Physical Optics and Light Measurements, Volume 26 (Methods in Experimental Physics)

Methods of Experimental Physics VOLUME 26 PHYSICAL OPTICS AND LIGHT MEASUREMENTS

METHODS OF EXPERIMENTAL PHYSICS Robert Celotta and Judah Levine, Editors-in-Chief

Founding Editors

L. MARTON

C. MARTON

Volume 26

Physical Optics and Light Measurements Edited by

Daniel Malacara Centro de lnvestigaciones en Optice Leon, Gto Mexico

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers

Boston

San Diego New York Berkeley London Sydney Tokyo Toronto

Copyright 0 1988 by Academic Press, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information qtorage and retrieval system, without permission in writing from the publisher. ACADEMIC PRESS, INC. 1250 Sixth Avenue, San Diego, CA 92101 United Kingdom Edifion publid~edbv ACADEMIC PRESS INC. ( L O N D O N ) LTD. 24-28 Oval Road. London NWI 7DX

Library of- Congress Cbraloging-in-Publication Data

Physical optics and light measurements. (Methods of experimental physics; v. 26) Includes bibliographies and index. 1. Optics, Physical. 2. Optical measurements. 1. Malacara, Daniel, Date11. Series. QC395.2.P49 1988 535’.2 88-978 ISBN 0-12-47597 1-8

Printed in the United States of America 88 89 90 91 9 8 7 6 5 4 3 2 1

CONTENTS LIST OF CONTRIBUTORS . .

. . . . . . . . . . . . . . . . . ix PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . xi ... LIST OF VOLUMESI N TREATISE . . . . . . . . . . . . . . . xi11

1 . Interference

by D A N I E LMALACARA 1.1 Introduction

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . 1.3. Multiple-Beam Interferometers . . . . . . . . . . . . . 1.4. Multiple-Pass Interferometers . . . . . . . . . . . . . 1.2. Two-Beam Interferometers

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5. Applications of Interferometry References

1

7 25 37 38 45

2 . Diffraction and Scattering by D A N I E LMALACARA

..................... 49 2.2. Fresnel Diffraction . . . . . . . . . . . . . . . . . . 53 2.3. Fraunhofer Diffraction and Fourier Transforms . . . . . 62 2.4. Diffraction Gratings . . . . . . . . . . . . . . . . . 70 2.5. Resolving Power of Optical Instruments . . . . . . . . . 79 2.6. The Abbe Theory of the Microscope . . . . . . . . . . 86 2.7. Scattering . . . . . . . . . . . . . . . . . . . . . . 92 ..................... References 102 2.1. Diffraction

V

vi

CONTENTS

3. Optical Polarization by FREDERIC R . STAUFFER

3.1. Introduction

. . . . . . . . . . . . . . . . . . . . .

107

3.2. Electromagnetic Description of Light

108

3.3. Wave Propagation in Isotropic

. . . . . . . . . . Media . . . . . . . . . . . . . . . . . . . . . . .

113

3.5. Thin Films

125

3.6.

..................... Wave Propagation in Anisotropic Media . . . . . . . .

133

. . . . . . . . . . . . . . . . .

157

3.4. Wave Propagation for Metals

3.7. Slits, Gratings, and Metal Grid Polarizers 3.8. Light Source and Detector Polarizations

3.9. Polarization Determination and Mathematical Description

References

. . . . . . . . . . . . . . . . . . . . .

120

158 159 162

4 . Holography

by R. D. BAHUGUNA A N D D . MALACARA

. . . . . . . . . . . . . . . . . . . . . Holography . . . . . . . . . . . . . . . .

4.1. Introduction

167

4.2. Theory of

168

. . . . . . . . . . . . . Some Applications of Holography . . . . . . . . . . . Experimental Procedures in Holography . . . . . . . . References .. .. .. .. .. .. . . . . . . . . .

4.3. Different Types of Holograms

175

4.4.

191

4.5.

199 206

5 . Photometry and Radiometry by WILLIAM L. WOLFE 5.1. Introduction

.....................

5.2. Symbols, Units. and Nomenclature

5.3. Formulas for Blackbody Radiation

. . . . . . . . . . . . . . . . . . . . . .

213 214 219

vii

CONTENTS

. . . . . . . . . . . . . . Radiometric Temperature Measurements . . . . . . . Radiometric Instruments . . . . . . . . . . . . . . Measurements . . . . . . . . . . . . . . . . . . . . Photometry: Radiometry of Visible Light . . . . . . . References .....................

5.4. Simple Radiative Transfer

5.5. 5.6. 5.7. 5.8.

. . .

223 239 246 263

.

284 287

6. Detectors

by T. 0. POEHLER

........... 6.2. Figures of Merit . . . . . . . . 6.3. Thermal Detectors . . . . . . . 6.4. Photon Detectors . . . . . . . . 6.1. Introduction

6.5. Noise

. . . .

......... 291 . . . . . . . . . . 292 . . . . . . . . . . 293 . . . . . . . . . . 303

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .....................

6.6. Optical Window Material References

328 331 332

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CONTRIBUTORS Numbers in parentheses indicate pages on which the authors’ contributions begin.

RAMENDRADEOBAHUGUNA(167), Centro de Znvestigaciones en Optica, Apartado Postal 948, 37000 Leon, Gto. Mexico DANIELMALACARA ( 1 , 49, 167), Centro de Investigaciones en Optica, Apartado Postal 948, 37000 Leon, Gto. Mexico THEODORE 0 . POEHLER(291), Applied Physics Laboratory, Johns Hopkins Road, Laurel, Maryland 20707 FREDERICR. STAUFFER(107), Sacramento Peak Observatory, Sunspot, New Mexico 88349 WILLIAM L. WOLFE(213), Optical Sciences Center, University of Arizona, Tucson, Arizona 85721

ix

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PREFACE Two books covering the field of modern optics have been prepared in this series “Methods of Experimental Physics”, separating the material into two parts, one with the title “Geometrical and Instrumental Optics”, and the other with the title “Physical Optics and Light Measurements”. The purpose of these books is to help the scientist or engineer who is not a specialist in optics to understand the main principles involved in optical instrumentation and experimental optics. Our main intent is to provide the reader with some of the interdisciplinary understanding that is so essential in modern instrument design, development, and manufacture. Coherent optical processing and holography are also considered, since they play a very important role in contemporary optical instrumentation. Radiometry, detectors, and charge coupled imaging devices are also described in these volumes, because of their great practical importance in modern optics. Basic and theoretical optics, like laser physics, nonlinear optics and spectroscopy are not described, however, because they are not normally considered relevant to optical instrumentation. In this volume, “Physical Optics and Light Measurements”, Chapter One describes the theory and applications of interference and interferometers. Chapter Two studies diffraction, its basic theoretical fundamentals, and some practical applications. Polarized light and its uses are considered in Chapter Three. Holography and holographic methods are studied in detail in Chapter Four. The photometric and radiometric principles are covered in Chapter Five. Finally, Chapter Six considers detectors. There might be some overlapping of topics covered in different chapters, but this is desirable, since the points of view of different authors, treating different subjects, may be quite instructive and useful for a better understanding of the material. This book has been the result of the efforts of many people. Professor H. W. Palmer started this project and spent many fruitful hours on it. Unfortunately, he did not have the time to finish his editorial work due to previous important commitments. I would like to express my great appreciation of and thanks to Professor Palmer and all of the authors, without whom this book could never have been finished. I also thank Dr. R. E. Hopkins and many friends and colleagues for their help and encouragement. Finally, I appreciate the great understanding of my family, mainly my wife Isabel, for the many hours taken away from them during the preparation of these books. DANIEL MALACARA Leon, Gto. Mexico.

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METH0DS OF EXPERIMENTAL PHYSICS Editors-in-Chief

Robert Celotta and Judah Levine Volume 1. Classical Methods Edited by lmmanuel Estermann Volume 2. Electronic Methods, Second Edition (in two parts) Edited by E. Bleuler and R. 0. Haxby Volume 3. Molecular Physics, Second Edition (in two parts) Edited by Dudley Williams Volume 4. Atomic and Electron Physics-Part A: Atomic Sources and Detectors; Part B: Free Atoms Edited by Vernon W. Hughes and Howard L. Schultz Volume 5. Nuclear Physics (in two parts) Edited by Luke C. L. Yuan and Chien-Shiung W u A: Preparation, Structure, Volume 6. Solid State Physics-Part Mechanical and Thermal Properties; Part B: Electrical, Magnetic, and Optical Properties Edited by K. Lark-Horovitz and Vivian A. Johnson Volume 7. Atomic and Electron Physics-Atomic Parts) Edited by Benjamin Bederson and Wade L. Fite

Interactions (in two

Volume 8. Problems and Solutions for Students Edited by L. Marton and W. F. Hornyak Volume 9. Plasma Physics (in two parts) Edited by Hans R. Griem and Ralph H. Lovberg Volume 10. Physical Principles of Far-Infrared Radiation By L. C. Robinson Volume 11. Solid State Physics Edited by R. V. Coleman

xiv

M E T H O D S OF EXPERIMENTAL PHYSICS

Volume 12. Astrophysics-Part A: Optical and Infrared Astronomy Edited by N. Carleton Part B: Radio Telescopes; Part C: Radio Observations Edited by M. L. Meeks Volume 13. Spectroscopy (in two parts) Edited by Dudley Williams Volume 14. Vacuum Physics and Technology Edited by G. L. Weissler and R. W. Carlson Volume 15. Quantum Electronics (in two parts) Edited by C. L. Tang Volume 16. Polymers-Part A: Molecular Structure and Dynamics; Part B: Crystal Structure and Morphology; Part C: Physical Properties Edited by R. A. Fava Volume 17. Accelerators in Atomic Physics Edited by P. Richard Volume 18. Fluid Dynamics (in two parts) Edited by R. J. Emrich Volume 19. Ultrasonics Edited by Peter D. Edmonds Volume 20. Biophysics Edited by Gerald Ehrenstein and Harold Lecar Volume 21. Solid State: Nuclear Methods Edited by J. N. Mundy, S. J. Rothman, M . J. Fluss, and L. C. Smedskjaer Volume 22. Solid State Physics: Surfaces Edited by Robert L. Park and Max G. Lagally Volume 23. Neutron Scattering (in three parts) Edited by K. Skold and D. L. Price Volume 24. Geophysics-Part A: Laboratory Measurements; Part B: Field Measurements Edited by C. G. Sammis and T. L. Henyey Volume 25. Geometrical and Instrumental Optics Edited by Daniel Malacara Volume 26. Physical Optics and Light Measurements Edited by Daniel Malacara

1. INTERFERENCE Daniel Malacara Centro de lnvestigaciones en Optica A.C. Apdo. Postal 948 37000 Leon, Gto. Mexico.

1 .l.Introduction The luminous phenomenon called interference is a direct consequence of the wave nature of light. Using the interference of light, we can make interferometers, which are instruments that use this phenomenon to measure very accurately many physical parameters. The general subject of interference has been treated extensively in many classical textbooks on optics like those by Born and Wolf,’ Cook,’ Franqoq3 Candler,4 Steel,’ and Tolansky.6 There are also special chapters on the subject of interference in many advanced books like those by Baird,’ Baird and Hanes,’ and Dyson: and others. This chapter describes very briefly the interference phenomenon and some interferometers. Special emphasis is placed on the applications of these useful instruments, which have played a very important role in the development of physics due to their extremely high accuracy. 1.1 .l. Methods to Obtain Interference Fringes

To obtain interference fringes, the phases of the two interfering waves must be synchronized, that is, they must be coherent. Before the advent of lasers, this was possible only if both waves originated from the same light source. In order to produce two waves from a single source, we must have either a division of the wave front or division of its amplitude. Division of Wave Front. This class of interference is produced when the two interfering wave fronts are taken from different portions of the original wave front. The typical examples are Young’s experiment, the Fresnel biprism, and Lloyd’s mirror, but there are many others. Young’s double-slit experiment is performed as shown in Fig. 1. The lenses are not strictly necessary, but their addition makes the theory easier. The light source S must be either a very narrow slit or a point, and lens L collimates the light to obtain an approximately flat wave front. Normally, 1 M E T H O D S OF E X P E R I M E N T A L PHYSICS Val. 26

Copyright 0 1988 by Academic Pres,. Inc. All rights of reproduction in any form reserved ISBN 0-12-475971-8

2

INTERFERENCE

screen

1.0-.8-

(b)

6.4 -

.2-

FIG. 1. Young's experiment. (a) Experimental arrangement and (b) Diffraction pattern.

the interference pattern would be observed at infinity, but lens L serves to bring the pattern closer over the screen. It can be found in almost any good textbook on optics' that the interference pattern is given by 7T-1

I

= I,,,

cos'{

A sin e ]

sin{ ?sin 7TD

e] 9

(1.1)

sin 0

A

where I,,, is the irradiance at the center of the pattern, and 8 is the angular deviation from the optical axis. This radiation pattern also appears with

3

INTRODUCTION

4

----

&l--

screen

s* (a)

SO"

rce

FIG.2. ( a ) Lloyd's mirror and (b) Fresnel biprism.

dipole antennas and radio waves. The total width of the central maximum is given by sin 0 = A l l . The Fresnel biprism and Lloyd's mirror also produce interference fringes by division of the wave front, as shown in Fig. 2. If two waves are to interfere producing fringes with good contrast, the polarization states of both waves must be the same. This condition is always satisfied in the Fresnel biprism. However, in Lloyd's system, only one beam is reflected. Therefore, the reflection coefficients and the phase shifts under reflection must remain nearly constant over the range of incident angles used. This is possible only near grazing incidence.

4

INTERFERENCE

A second reason for using grazing incidence in the Lloyd system is that the spacing between fringes decreases rapidly as the separation between the virtual sources S, and S2 is increased. The sources must be quite close to each other to make fringes visible, and this is possible only near grazing incidence. If the screen in the Lloyd system is placed near the edge of the mirror, we can observe that a dark fringe appears at this edge. This happens because there is a phase shift upon reflection with grazing incidence. Wolfe and Eisen" have studied the coherence requirements in Lloyd's mirrors. Division of Amplitude. This class of interference occurs when both interfering beams are obtained by division of the amplitude of the original wave front by means of a partially reflecting optical surface. Then, both beams travel different paths, and interference occurs when they are recombined. Typical examples are Newton rings and the Michelson interferometer described in the next section. 1.1.2. Classification of Interference Fringes

A classification for interference fringes can be made according to the way they are observed, namely, fringes of equal thickness and fringes of equal inclination. Fringes of Equal Thickness. Each fringe represents the locus of all points in which two optical surfaces (or wave fronts) have a constant separation. This is much better understood by means of the following example. Consider the optical arrangement in Fig. 3 where the flat or convex surface of one lens is placed against the flat surface of another lens. Monochromatic light enters the first lens and impinges upon the second surface from which

observing eye

observed

FIG. 3. Arrangement to observe Newton fringes.

fringes

INTRODUCTION

5

some of the light is reflected. Some of the light that goes through the first lens is reflected from the upper flat surface of the second lens. The two reflected beams interfere constructively when the phase difference is an integral multiple of 27~.We can see that the optical path difference is twice the surface separation. If one of the surfaces is spherical, each fringe is a ring that represents the locus of points with equal surface separation. These circular fringes, also called Newton rings, are a particular case of fringes of equal thickness. Fringes of Equal Inclination. I n this case, each fringe is the locus of points in the field of view with the same angle of the incidence 8 at the interferometer. As an example, let us consider two parallel reflecting surfaces as in Fig. 4. As in the previous example, the two interfering beams are produced by division of amplitude. Fringes of equal thickness cannot be formed because the optical path difference (OPD) is the same for the entire field. One way to change the phase difference and hence to observe fringes is to introduce a range of angles of incidence by using an extended light source. Circular fringes are observed with an angular radius 8 given by mA cos 8 =-) 2d

where rn is an integer smaller than or equal to 2 d / A .

FIG. 4. Arrangement to produce fringes of equal inclination.

6

INTERFERENCE

To directly observe these fringes, the lens and the screen may be replaced by the unaided eye (focused at infinity). Then, the observation must be made almost perpendicularly to the optical surfaces, and the eye must be placed very close to them, because of the small diameter of the pupil of the eye. If the observing distance d between the surfaces is so large that the fringes cannot be resolved with the naked eye, a telescope must be used. 1 .I .3. Light Sources for Interferometers

If two light waves are to interfere, they must have their phase synchronized. Then, the two waves are said to be coherent to each other. When the two interfering waves are taken from the same light source, either by division of the amplitude or by division of the wave front, this condition is satisfied under certain conditions that we will describe. Let us first assume that the light source is perfectly monochromatic, but that the light source is extended. Each point of the light source may produce a complete interference pattern. Good fringe contrast is obtained only if the phase difference for both waves is the same for all points over the extended light source. If the above condition is not satisfied, the light source has to be very small. In that case, it is said that the interferometer has to be illuminated by a spatially coherent beam. If the condition is satisfied, the light source can be extended, and the interferometer is said to be source-size compensated. Let us now assume that the illuminating beam is spatially coherent or that the interferometer is source-size compensated. The light, however, may not be perfectly monochromatic. Then, each wavelength component of the light produces an interference pattern. Good fringe contrast is obtained only if all patterns produced by the interference between both light beams are the same for all wavelengths. In general, this condition can be achieved only if this OPD is zero for all wavelengths, the interferometer is then said to be white-light compensated. Otherwise, the light has to have a narrow bandwidth whose maximum permissible value depends on the particular interferometer. A light beam with a narrow bandwidth is said to have light-temporal coherence. Very few interferometers are white-light compensated, and, therefore, their light sources have, in general, to be quite monochromatic. The monochromaticity of a light beam can be measured either by its bandwidth or by its coherence length, which is the maximum optical path difference that can be introduced before the fringe contrast is completely lost. The coherence length of light sources varies from a fraction of a millimeter to as much as one meter for the KrR6standard lamp (Baird and

TWO-REAM INTERFEROMETERS

7

Howlett”), or even several hundreds of meters for a single-frequency gas laser.

1.2. Two-Beam Interferometers Another possible classification for interferometers takes into account the number of light beams that form the interference pattern. Then, we may have either two-beam o r many-beam interferometers. In this section, the most common two-beam interferometers will be briefly described.” 1.2.1. The Newton Interferometer

We call a Newton interferometer any arrangement of two surfaces in contact with, and illuminated by, a monochromatic source of light, as shown in Fig. 5 . This interferometer has been described in detail by Murty.” monochromatic source of light

4

in contact

FIG.S . Newton interferometer.

8

INTERFERENCE

This instrument forms equal-thickness fringes; and a bright fringe is formed whenever the phase difference is equal to an integral multiple of 2 7 ~If . we take into account the fact that there is a phase shift under reflection for one of the two beams, we can say that a dark fringe occurs when

2nd

= mA,

(1.3)

where m is an integer, d is the thickness of the gap between the two reflecting surfaces, and n is the refractive index of the gap medium. This expression assumes that the fringes are observed normally to the reflecting surfaces, and hence the two following rules should be followed to obtain good accuracy: (a) If the surfaces are flat, a collimating lens is required in order to observe the fringes from the focus ofthis lens. Another acceptable alternative is to use a large extended source of light and then to observe the fringes from a minimum distance of about five times the diameter of the flat surface under test. (b) If the surfaces are spherical, the observation point should effectively be placed at the center of the curvature, as shown in Fig. 6. Figure 7 shows some typical defects that may be observed with this interferometer. 1.2.2. The Fizeau Interferometer

This instrument, shown in Fig. 8, has also been described in detail by Murty.” Equal-thickness fringes are formed by the two reflected beams. The lens is used to collimate the light and to optically place the observing eye at infinity. The optical path difference between the two beams is

OPD = 2nd,

( 1.4)

where d is the thickness of the gap, and n is its refractive index. If the field is free of fringes, we infer that the product nd is constant over the gap. We can conclude that d is a constant only if n is assumed to be constant. If straight and parallel fringes with separation S are observed, the two faces forming the gap are flat and with an angle 8 in radians between them, given by

e=-.

A

2 nS

This makes this interferometer quite useful for measuring small wedges in glass plates.

9

TWO- BEAM INTERFEROMETERS

k

I

surface

observation point

(a 1

under t e s t

observat Ion polnt

k

't2

(b)

urf ace under test

F I G .6. Testing of spherical surfaces with Newton interferometer.

1.2.3. The Michelson Interferometer

Probably the most famous of all interferometers was invented by Michelson for his well-known experiment. Figure 9 shows this interferometer. An extended light source sends the light to the beam splitter P which divides the beam in two, with one going to mirror MI and the other going to mirror Mz. Both mirrors send the beams back to the beams splitter and from there to the observing eye. When the mirrors M , and M2 are exactly perpendicular to their optical axes, the fringes are of the equal-inclination type. In this case, the fringes

10

INTERFERENCE

FIG. 7. Typical interference patterns with Newton interferometer.

are circles with angular radii given by Eq. (1.2), where d is given by

d = -OPD, 2 ’ and OPD, is the optical path differences along the optical axis.

11

TWO-BEAM INTERFEROMETERS

t-

c

light source

FIG.

8. Fizeau interferometer.

The interfering beam coming from mirror MI traverses the beam splitter three times, while the beam from mirror M 2 passes through the plane only once. To equalize the thickness of glass that each beam traverses, a plate P2 of the same thickness and orientation as P, is inserted into the beam to M 2 . This extra plate is called a compensating plate, and the interferometer containing one is said to be compensated. When the interferometry is not compensated, only one beam goes through the inclined beam splitter, and thus the optical path difference for a given value of 8 is different in different planes. In that case, the Haidinger fringes are not circular but elliptical in shape, as described by Guild.I3

light source

4observed pattern

F i c . 9. Michelson interferometer.

12

INTERFERENCE

Fic;. 10. Three kinds of Michelson fringes.

The fringes have good contrast only if the OPD, is smaller than the coherence length. Therefore, the fringes are observed with white light only if the variation of the OPD within the spectral region being observed becomes smaller than about A/4. Since the index of refraction is a function of the wavelength, the former condition cannot be satisfied unless the total amount of the glass traversed by both beams is the same, and then the OPD, is adjusted to be smaller than a few wavelengths. Hence, white-light fringes can be observed only in a compensated interferometer. If the mirrors MI and M 2 are not perpendicular to their optical axes, equal-thickness fringes will be observed instead of equal inclination fringes. However, they are exactly equal-thickness fringes only if the observing eye is very far from the interferometer, or a lens is used as in Fig. 9 to optically place the eye at an infinite distance from the interferometer. If the eye is close to the interferometer and if the mirrors are not perpendicular to their optical axes, intermediate fringe shapes are observed. These fringes are curved, with the convexity directed toward the narrower part of the wedge formed by the two mirror images. The three kinds of Michelson fringes are shown in Fig. 10. The extended light source permits the fringes to be observed only at a well-defined plane near the virtual images of the light source. These fringes are said to be “localized.”

1.2.4. Modifications of t h e Michelson Interferometer

Several modifications have been made of the Michelson interferometer. The most important ones are the Mach-Zehnder, Jamin, and TwymanGreen interferometers.

TWO-BEAM INTERFEROMETERS

13

r-i

I

t

1

I 1 sample 1 I I 1

FIG. 11. Jamin interferometer.

The Jamin interferometer is shown in Fig. 11. The light from the light source is split at the surfaces of a thick glass plate. Then, the two beams are recombined at a second identical glass plate. For some applications, the advantage of this arrangement over the Michelson interferometer is that the light passes through the sample only once. The plates have a large thickness so that enough room is left for the sample. This interferometer was used for refractometry, but it has now been replaced by other kinds of interferometers. The Mach-Zehnder interferometer, shown in Fig. 12, can be considered as a Jamin interferometer in which the four reflecting surfaces have been replaced by four plates. This instrument is generally used to detect small changes or space variations in the refractive index of the samples. Both, the Jamin and the Mach-Zehnder interferometers, are compensated if correctly adjusted, so that white-light sources can be used if desired. However, it must be pointed out that this adjustment is particularly difficult in the case of the Mach-Zehnder. The Twyman-Green interferometer14*" is one of the most useful modifications of the Michelson interferometer, used mainly to test the quality of optical components. The system is illuminated with a collimated

14

INTERFERENCE

,attern 2

9 p2

pattern I

2

+

3

\

Q

F I G . 12. Mach-Zehnder interferometer

beam of light, as shown in Fig. 13. The fringes that are observed by placing the eye at the focus of lens L are of equal-thickness. The Twyman-Green interferometer is not generally compensated, and large values of the OPD are frequently obtained, so that the light source

Y F I G . 13. Twyman-Green interferometer.

15

TWO-BEAM INTERFEROMETERS

--(C)

FIG. 14. Testing optical components with a Twyman-Green interferometer.

mu t be monochromatic. Many different kinds of optical components ca be tested with this interferometer as described by Malacara.I6 The simplest one to be tested is a plate of glass with plane parallel faces, which is inserted into one of the beams as shown in Fig. 14(a). The optical path difference introduced by the presence of the glass plate may be found considering that in the space with thickness d occupied by the plate, the index of refraction is changed from 1 (without the plate) to n (with the plate). A factor of 2 appears because the light passes twice through the plate. Thus,

O P D = 2 ( n - 1)d.

(1.7)

If the interferometer is adjusted so that no fringes are observed before introducing the plate into the beam, and so that fringes are observed after the plate is inserted, we establish that the quantity ( n - l ) d is a constant throughout the plate. Assuming that the index n does not vary over the plate, we may conclude that d is a constant. The variations in d and in n can be obtained independently only if another measurement is made with a Fizeau interferometer.

16

INTERFERENCE

If the plate faces are flat but not parallel and n is constant, straight and parallel fringes appear. Using the arrangements in Figs. 14(b) and 14(c), the quality of a lens or photographic objective can be tested. It is assumed in this test that the convex or concave mirrors are perfectly spherical. If the lenses are assumed to be perfect, the quality of the spherical mirror is tested. Interference patterns associated with various lens aberrations are shown in Fig. 15. Many

FIG. 15. Some Twyman-Green interferograrns.

TWO-BEAM INTERFEROMETERS

17

other types of optical elements, such as prisms and diffraction gratings, can also be tested with this interferometer. 1.2.5. Shearing Interferometers

These are a class of interferometers in which a perfect reference beam is not needed. The reference wave front is the same as the one under test but with a different orientation, position, or lateral extension. Basically, there are four types of shearing interferometers as illustrated in Fig. 16. Lateral shearing interferometers are probably the most popular of the four and have been extensively described by Murty.” Radial, rotational, and reversal shearing interferometers have also been described in detail by Malacara” and Bryngdahl.” The lateral shearing interferometer measures the wave-front deformation by comparing the wave front with a laterally displaced reproduction of itself, thus avoiding the need for a flat reference wave front. Probably, the first interferometer of this kind was designed by Bates.*’ This is basically a Mach-Zehnder interferometer used with a convergent wave front and with the plate P2 tilted to produce two laterally displaced

I ate ra I

rotational shear

radial

reversal shear

FIG.16. Four types of shearing interferometer operations.

18

INTERFERENCE h

front sheared wavef ront s microscope objective

gas LASER

rl

lens

under test

FIG. 17. Two lateral shearing interferometers.

wave fronts as shown in Fig. 17(a). The lens under test is placed between the light source and the interferometer. Murty” has proposed a simpler version of a lateral shearing interferometer in which the large temporal coherence of laser light permits large values for the OPD [see Fig. 17(b)]. Lateral shearing interferometers provide rapid and easy identification of lens aberrations as shown in Fig. 18. Given an interferogram, the wave front that produced it can be computed by using several methods, such as those described by Saunders,” M a l a ~ a r a , ’and ~ Rimmer and W ~ a n t . * ~ Many radial shear interferometers have been designed, but the first one was invented by Brown.25Basically, this is a Jamin interferometer that works in convergent light and has a small meniscus lens in one of the beams. To compensate the interferometer, a small parallel plate is placed in the other beam [see Fig. 19(a)]. Another interesting radial shear interferometer, shown in Fig. 19(b), was designed by M ~ r t y . ~ ~ Radial shear interferometers are sensitive to all types of aberrations, as the Twyman-Green interferometers, but to a lesser degree.

TWO-REAM I N T E R F E R O M E T E R S

19

FIG. 18. Lateral shearing interferometer patterns.

Rotational shear interferometers are not sensitive to rotationally symmetric aberrations like spherical aberration but are quite useful for the detection of nonrotationally symmetric aberration, like coma or astigmatism. These interferometers have been described by Murty and Hagerott2’ (see Fig. 20). Reversal shear interferometers are insensitive to rotationally symmetric aberrations and also to astigmatism if the axis of reversion coincides with the x or y axes. It is quite sensitive, however, to coma. Common examples of this kind of interferometers are the Koester prism interferometers shown in Fig. 21, and described by Gates28 and Saunders.” 1.2.6. Common-Path Interferometers

Where the reference and test beams follow separated paths in an interferometer, they can be differently affected by vibrations, turbulence, and

20

INTERFERENCE

RADIAL L I SHEARED WAVEFRON rs

BROWN'S

INTERFEROMETER

WAVEFRON r

UNDER T E S r I

.

