Topic 1 Interference and Diffraction (Halliday/Resnick/Walker Ch.35, 36)
Interference and Diffraction • Introduction to Interference • Young’ oung’s s Interfe Interferen rence ce Exp Experi erimen mentt • Intensity of Interference Pattern • Thin Film Interference • Optical Interferometers • Introduction to Diffraction Diffraction • Single Slit and Double Slit Diffractions • Diffraction Grating • Diffraction Resolution Limit
Interference and Diffraction • Introduction to Interference • Young’ oung’s s Interfe Interferen rence ce Exp Experi erimen mentt • Intensity of Interference Pattern • Thin Film Interference • Optical Interferometers • Introduction to Diffraction Diffraction • Single Slit and Double Slit Diffractions • Diffraction Grating • Diffraction Resolution Limit
Interference in Nature
butterfly CD/DVD peacock
soap film and bubble
beetle
What is Interference? Interference: a physical phenomenon in which two or more waves superimpose to form a resultant wave. It usually refers to the interaction of waves that are coherent/correlated with each other. Optical interference is the interference
between light waves and is applied in many branches of science and engineering. The resultant wave is obtained by the superposition principle. Its amplitude depends on the phase difference between the interacting waves,
In-phase interference → amplitude doubling Out-of-phase interference → amplitude cancelling
The blue colour on the top surface of a Morpho butterfly wing is due to optical interference. It shifts in color as you change your viewing angle.
Young’s Interference Experiment Young’s interference experiment (Young’s double slit interference expt): ~1801 Significance: proved that light is a wave, contradicted Isaac Newton’s view
Thomas Young (1773-1829)
Optical interference fringes Formation of wave interference pattern
Optical Path Difference For D d , rays r 1 and r 2
L d sin
parallel
(1.1)
Constructive and Destructive Interferences Bright fringes (constructive interference, maxima): L must be either zero or an integer number of wavelengths. Appear at angles satisfying d sin m , m
0, 1, 2,...
(1.2)
Dark fringes (destructive interference, minima): L must be an odd multiple of half a wavelength. Appear at angles satisfying
1 d sin m , m 0, 1, 2,... 2
(1.3)
Fringe Spacing Consider near pattern center, i.e. small
sin tan
y D
(1.4)
Spacing between 2 bright fringes:
ym
1
ym
D sin m D
1
m 1
D d
d
D sin m
D
m
Conclusion: The spacing between
d
neighboring bright fringes remains
(1.5)
unchanged when and d are small.
Example: Measuring Plastic Thickness Given that a plastic block of refractive
index 1.5 is placed on the upper slit, determine the plastic thickness.
Plastic block inserted → 1 (first order) bright fringe moves to center for = 600nm Let L be the plastic block thickness, the path difference at the center bright fringe is and is equal to L x (1.5-1)
9
0.5 L 600 10 m L
6
1.2 10 m
Wave Coherence For the interference pattern to appear on the screen , the light waves reaching any point on the screen must have a phase
difference that does not vary in time. When the phase difference remains constant, the light from slits S 1 and S 2 are said to be completely coherent. If the phase difference constantly changes in time, the light is said to be incoherent.
Incandescent light – incoherent light source Laser – highly coherent light source Sunlight – partially coherent, i.e., phase difference is constant only if the points under consideration are very close
Intensity of Interference Pattern Total energy is conserved
Consider electric field components of the light waves at point P on the screen: E 1
E 0
sin ωt
(1.6)
E 2
E 0
sinωt
(1.7) Constant phase difference → coherent
It can be shown (proof to follow) that the intensity (power/area) at P is
2 d I 4 I 0cos (1.8) where sin (1.9) 2 2
Intensity of Interference Pattern Proof: Recall
sin A sin B
A B A B 2 cos sin 2 2
Hence, the electric field at point P is E E 1
E 2 E 0 sin t E 0 sin t 2 E 0 cos sin t 2 2 Constant amplitude
(1.10)
Thus, the electric field amplitude is changed by a factor of 2 cos . 2 2 Intensity (field amplitude)2 → changed by 4 cos . I 4 I 0cos (1.8) 2 2 2
Phase difference
2
d sin
Path difference (1.1)
2 d
sin (1.9)
Thin Film Interference
Ray representation of incident, transmitted, and reflected light waves. Phase difference of the reflected waves is determined by: • phase shift by reflection • different physical path lengths • different RI
different optical path lengths
Phase Shift by Reflection Phase shift on normal reflection (RI = refractive index): Incident from high RI to low RI (e.g from water/glass to air) → no phase shift Incident from low RI to high RI (e.g from air to water/glass) → phase shift i.e., /2
http://www.schoolphysics.co.uk
Maxima and Minima Air
/2 reflection shift
Water
Assume normal incidence,
Air
no phase shift
Constructive interference (Maxima): 2 L n2
Destructive interference (Minima):
2 L
n2
1 m , m 0, 1, 2,... (1.11) 2
m , m
0, 1, 2,...
