin + cos + cos
12 22
ϕy12 2 ϕy22 2 2
ϕy12 ϕy22
.
, (4.21) , (4.22)
(4.23)
Representation B
After abbreviating the rotation matrix as R
=
cos ϕ sin ϕ
−
sin ϕ cos ϕ
,
(4.24)
we might reformulate Eq. (4.9) in the following way:
x y α β
=
R 0
0 R
x y α β
.
(4.25)
The transformation matrix T is therefore formed by two identical block matrices, T
=
R 0
0 R
.
(4.26)
Chapter 4
64
Described by representation B, an orthogonal anamorphic system has a system matrix E of the following form:
=
0 ayy 0 cyy
axx 0 cxx 0
E
0 byy 0 d yy
bxx 0 d xx 0
.
(4.27)
Unlike in the case for representation A, there are no zero block matrices, but an orthogonal anamorphic arrangement is characterized by diagonal block matrices. At this point, we can apply the transformation law (Hodgson and Weber, 1997) to find the system matrix E (ϕ) of the component or system rotated by an angle ϕ as E (ϕ) T −1 ET T T ET . (4.28)
=
=
Before specializing E to the case described by Eq. (4.27), it is convenient to have a look at the more general case because it will bring out an advantage of the form that the transformation matrix takes in representation B. Let us therefore consider the more general anamorphic system with E
=
A C
B D
.
(4.29)
The transformation law can directly be applied to this general matrix as E (ϕ)
=
R T 0
It follows that E (ϕ)
0 R T
=
A C
R T AR R T CR
B D
R T BR RT DR
R 0
0 R
.
(4.30)
.
(4.31)
The calculation of the matrix entries in the special case of an orthogonal anamorphic system is straightforward, i.e., R T AR
=
cos ϕ sin ϕ
− sin ϕ cos ϕ
axx 0
0 ayy
cos ϕ sin ϕ
−
sin ϕ cos ϕ
(4.32)
.
Multiplication of the matrices leads to R T AR
=
cos2 ϕa xx sin2 ϕa yy sin ϕ cos ϕ(a xx ayy )
+
−
−
.
(4.33)
−
,
(4.34)
sin ϕ cos ϕ(a xx ayy ) sin2 ϕa xx cos2 ϕa yy
+
The corresponding result for the upper-right entry is
=
R T BR
cos2 ϕb xx sin2 ϕb yy sin ϕ cos ϕ(b xx byy )
+
−
sin ϕ cos ϕ(b xx byy ) sin2 ϕb xx cos2 ϕb yy
+
and the other two entries follow on the same footing. Equation (4.31) is useful in studying anamorphic instruments or setups and will be used in illustrative examples.
Anamorphic Optics
65
4.4.2 Rotation of a “nonorthogonal” system Now, we will turn to the more general case of a nonorthogonal anamorphic component or system. Before rotation, its optical properties might be described by the system matrix E , X U E , (4.35) V Y
=
wherein U and V are nonvanishing block matrices. As before, the transformation formula is applied, in which ϕ appears as the parameter for the rotation angle, E (ϕ)
= T − E T . 1
ϕ
ϕ
(4.36)
After multiplying the matrices and using the diagonality of the block matrices Cϕ and S ϕ , we find E (ϕ)
=
Cϕ2 X
S ϕ2 Y Cϕ2 V
− C S (U + V ) + C S (X − Y ) − S U + ϕ ϕ
ϕ ϕ
2 ϕ
Cϕ2 U
− Y ) + − S X + C S (U + V ) + Cϕ S ϕ (X 2 ϕ
ϕ ϕ
S ϕ2 V Cϕ2 Y
.
(4.37)
This can be decomposed into two matrices, F (ϕ) and G (ϕ), E (ϕ)
= F (ϕ) + G(ϕ),
(4.38)
with F (ϕ) G (ϕ)
Cϕ2 X S ϕ2 Y S ϕ Cϕ (X Y )
= = −
+
− C S (U + V ) −S U + C V ϕ ϕ 2 ϕ
2 ϕ
−
S ϕ Cϕ (X Y ) S ϕ2 X Cϕ2 Y
+
Cϕ2 U S ϕ2 V Cϕ S ϕ (U V )
− +
,
.
(4.39)
(4.40)
The first matrix appearing in Eq. (4.38) corresponds to the orthogonal anamorphic case and the second matrix describes the deviation from it. Please note that—apart from the angle-dependent matrices—the matrix F (ϕ) only contains the matrices X and Y , whereas the matrix G (ϕ) only contains the matrices U and V . The second matrix has the interesting property that its trace is equal to zero.
4.5 Examples 4.5.1 Rotated anamorphic thin lens A common problem in building up an instrument with anamorphic components is to determine the system matrix E (ϕ) of an anamorphic lens that has been rotated by an angle ϕ about the optical axis of the device. This can be of interest as part of a sensitivity analysis or because the component is intentionally positioned at this angle. We will treat the lens in the thin-lens approximation.
Chapter 4
66
Using matrix representation B, the system matrix of the unrotated thin lens can then be stated as follows:
=
A C
E
= −
1 0
B D
0 1 0
1 f x
−
0
1 f y
0 0 1
0 0 0
0
1
.
(4.41)
Applying the transformation law [Eq. (4.31)], leads us to E (ϕ)
=
R T AR R T CR
RT BR R T DR
=
I R T CR
O I
,
(4.42)
where
= 1 0
I
0 1
and
= 0 0
O
0 0
.
Therefore, the system matrix of the anamorphic thin lens rotated by an angle ϕ about the optical axis is
E (ϕ)
=
1 0
−
cos2 ϕ f x
0 1 sin2 ϕ f y 1 1 f x f y
− sin ϕ cos ϕ − +
sin ϕ cos ϕ
−
sin2 ϕ f x
− −
1 1 f x f y cos2 ϕ f y
+
0 0
0 0
1
0
0
1
. (4.43)
4.5.2 Rotated thin cylindrical lens In workshop practice and industry, cylindrical lenses are often applied. The system matrix of a thin cylindrical lens rotated by an angle ϕ about the optical axis follows from the corresponding result for the thin anamorphic lens by letting either 1 /f y 0 or 1/f x 0. The system matrices that are obtained in this way for the cylindrical lenses with different initial orientations are
=
=
E (f y
E (f x
→ ∞, ϕ) → ∞, ϕ)
= =
1 0
−
cos2 ϕ f x
sin ϕ cos ϕ 1 0
−
0 1 sin ϕ cos ϕ 1 f x
−
sin2 ϕ f y
sin ϕ cos ϕ
−
1 f y
0 0
−
1
0
0
1
0 0
0 0
1
0
0
1
sin2 ϕ f x
0 1 sin ϕ cos ϕ
0 0
−
1 f y
cos2 ϕ f y
1 f x
.
, (4.44)
(4.45)
Anamorphic Optics
67
4.5.3 Cascading two rotated thin cylindrical lenses A typical arrangement encountered to analyze anamorphic optical devices is a setup of two cylindrical lenses. These lenses might have different focal lengths, f 1 and f 2 , and in the more general situation they are rotated by different angles α and β with respect to the optical axis. The two lenses are positioned with a distance d between them. Assuming a paraxial situation and assuming the thin-lens approximation to be a sufficient description for the given purpose, we can analyze this arrangement making use of Eq. (4.44). The paraxial optical properties of the two-lens arrangement in terms of the five parameters that intervene will then be expressed by the system matrix that is formed by matrices for the two anamorphic lenses and a translation matrix. We might note the first cylindrical lens with a block matrix F 1 as
=
F 1
I P 1 (f 1 , α)
O I
,
(4.46)
where
=
I
1 0
0 1
P 1 (f 1 , α)
=
,
0 0
O
cos2 α f 1 sin α cos α f 1
= −−
0 0
sin α cos α f 1 sin2 α f 1
− −
and
,
.
The spacing between both cylindrical lenses is taken into account by using the translation matrix D , I dI (4.47) D . O I
=
The second cylindrical lens has the block matrix F 2 ,
= −
I P 2 (f 2 , β)
F 2
with P 2 (f 2 , β)
=
−
cos2 β f 2 sin β cos β f 2
O I
sin β cos β f 2 sin2 β f 2
− −
,
(4.48)
.
These three matrices are now concatenated to find the system matrix S of the anamorphic two-lens arrangement as
= F DF .
S
2
1
(4.49)
It is advantageous to use the block-matrix description to perform the matrix multiplications instead of operating on the matrix entries of P 1 or P 2 , i.e.,
=
S
I P 2 (f 2 , β)
O I
I O
dI I
I P 1 (f 1 , α)
O I
.
(4.50)
Chapter 4
68
After multiplying the block matrices, we have
=
S
+
P 1 (f 1 , α)
+
I dP 1 (f 1 , α) P 2 (f 2 , β) dP 1 (f 1 ,α)P 2 (f 2 , β)
+
+
I
dI dP 2 (f 2 , β)
.
(4.51) To trace rays through the anamorphic arrangement, it is then necessary to make reference to the explicit form of the two matrices P 1 and P 2 in terms of focal lengths and rotation angles as stated above. In conjunction with these two equations, Eq. (4.51) allows us to get a convenient overview of the five-dimensional parameter space of this anamorphic optical setup.
4.5.4 Cascading two rotated thin anamorphic lenses The generalization of the considerations made for cylindrical lenses to the analysis of anamorphic lenses of a more general form is straightforward. But the notation becomes more involved because we have to deal with a seven-parameter problem instead of a five-parameter problem now. In place of making use of one of the matrices of a rotated cylindrical lens, we will make reference to the matrix given for a rotated thin anamorphic lens [Eq. (4.43)]. We will describe the first anamorphic y lens by its two focal lengths f 1x and f 1 and the angle of rotation α . The corresponding submatrix then reads as P 1 f 1x ,
y f 1 ,
= =
α
−
sin2 α y f 1 1 1 y f 1x f 1
cos2 α f 1x
− sin α cos α − +
sin α cos α
−
sin2 α f 1x
− −
1 1 y f 1x f 1 cos2 α y f 1
+
. (4.52)
The submatrix for the second anamorphic lens has the analogous form y
P 2 f 2x , f 2 , β
−
sin2 β y f 2 1 1 y x f 2 f 2
cos2 β f 2x
− sin β cos β − +
sin β cos β
−
sin2 β f 2x
−
1 f 2x
−
+
cos2 β y f 2
1 y f 2
. (4.53)
These two lenses are placed at a distance d from each other. The system matrix is determined as before by concatenating the two matrices for the lenses and a matrix D that describes the transfer through the spacing between these lenses, i.e.,
= F DF .
S
2
1
(4.54)
In block-matrix notation, this equation reads as
=
S
I y x P 2 (f 2 , f 2 , β)
O I
I O
dI I
I y x P 1 (f 1 , f 1 , α)
O I
. (4.55)
Finally, we have for the system matrix of the two rotated anamorphic lenses,
=
S
P 1 (f 1i , α)
+
I dP 1 (f 1i , α) P 2 (f 2i , β) dP 1 (f 1i ,α)P 2 (f 2i , β)
+
+
wherein i takes the values 1 and 2.
+
I
dI dP 2 (f 2i , β)
,
(4.56)
Anamorphic Optics
69
Please note that the concatenated block matrices of representation B take a familiar form that is reminiscent of the nonanamorphic matrix description when the matrix entries are scalars and no submatrices.
