physical optics and light measurements

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~ed bv ACADEMIC PRESS INC. (LONDON) 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 measure-

ments. 1. Malacara, Daniel, Date- 11. Series. QC395.2.P49 1988 535’.2 88-978 ISBN 0-1 2-47597 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 VOLUMES I N TREATISE . . . . . . . . . . . . . . . xi11 ...

1 . Interference by DANIEL MALACARA

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . 1

1.2. Two-Beam Interferometers . . . . . . . . . . . . . . 7

1.3. Multiple-Beam Interferometers . . . . . . . . . . . . . 25

1.4. Multiple-Pass Interferometers . . . . . . . . . . . . . 37

1.5. Applications of Interferometry . . . . . . . . . . . . . 38

References . . . . . . . . . . . . . . . . . . . . . 45

2 . Diffraction and Scattering by DANIEL MALACARA

2.1. Diffraction . . . . . . . . . . . . . . . . . . . . . 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

V

vi CONTENTS

3 . Optical Polarization by FREDERIC R . STAUFFER

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . 3.2. Electromagnetic Description of Light . . . . . . . . . . 3.3. Wave Propagation in Isotropic Media . . . . . . . . . . 3.4. Wave Propagation for Metals . . . . . . . . . . . . . 3.5. Thin Films . . . . . . . . . . . . . . . . . . . . . 3.6. Wave Propagation in Anisotropic Media . . . . . . . . 3.7. Slits, Gratings, and Metal Grid Polarizers . . . . . . . . 3.8. Light Source and Detector Polarizations . . . . . . . . . 3.9. Polarization Determination and Mathematical Description

References . . . . . . . . . . . . . . . . . . . . .

107

108

113

120

125

133

157

158

159

162

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

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . 167

4.2. Theory of Holography . . . . . . . . . . . . . . . . 168

4.3. Different Types of Holograms . . . . . . . . . . . . . 175

4.4. Some Applications of Holography . . . . . . . . . . . 191

4.5. Experimental Procedures in Holography . . . . . . . . 199

References . . . . . . . . . . . . . . . . . . . . . 206

5 . Photometry and Radiometry by WILLIAM L . WOLFE

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . 213

5.2. Symbols, Units. and Nomenclature . . . . . . . . . . . 214

5.3. Formulas for Blackbody Radiation . . . . . . . . . . . 219

CONTENTS vii

5.4. Simple Radiative Transfer . . . . . . . . . . . . . . . 223

5.5. Radiometric Temperature Measurements . . . . . . . . 239

5.6. Radiometric Instruments . . . . . . . . . . . . . . . 246

5.7. Measurements . . . . . . . . . . . . . . . . . . . . 263

5.8. Photometry: Radiometry of Visible Light . . . . . . . . 284

References . . . . . . . . . . . . . . . . . . . . . 287

6 . Detectors by T . 0 . POEHLER

6.1. Introduction . . . . . . . . . . . . . . . . . . . . . 291

6.2. Figures of Merit . . . . . . . . . . . . . . . . . . . 292

6.3. Thermal Detectors . . . . . . . . . . . . . . . . . . 293

6.4. Photon Detectors . . . . . . . . . . . . . . . . . . . 303

6.5. Noise . . . . . . . . . . . . . . . . . . . . . . . . 328

6.6. Optical Window Material . . . . . . . . . . . . . . . 331

References . . . . . . . . . . . . . . . . . . . . . 332

This Page Intentionally Left Blank

CONTRIBUTORS

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

RAMENDRA DEO BAHUGUNA (167), Centro de Znvestigaciones en Optica,

DANIEL MALACARA ( 1 , 49, 167), Centro de Investigaciones en Optica,

THEODORE 0. POEHLER (291), Applied Physics Laboratory, Johns Hopkins

FREDERIC R. STAUFFER (107), Sacramento Peak Observatory, Sunspot,

WILLIAM L. WOLFE (213), Optical Sciences Center, University of Arizona,

Apartado Postal 948, 37000 Leon, Gto. Mexico

Apartado Postal 948, 37000 Leon, Gto. Mexico

Road, Laurel, Maryland 20707

New Mexico 88349

Tucson, Arizona 85721

ix

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 appreci- ation 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.