."&

.^CC."

EYlSPHERlCAL

AM

SPLITTER

MURTY'S INTERFEROMETER

FIG. 19. Brown's and Murty's radial shear interferometers.

temperature variations. An interferometer in which both beams follow the same path is called a common-path interferometer. They have been extensively treated by M a l l i ~ k . ~ ' Common-path interferometers have zero optical path difference for all wavelengths and can therefore be used with white light. They are also very stable. Most of these instruments are either of the lateral shearing or radial shearing types. A large radial shearing is obtained when one of the beams is made to traverse a small area of the system under test, while the other beam traverses the whole aperture. The most popular instrument of this kind is the so called scatter-plate or Burch interfer~meter,~' illustrated in Fig. 22. After passing the light through the first scatter plate, there will be two beams: a diffracted one, which we

TWO-BEAM INTERFEROMETERS

21

FIG. 20. Murty and Hagerott's rotational shear interferometers.

FIG. 21. Koester's prism interferometers. (a) Gates interferometer and ( b ) Saunders interferometer.

22

INTERFERENCE

FIG. 22. Burch’s scatter plate interferometer.

call DF, and an undiffracted one (zero order), which we call DO. The undiffracted beam DO covers only a small area near the center of the surface under test, while the diffracted beam covers the whole surface. The first scatter plate is imaged onto the second plate, which is a mirror image of the first. The beam DO produces two beams upon passing through the second plate, one again undiffracted (DOO) and one diffracted for the first time (DOF). The beam D F also originates two beams, one undiffracted (DFO) and one diffracted a second time (DFF). The beam DO0 is observed as a small bright spot near the center of the mirror. The beam D F F is very faint and unobserved. The beams DOF and DFO form a fringe pattern. The pattern is sometimes interpreted as a radial shear interferogram with a very large amount of radial shear; however, it is best to think of it as a Twyman-Green pattern. There i s a large variation in common-path interferometers, many of them using birefringent materials, as described by FranGon’ and Mallick.”’ 1.2.7. Other Two-Beam Interferometers

There are many other two-beam interferometer designs as, for example, those that involve division of the wave front rather than division of the amplitude. Two very important instruments of this kind are the Rayleigh and the Michelson stellar interferometers. M i ~ h e l s o nwas ~ ~the first to apply interference phenomena to the measurement of refractive indices. Later, Lord Rayleigh designed an interferometer for this purpose that has been

23

TWO-BEAM INTERFEROMETERS

fringes

FIcj. 23. Rayleigh interferometer.

well described by Candler.4 It is very useful for the measurement of refractive indices of gases or liquids (see Fig. 23). The light from a narrow slit is collimated by means of a lens, it then goes through two glass tubes containing two gases whose indices are to be compared. The beams are then recombined by another lens in order to form a set of interference fringes. The working principle is the same as that of Young's double-slit interferometer. When the gases or liquids in the two tubes have different refractive indices, the fringes are laterally displaced. The Rayleigh interferometer (or refractometer) has more accuracy in the measurements of small refractive indices variations than any other instrument. It can detect changes in the refractive index of the order of lo-', or one unit in the sixth decimal place. The Michelson stellar interferometer ( Michelson6") is also based on the Young interferometer, used in order to measure the angular diameter of stars. The measurement is indirectly made by measuring the degree of spatial coherence of the incoming light. This interferometer is illustrated in Fig. 24.

f l a t mirrors

s c r e e n or d

photographic plate --La"

'IrnI" flat mlrrors

F I G . 24. Michelson stellar interferometer.

24

INTERFERENCE

The light received from opposite sides a and b of the star disc has an angular separation 8. The optical path difference from point a assumed to be on the axis is represented by OPD,. Thus, the optical path difference from point b, off-axis, is given by OPD,,

= OPD,

-

ed,

(1.8)

where d is the separation between the two small mirrors receiving the light from the star. The irradiances produced by the two sources at points a and b can be obtained from Eq. (1.1) as I, =21[1 +cos(koOPD,)], and Ih

= 21[ 1

+ COS( koOPDh)].

(1.9)

Since the two point sources are not coherent with respect to each other, the total irradiance on the pattern is 1, = 1,

+Ib

=41

+ ~ I [ c okoOPD,) s( + cos ko(OPD, - Od )].

(1.10)

The fringe visibility versus the slit separation is plotted in Fig. 25. The angular diameter of the star may be calculated from the measurement of the distance d, which produces the first minimum in the visibility. As an example, the star Betelgeuse has an angular diameter of about 0.05 sec of arc, which lets d equal about three meters for green light. A modern version of the Michelson stellar interferometer is the irradiance interferometer in which the light from each slit is not made to interfere. Instead two irradiance detectors are used, and then the irradiance variations in the two signals are electronically correlated. This instrument has a

t"

t"

1.22 X/8

X./28

01 Two point source8

b) C l r c u l a r

rxtrndrd

FIG.25. Fringe visibility in Michelson stellar interferometer.

SOUICI

M ULTIPLE-BEAM I N T E R F E R O M E T E R S

25

quantum theory explanation, after an experiment devised by Brown, Hanbury, and Twiss.3’

1.3. Multiple-Beam Interferometers Some interferometers form their interference patterns with the superposition of more than two wave fronts. These interferometers have many advantages, as we shall see in this section. 1.3.1. The Fabry-Perot Interferometer

Multiple-reflection interferometers have been described in detail by Royc h ~ u d h u r and i ~ ~ Polster er a/.” The most popular of this class of interferometers is the F a b r y - P e r ~ t .The ~ ~ irradiance pattern of a two-beam interferometer is sinusoidal, and therefore the exact position of a fringe maximum cannot visually be located with an accuracy greater than about one tenth of the fringe spacing. The Fabry-Perot interferometer produces extremely narrow equal-inclination fringes, and thus the fringe maxima can be located with an accuracy better than one hundredth of a wavelength. The principle involves the use of multiple reflections, so that interference is produced by an almost infinite number of beams as shown in Fig. 26.

Sl FIG.

s2

26. Multiple reflection in a Fdbry-Perot interferometer.

26

INTERFERENCE

(b) FIG. 27. Fabry-Perot configurations.

The two parallel surfaces may be in two plates as in Fig. 27(a), or in one plate as in Fig. 27(b). Such plates are sometimes called etalons, from the French word for "standard." The transmitted irradiance 1 ( 6 ) as a function of the phase difference, assuming the amplitude transmission and reflection coefficients in Fig. 26, may be given by (1.11)

where it has been assumed that rl and rz are equal, in other words, that the phase shift upon reflection is either 0" or 180". Here, I is the incident irradiance, the asterisk denotes the complex conjugate, and the phase difference is given by

477

6 =-

A

nd cos 0 2 ,

(1.12)

where d is the separation between the reflecting surfaces, n is the refractive index of the material between the reflecting plates, and O2 is the angle of incidence of the rays inside the gap. If the reflectances R , and R2 and the transmittances T, and Tz are always measured in air, not inside of the glass plates as sometimes defined, Eq.

MULTIPLE-BEAM I N T E R F E R O M E T E R S

27

( 1 . 1 1 ) reduces to

I ( 6 ) = I”

T , T2 6’ ( 1 - R,)2+4R, sin22

(1.13)

as shown by Ba~meister,~’where the average reflectance R, is defined as (R~R~)~’~. This expression can be generalized to include the case in which there are phase shifts E , and E~ upon reflection as measured in the medium of the spacer (shown by Baumeister and Jenkins3’) by writing Eq. (1.12) as 47T 6 = - nd cos O2 -$( A

+

E ~ ) .

(1.14)

As pointed out before, the great advantage of multiple-reflection over two-beam interferometers is the increased sharpness of the fringes. The width of the fringes depends very much on the reflectances of the plates as illustrated in Fig. 28. If we define the half-width A6 of the fringe as the width at half the maximum fringe irradiance, it is given by

A6=2-

1-R, R;I2

(1.15) ’

The “finesse” N R is a number used to specify the sharpness of the fringes. It is defined as the ratio of the fringe spacing to the width in Eq. (1.15). Thus, (1.16) As shown in Fig. 28, the finesse can be extremely large even for relatively small values of R,.

FIG. 28. Fringe profile in a Fabry-Perot interferometer.

28

INTERFERENCE

As pointed out by Baird and Hanes,' with a Fabry-Perot interferometer we can observe fringes of equal inclination, fringes of equal thickness, and fringes of equal chromatic order, which will be described in this section. Fringes of equal inclination are formed when the light source is extended or convergent, so that a wide range of angles of incidence are present and the two plates are parallel to each other. The fringes are observed at an infinite plane by means of a lens with a screen at its focus or a small telescope focused at infinity. The fringes are circles located at positions such that

OPD = 2nd cosz 8 = mA,

(1.17)

where m is an integer. The smallest circular fringe corresponds to the largest value of m, given by 2 nd m,,,=-- A

e,

(1.18)

where e is the fractional value of ( 2 n d l A ) . Thus, the value of m for the pth fringe is (1.19) hence the angular radius of this fringe is given by cos

A

e2= 1 --2 nd ( p - 1 + e ) .

( 1.20)

Two rings with different wavelengths may have the same angular diameters, because their value of m may also be different. However, this ambiguity is eliminated by measuring two sets of rings with different values of d. Fringes of equal thickness may also be formed with a Fabry-Perot interferometer by forming a small wedge with the two reflecting surfaces. In this case, an extended source is not needed, and the light is normally collimated by a lens. The instrument may now be called a multiple-reflection Fizeau interferometer. Fringes of equal chromatic order appear when a white light source is used to illuminate the interferometer adjusted with parallel faces, and then the outgoing light is seen through a spectrograph. Narrow bright bands are seen in the spectrum. This effect is called a channeled spectrum, and the fringes are called fringes of equal chromatic order or Edser- Butler fringes. Tomkins and Fred39 have used these fringes to calibrate a spectrograph by using the fact that they are located at positions A N = 2 n d / ( N - 4 / 2 ~ ) ,

MULTIPLE-BEAM INTERFEROMETERS

29

where N is an integer, and C#I is the total phase change on the two internal reflections. T ~ l a n s k y ~ has ~ . ~also * ~used ' these fringes to measure the microtopography of crystal surfaces and films. 1.3.2. The Spherical Fabry-Perot Interferometer

The optical path diff erence in a Fabry-Perot interferometer changes with the angle of incidence. This property is not desirable in photoelectric spectrometry where a large solid angle of acceptance of the light is required together with a small variation in the optical path difference. In this technique, the light spectrum is analyzed not by changing the angle of incidence to form rings but by changing the plates' separation by a small amount, usually of the order of a wavelength or less. An interferometer in which the optical path difference does not change with small variations in the angle of incidence was designed by C ~ n n e s . ~ * It consists of two spherical confocal surfaces as shown in Fig. 29. The two spherical surfaces have their focal planes coincident at some plane between them. If the two mirrors have the same radius of curvature, the center of curvature of each mirror is located on the opposite mirror. An incident ray reaching mirror M , at point A travels to point B, where it is partially reflected. The reflected ray then follows path BDCAB, so that it comes back to the original trajectory AB at A, where it is again partially reflected. The optical path difference between two consecutive rays is

OPD = 4t,

(1.21)

where r is the separation between the vertices of the two mirrors. The lower half of each mirror is totally reflecting, while the upper half is partially reflecting. We can see that within the limits of the paraxial optics (small angles of incidence), the phase difference is independent of the angle of incidence.

/

\

FIG. 29. Connes's confocal interferometer.

30

INTERFERENCE

We should also notice that the emerging rays are all coincident along the incident ray and not the parallel and laterally separated rays as in the Fabry-Perot interferometer. To keep the rays within the limits of the paraxial optics, two small circular diaphragms are frequently used in front of the mirrors. The only adjustment that this interferometer needs is to fix the correct distance between the mirrors with a tolerance of a few microns. Since the mirrors are spherical, there is no need to adjust their parallelism. The main use of this instrument is to examine the spectrum of lasers, as described by Herriott4' and Fork et al.,44by oscillating one of the mirrors a small distance (usually about h / 4 by means of an electromechanical device). 1.3.3. Other Multiple-Reflection Interferometers

In some other interferometers, the principle of multiple reflection is also used in order to increase the accuracy of the instrument. These interferometers are not used much and are mainly of historical interest. An example is the Lummer-Gehrke plate4 shown in Fig. 30(a): It may be

FIG. 30. Other multiple-beam interferometers. ( a ) Lummer-Gehrke plate and ( b ) Echelon.

M ULTIPLE-BEAM INTERFEROMETERS

31

considered as a Fabry-Perot etalon using a very large angle of incidence. The angle of incidence of the light inside the plate is very near the critical angle, so that a large reflectivity is obtained without any kind of coating on the surfaces. Light enters the system through a small prism cemented onto one end of the plate. This system was very much used4’ at the beginning of the century for spectrographic analysis, but is has now been largely replaced by the Fabry-Perot interferometer. The e ~ h e l o n ~ ~of. ~Michelson .~’ [Fig. 30(b)] is another multiple-beam device, formed by a stack of flat parallel plates with constant thickness t. The plates are arranged so that steps with approximate height of 1 cm and width of 0.1 cm are formed. In the transmission echelon, the wave front is divided in M portions with the relative optical path difference as a coarse grating, blazed for a very high order. A complete description of transmission and reflection echelons is found in the book by Candler.4 1.3.4. Interference Filters

A common application of the Fabry-Perot interferometer is to isolate narrow spectral regions. Used in this way (Fig. 31), it is called an interference filter. It should be used only with collimated light. The wavelength transmitted by the filter is given by

A=

2nd cos e m

(1.22)

If a filter is to isolate a line with wavelength A, a very low order of interference, m about 2, is chosen so that there is a large wavelength difference between the different orders. The undesired orders are then eliminated by means of any broad-band absorptive filter. An interference filter can be made by evaporating a thin reflective film, like silver or aluminum, on a piece of glass. Then a transparent film, like magnesium fluoride, is evaporated on the metal. On this layer, another film of metal is evaporated to complete the films. It is interesting to point out that the glass substrate does not have to be extremely flat, since the important thing is only the constant thickness of the evaporated spacer.

-glass m -c-i-elatl

coating dielectric spacer -glass

Fic;. 31. Interference filter.

32

INTERFERENCE

ANGLE OF INCIDENCE

F l c . 32. Wavelength change with orientation in an interference filter.

An interference filter may be slightly tunned by changing the filter orientation with respect to the light beam. Keeping in mind that 8 is the angle inside the spacer with refractive index n, and 4 is the angle of incidence outside the filter, we can see that ( 1.23)

where A, and A. are the wavelengths for an angle of incidence 4 and for normal incidence, respectively. Figure 32 shows a plot of (Ao- & ) / A o versus 4. A consequence of this angle sensitivity is that the narrower the passband, the more accurate the angle of incidence and collimation have to be. Half-widths of interference filters can have a wide range of values, between about 80 A and about 1 A. In order to reduce the passband width (or half-width), interference filters are often made with more than one cavity, that is, with several spacers with reflecting films made of dielectric multilayers instead of metal. 1.3.5. Thin Films A multilayer film consists of thin layers of different dielectric materials deposited by evaporation, one on top of the other. Table I lists a few of the materials used. A good review of this subject was written by Ba~meister,~’ and a more complete description is given in Chapter 13 in this book. Due

33

M ULTI PLE-BEAM I N T E R F E R O M E T E R S

TABLE1.

Some Dielectric Materials for Thin Films

Formula

Name

Refractive index A = 550 nrn

Transparency range in pm

MgF, ZnS Ti,O, SiOz NalAIF,

Magnesium fluoride Zinc sulfide Titanium oxide Silicon dioxide Cryolite

1.38 2.3 2.3 1.46 1.35

0.13-8 0.39-14.5 0.39-12 0.2-9 0.2-14

to the interference produced by multiple reflections at the interfaces between layers, these filters are highly color selective in both transmission and reflection. There are many uses for these films, for example: (a) Antireflection coatings. A lens can be coated with one or more dielectrics for the purpose of minimizing reflected light, thereby enhancing the transmitted light. The flare around the image is reduced by eliminating the unwanted light that would otherwise come from reflections at the lens surfaces. (b) Enhancement of the reflection and/or protection of metal mirrors. (c) Beam splitters and semitransparent mirrors for interferometers and other kinds of optical instruments. For example, partially transmitting mirrors are used in lasers. These mirrors have an advantage over the metallic ones in that they do not absorb energy. (d) Band-pass filters. These filters reflect all the colors that are not transmitted, as opposed to the gelating or glass filters that absorb all the energy that is not transmitted. They are often called dichroic mirrors, but the term dichrom strictly applies to materials that absorb two polarizations differently. Among the uses for these filters are cold mirrors for movie projectors that do not reflect infrared but only visible light; color beam splitters for cold television cameras; and others. Single-layer Coatings. A single-layer coating deposited over a substrate can be used to reduce the reflectance of a dielectric surface or to enhance the reflectance of a metallic surface. Let us consider the case illustrated in Fig. 33. The minimum reflectivity is obtained when the optical thickness of the layer is h / 4 and its refractive index is - nl/2

2-

s

9

(1.24)

where n , is the substrate refractive index. This condition cannot be exactly

34

INTERFERENCE

surface surface 2

+

V/II

I coating Y ' ' H

I

I

I

glass

@

fl

substrate

@

f

N 2 I

,&

d

N,

FIG. 33. Single-layer coating.

satisfied with practical dielectrics. It may, however, be approached very closely, for example, by the magnesium fluoride. The reflectance of the system varies with the optical thickness of the coating as shown in Fig. 34. We can see that the maximum reflectance that can be obtained for any thickness is the reflectance value with no coating. In other words, the reflectance can be reduced, but never increased, when this coating is deposited on a dielectric surface. This result can be shown to be true whenever the coating has a refractive index smaller than that of the substrate. The minima have zero reflectance only when the condition in Eq. (1.24) is exactly satisfied. In order that the wavelength interval between the selected minimum at A,, and its adjacent maxima be as large as possible, the minimum optical thickness that produces a minimum is used, that is, Ao/4. Then, if R is the ,reflectance

uncoated

0

x/4

substrate

A/

2

3x/4

OpllCOl thickness N d

FIG. 34. Change in the reflectance of a single antireflecting layer with the optical thickness nd.

35

MULTIPLE-BEAM I N T E R F E R O M E T E R S

0

’0°

600

wavelength

700nm

FIG. 35. Reflectance for a single antireflecting film.

reflectance of the uncoated substrate, the final reflectance becomes

R

h).

= Ro cos2( .rrA 0

(1.25)

as shown graphically in Fig. 35. If the reflectance is to be increased rather than decreased, the index of refraction of the coating should be as high as possible with respect to that of the substrate. In this case, there is a phase change under reflection at one of the interfaces. Therefore, when increasing the optical thickness, the first maximum of the reflectance occurs at Ao/4 (see Fig. 36). Multilayer Films. Several films with different thickness and alternatively of high and low index of refraction can be superimposed by evaporation. By selecting the appropriate thickness and number of layers, it is possible to obtain almost any desired bandwidth, reflectance, and transmittance of the multilayer. A particularly interesting multilayer is the “quarter-wave stack.” As shown in Fig. 37, this multilayer consists of an even number of layers of alternatively low and high refractive indices. All layers have the same optical thickness Ao/4. Figure 38 shows the reflectance curve of a quarter-wave stack with four periods of high and low index layers.

36

INTERFERENCE

.3c

.2c

u

0

-" -

.I 0

(

4 1

I

I

so0

wavelength

600

FIG.36. Reflectance of a single reflecting layer.

FIG.37. A quarter-wave stack.

I

reflectance

F I G .38. Reflectance curve for a quarter-wave stack.

I

700nrn

37

M ULTI PLE-PASS I N T E R F E R O M E T E R S

-==--+I ,test

l

M

lens

3

FIG. 39. Double-pass Twyman-Green and Fizeau interferometers.

1.4. Multiple-Pass Interferometers These are interferometers in which at least one of the wave fronts traverses the normal trajectories more than once. The instruments, which have been described in detail by H a r i h a ~ - a nand ~ ~ by Langenbe~k,~’ have some advantages, as we shall see in this section. A double-pass interferometer is designed with the purpose of separating the symmetrical and the antisymmetrical parts of the wave aberration and to display them in separate interferograms. Figure 39 shows the double-pass Twyman-Green interferometer described by Hariharan and Sen.” The two beams going out from the interferometer are reflected back to it by a small mirror M 3 . Then, the patterns are observed by means of the beam splitter S 2 , where there are four wave fronts. By moving the light source slightly off-axis, two different patterns are observed. One pattern represents the symmetrical component and the other one the antisymmetrical component of the wavefront. Double-pass Fizeau interferometers have also been designed by Sen and Pu~~tambekar’”’~ and by Puntambekar and Sen” in order to reduce the coherence requirements of the light source. The multiple-pass principle can also be used in order to increase the sensitivity of an interferometer. Two examples are the multiple-pass Twyman-Green and Fizeau4’ interferometers shown in Fig. 40.

38

INTERFERENCE

aux i lio r y be a rn divider

test

surf ace

coated mirror

surface

FIG.40. Multiple-pass (a) Twyman-Green and (b) Fizeau interferometer.

1.5. Applications of Interferometry Interferometric techniques are very powerful tools in many different branches of science for very accurate measurements of a great number of parameters. A complete list of all applications of interferometry would be almost infinite,47~s4*ss~sh but a brief description of the main uses of these techniques is given in this section. 1.5.1. Relativity Measurements

One of the crucial experiments in modern physics was the MichelsonMorley e ~ p e r i m e n t , 'which ~ is now described in many modern textbooks of p h y s i ~ s . ~The ' experiment was performed with the famous Michelson interferometer in order to determine the frame of reference (the aether) with respect to which the light moves. It was assumed that the light had a constant speed with respect to the aether and that the earth moves in space,

APPLICATIONS OF INTERFEROMETRY

39

with the aether being stationary. Hence, if the interference pattern is observed with one arm of the interferometer being parallel to the earth's direction of movement, a small movement of the fringes is to be expected when the interferometer moves with the earth at 90" with respect to the former direction of movement. The negative result of the experiment led to many hypotheses trying to explain it,SXbut the satisfactory explanation came only with the special theory of relativity. Modern versions of this experiment have recently been performed with the same results.

1.5.2. S t a r Diameter Measurements

Michelson used his stellar interferometerh0in order to measure the angular diameter of stars (see Section 1.2.7). Although this instrument has been successful in measuring stellar diameters, it has many stability problems that limit its accuracy.

1.5.3. Refractometry

Interferometry can also be used to measure very small changes in the refractive index of a gas or to compare the refractive indices of two liquids. As mentioned before, Michelson3' applied interference for the first time for this purpose. The most commonly used instrument for this application is the Rayleigh interferometer4 described in Section 1.2.7. A Jamin or a Fabry-Perot interferometer can also be used to measure refractive indices.

1.5.4. Schlieren Techniques

The well-known Foucault or knife-edge test and many variations described under the general name of schlieren techniques can be considered to be interferometric. A good description of this test can be found in many books" and papers6' As shown in Fig. 41, the general idea behind this test is to detect small lateral displacements of rays focused to a point. This test can be applied to measure wave-front or surface deviations in spherical mirrors or lenses. Another application is to study the turbulence in a cell placed between the wave-front generator and the knife-edge. The Ronchi testh3 could be considered as a variation of the classical schlieren techniques.

40

INTERFERENCE

FIG.41. Foucault test.

1.5.5. Wave-front Topography

Almost all interferometers can be applied to the study of the topography of wave fronts or optical surfaces. An extensive bibliography on this subject is found in a paper by Malacara et ~ 1 . "Brief ~ reviews are given by Briers" and by Schulz and Schwider.66 An extensive study is presented in the book edited by M a l a ~ a r a , ~where ' even modern techniques such as fringe-scanning interferometers are described by Bruning."' It would be impossible to describe the numerous testing techniques in this chapter, and so the reader is referred to the specialized publications. 1.5.6. The Transfer Function of Lenses A way to specify the resolving power of a lens is by means of the optical transfer function (OTF). This function gives the contrast modulation of the phase shift for each spatial frequency in lines per unit length. There are a large number of ways to measure the optical transfer function of a lens, as explained by R o s e n h a ~ e r . "Some ~ of these methods are interferometric. The obvious way is to measure the wave-front deviations by means of any interferometer, as for example the Twyman-Green interferometer, and then to compute the transfer function. A more direct way is to use a lateral shearing interferometer and to make use of the fact that the transfer function can be mathematically obtained from a convolution of the wave front, as

APPLICATIONS OF INTERFEROMETRY

41

shown by hop kin^.^' Some lateral shearing interferometers made for this purpose are described by Fran~on.' 1.5.7. Length and Angle Measurements

Very precise length measurements can also be made by means of interferometry. For example, the separation between the two reflecting flats of a Fabry-Perot interferometer can be accurately measured. From Eq. (1.17), the order of interference m is given by m=

2nd cos 8

(1.26)

A

If this order of interference m is somehow determined and the wavelength is known, the distance d can be determined. A shown by F r a n ~ o n ,there ~ are several methods to determine the order of interference. M i ~ h e l s o n , ~at' the Bureau International des Poids et Mesures, was the first one to make an experimental determination of the length of the meter by means of interferometry. He used a Michelson interferometer with the device in Fig. 42 instead of one with flat mirrors, and found that the meter was equal to n, where n = 1,553,165.3 and h o = 6438.472 8, for the red cadmium line. A more recent determination of the length of the meter was made by Baird and H ~ w l e t t , ~after * the adoption in 1960 of the new international meter as 1,650,763.73 vacuum wavelengths of the 6057 8, line of Kr.x6The precision with this method is about lo8. Modern definitions of the meter, however, involve the use of lasers, with the advantage of their great coherence length.5 The frequency of lasers is calibrated against the cesium standard, thus defining the unit of time. The

I

I

FIG. 42.

Michelson etalon.

42

INTERFERENCE

second step was to measure the wavelength of the laser line by interferometric comparison with the Krypton line. Next, the vacuum speed of light was computed from the product of the wavelength and the frequency, and this result, equal to c = 299,792,498 meters/sec. was defined to be exact. In conclusion, the definition of the meter fixes the speed of light, and any source of known frequency may become a reference standard for length. An obvious application of these techniques is the calibration of scales in many instruments, for example in the fabrication of diff raction gratings.73 Interferometry using amplitude-modulated beams can be used to measure longer distances. It is like having a much longer wavelength, but the accuracy of the measurements can be much greater than that obtained by other methods. Instruments using this principle in order to measure long distances of the order of several meters have been made by Dukes and Gordon.74 A good description of these instruments can be found in a paper by Bruning.68 Even the distance from the earth to the moon has been measured with this principle by H a m m ~ n d . ~ ' as well as l a ~ - g e ~angles ~ * ~ ' can also be measured very accurately by interferometric procedures. 1.5.8. Microscopy

The principle of multiple-beam interference in thin films can be used with great advantage to study the microtopography of small crystal or glass surfaces. This subject has been fully described by Tolansky4' and by Krug et al." One of the many applications of this method is the observation of the polishing defects of a metal or glass surface. 1.5.9. Doppler Interferometry If a light source is moving with a velocity u with respect to the observer, the observed wavelength and frequency shift by an amount given by AA A

-

Av - u v

c

(1.27)

These small wavelength and frequency shifts can be measured very accurately by means of interferometry as described by Foreman er a/.''' An interesting application is the measurement of fluid-flow velocities by using small light-scattering particles in colloidal suspension.".'* Another application of the Doppler effect to interferometry is the measurement of slow rotations, as for example the earth rotation, with very high accuracy. This measurement is performed with the so-called Sagnac or ring interferometer," shown in Fig. 43. Two beams travel in the interferometer,

APPLICATIONS O F I N T E R F E R O M E T R Y

43

MONOCROMATIC COLIMATED LIGHT SOURCE OBSE LIGH'

E R OR ETECTOR

FIG. 43. Sagnac interferometer

one moves clockwise, and the other one counterclockwise. The result is that there is a fringe shift proportional in magnitude to the angular rotation w , as follows: 4Aw

AN=-

CA '

( 1.28)

where A is the area of the interferometer ( A = 2R'). M i c h e l ~ o n ~ and ~ , ' ~Michelson and Gale'' were the first to use this interferometer to measure the earth's rotation. Macek and Davisx7 designed a modern version of this instrument, using four laser tubes with a cavity in a ring configuration. With lasers, a stronger signal is obtained, but the two counter-running methods tend to lock together at low speeds, but with special methods this can be avoided, as shown by Aronowitz.xx The cyclic interferometer as well as the ring laser may be used as gyroscopes, as described by Rowland and A g r a ~ a l . ' ~

44

1 NTERFERENCE

1.5.10. Spectroscopy Probably one of the most popular and oldest applications of interferometers is the measurement of the spectral components of light beams, as was already mentioned in Section 1.3. The resolution of interferometric methods is so good that the hyperfine structure46and the wavelength values3' can be measured with high accuracy. Another very important and relatively new spectroscopic method that we want to describe here is called Michelson-Fourier spectroscopy. This method has been treated by P. Connes'" and Vanasse and Strong." It works on the principle that the temporal coherence or wave-train shape determines the frequency spectrum and vice versa. A Twyman-Green interferometer can be used to measure the wave-train shape, and then the spectrum can be computed mathematically. Suppose that we want to determine the spectrum of the light source illuminating the interferometer in Fig. 13. This is done by moving one of the flat mirrors along its optical axis to change the OPD. A light detector then replaces the eye in order to permit the average irradiance of the interference pattern to be recorded. The irradiance measured by the detector is a function of the optical path difference. Defining (1/2)Zl(k)dk as the average irradiance due to either of the two beams on a narrow band dk centered at k, the irradiance d l at the detector will vary sinusoidally with the OPD as follows: d l = 11(k ) [1+ COS( kOPD)] dk.