4.5.5 “Quadrupole” lens With the new prerequisites of this chapter, we can study interesting optical devices as the so-called quadrupole lens. It is described in Hodgson et al. (1992), and Hodgson and Weber (1997), as a convenient tool for an experimental phase-space analysis. A look at the anamorphic matrix will give us insight into why this is so. The quadrupole lens mainly can be modeled by two thin cylindrical lenses of opposite focal length. These lenses are rotated at 90 deg with respect to each other and the two-lens arrangement is rotated with respect to the optical axis. Using symbols as in Eq. (4.51), these angles of rotation are α 45 deg and β 45 deg. We can assume a vanishing distance d 0 between the two lenses. In fact, the effect of the two crossed lenses can be incorporated in a single anamorphic lens to form the “quadrupole” lens (Hodgson et al., 1992; Hodgson and Weber, 1997). The system matrix for two rotated thin cylindrical lenses as given by Eq. (4.53) then reduces to I O S (4.57) . P 1 (f 1 , α) P 2 (f 2 , β) I
= −
=
=
= +
+
Incorporating the special features of the quadrupole lens, we have
=
I 45 deg) P 2 (f,
O I
. (4.58) − − + +45 deg) √ √ Using sin(±45 deg) = ±1/ 2 and cos(45 deg) = 1/ 2, we have for the sum of S
P 1 ( f,
the two submatrices
1
− −45 deg) + P (+f, +45 deg) = f
P 1 ( f,
2
1 2
1 2
1 2 1 2
− −1 − f 1 2
1 2
1 2 1 2
. (4.59)
Equation (4.58) can therefore be written in a surprising mathematical form, i.e.,
=
1 0 0
=
1 0 0
S
1 f
0 1
−
1 f
0
0 0 1
0 0 0
0
1
.
(4.60)
Looking at the corresponding input–output relation for the quadrupole lens,
out
x y out α out β out
1 f
0 1
−
1 f
0
0 0 1
0 0 0
0
1
x in y in α in β in
,
(4.61)
Chapter 4
70
we see that a direct link has been established between y in and α out and between x in and β out . To understand why this optical property of the quadrupole lens is advantageous in a so-called phase-space beam analyzer, we will analyze such a device using the description by anamorphic matrices in representation B. The purpose of this instrument is to create a phase-space representation of the light distribution in a given object plane. On a screen in the image plane of the instrument, a typical representation is created with one axis of the screen representing the space coordinate of a one-dimensional distribution in the object plane and the other axis representing the angular coordinate of a one-dimensional distribution in the object plane. A onedimensional distribution is selected by using one slit close to the object plane to restrict the spatial extension of the light distribution and another slit close to the imaging lens to restrict the angular distribution accordingly. A complete analysis can be performed by rotating the analyzing device, i.e., the two slits, the imaging lens, and the quadrupole lens, about the optical axis. The quadrupole lens is positioned at the focal distance f 0 of the imaging lens behind that lens. The quadrupole lens has the length f . Naming the distance from the object plane to the imaging lens a and the distance from this lens to the image plane b, one can write the following matrix chain to characterize the experimental situation (Hodgson and Weber, 1997): S T b−f 0 Qf T f0 F f0 T a , (4.62)
=
wherein three translation matrices appear and Qf stands for the quadrupole lens we obtained before,
= = − −
1 0 0
Qf
0 1 1 f
−
1 f
0
and F f0 represents a standard lens, 1 0
F f0
0 1 0
1 f 0
−
0
1 f 0
= = 0 0 1
0 0 0
0
1
I Q
0 0 1
0 0 0
0
1
O I
I P
,
O I
(4.63)
.
(4.64)
Rewriting the matrix chain in block-matrix form, we have
=
S
I O
(b
f 0 )I I
I Q
O I
I O
f 0 I I
I P
O I
I O
aI I
.
(4.65)
After multiplying the submatrices, we obtain
S = + I
+ (b − f )Q + (b − f )f QP P + Q + f QP
bP
0
0
0
0
(a
+ b)I + bP + (b − f )(a + f )Q + (b − f )f abQP I + aP + (a + f )Q + f aQP 0
0
0
0
0
0
.
(4.66)
Anamorphic Optics
71
The product of the two submatrices takes diagonal form
=
QP
−1/f
0 1/f
−
0
−
1/f 0 0
=
0 1/f 0
−
1 ff 0
I.
(4.67)
For the given purpose, we are interested in the functions that determine (x out , y out ). It is therefore sufficient to have a look at the upper-left and the upper-right submatrices appearing in the system matrix. The upper-left submatrix can be evaluated as follows:
− + − + −− − b f 0
I
0
b f 0
0
=
b f 0
1
b f 0 f
0
0
−
0
1
b f 0
−
−
b f 0 f
+
0
−
0
b f 0 f
−
0
b f 0 f
.
(4.68)
We observe the cancellation that takes place and causes that this submatrix has an advantageous diagonal form. For the upper-right submatrix, we find (a
= +
+ b)I + abP + (b − f )(a + f )Q + (b − f )f aQP 0
0
0
0
a
=
At this point, we can use the imaging condition 1 /f 0 advantage. It implies that the following equation holds: (a
b
−
ab f 0
(1/a)
− I
+
−
f 0 (b f 0 ) I. f
(4.69) (1/b) to our
+ b)I + abP + (b − f )(a + f )Q + (b − f )f aQP = − f (bf − f ) I. (4.70) 0
0
0
0
0
0
Putting these results together, we have the following input–output relation: x out y out
x in y in
= − − 1
b
f 0
1 0
0 1
f 0 (b
− f ) 0
f
α in β in
0 1
1 0
. (4.71)
If we assume the input light distribution to be restricted to one dimension and choose an orientation of the slits along the x -axis, for example, we may assume that y in 0 and β in 0. It follows that
=
=
b f 0
x in
= − x out y out
1
−
−
f 0 (b f 0 ) in β f
.
(4.72)
This reflects that the instrument indeed operates as a phase-space analyzer: the two phase-space coordinates (x in , α in ) are imaged on the analyzing screen (with different scaling factors).
Chapter 4
72
4.5.6 Telescope built by cylindrical lenses In industrial inspection devices, anamorphic arrangements are used that have a telescoping effect in one principal plane. One way to realize them is by mounting cylindrical lenses. To describe such a setup, we can use the system matrix of two cylindrical lenses. They are mounted to have a refracting effect in the same principal plane. Therefore, we can set α 0 deg and β 0 deg. In order to have the wanted telescoping effect in that plane, the condition d f 1 f 2 has to hold, where d is the distance between the two lenses. Depending on whether a Galilean telescope (f 1 < 0 , f 2 > 0 ) or a Keplerian telescope ( f 1 > 0 , f 2 > 0 ) is considered, the signs of the focal lengths might differ. For both cases, the submatrices in Eq. (4.53) take the following form:
=
P 1 (f 1 , 0 deg)
=
− 1 f 1
0
0
=
and
0
P 2 (f 2 , 0 deg)
= +
=
− 1 f 2
0
0
0
. (4.73)
With these submatrices and the telescoping condition, the system matrix of the compound anamorphic telescope reads, using representation B, as
= −
S
1 1 f 1
−
+
f 1 f 2 f 1
0 1
0
1 f 2
− +
+
f 1 f 2 f 1 f 2
0 0
0
f 1
+ f
0
2
0
1
−
f 1
+
f 1 f 2 f 2
+ f
2
0 1
0
4.5.7 Anamorphic collimation lens
.
(4.74)
Representation A also has its advantages and can be convenient if a decoupling of both orthogonal planes can be exploited during the calculation. This might be illustrated using an example from LED technology where anamorphic lenses are used to project the light from the LED chip onto the road or on the façade of a building. Let us consider a lens that has a collimating effect in the y –z plane and that diverges the emitted light in the x –z plane. First, we note the matrix of an anamorphic lens with two different focal lengths in representation A, i.e.,
− =
1
F
1 f x
0 0
0 1
0 0
0 0
0 0
1
0 1
−
1 f y
.
(4.75)
The lens is positioned at a distance d from the LED. We can express this using the matrix G , 1 d 0 0 0 1 0 0 G . (4.76) 0 0 1 d 0 0 0 1
=
Anamorphic Optics
73
The system matrix for an anamorphic lens at the given distance then follows directly by concatenating both matrices,
− =
1
E
= F G
d
1 f x
1
0 0
−
f x d
0 0
0 0
0 0
1
d
−
1 f y
1
f y d
−
.
(4.77)
At this point, we can introduce the condition for collimation in the y –z plane by setting d f y in Eq. (4.75),
=
= −
1
E
f y
1 f x
1
0 0
−
f x f y
0 0
0 0
0 0
1
f y 0
−
1 f y
.
(4.78)
The last entry of the matrix reflects the collimation condition. The entry in the second row and the second column shows how the diverging effect in the orthogonal plane scales with the two focal lengths.
4.6 Imaging Condition In Section 4.5, examples of an anamorphic telescoping arrangement and a matrix description of a lens that has a collimating effect in one plane were given. Here, we will turn to the question how the imaging condition is reflected in this formalism. We consider a system matrix expressed in representation B,
x out y out α out β out
=
a11 a21 c11 c21
a12 a22 c12 c22
b11 b21 d 11 d 21
b12 b22 d 12 d 22
x in y in α in β in
.
(4.79)
For imaging, the relation between the transversal spatial coordinates in the object plane and in the image plane is of importance, i.e., x out y out
x in y in
α in β in
= + A
B
.
(4.80)
Rays leaving at (x in , y in ) should all arrive at (x out , y out ) in the paraxial approximation. This implies that (x out , y out ) should not depend on B ϕ in , where (x in , y in ). We might state this as 0 α in (4.81) B . β in 0
=
Chapter 4
74
In order to proceed, we might interpret this equation as a characteristic equation for the matrix B with an eigenvalue 0 (van den Eerenbeemd, 2001). Because the determinant of a matrix is the product of its eigenvalues, it follows that the determinant is zero in this case, i.e.,
= 0.
det B
(4.82)
Equation (4.82) is the analytical expression of the imaging condition in the anamorphic case. Example
To illustrate its application and its physical content, we will have a look at a simple example. Let us consider a thin anamorphic lens between two spacings a and b. This situation can be described by the following matrix product in representation B:
= T F T
S
b
=
a
I O
bI I
I
−
O
1 f x
0
−
0
I
1 f y
I O
aI I
.
(4.83)
Multiplication of the block matrices gives
= + I
S
bP P
(a
+ b)I + abP I + aP
,
(4.84)
with
=
P
−
1 f x
0
−
0
1 f y
.
At this point, we can identify the upper-right submatrix with the matrix B ,
= (a + b)I + abP.
B
(4.85)
Therefore, the following equation corresponds to the anamorphic imaging condition: ab a b 0 f x det 0. (4.86) ab a b 0 f
+ −
+ −
y
=
For a given a , this is a quadratic equation in b , i.e.,
+ a
b
ab
− f
x
+ a
b
ab
− f
y
=
0.
(4.87)
The imaging equation has two solutions in this anamorphic case. They determine image positions that correspond to the two orthogonal planes. The positions of the focal lines of anamorphic systems can be determined with a similar approach.
Anamorphic Optics
75
4.7 Incorporating Sensitivities and Tolerances in the Analysis In this chapter, we put some effort into understanding the effect of a rotation about the optical axis in an anamorphic system. In Chapter 3, we considered tilt and decentering using homogeneous coordinates. A question that is near at hand at this stage is whether both mathematical descriptions might be combined in order to study tolerance anamorphic systems in more detail. This can in fact be done by extending the transfer matrices of representation B by homogeneous coordinates,
x out y out α out β out 1
=
a11 a21 c11 c21 0
a12 a22 c12 c22 0
b11 b21 d 11 d 21 0
b12 b22 d 12 d 22 0
x y α β 1
x in y in α in β in 1
.
(4.88)
The misadjustment vectors for decentering and tilt appear in the fifth column of this 5 5 matrix. For concatenating such matrices it is convenient to write them in block-matrix form,
×
r out ϕ out 1
=
A C 0 T
B D 0 T
r ϕ 1
r in ϕ in 1
.