M ETH 0 DS 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

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

Volume 6. Solid State Physics-Part A: Preparation, Structure, Mechanical and Thermal Properties; Part B: Electrical, Magnetic, and Optical Properties

Detectors; Part B: Free Atoms

Edited by K. Lark-Horovitz and Vivian A. Johnson

Volume 7. Atomic and Electron Physics-Atomic Interactions (in two

Edited by Benjamin Bederson and Wade L. Fite

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

Parts)

xiv METHODS 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

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

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

B: Crystal Structure and Morphology; Part C: Physical Properties

Measurements

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.

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,

Division of Wave Front.

1

METHODS OF EXPERIMENTAL 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

1.0

.8-

6-

.4 -

.2-

--

screen

( b )

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

7 T - 1 sin{ ?sin e ] I = I,,, cos'{ A sin e ] 7TD 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

INTRODUCTION 3

4 ---- screen

&l--

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.

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.

Division of Amplitude.

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.

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

Fringes of Equal Thickness.

observing eye

observed fringes

FIG. 3. Arrangement to observe Newton 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.

In 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.

Fringes of Equal Inclination.

Circular fringes are observed with an angular radius 8 given by mA 2d

cos 8 =-)

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

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

6 I N T E R F E R E N C E

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 KrR6 standard lamp (Baird and

T W O - R E A M I N T E R F E R O M E T E R S 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

A

2 nS e = - .

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

TWO- BEAM INTERFEROMETERS 9

I surface

under t e s t

observat ion k point

(a 1

observat Ion polnt

k

't ( b ) 2 urf ace

under test

FIG. 6. Testing of spherical surfaces with Newton interferometer.

1.2.3. The Michelson Interferometer

Probably the most famous of all interferometers was invented by Michel- son 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

OPD, 2 ’

d = -

and OPD, is the optical path differences along the optical axis.

TWO-BEAM INTERFEROMETERS

light source t- 11

c

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

4 observed pattern

F i c . 9. Michelson interferometer.


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 Twyman- Green interferometers.

TWO-BEAM INTERFEROMETERS

sample 1

t

13

r - i I 1 I 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

2 3

Q + \

INTERFERENCE

,attern 2

9 - p2 pattern I

F IG. 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 FIG. 13. Twyman-Green interferometer.

TWO-BEAM INTERFEROMETERS 15

mu

- - ( C )

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

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,

OPD=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.


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

rad ia l I ate ra I

rotational shear reversal shear

FIG. 16. Four types of shearing interferometer operations.

18 INTERFERENCE

h

front

microscope objective

gas LASER

rl

sheared wavef ront s

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,” Mala~ara , ’~ and Rimmer and W ~ a n t . * ~

Many radial shear interferometers have been designed, but the first one was invented by Brown.25 Basically, 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 INTERFEROMETERS 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 S P L I T T E R

M U R T Y ' S I N T E R F E R O M E T E R

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 Mall 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


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 DFF 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 n ~ ~ was the first to apply interference phenomena to the measurement of refractive indices. Later, Lord Rayleigh designed an interferometer for this purpose that has been


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

d

flat mlrrors

F I G . 24. Michelson stellar interferometer.

screen or

photographic

plate

- - L a "

'IrnI"

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

I h = 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, + I b = 4 1 + ~I[cos( koOPD,) + 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 "

X./28 1.22 X/8

01 Two point source8 b) Cl rcular rxtrndrd SOUICI

FIG. 25. Fringe visibility in Michelson stellar interferometer.

M ULTIPLE-BEAM INTERFEROMETERS 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 Roy- c h ~ u d h u r i ~ ~ and 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.