( 1.29)

Since different wavelength light beams are mutually incoherent, the irradiance I due to all colors acting simultaneously is

I= If we now define

lom lom I,(k)[l+cos(kOPD)] dk.

( 1.30)

I,=

(1.31)

I,(k) dk,

and I(OPD)=

I,'

I , ( k ) c o s ( k . O P D ) dk,

(1.32)

the irradiance at the detector can be written

I = I"+ I ( 0 P D ) .

(1.33)

After I ( 0 P D ) has been experimentally measured, the irradiance spectrum I ( k ) can be obtained from the cosine Fourier transform of I(OPD), which

45

APPLICATIONS O F I N T E R F E R O M E T R Y

gives d(0PD). Z(0PD) C O S ( ~ OPD)

( 1.34)

In practice, the mirror displacements cannot have infinite magnitude, and thus this integral is approximated by

I , ( k ) =-

Z(0PD) cos(k- OPD) d(OPD),

(1.35)

where the value of OPD,,, is twice the maximum mirror displacement. The larger this OPD,,, , the better the spectral resolving power. There are two clear advantages of interference spectroscopy over conventional dispersing spectroscopy. The first advantage is called Jucquinot and Dufour” or etendue advantage. The spectroscopic field of view of an interferometer is circular and much greater than the narrow and long field given by the slit of a spectrometer. For instance, with a Fabry-Perot interferometer, the spectrum of the whole image of the sun can be simultaneously studied, while with a spectroheliograph, this has to be done by dividing the image of the sun into slits. The second advantage is called Fellgett9’ or multiplex advantage. In a spectroscope, different wavelengths are studied either at different times if a photoelectric detector is used or with different photographic regions if a photographic plate is used. In interference spectroscopy, on the other hand, all spectral lines are measured simultaneously with a single photoelectric detector. This is equivalent to coding each optical frequency by an electrical cosine signal with different frequency, and then decoding numerically by Fourier analysis. In this manner, the signal-to-noise ratio is much greater than in conventional spectroscopy. We could describe many other useful applications of interferometry, but it would be impossible to mention all of them in a single chapter. Acknowledgment The author wishes to express his gratitude to Professor W. H. Steel for his many valuable suggestions.

References M . Born and E. Wolf, Principles ofoptics, Pergamon Press, New York, 1959. A. H . Cook, Interference of Electromagnetic Waves, Clarendon Press, Oxford, 1971. M . Fraqon, Optical Interferometry, Academic Press, New York, 1966. C. Candler, Modern Interferometers, Hilger and Watts, London, 1951. 5. W. H. Steel, Inferferomerry, 2nd ed., Cambridge University Press, London, 1983.

1. 2. 3. 4.

46

INTERFERENCE

6. S. Tolansky, An Introduction to Inrerferometry, Longmans Green, London, 1955. 7. K. M. Baird, “Interferometry: Some Modern Techniques,’’ in Advanced Optical Techniques (A.E.S. Van Heel, ed.), p. 123, North-Holland Publ., New York, 1967. 8. K. M. Baird, and G. R. Hanes, “Interferometers,” in Applied Optics and Optical Engineering, Vol. 4, p. 309 (R. Kingslake, ed.), Academic Press, New York, 1967. 9. J. Dyson, “Interferometers,” in Concepts ofClassical Optics (J. Strong, ed.), p. 377, W. H. Freeman, San Francisco, 1958. 10. R. N. Wolfe and F. C. Eisen, “Irradiance Distribution in a Lloyd Mirror Interference Pattern,” J. Opt. SOC.Am. 38, 706 (1948). 1 1 . W. H. Steel, “Two Beam Interferometry,” in Progress in Optics, Vol. 5, Chap. 3 ( E . Wolf, ed.), North-Holland Publ., Amsterdam, 1966. 12. M. V. R. K. Murty, ”Newton, Fizeau and Haidinger Interferometers,” in Optical Shop Testing ( D . Malacara, ed.), Chap. 1, Wiley, New York, 1978. 13. J. Guild, “Fringe Systems in Uncompensated Interferometers.” Proc. Phjjs. Soc. (London) 33, 40 (1920-1921). 14. F. Twyman, “Camera Lens,” British Patent 130224 (1919). 15. F. Twyman, “The Testing of Microscope Objectives and Microscopes by Interferometry,” Trans. Faraday Soc. 16, 208 (1920). 16. D. Malacara, “Twyman-Green Interferometer,” in Optical Shop Testing ( D . Malacara, ed.), Chap. 2, Wiley, New York, 1978. 17. M. V. R. K. Murty, “Lateral Shearing Interferometers,” in Optical Shop Testing (D. Malacara, ed.), Chap. 4, Wiley, New York, 1978. 18. D. Malacara, “Radial, Rotational. and Reversal Shear Interferometers,” in Optical Shop Testing (D. Malacard, ed.), Chap. 5, Wiley, New York, 1978. 19. 0. Bryngdahl, “Applications of Shearing Interferometry,” in Progress in Optics Vol. 4, Chap. 2 (E. Wolf, ed.), North-Holland Publ., Amsterdam, 1965. 20. W. J . Bates, “A Wavefront Shearing Interferometer,” Proc. Phys. Soc. 59, 940 (1947). 21. M. V. R. K. Murty, “The Use of a Single Plane Parallel Plate as a Lateral Shearing Interferometer with a Visible Gas Laser Source,” Appl. Opt. 3, 531 (1964). 22. J . B. Saunders, “Measurement of Wavefronts without a Reference Standard. I: The Wavefront Shearing Interferometer,” J. Res. Nut. Bur. Stand. 65b, 239 (1961). 23. D. Malacara. Testing of Optical Surfaces, Ph. D. Thesis, Institute of Optics, University of Rochester, New York, 1965. 24. M. P. Rimmer and J. Wyant, “Evaluation of Large Aberrations Using a Lateral Shear Interferometer Having Variable Shear,” Appl. Opt. 14, 143 (1975). 25. D. S. Brown, Interferometry N.P.L. Symposium No. I / , p. 253, Her Majesty‘s Stationery Office, London, 1959. 26. M. V. R. K. Murty, “A Compact Radial Shearing Interferometer Based on the Law of Refraction,” Appl. Opt. 3, 853 (1964). 27. M. V. R. K. Murty and E. C. Hagerott, “Rotational Shearing Interferometer,” Appl. Opt. 5, 615 (1966). 28. J. W. Gates, “The Measurement of Comatic Aberrations by Interferometry,” Proc. fhvs. Soc. B 68, 1065 (1955). 29. J . B. Saunders, “Inverting Interferometer,” J. Opt. Soc. Am. 45, 133 (1955). 30. S. Mallick, “Common-Path Interferometers,” in Optical Shop Testing ( D . Malacara, ed.), Chap. 3, Wiley, New York, 1978. 31. J. M. Burch, “Scatter Fringes of Equal Thickness,” Nature 171, 889 (1953). 32. A. A. Michelson, “Interference Phenomena in a New Form of Refractometer,” Phil. Mag. 5, 236 (1882). 33. Hanbury R. Brown, and R. Q. Twiss, “A new Type of Interferometer for Use in Radio Astronomy,” Phil. Mag. 45, 663 (1954).

REFERENCES

47

34. C. Roychoudhuri, “Multiple Beam Interferometers.” in OpficalShopTesfing ( D . Malacara, ed.), Chap. 6, Wiley, New York, 1978. 35. H. D. Polster, R. M. Scott, R. L. Crane, P. H. Langenbeck, R. Pilston, and G. Steinberg, “New Developments in Interferometry,” Appl. O p f .8, 521 (1969). 36. C. Fabry and A. Perot, “Thtorie et Applications d’une Nouvelle Mithode de Spectroscopie Interferentielle,” Ann. Chim. Phys. 7, 115 (1899). 37. P. W. Baumeister, “Interference and Optical Interference Coatings,” in Applied Optics and Opfical Engineering, Vol. 1, p. 285 (R. Kingslake. ed.), Academic Press, New York, 1965. 38. P. W. Baumeister and F. A. Jenkins, “Dispersion of the Phase Change for Dielectric Multilayers. Application to the Interference Filter,” J. Opt. Sac. Am. 47, 57 (1957). 39. F. S. Tomkins and M. Fred, “Wavelength Measurements with a Concave Grating Spectrograph,” Appl. O p f . 2, 715 (1963). 40. S. Tolansky, Multiple-beam Interferomefry of Surfaces and Films, Clarendon Press, Oxford, 1948. 41. S. Tolansky, Surjace Microtopography, Wiley Interscience, New York, 1960. 42. P. Connes, “L‘italon d e Fabry-Perot Sphirique,” J. Phvs. Radium 19, 262 (1958). 43. D. R. Herriott, “Spherical-Mirror Oscillating Interferometer,” Appl. Opt. 2, 865 (1963). 44. R. L. Fork, D. R. Herriott, and H. Kogelnik, “A Scanning Spherical Mirror Interferometer for Spectral Analysis of Laser Radiation,” Appl. Opt. 3, 1471 (1964). 45. K. W. Meissner, “Interference Spectroscopy. Part I,” J. Opt. Soc. Am. 31, 405 (1941). 46. K. W. Meissner, “Interference Spectroscopy. Part 11,” J. O p f . Sac. A m . 32, 185 (1942). 47. W. E. Williams, Applicafions of Inferferomefry,4th ed., Methuen, London, 1950. 48. P. Hariharan, “Multiple Pass Interferometers,” in Optical Shop Tesfing, ( D . Malacara, ed.), p. 217, Wiley, New York, 1978. 49. P. Langenbeck, “Multipass Twyman-Green Interferometer,” Appl. O p f . 6 , 1425 (1967). 50. P. Hariharan and D. Sen, “The Separation of Symmetrical and Asymmetrical Wavefront Aberrations in the Twyman-Green Interferometer,” Proc. Phys. Sac. (London) 77, 328 (1961 ). 51. D. Sen and P. N. Puntambekar, “An Inverting Fizeau Interferometer,” O p f . A c f a 12, 137 (1965). 52. D. Sen and P. N. Puntambekar, “Shearing Interferometers for Testing Corner Cubes and Right Angle Prisms,” Appl. O p f . 5, 1009 (1966). 53. P. N. Puntambekar and D. Sen, “A Simple Inverting Interferometer,” Opt. Acta 18, 719 (1971 ). 54. W. E. Williams, “Applications of Interferometry,” in Concepfs of Classical Opfics (J. Strong, ed.), p. 373, W. H. Freeman, San Francisco, 1958. 55. J. Dyson, Inferferomefry as a Measuring Tool, Machinery Publishing Co., Brighton, 1970. 56. P. Mollet, Opfics in Mefrology, Pergamon Press, Oxford, 1960. 57. A. A. Michelson and M. Morley, Silliman J. 34, 333 (1887). 58. R. B. Leighton, Principles of Modern Physics, Chap. 1, McGraw-Hill, New York, 1959. 59. T. S. Jaseja, A. Javan, J. Murray, and C. H. Townes, Phys. Rev. 133A, 1221 (1964). 60. A. A. Michelson, “Measurement of the Diameter of Orionis with the Interferometer,” Asfrophys. J. 53, 249 (1921). 61. J. Ojeda-Castatieda, “Foucault, Wire and Phase Modulation Tests” in OpticalShop Tesfing, ed. D. Malacara, John Wiley and Sons, New York, 1978. 62. V. Grigull and H. Rottenkolber, “Two-Beam Interferometer Using a Laser,” J. Opf. Soc. Am. 57, 149 (1967). 63. A. Cornejo-Rodriguez, “Ronchi Test’’ in Optical Shop Tesfing (D. Malacara, ed.), Chap. 8, Wiley, New York, 1978. 64. D. Malacara, A. Cornejo, and M. V. R. K. Murty, “Bibliography of Various Optical Testing Methods,” Appl. Opf. 14, 1065 (1975).

48

INTERFERENCE

65. J. D. Briers, “Interferometric Testing of Optical Systems and Components: A Review,” Opt. Laser Technol. 4, 28 (1972). 66. G. Schulz and J. Schwider, “lnterferometric Testing of Smooth Surfaces,” in Progress in Optics, Vol. 13, Chap. 4, (E. Wolf, ed.), North-Holland Publ., Amsterdam, 1976. 67. D. Malacara, ed. Optical Shop Testing, Wiley, New York, 1978. 68. J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing (D. Malacara ed.), Chap. 13, Wiley, New York, 1978. 69. K. Rosenhauer, “Measurement of Aberrations and Optical Transfer Functions of Optical Systems,” in Advanced Optical Techniques, Chap. 18, Wiley, New York, 1967. 70. H. H. Hopkins, “Interferometric Methods for the Study of Diffraction Images,” Optica Acta 2, 23 (1955). 71. A. A. Michelson, “Ditermination Expirimentale de la Valeur du Metre en Longeurs d’Ondes Lumineuses,” Trau. Mem. Bur. Int. Poids Mes. 11, 1 (1895). 72. K. M. Baird and L. E. Howlett, “The International Length Standard,’. Appl. Opt. 2, 455 (1963). 73. G. R. Harrison, N. Sturgis, S. P. Davis, and Y. Yamada, “Interferometrically Controlled Ruling of Ten-Inch Diffraction Grating,” J. Opt. SOC.A m . 49, 205 (1959). 74. J. N. Dukes and G. B. Gordon, Hewlett-Packard Journal 21, 2 (1970). 75. A. L. Hammond, Science 170, 1289 (1970). 76. V. Met, “Determination of Small Wedge-Angles Using a Gas Laser,” Appl. Opt. 5, 1242 (1966). 77. D. Malacara and 0. Harris, “Interferometric Measurement of Angles,” Appl. Opt. 9, 1630 (1970). 78. D. T. Tentori and M. Celaya, “Continuous Angle Measurement with a Jamin Interferometer,” Appl. Opt. 25, 215 (1986). 79. W. Krug, J. Rienitz, and G. Schultz, Contributions to Interference Microscopy (English trans.), Hilger and Watts, London, 1964. 80. J. W. Foreman, Jr., E. W. George, and R. D. Lewis, Appl. Phys. Lett. 7 , 77 (1965). 81. C. P. Wang and D. Snyder, “Laser Doppler Velocimetry: Experimental Study,” Appl. Opr. 13, 98 (1974). 82. Y. Yeh and H. Z. Cummins, Appl. Phys. Lett. 4, 176 (1964). 83. G. Sagnac, 1. Phys. Radium 4, 177 (1914). 84. A. A. Michelson, Phil. Mag. 8, 716 (1904). 85. A. A. Michelson, Astrophys. J. 61, 137 (1925). 86. A. A. Michelson and H. G. Gale, Asrrophys. J. 61, 140 (1925). 87. W. M. Macek and D. T. M. Davis, Jr., Appl. Phys. Lett. 2, 67 (1963). 88. F. Aronowitz, Laser Applications, Vol. 1, p. 133, (M. Ross, ed), Academic Press, New York, 1971. 89. J. T. Rowland and Agrawal, Opt. Laser Technol. 13, 239 (1981). 90. P. Connes, “Recherches sur la Spectroscopie par Transformation de Fourier,” Rev. d’Optrque 40, 116 (1961). 91. G. A. Vanasse and J. Strong, “Application of Fourier Transformations in Optics: Interferometric Spectroscopy,” in Concepts of Classical Optics (J. Strong, ed.), p. 419, W. H. Freeman, San Francisco, 1958. 92. P. Jaquinot and C. Dufour, J. Rech. Cent. Natl. Rech. Sci. 2, 91 (1948). 93. P. Fellgett, “A Propos de la Thiorie du Spectromitre Interfirentiel Multiplex,” J. Phys. Radium 19, 187 (1958).

2. DIFFRACTION A N D SCATTERING Daniel Malacara Centro de lnvestigaciones en Optica. A.C Apdo. Postal 948 37000 Leon, Gto. Mexico

2.1. Diffraction Diffraction phenomena have been some of the most interesting and most thoroughly studied manifestations of light in the history of physics. Many good textbooks present this subject in detail like those by Born and Wolf,’ Meyer,2 Marechal and F r a n ~ o n ,and ~ Ra~leigh.~ The , ~ earliest known mention of this phenomenon appears in the works of Leonard0 da Vinci (1452-1519).However, Grimaldi7 is the first one to seriously treat the subject in his book. It is interesting to know that Newton* was aware of Grimaldi’s observations. Huygens’ was the first one to develop a wave theory (see Fig. l), although he also seems to have been aware of Grimaldi’s observations, since he did not attempt to explain the existence of diffraction bands. An attempt to explain diffraction qualitatively was made by Young, 1 0 * ’ 1 - 1 2 who assumed that the wave passes undisturbed through the aperture, except at the edges, where the edge wave is formed. In this case, he thought that the interference between the direct wave and the edge waves formed the diffraction pattern. It was not until 1818 that FresnelI3 used the wave theory together with the principle of interference to explain diffraction quantitatively. He assumed mutual interference between all secondary waves, adding on the screen the amplitudes of each Huygens wavelet, and taking into account any phase differences. The Huygens principle, modified in this way, is called the HuygensFresnel principle; it leads to quantitatively good predictions of the light distribution on the screen. This theory, nevertheless, is not perfect. One of the problems is that there is no explanation as to why the wavelets do not travel backwards to the source, and another is that the resultant phase on the screen, computed by this theory, is 7r/2 less than the experimental value. 49 M E T H O D S O F E X P E R I M E N T A L PHYSICS Vol. 26

Copyright 0 1988 by Academic Press. Inc. All rights of reproduction in any form reserved ISBN 0- 12-475971-8

50

DIFFRACTION A N D SCATTERING

FIG. 1. Huygen's explanation of diffraction.

2.1.1. Kirchhoff's Diffraction Theory

In 1876 Kirchhoff 14.15,'6.17 established a complete mathematical theory of diffraction. He then proved that the Huygens-Fresnel principle could be regarded as an approximate form of his theory. Only a brief description will be given here, since very complete and detailed accounts can be found in several books.' Kirchhoff's theory starts with the so-called HelmholtzKirchhoff theorem derived from the scalar differential wave equation. He considers an arbitrary closed surface and an observation point P inside it. The idea is to evaluate the field at the internal point P, from a knowledge of the field at the closed surface (see Fig. 2). A point light source is placed outside the closed surface, at a distance r much greater than A. Kirchhoff then shows that the amplitude U ( P )at this point is given by:

This expression may now be applied to the solution of diffraction problems by considering the surface S as formed by three parts: portion A, the aperture through which the light passes, portion B, the diffracting flat screen,

51

DIFFRACTION

FIG.2. Helmholtz-KirchoH theorem

and portion C, a spherical shell which intersects the diffracting screen and which has its center at the observation point P. The amplitude over the diffracting screen must be zero. Therefore the integral over portion B must also be zero. The integration over the sphere C also goes to zero when its radius goes to infinity. If the angle between the incident and the diffracted rays is called 8, it may be shown that: cos( n, s) + cos( n, r ) = 1 + cos 8.

(2.2)

Hence, Eq. (2.1) becomes rs

(2.3)

which is the well-known Kirchhoff diffraction integral, where u is the diffracting aperture. We notice here a few differences between the results of this theory and those of Fresnel. The first important difference is the presence of the complex number i in front of the integral. This means that the phase of the observation point P, obtained with the Kirchhoff theory, has a difference 7r/2 with respect to the phase obtained with the Huygens-Fresnel theory. This also means that the phase of the illumination at point P, using a very small diffraction pinhole (only one Huygens wavelet), and the phase of the illumination at the same point with no diffraction, differ by 7r/2.

52

DIFFRACTION A N D SCATTERING

Second, we have the presence of the factor (1 + cos 0)/2. Usually this is called the inclination factor, first derived by Stokes. It implies that the amplitude of each secondary wave decreases when the inclination angle increases. This inclination factor appeared naturally in Kirchhoff’s theory and is only postulated in the Huygens-Fresnel theory. The third difference is the presence of the product l / r s in the integral in Eq. (2.3) which would appear in the Huygens-Fresnel theory only when the decrease of the amplitude, as l / r s , is considered. Experimental verifications of Kirchhoff’s scalar theory of diffraction were made by Silver” in 1962. Even Kirchhoff’s theory is incomplete, not only because of the approximations made, but mainly because the vectorial transverse nature of light waves is ignored. The scalar theory of diffraction produces, in general, very accurate results, but a further refinement is achieved if the vectorial nature of light20.2’.22.23324.2s and the optical properties of the screen or stops are considered by means of the electromagnetic theory. 2.1.2. Other Theories

As pointed out before, Kirchhoff’s theory is not perfect. PoincarC26and S ~ m m e r f e l dshowed ~~ that Kirchhoff’s hypotheses were inconsistent, because the theory does not reproduce the assumed values on the screen and in the plane of the aperture. As described by Bouwkamp28 (including numerous references), many modifications to Kirchhoff’s theory were made. One of the most important modifications of Kirchhoff’s hypotheses was made by K ~ t t l e r , ~ who ~ . ~ ’showed that it is more appropriate to prescribe some discontinuities at the edge of the aperture rather than continuous boundary values. This is called a saltus problem, and it is especially interesting when considering the diffraction at a black, completely absorbing screen. Another interesting theory again concerns the existence of Young’s edge waves. Young postulated these waves when noticing that the edge of the aperture appeared luminous when observed from points within the geometrical shadow. It is interesting to know that Fresnel considered these edge waves but finally discarded them. The first one to seriously consider the edge waves again was M a e ~ , but ~’ he did not go very far. Sommerfeld” used the edge waves to find an exact solution to the problem of diffraction by an infinite half plane. The problem was further studied by R u b i n o ~ i c z ,confirming ~ ~ . ~ ~ the existence of Young’s edge waves for flat or spherical diffracted wave fronts. Miyamoto3’ and Miyamoto and developed even further the Rubinowicz theory showing that Kirchhoff‘s diffraction integral can be

FRESNEL DIFFRACTION

53

FIG.3. Some Fresnel diffraction patterns of recognizable objects.

separated into two waves, one coming from the aperture and another one coming from its edge. In other words, they showed that a diffraction wave appears as arising from the scattering at the edge of the aperture. Descriptions of the Miyamoto-Wolf theory can be found in R ~ b i n o w i c z . ~ ~ . ~ ~ Even a corpuscular quantum approach to the diffraction problem in gratings has been given, namely by Duane.40

2.2. Fresnel Diffraction Depending on the positions of the light source and the observation plane, we may consider two types of diffraction. Fresnel diffraction occurs when either the light source or the observation plane or both are at finite distances from the diffracting aperture or object. Fraunhofer diffraction occurs when both the light source and the observation plane or both are at finite distances from the diffracting aperture. In the following sections, we will consider some selected configurations for Fresnel diffraction. Figure 3 shows some interesting Fresnel diffraction patterns. 2.2.1. Single-Slit Diffraction

Let us consider the configuration shown in Fig. 4, with a point source at 0 and the observing point at P. A direct ray OFP and a diffracted ray O Q P meet at P, with an optical path difference (OPD) equal to the small line segment Q R .

54

DIFFRACTION A N D SCATTERING

-

c

a

c

b

l

\

\ \ \

\

FIG;. 4.

Fresnel diffraction geometry.

Assuming that the distances a and b are very large compared to the height S, the segment QR may be approximated by the sum of the sagittas of the two arcs passing through Q and R, respectively. S 2 S2 a + b OpD=QR=-+-=2a 2b 2ab

s2,

(2.4)

giving a phase difference 6: S=kOPD=

n( a

+b ) S2.

abh

(2.5)

To simplify the mathematics, we consider this as a two-dimensional problem. Defining a nondimensional variable u as:

we can write the phase difference 6: (2.7)

If we divide the wave front into narrow parallel strips of width ds, we may use the Huygens-Fresnel principle and add the contribution of all the strips to the amplitude at point P. This sum is made by considering the

55

FRESNEL DIFFRACTION

FIG.5. Total amplitude at point P.

phase differences between the different strips and by adding them vectorially as shown in Fig. 5. Each vector, representing the contribution of each slit, has a magnitude directly proportional to the width ds and hence to dv. Each vector forms an angle 6 with the x axis. The resultant amplitude at the point P is directly proportional to the resultant vector R. We can mathematically generate a curve, as in Fig. 5, that can be used to find the amplitude at point P for any width and position of the diffracting slit. Using Eq. (2.7), we find: dx = dv cos 6 = cos dy = dv sin 6 = sin

and by integrating, we obtain: x= y=

(q)

dv,

(G) dv,

lo" ($) lo" ($)

(2.9)

cos

dv,

(2.10)

sin

dv.

(2.11)

56

DIFFRACTION A N D SCAITERING

f

q

A2

FIG.6 . Cornu spiral.

Equations (2.10) and (2.1 1 ) are the Fresnel integrals, whose numerical values are given in table form in many book^.^'.^^ The curve produced with these integrals is the Cornu spiral shown in Fig. 6. Strictly speaking, we should have used the inclination factor (1 + cos 8 ) / 2 here, since the amplitude of each secondary wavelet decreases when the angle 8 increases. We assumed, however, that 8 remains very small everywhere, and therefore the inclination factor is always very close to 1. The Corm spiral allows us to find the amplitude of the illumination at the point P on the axis. If the diffracting slit is centered on the optical axis and has a width S, we first compute u by using Eq. (2.6). Then we choose two symmetrical points A , to A, on the Cornu spiral, such that their separation measured along the spiral be equal to u. The distance from A , and A * , measured along a straight line, represents the amplitude of the illumination at P. If the diffracting slit is decentered off the optical axis, we again use Eq. (6.6) to compute the decentering and the slit width in terms of u. Then two

57

FRESNEL DIFFRACTION

points A; and A2 are chosen such that their positions as measured along the curve corresponding to the slit width and to the decentering. Again, the separation between these two points, measured along a straight line, represents the amplitude at P. The whole diffraction pattern for a diffracting slit, with a width corresponding to a certain value of u, is found by moving two points, A; and A;, along the curve in such a way that the distance between these two points along the curve remains constant and equal to u. The distance between the two points in a straight line will change continuously to give the amplitude of the whole diffraction pattern. The irradiance which is the square of the amplitude is shown in Fig. 7(a). In order to relate the resultant irradiance to the value of the unobstructed irradiance, the squares of amplitudes obtained from Fig. 6 must be divided by 2. If we have only straight edges, we may consider that we have a decentered slit, with one edge on the optical axis and the other edge at infinity. In this case, we have one fixed point 2, and a moving point A; on the Curnu spiral. We then obtain the diffraction pattern in Fig. 7(b). If there is no INTENSITY

I

I

I

r

GEOMETRICAL

LIMITS

(A)

INTENSITY

G

GEOMETRICAL EDGE

(B) FIG.7. Diffraction light distribution in (a) a wide slit, (b) a straight edge.

58

DIFFRACTION A N D SCATTERING

diffraction obstacle, the intensity everywhere on the screen is represented by the distance along a straight line from 2, to 2,. In this case, we can see that the phase at the point P, with no diffraction, is ~ r / 4(45") ahead of the phase at the same point P, produced with the infinitely narrow diffracting slit, and is 7r/4 ahead of the phase at P produced with an infinitely small diffracting pinhole. 2.2.2. Circular Aperture Diffraction

Fresnel diffraction by a circular aperture can be easily studied for a point on the optical axis. In doing so, we can obtain very interesting results, as we shall see. Using the same geometry as in Fig. 2, we can find the amplitude at the point P by dividing the wavefront into annular zones concentric with the point F, where each zone has a radius S, and then by adding the contributions of each zone to the final amplitude at P. From Eq. (2.5), the phase difference between the light on a ring with radius S and the light through the center is 6 = KS2,

(2.12)

where K=

v(a

+b )

abh

.

(2.13)

The contribution of each zone is directly proportional to its area, and this area in turn is directly proportional to its width ds. Taking this into account, we can find a graph analogous to the Cornu spiral, which can be easily shown to be given by:

[

x + y--

;I*=[&l'~

(2.14)

This is a circle with its center on the y axis, tangent to the x axis, and with radius A/2K. This means that, if the radius S of the diffracting circle is increased continuously, the amplitude on the axis at the point P oscillates, as shown in Fig. 8(a). This is physically unacceptable, because these oscillations in the amplitude do not occur when the diffracting aperture is very large. The problem disappears when taking into account the inclination factor (1 +cos 8)/2, because the radii of the circles will become smaller when the aperture grows, increasing the angle 8. The curve is now a spiral, as in Fig. 9. When the aperture becomes very large, the amplitude reaches the value A/2K at the point P on the axis. The color at the center of the diffraction pattern produced by a circular aperture and illuminated with white light has been studied by Bergsten and H ~ b e r t y . ~ ~

59

F R E S N E L DIFFRACTION AMPLITUDE AT POINT P

A 2K

S

AMPLITUDE AT POINT P

FIG.8. Amplitude variation on axis with a diffracting circle, (a) without obliquity factor, ( b ) with obliquity factor.