(4.89)
The concatenation of two anamorphic subsystems I and II can then be stated as AII C II 0 T
B II D II 0 T
r II ϕ II 1
=
AII AI C II AI
+B C +D C 0
II
II
T
AI CI 0 T
I
I
BI DI 0 T
r I ϕ I 1
A B +B D C B +D D 0 II
I
II
I
II
I
II
I
T
AII r I C II r I
II
I
II
II
I
II
+ B ϕ + r + D ϕ + ϕ 1
. (4.90)
Chapter 5
Optical Systems 5.1 Single-Pass Optics 5.1.1 Triplet synthesis A triplet is a system of three lenses. In a straightforward way, one might derive the matrix of a three-lens arrangement by concatenating the matrices of thin lenses and air spacings as
S =
1 − f 1
3
0 1
1 0
d 2 1
1 − f 1
0 1
2
1 0
d 1 1
1 − f 1
0 1
1
. (5.1)
The matrix obtained in this way has the disadvantage of being relatively lengthy, i.e., S =
1− − f 1 − 3
1 f 12
d 1 f 1
d 2 f 12
−
+
d 1 f 1 f 3
+
d 1 + d 2 − d 2 f 12 f 3
1−
d 2 f 3
−
d 1 d 2 f 2 d 1 f 23
,
(5.2)
with 1 f 12
=
1 f 1
+
1 f 2
−
d 1 f 1 f 2
and
1 f 23
=
1 f 2
+
1 f 3
−
d 1 f 2 f 3
.
Again, it is advantageous for the discussion to reference the spacings that intervene to the focal planes of the lenses. We saw already that this facilitated the discussion of the doublet,
S = −
1 f 1 f 2 f 3
−E2 f 32 −f 22 + E2 E3
f 12 f 32 −E3 f 12
.
(5.3)
With this prerequisite, we can now state the design approaches for a focusing triplet, an imaging triplet, an afocal triplet, or a collimating triplet. Making reference to the consideration on the matrix entries, we can state the points of departure for the triplet synthesis in these four cases.
78
Chapter 5
For a focusing arrangement:
s11 0
s12 s22
= T 4 F 3 T 3 F 2 T 2 F 1 1 −E2 f 32 − E4 f 22 + E2 E3 E4 =− −f 22 + E2 E3 f 1 f 2 f 3
−E1 E2 f 32 − E3 E4 f 12 + f 12 f 32 −E3 f 12
.
(5.4)
For an imaging arrangement:
s11 s21
0 s22
= T 4 F 3 T 3 F 2 T 2 F 1 T 1 1 −E2 f 32 − E4 f 22 + E2 E3 E4 =− −f 22 + E2 E3 f 1 f 2 f 3
with h 12 = −E1 E2 f 32 − E1 E4 f 22 −
h12 2 −E1 f 2 − E3 f 12 + E1 E2 E3 E3 E4 f 12 + f 12 f 32 + E1 E2 E3 E4 .
,
(5.5) For an afocal arrangement:
s11 0
s12 s22
= F 3 T 3 F 2 T 2 F 1 = −
1 f 1 f 2 f 3
−E2 f 32 −f 22 + E2 E3
f 12 f 32
−E3 f 12
. (5.6)
For a collimating arrangement:
s11 s21
s12 0
= F 3 T 3 F 2 T 2 F 1 T 1 =−
1 f 1 f 2 f 3
−E2 f 32 −f 22 + E2 E3
−E1 E2 f 32 + f 12 f 32 −E1 f 22 − E3 f 12 + E1 E2 E3
.
(5.7)
As an example of how this synthesis approach can be applied, let us consider the afocal arrangement of three lenses. To fulfill the condition given by Eq. (5.6), the following equation should hold: f 22 = E2 E3 .
(5.8)
This is the imaging equation in Newtonian form. It expresses that an imaging condition has to be realized for the central lens in order to have the given three-lens arrangement operate as an afocal system. Such an arrangement is well known as the terrestrial telescope. In the literature, triplet synthesis is quite often considered for the first case, i.e., for three lenses arranged in such a way that an object at infinity is imaged at a finite distance (Berek, 1930, Malacara and Malacara, 1994).
Optical Systems
79
The synthesis approach with concatenated matrices is not restricted to the four types of matrices considered as examples. Depending on the application, other target matrices can be chosen for the system matrix. Several examples of system matrices are given by Goodman (1995).
5.1.2 Fourier transform objectives and 4f arrangements Here, we will consider an example that features unique properties, namely, the matrix 1 0 f (5.9) S = . 0 −f
A way to obtain it with a single lens is by choosing both the input reference plane and the output reference plane at the distance of the focal length, S =
1 0
f 1
1 −
0 1
1 f
1 0
f 1
.
(5.10)
This matrix has the property of mapping incident ray heights on ray angles and incident ray angles as ray heights and is referred to as the “Fourier transform” arrangement (Goodman, 1995). It is central to Fourier transform objectives and socalled 4 f arrangements, which play a prominent role in spatial filtering and optical information processing (Goodman, 1968; Yu and Khoo, 1990). In the paraxial approximation, we can model them by concatenating two subunits of the abovementioned type (Fouckhardt, 1994), S = S 2 S 1
=
0 −
f 2 0
1 f 2
0 −
1 f 1
f 1 0
=
f 2 − f 1
0
0 −
f 1 f 2
.
(5.11)
After the two transformations, ray heights in the input reference plane are now imaged on ray heights in the output reference plane, and input ray angles on output ray angles. Figure 5.1 shows typical rays in an arrangement with equal focal lengths. This 4f scheme is applied in order to influence the output ray parameters in a controlled way, for example, by placing a mask structure in the intermediate plane, y out β out
=
=
0 −
1 f 2
f 2 0
f 2 − m22 f 1 m12 f 1 f 2
m11 m21
f 1 f 2 m21 −
f 1 m f 2 11
m12 m22
0 −
y in β in
.
1 f 1
f 1 0
y in β in
(5.12)
While the wave-optical analysis of these systems is performed in terms of Fourier transforms (Goodman, 1968, Yu and Khoo, 1990), the matrix method provides a very convenient way to analyze processing devices of this kind in the first order of approximation.
Chapter 5
80
Figure 5.1 4f arrangement. The figure shows schematically how typical rays of light are transformed by the 4f arrangement. The vertical lines represent the positions of the two lenses.
5.1.3 Telecentric lenses In many optical applications, telecentricity of a lens arrangement is an important feature to make the device relatively insensitive against axial changes. We use the definition of telecentricity given by Goodman (1995): “A lens is telecentric if the chief rays are parallel to one another. Most commonly, they are also parallel to the lens axis and perpendicular to the object and/or image planes that are perpendicular to the axis. . . ” Therefore, the following three types of telecentric systems can be distinguished: 1. Telecentric in object space. 2. Telecentric in image space. 3. Doubly telecentric. We considered the principal ray as a chief ray descriptive of the system in Chapter 1. Making reference to the decomposition used there, we can describe the system by a matrix mapping from the object plane to the plane of the aperture stop and another matrix that maps from the aperture stop to the image plane,
S = QP .
(5.13)
We can now state the condition of telecentricity in image space by the mapping y out 0
=
Q
0 β
.
(5.14)
The condition that 0 = q22 β can only be fulfilled if q 22 = 0 holds. In this way, the telecentricity condition in image space imposes a condition on the matrix Q . In an analogous way, the condition of telecentricity in object space can be formulated. Starting from the mapping from the object plane into the plane of the
Optical Systems
81
aperture stop, y in 0
0 β
=
P
,
(5.15)
we see that 0 = p11 y in implies p11 = 0. This is the condition imposed on the matrix P . Finally, the question is near at hand: What would a doubly telecentric lens arrangement look like? In the framework of the matrix description, the answer is found by concatenating the submatrices that express the telecentricity conditions in object space and in image space to determine a system that fulfills both conditions, i.e.,
S = QP =
q11 q21
q12 0
0 p21
p12 p22
=
q12 p21 0
q11 p12 + q12 p22 q12 p12
. (5.16)
The system matrix of a doubly telecentric system is therefore characterized by the entry s21 = 0. This implies that such a telecentricity condition can only be realized if an afocal arrangement is chosen as the basis of the optical design.
5.1.4 Concatenated matrices for systems of nlenses At the beginning of the chapter, the triplet was considered. For the purpose of reference, four cases are listed here that correspond to concatenating matrices of the form F k T k with
F k
=
1 −
0 1
1 f k
and
T k
=
1 0
f k−1 + Ek 1
+
f k
.
(5.17)
The special form of the translation matrix is chosen to simplify the product matrices.
n = 1: n = 2:
F 1 T 1
=−
F 2 T 2 F 1 T 1
1 f 1 =
0 1 1
f 1 f 2
n = 3: F 3 T 3 F 2 T 2 F 1 T 1
=−
2
−f 1
E1
1 f 1 f 2 f 3
2
−f 2
2
−E1 f 2 2
E2
,
−f 1 +
E1 E2
2
−E2 f 3 2
−f 2 + E2 E3
(5.18)
, 2
(5.19)
f 12 f 32 2 2 −E1 f 2 − E3 f 1 + E1 E2 E3 −E1 E2 f 3 +
,
(5.20)
82
Chapter 5
n = 4: F 4 T 4 F 3 T 3 F 2 T 2 F 1 T 1 1 = f 1 f 2 f 3 f 4 ×
f 22 f 42 − E2 E3 f 42 −E2 f 32 − E4 f 22 + E2 E3 E4
+E1 f 22 f 42 + E3 f 12 f 42 − E1 E2 E3 f 42 −E1 E2 f 32 − E1 E4 f 22 − E3 E4 f 12 + f 12 f 32 + E1 E2 E3 E4
.
(5.21)
5.1.5 Dyson optics Dyson (1959) presented a system with a Petzval radius that is advantageous for flattening the image plane. Figure 5.2 shows the concentric optical layout. The formula that establishes the relationship between the radii and the refractive index of the lens is Rg n (5.22) = . R+ n−1 Using this in the equation for the inverse Petzval radius [Eq. (1.96)] gives a value of zero. The Dyson optics influenced optical systems used for lithography (Grenville et al., 1991, 1993; Hsieh et al., 1992; Owen et al., 1993). New et al. (1992) presented a modified version of the optics in which the lens and mirror are arranged nonconcentrically. It will now be shown how the systems matrix of the Dyson optics can be obtained. Starting from Fig. 5.2, we can unfold the optics along the apex of the mirror and consider three separate steps of the ray propagation: (1) propagation to the reference plane at the apex of the mirror, (2) reflection by the mirror, and (3) propagation from the reference plane at the apex of the mirror to the image plane. A reference plane directly in front of the plane part of the lens is chosen to start. The ray will undergo a refraction at this plane and then pass through the lens. To determine the refraction at the curved side of the lens, it is important to apply the
Figure 5.2 Dyson optics. The two spherical surfaces are separated by a distance z .
83
Optical Systems
sign convention consistently. The convention used here is that the radius is negative if the surface is concave. To have a quantity for the distance that is counted positive as the ray propagates ∼
∼
along the optical axis, R + = − R is introduced, where R is the usual radius in the refraction matrix. The matrix for the ray transfer through the lens can then be written as follows: A=
1
0 n
1−n R+
1 0
R+ 1
1 0
0
=
1 n
R+ n 1 n
1 1−n R+
.
(5.23)
The next step is a propagation by a distance z. Having a look at Fig. 5.2, we see that this lens should focus a parallel beam on the apex of the mirror. We can identify the component a21 as the negative reciprocal of the focal length of the plano-convex lens, 1 1−n − = . (5.24) f R+ This suggests the choice R+
(5.25) 1−n in the propagation matrix. [At this point, it is interesting to note that in combination with the concentricity condition ( R+ + z = Rg ), Eq. (5.22) follows directly.] The ray transfer matrix to the reference plane of the mirror reads as z=−
1 0
R − 1−+n
1
1
R+
1−n R+
1 n
0
=
R+ n
1−n R+
1−
1 1−n
1 n
.
(5.26)
Making use of Eq. (5.22), the reflection matrix is
1 − R2
g
0 1
=
1 −1) − 2(n nR +
0 1
.
(5.27)
In analogy to the first step, the matrix that describes the third step of the propagation reads as
1 n n−1 −R +
R+ n
1
1
R − 1−+n
0
1
=
1 n − nR−1 +
R+ n
1− 0
1 1−n
.