(1.12)
Soap Film and Bubble Different soap film thicknesses → maxima at different wavelengths Dark region: L<< (about to burst) Path difference: negligible Phase difference dominated by phase shift in reflection =
m
=1
m
=2
m
=3
Thin Film Coating Thin film coatings are widely used in glasses, camera lens, optical instruments, semiconductor optical devices, fiber devices, etc. to improve their performances through reduction or enhancement of optical reflections/transmission of specific wavelengths.
CUHK Photonic Packaging Lab Coating Machine
Example: Antireflection Coating A glass lens (RI=1.50) is coated with MgF2 (RI=1.38) to reduce optical reflections. Determine the minimum coating thickness required to eliminate reflections at the center of the visible spectrum, i.e. =550 nm. Assume
incident light is perpendicular to the lens surface.
Reflection is minimized if we choose coating thickness L s.t.
waves reflected from the interfaces are exactly out of phase.
2 L n2 m
1
, m 0, 1, 2,... 2
For minimum thickness, m = 0 L
4n2
550 nm 4 1.38
99.6 nm
Optical Interferometer Interferometer: an optical instrument/device that measures lengths or changes in length (or RI) with great accuracy by means of interference fringes. Common types of Interferometer: Beam splitter
mirror
Mach-Zehnder Interferometer
Michelson Interferometer
Fabry-Perot Interferometer
Sagnac Interferometer
Michelson Interferometer The Nobel Prize in Physics 1907 was awarded to Albert A. Michelson "for his optical precision instruments and the spectroscopic and metrological investigations carried out with their aid". He was the first American to receive the Nobel Prize in Sciences.
http://en.bj-force.net/
MIPAS Interferometer Assembly (Michelson Interferometer for Passive Atmospheric Sounding) https://earth.esa.int/web/guest/missions/esa-operational-eomissions/envisat/instruments/mipas
Michelson Interferometer Path difference of the interfering branches is 2d 2-2d 1. When one of the mirrors is moved by /2, the path difference is changed by and the interference pattern will be shifted by one fringe. Similarly, insertion of a transparent material of thickness L and RI n on one of the paths will cause a phase change equal to
2 L
n 1
(1.13)
For each path change of (phase change = 2), the pattern is shifted by one fringe. Thus, by counting the number of fringes through which the material causes the pattern to shift, the thickness L can be determined in terms of . Animation on shifting of interference fringes http://skullsinthestars.com/2008/10/16/fabry-perot-and-their-wonderfulinterferometer-1897-1899/
Diffraction
circular disk
square aperature
razor blade
sunlight diffraction
vertical slit diffraction photography
Basics of Diffraction Light diffraction: light passes through a narrow slit (dimension ~ wavelength) or an edge interferes with itself and produces an interference
pattern called the diffraction pattern. It is a wave nature of light.
Water wave encounters a barrier with an opening of dimensions similar to the wavelength, the part of the wave that passes through the opening will flare (spread) out—will diffract—into the region beyond the barrier. The flaring is consistent with the spreading of wavelets according to Huygens principle.
Interference pattern from twoslit diffraction http://en.wikipedia.org/wiki/Diffraction
Dependence on Slit Width
The narrower the slit, the stronger the diffraction.