2.2.3.Fresnel Zone Plate Diffraction The amplitude on the axis of a diffracting circular aperture oscillates because some annular zones contribute constructively to the interference, while others contribute destructively. If we mask all the zones that contribute destructively, the irradiance at the point P is drastically enhanced. Then the diffracting hole becomes a Fresnel zone plate, which is illustrated in Fig. 10.

60

DIFFRACTION A N D SCATTERING

FIG.9. Spiral for a circular aperture.

F I G .10. Fresnel zone plate

FRESNEL DIFFRACTION

61

Since the irradiance is greatly enhanced at the point P, it must be greatly reduced at all other points by the energy conservation principle. This means that a Fresnel zone plate acts like a lens and can even form images. The outer edge of a dark ring occurs for 6 = nm, while the inner edge of the same ring occurs for 6 = (n - 1/2)7r. Thus, from Eqs. (2.12) and (2.13), it may be shown that the radii S of the limits of a dark ring n are given by (2.15)

and ab s . =[ h ( n - 1/2) a+b

(2.16)

n'n

Defining So= SIin a n d f = 2Si/h (for a zone plate with a transparent center zone), we may show that (2.17)

which is the equivalent of the thin-lens formula for the positions of the object and the image, at a and b, respectively. In spite of the close similarity between a thin lens and zone plate, there are some important differences. The first difference for a lens is that the focus is not unique, but there are an infinite number of them with focal lengths fm, given by 2s; mh '

fm =-

m =integer,

(2.18)

where rn is any positive or negative odd integer. It should also be noticed that for every convergent focus there is a corresponding diverging focus. In other words, a zone plate acts simultaneously as a convergent and a divergent lens. A very useful application of zone plates is for alignment, as described by Van A pinhole camera uses a small pinhole, instead of a lens, to form the image. A consequence of Eq. (2.17) is that in order to optimize the resolution and light-collecting power, the pinhole must have a radius S, and the photographic-plate-to-pinholedistance should be f: A Gabor plate is quite similar to a zone plate, with the difference being that its transmittance does not change in steps but in a quasi-sinusoidal manner. It is normally produced by the interference between a flat and a spherical wave front.

62

D I F F R A C T I O N A N D SCATTERING

I/ I

/-

-I+-POINT COLLIMATOR

I

\‘I

DIFFRACTING APERTURE

FIG.1 I . Arrangement to observe Fraunhofer diffraction.

2.3. Fraunhofer Diffraction and Fourier Transforms When the light source and the observing screen are both at infinite distances from the diffracting aperture, we say that we are observing Fruunhofer difruction. The light source may really be at infinity as in the case of a star, but more frequently the light source is placed “optically at infinity” by means of a collimating lens. In this case the light source is at the focus of a convergent lens, as shown in Fig. 1 1 . The observing screen cannot be really placed at infinity, but it is again done optically by placing it in the focal plane of a convergent lens. The fact that the observing screen is at infinity may be interpreted by saying that what we observe on the screen in Fraunhofer diffraction is the angular distribution of the light after the diffracting aperture. Figure 12 shows the slow transition from Fresnel to Fraunhofer diffraction, with a point source at infinity, by increasing the distance b of the observing screen from the diffracting aperture. We say that we have reached the far field or Fraunhofer diffraction region when b >> D‘/A, where D is the diameter of the diffracting object. The inclination factor is not important in the case of Fraunhofer diffrdction, and if we also take the case of normal incidence ( r = constant), the

F R A U N H O F E R DIFFRACTION A N D FOURIER T R A N S F O R M S

0 44

63

64

DIFFRACTION A N D SCATTERING

diffraction integral of Eq. (2.3) becomes

U(P)=i{

{

AcdS, S

(2.19)

where A is a constant whose magnitude is directly proportional to the amplitude on the diffracting aperture plane. If the aperture is on the x, y plane, and the amplitude on this plane is a function of x and y , we consider A also to be a function of x and y. The distance s is variable enough to produce significant changes in the phase ks, but constant enough to include l / s within the value of A. Thus, U (P ) =

J J ,~ ( xy ,) eiksdx dy.

(2.20)

We may show that s = x s i n O , + y s i n O,.;

(2.21)

and by defining

k,

=k

sin 0,

and

k,, = k sin O v ,

(2.22)

we can write Eq. (2.20) in the form

U(k,, k,) = i

{

A(x, y ) e’(k\r+k”’) dx dy.

(2.23)

On the other hand, from Fourier theory, the function A(x, y ) can be given by: (2.24) The functions A(x, y ) and U (k,, k,) are two-dimensional Fourier transforms of each other. Thus, except for unimportant constants, we can say that the angular diffraction pattern is the Fourier transform of the amplitude function on the diffracting aperture. In a similar way, for diffraction apertures in one dimension, we may write the one-dimensional Fourier transforms as (2.25)

FRAUNHOFER DIFFRACTION A N D FOURIER TRANSFORMS

65

and U ( k , ) e - l k \ ' dk,.

(2.26)

Fraunhofer diffraction has been widely studied by many authors, among whom we must mention S c h ~ e r d , ~whose ' excellent work includes many fine handmade color diffraction pictures. This work was reviewed by Hoover and Harris.46Harris4' has computed diffraction patterns for many different apertures. The colors observed when the aperture is illuminated with white light have been studied by Bergsten and P ~ p e l k a . ~ '

2.3.1. Single-Slit and Rectangular Aperture Diffraction

Let us take a slit on the x, y plane centered on the y axis. If the slit width is 2 a , we can write Eq. ( 2 . 2 5 ) in the form:

U ( k , )= A

['

J

(2.27)

elk\' dx,

-0

assuming that the amplitude A ( x ) remains constant inside the slit. Integrating, and using Eqs. ( 2 . 2 1 ) and ( 2 . 2 2 ) , we obtain:

U (k ) = U,

I

sin( k a sin 6) k a sin 6 '

(2.28)

where U, is a constant; Fig. 13 shows this amplitude and its corresponding irradiance versus k a sin 6. The first dark zone in the irradiance pattern can be shown to be at an angle 6 given by A

sin 6 =-. 2a

(2.29)

Therefore, the diffraction pattern becomes narrower when the slit width 2 a increases. For a rectangular aperture, with width 2 a and length 2b, we can use Eq. ( 2 . 2 3 ) to obtain U(O,, 4.) = U"

sin( k a sin 0,) k a sin Ox

sin( k b sin Or) k b sin 6,

][

3.

(2.30)

66

DIFFRACTION A N D SCATTERING

k o sin

(B)

6

FIG. 13. Fraunhofer diffraction pattern for single slit. ( a ) Amplitude distribution. ( h ) lrradiance distribution.

2.3.2.Circular Aperture Diffraction The diffraction pattern of a circular aperture may be found by integrating Eq. (2.23) inside the aperture, thus obtaining sin 0 ) u(e)=2u0 J,(ka ka sin 0

[

1’

(2.31)

where a is the radius of the aperture and J , ( x )is a first-order Bessel function

FRAUNHOFER DIFFRACTION A N D FOURIER TRANSFORMS

-

AIRY DISC

--

67

ka sin 8

(6) F l c . 14. Fraunhofer diffraction pattern for a circular aperture. ( a ) Amplitude aperture. ( b ) Irradiance distribution.

of x. The function U ( 0 ) and its corresponding irradiance U 2 (0 ) are plotted in Fig. 14. The first dark ring, which outlines the central spot or Airy disc, studied by Airy in 1835,49 can be seen to have a direction given by h

sin 0 = 1.22 -= 2a

where D is the diameter of the aperture.

A 1.22 D’

(2.32)

68

DIFFRACTION A N D SCATTERING

If the pattern is formed at the focal plane of a lens of clear aperture D with focal length A the radius r of the Airy disc is:

f

r = 1.22h -. D

(2.33)

If a collimated beam of light illuminates a well-designed convergent lens, the image is not a point but a Fraunhofer diffraction pattern of the free aperture. This is the reason why the resolving power of any perfect optical instrument (an instrument with no geometrical aberrations) is not infinity, but is limited by the size of the diffraction image. Sletten and Blacksmithso have shown that a paraboloidal microwave antenna has also an image at the focus, limited in size by Fraunhofer diffraction. It should be pointed out, however, that in a paraboloidal antenna, the f / D ratio is between 0.2 and 0.5, and thus the image structure differs from the Airy pattern because of the large angle of convergence, as described by Minnett and Thomas.’’ Figure 15 shows the Fraunhofer diffraction patterns produced by four different aperture shapes.

2.3.3.Babinet Principle Two pupil functions, A l ( x , y ) and A 2 ( x , y ) are complementary to each other if A,(x,y)+A,(x,y)=constant.

(2.34)

If the amplitude is constant over the whole x, y plane, then we say that two diffracting apertures are complementary if the transparent parts in one are the opaque parts in the other, and vice versa. For example, the complementary aperture of a circular aperture is the whole x, y plane, excluding only the opaque disc of the same size as the circular aperture. Let us assume that we have two arbitrary complementary apertures, giving diffraction pattern U , ( P ) and U , ( P ) . If U ( P ) is the amplitude of the illumination produced without any diffraction apertures, the Babinet principle’’ says that (2.35)

which can be very easily shown to be true. Applying this principle to the Fraunhofer diffraction patterns produced by a circular aperture and an opaque disc, we can see that U ( P ) is a very high and narrow peak at the origin, because all the light will travel in a single direction if there is no diffracting aperture.

FRAUNHOFER DIFFRACTION A N D FOURIER TRANSFORMS

69

FIG. 15. Fraunhofer diffraction pattern for four different apertures.

Thus, except for the point at the origin, the Fraunhofer diffraction patterns of complementary apertures are identical in shape but with amplitudes of opposite sign. 2.3.4. Parseval‘s Theorem As in any other physical process, energy is preserved, but it is probably more evident in Fraunhofer diffraction, where all computations are made by means of Fourier transforms as given by Eqs. (2.23) and (2.24). The following result we quote here can be mathematically proved from the definition of the Fourier transform.

(2.36)

70

D I F F R A C T I O N A N D SCATTERING

This equation is known as Parseval’s or the energy theorem, because it may be used to show that energy is always conserved in Fraunhofer diffraction experiments.

2.4. Diffraction Gratings A diffraction grating is just a diffraction screen formed by many parallel and equidistant slits. The principle of the diffraction grating was discovered by Rittenhouse” and later rediscovered by F r a ~ n h o f e r A. ~good ~ modern review of diffraction gratings is found in the book by Hutleyss and in the papers by Richardson” and Welford.” We could study the diffraction gratings by applying Kirchhoff’s integral directly to obtain all their properties. However, by studying them using more elementary principles, we gain better physical insight. 2.4.1. Angular Distribution of Light As shown in Fig. 16, the maxima of irradiance occur at angles such that the interference between the different slits is constructive. The interference is assumed to take place at infinity, since we are considering Fraunhofer diffraction. In Fig. 16, we have represented only two consecutive slits. It is easy to see that the maxima of the intensity are at an angle such that the optical path difference between two consecutive slits is given by

OPD = CA - BD = mA,

FIG. 16. Maxima of irradiance in a diffraction grating.

(2.37)

DIFFRACTION GRATl NGS

71

where m is any integer including zero, commonly called the order of interference. If d is the separation between the centers of the slits, and 8 , and Or are the angles of incidence and diffraction, we can see that d(sin

-sin 8,)

= mh.

(2.38)

The angles 8 , and O2 are both positive, as shown in Fig. 16, because they are each measured counterclockwise from the normal to the grating. The complete angular distribution of the light may be obtained by adding the amplitude contributions of each of the slits at a given angle 0,. Assuming that the grating has N slits and that 6 is the phase difference between two consecutive slits, the total amplitude at an angle 02 is:

E

= A[1

+ elS+ e2"+.

. . + e ( N p l ) 'I,S

(2.39)

which can be transformed to give (2.40)

Irradiance is then given by (2.41)

which can be shown to be equal to

(2.42)

i n order to visualize the form of this function, we have plotted the numerator, the denominator, and their quotient for a grating with five slits ( N = 5) in Fig. 17. The high peaks of the irradiance occur at values of 6, such that

_6 -- mr,

m =integer,

2

since at those points,

[

sin'(

lim S/2-mrr

y) (I")

and hence the peaks have a maximum irradiance A 2 N 2 .

(2.43)

(2.44)

DIFFRACTION A N D SCATTERING

I

I

I

I

I

I

I

I

I

I

m=O

m=l

I

A

A

m=2

m =3

FIG.17. Irradiance in a grating with five slits.

The phase difference 6 is given by:

S=kOPD=

2~rd(sinO1 -sin 0,)

A

(2.45)

Therefore, from Eq. (2.43) we may obtain Eq. (2.38) as expected. There are ( N- 2) secondary peaks with much lower irradiance between any two of the main peaks. If N is increased, the number of secondary maxima increases, but their height decreases very rapidly. Figure 18 shows the diffraction patterns produced by four gratings with a different number of slits. 2.4.2. Chromatic Dispersion and Resolving Power

If light of several wavelengths (several colors) impinges upon a diffraction grating at a given angle of incidence O,, there will be a different angle of diffraction O2 for each color. This property is used to make spectrographs and spectrometers using diffraction gratings instead of prisms. From Eq.

DIFFRACTION GRATINGS

73

FIG. 18. Diffraction patterns for four different arrays of slits.

(2.38), the chromatic dispersive power can be given by:

m A AA - d cos 02'

02 --

(2.46)

The chromatic dispersive power increases linearly with the order of interference. In a spectrograph using gratings, the angle O2 does not change much within a given order; therefore, cos O2 may be considered to be a constant. Thus A 0 J h A is approximately constant. In other words, the chromatic dispersion is approximately linear with A, as opposed to that for a dispersing prism. The chromatic resolving power of a grating may be computed by means of the Rayleigh criterion illustrated in Fig. 19. This criterion may be stated by saying that two spectral lines with slightly different wavelengths are just resolved when the maximum of one of the lines coincides with the first minimum of the other. We can see that the phase difference between the peak of the line and the first minimum is given by: (2.47) but from Eq. (2.45), this difference in phase corresponds to a difference

74

DIFFRACTION A N D SCATTERING SUM OF

TWO LINES

FIG. 19. Two lines just resolved by a diffraction grating.

he2 in the angle 0 2 ,given by 2 .rrd A~=--cos A

8,AO2.

(2.48)

From these two equations, the angular difference between the line maximum and its first minimum is

A = - Nd cos B2’

(2.49)

If two spectral lines with wavelength difference AA are separated by just this angle, from Eqs. (2.49) and (2.46), we obtain:

A _ - mN. AA

(2.50)

This is the well-known expression for the resolving power of a grating. If we define the effective width We, of a grating as

we,,= w cos 82 = Nd cos 8 2

(2.51)

and use Eq. (2.49) we find a n alternative expression showing that the resolving power of a grating is limited only by its effective width and the wavelength, as shown by Michelson,5s (2.52)

DIFFRACTION GRATINGS

15

We may also write: (2.53) which shows that the chromatic resolving power of a grating is directly proportional to its effective diameter times its dispersive power.

2.4.3. Energy Distribution Among Different Orders

Until now, we have been considering the grating formed by infinitely thin slits. If the width of each slit is 2a, we cannot assume that the light diffracted by each slit travels in all directions with equal irradiance but according to the diffraction pattern for a single slit [Eq. (2.28)]. Thus, the irradiance in a direction O will be the product of Eq. (2.42) times the square of Eq. (2.28): sin'( sin

y) (Ifi)

sin ka(sin O1 -sin 0,) [ka(sin O1 -sin O,)] '

(2.54)

We have plotted this irradiance in Fig. 20(a) for a grating with five slits of widths 2a = d/3. A line with some order of interference does not appear when its direction coincides with the direction of a zero of the second factor in Eq. (2.54). The zeros of the second factor may be shown to occur at angles such that 2a(sin O1 -sin Oz) = nh,

n = a n y integer excluding zero.

(2.55)

The line positions are given by Eq. (2.38). Thus, the missing order numbers are: d 2a

m = n--,

(2.56)

where n is any integer, excluding zero, such that m is also an integer. The influence of the slit width is illustrated in Fig. 21, which shows the diffraction patterns for two slits with different widths but with the same separation. 2.4.4. Ghost Lines in Diffraction Gratings

Often spurious spectrum lines are seen in grating spectra, due to nonconstant separation in the slits. There are two main types of these so-called ghost lines.

76

DIFFRACTION A N D SCATrERING

-6

-4

-2

I

0

2

4

6

m

(B) FIG.20. lrradiance distribution in a grating with slits with finite width. ( a ) Finite-size grating. ( b ) Infinite-size grating.

Rowland ghosts,59 which have been studied by Michelson"" and Anderson,6' are due to periodic errors in the spacing of the slits because of imperfections in the ruling engine. These spurious lines appear as two faint lines symmetrically located with respect to the main or parent line. Lyman ghosts have also been studied, among others, by Anderson6' and by Runge.62They are due to nonperiodic errors in the line spacing. These lines are usually observed at large distances from the parent lines.

DIFFRACTION GRATINGS

0

5 LI1

m

rn

5

5 0

77

78

DIFFRACTION A N D SCATTERING

A ) T R A N S M I S S I O N TYPE

B ) REFLECTION TYPE

FIG. 22. Basic types of diffraction gratings.

There are some other types of spurious lines and ghosts that may appear in a diffraction grating spectrum, but all of them are due to ruling imperfections. 2.4.5. Different Types of Diffraction Gratings

There are reflecting as well as transmission diffraction gratings, as shown in Fig. 22. Both of these gratings usually have the groove faces made at an angle (called the blaze angle) from the plane of the grating. The purpose of the blaze angle is to shift the maximum of the light distribution among the different orders, from the zero order to some other desired order, as first described by W00d63.h4and by Babco~k.'~ To avoid absorption and other undesirable effects due to the light transmission through glass, the grating type used most is the reyedon grating. The blaze angle a in this case is given by 6 ,- 6 2 2 '

(2.57)

y = o1 62.

+

(2.58)

mh - 2 cos Y sin a. _ d 2

(2.59)

a=-

and the total deflection angle is

Then, from Eq. (2.38), we find:

A case of special interest arises, y

= 0;

( a = el = &), reducing Eq. (2.59) to

mh = 2 d sin a = 2 h ,

(2.60)

RESOLVING POWER O F OPTICAL I N S T R U M E N T S

79

GROOVE

\

.

B L A Z E D GRATING

I / ECHELLE GRATING

FIG.23. Blazed reflection grating and echelle grating.

where h is the step height as shown in Fig. 23. The blaze angle is often specified as the wavelength that fulfills this condition in the first order. WoodG6pointed out that very efficient gratings for the infrared could be obtained by using coarse-blazed gratings with this condition ( y = 0). Since the light is reflected on the grooves' faces in a mirror like manner, he called them echellettes, Harrison6736x considered that since resolving power depends only on the total grating width, the echellette could be made with spacings as large as about 100 grooves/mm. The typical blaze angle for these gratings, named echelles by Harrison, is 63"26'. Echelles are very useful in order to achieve high resolving powers in the infrared. Gratings can also be concave and can be used in many configurations that will not be described here. The reader is referred to the book by Sawyer69 for a more complete description. The art of grating manufacture is a very complicated and interesting one. The reader can consult the excellent papers by Harrison,67768S t r ~ n g , ~ and " Stroke7' for details. In closing the subject of diffraction gratings, it should be mentioned that there are also two- and three-dimensional gratings. Figure 24 shows the diffraction patterns for two two-dimensional arrays of apertures.

2.5. Resolving Power of Optical Instruments Because of diffraction effects, the resolving powers of all optical instruments, even if they have no geometrical aberrations, are limited by the sizes of their apertures.

80

DIFFRACTION A N D SCATTERING

FIG.24. Diffraction of a two-dimensional array of holes. (a) A regular array and (b) its diffraction pattern. (c) An irregular array, and (d) its diffraction pattern.

If a lens is free of aberrations of any kind, its resolving power is determined by the size of the Fraunhofer diffraction pattern produced by the lens source in the Airy disc. We can see that the angular resolution of a lens in a telescope is limited only by the diameter D of the lens. If we define the image distance ratio ( L ' /T )as (L'/ D ) , which becomes equal to the focal ratiof/T when L ' = J we may write the linear radius of the Airy disc on the image plane as: r = 1.22A

(b).

(2.61)

Let us consider two very close stars, with their Airy discs overlapping in the image plane. The light of the two stars is mutually incoherent, so that there is no fixed phase relation between the two images. The resultant combined image is then obtained by adding the intensities of the two images.

RESOLVING POWER O F OPTICAL INSTRUMENTS

SUM

81

OF TWO IMAGES

-

RADIUS OF AIRY DISC

FIG. 25. Two point images just resolved.

Rayleigh established that under these conditions of incoherent illumination, the two images are just resolved when they are separated by a distance equal to the radius of the Airy disc, as illustrated in Figs. 25 and 26. Until now, we have considered images produced by lenses with no geometrical aberrations that produce perfectly spherical wave fronts. When the wave fronts have deformations smaller than A / 4 (Rayleigh criterion), the image is not much different from that produced by a perfect lens, as shown by Marechal.’* If the wave-front deformations are large compared with the wavelength, the image is that predicted by geometrical optics. The intermediate case, however, is extremely complicated, as described by Marechal and F r a n ~ o n Figure .~ 27 shows the appearance of an image of a point source in the presence of aberrations. The angular resolution of a telescope increases, that is, it resolves a smaller angle when the diameter is increased. However, a telescope with a resolution greater than that permitted by the atmospheric “seeing” or turbulence is unnecessarily precise, because in this case the resolution limiter will be the seeing and not the telescope. Depending on the geographic place, the seeing is of the order of a few tenths of a second of arc, which may be matched by a telescope with a diameter of about 40cm. To take full advantage of the resolution of a telescope with diameters much larger than 40 cm, we have to use it outside of the terrestrial atmosphere. However, very large telescopes are made because the light-gathering power increases with the square of the diameter.

82

DIFFRACTION A N D SCATTERING

F i c . 2 6 . Two point images with equal irradiance. ( a ) Unresolved. ( b ) Just resolved. (c) Completely resolved.

2.5.1. Optical Transfer Function

The different points in the image formed by a photographic lens are also mutually incoherent. Therefore, we say that the object is incoherently illuminated. The quality of the image in a photographic lens is seldom completely determined by the maximum resolving power of the lens. For example, two lenses might be able to resolve the same fine detail, but with very different contrast. As in electronic amplifiers, it is not enough to know

RESOLVING POWER O F OPTICAL I N S T R U M E N T S

83

ill

0)

C

.-.g D 0

L

d

P N

! i

84

DIFFRACTION A N D SCATTERING

the high-frequency cutoff of an audio amplifier, but the complete frequency response curve must also be known. The optical equivalent for lenses of the frequency response curve for electronic amplifiers may be found by defining the spatial frequency content of an image. To illustrate this concept, we may take any picture and take a measurement of the irradiance along a straight line. The graph of the irradiance, with respect to x, may be considered, by means of Fourier analysis, as the superposition of many sine waves as shown in Fig. 28, with different wavelengths L. Then, we define the spatial frequency w as: w=-

2T L’

(2.62)

The frequency response curve of a lens described first by O’Nei1173 in 1954 is called the optical transferfunction (OTF) and is defined as the ratio of the harmonic components in the image to those in the object. It has two

m

-wK 3 0 Lb

FIG. 28. Spatial frequency content of an image.

85

RESOLVING POWER O F OPTICAL I N S T R U M E N T S

parts. The plot of the amplitude ratio versus spatial frequency is often called the modulation transferfunction (MTF). The second part, the plot of the phase ratio, is less frequently used. The best way to measure the MTF is to form the image of an object that contains all spatial frequencies in equal amounts. The plot of the amplitudes of the spatial frequency components in the image thus formed would then represent the frequency response of the lens. An object that contains all spatial frequencies in equal amounts is a point source, as may be proved without much difficulty. The image of this point source is called point spreadfunction. The point spread function is the same as the Fraunhofer diffraction pattern of the lens aperture only if the lens is perfectly free of geometrical aberrations. We shall assume a point source at infinity. The point spread function in the focal plane of a lens with focal length F, with coordinate x F and y F can be obtained from Eq. (2.23) if we first obtain the amplitude as follows: A(XF, yF) =

II-:

T ( ~y ), e ~ ( k / F ) ( r ~ f i + i ,)i ldx dy,

(2.63)

where T ( x ,y ) represents the pupil function, including any deformations of the wave front due to aberrations of the lens. The point spread function S ( x F ,y F ) is then given by:

and therefore, from Eq. (2.63):

.

[I

1

T * ( x ,y ) e - l ( k I ~ ) ( X X f i + Y v F ) d x d y . -‘m

(2.65)

The variables x and y in these integrals do not remain after the integration and, therefore, may be replaced by the symbols 2 and J in the second integral, so that we may move the entire expression under the same integral signs.

(2.66)

Once the point spread function is computed from this expression or becomes known by some other means, the optical transfer function F ( w , , w , , ) may be computed by using the Fourier transform of S ( x F ,y F )

86

DIFFRACTION A N D SCATTERING

as follows: F(w,,w,) =

I:I

s ( x F y, F )e - “ W \ X F + ’ > ” ,

)

dXF dYF.

(2.67)

There is a simpler way to compute the OTF when T ( x , y ) instead of S ( X , , y F )is known, as will be described here. If the function F ( w , , w , ) is known, the point spread function can be computed by taking the Fourier transform of F ( w , , w ) ) , as follows:

Then, except for an unimportant constant, Eq. (2.66) becomes identical if we define: w,

k =-(x-Z) F

(2.69)

and

k F

w , =- ( Y

-?I,

(2.70)

obtaining thus:

If a lens is perfectly free of aberrations, we can set T ( x , 1 ) = T*(x,y ) = 1 inside the clear parts of the aperture. Then F ( q , w.,.) is the common area of two sheared images of the aperture, as in Fig. 29. Summarizing, the optical transfer function is the two-dimensional Fourier transform of the point spread function and also the autocorrelation of the pupil function. In Fig. 30, we have the optical transfer function of a perfect circular lens for various defocusing magnitudes. The cutoff frequency in focus is given by: kD F

w, =-.

(2.72)

2.6. The Abbe Theory of the Microscope When we defined the resolution of a lens, we assumed that the two neighboring points to be resolved were perfectly incoherent to each other.

87

T H E A B B E T H E O R Y OF T H E MICROSCOPE

COMMON AREA: F(wx,wy)

FIG.29. Computation of the optical transfer function of an aberration-free lens.

\ \ \

INFOCUS-,

\

h I

0

-.2

0

I

0.2

1

I

0.4

I

I

I

0.6

dr

0.0

(i&P FIG.30. Some optical transfer functions.

88

DIFFRACTION A N D SCATTERING

FIG.

31. Abbe theory of the microscope.

The appearance of the image is quite different if the two points are coherent. This is, for example, the case if the object is illuminated by a single point source or a laser. Ernest Abbe74 developed a theory, usually called the Abbe theory of the microscope, which allows the computation of the resolving power of a lens when the object is illuminated with coherent light. His main assumption was that the spatial frequency structure of the object to be imaged acts like a diffraction grating when illuminated by a coherent light source, as shown in Fig. 31. If the object acts as a diffraction grating, the light will be diffracted into many beams with different orders of diffraction. In this case the image may be regarded as the interference pattern arising from the superposition of all diffracted beams. If the lens is so small that only the zero order of diffraction passes through the lens, no interference takes place at the image plane, and therefore no image is formed. The object structure is first detected at the image plane when the lens is large enough to let the two first orders of diffraction pass through the lens. Under these conditions, the image is not an exact reproduction of the object, because the orders of diffraction with magnitude greater than 1 are not reaching the image plane. The image, in this case, has a sinusoidal profile with a spatial frequency equal to that of the object (see Fig. 32). When the lens is larger, so that the order of diffraction + 2 and -2 passes through the lens, the image will more closely resemble the object, as shown in Fig. 32.

89

T H E A B B E T H E O R Y O F T H E MICROSCOPE

FIRST ORDER IMAGE

OBJECT

(A)

THIRD ORDER IMAGE

(B)

(C)

FIG.32. Explanation of Abbe theory.