(5.28)
At this point, we use the following abbreviation: h := −
R+
1−n
.
(5.29)
If we now collect the matrices for the three steps, we get the following product for the system matrix:
1 n
− h1
h 0
1 2 nh
0 1
0 − h1
h 1 n
=
−1 0
4 hn −1
,
(5.30)
84
Chapter 5
and after using Eq. (5.28), the system matrix takes the form S =
4R
−1
− n(1−+n)
0
−1
.
(5.31)
In some way, this optics is similar to optics that will be treated as double-pass optics in Chapter 5.2. With the matrix method, we can check another useful property of the Dyson system (Malacara and Malacara, 1994): It is both front and back telecentric.
5.1.6 Variable single-pass optics The usage of the word zoom lens may vary from author to author. The definition that will be applied here was given by Clark (1973): “A zoom lens is an image-forming optical system with such properties that an axial movement of certain components will produce a change in the equivalent focal length of the system, while keeping the resultant image fixed with respect to the desired image plane.” These systems will be examined first. Later, we will enlarge our view and also consider varifocal systems that form no image in the proper sense. Designing a zoom device, it is important to distinguish between two possible approaches, namely, mechanical compensation and optical compensation (Back and Lowen, 1958; Johnson and Feng, 1992). In a mechanical zoom lens arrangement, the movement of one lens (or lens group) is compensated by the movement of another lens (or lens group) in order to keep the position of the image fixed. Generally, the movement of the second lens (or lens group) has to be nonlinear in order to obtain compensation. In an optically compensated zoom lens device, on the other hand, only one lens (or lens group) is moved and the optical layout is devised in such a way that (approximate) constancy of the position of the image plane is obtained. Both cases can be studied using the triplet as an example. To consider a triplet objective that creates an image at a finite distance and that has its object at infinity, we can make use of the systems matrix stated before in Eq. (5.2), A=
with
1 f 12
=
1 f 1
1− − f 1 − 3
+
1 f 2
1 f 12
−
d 1 f 1
−
+
d 1 f 1 f 3
d 1 f 1 f 2
d 2 f 12
+
and
d 1 + d 2 − d 2 f 12 f 3
1 f 23
1−
=
1 f 2
d 2 f 3
+
−
d 1 d 2 f 2 d 1 f 23
1 f 3
−
,
d 1 f 2 f 3
(5.32)
.
On this matrix, the focusing condition is imposed, namely, a11 = 0. This leads to the following equation, which establishes the relation that has to be fulfilled between the focal lengths of the individual lenses and the spacings between them in order to have a lens arrangement with the desired property:
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Optical Systems
d 1 + d 2 + d 3
1−
f 1
+ d 2 d 3
1 f 1
+
d 2 + d 3
−
1 f 2
f 2
−
−
d 3
+
f 3
d 1 d 2 d 3 f 1 f 2 f 3
d 1 f 1
d 2 + d 3 f 2
= 0.
d 3
+
f 3
(5.33)
Now, the focal length of the lens arrangement has to be changed while keeping this condition. As mentioned above, there are two ways to achieve this aim, namely, mechanical or optical compensation. 5.1.6.1 Mechanical compensation of the triplet
We assume that a starting configuration has been found for which Eq. (5.33) holds, and we consider that a lens is moved linearly. To realize that Eq. (5.33) also holds for these new positions, an additional lens can be moved. The movement that is necessary for proper compensation can be calculated introducing shifts l1 and l2 of the first and second lens as d 1 = d 10 − l1 + l2 ,
d 2 = d 20 − l2 .
(5.34)
(5.35)
By combining the three equations, we see that the movement of the first lens must be nonlinear if the second lens is moved in a linear way. 5.1.6.2 Optical compensation of the triplet
For optical compensation of the triplet, we can either move only a single lens or link two lenses together and move them as a lens group. Let us first consider the case of a single moving lens. As an example, the central lens of the triplet is chosen and moved by a distance d along the optical axis. The “unzoomed” start configuration is designated by d 10 , d 20 , and d 30 . This corresponds to setting d 1 = d 10 + d,d 2 = d 20 − d and d 3 = d 30 in Eq. (5.32), i.e., 1−
d 10 + d 20 + d 30
+
f 1 d 20
−
d 30
− d
d 20 + d 30 − d f 2
1 f 1
+
1 f 2
−
−
d 30 f 3
+
d 10 + d d 20 + d 30 − d
f 1
(d 10 + d)(d 20 + d)d 30 f 1 f 2 f 3
f 2 = 0.
+
d 30 f 3
(5.36)
This will now be interpreted as a polynomial in d , a0 + a1 (d) + a2 (d)2 = 0,
(5.37)
with a0 = 1 − +
d 10 + d 20 + d 30 f 1
d 20
d 30
1 f 1
+
−
1 f 2
d 20 + d 30 f 2 −
−
d 30 f 3
(d 10 )(d 20 )d 30 f 1 f 2 f 3
+
d 10 f 1
= 0,
d 20 + d 30
+
f 2
d 30 f 3
(5.38)
86
Chapter 5
a1 = a2 =
1 f 2
+
d 20 + d 30 f 1 f 2 d 30
1 f 1 f 2
+
d 30 f 1 f 3
−
d 30
f 3
1 f 1
+
1 f 2
−1 .
−
(−d 10 + d 20 )d 30 f 1 f 2 f 3
, (5.39)
(5.40)
The term a 0 is equal to zero if the triplet complies with the focusing condition in its starting configuration (characterized by d = 0). This is what we assume in this derivation. For nonvanishing a 2 , the polynomial is of second degree. This implies that the focusing condition can be exactly fulfilled for two zoom positions only. If we decide to move the first lens, the degree of the polynomial would be even lower, namely, one. This is because only the spacing after this lens is then effectively changed according to d 1 = d 10 + d , while the spacings that follow remain unchanged. The spacing before the lens does not contribute because the object is at infinity and this spacing is therefore related to the passage of a collimated beam. Moving the first lens would therefore not be a good choice to fulfill Eq. (5.32) using optical compensation. Now, the case of two coupled moving lenses is considered. A common choice is to move the outer lenses (Back and Lowen, 1954, 1958). They are moved by the same distance d along the optical axis. The variable spacings between the lenses then change according to d 1 = d 10 − d , d 2 = d 20 + d and d 3 = d 30 − d . Equation (5.32) then takes the form 1−
d 10 + d 20 + d 30 − d
+ −
f 1
−
d 10 + d d 20 + d 30 f 1
f 2
+
d 20 + d 30 f 2
−
d 30 − d f 3
f 3
(d 10 + d)(d 20 + d)(d 30 + d) f 1 f 2 f 3
d 30 − d
+ d 20 + d d 30 + d
= 0.
1 f 1
+
1 f 2
(5.41)
This can be rearranged as a0 + a1 (d) + a2 (d)2 + a3 (d)3 = 0.
(5.42)
In this polynomial, a0 also vanishes because we have assumed that the start configuration fulfills the focusing condition, a1 (d) + a2 (d)2 + a3 (d)3 = 0.
(5.43)
From the term on the right-hand side, we see that this is a polynomial of third degree in d . This implies that exact compensation can be realized for three zoom positions (Fig. 5.3). Alternatively, we might consider coupling the two lenses on the right-hand side and moving them a distance d along the optical axis. This leads to d 1 = d 10 + d,d 2 = d 20 − d + d = d 20 , and d 3 = d 30 + d . The interesting point here is
Optical Systems
87
Figure 5.3 Variable optics: exact compensation at three points. The deviation from compensation is shown as a function of the displacement d .
that the movements of the two lenses that enclose the spacing d 2 leave its effective value unchanged. Therefore, if we substitute into Eq. (5.33) and take the same steps as before, we are led to a polynomial of degree two. This suggests that letting the positions of the outer lenses vary is a better choice for a triplet when optical compensation has to be achieved. It is observed that the number of positions for which exact compensation can be achieved depends on the variable spacings. They contribute to the number if they are not in a parallel beam and if the two lenses that they separate are not moved in the same way. In designing a zoom lens, it is therefore important to make this choice carefully in order to obtain the best compensation possible. In the examples discussed until now, the object was at infinity. The same approach can also be applied to zoom objectives that perform point-to-point imaging. In this case, we make reference to the imaging condition [Eq. (1.48)] and apply it to the system matrices. On the same footing, other instruments can be treated that are not zoom lenses in the proper sense of the above-mentioned definition, but that are also varifocal devices. They have applications as variable beam expanders or adaptive collimators, to name just a couple. In the design of all these devices, the following theorem, which generalizes the observations made in this chapter, can be advantageously applied. 5.1.6.3 Theorem on optically compensated adaptive lens arrangements
The number of positions for which rigorous compensation can be obtained equals the number of variable spacings that are not in a collimated beam and that are not between two lenses moving in the same way.
88
Chapter 5
Figure 5.4 Theorem on optically compensated systems: application to two lenses. The horizontal bars indicate the components that are moved. They might be moved together, indicated by a bar that links two lenses, or separately.
The application of this theorem is illustrated in table form for different lens numbers and different arrangements in Figs. 5.4–5.6. 5.1.6.4 Adaptive optics
Adaptive devices (Tyson, 1998) are constantly gaining ground in scientific and industrial applications. A basic problem of adaptive optics might be stated as follows: the output of an optical instrument is deteriorated by an error source that introduces aberrations into the system. As a countermeasure to compensate these aberrations and to keep the output of the instrument stable, an adaptive subsystem is added. We might designate an unwanted change in optical power introduced to the optical system by 1/f aberr , the focal length of the undisturbed system by f stat , and the focal length of the subsystem used for compensation by f comp .
89
Optical Systems
Figure 5.5 Theorem on optically compensated systems: application to three lenses. As in Fig. 5.4, the horizontal bars indicate the components that are moved.
Considering this example, we can use Eq. (5.33) again but read it differently. It now gives the focal length f comp , which is necessary to keep the image position as a function of an unwanted change in optical power due to aberrations. Substituting f 1 = f aberr , f 2 = f stat , and f 3 = f comp in Eq. (5.33), one gets f comp = d 3 + (d 1 d 3 /f aberr )[(d 2 /f stat ) − 1]
1 − [(d 1 + d 2 + d 3 )/f aberr ] − [(d 2 + d 3 )/f stat ] + {(d 1 /f aberr )[(d 2 + d 3 )/f stat ]} + d 2 d 3 (1/f aberr + 1/f stat )
.
(5.44)
90
Chapter 5
(a) Figure 5.6 Application to four lenses. (a) Theorem on optically compensated systems—the case of a vanishing upper-left matrix entry is represented.
In the case of no disturbing aberrations ( f aberr → ∞), the focal length of the compensating device would take the following value: f comp =
d 3
1 − [(d 2 + d 3 − d 2 d 3 )/f stat ]
.
(5.45)
In this example, f comp can be realized by a zoom device as treated before, or by a liquid lens (Berge and Peseux, 2000). In the case of a zoom device, the next step in the design would be to model the zoom optics in the thick-lens approximation. The thin-lens approximation used here gives a guideline on what zoom range has to be realized.
91
Optical Systems
(b) Figure 5.6 (Continued.) (b) Theorem on optically compensated systems—the case of a vanishing upper-right matrix entry is represented.