Diffraction limits geometrical optics (ray optics) since a narrow slit will cause the light to spread. Geometrical optics holds only when slits or apertures have dimensions much greater than the wavelength of light.
In projecting an image of tiny features on a photo film, the image will become blurred through optical diffraction.
Single Slit Diffraction: Locating Minima This pair of rays r 1 and r 2 cancel each other at P1. Similarly, we can pair up other rays in the two zones.
For D>>a, rays r 1 and r 2 can be treated as parallel and inclined at angle to the central axis.
This path length difference shifts one wave from the other, which determines the interference.
First, if we divide the slit into two zones of equal widths a / 2, and then consider a light ray r 1 from the top point of the top zone and a light ray r 2 from the top point of the bottom zone. For destructive interference at P 1, a / 2sin / 2.
Hence, the first minimum is located at
a
sin
(1.14)
Single Slit Diffraction: Locating Minima After locating the first dark fringes, one can find the second dark fringes above and below the central axis. The slit is now divided into four zones of equal widths a / 4.
The second minimum is located at
/ 4sin
a
/ 2, i.e.
a
In general, the minima (dark fringes) are located at a sin m ,
for m 1,2,3,...
(1.15)
sin
2
Single Slit Diffraction: Intensity Pattern The slit is divided into 18 zones. Resultant amplitudes E are shown for the central maximum and for different points on the screen.
(d) With even larger phase difference, the waves add to give a small net amplitude.
(c) With a sufficiently large phase difference, the waves can add together to give zero amplitude.
(b) Light waves have a small phase difference and add to give a smaller amplitude
(a) Light waves from the zones are in phase and add constructively to
ENGG 2520 Engineering Physics II
Double Slit Diffraction For slit width a << central maximum of the diffraction pattern of each slit covers the entire screen. Interference leads to fringes of same intensities. →
If a << is NOT satisfied each slit produces its own diffraction pattern described on p.27 with the first minima close to the center. →
Hence, interference of light from the 2 slits produces fringes of different intensities. The diffraction pattern in (b) sets an envelope for the intensity plot in (a) to produce plot (c). Double slit Spring 2013
ENGG 2520 Engineering Physics II
Single slit
Diffraction Gratings Diffraction gratings contain a large number of slits, often called rulings. They can be made on a opaque surface with narrow parallel grooves arranged like the slits, forming a reflection grating instead of a transmission grating. The fine rulings of ~0.5 micron width on a CD function as a reflection grating Interference fringe width is inversely proportional to the number N of rulings for a given ruling separation d and wavelength. Hence, a diffraction grating with large N can be used to determine the wavelength of a monochromatic light. In contrast, the bright fringes from double slit interference are broad and different wavelengths will overlap too much to be distinguished. Constructive interference (maxima) occurs at d sin
Spring 2013
m ,
for m
ENGG 2520 Engineering Physics II
0, 1, 2, ... (1.16)
Visible emission lines of Cadmium resolved with a grating spectroscope
Diffraction by Circular Aperture A tiny circular aperture such as a circular lens can result in an optical diffraction pattern.
Diffraction pattern of a circular aperture showing the central maximum and the interference fringes. For an aperture of diameter d , the first minimum is located at an angle given by
sin
1.22
d
(1.17)
Compared to (1.14), the factor 1.22 here is caused by the circular shape of the aperture. In photography, a larger aperture (smaller f-number) gives a better resolved image due to weaker diffraction effect. You may take a look on photos taken with different apertures in the following link: http://www.luminous-landscape.com/tutorials/understanding-series/u-diffraction.shtml
Resolvability Consider the lens image of two distant point objects (e.g. stars) with a small angular separation. They cannot be resolved (distinguished) if their diffraction patterns overlap. Two objects are barely resolvable when their angular separation
is given by R
sin R
1.22
d
.
That is, the central maximum of one source
coincides with the first minimum of the other source in the diffraction pattern. This is called the Rayleigh’s criterion.
Indistinguishable
marginally distinguishable
clearly distinguishable