The resolving power limit may be easily found by using the diffraction grating equation, (2.38) for the first order rn and normal incidence:

d=- A sin 8’

(2.73)

where A, is the wavelength in the object medium, d is the minimum object diameter, and 8 is the angular radius of the lens, as seen from the object. Alternatively, we can write this diameter d as (2.74) where A. is the wavelength in vacuum, N is the refractive index in the object medium, and NA is the numerical aperture of the lens. We can see that the resolution is greater when the object is illuminated with incoherent light. The resolution with coherent illumination is almost half of that obtained with incoherent illumination. Another difference between illumination by coherent and incoherent light is that the optical transfer function in the coherent case is constant for all spatial frequencies, dropping suddenly to zero at the cutoff frequency, while in the incoherent case, it decreases continuously as the frequency increases. It must be pointed out, however, that the differences between the two types of illumination are so complex that a superficial examination must be avoided. Comparison should be limited only to certain particular aspects. The resolution in the case of partial coherence (illumination with a mixture of coherent and incoherent light) is very interesting and has been studied by many authors, like

90

DIFFRACTION A N D SCATTERING

Ar2

A6 2

A) NORMAL MICROSCOPE

B ) P H A S E C O N T R A S T MICROSCOPE

FIG.33. Amplitudes in a phase-contrast microscope.

2.6.1. The Phase-Contrast Microscope

A logical application of the Abbe theory of the microscope is the phasecontrast microscope developed by Zer~~ike,’~.’’ (he received the 1953 Nobel Prize in Physics for this development). This microscope permits the observation of transparent objects without staining. It is only required that the object’s index of refraction be different from that of the medium in which it is immersed. If the object has a refractive index different from that of the surroundings, a wave front passing through it becomes deformed because of the different optical paths. Thus the light becomes diffracted as in any other kind of object. No image is visible, however, because the interference at the image plane produces a constant amplitude everywhere. To understand this, we might consider (see in Fig. 33) that the amplitude A, at a given point on the image is the sum of the nondiffracted amplitude A,, (zero order) and the diffracted amplitude Ad (nonzero order). The diffracted amplitude changes from point to point on the image, because it contains the information about the object structure. The nondiffracted amplitude is constant in phase and amplitude everywhere on the image plane. The resultant amplitude A, has the same magnitude for all image points, even though Ad and the phase difference A 6 are variables. The object is transparent, but by shifting the phase of the nondiffracted wave by an angle 7r/2, as we see in Fig. 33, the resultant amplitude A, will be a function of the phase shift As. The object then becomes visible, especially at the edges, where the light is diffracted. As shown in Fig. 34, the phase shift in the zero order of diffraction is obtained by means of a “phase plate” placed at the back focus of the microscope objective. The phase plate is a thin glass plate, on which a ring

91

THE ABBE THEORY O F THE MICROSCOPE

CON DENSER

ANULAR DIAPHRAGM

OBJECTIVE

OBJECT

PHASE PLATE

IMAGE PLANE

FIG. 34. Optical arrangement in a phase-contrast microscope.

of some dielectric material has been evaporated to produce the phase shift. The illuminating system has a diaphragm with an annulus aperture and a condenser. This illuminating system is designed so that all the nondiffracted light passes through the ring on the phase plate. The diffracted light tends to pass out of the ring on the phase plate. Figure 35 shows the objected observed in a normal and in a phase-contrast microscope. 2.6.2. Spatial Filtering of Images

This is a technique based on the Abbe theory of the microscope. We have seen before that any irradiance distribution may be considered as the

(A)

( B)

FIG. 35. Pictures taken using a microscope. (a) Normal. (b) Phase contrast.

92

DIFFRACTION A N D SCATTERING

FRONT FOCAL PLANE OF FOURIER L E N S

BACK FOCAL PLANE OF FOURIER L E N S

FIG.36. Spatial filtering of images.

superposition of the light diffracted by the many spatial frequencies contained in the object. Alternatively, it can be said that each order of diffraction contributes to the formation of the image with a different spatial frequency. From the point of view of Fraunhofer diffraction, it is important to remember that the angular distribution of the light diffracted by an aperture is the two-dimensional Fourier transform of the amplitude distribution over the plane of the diffracting screen. Because of this property, if a convergent lens is illuminated with a collimated beam of light, the light distribution on the back focal plane is the two-dimensional Fourier transform of the amplitude distribution over the lens plane, or pupil function, except for a phase factor. The phase factor can be eliminated in order to obtain the exact Fourier transform if the diffracting object is not in contact with the lens, but in front of it on the front focal plane, as shown in Fig. 31. The spatial filtering of images is used to analyze the spatial frequency content of any image and even to remove undesired frequencies by means of appropriate stops at the back focal plane of the Fourier lens. This is usually done with the optical arrangement in Fig. 36. The object to be filtered is at the front focal plane of the lens and the two-dimensional Fourier transform is at the back focal plane, where the filtering stops are placed. Examples of spatial filtering are shown in Figs. 37 and 38. There have been a great number of applications for the use of this technique in the analysis of many graphic results. For example, in seismic studies where a Fourier transform of the picture is desired. It is often used to remove the noise from a picture if the spatial frequency of the noise is known.

2.7. Scattering Most of the light that our eyes receive comes not directly from the light source but through scattering. Looking at almost any object, what we see

SCATTERING

93

FIG.37. Spatial filtering of a two-dimensional grid.

is the light scattered from its surface. If we look at the blue sky or at clouds, we also see scattered light. These examples give us an idea of the great importance of scattering phenomena. Furthermore, the subject is very complicated, and many good books have been written covering it, for example, those by Van de Hulst” and Kerker.79 Scattering phenomena may be classified according to the size or shape of the scattering object or to the wavelength. In general, scattering is an anisotropic phenomenon, depending on the shape and orientation of the

94

DIFFRACTION A N D SCATTERING

FIG.38. Spatial filtering of a picture with TV scanning lines

object and on the orientation of the plane of polarization. The shapes of scattering objects which are most easily analyzed mathematically are spheres, ellipsoids, and cylinders. Miex" developed a very complete theory for scattering by spheres of arbitrary size, using electromagnetic theory. 2.7.1. Scattering by Large Objects

Depending on the size of the object, several models could be used to explain scattering. If the object is very small compared with the wavelength, the electric field of the wave forms electric dipoles with each particle. This theory will be considered in Section 2.7.2. If the size of the object is of the order of the wavelength or larger, the main mechanism producing scattering is diffraction. As shown in Fig. 24, if a scattering cloud is formed of many spherical particles of the same diameter, their size can be found by examination of the diffraction pattern. Using this idea, Young designed an instrument called eriometer to measure the size of blood corpuscles.

SCAnERING

95

Young's eriometer does not work for particles of varied sizes, because the Fraunhofer diffraction pattern is formed from the irradiance sum of all the individual patterns. Parrent and Thompson" and Silverman et aLX2 pointed out this problem and proposed to observe the pattern in the far field (s >> n 2 / A ) of the individual particles but close enough to be in the near field of the whole sample. Under those conditions, the diffraction patterns of each particle will be in different places instead of all being superimposed. It is interesting that the diffraction patterns are not Airy patterns, because they may be interpreted as Fresnel patterns, or alternatively, as Fraunhofer patterns with a coherent background. If the object is very large compared with the wavelength, the mechanisms producing scattering are diffraction on the edge of the object and refractions and reflections inside it. These particles may be water drops, such as in fog, rain, mist, and haze. The rainbow is one example of scattering by water drops. This phenomenon has been considered by poets as well as scientists. The scientific description is often assumed to be an exercise in geometrical optics, but it is much more complicated than that, as shown by Khare and Nussenzveigs3 and Van de H ~ l s t . ' ~ The best approximate theory to explain the rainbow was formulated by Airy. An exact but complicated theory is obtained applying the general Mie theory. The glory is another beautiful atmospheric phenomenon due to scattering by water drops, as described in the book by Mir~naert.'~ It is observed as a rainbow surrounding the sun or moon, covered by a thin veil of clouds. More often, persons standing on a high point, observe their shadows projected onto low clouds, showing a colored ring around each of their heads. The glory can be seen by each observer, with the ring only around his or her own head. 2.7.2. Scattering by Small Particles

When an electromagnetic wave passes through matter or a cloud of particles much smaller than the wavelength, its electric field acts on each molecule or particle, displacing the positive charges in the opposite direction to the negative charges. The resulting deformed molecule or particle may thus be considered as an electrical dipole, formed by two charges of opposite sign joined by an elastic force. An electrical dipole is the system formed by two charges to a -9 with masses M and m, respectively, and joined by an elastic force ky, where k is a constant, and y is the separation of each charge from the equilibrium position. We shall assume that M >> m.Thus, if the electric charges are free

96

DIFFRACTION A N D SCATTERING

of any friction or driving force, their movement equation would be m -d=2Y- k dt2

(2.75)

’

with the solution y

= A e-‘wo‘

(2.76)

which, substituted into Eq. (2.75), gives the characteristic vibration frequency w =27ruo=

&

(2.77)

where vo is called the resonance frequency. Let us assume that an electromagnetic wave with the electric field

acts as a driving force for the dipoles. With this force, the movement equation would become (2.79)

The choice of sign for the driving force is made by taking into account that the electron feels the force in the direction opposite to E, because of its negative charge -9. However, this equation is incomplete, because we know that any accelerated charge radiates, and therefore the dipole charges will emit an electromagnetic wave. Due to this radiation, the electrons experience some kind of reaction force that opposes their movement. This reaction force has been shown by Lorentz to be d3y F , = - . -2q2 3~x10’ dt3’

(2.80)

Thus, the complete movement equation is d3y m -d- 2Y - = - k y + - 2q2 --qEo dt2 3cx lo7 dt3

e-””.

(2.81)

The effect of the magnetic field of the driving wave is negligible compared with that of the electric field and thus it is ignored. By proposing again a solution of the type Y = A e-‘”‘

(2.82)

97

SCATTERING

and by substituting the value for the amplitude, A is found to be equal to

A=

-9Eo

m[wi-w2- iyw3/oi]'

(2.83)

where y is given by (2.84)

Therefore, we see that the dipole oscillates with a frequency equal to that of the driving wave. In order to study the amplitude and phase of the dipole oscillation, we shall consider three different cases. We assume that y w 3 / w i is very small compared with ( w 2 - w ' ) . (1) If w << w o , as is the case for most materials at frequencies in the visible range, A is negative, and hence y is opposite in sign to E. Thus, the phase of the dipole vibration is the same as that of the driving wave, as shown in Fig. 39. We say that the electrons vibrate in phase with the driving wave if they always move following the force they feel, that is, opposite to the driving electric field. (2) If w >> w o , as in the case of free electrons or most materials at X-ray frequencies, A is positive. Thus, the dipole vibration is 180" out-of-phase with respect to that of the driving wave, as shown in Fig. 39. DRIVING ELECTRIC FIELD

W<<Wo

w 0

a

FIG.39. Phase of the vibration of dipoles driven by an electromagnetic wave.

98

DIFFRACTION A N D SCATTERING

I DIPOLE I

1 FIG.40. Radiation pattern for a dipole.

( 3 ) If w = w o , that is, near the resonance frequency, the value of A becomes purely imaginary. Thus, the dipole vibration has a very large amplitude and is 90" out-of-phase with respect to the driving wave. When the dipole vibrates, it emits an electromagnetic wave with an irradiance ( S ) that depends on the direction being considered. The symbol ( S ) represents the time average of the Poynting vector, in a direction as follows:

(S)=

x 10-~

q2w4 8.rrcr2

- A: sin2 6,

(2.85)

where 6 is the angle between the given direction and the dipole orientation. The angular distribution of the dipole radiation is plotted in Fig. 40. The complete radiation pattern would be obtained by revolving the pattern about the dipole axis, thus obtaining a toroidal distribution. The irradiance ( S ) is inversely proportional to the square of the distance r from the dipole to the point under consideration. The symbol A represents the maximum amplitude of oscillation of the dipole, as given in Eq. ( 2 . 3 8 ) ; thus, if A,?,= A . A*,

q4E:x (S)=

8.rrm2cr2

.

w 4 sin2 6

(2.86)

99

SCAITERING

In the vacuum, using the values of the refractive index n = 1 a n d po= 47r x lo-', the time average of the Poynting vector may be written as:

(2.87)

Therefore, Eq. (2.86) may be transformed into

The emitted wave has a phase difference kx with the electron vibration only at distances from the dipole that are greater than the wavelength. At very small distances from the dipole, the emitted wave has a phase shift of 180" with respect to the dipole vibration. The reason for this phase difference at small distances is that at low frequencies, the charges in the dipole move in a direction such that the electric field produced by the dipole tends to cancel that of the driving wave. 2.7.3. Rayleigh Scattering

Let us think of a light wave entering a gas where the individual molecules are randomly distributed with a n average distance between them that is larger than the wavelength of the light. If the size of the particles is much smaller than the wavelength of the light, we have Rayleigh scattering.' In this case, we may think that each molecule o r particle absorbs the luminous energy a n d then reemits it in all directions. Due to the random distribution of the molecules, the light emitted by each particle is assumed to be incoherent with the light emitted by the others. The energy scattered in a direction 0 by each particle is found by means of Eq. (2.89). The total energy W scattered per unit time by the particle may be found by integrating ( S ) over a sphere with radius r, in the center of which is the particle, as shown in Fig. 41. The total energy W emitted in all directions, per unit time, is given by:

w=

I,:

(S)(27rr sin e ) r do,

(2.89)

and, using Eq. (2.88), we obtain by integrating 8 T q 4 w 4 10-'~(s~) ~

W=

3m2[ ( w t - w ' ) ' +

I)$(

2,

(2.90)

100

DIFFRACTION A N D SCATTERING

r sin t? 4

-1

I I I SCATTERED LIGHT

INCIDENT LIGHT

FIG.41. Total scattering by a molecule.

which may be written in the form:

where (2.92)

The quantity u represents the units of area. Therefore, we could think of the molecule as having an apparent area u that scatters only the light that falls on it. For this reason, the quantity u is called the scattering cross section. Here we can actually distinguish the following three kinds of scattering: (1) If w = w " , as in the case of most materials illuminated with visible light, (T can be approximated by U =

8Tq4w4 10-1~ 3m'w:

(2.93)

This Rayleigh scattering is directly proportional to the fourth power of the frequency.

SCATTERING

101

( 1 ) If w >> w o , as in the case of most materials illuminated with X-rays or free electrons illuminated with visible light, we can approximate u by

(2.94)

This is called Thompson scattering, and the magnitude of u is independent of the frequency of the light in this case.

FIG.42. Two pictures taken with (a) normal black-and-white film, (b) infrared film and filter.

102

DIFFRACTION A N D SCATTERING

(3) If o = w o , a condition that occurs only if the frequency of the light is near the resonance frequency of the material, we approximate by writing W’-W;=~O~ (w-wO), thus: 2.rrq4w2x 1 0 - 1 ~

(2.95)

We see that if u = w o , the magnitude of the scattering becomes very large. The resonance frequencies for most materials are in the ultraviolet region. This kind of scattering is called resonance scattering. The fact that the sky is blue is due to the o4 dependence of Rayleigh scattering of visible light by the air molecules in the atmosphere. In other words, the blue light is scattered much more than the red light. For the same reason, the sun or the moon look red near the horizon, because the blue light emitted by the sun or moon is scattered away and only the red light reaches the observing eye. Any distant object, for instance a mountain, is seen much more clearly through a red filter than through a blue filter. This is the reason for using infrared photography instead of the normal visible photography for distant landscapes (see Fig. 42). In closing this chapter, we must mention that the measurement of scattering are very useful tools that permit us to obtain interesting information about the structure of the scattering particles. This tool is especially useful in the fields of chemistry, meteorology, and astronomy, as is shown by Van de Hulst.’’ Acknowledgment The author wishes to thank Professor W. H. Steel for his very valuable comments.

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DIFFRACTION AND SCATTERING

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REFERENCES

105

81. G. B. Parrent, Jr. and B. J . Thompson, “On the Fraunhofer (Far Field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background,” Oprica Acra 11, 183 (1964). 82. B. A. Silverman, B. J. Thompson, and J. H . Ward, J. Appl. Meteor. 3, 792 (1964). 83. V. Khare and H . M. Nussenzveig, Phys. Rev. Lert. 33, 976 (1974). 84. M. Minnaert, The Nature of Light and Color in rhe Open Air, Dover, New York, 1954.

This Page Intentionally Left Blank

3. OPTICAL POLARIZATION Frederic R. Stauffer Sacramento Peak Observatory Sunspot, New Mexico 88349

3.1. Introduction Polarization effects are a result of the wave interaction of light with the physical world and have long been demonstrated with a double refractive index crystal known as calcite. Light waves in two orthogonal planes are laterally separated after passing through calcite, resulting in two polarized images. Christian Huygens in the late seventeenth century experimented with and successfully described calcite’s double images by using wave-front constructions. But, at this time, light was thought to be a longitudinal wave like sound, and polarized light remained undiscovered. In 1808, E. L. Malus discovered that reflected light viewed with a properly oriented calcite crystal passed a single image and extinguished the other image. Malus concluded that the reflected light was polarized. In 1815 Brewster showed the significance of the incidence angle on the polarization of the reflected beam. Maximum polarization (lOOo/~ for dielectrics) occurs at an incident angle, now termed Brewster’s angle, depending on the material’s determined optic parameters. In 1816, Fresnel and Arago demonstrated with Brewster angle polarizers that two orthogonal polarized beams did not interfere, which led to the important conclusion that light is a transverse wave propagation. Fresnel proceeded to simply state the polarization effects produced by light’s reflection and refraction at boundaries between materials further supporting light’s wave nature. Later in the century, Maxwell’s and Faraday’s efforts produced Maxwell’s equations describing the fundamental interrelationship between the basic electromagnetic vectors. The necessary theories to understand polarization were complete.’.’ The theories are based on observations. To understand polarization, it is best to experiment with its different aspects. Computer optical design programs have also made the polarization equations easier to use.

107 METHODS OF EXPERIMENTAL PHYSICS Vol. 26

Copyright 63 1988 hy Academic Press. Inc. All rights of reproduclion in any form reserved I S B N 0-12-475071-8

108

OPTICAL POLARIZATION

3.2. Electromagnetic Description of Light 3.2.1. Field Equations and Wave Equations

Maxwell’s equations are vector and scalar products written in terms of the basic fields, the electric field E and the magnetic induction B, and in terms of the derived fields (due to the fields’ effects on matter), the electric displacement D and the magnetic field H, taking into account the average contribution of the charge distribution p and the electric current density J. To determine the interaction of light with matter, Maxwell’s equations need to be supplemented with equations describing the effect of an electromagnetic field on matter known as the material equations. Generally, these equations can be written for complicated three-dimensional anisotropic and/or relativistic situations. But, assuming static and isotropic conditions makes the problem simpler and realistic, and the material equations reduce to constant relations between the electromagnetic vectors. The quantities needed to describe the effect on matter are the dielectric constant or permittivity E, permeability p, and conductivity u. VxE+B=O VXH-D=J

Maxwell’s equations

(3.1)

V-D=p

V.B=O

D = EE B=pH

material equations

J=oE where E , p and u may vary with wavelength. Using the material equations and vector identities to separate the cross product equations and assuming the medium is homogeneous results in a second-order differential wave equation describing the space-time distribution for an electromagnetic field. V2E= PEE+ puE.

(3.2)

Plane and spherical wave solutions exist for this differential equation. Plane waves in Cartesian coordinates are useful in describing electromagnetic effects and are used as the example. For dielectrics, the conductivity is zero,

ELECTROMAGNETIC DESCRIPTION O F LIGHT

109

and the wave equation simplifies to

1

=C

6= -V1

for a vacuum for a dielectric

A =wavelength

s = propagation direction unit vector

k = ks S

= constant

phase term V * E=O V * B=O

B=JCLESXE E - volts/meter D - coulombs/(meter)2

B - tesla

H - amps/ meter The plus or minus exponential of the wave solution means that waves can propagate in two directions. It is important to define and use a sign convention for consistent results. The sign convention in the literature varies, but a standard for polarization work has been defined. The convention used is light traveling in the negative direction or a positive time d e p e n d e n ~ e . ~ The propagation direction for the electromagnetic field is s, and r is the vector distance from an arbitrary origin to a plane in space of constant phase. The phase term 6 offsets the wave front in space for a fixed time. Applying Maxwell’s equations to the solutions for E and B shows their divergence equal to zero and thus oscillating in a plane perpendicular to the propagation direction.

110

OPTICAL POLARIZATION

The constant electric field plane intersecting at r is defined by k * r, and the planes with equal electric fields are spaced apart by the wavelength A. A snapshot of a plane wave or a fixed position in time shows the electric field constant across planes perpendicular to the propagation direction; and the electric field would be a sinusoidal modulation along the propagation direction. Another view is to fix a point in space and to allow the time to vary. Then, the electric field would be constant across the plane perpendicular to the propagation direction, and it would be time-dependent and modulated by w t as the plane waves go by. A coordinate system about the propagation direction and a plane projection can be used to describe the resultant electric field. A linear resultant oscillation is termed linear polarization. Linear polarizations with varying azimuth can be constructed by combinations of linear orthogonal components shifted by zero or multiples of half-wavelengths corresponding to a phase shift of zero or integral multiples of r. For example, equal amounts of horizontal, E H , and vertical, Ev, linear polarizations with zero phase difference give a resultant linear polarization with a 45" azimuth, A. The linear polarizations add together as a planar electromagnetic field distribution along the propagation direction propagating at light's velocity. The resultant plane projection of the light beam results in a linear vector.

E = EH e - J ( k

r-wr)

+ E~

e-J(k

e") e

E =(E H E H +

r-wr-8)

"k'r-"'

S=O

(3.4)

A = arc tan( Ev/ E H )

where E~ and E~ are the dialectric constants in the horizontal and vertical directions. Now, if the phase difference is changed to an odd multiple of 7r/2, equal intensity orthogonal linear polarizations add together, resulting in a helical electromagnetic field distribution along the propagation direction propagating at light's velocity. The resultant plane projection of the light beam results in a circular spinning vector produced by the helical distribution moving through space. E = ( E H E+~E , , E ~e J * )e-J(k r - w r ) S=~ / 2 ,

E,

EH

= Ev

for circular polarization,

=E ~ ( f E E~V eJTI2) e-J(k r - w r )

(3.5)

A right-handed helix, E-, produces a clockwise rotating plane projection looking towards its source, and vice versa for a left-handed helix, E,. These spinning projections define right- (R)and left- (L) handed circular polariz-

ELECTROMAGNETIC DESCRIPTION OF LIGHT

111

ations, re~pectively.~ Physically, Ev is advanced or retarded with respect to EHby an odd multiple of A/4,for example by an anisotropic medium such as calcite. Right and left circular polarizations describe another orthogonal coordinate system perpendicular to the propagation and give another perspective to view polarization. Normalized circular polarization vectors are, &,

E(r,

= (2)l/z(EHfjEV)

e-~lh.r-uf)

+E-E-) epJ'h'r-wf)

1 ) = (E+&+eJd

(3.6)

The oppositely spinning E+ and E- vectors add around their paths, resulting in an ellipse whose major axis, a, is the sum, and whose minor axis, b, is the difference of the intensities. The major axis is tilted at an angle A, which is half of the phase difference between the vectors. Figure 1 shows various combinations of L and R and phase. a=E++E

b=E+-E ellipticity = b / a handedness:

E- > E,

right handed

E-

indeterminate

= E,

E- < E,

left handed

azimuth = A = 6/2

L/E, = 1.0

E,

E_

Result.

00- 00./00.1

6 = 0.0'

6

= 90.0'

6 = 180.0'

FIG. 1. Plot of L and R polarizations with different phases and amplitudes and the resultant vector. The polarized light sources are in the page, and the resultant vector is produced by adding the sources.

112

OPTICAL POLARIZATION

3.2.2.Poincare Spherical Description It takes three independent values to determine the polarization state of the monochromatic wave. For example, the amount of left and right circular polarizations and their phase relationship describes the ellipticity, azimuth, and the handedness (the resultant vector’s rotation direction). A threedimensional spherical surface for a given radius or beam intensity can be used to map the three values needed to describe polarization. Orthogonal polarizations, such as L and R or H and V, are at opposite ends of the sphere’s diameters. The poles of a sphere are taken to represent pure L or R polarization with increasing amounts of the other component added until equal amounts of L and R polarization meet at the equator, producing Linear polarization with a varying azimuth, as in Fig. 2. This results in ellipticities ranging from - 1 to 0 to +1, or from pure R at one pole, to linear polarization at the equator, to pure L at the other pole. The sphere’s latitude, LAT, is related to the ellipticity and the handedness. The longitude, LONG, is equal to twice the azimuth of the elliptical polarization or the phase difference between the L and R components.

tan

(y)i, a =

LONG = 2A.

L

+45.0’

-45.0

R FIG. 2. Polarization ellipticity and handedness at positions as mapped on the Poincare sphere. The handedness is L for the upper and R for the lower hemisphere. The sense of rotation for handedness is as in Fig. 1, with the polarized light source in the page.

WAVE PROPAGATION IN ISOTROPIC MEDIA

113

An elliptical polarization is a point on the sphere, P(LAT, LONG). Cartesian coordinates can describe the sphere’s latitude and longitude, too. Taking the axes z through R, x through H, and y through +45”, linear polarization gives three equations to mark a polarization, P ( x , y, z). x = cos( LAT) sin( LAT) y = cos( LAT) sin(LONG)

(3.7)

z = sin( LONG)

PoincarC first suggested the description, and the sphere is referred to as the Poincare‘ sphere.’

3.3. Wave Propagation in Isotropic Media 3.3.1. Fresnel Equations

When light is incident on a dielectric medium with different constants r) and p ’ ( w , r), Maxwell’s equations and their continuity at the boundary need to be considered for waves parallel to the plane defined by the boundary’s normal and the incident beam’s propagation direction (also known as P, T , or TM-transverse magnetic waves), and for waves perpendicular to this plane (S, a,or TE waves). Recall that the tangential components of the electric vector are continuous across the boundary, the normal component of the electric displacement changes across the boundary due to surface charges, the normal component of the magnetic induction is E’(o,

FIG.3. Sign convention for the Fresnel equation interpretation and derivation. r, is perpendicular and r,, is parallel to the page. 0 is a vector out of, and x is a vector into, the page.

114

OPTICAL POLARIZATION

continuous, and the tangential component of the magnetic vector changes across the boundary due to a surface current density. From the boundary conditions and measuring angles about the boundary’s normal, it is found that for a beam incident at an angle equal to di,( 1 ) a reflected beam leaves at an angle equal in magnitude to the incident angle (4, = -bi),(2) a refracted beam proceeds at an angle 4, obeying Snell’s law, and (3) the perpendicular and parallel components are described by Fresnel’s equations6 ( p ~ / p ~ ~ ~ J ” 4i ’ s=i n( ~

sin 4,

L ’ E ‘ / ~ ~ E ~ ) ’ ’ = ~

n

= refractive

n cos EsreAected

-

index

Snell’s law

= c/ o

4 i - Py n’cos cpl P

- rq=

bsincident

n sin +i = n’sin 4t,

n cos

9

P 4i+ y n’ cos P

Estransmitted

2n cos 4i

= t, =

sincident

n cos

c$~

, Fresnel equations

P

+ y n‘ cos

(3.8)

C#I~

P

Epretlected

6n’ cos 4I - n cos 4t , 6 n’cos 4 , + n cos 4t P

P - rp =

pincident

Eptransmitted

E p incident

- tp=

2n cos 4,

6 n‘cos + , + n cos P lrs12+lt,12= 1,

R, = rsr?,

Irp12+lfp12= 1,

R , = rprp*,

where R, and R , are the reflectance and are related to the energy flow by Poynting’s equation. In most cases, p / p ’ - 1 , simplifying the equations. The Fresnel equations are derived according to Fig. 3 and predict the phase and amplitude for S and P polarizations. The incidence plane is the page, and it is also the rp plane. The vector perpendicular to the page is r,, and it is an x or 0 into or out of the page. The equations’ phase shifts are interpreted in Fig. 3. For example, at normal incidence, rs and rp are indistinguishable, and the minus sign difference is because the rp vector is antiparallel to r, in the figure.

WAVE PROPAGATION I N ISOTROPIC M E D I A

115

The vectors r, and rp have some interesting and useful mathematical relations that simplify the equations. Effective indices are an example. The equation form for S and P polarizations for the lim 4i+ 0 reduces to the normal incidence coefficients even though there is no physical difference between S and P.

n-n’ r, = n + n”

n’- n rp = n + n”

(3.9)

For non-normal incidence, the reflection coefficient equations can be rewritten with effective indices p and p ’ to reproduce the forms of the normal incidence equations.’ p s= n cos c$~,

p:=

n‘cos 41, n‘ cos cpl ’

CL; =CL,-cL:

r, = -

rp = CL; - CLP

(3.10)

CLP+CLb’

C L S + d ’

The reflection coefficients can also be written in terms of Snell’s law and incidence angle only. v=-

r, =

n’ n

cos 4, - ( v’ - sin’ 4i)”’ cos 4, + ( Y’ -sin2 4 ~ ” ”

v’ cos bi- ( v 2 - sin’ 4 ~ ” ’ rp = v’ cos 4i ( v’ - sin’ 4i)’/2*

(3.11)

+

The positive sign of the square root term was chosen because the replaced cosine term is positive real valued. When the square root goes to zero and then becomes imaginary, the negative root is chosen because it predicts an exponential decrease in the refracted beam’s amplitude, which is the real situation. Eliminating v from equation (3.11) for rp gives an equation for rp in terms of r, and the incident angle only.’ (3.12)

There are two cases for v, IvI > 1 and Ivl< 1 . For IvI > 1, for example a beam incident from air to glass, the value of the square root term is always positive, real, and less than cos so that r, is negative. The negative sign is interpreted as a r phase shift for 8 , . The rp goes from a positive value,

116

OPTICAL POLARIZATION

.om

-

-

4 L -

10 n

___________________ -

..-.-.-._._,

6, = 0, for 8, = .rr, for for Ivl< 1.