To generalize this approach to adaptive optics using the matrix method, we might consider the adaptive feedback loop illustrated in Fig. 5.7. The matrix of the adaptive system is S = C T A ,
(5.46)
wherein A describes the aberrations introduced to the system, T represents the static part of the system, and C describes the subunit of the system used for compensation. Depending on the application, we might impose the condition s11 = 0 as the target state of the adaptive system. In this example, the following equation would
92
Chapter 5
(c) Figure 5.6 (Continued.) (c) Theorem on optically compensated systems—the case of a vanishing lower-left matrix entry is represented.
give a condition for two entries in the matrix C of the subsystem used for compensation: c11 (t 11 a11 + t 11 a21 ) + c12 (t 21 a11 + t 21 a21 ) = 0. (5.47)
5.2 Double-Pass Optics 5.2.1 Autocollimator Autocollimators are optical measurement instruments used in tooling. A typical measurement task is to quantify a slight tilt of a reflecting plate. This can be done with an autocollimator setting as depicted in Fig. 5.8. The divergent beam from the light source is collimated by the lens and directed to the target. Because of
93
Optical Systems
(d) Figure 5.6 (Continued.) (d) Theorem on optically compensated systems—the case of a vanishing lower-right matrix entry is represented.
the slight tilt of the target, the reflected beam shows a slight deviation as it hits the detector. Assuming that the autocollimator is well aligned, no tilt of the plate corresponds to a spot at the center of the detector, i.e., at the position were the detector has its intersection with the optical axis. The deviation from this target position is a measure of tilt. For the purpose of the matrix description, we unfold the optical arrangement of the double-pass optics. We make a coordinate break at the position of the beamsplitter and reference the matrices to the corresponding optical axes before and after the coordinate break. The passage through the beamsplitter is then described by the unity matrix, but we have to keep in mind that we changed the coordi-
94
Chapter 5
Figure 5.7 Adaptive feedback loop. The unit A describes the aberrations introduced in the system, T represents the static part of the system, and C describes the subunit of the system used for compensation.
Figure 5.8 Autocollimator. The rays emitted from the light source are reflected from the slightly tilted mirror and pass through the beamsplitter to the detector element of the autocollimator.
nate system. A similar coordinate break is made to model the reflecting plane. Here, we do not assume that it is well aligned, but we want to take its tilt angle γ into account. This could be done making reference to homogeneous coordinates. At this point, the autocollimator is described in terms of 2 × 2 matrices (Kloos, 2007). Using the unity matrix of the unfolded beamsplitter implicitly, the matrix chain from the light source to the target is A=
=
1 0
g 1
1−
g f
− f 1
1 − f 1
0 1
1 0
h + d 1
g
h + d + g − f (h + d) − f 1 (h + d) + 1
,
(5.48)
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Optical Systems
where f is the focal length of the thin lens and h , d , and g are the distances appearing in Fig. 5.8. The ray is then affected by the tilted target and we use the following relation for a tilted plane mirror to describe it: y (2) β (2)
y (1) β (1)
1 0
=
0 1
+
0 2γ
.
(5.49)
The matrix chain for the passage from the tilted target to the detector reads as follows: B =
=
1 0
d + e 1 d +e f 1 − f
1−
1 − f 1
0 1
1 0
g + (d + e) 1 −
1−
g 1
g f
,
g f
(5.50)
where e is the distance from the beamsplitter to the detector. Additionally, the setting of the autocollimator has to be taken into account. The lens should direct collimated light onto the target plane and it should focus the reflected light onto the detector surface. This is expressed by the following two equations: h + d = f,
(5.51)
d + e = f.
(5.52)
Combing these equations and the relations stated above, we find the relation for the autocollimator as y out β out
y in β in
=
−1 g 2 f 2 −
1 f
0 −1
+ γ
2f 2 1 − f g
.
(5.53)
As read on the detector, the difference between the y value of the tilted plate and the y value that would correspond to a perfectly aligned plate is of importance, i.e., yout = y out (γ ) − y out (0) = 2f γ .
(5.54)
It is this relation that reflects how the tilt of the target is translated to a difference in position by the autocollimator. Furthermore, a look at the input–output relation of the collimator suggests that g = f might be a convenient choice for the corresponding distance.
5.3 Multiple-Pass Optics Multiple-pass optics are encountered in spectroscopy as absorption cells and in laser technology as optical resonators. To describe the multiple passage of a ray through the same optical system, we could apply the same system matrix N times. This corresponds to raising the system matrix to its N th power and applying it to
96
Chapter 5
the vector of the input ray. Therefore, the N th power of the ray transfer matrix is an important tool for studying multiple-pass optics. In this chapter, we will use some basic results from the eigenvalue theory of linear algebra. To have explicit formulas for the matrix A raised to its N th power, it is useful to have a look at the eigenvalues of the matrix. They are helpful in order to diagonalize the matrix and they are also closely related to the optical problem of tracing a ray through a periodic system. The eigenvalues λ of a 2 × 2 matrix are calculated from its characteristic equation, a11 − λ a12 det (5.55) = 0, a21 a22 − λ
where det A stands for the determinant of the matrix A . This is equivalent to λ2 − (a11 + a22 )λ + a11 a22 − a12 a21 = 0.
(5.56)
This quadratic equation has the two solutions, a11 + a22
λ1/2 =
2
±
a11 + a22
2
2
− a11 a22 + a12 a21 .
(5.57)
For a system matrix, det A = n1 /n 2 holds. Here, we will assume that both indices of refraction are equal. Therefore, we have the simpler equation,
λ1/2 =
a11 + a22
2
±
a11 + a22
2
2
− 1.
(5.58)
It is the discriminant dis =
a11 + a22
2
2
−1
(5.59)
that determines the character of the eigenvalues. This discriminant provides a distinction that is of fundamental importance to the optics of multipass systems. If dis < 0, the eigenvalues have an imaginary part, i.e., they have to be described by complex numbers. If dis > 0, the eigenvalues are given by real numbers. In the limiting case of dis = 0, there is only a single, real eigenvalue, namely, λ=
a11 + a22
2
.
(5.60)
From Eq. (5.58), we see that the trace a 11 + a22 of the matrix A determines the sign of the discriminant. Because the assumption det A = 1 was made, all eigenvalues satisfy the condition λ1 λ2 = 1.
(5.61)
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Optical Systems
We have to determine the corresponding matrices A N . To find them, we will make a theorem from linear algebra. It states that A N can be calculated as AN = F D N F −1
(5.62)
if A can be transformed to the diagonal matrix D by a transformation matrix F and its inverse matrix F −1 by the transformation A = F DF −1 .
(5.63)
To understand this theorem, we can concatenate the matrix A of Eq. (5.63) N times and see how the inverse matrices cancel out in the product matrix. Now, we will have a look at the case where dis < 0 and calculate the matrix A N that corresponds to this case. First, the transformation matrix F has to be determined. From linear algebra, we know that the rows of this matrix are calculated from the eigenvector spaces of the matrix A that correspond to the eigenvalues λ1 and λ 2 . The eigenvector space eig (A,λ1/2 ) follows as the solution space of the equation, i.e.,
a11 − λ1/2 a21
a12 a22 − λ1/2
x1 x2
0 0
=
.
(5.64)
Knowing that both eigenvalue are complex and that λ1 λ2 = 1 holds, we can set λ1/2 = e±iϕ with i as the imaginary unit, for which i 2 = −1 holds. Solving Eq. (5.31) for these eigenvalues, we find e±iϕ − a22 a21
eig A, e±iϕ =
.
(5.65)
With this result, the transformation matrix can now be composed as F =
e
iϕ
−iϕ
− a22 a21
.
(5.66)
−e −iϕ + a22 eiϕ − a22
.
(5.67)
− a22 a21
e
The inverse matrix reads F −1 =
a12 −a12
Using these transformations, the matrix D takes the diagonal form D=
λ1 0
0 λ2
=
e+iϕ 0
0 e−iϕ
.
(5.68)
We can now apply the theorem to find the matrix A N . Most entries follow directly by performing the matrix multiplication. Calculating the upper-right entry of the
98
Chapter 5
matrix, the equations sin x sin y = 2sin a11 + a22
x+y
cos
x−y
, 2 2 = trace(A) = 2cos ϕ, and
(5.69) (5.70)
det A = 1
(5.71)
have to be used. Finally, we have 1
sin[(N + 1)ϕ] − a22 sin(N ϕ) a21 sin(N ϕ)
a12 sin N ϕ AN = . a22 sin(Nϕ) − sin[(N − 1)ϕ] sin ϕ (5.72) In this case, the matrix AN is described by trigonometric functions. Equation (5.72) is designated as Sylvester’s theorem in the scientific literature (Gerrard and Burch, 1975). To complete the picture, we will now have a look at the case where dis > 0. We will proceed in an analogous way, but for a different pair of eigenvalues. Taking into account that both eigenvalues are real and that λ1 λ2 = 1 holds, we can use λ1/2 = e±α as a starting point. Using this input, we then calculate
eig(A,e±α ) =
e±α − a22 a21
.
(5.73)
With this result, we can now build the rows of the transformation matrix. In this case, it reads e+α − a22 e−α − a22 F = , (5.74) a21 a21
and its inverse matrix is F −1 =
1 a21 (e+α − e −α )
a21 −a21
−e −α + a22 eα − a22
.
(5.75)
Using 2 sinh(α) = eα − e−α and applying the theorem A n = F D n F −1 , also in this case, we have An =
1
sinh[(n + 1)α] − a22 sinh(nα) a21 sinh(nα) sinh α
a12 sinh(nα) a22 sinh(nα) − sinh[(n − 1)α]
.
(5.76)
Also here, the upper-right entry of the matrix needed special attention and a11 + a22 = trace(A) = 2 cosh α and det A = 1 had to be used. The interesting point now is to compare both results. In the first case, we have a matrix made up of periodic functions that describes a periodic passage of the ray through the multipass optics. This is a stable solution. In the second case, on the other hand, exponential functions are encountered. The ray parameters diverge and
Optical Systems
99
this solution is therefore unstable. From this, we can conclude that the trace of the system matrix is a decisive quantity to characterize the function of multipass optics. Stable passages of the rays are desired in spherical mirror interferometers (Herriott et al., 1964) and optical delay lines (Herriott and Schulte, 1965). The Herriott cell found applications in spectroscopy (Altmann et al., 1981, Hovde et al., 2001) as an absorption cell and in interferometry (Antonsen et al., 2003a, 2003b). It consists of two spherical mirrors separated by almost their radius of curvature. Considerations on multipass cavities of this kind are given by Sennaroglu and Fu jimoto (2003). The distinction between stable and unstable periodic systems is fundamental in the theory of laser resonators (Siegman, 1986, Hodgson and Weber, 1997, Svelto, 1998). The second case is also of importance in laser technology because so-called unstable optical resonators find broad application in high-power laser systems. The literature on optical resonators is abundant; there are many studies that make use of the matrix method and apply it to the design and analysis of lasers. Generalizations of Sylvester’s theorem to the case where det A = 1 and/or incorporating error vectors in homogeneous coordinates are given by Tovar and Casperson (1995).
5.4 Systems with a Divided Optical Path The method of ray-transfer matrices as presented here is not appropriate to describe the phenomenon of interference. But it is a useful tool for analyzing problems that arise while setting up and operating interferometers. It gives quantitative answers on how to balance the optical beam paths of two-beam interferometers and on the adjustment sensitivity of these instruments. The method can be used to derive guidelines on how to make interferometric devices more robust. The results obtained in this chapter can also be used in spectrometer design because the key optics of these instruments is often based on principles of interferometry. A typical example is a Fourier spectrometer based on a Michelson interferometer arrangement (Baker, 1977). An interesting account on designing spectrometers that are robust with respect to tilt was given by Breckinridge and Schindler (1981).
5.4.1 Fizeau interferometer Figure 5.9 shows the optical layout of a Fizeau interferometer used for lens testing. We will have a look at the measurement arm of the interferometer in order to analyze its sensitivity with respect to the tilt of the spherical mirror. The entrance plane of the measurement arm is chosen directly behind the beamsplitter. From this plane, the ray first propagates along a distance e before it is refracted by the lens under test. The optical unit formed by the lens and the concave spherical mirror corresponds to an arrangement that was considered in Section 2.1,
100
Chapter 5
Figure 5.9 Fizeau interferometer. The beamsplitting element is represented by the horizontal bar. It separates the light from the laser into a reference beam and a beam that passes through the measurement arm. Both beams are then directed by an additional beamsplitter mounted at 45 deg to the detector or a camera.
i.e., y β
wherein A=
and
1−
2d f
− f 2 1 −
−
d f
=A
2d R
−
1−
2 R
b1 b2
d f
1−
y β
d 2 f
b1 b2
+
1−
2d 1 −
d R
2d f
1−
−
d d 1 − f
= 2γ
,
2d R
d f
(5.77)
,
.