S l s2

FIG. 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

A 6 =- 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 O 2 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 INTERFEROMETERS 27

( 1 . 1 1 ) reduces to

(1.13) T, T2

6’ ( 1 - R,)2+4R, sin2-

2

I ( 6 ) = I”

as shown by Ba~meister ,~’ where the average reflectance R, is defined as

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

( R ~ R ~ ) ~ ’ ~ .

47T

A 6 = - nd cos O2 -$( + 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

1 - R , A 6 = 2 -

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 A m,, ,=- - e, ( 1 . 1 8 )

where e is the fractional value of ( 2 n d l A ) . Thus, the value of m for the pth fringe is

hence the angular radius of this fringe is given by

A cos e2 = 1 -- ( p - 1 + e ) .

2 nd

(1.19)

( 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 ~ ) ,


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.


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.,44 by 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): I t 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

2nd cos e m

A = (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 ------metallic 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 ( A o - & ) / 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

M ULTI PLE-BEAM INTERFEROMETERS 33

TABLE 1. Some Dielectric Materials for Thin Films

Refractive index Transparency Formula Name A = 550 nrn range in pm

MgF, Magnesium fluoride 1.38 0.13-8 ZnS Zinc sulfide 2.3 0.39-14.5 Ti,O, Titanium oxide 2.3 0.39-12 SiOz Silicon dioxide 1.46 0.2-9 NalAIF, Cryolite 1.35 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.

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

Single-layer Coatings.

- n l / 2 2 - s 9

(1.24)

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

34 INTERFERENCE

V / I I I coat ing Y ' ' H + @ N 2 I f d sur face I

s u r f a c e 2 I I

f l @ ,& N, g l a s s

s u b s t r a t e

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

, r e f l e c t a n c e

u n c o a t e d s u b s t r a t e

0 x/4 A/ 2 3x/4 OpllCOl thickness N d

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


0 ’0° wavelength

60 0

FIG. 35. Reflectance for a single antireflecting film.

reflectance of the uncoated substrate, the final reflectance becomes

R = Ro cos2( h). .rrA 0

700nm

(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).

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.

Multilayer Films.

36 INTERFERENCE

.3c

.2c

u

0 - - - "

.I 0

( 4

I I I

1 so0 600 700nrn wavelength

FIG. 36. Reflectance of a single reflecting layer.

FIG. 37. A quarter-wave stack.

reflectance I

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

M ULTI PLE-PASS INTERFEROMETERS 37

-==--+I ,test lens

l M 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 Har iha~-an~~ and 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 Michelson- Morley 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,SX but 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. Sta r Diameter Measurements

Michelson used his stellar interferometerh0 in 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 Mala~ara ,~ ' 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 Rosenha~er ."~ 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


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

2nd cos 8

A m = (1.26)

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.x6 The 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.


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

the observed wavelength and frequency shift by an amount given by If a light source is moving with a velocity u with respect to the observer,

AA Av u

A 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,


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 inter-

ferometer 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 a t 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 structure46 and 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 dl 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 = lom I ,(k)[l+cos(kOPD)] dk. ( 1.30)

If we now define

I ,= lom I , (k) dk,

and

I (OPD)= I , (k)cos(k .OPD) dk, I,'

(1.31)

(1.32)

the irradiance at the detector can be written

I = I"+ I (0PD) . (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

APPLICATIONS O F INTERFEROMETRY 45

gives

Z(0PD) C O S ( ~ - OPD) d(0PD) . ( 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 1 . M . Born and E. Wolf, Principles ofopt ics , Pergamon Press, New York, 1959. 2. A. H . Cook, Interference of Electromagnetic Waves, Clarendon Press, Oxford, 1971. 3. M . Fraqon, Optical Interferometry, Academic Press, New York, 1966. 4. C. Candler, Modern Interferometers, Hilger and Watts, London, 1951. 5 . W. H. Steel, Inferferomerry, 2nd ed., Cambridge University Press, London, 1983.