I I

-

4

y.-

,

,

,

4i = 0, through zero for Brewster’s angle, 4B,to a negative value 4i > c#J~. Figure 4 is a plot of r, and rp versus the incidence angle

Brewster angle conditions are, v 2 cos 4

B = ( v 2 -sinZ 4B)’”, rs = cos 2 4 B .

(3.13)

At Brewster’s angle, the reflected polarization parallel to the incident plane is zero and produces a highly polarized reflected light beam perpendicular to the incident plane. For incident angles less than Brewster’s angle, the S and P reflections have a n phase difference, and the reflected polarization is orthogonal to the incident polarization. A light beam can be optically isolated with a circular polarizer and a reflection, assuming there are no other polarization effects. For IvI < 1, a beam traveling from glass to air, for example, r, is always positive and reaches unity at a critical angle, &, which gives total reflection and an imaginary phase related component due to the square root term. sin’ 4, = v2.

(3.14)

An S-beam incident from glass to air does not have a phase shift upon reflection until it is incident at c#+ > &. For +i 2 &, Ir,l= 1. For r p , its value is less than zero up to Brewster’s angle, where rp equals zero and then is positive real valued until the critical angle where the imaginary phase related term influences r p , and lrpl = 1. Figure 5 is a plot of r, and rp versus the incidence angle for I vI > 1. We can write r p and r, as complex expressions. r P = p P ej S p , r s = p s e jSs .

(3.15)

117

WAVE PROPAGATION IN ISOTROPIC M E D I A

10 n 4

4 4 L -

0.0

0.u

v@ ‘pe

9o.a

Incidence angle, 9

FIG. 5. Absolute magnitude for the reflection coefficients r,,p and phase shift S,,,, from a more to a less dense medium.

In ellipsometry, the following notation is also used: (3.16)

For total internal reflection, expressions for the phase changes

a,,

6, are,

(3.17)

Figure 6 is a A ( 8 p - 6,) plot, and the extremum in the curve is found by maximizing A with respect to the incident angle. (3.18)

Also, [A( 6,- &)/A&] slowly varies about the extremum, and this fact can be used to introduce a phase shift between P and S which is fairly insensitive to the incident angle. A Fresnel rhomb is a total internal reflection device to produce a phase shift between P and S. The incident beam totally internally reflects for both r, and rpoff of the two sides inclined so that the incidence angle gives a ?r/4 phase difference between P and S on each reflection. This gives a total retardation between P and S of ~ / 2 A. linearly polarized beam with equal P and S components incident on the Fresnel

118

OPTICAL POLARIZATION

n

0

0.d

9O.U

Incidence angle,

(pl

- - - - - - -. FIG.

6. A for u < 1 and

u>1

AV>I &
as a function of incidence angle

rhomb will become a L or R circularly polarized beam depending on the incident beam's orientation. The A phase shift between the reflected S and P components translates to a A rotation of the PoincarC sphere about the axis through the S and P linear polarizations. For example, linear polarization incident with equal components in the S and P planes will reflect with a different ellipticity. 3.3.2. Complex Plane Representation

Another way to view Fresnel's coefficients is to map relations in the complex plane. For example, the material constants determine r,, which, with the incidence angle, solely determine the behavior of r p . If r, is described as the upper half-plane area (positive time dependence in the wave equation) bounded by the unit circle about the origin, then the values of r, map into r, an area bounded by a unit circle in the complex plane. Constant phase lines radiate from the origin, and constant amplitude lines are circles about the origin bounded by the unit circle.'" Figure 7 is a plot of r, and several plots of rp for different incidence angles. A noteworthy feature of Fig. 7 and the equation relating r p to r, is that bi= 45". At this angle, rp = r f , 6, = 26,, and a known polarization is produced, which can calibrate an instrument in terms of phase and amplitude changes through the entire system for both P and S polarization. The P polarization reflected at 45" is unaffected by small changes in the media due to temperature, stress, and electromagnetic fields. This result works for a film-free substrate, since P is written only in terms of S and the incidence angle. Another complex transformation is from a normal incidence reflection coefficient, z, to a coefficient, w,at other incidence angles."-'2 The coefficient w is an analytic function of z and is dependent on the incident angle. The

119

WAVE PROPAGATION I N ISOTROPIC M E D I A

l5 .

3

(s=45.0'

rP

, q=75.0'

rP

FIG. 7. Complex plane representation for r, in terms of constant phase (radial lines) and amplitude (semi-circles), and r p as a function of r, and incidence angle.

normal incidence equation form is, r =-r

1-v

=--

l+v

- z,

(3.19)

1-z v=l+z'

The transformation from z to w, and the inverse transformation from w to z for S polarized light are, Z=

( a +2p)

* [(a' - 4 ) + 4 ( a +2)p]'12 2(1 - P )

+ z ) - ( z 2 - a z + 1)'12 ' ( 1 + z)(z' - az + 1

(1 W =

9

(3.20)

a=4tan2cpi+2, p = [ 'I+w -"]'.

This transformation is particularly useful. If an S reflectance measurement is made at normal incidence R,( &) and at a known incidence angle R,( cpi), then the complex normal reflection coefficient, z, can be determined. In Fig. 8(d), a circle with Izl= R , ( I $ ~ ) " *is plotted on the complex plane with the image for IwI = R S ( ~ J ' l Where 2. the two curves meet determines v and z.

120

OPTICAL POLARIZATION

...'..,

[O .'......'.

(4 FIG. 8. (a) Constant amplitude circles representing either IzI or IwI, (b) IzI mapped into w for 4, = 30". (c) IwI mapped into z for +a= 30". and (d) IzI normal incidence measurement overplotted with a I wI measurement for 4, = 30" mapped into z.

3.4. Wave Propagation for Metals 3.4.1. Polarization Effects

Metals include the conductivity coefficient which affects the Fresnel equations. The damping term due to conductivity in the wave equation is observed as Joule heating. The wave equation solution is written as a monochromatic wave as for dielectrics with E satisfying the wave equation with conduction included.

V2E+ k2E = 0,

k 2 = 0 2 p (E

,):-

(3.21)

whe k is the complex wave number. A solution for a conducting medium wave equation is found by substituting the complex dielectric constant i, which expresses the attenuation due to the conductivity term, into the nonconductive medium solution and by making the appropriate changes in phase velocity, wave number, and refractive index. This also carries over into the Fresnel equations and Snell's law.

121

WAVE PROPAGATION FOR METALS

(3.22) where K is the absorption index, n and K are real, and n' is the complex refractive index. The absorption coefficient k = nK is most often used. It is important to remember the distinction between k and K . Snell's law provides a complex-value refracted sine term when light is incident from a dielectric to a metal, i.e., n , sin 4i= nl2 sin 4,. This means that for a refracted wave in a conductor, the complex wave number produces an imaginary term which represents constant beam amplitude surfaces parallel to the boundary and a real term depending on the incidence angle which relates constant phase surfaces. Since the constant amplitude and constant phase surfaces do not necessarily agree, the electromagnetic wave is said to be inhomogeneous (see Ref. 2, p. 611). In Fig. 9, Eq. (3.11) is used to plot the reflection coefficients for aluminum and silver as a function of the incidence angle. The effect of the attenuation term smoothes the extremum termed the second Brewster angle. This also means that the rp term is not extinguished as for a dielectric and increases

*T - - - - - -

\

\

4

\

Lo

0

.

6

acr

"

'

'

"

"

Incidence angle, 9

'

>

9O.U

FIG.9. Absolute reflection amplitude coefficients and the phase shift for S and P polarizations as a function of incidence angle for aluminum (top) and silver (bottom).

122

OPTICAL POLARIZATION

0

90.0-

0.0‘

Incidence angle, AAI

_------- AA9 FIG. 10. A as a function of incidence angle for aluminum and silver.

with K . Silver has a large absorption coefficient and consequently has only a small variation in reflectance as a function of the incidence angle. However, aluminum or rhodium have a lower absorption coefficient and produce relatively large polarization effects compared to silver. The difference between S and P phase shifts is plotted in Fig. 10. An incidence angle with a 7r/2 phase shift in the plot reflects a linearly polarized beam as an elliptically polarized beam whose axes are along rp and r,. If rp = r,, circular polarization is produced. This is called the principal angle, and it is a way to produce a known elliptical polarization. 3.4.2. Reflection and Transmission Polarizers

Reflection or transmission polarizers can be made by using the fact that R , is close to zero ( Tp 1) at Brewster’s angle.’ For transmission polarizers, a pile of plates is needed to eliminate the S component. The angular acceptance has to be limited, because R,/ R , increases away from Brewster’s angle.13 Their advantages, especially for reflection polarizers, are that they cover wavelength ranges not covered by crystal polarizers, that they are generally achromatic, have wide apertures, have uniform polarization across the aperture, have high extinction ratios, have relatively high transmission, and are easily fabricated. Their disadvantages are limited angular acceptance (depending on tolerances and polarizer throughput), physical size of the polarizer, possible beam deviation, and multiple reflections. Reflection polarizers are used primarily in the vacuum ultraviolet (VUV), ultraviolet (UV), and infrared (IR) spectral regions. 14-18 A good polarizer needs to have a high R , / R , value which is dictated by the surface’s optic constants. Metal coatings are most often used and are sometimes overcoated with a protective layer whose thin-film interference effects need to be considered and are discussed later. The metal coatings are to increase the

-

WAVE PROPAGATION FOR METALS

123

r, reflection. According to Fresnel's equations, r, increases with the refraction index of the reflecting medium, and the Brewster angle or minimum rp angle shifts towards increasing incidence angles, as seen in Fig. 9. With increasing absorption, rp increases and R,/ R , decreases. This suggests a material with a large n and small K in order to get the largest values of R , / R , . It is also important to consider how the material properties change with wavelength for VUV and UV applications. Magnesium fluoride, for example, becomes absorptive around 1140 A, and the choice of material for the spectral region is important. The literature gives a fairly complete description and the ( R , / R p ) m a xR, , transmittance, and angle of incidence for ( R , / R p ) m a xfor different materials from 300b; to 2000A which can possibly be used as mirror coatings: gold, iridium, osmium, rhodium, and rubidium." Gold is the best coating for a polarizer to cover from 300 b; to 2000 A, because its ( R s / R p ) m astays x fairly uniform and is generally the largest. If a selected wavelength range is used, the other materials, such as rhodium from 900 A to 1300 A will give better performance (higher R,/ R,) than gold. For the IR, the choice of metal coatings is not as difficult as for the VUV, and overcoated aluminum mirrors have been used as polarizers. The angular acceptance and R J R , values can be increased by multiple reflections from more than one mirror (at the same time decreasing throughput). Single Brewster-angle polarizer plates deviate the beam which might be unwanted, and three- or four-mirror arrangements can be used to bring the beam back in the original d i r e c t i ~ n . ' ~ , ' " . 'This ~ * ' ~increases R,/ R , versus the incidence angle by multiple reflections at Brewster's angle, because the values ( R J R , ) " and (R,)" ( m is the number of reflections at Brewster's angle) increase and decrease accordingly. The acceptance angle and polarizance can be increased, while sacrificing throughput by using multiple reflections at Brewster's angle. A three-mirror polarizer is drawn in Fig. 11. The middle mirror is set with p = Brewster's angle; LY is the entrance incidence angle and is greater than p, resulting in a higher R , and R , than for the middle mirror, with R , being significantly diminished before the middle mirror reflection. Two disadvantages of the three-mirror arrangement are (1) the image is rotated, and (2) the polarization varies across the acceptance angle in a complex I

I I

FIG. 1 1 . S polarizer made of three mirrors.

124

OPTICAL POLARIZATION

FIG. 12. S polarizer made of four mirrors.

manner, because the incidence angle is not the same for the three mirrors. The four-mirror arrangement in Fig. 12 has all the mirrors at Brewster's angle, thus a higher polarizance, low output of R,, the image is not rotated, and polarization varies across the acceptance angle as for a single mirror. Transmission Brewster-angle polarizers or pile-of-plane parallel plates transmit the P component according to the amount of absorption and therefore, theoretically, 100% for no absorption. Realistically, though, there exist scattering, surface films, and crystal imperfections. Also, the beam is deviated perpendicular to its propagation direction, and multiple reflections make T, difficult to extinguish. These two problems can be corrected by proper arrangement of the elements and by making them wedge-shaped to reduce multiple reflection effects. These polarizers are useful in the IR with materials such as selenium, silver chloride, as shown for Saran Wrap," and in the UV with calcium fluoride, lithium fluoride, and magnesium fluoride. There are several advantages of transmission polarizers: an undeviated beam, high extinction, and high energy use. They are shorter than reflection type polarizers and are an easy way to produce polarized light.2' Reflection circular polarizers have been demonstrated, particularly for the UV. As was shown above for metals, there is a principal angle where the phase shift between r, and r p is 7712, so that an incident linearly polarized beam adjusted to give r, = rp produces a circularly polarized beam. In a paper by McIlrath et al., the authors show results for an aluminum (Al) mirror exposed to air (overcoated by an aluminum oxide (AI,O,) layer of about 25 A ) which produces /rsl = Irpl over all incidence angles and thus can be used as a phase shifter between r, and rp. At an incidence angle . experimental equal to 52" at Lyman-a, the phase difference is ~ / 2 Their arrangement was an A1203 linear polarizer at Brewster's angle which fed the A1 phase shifter mirror, and then analyzed by an LiF Brewster-angle analyzer in combination with a detector. They produced a circular polarizer which transmitted 12% with E - / E + = 6 to 10 at A =1216A with a 5" acceptance angle at c#+ = 52". Circular polarizers are considered later in discussing thin films and crystal polarizers.22

125

THIN FILMS

3.5. Thin Films 3.5.1. Single-layer Films 3.5.1.l.Polarization Effects. In general, most optics have thin films on their surfaces, and these introduce, via Fresnel's equations and phase interference, an effect on the P and S components. Figure 13 is an example of a light beam in a medium ( n l )incident at an angle 4, which is refracted at an angle 4, in a film ( n 2 ) d thick on a substrate ( n 3 ) . Multiple substrate reflections interfere with front surface reflections to produce minima and maxima in the reflectance depending on the incidence angle, film thickness, and material constants.23 The phase shift added to a beam after traveling through the film thickness once compared to a beam traveling in the medium is,

phase thickness = /3

= Wr =2

cos 4, m 2 d -. A

(3.23)

The reflected and transmitted beams are written in terms of Fresnel's reflection, r m , n , and transmission, r m , n , where the coefficients with the subscripts m and n denote the boundary media, with m being the incident medium. For the reflected beam, the components in order of intensity are: r I 2 ,r12r23r21 e-''@, t12r23r21r23t21 e-'J@,.. . .

The total reflection coefficient is: rs,p= r 1 2 +r23rlzr21 e-2Jp+ r21r:3r12r:1 e-'J@+

- . *

The reflection coefficient, rs.p, can be rewritten with relations between the transmission and reflection coefficients.

+

'S.P

- r I 2 r2, e-J2@ - 1 + r12r23 e - i 2 ~ .

(3.24)

This is written for complex P or S coefficients and is a description for an absorbing substrate and film. Beam splitters, overcoated mirrors, interference filters, etc., have variations in polarization due to thin films which need to be considered. Rewriting the reflection coefficient between a dielectric film and a metal substrate in complex exponents produces a single equation for the reflectance. r23= pZ3eJ'23, (3.25)

126

OPTICAL POLARIZATION

FIG. 13. Multiple reflections in a thin film.

The reflectance thus varies periodically with thickness with extrema given by d R / d d =O. This occurs when sin(S2,+2P)=0,

S2,+2P=rn7r,

rn=0,*1,*2 , . . . ,

The phase term (S2,+2P) can be adjusted with /3 either by film thickness or by the incidence angle for a specific ~ a v e l e n g t hFigure . ~ ~ 14 is a plot of r, and rp versus film thickness for two incidence angles showing the periodic nature due to interference. The maximum and minimum values are determined by the reflection coefficients and the material. For a dielectric film and substrate, the maxima and minima occur at phase thickness equal to odd and even multiples of a quarter wave. Figure 15 is a plot of the reflection coefficients versus the incidence angle for two film thicknesses. In Fig. 16 the incidence angle and the film thickness are held constant, making the S and P reflections vary periodically with wavelength. 3.5.1.2. Linear Polarizers and Phase Retarders. Writing the ellipsometric equation r p / r s=tan Ic/ e J Afor the general case of an absorbing film and absorbing substrate gives an equation nicely mapped in the complex plane. The points where tan II, equals zero or infinity are possible designs for a P or S extinguishing polarizer, while points tan Ic/ = 1 and A = *77/2 produce a circular polarizer. This suggests that the phase difference can be controlled between r,* and r2, via the incident angle, refractive indices, and film thickness to make P or S reflective polarizers, linear partial polarizers (which differentially attenuate the P and S components without shifting the phase between the components), and wave retarders.25-28 The P or S extinguishing polarizers with a single dielectric film on a metal substrate require large incidence angles for the higher reflecting

127

THIN F I L M S

OA

la

Film thickness, d

1.0

FIG. 14. rr and r,, for reflections from air to glass film to glass substrate showing the effect on r, and r,, as a function of film thickness, n , = 1.0, n, = 2.0, and n, = 1.5.

1.0 r

I

....

Incidence angle,

(h

Ir, I

-.-.-.-

0.u

Incidence angle,

Irpl

90.0-

F I G . 15. r\ and rp for reflections from air to glass film to glass substrate with a 2h and a Oh film, showing the film's effect as a function of incidence angle.

128

OPTICAL POLARIZATION

-1

p = 30'

4 L-

0.0 5000 A

5500 A

Wavelength, h

1.0

t

= 60'

FIci. 16. rs and rp for reflections from air to glass film to glass substrate showing the film's effect as a function of wavelength about a central wavelength A o .

substrates, since the reflectance required from the dielectric's surface to match the substrate's reflectance means large incident angles (> 80°).2y~'0 Also, the slope for the reflectance versus the incidence angle for the dielectric is very steep, thus making the angular acceptance very sensitive. A gold substrate with its lower reflectance can be used with a high-index film to produce a reasonable P or S extinguishing polarizer which has high extinction, smaller incidence angle, and better angular sensitivity. Another solution is to add some absorption to the film (this cannot be used in high-energy beams due to the absorption) which produces multiple incidence angles (75'-85') at which the P or S component is extinguished and the sensitivity to angular acceptance is d e ~ r e a s e d . There ~' is a trade-off of throughput for angular insensitivity and the unwanted component's intensity. These are simple, single-film, wavelength-dependent reflection polarizers which are useful when used with low-power densities. There also exist multilayer dielectric films on a transparent substrate which are used as reflection/transmission polarizers for higher-power densities.

3.5.2. Multilayer Films 3.5.2.1.Polarization Effects. Another way to make thin-film polarizers is to use periodic multilayer dielectric films of high and low in dice^.^*-^^

THIN FILMS

129

These are suitable for beams with high-energy densities. The time dependence of the wave equation, wt, is assumed fixed, and the equations are expressed as second-order differential equations in space for the magnetic and electric wave propagation. The multilayer’s effect on the incident beam’s magnetic and electric field is expressed as a 2 x 2 unimodular transformation matrix. The new beam produced from the first layer is applied to the second layer, producing a modified beam for the third layer, etc. The total thin-film stack can thus be expressed as a single unimodular transformation matrix. To see the effect of the dielectric multilayer on the S and P polarization, first consider the mth thickness z, at normal incidence. The matrix transformation of the electric and magnetic fields is

The diagonal and cross-diagonal terms are pure real cosine and pure imaginary sine terms with arguments which are the phase thickness,

A,=-.

2rn,z,

(3.28)

A0

To apply this at oblique incidence angles, the effective phase thickness and effective index (a, p m )are used to describe the uniform plane wave at an oblique incidence as a nonuniform plane wave at normal incidence, via 6, = A, cos 4,

and

where 4, is the refracted angle through the layer. The last two relations were introduced in Fresnel’s equations earlier and are substituted into normal incidence Fresnel equations to produce nonnormal incidence equations. Periodic multilayer dielectric thin films have a transformation matrix, also known as a Herpin matrix, which is equivalent to a two-film combination for a particular wavelength. A symmetric three-layer film is a special case which simplifies to an equivalent single-film matrix for all wavelengths. The single film is described by an equivalent index, N, and an equivalent phase thickness, r.

(3.29)

130

OPTICAL POLARIZATION

A symmetric three-film combination with an effective phase thickness (Y + /3 can be made by sandwiching one layer between the halves of the other layer. Three films with index layers n,,, n p , n, and phase thicknesses a / 2 , p / 2 , a / 2 result. Equating the three-film transformation matrix to the singlefilm equivalent matrix gives an equation for the equivalent phase thickness,

cosh

r = cos

(Y

cos /3

(3.30)

For values of lcosh rl> 1, the effective index N is complex in order to preserve the pure imaginary cross diagonal terms for the equivalent single film. This means that there is attenuation of the beam propagating through the medium (a stop band), while for lcosh I'l< 1, N is real and the beam propagates through the medium (pass band). The number of periodic layers increases the attenuation and improves performance. Since the effective index for the S and P components are functionally different in 4, an incidence angle and phase thickness can be found for a given wavelength which will give lcosh rl> 1 for the S component and lcosh rl< 1 for the P component, thus making an S-reflecting, P-transmitting polarizer. If it is possible, N is adjusted to the medium surrounding the film, or else an antireflection coating is needed. In practice, there is more to multilayer polarizer design, such as antireflection coatings to index match the multilayer to the air and/or glass interface, or to design the multilayer to take into account the electric field distribution through the stack to reduce laser damage.37-41 3.5.2.2. Linear Polarizers and Phase Retarders. Mahlein has published articles on quarter-wave multilayer dielectric stacks for laser mirror His graphs represent various thin-film combinations of n, versus np ranging from 1.3 to 3.0 with various incidence angles for P or S extinguishing reflection/transmission polarizers. He also shows possible designs for nonpolarizing beam splitters and polarizing interference filters. As for reflection quarter-wave retarders, Southwell has designed a dielectric multilayer on a silver substrate with a reflectance equal to 99'/0.~~His design starts with a quarter-wave-thick periodic multilayer stack, and the film thicknesses is adjusted according to a merit function which maximizes the desired characteristics-90" retardance *3' over a 5% range about the design wavelength and a reflectivity of 99.8%. These coatings can be used to reduce polarization effects in corner cube reflectors. 3.5.2.3. MacNeille Polarizing Prisms. Another application of thin films as a polarizer is in beam-splitting glass cubes first described by M a ~ N e i l l e . ~A' *glass ~ ~ cube is cut along a diagonal, and a multilayer stack is deposited such that the Brewster angle condition is met at all the diagonal interfaces, as drawn in Fig. 17. If the effective indices for the high- and

THIN FILMS

131

FIG. 17. A P-polarizing MacNeille prism using multiple reflections at Brewster's angle &.

low-index films are set equal, the parallel Fresnel-reflection coefficient is zero. This condition with the Brewster angle condition produces an equation in terms of the refractive indices and the incident angle in the prism. nG sin

cPG = n H sin 4H= nL sin 4L, (3.31)

Commercially-made, broad-spectral-band, thin-film-polarizing cubes (4000 A to 7000 A) are available with an extinction ratio of 100 to 1, and narrow-band cubes are available with an extinction ratio of 1000 to 1. Optically coupling the prism halves for high-power densities is difficult, and residual birefringence exists in the glass. Liquid prisms with immersed interference-coated substrates have been demonstrated as practical to overcome these problems and can have a wide spectral range from the UV through the ~ i s i b l e . ~ ' 3.5.2.4. Total Internal Reflection Devices. Total internal reflection phase-retarding devices, such as the Fresnel rhomb mentioned earlier, control the phase difference introduced on reflection to make quarter- and half-wave retarder^.^"'^ The phase difference between the P and S components, A, for most glasses in air does not occur at the maximum, A,,,, in the plot A versus the incidence angle (Fig. 6). Usually the larger incidence angle is used to make the rhomb, since its angular sensitivity is not as large as the smaller incidence angle. For the least angular sensitivity, Amax would equal the desired phase retardation. Increasing the refractive index increases A,,,, so a film with a smaller refractive index is deposited, and its thickness is adjusted to give A,, = 45". For a *3" incidence angle variation over the

132

O P T I C A L POLARIZATION

visible spectrum, the retardation variation is 0.4" for a coated rhomb and 5.0" for an uncoated version. Imperfections in total internal reflection devices also are due to strain birefringence which is inversely proportional to wavelength, and for applications in the ultraviolet it becomes critical, especially if the retarder is to be achromatic. Also, it is reported that polishing the faces that are not used reduces strain 10-15 mm inside the glass by a factor of two over that for a fine-ground face. Attention should be given to mounting the retarder, and silicone rubber sealant as an adhesive does not introduce any measurable strain. Another error source is the refractive index of the glass and thin film. For the glass, the surface is slightly different in phase retardation, about 0.5" more than the theoretical values, because surface shaping and polishing changes the refractive index. Also, generally for most materials, the refractive index increases with a decrease in wavelength, except for polyvinyl alcohol. Retardation error sources can be corrected by adjusting the reflected beam's phase difference experimentally with film thickness. The film can increase achromatism, because the increase in refractive index with decrease in wavelength can be balanced by the thin film's increased phase thickness due to p. King has made Fresnel rhombs with either 144A magnesium fluoride films on each reflecting surface or a 268A film on one surface leaving the other uncoated, which produce a retardation of 90" f 0.2" from 3440 A to 6680 A compared to 90"*2.5" for an uncoated rhomb. In Fig. 18, various other retardation devices are shown. They have about the same chromaticity but various other advantages and disadvantages with regard to beam displacement, physical size, and angular acceptance. A Fresnel rhomb displaces the beam; and, if not optimized by thin films, it has a 9" variation in retardation with a *7" incidence angle. Achromatic device one, ADl, is self-compensating in incidence angle, because the first two reflections on the minus side of the optimum angle will have two reflections on the positive side, therefore almost canceling the off -angle effects. It is a very long prism, 175 mm long for a 10-mm beam, and it would be hard to meet theoretical predictions due to strain birefringence. It has a 0.7" retardance variation for a *7" incidence-angle variation. The Mooney rhomb is made of two dense flint 60" prisms and is self-compensating in incidence angle with a performance similar to AD1. Also, its physical size is small. Achromatic device two, AD2, is physically small and does not deviate or displace the beam, but its incidence angle sensitivity is terrible with a 13.2" retardation variation for a *3" incidence-angle variation.'' Half-wave or full-wave rhombs have also been designed to deviate the light beam 90" without changing the polarization. A simple design is the dove prism.

WAVE PROPAGATION I N A N I S O T R O P I C M E D I A

133

FIG. 18. Total internal reflection devices. (a) Fresnel rhomb, ( b ) Mooney rhomb, (c) achromatic device 1 ( A D l ) , and (d) achromatic device 2 (AD2).

3.6. Wave Propagation in Anisotropic Media 3.6.1. Birefringence in Crystals 3.6.1.1. Dielectric Tensor and Energy Distribution. Another way to separate orthogonal polarizations is by dielectrics with anisotropic optical constants either naturally occurring in crystals or induced by an electromagnetic field or stress and compression. This produces different velocities for orthogonal polarizations depending on the travel direction in the crystal.” The medium can be described in terms of its dielectric constants as a nine-element tensor.

(3.32)

where D is now not necessarily parallel to E as for an isotropic media, and the angle difference is between the wave front’s normal, s, and Poynting’s energy propagation vector, S. This means that there is a wave-front phase velocity and an energy ray velocity as shown in Fig. 19. The phase velocity is a projection of the ray velocity onto the wave front’s normal. Maxwell’s equations and the material equations with the dielectric tensor give an

134

OPTICAL POLARIZATION

equation relating the phase velocity to the wave normal direction and the material constants. Considering the functional form of the energy density, the dielectric tensor is seen to be symmetric, and at most six independent values need to be determined. A coordinate system, termed the principal axes, can be fixed to the tensor’s diagonal. In this coordinate system, the material equations simplify to the diagonal terms. Eyy+Ey,

E,,+Ex,

EZZ+EZ,

i = x, y , x.

Assume the media is homogeneous, nonconductive, nonmagnetic, and electrically anisotropic. A plane wave, Eq. (3.4) simplifies the derivatives in Maxwell’s equations.

V x H =j w D = -jks x H V x E = -jpwH

= -jks x E

(3.33)

:);(

= phase

velocity = V,

(kY2

=velocity in i direction = (3.34)

where s is the unit vector for propagation direction and k defines equal phase planes A apart. D and H are perpendicular to s, but E is not. E is perpendicular to the Poynting energy flow vector, S.