(5.78)
(5.79)
We can therefore take its system matrix and error vector and apply it here to analyze the Fizeau interferometer. Doing this, the measurement arm is implicitly unfolded with respect to the apex of the spherical mirror. The first propagation is described by a simple transfer matrix, y (1) β (1)
y in β in
=
1 0
e 1
.
(5.80)
During the second step, we make reference to the result of Sec. 2.1, i.e., y (2) β (2)
y (1) β (1)
=A
+
b1 b2
.
(5.81)
101
Optical Systems
The ray finally returns to the reference plane chosen directly behind the beamsplitter and we have 1 e y out y (2) (5.82) = . 0 1 β out β (2)
When these three steps are combined, they result in y out β out
y in β in
1 0
=
e 1
1 0
A
e 1
1 0
+
e 1
b1 b2
,
(5.83)
or
out
y β out
=
1 0
e 1
A
1 0
in
e 1
y β in
+ 2γ
d + e(1 −
1−
d f
d ) f
. (5.84)
Having a look at the sensitivity, the γ -dependent vector on the right-hand side is of interest. The sensitivities with respect to mirror tilt of the ray leaving the measurement arm now follow immediately as ∂y out
d
= d + e 1 −
∂γ
∂β out
=1−
∂γ
d f
f
,
.
(5.85) (5.86)
This sensitivity consideration would imply that d = f would be a convenient choice to reduce the tilt error. In contrast to this finding, experimental arrangements corresponding to d = f + R are also often encountered. The reason for this is that such an interferometric configuration allows for the measurement of odd and even terms of the aberrations, while the first-mentioned arrangement allows for the measurement of only the even terms. This is because the reflected beam passes through the opposite half of the lens after reflection in the case of d = f and is not reflected into itself. For the arrangement that is suitable to measure odd aberrations, the sensitivity analysis provides hints to minimize the tilt sensitivity of the interferometer: ∂β out
∂γ
d =f +R
=−
R f
.
(5.87)
This implies that it is advantageous to choose a radius of the spherical concave mirror that is small compared to the focal length under test.
5.4.2 Michelson interferometer The Michelson interferometer is a generic type of interferometer and can be considered as a kind of workhorse of interferometry. We will consider a simple problem that is encountered, for example, while adjusting such an interferometer for the
102
Chapter 5
Figure 5.10 Michelson interferometer. A beamsplitting cube with a thickness h serves to separate and recombine reference and measurement beams of this Michelson interferometer.
Figure 5.11 Michelson interferometer, and measurement arm.
measurement of small vibrations of a sample provided with a mirror. The experimental situation is schematically shown in Fig. 5.10. The light beams originating from both arms of the interferometer have to be superimposed carefully on the detector surface in order to observe and evaluate an interference pattern. It is then of interest to know how the position of a laser beam that enters the beamsplitter cube parallel to the optical axis would be changed by an unintentional tilt of the sample. Using the ray-transfer matrix method, this and similar questions can be answered in a rather straightforward way. To this end, the measurement arm of the interferometer is first “unfolded” as shown in Fig. 5.11. Using the matrix derived for a plane-parallel plate to describe the passage through the beamsplitter cube, the ray transfer to the reference plane of the sample under test is expressed by the following product matrix:
1 0
t 1
1 0
h n
1
1 0
a 1
=
1 0
a+
h n
+ t
1
.
(5.88)
This implies that
(1)
y β (1)
=
1 0
a+
h n
1
+ t
y in β in
.
(5.89)
The effect of an unintentional tilt of the sample is expressed by an error vector
103
Optical Systems
added to the term for an unfolded plane reflector, i.e., y (2) β (2)
y (1) β (1)
1 0
=
0 1
0 2γ
+
.
(5.90)
The propagation from the reference plane of the sample to the detector can be written as a product of matrices in an analogous way as in Eq. (5.88), i.e.,
1 0
e 1
h n
1 0
1 0
1
t 1
1 0
=
t +
h n
+e
1
.
(5.91)
In this way, an equation for the output ray vector is obtained as y out β out
=
1 0
t +
h n
+e
1
x (2) β (2)
.
(5.92)
By combining Eqs. (5.89), (5.90), and (5.92), the transformation that describes the measurement arm of this Michelson interferometer is obtained as
out
y β out
=
1 0
a+2
h n
+ e + t
1
in
x β in
t +
+ 2γ
h n
+e
1
. (5.93)
The answer to the initial question can now be directly concluded from this equation. For a laser beam that is first parallel to the optical axis ( β in = 0), the change in position as it impinges on the detector is as follows:
y = y out − y in = 2γ t +
h n
+e .
(5.94)
5.4.3 Dyson interferometer The Dyson interferometer (Dyson, 1968, 1979; Steel, 1983; Heavens and Ditchburn, 1991, p. 197) is an example of a measurement head that is relatively insensitive with respect to tilt. This layout had some influence on later work in the field of optical interferometry (Roberts, 1975). Figure 5.12 shows a schematic layout of the interferometer. It is interesting to analyze the device in order to understand the underlying compensation principle. To this end, the optical path is unfolded in order to determine an appropriate representation in terms of transfer matrices. If the thicknesses of the quarter-wave plate and of the reflecting element on the Wollaston prism are neglected, Fig. 5.13 represents the central beam. The transfer matrix for this beam takes the following form:
yout βout =
a b 8 ( 1 − f )( 1 − f )(−(a + b) + 1 + f
×
a 4 1 ( 1 − f )[1 − f (a + b − − f
ab f
ab ) f
)]
a b 4 ( 1 − f )( 1 − f )(a + b − − f
yin βin
+ 2γ
4 a+b− 2 1−
a f
ab f
b 4 ( 1 − f )(a + b − − f
1 − f 1 (a + b) +
ab f 2
ab − f 2 (a + b) + 2 f 2 + 1
ab ) f
.
+
ab 2 ) f 2 (1 − f (a
+ b) +
2ab 2 ) f 2
(5.95)
104
Chapter 5
Figure 5.12 Dyson interferometer. A Wollaston prism serves as the beamsplitting element. A quarter-wave plate near a small mirror on (or near) the Wollaston prism changes the polarization in both paths of the interferometer accordingly, so that the light beams are recombined when passing through the Wollaston prism again.
Figure 5.13 Dyson interferometer, and the unfolded path of the central beam.
For the purpose of tilt compensation, we are looking for a system with a vanishing tilt-error vector. From Eq. (5.95), it is obvious that the second component of this vector will vanish if a = f is chosen. The first component will then also be equal to zero for any choice of b . A look at the systems matrix A suggests that b = f would be a good choice to let the matrix entry a 12 vanish. Therefore, the following settings are made: a = f,
b = f.
(5.96)
For this arrangement, the transfer mapping reads as
yout βout
=
1 0
0 1
yin βin
.
(5.97)
Using the concepts of Secs. 1.1 and 5.1, the stages of the unfolded optical path can be interpreted as a Fourier lens arrangement, followed by a tilted mirror and a cat’s eye that is again followed by a tilted mirror and a Fourier lens arrangement. While the equation for the central beam was given without detailed derivation, we will have a closer look at the outer beam because its compensation is more
105
Optical Systems
critical and its analysis will contain the central beam as a special case. Looking at the outer beam, it is necessary to take into account the difference in beam path that is illustrated in Fig. 5.14. This leads to an unfolded optical path as shown in Fig. 5.15. We directly consider the case for which we expect that compensation is achievable, namely, a = f , b = f . Keeping general values for a and b , as in Eq. (5.95), would be useful when defocusing errors have also to be analyzed. Separate steps of the ray transfer are considered separately first, and then lumped together in order to facilitate the computation. The propagation from one focal plane of the lens to the other focal plane is described by the following matrix: G=
1 0
f 1
1 − f 1
0 1
1 0
f 1
=
0 − f 1
f 0
.
(5.98)
For the central part of the unfolded optical path, the square of this matrix will be convenient, that is, 0 −1 (5.99) G2 = . −1 0
This establishes a link to the retroreflector discussed in Sec. 2.1.
Figure 5.14 Dyson interferometer, outer beam.
Figure 5.15 Dyson interferometer, and the unfolded path of the outer beam.
106
Chapter 5
To incorporate the effect of the tilted mirror on the ray vector, the matrix C and a corresponding error vector are defined for the upper side of the reflector, and the matrix D and its corresponding error vector for the opposite side. They contain the propagation by a distance of cγ or dy , respectively, the addition of the tilterror vector, and, again, a propagation by a distance of cγ or dy , respectively. Put together, this gives
1 0
cγ 1
D
y β
cγ 1
1 0
with C = and
1 0
+
y β
2cγ 1
2cγ 2 2γ
+
0 2γ
y β
=C
+
−2dγ 2 2γ
with D =
1 0
(5.100)
−2dγ 1
.
(5.101)
Please note the different signs that follow from Fig. 5.14. Now, a first step of the propagation can be written as follows: (1)
y β (1)
=
0 − f 1
f 0
y in β in
.
(5.102)
The following ray vector is one changed by the tilted mirror: y (2) β (2)
y (1) β (1)
2cγ 2 2γ
=C
+
.
(5.103)
Using Eq. (5.102), this gives 2cγ − f − f 1
(2)
y β (2)
=
f
0
y in β in
+
2cγ 2 2γ
.
(5.104)
In order to describe the propagation through the lens, a reflection by a mirror, which is assumed to be untilted, and an additional propagation back through the lens, the use of the square of the matrix G is now convenient, i.e., y (3) β (3)
=
−1 0
0 −1
y (2) β (2)
.
(5.105)
To this result, the transformation involving the matrix D will then be applied as (4)
y β (4)
=
1 0
−2dγ 1
(3)
y β (3)
+
2
−2dγ 2γ
.
(5.106)
After performing the multiplications, it follows that
(4)
y β (4)
=
2γ (c − f 1 f
d)
−f
0
y in β in
−
2cγ 2 − 2dγ 2 2γ
+
−cγ 2 2γ
.
(5.107)
107
Optical Systems
This intermediate step is instructive because the compensation occurs here as
(4)
y β (4)
=
2γ (c − f 1 f
−f
d)
y in β in
0
−
−2dγ 2 0
.
(5.108)
The term that is quadratic in γ will be neglected because it is beyond the scope of the linear approximation used here. To transform to the output ray vector, the matrix G is applied again as out
y β out
=
0 − f 1
f 0
y (4) β (4)
.
(5.109)
This can be written as follows in terms of the input vector:
out
y β out
=
1 2γ − f 2 (c − d)
0 1
y in β in
.
(5.110)
It follows from this equation that it is appropriate to adjust the measurement head in a symmetric way so that the condition c = d is fulfilled. In this case, the tiltdependent entry in the system matrix is brought to zero.
5.5 Nested Ray Tracing Köhler’s illumination system provides a typical example for nested ray tracing. This illumination system is widely applied in optical microscopy. Other fields of application are microcamera systems (Reynolds et al., 1989) and the illumination units of systems used in optical lithography (Wong, 2001). The optical arrangement is depicted in Fig. 5.16. It consists of a collector lens and a condenser lens. A field stop is situated in the immediate vicinity of the collector lens. The condenser images the field stop onto the target plane that has to be illuminated. The collector lens, on the other hand, images the light source onto the plane of the aperture stop. Attributing the focal length f 1 to the collector lens and the focal length f 2 to the condenser, the matrix chain can be written as follows: S =
×
1 0
f 2 + c 1
1 − f 1
1
0 1
1 − f 1
2
1 0
0 1
f 1 + a 1
1 0 .
f 2 1
1 0
f 1 + b 1
(5.111)
The distances appearing in this equation are related by the imaging conditions imposed on the nested optical arrangement. To apply the imaging condition for the collector, we have a look at the matrix A that describes the passage of a ray from
108
Chapter 5
Figure 5.16 Köhler’s illumination system.
the light source to the aperture stop, i.e., A=
1 0
f 1 + b 1
1 − f 1
0 1
1
1 0
f 1 + a 1
=
− f b
f 1 −
− f 1 1
− f a
1
ab f 1 1
.