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

22. J . B. Saunders, “Measurement of Wavefronts without a Reference Standard. I: The

23. D. Malacara. Testing of Optical Surfaces, Ph. D. Thesis, Institute of Optics, University of

24. M. P. Rimmer and J. Wyant, “Evaluation of Large Aberrations Using a Lateral Shear

25. D. S. Brown, Interferometry N.P.L. Symposium No. I / , p. 253, Her Majesty‘s Stationery

26. M. V. R. K. Murty, “A Compact Radial Shearing Interferometer Based on the Law of

27. M. V. R. K. Murty and E. C. Hagerott, “Rotational Shearing Interferometer,” Appl. Opt.

28. J. W. Gates, “The Measurement of Comatic Aberrations by Interferometry,” Proc. fhvs .

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.),

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.

33. Hanbury R. Brown, and R. Q. Twiss, “A new Type of Interferometer for Use in Radio

Interferometer with a Visible Gas Laser Source,” Appl. Opt. 3, 531 (1964).

Wavefront Shearing Interferometer,” J. Res. Nut. Bur. Stand. 65b, 239 (1961).

Rochester, New York, 1965.

Interferometer Having Variable Shear,” Appl. Opt. 14, 143 (1975).

Office, London, 1959.

Refraction,” Appl. Opt. 3, 853 (1964).

5, 615 (1966).

Soc. B 68, 1065 (1955).

Chap. 3, Wiley, New York, 1978.

5, 236 (1882).

Astronomy,” Phil. Mag. 45, 663 (1954).

REFERENCES 47

34. C. Roychoudhuri, “Multiple Beam Interferometers.” in OpficalShop Tesfing (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 Spectro- graph,” Appl. Opf . 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

45. K. W. Meissner, “Interference Spectroscopy. Part I,” J. Opt. Soc. Am. 31, 405 (1941). 46. K. W. Meissner, “Interference Spectroscopy. Part 11,” J. Opf . Sac. Am. 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,

49. P. Langenbeck, “Multipass Twyman-Green Interferometer,” Appl. Opf . 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,” Opf . Ac fa 12, 137 (1965).

52. D. Sen and P. N. Puntambekar, “Shearing Interferometers for Testing Corner Cubes and Right Angle Prisms,” Appl. Opf . 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,”

61. J. Ojeda-Castatieda, “Foucault, Wire and Phase Modulation Tests” in OpticalShop Tesfing,

62. V. Grigull and H. Rottenkolber, “Two-Beam Interferometer Using a Laser,” J. Opf. Soc.

63. A. Cornejo-Rodriguez, “Ronchi Test’’ in Optical Shop Tesfing (D. Malacara, ed.), Chap.

64. D. Malacara, A. Cornejo, and M. V. R. K. Murty, “Bibliography of Various Optical Testing

for Spectral Analysis of Laser Radiation,” Appl. Opt. 3, 1471 (1964).

ed.), p. 217, Wiley, New York, 1978.

Asfrophys. J. 53, 249 (1921).

ed. D. Malacara, John Wiley and Sons, New York, 1978.

Am. 57, 149 (1967).

8, Wiley, New York, 1978.

Methods,” Appl. Opf. 14, 1065 (1975).

48 INTERFERENCE

65. J. D. Briers, “Interferometric Testing of Optical Systems and Components: A Review,”

66. G. Schulz and J. Schwider, “lnterferometric Testing of Smooth Surfaces,” in Progress in

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. Am. 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 Inter-

ferometer,” 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.

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

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: Inter-

ferometric Spectroscopy,” in Concepts of Classical Optics (J. Strong, ed.), p. 419, W. H. Freeman, San Francisco, 1958.

Opt. Laser Technol. 4, 28 (1972).

Optics, Vol. 13, Chap. 4, (E. Wolf, ed.), North-Holland Publ., Amsterdam, 1976.