WAVE PROPAGATION I N ANISOTROPIC M E D I A

135

S

E FIG. 19. The orientation of the electromagnetic vectors H, D, and E, the wave normal s, and the energy propagation vector S in an anisotropic medium.

The Ei can be eliminated, leaving an equation of the material constants, the propagation direction, and the phase velocity.

(3.35) The result is known as Fresnel’s equation of wave normals. V ; has two values and implies with Eq. (3.34) that the medium has two linearly polarized beams with different phase velocities. It can also be shown that the two electric displacement vectors are perpendicular. 3.6.1.2. Crystal Types by Optical Properties. The dielectric tensor is a second-order symmetric tensor, and therefore it needs six parameters to describe it. These parameters can be interpreted as the principal axes’ lengths and angular orientations with respect to the crystallographic axes. The number of parameters to describe the tensor is dependent upon their optical anisotropies which are subdivided according to crystallographic structure.” (1) Optically isotropic crystals are called cubic crystals. Three crystallographically, mutually orthogonal axes can be found, and the diagonal terms of the dielectric tensor are equal. Only one parameter, the refractive index, is needed to describe the dielectric tensor. (2) Uniaxial crystals have a plane with equal optic constants and a direction normal to the plane with a different optic constant called the optic axis. Two parameters, the lengths of the two principal axes, are needed to describe the dielectric tensor. A set of crystallographically equivalent directions are in the plane about an axis of symmetry aligned with the optic axis. Crystals can have three, four, or six crystallographically equivalent directions in a plane or three-, four-, or sixfold symmetry about the optic axis. Trigonal, tetragonal, and hexagonal crystals are in this group. Calcite is an example. It is a rhombohedra1 crystal, and the optic axis is a threefold

136

0 PTlC AL POLAR1 ZATIO N

symmetry axis directed into the rhomb’s corner. A surface cut perpendicular to the optic axis on the corner accurately determines the crystal’s principal axes.’ Unpolarized light incident on a uniaxial crystal produces two linearly polarized beams determined by the optic constants for the plane and the optic axis. For all field orientations with respect to the optic axis, an electric vector parallel to the plane exists and behaves as for an isotropic medium. This is termed the ordinary beam, 0, and is described by a spherical velocity surface. The wave perpendicular to the 0 beam depends on the incident beam’s orientation and varies from the optic constants for the plane to the optic constants perpendicular to the plane. This is the extraordinary beam, E, and its phase velocity as a function of propagation direction is a spheroidal surface. The phase velocity surfaces for a uniaxial crystal in terms of the principal axes are found by applying Fresnel’s equation of wave normals. The z axis is made parallel to the optic axis, and the angle from the wave normal to z is 9.

v, = v, = v,, s: + s: = sin2 9, ( V ; - vf)[(V ; - ~ f sin’ ) 9

v, = v,, s t = cos2 8,

+ ( V : - v;)cos2 e] = 0,

(3.36)

v;,= vf, v;*= vf sin2 e + v’,cos2 e. Figure 20 is a plot of the two phase velocity surfaces and an arbitrary wave normal, s, with its two phase velocities. When s is in the optic axis direction, the phase velocities are equal, and when s is perpendicular, the difference between phase velocities is maximum.

z O.A.

z O.A.

FIG. 20. Cross sections of phase-velocity surfaces for a negative and a positive uniaxial crystal predicted by the Fresnel equation for wave normals. The phase velocities are determined by the wave normal’s orientation.

WAVE PROPAGATION IN ANISOTROPIC MEDIA

137

Birefringence is the difference between the 0 and E refractive indexes. birefringence = nE - no

(3.37)

The sign means that the ordinary velocity surface is ahead of or behind the extraordinary surface. (3) Crystals with principal axes as a function of three dielectric constants and angle orientation with the crystal axes are called biaxial crystals. TE waves are refracted at an angle depending on orientation with respect to the crystal's axes and wavelength. Both waves perpendicular to s act as extraordinary waves, and the velocity surfaces are general ellipsoids. Orthorhombic crystals have their principal axes aligned with the crystallographic axes, and so only three parameters, namely, the lengths of the three principal axes, are needed. Monoclinic crystals have a fixed principal axis parallel to a crystal axis and need four parameters for the dielectric tensor. Triclinic crystals need all six parameters to describe the dielectric tensor. Applying the Fresnel equation of wave normals to biaxial crystals has three dimensions that can vary. Three cross-sections, each about a different principal axis, simplify the problem. Figure 21 shows three cross sections for a biaxial crystal. One cross section has the curves touching at four points, and the two lines connecting the origin with the points are the directions of the optic axes. With the wave normal in these directions, the two phase velocities 'are equal. Quartz and calcite are the most commonly used crystals for crystalretarding plates and prism polarizers and beamsplitters. Quartz is a biaxial crystal. A beam propagating along the optic axis separates into L and R circular polarizations due to a weak circular birefringence. for

nL- n R = *0.000071

A = 5892 A.

Separating L and R polarization in phase is equivalent to rotating the PoincarC sphere about the L and R poles, and the resulting elliptical 2

Y

Z

vx

>

vy

>

vz

FIG. 21. Three cross sections of phase-velocity surfaces described by the Fresnel wave normal equation for a biaxial crystal.

138

OPTICAL POLARIZATION

polarization azimuth is rotated according to the phase shift. This phenomenon is termed optical activity. The beam’s ellipticity changes quickly to linear polarization for angles not far from the optic axis, and quartz behaves almost as a uniaxial crystal. Quartz’s weak linear birefringence makes it useful for retardation plates. nE- no = 0.0091

for

A = 5892 A.

Calcite is a negative uniaxial crystal. Prism polarizers and beamsplitters are usually made from calcite, because the transmission and birefringence are large, from 2000 to 12,000 A. nE = 1.486, no

= 1.658

for

A = 5892 A.

The literature has references with material constants given for the more commonly used crystals and wavelength ranges.

3.6.2.Crystal Polarizers 3.6.2.1. Prism Beamsplitters. Beam-splitting polarizers, such as the Rochon, SCnarmont, and Wollaston are biprisms which separate the 0 and E beams by introducing a discontinuity in the refractive index for either the E beam (Rochon and Sknarmont) or for both the E and 0 beams (Wollaston). Figure 22 shows the E and 0 rays for a Rochon and a Wollaston. The Rochon and SCnarmont are basically the same with their first prisms’ optic axis perpendicular to the entrance face. The Rochon gives a slightly larger E-beam deviation than the SCnarmont, because the beam sees the maximum birefringence. These prisms can be used backwards, but the E-beam deviation will be slightly less. Snell’s law at the prism discontinuity, now from nE to no, reverses the incident beam’s refracted angle from greater to smaller deviation. The beam deviation and

FIG.22. ( a ) Rochon and ( b ) Wollaston polarization beam splitters. The optic axis direction is oriented parallel I or perpendicular 0 to the page.

WAVE PROPAGATION IN ANISOTROPIC MEDIA

139

accepted field angle are a p p r ~ x i m a t e l y ~ ~

(3.38)

A Wollaston polarizing beamsplitter has the optic axis of its first prism parallel to the face to produce an E and an 0 beam which sees the second prism's optic axis rotated 90" about the beam axis to produce a maximum birefringence and deviation for both beams. Wollaston prisms can produce angular separations up to 20" with a slight asymmetry, and triple-element Wollastons produce larger angular separations. At large angles through crystal polarizers, image distortion occurs due to nE varying in angle with respect to the optic axis. Since the birefringence of Rochon's first prism is not used, it can be substituted with strain-free glass of either index, nE or no. Glass eliminates image distortion due to the effects of the birefringent crystal and is easy to manufacture cheaply with a good tolerance. The residual birefringence in glass needs to be c o n ~ i d e r e d . ~ ~ 3.6.2.2.Prism Polarizers. Polarizing prisms are birefringent crystals cut into two pieces with an interface of an index which totally internally reflects the higher-index beam, for calcite the ordinary beam.' A Glan-Thompson prism is illustrated in Fig. 23. A Nicol prism was the first polarizing prism design, and it is essentially a calcite rhomb cut along a diagonal and cemented together. Glan-type prism polarizers arrange the optic axis to give

i"

i i i I

i

i

,-I

_._.-.- 0 - - - - - - -. E

FIG. 23. Clan-Thompson prism drawn with normal and extreme rays. The optic axis is perpendicular to the page, 0.

140

OPTICAL POLARIZATION

maximum use of birefringence, thus increasing acceptance angle. Beam distortion and polarization uniformity are better controlled for Glan prisms. The prism’s angular acceptance is limited by either the critical angle of the larger-index O-rays, or by the lower-index transmitted E-ray being refracted along the prism’s cut angle. The refractive index discontinuity, n,, at an angle S in Fig. 23 has its normal rotated ad angle ( 9 0 ” - S ) with respect to the entrance normal. Applying Snell’s law to the 0 beam at surface 1-2 gives an equation relating the maximum incidence angle for total internal reflection to the angle S and the cement’s refractive index, n,, n, sin di= ( n i - n:)”, cos s - n, sin S.

(3.39)

For the E beam’s limit, the total internal reflection case n E > n, can be described with the 0 beam’s equation and the appropriate substitutions, n , sin

di= n2 sin s - ( n i- n:)”, cos S.

( 3.40)

To make the maximum usable acceptance angle for both 0 and E beams, the two incidence angle equations are made equal in magnitude by adjusting the cut angle and cement. This gives, tan S =

( n h- n:)”,

(3.41)

2n2

The acceptance angle reaches a maximum for n2 = n,. For n E < n,, the maximum accepted incident angle for the transmitted beam is refracted along the cut. n, sin d ,,

= n2 sin

S.

By equating the maximum incident angle for the transmitted beam to the totally reflected beam gives an equation relating the cut angle and the (3.42)

Considering the totally reflected beam, the maximum incident angle to be reflected increases with the smallest cut angle which is achieved with n, = n E . The tan S equations reduce to the same form for n2 = nE. Glan-type prisms are sold with the length-to-aperture ratio assuming a square aperture, and this gives an idea of its acceptance angle. L A

1 tans’

(3.43)

Since optical cements have absorption, air-spaced calcite prism polarizers are made for use up to calcite’s ultraviolet limit and for high-energy densities.

WAVE PROPAGATION I N ANISOTROPIC MED IA

141

The usable maximum (symmetric about the prism's normal) incident angle or semifield angle is about 8" in the ultraviolet, decreasing for longer wavelengths, and is limited by the transmitted E beam being totally internally reflected at the cut. Large prism apertures can be made, since most of the crystal is used. An air-spaced Glan-Thompson is termed a Clan- Foucault The optic axis is parallel to the cut surface, so that the transmitted beam has its electromagnetic vibration oriented parallel to the cut surface, which has the higher Fresnel reflection coefficient, r,. If the optic axis is rotated 90" about the beam axis, the reflection coefficient is now rp,which is significantly reduced. The transmission is increased in the ultraviolet by 10% for this design, which is called a Clan- Taylor. The optic axis orientation decreases the useful birefringence which reduces the semifield angle. The cement n, is unity, and the angular acceptance primarily depends on the cut angle and the crystal's wavelength-dependent birefringence. The L / A ratio is about equal to one. To increase the semifield angle or to decrease the L / A ratio for a given maximum incident angle, double prisms are constructed.' Examples are the Ahrens prism (two Glan-Thompson prisms next to each other), the Grosse prism (an air-spaced Ahrens), and the Marple-Hess prism (a double GlanTaylor back to back). Since the number of components increases in double prisms, the manufacturing difficulty is increased, and extinction performance is about an order of magnitude worse than for single polarizing prisms. The Ahrens and Grosse prisms have half the L / A ratio of their Glan single-prism counterparts. For example, an Ahrens prism with an L / A of 1.8 has been made with a 26" semifield angle. The Grosse prism has an L / A about equal to one-half and is useful for tight physical limitations. MarpleHess prisms have definite advantages due to the cut angles S and -S. 0-rays below the axis will be totally reflected by the first surface, and 0-rays above the axis will be totally reflected by the second surface. The axial 0-ray and its total internal reflection due to n , ( A ) decreasing in A is the design parameter for the ordinary beam. The prism's limiting field angle is determined by the E-ray's total reflection, and the limiting wavelength depends on the axial 0-ray's transmission. These prisms have a lower transmission in the ultraviolet because of their increased L / A ratio. Glass-calcite Glan-type prism polarizers are possible, because the second prism is used to transmit the E-ray, and its birefringence is not used. The second crystal prism can be replaced with an index n, glass prism. Or, the first prism can be replaced with an no index glass, and an n, cement used to assemble the prism. The E-ray is totally reflected in this case. These prisms are usually not as good as all-crystal polarizers, because the greater L/ A ratio and optical cement enhance the resulting residual strain.s3

142

OPTICAL POLARIZATION

As with prism beamsplitters, image distortion in prism polarizers occurs for off-axis beams. Other contributions to degradation in extinction ratio is scattered light and the matter of eliminating the unwanted ray. One solution is to blacken the crystal’s sides. Another method is to cut and polish the prism side to make the unwanted beam exit normal to the exit face. The unwanted beam is saved. Prism performance studies show an order of magnitude increase in extinction ratio when the detector is moved farther from the prism, and entrance and exit apertures are used.s4 The crystal grades, optical finish, strain birefringence in contacting the two prisms, and optical alignment affect the prism’s p e r f ~ r m a n c e . ~ ~

3.6.3. Retarding Waveplates 3.6.3.1. Crystal Retardation Plates. A quartz or calcite plate with its optic axis parallel to the entrance surface introduces a phase shift between linear orthogonal polarizations, d 6 = 274 n E - n,) -. A

(3.44)

For n E > no, linearly polarized light travels faster parallel to the optic axis, and for nE < no, the fast direction is perpendicular to the optic axis. A fast (F) and a slow (S) coordinate system oriented with the optic axis describes the retarder’s orientation. The PoincarC sphere is useful to describe retardation effects. Incident and resultant elliptical polarizations are points on the sphere which are transformed due to phase shifts. A retarder’s phase shift between F and S rotates the incident point about the point marking F’s orientation. A positive phase shift is a clockwise rotation about F equal to the phase shift. For example, a quarter-wave retarder with its fast axis at +45” will pass H as R and V as L polarizations. The spectrum viewed with a retarding plate between crossed polarizers is fringed with the polarizations’ cyclical wavelength retardation. The wavelength interval between equal polarization states decreases for the thicker plates and can be used to filter or select a wavelength by selecting the corresponding polarization ellipticity. Lyot birefringent filters use this method to make typically a 1/4-A pass band isolating the fringe from its neighbors by cascading plates with double the fringe spacing in the spectrum or half the previous element’s thickness until the final element’s fringe spacing is 128 or 256 A. Retardation plates are usually made of quartz, mica, or plastic for the near ultraviolet to the near The spectral range is expanded by using other birefringent crystals. Calcite’s birefringence is very high, thus

WAVE PROPAGATION I N ANISOTROPIC M E D I A

143

enhancing its dependence on thickness and temperature. A quarter-wave plate for a particular wavelength would be more difficult to obtain and maintain than for quartz. The thickness of a retardation plate can be a single-order retardation period (which can be achieved with split mica or quartz in the infrared) or multiple orders thick. d

= mh.

(3.45)

Thick multiple-order plates are sensitive to changes in the retardation equation. Wavelength, birefringence, refracted angle, and temperature variations are factors in determining the resulting retardation. The sensitivity can be decreased by using thick plates by orienting one plate’s optic axis perpendicular to the other plate and by adjusting their thickness difference to a single-order thick. This combination acts as a single-order plate and is called a jirst-order plate. 3.6.3.2.Compensating Retardation Plates. All simple retardation plates still depend on wavelength. The retardation’s dependence on wavelength can be corrected for either with variable thickness plates called compensators, or with composite plates to produce the desired retardation characteristics. Compensators change the net thickness experiencing a birefringence which directly changes the phase difference between the two orthogonal polarizations, as demonstrated by Poincare’s sphere. The Babinet compensator is a fixed and a movable quartz wedge with the optic axis parallel to the entrance and exit surfaces, but crossed to each other. The field’s center is marked with a reference line on the fixed wedge, and the relation between the movable-wedge displacement and retardation can be empirically determined by using crossed polarizers as a polarizer and an analyzer. White light viewed with a Babinet between crossed polarizers produces colored bands and extinction indicating changing retardation orders. Zero phase difference at the center will produce a dark line with the reference line. Introducing a retarding plate will shift the fringes, and a correction opposite the retarding plate’s phase shift is needed to extinguish the center again. The distance moved by the wedge determines the introduced phase shift. To increase the useful field, a Soleil compensator is used. It is a fixed retarding plate and a variable-thickness plate with their optic axis crossed. The variable thickness plate introduces a uniform variable phase shift across the viewing aperture. If white light is viewed with a Soleil between crossed polarizers, the field is tinted with phase-shifted wavelengths. Zero net thickness between the plates results in extinction. A retarding plate introduced between the polarizers will transmit a particular wavelength, and a

144

OPTICAL POLARIZATION

phase correction made by the variable-thickness plate will again extinguish that wavelength. The Senarmont compensator is an elliptical polarization analyzer consisting of a quarter-wave plate and a polarizer. Elliptical polarization is incident with its major axis aligned with the wave plate's fast and slow axes. The wave plate will retard either the major or minor axis by a quarter wave, resulting in a linear polarization with an azimuth determined by the ellipticity. An elliptical polarization is selected with the azimuth of a following linear polarization analyzer with respect to the wave plate's axes. If the elliptical polarization analyzer is turned around with light now incident on the linear polarizer, then a known elliptical polarization with its axes aligned with the wave plate is produced. For example, consider how circular polarization is produced. The linear polarizer oriented parallel to the wave plate's axes produces linear polarization. The linear polarizer oriented 45" to the wave plate will produce either L or R circular polarization. 3.6.3.3.Achromatic Wave Plates. Composite achromatic wave plates can be divided into composites of the same or into composites of positive and negative birefringent crystals similar to focus correction for lenses.60 Possible positive and negative birefringent wave-plate composites for visible applications are magnesium fluoride and ammonium deuterium phosphate or magnesium fluoride and potassium deuterium phosphate designs with a 0.5% deviation; a calcite, quartz, and magnesium fluoride triplet with a 0.5% deviation; a magnesium fluoride and quartz doublet with a 5% deviation over the spectral region. Composite wave plates with the same material can be made with zero first-order variations in retardation and wave plate axis. This is demonstrated using Jones' matrix techniques. Three- and nine-plate combinations act as pure wave plates, and four- and ten-plate combinations behave as a wave plate plus a rotator. A three-element achromatic wave plate made of the same material with the first and third elements identical leads to the consequence that they are parallel. Various achromatic combinations of three-element wave plates for a desired retardation can be made using these assumptions, but the retardation axis may vary to first order in wavelength. Constraining the retardation axis leads to the middle plate being a half-wave plate, which is rotated by an angle 0 with respect to the outer elements. The total retardation is 26. COS~O=-T~~

3

cos 6 +-

(

= dlsin(26,),

where *A6 is the tolerance as a function of wavelength.

(3.46)

WAVE PROPAGATION I N ANISOTROPIC M E D I A

145

For three-plate combinations, some chromaticity is left, due to a quadratic term. A central-design wavelength needs to be chosen, and the allowable deviation from the desired retardation gives the useful spectral range. The correct retardation occurs at a wavelength above and below the central wavelength for all but the half-wave combination which varies cubicly. Combinations for the nine-element retarder (a threefold three-element retarder) are possible to make a retarder's retardation vary about 1% from 3514 to 10,000 A. For example, the three-element quarter-wave plate consists of two outer plates with a 115.5" retardation and a half-wave plate. A half-wave plate is made of three half-wave plates. A small additional rotation of the central half-wave plate introduces a first-order term which cancels the third-order term, improving the retarder's performance. If a three-element half-wave plate is split in the middle and the halves are then rotated, retardation is equal to twice the complement of the angle rotated. Also, the polarization is rotated by the complement of the separation angle. This combination of quarter-wave and half-wave plates is a fourelement variable-retardation plate which rotates the polarization axis. Nineelement half-wave combinations split in the middle give a ten-element variable wave plate with a performance similar to the nine-element one and in addition, it rotates the polarization as the four-element wave plate does. Two-element composite retarding plates have been mentioned in the literature, but upon considering the Jones' matrix calculation for two wave plates, no degree of achromaticity can be obtained, and the plates need to be tuned for the wavelength. Wave plates are usually made of quartz, mica, or stretched polyvinyl. alcohol. Since the birefringence of mica and polyvinyl alcohol is variable, wave plates need to be selected for the application. Some of the plastic laminates have a curl to them, and it is best to leave the curl alone. Mica sheets, usually muscovite, can be split into large apertures to the desired retardation by trial and error to account for the birefringence differences for different crystal samples.6' This can be accomplished with adhesive tape, which can remove layers that are 5-micron-thick or less and can remove high spots in large apertures. Since the wave plate is thin, little beam deviation and wave-front distortion occurs, but multiple coherent reflections can occur which can be eliminated by cementing thin films into an optically thick medium. Ultraviolet light curing cement is suitable for this purpose, taking care that the illumination is uniform across the aperture to minimize stress. 3.6.3.4. Measuring Wave Plates. To determine the optic axis for a wave plate, it is placed between crossed polarizers and rotated for maximum intensity. The entrance polarizer and wave plate act as an elliptical polarizer

146

OPTICAL POLARIZATION

with the resultant polarization ellipse axes aligned 45" to the wave plate's extraordinary and ordinary axes. The analyzer is then oriented 45" to the ellipse axes. Rotating the wave plate about the optic axis direction has no effect on the birefringence. The light-path length in the crystal and retardation will change, but the analyzer will transmit. Rotating the wave-plate surface perpendicular to the optic axis changes the birefringence from a maximum to a zero value for a 90" rotation, and so extinction occurs.62 The wave plate's retardation is related to the intensity of the elliptical polarization's major and minor axes, as seen earlier. It is a matter of measuring the maximum intensity of the above crossed polarizers and wave-plate arrangement and rotating the analyzer 90" to measure the minimum or minor axis intensity, (3.47)

The error in measuring the wave plate's retardation is determined by the intensity's error measurement and the crossed polarizer's extinction ratio. A 0.001YO intensity uncertainty produces less than 0.1" retardation measurement error.63 3.6.4. Induced Birefringence and Phase Modulation

Anisotropies can also be induced in a material's dielectric tensor by elastic stresses, electric fields, and magnetic fields, divided into elasto-optic, electrooptic, and magneto-optic effect^.^^'^^ Linear electro-optic and elasto-optic effects are due to nonlinearities between the material polarization vector P and the electric field. The material effects are described in terms of the impermeability tensor, B = E - ' . Small changes in the tensor due to linear changes in the electric field ( E ) and strain (S) is A B,

= rzkEk

+ pfklsk[= A B z -k A B: .

(3.48)

where rllk is the electro-optic coefficient, and P Y k l is the elasto-optic coefficient. The superscript implies holding that variable constant while determining the other variable. Summation is implied for indices which appear in both terms of a product of matrix elements. For example, the term r,,kEk means to sum the elements varying k to account for all k contributions. Also, symmetric tensors can be expressed in a compressed matrix notation to simplify the matrix element numbering from row and column to an ordering as follows:

r I 3= r3, = r s ,

r,* = rz, = r,.

WAVE PROPAGATION IN ANISOTROPIC M E D I A

147

Crystals with a linear electro-optic effect are also piezoelectric. Strained piezoelectric crystals produce an electric displacement related to the strain, so that the electro-optic and elasto-optic coefficients are defined while holding the other effect constant. The variable that is held constant, either E or S, is denoted by a superscript. The effect on a strained piezoelectric crystal is,

(3.50) where C:k/ is the elastic tensor element, ekliis the piezoelectric tensor element, and Tj is the strain tensor element. Conversely, an applied electric field in a properly cut piezoelectric crystal can generate shear or longitudinal waves which are used for transducers. The change in refractive index can be found by considering how the impermeability matrix changes for either effect. Under the assumption that the index change is small to the material’s initial index, the index change is

nips Ans = -2 ’ (3.51)

where p and r are chosen for a particular crystal orientation or application, and p S and rE are typically less than lop4. 3.6.4.1. Electro-optic Effect. Electro-optic modulators use either the linear Pockels electro-optic effect or the quadratic Kerr electro-optic effect, and the light beam travels either parallel to (longitudinal mode) or perpendicular to (transverse mode) the applied electric field. The retardation across an electro-optic modulator is limited by the beam’s maximum incident angle, electric field uniformity, scattered light, and refractive index variation. These modulators work best in collimated light, and less than 5% retardation variation across the aperture can be attained. The aperture should be checked between crossed polarizers in the on and off state. Extinction ratios between the on and off state are determined by the crossed polarizers extinction and the modulator’s retardation uniformity; extinction values range from 100 : 1 to better than 1000: 1. Some useful materials for Pockels cells are ammonium dihydrogen phosphate (ADP), potassium dihydrogen phosphate (KDP), potassium dideuterium phosphate (KD*P), lithium tantalate, lithium niobate, hexamine, and lead lanthanum zirconate titanate (PLZT) ceramic.66-68Spectral transmission, chemical stability, optical constants, and modulation frequency determine the material’s application. The natural birefringence

148

OPTICAL POLARIZATION

can be either offset with a bias voltage or can be eliminated by using crossed crystal pairs to cancel each other’s birefringence (this also reduces temperature-birefringence effects). The voltage applied to produce half-wave retardation is in the kilovolts range. The drive voltages for modulators depend also on the path length through the crystal and on electrode configuration. With many passes through a crystal or with cascaded crystal pairs, the drive voltage is reduced. High-frequency modulation is limited by capacitance, mechanical stress, and heating; gigahertz frequencies can be achieved. Low-frequency modulation on the order of a second or longer has the problem of current leaking across the crystal. The charge needs to be replaced. This is accomplished with a slightly conducting grease for contacting the electrodes to the crystal.69 Longitudinal Pockels cells have either transparent electrodes or ring electrodes on the entrance and exit surfaces. The phase shift is

(3.52) where I is the distance through the crystal, and E is the electric field. A specification for longitudinal . modulators is the voltage for half-wave retardation, VAI2, which is needed to produce maximum transmission between crossed polarizers. Large apertures can be made. Transverse Pockels cells have electrodes on the crystal’s sides, and the light beam passes perpendicular to the applied field. Retardation increases with an increased path length through the crystal and with a decreased entrance aperture or electrode gap. Capacitance increases with increased crystal length so that the frequency bandwidth decreases. The retardation is

(3.53)

where I is the crystal length, a is the electrode gap, and is the half-wave field distance product defined for a particular crystal, its orientation, wavelength, and temperature. The quadratic Kerr effect aligns the material’s molecular polarization vectors by an applied electric field. The time to orient and induce birefringence with a high voltage and to disorient it again determines the operating

WAVE PROPAGATION I N ANISOTROPIC M E D I A

149

frequency for an electro-optic Kerr cell. The retardation is S = 2.rrKE21,

K = Kerr constant

(3.54)

I = length Electro-optic Kerr cells use a variety of gases, or liquids, or PLZT. For liquids and gases, the drive voltages are typically tens to hundreds of kilovolts, and the switching times are nanoseconds. For PLZT, the drive voltages are hundreds of volts and the switching speeds are microseconds. Also, liquid and gas Kerr cells do not have any residual natural birefringence. Subpicosecond laser pulses and high-power densities can be used to induce birefringence in Kerr materials, This is the optical Kerr efect. The optical Kerr cell can be either transverse or longitudinal; the type determines the modulated beam’s direction with respect to the gating laser pulse. The transverse cell has a long interaction path with the modulated laser beam, and self-focusing problems can be induced for crystal lengths greater than one or two centimeters thick. Usually, the optical Kerr cell is a longitudinal cell with the gating laser pulse colinear with the light beam to be modulated. The limiting switching speed is the time to randomize the molecular polarization after a gating pulse has passed. For nitrobenzene, the relaxation time is 32 ps and for carbon disulfide about 2 ps. The induced birefringence due to the electric field E ( t ) from the laser is 6nll- Sn, = n28

I:,

E 2 ( t ) exp[-(t - r’)]

dt’

-, T

(3.55) where n is the refractive index, n28 is the ac Kerr coefficient, Sn, and Snll are the refractive index changes perpendicular and parallel to the E field, T is the relaxation time, t is the gating pulse length, I?( t ) is the time average over one period, and P ( t ) is the power density. For t > T, the birefringence ~ n ,-,~ n = , n,,E2( t ) .

(3.56)

The sample beam’s retardation is 6 = 27rn,,P( t )

I ~

(Acne,) ‘

(3.57)

The on/off extinction ratio is typically about 1000: 1, determined mostly by the extinction ratio of the entrance and exit polarizer, and the aperture is the gating beam diameter.