(5.112)
The imaging condition implies that a 12 = 0. Therefore, we have f 12 = ab,
(5.113)
and can express one of the distances appearing in Eq. (5.112) in terms of the other distance. In this way, the matrix A takes the form A=
− f b
0
1
− f 1 1
−
f 1 b
,
(5.114)
and the system matrix changes accordingly to S =
1 f 1
bc f 2
− f 2
b f 1 f 2
−
f 1 f 2 b
0
.
(5.115)
Now, the other condition has to be incorporated into the system matrix. To this end, we consider the matrix B that is an expression for the transit of a ray from the field stop to the target that has to be illuminated, i.e., B =
=
1 0
f 2 + c 1
− f c
2
− f 1
2
f 2 −
1 − f 1
2
c (f 1 + f 2 f +b − 1f 2
0 1 b)
.
1 0
f 2 1
1 0
f 1 + b 1
(5.116)
Specializing this matrix of a subsystem to the case where an imaging condition holds between the plane of the field stop and the target plane is equivalent to setting
109
Optical Systems
b21 = 0. This implies
f 22
c=
f 1 + b
.
(5.117)
Using this in Eq. (5.114), the system matrix of a paraxial model for Köhler’s illumination system reads as S =
f
− f +2 b
−
1
b f 1 f 2
f 1 f 2 b
0
.
(5.118)
We can now draw conclusions from these considerations that characterize the illumination scheme. First, we consider the impact of changes of the aperture stop. The passage of a ray from the plane to the aperture stop to the target plane is represented by the matrix C=
1 0
f 2 + c 1
1 − f 1
0 1
2
1 0
f 2 1
=
− f c 2 1 − f 2
f 2
0
.
(5.119)
Equation (5.117) might also be used at this point, and we have the mapping
f
out
y β out
− f +2 b
f 2
− f 1
0
1
=
2
y ap β ap
,
(5.120)
which implies that the angle β out in the plane of the target is controlled by the diameter y ap of the aperture stop. The corresponding mapping for the field stop follows from combining Eqs. (5.116) and (5.117), i.e.,
out
y β out
=
f
− f +2 b
0
1
− f 1
2
−
f 1 +b f 2
y field-stop β field-stop
,
which shows that adapting the diameter y field-stop of the field stop directly changes the field on the target plane.
Chapter 6
Outlook The method of Gaussian matrices is useful for finding a paraxial solution to a given optical-design problem. This solution can then be refined by analytical methods of aberration theory or by optical-design software. A generic way to proceed is as follows: a paraxial start layout is determined using the matrix approach. Its properties and parameter dependencies can then be analyzed in a first order of approximation with this method. The paraxial model can be implemented as a start configuration in an optical-design program. The necessary paraxial element is included in most optical-design packages. Let us say that we started with a lens in the thin-lens approximation to find and understand the first layout. Then we can make the transition to a thick spherical lens with the same focal length and extend the validity of the solution from the paraxial domain to the complete aperture while trying to keep the main properties of the layout. The control features of the software can be used for this purpose. Depending on the degrees of freedom available for the design task, we might improve the aberration control in further steps of optimization by allowing for an aspheric lens. This method also proved efficient in designing variable optical devices such as zoom lenses. There are some approaches in the literature for extending the matrix theory that were not treated in this book: To overcome the restriction to the paraxial region, trigonometric expressions as matrix entries are used and the concept of the so-called exact matrices are introduced in Das (1991). The relation to Seidel aberrations is described by Guillemin and Sternberg (1984). And an extension of the matrix method is also proposed in Brouwer’s book on instrument design (Brouwer, 1964). In this book, the focus is on optical design, and a most natural step from the practical point of view is to use the matrix description as groundwork (and, in a way, as a framework) and then turn to aberration control for refinement of the design obtained. But the theory of Gaussian matrices also lays a solid groundwork for some in-depth theoretical studies of subjects such as Fourier optics and symplectic systems (Guillemin and Sternberg, 1984). The matrix description is considered a good starting point to dive into the theoretical description of light propagation through optical systems by integral transformations (Guillemin and Sternberg, 1984; Hodgson and Weber, 1997). The Collins
112
Outlook
integrals can be classified in a straightforward way by making reference to Gaussian matrices (Hodgson and Weber, 1997). A similar statement holds for Wigner distribution functions. Considerations on ray matrices and on Wigner distribution functions on a high theoretical level can be found in the book of Torre (2005). In laser technology, extensive use is made of the theory of Gaussian matrices to design optical resonators. In fact, many results in matrix optics stem from research in that field. A surprising application of tools similar to those presented in Chapter 3 for sensitivity analysis and tolerancing is made by Duparré et al. (2005) and Schreiber et al. (2005). They apply this mathematical description to analyze lens arrays. There are many other interesting applications of the matrix method and related paraxial methods to optical design and instrumentation. This text could only contain a part of them, and the reader will presumably discover new fascinating optical results during her/his design work in the field of optics.
Bibliography Altmann, J., Baumgart, R., and Weitkamp, C., “Two-mirror multipass absorption cell,” Appl. Opt. , 20, pp. 995–999 (1981). Antonsen, E.L., Burton, R.L., Spanjers, G.G., and Engelman, S.F., “Herriott cell interferometry for millimeter-scale plasma measurements,” Rev. Sci. Instrum., 74, pp. 88–93 (2003a). Antonsen, E.L., Burton, R.L., Spanjers, G.G., and Engelman, S.F., “Herriott cell augmentation of a quadrature heterodyne interferometer,” Rev. Sci. Instrum. , 74, pp. 1609–1612 (2003b). Back, F.G., and Lowen, H., “The basic theory of varifocal lenses with linear movement and optical compensation,” J. Opt. Soc. Am., 44(9), pp. 684–691 (1954). Back, F.G., and Lowen, H., “Generalized theory of Zoomar systems,” J. Opt. Soc. Am., 48(3), pp. 149–153 (1958). Baker, D., “Field-widened interferometers for Fourier spectroscopy,” Spectrometric Techniques, G.A. Vanasse (Ed.), pp. 71–106, Academic Press, New York (1977). Barr, J.R.M., “Achromatic prism beam expanders,” Opt. Commun. , 51(1), pp. 129–134 (1984). Berek, M., Grundlagen der praktischen Optik—Analyse und Synthese optischer Systeme , Walter de Gruyter, Berlin (1930) (reprint 1970). Berge, B., and Peseux, J., “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E , 3, pp. 159–163 (2000). Besenmatter, W., “Designing zoom lenses aided by the Delano diagram,” Proc. SPIE , 237, pp. 242–255 (1980). Born, M., Optic—Ein Lehrbuch der Electro-magnetischen Lichttheorie , p. 102, Springer, Berlin (1933) (reprint 1985). Born, M., and Wolf, E., Principles of Optics—Electromagnetic Theory of Propagation, Interference and Diffraction of Light , pp. 225–226, Pergamon Press, Oxford (1980). Breckinridge, J.B., and Schindler, R.A., “First-order optical design for Fourier spectrometers,” Spectrometric Techniques , G.A. Vanasse (Ed.), Vol. II, Academic Press, New York (1981). Brouwer, W., Matrix Methods in Optical Instrument Design , Chap. 9, Benjamin, New York (1964). Casperson, L.W., “Synthesis of Gaussian beam optical systems,” Appl. Opt., 20, pp. 2243– 2249 (1981). Chang, C.-M., and Shieh, H.-P.D., “Design of illumination and projection optics for pro jectors with single digital micromirror devices,” Appl. Opt., 39, pp. 3202–3208 (2000). Clark, A.D., Zoom Lenses , Adam Hilger, London (1973).
114
Bibliography
Das, P., Lasers and Optical Engineering , Springer-Verlag, Berlin, p. 302 (1991). Delano, E., “First-order design and the y , y¯ diagram,” Appl. Opt., 2, pp. 1251–1256 (1963). Demtröder, W., Laserspektroskopie—Grundlagen und Techniken , Chap. 4, SpringerVerlag, Berlin (1999). DeWeerd, A.J., and Hill, S.E., “Reflections on handedness,” Phys. Teacher , 42, pp. 275– 279 (2004). Dickey, F.M., and Holswade, S.C., Laser Beam Shaping—Theory and Techniques , Marcel Dekker, New York (2000). Duarte, F.J., and Piper, J.A., “Dispersion theory of multiple-prism beam expanders for pulsed dye laser,” Opt. Commun., 43(5), pp. 303–307 (1982). Duarte, F.J., and Piper, J.A., Erratum, Opt. Commun. , 45, p. 420 (1983). Duarte, F.J., “Note on achromatic multiple-beam expanders,” Opt. Commun. , 53(4), pp. 259–262 (1985a). Duarte, F.J., Erratum, Opt. Commun., 54, p. 388 (1985b). Duparré, J., Schreiber, P., Matthes, A., Pshenay-Severin, E., Bräuer, A., and Tünnermann, A., “Microoptical telescope compound eye,” Opt. Express, 13 (3), pp. 889–903 (2005). Dyson, J., “Unit magnification optical system without Seidel aberrations,” J. Opt. Soc. Am. , 49, pp. 713–716 (1959). Dyson, J., “Optics in a hostile environment,” Appl. Opt., 7, pp. 569–580 (1968). Dyson, J., Interferometry as a Measuring Tool , pp. 96–98, Machinery Publ., Brighton (1979). Fantone, S.D., “Optical system with anamorphic prism,” U.S., Patent No. 4,627,690 (1986). Fantone, S.D., “Anamorphic prism: A new type,” Appl. Opt., 30(34), pp. 5008–5009 (1991). Fouckhardt, H., Photonik—Eine Einführung in die integrierte Optoelektronik und technische Optik , pp. 94–95, Teubner, Stuttgart (1994). Gerrard, A., and Burch, J.M., Introduction to Matrix Methods in Optics , Wiley, New York (1975). Goodman, J.W., Introduction to Fourier Optics , McGraw-Hill, New York (1968) (reissued 1988). Goodman, D.S., “General principles of geometric optics,” Handbook of Optics , M. Bass, E.W. Van Stryland, D.R. Williams, W.L. Wolfe (Eds.), Vol. I: Fundamentals, Techniques, and Design, pp. 1.1–1.109, McGraw-Hill, New York (1995). Grenville, A., Hsieh, R.L., von Bünau, R., Lee, Y.-H., Markle, D.A., Owen, G., and Pease, R.F.W., “Markle-Dyson optics for 0.25 µ m lithography and beyond,” J. Vac. Technol. B, 9(6), pp. 3108–3112 (1991). Grenville, A., Owen, G., and Pease, R.F.W., “Image monitor for Markle-Dyson optics,” J. Vac. Sci. Technol. B, 11(6), pp. 2700–2704 (1993). Grove, S.L., and Shuman, C.A., “Anamorphic achromatic prism for optical disk heads,” U.S. Patent No. 5,155,633 (1992). Guillemin, S., and Sternberg, S., Symplectic Techniques in Physics , Cambridge University Press, Cambridge, England (1984). Hanna, D.C., Kärkkäinen, P.A., and Wyatt, R., “A simple beam expander for frequency narrowing of dye lasers,” Opt. Quant. Electron. , 7, pp. 115–119 (1975). Heavens, O.S., and Ditchburn, R.W., Insight into Optics , Wiley, Chichester (1991).
Bibliography
115
Herriott, D., Kogelnik, H., and Kompfner, R., “Off-axis paths in spherical mirror interferometers,” Appl. Opt., 3, pp. 523–526 (1964). Herriott, D.R., and Schulte, H.J., “Folded optical delay lines,” Appl. Opt., 4 , pp. 883–889 (1965). Herzberger, M., “Gaussian optics and Gaussian brackets,” J. Opt. Soc. Am. , 33(12), pp. 651–655 (1943). Hodgson, N., Haase, T., Kostka, R., and Weber, H., “Determination of laser beam parameters with the phase space beam analyzer,” Opt. Quant. Electron. , 24, pp. S927–S949 (1992). Hodgson, N., and Weber, H., Optical Resonators—Fundamentals, Advanced Concepts and Applications, Springer-Verlag, Berlin (1997). Hovde, D.C., Hodges, J.T., Scace, G.E., and Silver, J.A., “Wavelength-modulation laser hygrometer for ultrasensitive detection of water vapor in semiconductor gases,” Appl. Opt., 40, pp. 829–839 (2001). Hsieh, R.L., Lee, Y.Y.-H., Maluf, N.I., Browning, R., Jerabek, P., and Pease, R.F., “Reflective masks for 1X deep ultraviolet lithography,” Proc. SPIE , 1604, pp. 67–77 (1992). Ichie, K., “Laser scanning optical system using axicon,” Eur. Patent No. EP 0 627 643 (1994). Johnson, R.B., and Feng, C., “Mechanically compensated zoom lenses with a single moving element,” Appl. Opt., 31, pp. 2274–2278 (1992). Kay, D.B., and Gage, E.C., “Heads and lasers,” T.W. McDaniel, and R.H. Victora, Handbook of Magneto-Optical Data Recording—Materials, Subsystems, Techniques , p. 62, Noyes Publications, Westwood, NJ (1997). Kloos, G., Entwurf und Auslegung optischer Reflektoren , Expert Verlag, Renningen (2007). Lam, J.F., and Brown, W.F., “Optical resonators with phase-conjugate mirrors,” Opt. Lett., 5, pp. 61–63 (1980). Latta, M.R., Strand, T.C., and Zavislan, J.M., “Effect of track crossing on focus servo signals: Feedthrough,” Proc. SPIE , 1663, 157163 (1992). Lipson, S.G., Lipson, H.S., and Tannhauser, D.S., Optik , Springer-Verlag, Berlin (1997). Luecke, F.C., “Achromatic anamorphic prism pair,” U.S. Patent No. 5,596,456 (1997). Magarill, S., “First order property of illumination system,” Proc. SPIE , 4768, pp. 57–64 (2002). Malacara, D., and Malacara, Z., Handbook of Lens Design , Marcel Dekker, New York (1994). Marchant, A.B., “Method and apparatus for anamorphically shaping and deflecting electromagnetic beams,” U.S. Patent No. 4,759,616 (1988). Marchant, A.B., Optical Recording—A Technical Overview , Addison-Wesley, Reading, MA (1990). McLeod, J.H., “The axicon: a new type of optical element,” J. Opt. Soc. Am., 44(8), pp. 592–597 (1954). McLeod, J.H., “Axicons and their uses,” J. Opt. Soc. Am., 50(2), pp. 166–169 (1960). Naumann, H., and Schröder, G., Bauelemente der Optik—Taschenbuch der technischen Optik , Carl Hanser Verlag, München, p. 187 (1992). New, R.M.H., Owen, G., and Pease, R.F.W., “Analytic optimization of Dyson optics,” Appl. Opt. 31, pp. 1444–1449 (1992). Niefer, R.J., and Atkinson, J.B., “The design of achromatic prism beam expanders for pulsed dye lasers,” Optics Commun. 67(2), pp. 139–143 (1988).
116
Bibliography
Okuda, I., Takishima, S., Oono, M., Maruyama, K., and Noguchi, M., “Optical system for optical disc apparatus including anamorphic prisms,” U.S. Patent No. 5,477,386 (1995). Owen, G., Grenville, A., von Bünau, R., Jeong, H., Markle, D.A., and Pease, R.F., “The effect of gaps in Markle-Dyson optics for sub-quarter-micron lithography,” Jpn. J. Appl. Phys. (Pt. 1), 32, pp. 5840–5844 (1993). Pegis, R.J., and Peck, W.G., “First-order design theory for linearly compensated zoom systems,” J. Opt. Soc. Am., 52(8), pp. 905–911 (1962). Pérez, J.-P., Optique—Fondements et applications , Masson, Paris (1996). Reynolds, G.O., DeVelis, J.B., Parrent, G.B., and Thompson, B.J., The New Physical Optics Notebook: Tutorials in Fourier Optics , p. 154, SPIE, Bellingham, WA, American Institute of Physics, New York (1989). Rioux, M., Tremblay, R., and Bélanger, P.A., “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. 17, pp. 1532–1536 (1978). Roberts, R.B., “Absolute dilatometry using a polarization interferometer,” J. Phys. E 8, pp. 600–602 (1975). Saito, K., Sato, S., Shino, K., and Taniguchi, T., “Device for magneto-optic signal detection with a small crystal prism,” Appl. Opt., 39(8), pp. 1315–1322 (2000). Schreiber, P., Kudaev, S., Dannberg, P., and Zeitner, U.D., “Homogeneous LEDillumination using microlens arrays,” Proc. SPIE , 5942, pp. 59420K-1–59420K-9 (2005). Sennaroglu, A., and Fujimoto, J.G., “Design criteria for Herriott-type multi-pass cavities for ultrashort pulse lasers,” Opt. Express , 11(9), pp. 1106–1113 (2003). Shack, R.V., “Analytic system design with pencil and ruler—The advantages of the y –y¯ diagram,” Proc. SPIE , 39, pp. 127–140 (1973). Siegman, A.E., Lasers, University Science Books, Mill Valley, pp. 582–583 (1986). Steel, W.H., Interferometry, Cambridge University Press, Cambridge, England (1983). Svelto, O., Principles of Lasers, Plenum Press, New York (1998). Tanaka, K., “Allgemeine Gausssche Theorie eines mechanisch kompensierten ZoomObjektives,” Opt. Commun. 29(2), pp. 138–140 (1979). Tanaka, K., “Paraxial analysis of mechanically compensated zoom lenses. 1: Fourcomponent type,” Appl. Opt. 21(12), pp. 2174–2183 (1982). Torre, A., Linear Ray and Wave Optics in Phase Space—Bridging Ray and Wave Optics in the Wigner Phase-Space Picture , Elsevier, Amsterdam (2005). Tovar, A.A., and Casperson, L.W., “Generalized Sylvester theorems for periodic applications in matrix optics,” J. Opt. Soc. Am. A, 12, pp. 578–590 (1995). Tyson, R.K., Principles of Adaptive Optics , Academic Press, New York (1998). van den Eerenbeemd, J., “Combining astigmatism of a plane parallel plate and a cylindrical lens,” ISOM ’01—International Symposium on Optical Memory 2001 Technical Digest , Washington, D.C.: Optical Society of America. pp. 196–197 (2001). Wolfe, W.L., “Nondispersive Prism,” Handbook of Optics , M. Bass, E.W. Van Stryland, D.R. Williams, W.L. Wolfe (Eds.), Vol. II: Devices, Measurements, and Properties, pp. 4.1–4.29 McGraw-Hill, New York (1995). Wong, A.K.-K., Resolution Enhancement Techniques in Optical Lithography , pp. 7–8, SPIE, Bellingham, WA (2001). Wooters, G., and Silvertooth, E.W., “Optically compensated zoom lens,” J. Opt. Soc. Am. , 55(4), pp. 347–351 (1965).
Bibliography
117
Yoder, P.R., “Non-image-forming optical components,” Proc. SPIE , 531, pp. 206–220 (1985). Young, M., Optik, Laser, Wellenleiter , Springer-Verlag, Berlin (1997). Yu, F.T.S., and Khoo, I.C., Principles of Optical Engineering , Wiley, New York (1990). Zissis, G.J., “Dispersive prisms and gratings,” Handbook of Optics , M. Bass, E.W. Van Stryland, D.R. Williams, and W.L. Wolfe (Eds.), Vol. II: Devices, Measurements, and Properties, pp. 5.1–5.16, McGraw–Hill, New York (1995).
Index 4f arrangements, 79 A absorption cell, 99 adaptive axicon, 48 adaptive feedback loop, 94 adaptive optics, 88 adjustment sensitivity, 99 afocal system, 11, 12, 20 afocal triplet, 77 anamorphic optics, 59 anamorphic thin lens, 65 angular magnification, 19 aperture stop, 16 autocollimator, 92, 94 axial misalignment, 56 axial ray, 16 axicon devices, 47 B balance the optical beam paths, 99 beam pointing error, 53, 57 beam shaping, 33 Brewster condition, 33 C cardinal elements, 7 cascading, 60, 67, 68 cat’s-eye retroreflector, 27, 56 central theorem of first-order ray tracing, 14 chief rays, 80 collimating system, 12, 20 collimating triplet, 77 collimation lens, 72 Collins integrals, 112 compensated adaptive lens arrangements, 87
compensation, 87, 103 cylindrical lens, 66 D decentering, 53 decomposition of matrices, 13 defocusing, 53 Delano diagram, 19 determinant, 5 dispersive prisms, 32 divided optical path, 99 double-pass optics, 92 doubly telecentric lens, 81 Dyson interferometer, 103 Dyson optics, 82 E eigenvalue theory, 96 error vector, 54 F feedback loop, 91 field stop, 17 Fizeau interferometer, 99 focal length, 8 focusing condition, 12, 20 focusing triplet, 77 Fourier optics, 111 Fourier transform objectives, 79 G Galilean telescope, 72 Gaussian brackets, 22 H Herriott cell, 99 homogeneous coordinates, 54, 75
120
I image field, 19 imaging, 10 imaging condition, 11, 12, 20, 73 imaging equation, 74, 78 imaging triplet, 77 integrating rod, 49 internal reflection, 45, 46 K Keplerian telescope, 72 Köhler’s illumination system, 107 L Lagrange invariant, 18 laser beam circularization, 33 laser resonators, 99 laser technology, 95 law of refraction, 1, 4 lens, 6 lens doublet, 12 light guide, 49 liquid lens, 90 M magnification, 19 matrix representations, 5 mechanical compensation, 84, 85 Michelson interferometer, 101 microscope, 13 microscopy, 107 misadjustment, 53 mixing rod, 52 multiple-pass optics, 95 N nested ray tracing, 107 nondispersive, 32 nonorthogonal anamorphic system, 60 O optical compensation, 84, 85 optical delay lines, 99 optical resonators, 95 optics, 84 orthogonal anamorphic system, 60 P partitioned matrix, 59 Petzval radius, 19
phase space, 20 phase-conjugate mirror, 26 phase-space analysis, 69 phase-space beam analyzer, 70 plane mirror, 25 plane-parallel plate, 29 principal planes, 8 principal ray, 17 prism for compression along one axis, 38 prism for expansion along one axis, 36 prisms, 31 Q quadrupole lens, 69 R ray slope, 6 recursion formula, 23 reflection, 5, 25 refraction, 3, 29 refraction matrix, 21 retroreflector, 26 roof mirror, 28 rotation of an anamorphic component, 61 S sensitivities, 53, 75 single-pass optics, 77 spatial filtering, 79 spherical mirror interferometers, 99 spherical surface, 4 stable, 99 Sylvester’s theorem, 98 symplectic systems, 111 synthesis of optical systems, 13 system synthesis, 13 T telecentric lenses, 80 telecentricity, 80 telescope, 13, 72 telescopes, 11 terrestrial telescope, 78 thin-prism approximation, 32 tilt, 53 tolerances, 53, 75 tolerancing, 39
121
transfer function, 35 translation matrix, 3, 21 trigonometric description of dispersive prisms, 33 triple mirror, 52 triplet, 77, 84 tunnel diagram, 31
V variable, 84 varifocal systems, 84
U unstable, 99
Z zoom lens, 84
W Wigner distribution functions, 112