150

OPTICAL POLARIZATION

3.6.4.2. Acousto-optic Effect. There are several ways to use the elastooptic effect depending on incident beam diameter and frequency modulat i ~ nA. uniaxial ~ ~ static strain deforms the isotropic media’s spherical phase velocity surface into a spheroidal phase velocity surface with an optic axis oriented along the strain direction with a birefringence,

Anli= - n o3- PSSC. 2 C = stress-optic constant

(3.58)

A useful wave plate can be made with a stressed plate and light traveling perpendicular to the stress. The equation for phase induced by stress is,

2 IT A nl

6=---

A

-

2 TSCl A

’

(3.59)

where 1 is the distance through the crystal. For quartz, the stress needed to produce quarter-wave retardation at 5500 A is an order of magnitude less than needed to destroy the plate. If the stress is a sound wave with a wavelength, A, much larger than the beam diameter, the phase modulation varies sinusoidally with the stress.

s = s,,, for

e-i(kr-RO

-

s,,,

COS(Rf)

(3.60)

A > than the beam diameter

6 = cos(Rr),

where R is the angular frequency, and k is the wave number for sound. If the sound wavelength is comparable to the beam diameter, the phase modulation or birefringence will vary across the aperture, and the intensity modulation contrast between crossed polarizers will not be as large. This limits the maximum frequency for resonating elasto-optic modulators to less than 1 MHz. Standing-wave photo-elastic modulators can drive in a self-resonating mode and can be made with zinc selenide, calcium flouride, KRS-5 infrared glass, germanium, and fused quartz. There are two types of construction, either the bar type73or an isometric-contour element.74 The bar type is the first design which is made of either one or two transducers connected to either side of the optical element to be modulated. The transducers and the optical element all resonate in a standing-wave mode with nodes at the joints. The elements can be cemented together with silicone cement to reduce strain birefringence. Bar-type modulators have two error sources which are eliminated by the isometric-contour-type modulator. The errors are, ( 1 ) a

WAVE PROPAGATION I N ANISOTROPIC M E D I A

151

nonlinear, anharmonic dc birefringence proportional to the driving amplitude squared (an error of about l % ) , and (2) the reflectance at the modulator’s surfaces is modulated by the modulated refractive index adding a small variation on the order of 0.001%. The isometric-contour-type modulator is a rectangular optical element supported at its corners along the length and modulated across the width. The light beam travels the long direction through the bar’s center. The usable entrance aperture is a little larger for these types. For photo-elastic modulators in general, their entrance apertures can be large, and the incidence angle can be as large as 50”. Modulators can be driven by transducers at sound frequencies with many sound wavelengths across the beam diameter.72s75 The compressions moving at sound’s velocity will appear to the light beam as an almost stationary phase grating with a grating spacing equal to the sound’s wavelength. If light’s interaction with the sound waves is small, it is diffracted into orders, N, with angle, + N , according to the thin phase grating equation, sin+N-sin+,,=-,

NA A

N=0,*1,*2

,....

(3.61)

The amplitude of the various orders is given by Bessel functions and the maximum phase shifts, 6, in the medium due to the sound wave. (3.62) where I,, is the incident intensity. Changing the sound frequency changes the orders’ angles, and changing the sound amplitude modulates the intensity in other orders. These effects can be used in a feedback loop as an active element for beam steering and modulating. Since the phase grating is traveling, the diffracted light in the N t h order will be frequency-shifted by an amount, (3.63) where V, is the material’s sound velocity, and & is the light frequency. For high sound frequencies or for a long interaction path with light, the behavior changes from a thin phase grating operating in the Raman- Nath regime to the thick phase grating behavior similar to X-ray diffraction called the Bragg regime. The light is diffracted into two orders, because the sound wave is a sinusoidally varying density function needing one frequency term to describe it, while for X-ray diffraction in particle distributions, higher spatial harmonics are needed to describe the particles’ positions producing a range of diffracted angles.

152

OPTICAL POLARIZATION

Light incident at the angle OB determined by the Bragg conditions will be diffracted into a zero-order beam and either the plus or minus first-order beam, depending on the incident angle. A

sin Be = 2n0A’

(3.64)

where no is the material’s refractive index. Modulators are usually Bragg-type modulators, and the on/off contrast in the first-order beams are limited by scattering. Extinction ratios of 1000 : 1 are possible in the first-order beams. But, the modulator is not able to diffract all of the energy out of the zero-order beam, because rediffraction of higher orders and the sound wave’s refractive effects add light to the zero-order beam. Contrast ratios of 10: 1 are typical. To improve the on/off contrast, spatial or polarization filtering is used to separate the zero-order beam. Both filtering methods use two modulators. Spatial filtering uses aperture stops. The first modulator has an aperture placed far behind it to isolate the zero order; the beam is then passed through a second modulator and aperture combination like the first. Contrast ratios of about 100: 1 are produced. Polarization filtering uses a shear-driven modulator which acts as a half-wave retardation for the first-order beams. The zero-order beam is selected by an entrance polarizer parallel to the transducer surface and a parallel exit polarizer. To improve contrast, the beam is passed through a second combination or reflected through the first combination. The contrast ratio is again 100 : 1 .76 The efficiency to drive an acousto-optic modulator is a measure of a material’s usefulness as a modulator and its frequency b a n d ~ i d t h . ~The ’ stress retardation for an elasto-optic material can be rewritten by using a relation between stress, velocity v, and mass density p ; this gives an expression for the strain now replaced with an expression from the sound power density P, of energy propagating at sound velocity under strain.

’= -[

‘

2 Ps1n:p2 (pv3hA2)]

(3.65)

where M is the acousto-optic figure of merit and is related to the power needed to produce a given retardation, h is the height perpendicular to the light and sound waves. Other figures of merit take into account refinements in the theory such as the bandwidth.

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153

Acousto-optic modulators recently have been used for spectral filters tuned by sound frequency and observing the first-order output as the sound frequency is swept. Rapid scans, of usually less than a second, can cover large spectral regions, and applications in the infrared are common.78 Another use for acousto-optic modulators is to impress light-phase information on electrical signals controlling a modulator in a phase-locked loop to measure without ambiguity light-phase shifts. Other uses are beam steering, light-frequency shifting, and m o d ~ l a t o r s . ~ The ~*~ useful ~ bandwidth is usually hundreds of MHz, and the maximum useful frequency modulation is limited by the number of sound waves across the beam diameter, usually tens of MHz. 3.6.4.3. Magneto-optic Effect. Magneto-optic effects are the Faraday, Kerr, and Cotton-Mouton effect; the Faraday effect is the most prominent effect and most often used. A helical electric current coiling along a material demonstrating the Faraday effect will rotate the plane of polarization along the current direction due to the phase retardation between left and right circular polarization. The retardation introduced is S

=

VHI.

(3.66)

V = Verdet constant H =field strength 1 = light path length

This also means that multiple reflections will show an accumulation of multiple retardations and rotations. Faraday rotating glasses are available, too. 3.6.4.4. Depolarizers and Polarization Scramblers. Polarization can be scrambled such that the detection system sees a net zero polarization created by either a time average, a spectral average, or a spatial average across the beam, or by all three effects. Incoherent multiple scattering randomizes the polarization to produce an unpolarized beam.7 The Lambertian sphere, an opal glass diffusing plate, powder-liquid mixtures, hot-pressed polycrystalline aluminum oxide or thorium oxide, cold-pressed magnesium oxide, and cloudy fused quartz are examples. The powder-liquid mixtures can effectivelydepolarize the light for very thin films. But, scattering depolarizers destroy the image and usually strongly attenuate the beam. The preservation of the image and intensity suggests using birefringence to phase average incident polarization either in wavelength, across the aperture, or in time.'' A birefringent plate with its optic axis oriented 45" to the azimuth of the incident polarization produces a cyclically varying retardation in wavelength increasing in frequency with plate thickness. A

154

OPT1C A L PO LA R I ZATIO N

thick plate with many retardation cycles in a small wavelength range produces a fairly unpolarized beam for a system with poor spectral resolution. To allow for a varying azimuth incident on a thick-plate pseudodepolarizing element, a Lyot pseudodepolarizer is made of two plates with a 2 : 1 thickness ratio and a 45" angle between their optic axes.82 The 45" angle is most critical to the depolarizer's performance, and the thickness ratio only needs to be different from unity. Lyot depolarizers also limit spectral resolution. Birefringent crystal wedges can be made to introduce a variable phase shift in one direction across the beam diameter for a particular wavelength and is a monochromatic depolarizer if the optical system's final image does not spatially resolve the phase variation. Two wedges with the optic axes oriented 45" to e8ch other are independent of the azimuth of the incident polarization. Double images and beam deviation occur for these depolarizer types. A varying driving function with a single optic modulator aligned with its axes horizontal and vertical produces a polarization intensity modulation expressed in four intensities, for example total intensity, horizontal linear, +45" linear, and right circular polarization intensities modulated by the driving function. driving function

(3.67)

Sb = so

total intensity polarized + unpolarized

S : = S, cos S - S3 sin 6

horizontal linear polarization

s; = s,

+45" linear polarization

S ; = S, sin 6 + S3 cos S

circular polarization,

where S' and S are the transformed and incident polarization vectors. The phase-modulated beam will always have one linear component unmodulated. If the modulator is rotated 45", then the modulated exiting linear polarization changes from horizontal to 45" linear.

s; = s,, S ; = S, cos S - S, sin 6.

If the driving function is a sinusoidally varying function, then the intensity phase modulation can be expressed with Bessel function expansions for the sine and cosine intensity modulation.

155

WAVE PROPAGATION I N ANISOTROPIC M E D I A

S = S,,, sin wt, u)

sin 6 =sin(S,,,

sin w t ) = 2

C

(3.68)

J2n+l(Smax) sin[(2n + l)ot],

n=O

m

C

cos 6 =cos(&,,, sin wt)=Jo(6,,,)+2

JZn(Smax) sin(2nwt).

fl=l

The time averages for the sine terms are zero, and the remaining term is J,( S,,,) = 0 for S,,, = 2.405, the first zero of J, . This means zero polarization can be produced in either horizontal linear and circular polarizations, or in 45" linear and circular polarizations, and can be used to calibrate instrumental polarization to an accuracy of 0.01 to o . o o ~ ~ ~ . ~ ~ If an optic modulator were driven by a sawtooth function such that the phase modulation varied linearly from 0 to 2rr, the time-averaged output light would have only a linear polarized component. If a second modulator is added with its optic axis 45" to the first modulator and driven at an integral multiple of the first frequency nw, then the resulting polarization is zero. Another method to produce depolarized light makes use of the timeaverage effects for rotating retardation plates.83

s=

plate retardation

+ unpolarized

st,= s,

total intensity polarized

S' -I - 2( 1 + cos 6)Sl

horizontal linear polarization

S ; = $(1 + cos 6)S2

+45" linear polarization

s; = s3 cos 6

circular polarization

A rotating achromatic quarter-wave plate will pass only unpolarized and linearly but not circularly polarized light. A rotating achromatic half-wave plate will pass only unpolarized and circularly but not linearly polarized light, and the circular polarization's handedness is reversed. A combination of two wave plates rotating in opposite directions produces an element passing only unpolarized light.

3.6.5. Liquid Crystals

Liquid crystals (LCD) are organic molecules in a state between liquid and solid and show electromagnetic anisotropies. They are structured in three basic geometries which are easily disrupted by electric fields, magnetic fields, or temperature, depending on the design. The liquid crystals can be

156

OPTICAL POLARIZATION

chosen such that the net molecular polarization lines up parallel or perpendicular to the crystal's long axis; this corresponds to positive or negative birefringence, A n 0.2. For example, tunable birefringence can be produced by treating the container's walls such that negative nematic crystals line up parallel to the light beam in the off state. An electric field applied to the walls will flip the liquid crystals perpendicular to the electric field and light beam and produce a retardation proportional to the voltage up to a threshold voltage. Or, the cell's interior wall can be treated to align the positive nematic crystal structure perpendicular to the beam; this is done by rubbing the glass to form parallel microgrooves and then treating the surface with a high-energy surfactant. Then, a 90" twist is introduced between the entrance and exit surface, rotating the beam 90". A voltage of 5-10 volts will completely disrupt the rotation and extinguish the beam between crossed polarizers. Liquid crystals are low-current, low-voltage, high-contrast, thindisplay p o l a r i z e r ~ . ~ ~

-

3.6.6. Dichroic Polarizers

Linear dichroic molecules and crystals absorb polarization with different absorption coefficients K, and K y . To reduce scattering effects, molecules of small dimensions are used, usually needle-shaped iodine molecules. To align the molecules, they are either applied to a burnished glass surface with parallel microgrooves and allowed to settle, or are placed on a polyvinyl alcohol or nitrocellulose base and stretched, which also introduces unwanted birefringence. Better, controllable performance is obtained with the plasticsheet p ~ l a r i z e r . ~ ~ Extinction ratios and transmission vary with the concentration of dichroic molecules; and concentration of molecules, standard densities, and extinction ratios are commercially available. Heavily dyed polarizers have a parallel transmission of 12-14% and a crossed transmission of 0.001%. As the dye is decreased, the short-wavelength extinction or optical density decreases, thus decreasing the spectral region with high uniform optic densities for crossed polarizers. Their parallel transmission is 30% or more. To extend the plastic's usefulness into the infrared, the polyvinyl alcohol sheet is catalytically dehydrated to polyvinylene. It is useful in the infrared from about 8000 A-20,OOO A, with a large ttansmission of the unwanted component at 9200 A. For near ultraviolet use, the polyvinyl alcohol/iodine polarizer can be purified. Its range is down to 2000A. The thin Polaroid sheets are usually cemented in glass or plastic laminates. King has reported on dichroic polarizers and crystal prism polarizers. The dichroic polarizer type has its place according to degree of p e r f o r m a n ~ e . ~ ~

SLITS, GRATINGS, A N D METAL GRID WLARIZERS

157

3.7. Slits, Gratings, and Metal Grid Polarizers An ideal slit is an infinite conductor and is very thin. Approximate examples are scratches on thin silver-coated mirrors. The light transmitted by narrow slits of width d satisfying A / d > 2 are completely S polarized and perpendicular to the slit. This is the electromagnetic theory region, and the effect described is Hertzian polarization. Either finite conductivity or a thick slit will decrease the slit’s transmission. When A / d < O S , then the polarization effect for P or S polarizations is negligible. This is called the scalar theory region described by infinite conductivity. For the region 2 > h / d > 0.5, the transmitted polarization is predominantly P polarization due to the slit’s finite conductivity and thicknesss6 Wire grids with a spacing of A / d > 0.5 are made to S polarize light. Successful wire grids are either gratings with their ridges metal-coated at a high-incidence angle or metal whiskers grown by a metal beam directed at a substrate at a high-incidence angle.87Grid spacings as small as visible-light wavelengths can be made for polarizers for wavelengths greater than 8000 A. Extinction ratios are greater than 100: 1 and improve with wavelength. Also, wire-grid polarizers can have a large acceptance angle and aperture.’ Diffraction gratings polarize the light according to the gratings material and overcoat, the groove shape and spacing d, the wavelength A, and the order N. As for slits, the P polarization is predicted fairly well with a scalar theory down into the ultraviolet.** The S polarization perpendicular to the slit is not as well behaved, and polarization anomalies in different orders and wavelengths are more pronounced than for P polarizations. Energy can be either diffracted into or out of the observed order, so the anomalies can appear either bright or dark. The influence of a dielectric coating is to broaden the anomaly and to shift it to longer wavelengths. For h / d < 0.2, the polarization effects become negligible, and scalar theory applies to both the P and S components. The grating’s efficiency in wavelength and diffracted angle is strongly influenced by anomalous behavior. The grating anomalies are either Wood’s anomalies or resonance a n ~ m a l i e s . *Wood’s ~ * ~ ~ anomalies occur in a grating order at a wavelength which is at the same time being diffracted at *90° in another order, passing onto or off the grating. Wood’s anomalies occur at Rayleigh’s wavelengths given by the grating equation nh = a(sin i * I ) ,

(3.69)

where n is the passing-off order.“ Resonance anomalies are leaky surface waves supported by the grating and depending on the grating surface structure. A grating with shallow grooves and a slightly modulated surface has little reactance and little

158

0 PTIC A L PO LA RI ZATlO N

anomalous effect. The resonance and Wood's anomalies are close together for this type of surface. For a strongly modulated surface, the reactance increases and the anomalies separate. Blazed gratings in the Littrow configuration (the incident and. diffracted beams have the same angle), with right triangle-shaped grooves and the 90" corner at the apex, can be described by the blaze angle only. First-order diffraction is most strongly influenced by anomalies which decrease for higher orders because A / d decreases. Higher-order anomalies and diffracted angle deviations can be deduced in terms of the first-order Littrow configuratiom9* First-order gratings with a blaze angle of less than 5" behave close to scalar theory and show few anomalies. The P and S polarizations have a peak in their efficiency at the blaze angle equal to 100%. For gratings blazed between 5" to 18", Wood's anomalies at the Rayleigh wavelength for the -1 and + 2 order at A/d = 2 / 3 , about 19.5", decrease in strength with the blaze angle. The S grating efficiency is 100% at the blaze angle, and the P efficiency peak is decreasing. For gratings blazed at 18" to 22", the anomalies are suppressed because the Rayleigh pass-off anomaly occurs in the same direction as the blaze angle. The S efficiency becomes more uniformly high for diffraction angles greater than about 19". The P efficiency is a minimum of about 80%. For blaze angles greater than 22", the S efficiency for diffraction angles greater than 19" is high up to 60" and is 100% at the blaze angle. The P efficiency peak is 100% at the Rayleigh wavelength A / d = 2 / 3 . Gratings with an angular deviation between the incident and reflected beams have lower P and S efficiencies, and the anomalies and peak efficiencies shift in wavelength. Gratings with known P and S efficiencies can be used as a polarizing element in an instrument.

3.8. Light Source and Detector Polarizations Light sources can be polarized for a number of reasons. Lasers are usually linearly polarized coherent emission sources. Incandescent and fluorescent emissions are polarized because the emitted radiation is refracted at the surface; and radiation refracted at a grazing angle to the surface is strongly P polarized.93 Metal filaments, ribbons, and Nernst glowers have demonstrated strong polarization effects for radiation from the highly angled cylindrical edges. Also, deuterium lamps have been studied, which demonstrate strong polarization effects for old lamps with a thin metal film deposited on the glass envelope.94 New lamps are not as strongly polarized across the light spot, but the polarization is still nonuniform away from the central light spot, which is possibly due to polarized reflections from the

POLARIZATION DETERMINATION A N D MATHEMATICAL DESCRIPTION

159

envelope. The envelope’s polarization effects have been demonstrated for tungsten lamps. Limiting light to the light spot reduces polarization effect^.^^'^^ Depolarizers can be used to remove the polarization effects if the effects are found to be significant enough. Detectors are also polarization sensitive for various reasons which sometimes cannot be neglected. For example, photomultipliers or photocathodes are sensitive to polarization due to their intrinsic characteristics. Also detectors and arrays can have windows with varying static birefringence. The polarization effects need to be considered for the particular application and can be removed, if necessary, by either calibration or a depolarizer.

3.9. Pola rizat ion Deter mi nation and Mat hemat i ca I Descri ption Polarization is either coherent and totally polarized or incoherent with an unpolarized and a polarized component. In either case, the PoincarC spherical surface description is applicable. An elliptical polarization is marked on the sphere and transformed to a new polarization after passing through an optical element. Coherent polarized light is known in terms of phase shift between orthogonal polarizations and their intensities, for example linear or circular polarizations. Incoherent polarized light is described by Cartesian coordinates as the amount of linear polarization in the x and +45” axes, and the amount of L or R polarization. This coordinate system plus intensity is known as Stokes’ parameters. Two- or four-dimensional transformation matrices describe the polarization vector and the optical element’s effects; and the cumulative effect is a matrix multiplication of all the effects in their order, resulting in a single transformation matrix describing the system. Coherent light can be written as a two-dimensional vector multiplied by a complex phase term and transformed with 2 x 2 matrices known as Jones’ matrices. Incoherent radiation can be described in real numbered Stokes’ parameters and transformed with 4 x 4 matrices known as Mueller’s matrices. Light represented in terms of Jones’ matrices is the vector sum of two linear orthogonal amplitudes E , and E,, and a phase shift 6. The fact that they have the same frequency can be implied and removed from the calculations. The modified form for an incident beam E and output beam E’ is,

E = E x e x+ EyeY=

(3 ,

(3.70)

160

OPTICAL POLARIZATION

Clear isotropic materials, linear polarizers, retarders, reflections, and coordinate rotations transform E to E’. E‘=M*E

transparent isotropic medium in x - y M = ( kx 0

0

k,

),

kx 1

Ex

k, E,

Eb

)

partial linear polarizer in x - y

6 phase retardation in x - y

reflection in x - y -sina

Ex cos a + E, sin a -Ex sin (Y + E, cos a

cosa

rotation of an angle a counterclockwise looking into the beam cos2 a

1 0

cos a sin a

general orientation of linear polarizer M

=R(7Y)(

‘6“

cos2a e i r r I 4 j ( 2 ) ” 2 cos a sin a e P * , 4 ) R ( a ) = ( j ( 2 ) ’ / 2cos a sin a sin2a ePJrrI4

general orientation of quarter-wave plate

(3.71)

Light represented in terms of Stokes’ vectors and Mueller’s matrices are related to the Jones’ matrices. The Stokes’ vector has several labeling schemes,

(I Q

u

V)

(I M C S) (So

s,

s 2

SJ

POLARIZATION DETERMINATION A N D MATHEMATICAL DESCRIPTION

161

The following relations hold between the Stokes' vectors and the timeaveraged electromagnetic vectors,

I:=(E:)+(E:)+I~, 2

Q 2 + U 2 + V2,

0 = ( E 3 -W;L U = 2( E,E,,) cos S

= Re( I?$,,),

V = 2( ExE,,)sin 6 = Im( ExEy).

(3.72)

The Stokes' vectors can also be written in terms of the light's polarization azimuth a,ellipticity e, and the percent polarization P.

P=

( Q 2 + U 2 + V2)'/2 3

IT

a = 0.5 arctan(

:),

e = 0.5 arcsin Q = ITP cos(2e) sin(2a),

U = ITP cos(2e) cos(2a), V = ITP sin(2e).

(3.73)

The Mueller transformation matrices are similar to the Jones' matrices and are determined by how they transform the incident light to output light. 1 1 0 0

0" linear polarization -1

0 0

0 0 0 90" linear polarization

162

OPTICAL POLARIZATION

1 0 1 0

45” linear polarization

right circular polarization 1 0 0 - 1

-1

0 0

left circular polarization 0

0

0 rotation of an angle

cy

counterclockwise looking into the beam.

The 4 x 4 transformation matrices can also be determined experimentally to take out an instrument’s polarization. A fairly complete list of matrices is given by Shurcliff,2’ and recent articles go into details for error analysis with Jones’ and Mueller’s matrices for polarization measurement^.^'-^^' References J. Strong, Concepts of Classical Optics, p. 110, W. H. Freeman, San Francisco, 1958. M. Born and E. Wolf, Principles ofOprics, p. xxi, Pergamon Press, New York, 1964. R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surface Sci. 16, 14 (1969). J. D. Jackson, Classical Electrodynamics, p. 269, Wiley, ,New York, 1975. Ramachendran and Ramaseshan, Encyclopedia of Physics, Vol. 2 5 , Part 1, S, p. 1 (Flugge, ed.), Springer-Verlag, 1961. 6. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, p. 270, NorthHolland Publ. Amsterdam, 1977. 7. J. M. Bennett and H. E. Bennett, “Polarization,” in Handbook ofoptics, Chap. 10, ( W . G. Driscoll and W. Vaughan, eds.), McGraw-Hill, New York, 1972.

1. 2. 3. 4. 5.

REFERENCES

163

8. R. M. A. Azzam, “Consequences of Light Reflection at the Interface Between Two Transparent Media Such that the Angle of Refraction is 45”;’ J. Opt. SOC.Am. 68, 1613 (1978). 9. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, p. 288, NorthHolland Publ., Amsterdam, 1977. 10. R. M. A. Azzam, “Direct Relation Between Fresnel’s Interface Reflection Coefficients for the Parallel and Perpendicular Polarizations,” J. Opt. SOC.Am. 69, 1007 (1979). 11. R. M. A. Azzam, “Transformation of Fresnel’s Interface Reflection and Transmission Coefficients Between Normal and Oblique Incidence,” J. Opt SOC.Am. 69, 590 (1979). 12. R. M. A. Azzam, “Mapping of Fresnel’s Interface Reflection Coefficients Between Normal and Oblique Incidence: Results for the Parallel and Perpendicular Polarizations at Several Angles of Incidence,” Appl. Opt. 19, 3361 (1980). 13. M. V. R. K. Murty and R. P. Shukla, “Extinction of Light in Brewster Polarizers: a Shadowlike Phenomenon,” Appl. Opt. 22, 1094 (1983). 14. G. Hass and W. R. Hunter, “Reflection Polarizers for the Vacuum Ultraviolet Using Al+MgF2 Mirrors and an MgF2 Plate,” Appl. Opt. 17, 76 (1978). 15. H. A. Van Hoof, “Imaging Polarizer for the VUV with Spherical Mirrors,” Appl. Opt. 19, 189 (1980). 16. G. Hass, “Reflectance and Preparation of Front-surface Mirrors for use at Various Angles of Incidence from the Ultraviolet to the Far Infrared,” J. Opt. SOC.Am. 72, 27 (1980). 17. J. 0. Stenflo, H. Biverot, and L. Stenmark, “Ultraviolet Polarimeter to Record Resonanceline Polarization in the Solar Spectrum Around 130-150 pm,” Appl. Opt. 15, 1188 (1976). 18. J . T. Cox and G. Hass, “Highly Efficient Reflection-type Polarizers for 10.6 p m CO, laser Radiation using Aluminum Oxide Coated Aluminum Mirrors,” Appl. Opt. 17,1657 (1978). 19. W. R. Hunter, “Design Criteria for Reflection Polarizers and Analyzers in the Vacuum Ultraviolet,” Appl. Opt. 17, 1259 (1978). 20. D. T. Rampton and P. W. Grow, “Economic Infrared Polarizer Utilizing Interference Effects in Films of Polyethylene Kitchen Wrap,” Appl. Opt. 15, 1034 (1976). 21. W. W. Shurcliff, Polarized Lighf,p. 78, Harvard University Press, Cambridge, Mass., 1964. 22. T. J. McIlrath, “Circular Polarizer for Lyman-Alpha Flux,” J. Opt. SOC.Am. 58,506 (1968). 23. R. M. A. Azzam and N. M. Bashara, EIlipsornefry and Polarized Light, p. 283, NorthHolland Publ., Amsterdam, 1977. 24. M. Ruiz-Urbieta, E. M. Sparrow, and E. R. G. Eckert, “Thickness and Optical Constants of Films,” J. Opt. SOC.Am. 61, 351 (1971). 25. R. M. A. Azzam, A.-R. M. Zaghloul, and B. M. Bashara, “Ellipsometric Function of a Film-substrate System: Applications to the Design of Reflection-type Optical Devices and to Ellipsometry,” J. Opt. SOC.Am. 65 252 (1975). 26. R. M. A. Azzam, A.-R. M. Zaghloul, and B. M. Bashara, “Design of Film-substrate Single-reflection Linear Partial Polarizers,” J. Opt. SOC.Am. 65, 1472 (1975). 27. A.-R. M. Zaghloul, R. M. A. Azzam, and N. M. Bashara, “Designs of Film-substrate Single-reflection Retarders,” J. Opt. SOC.Am. 65, 1043 (1975). 28. S. Kawabata and M. Suzuki, “MgF,-Ag Tunable Reflection Retarder,” Appl. Opt. 19, 484 (1980). 29. M. Ruiz-Urbieta and E. M. Sparrow, “Reflection Polarization by a Transparent Filmabsorbing Substrate System,” J. Opt. SOC.Am. 62, 1188 (1972). 30. M. Ruiz-Urbieta and E. M. Sparrow, “Wavelength Bandwidth and Other Design Aspects for Film-substrate Reflection Polarizers,” Appl. Opt. 12, 590 (1973). 31. M. Ruiz-Urbieta and E. M. Sparrow, “Effect of Film Absorption on a Film-substrate Reflection Polarizer,” J. Opt. SOC.Am. 63, 194 (1973). 32. N. Born and E. Wolf, cf. Ref. 2, Chap. 1.

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OPTICAL POLARIZATION

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53. E. 0. Ammann and G. A. Massey, “Modified Forms for Clan Thompson and Rochon Prisms,” J. Opt. SOC.A m . 58, 1427 (1968). 54. J. F. Archard, “Performance and Testing of Polarizing Prisms,” J. Scr. Instr. 26, 188 (1949).

55. R. J. King and S. P. Talim, “Some Aspects of Polarizer Performance,” J. Phys. E 4, 93 (1971). 56. W. C. Davis, “Simplified Wave Plate Measurements,” Appl. Opt. 20, 2879 (1981). 57. E. L. Gieszelman, S. F. Jacobs, and H. E. Morrow, “Simple Quartz Birefringent QuarterWave Plate for Use at 3.39km,” J. Opt. SOC.A m . 59, 1381 (1969).

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