13, 98 (1974).

York, 1971.

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 AND 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 R a ~ l e i g h . ~ , ~ 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, 10*’1-12 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 Huygens- Fresnel 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

METHODS OF EXPERIMENTAL PHYSICS Vol. 26

Copyright 0 1988 by Academic Press. Inc. All rights of reproduction in any form reserved

ISBN 0- 12-47597 1-8

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 Helmholtz- Kirchhoff 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,

DIFFRACTION 5 1

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:

(2.2) cos( n, s) + cos( n, r ) = 1 + cos 8.

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.


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. PoincarC26 and S ~ m m e r f e l d ~ ~ showed 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.

developed even further the Rubinowicz theory showing that Kirchhoff‘s diffraction integral can be

Miyamoto3’ and Miyamoto and

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. Descrip- tions 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 OQP meet at P, with an optical path difference (OPD) equal to the small line segment QR.


c - c

a l b

\ \

\ \ \

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.

S2 S2 a + b 2a 2b 2ab OpD=QR=-+-=- s2,

giving a phase difference 6:

n( a + b ) abh

S = k O P D = S2 .

(2.4)

( 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


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 (q) dv,

dy = dv sin 6 = sin (G) dv,

and by integrating, we obtain:

x = lo" cos ($) dv,

y = lo" sin ($) dv.

(2.9)

(2.10)

(2.11)

56 DIFFRACTION A N D SCAITERING

f q A 2

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


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

GEOMETRICAL EDGE

(B) G

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


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 = K S 2 ,

where

v ( a + b )

abh . K =

(2.12)

(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 oscilla- tions 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 . ~ ~


A 2K -

AMPLITUDE AT POINT P

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.


FIG. 9. Spiral for a circular aperture.

FIG. 10. Fresnel zone plate


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

and

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

(2.15)

(2.16)

Defining So = SI in 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; f =- m =integer,

mh ' m (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-pinhole distance 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.


/- 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 Fruun- hofer 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 Fraun- hofer 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 diffrdc- tion, and if we also take the case of normal incidence ( r = constant), the

FR

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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 ~ ( x , y ) eiks dx dy. ,

We may show that

s = x s i n O , + y s i n O,.;

(2.20)

(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 AND FOURIER TRANSFORMS 6 5

and

U ( k , ) e-lk\' dk, . ( 2 . 2 6 )

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.46 Harris4' 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

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

U ( k , ) = A [' elk\' dx, J - 0

( 2 . 2 7 )

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

ka sin 6 I ' sin( ka sin 6) U ( k ) = U, ( 2 . 2 8 )

where U, is a constant; Fig. 13 shows this amplitude and its corresponding irradiance versus ka sin 6.

The first dark zone in the irradiance pattern can be shown to be at an angle 6 given by

A

2 a sin 6 =-. ( 2 . 2 9 )

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

3. ( 2 . 3 0 ) sin( ka sin 0,) sin( k b sin Or)

ka sin Ox ] [ k b sin 6, U(O,, 4.) = U"


k o sin 6 (B)

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

2.3.2. Circular Aperture Diffraction

Eq. (2.23) inside the aperture, thus obtaining The diffraction pattern of a circular aperture may be found by integrating

J,(ka sin 0 ) [ ka sin 0 1’ u(e) =2u0 (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 67

- -- ka sin 8 A I R Y D I S C

(6)

Flc . 14. Fraunhofer diffraction pattern for a circular aperture. (a ) Amplitude aperture. (b) Irradiance distribution.

of x. The function U ( 0 ) and its corresponding irradiance U2( 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 A sin 0 = 1.22 -= 1.22 -

2a D’ (2.32)

where D is the diameter of the aperture.


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:

(2.33) f r = 1.22h -. D

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

other if Two pupil functions, A l ( x , y ) and A 2 ( x , y ) are complementary to each

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)


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. I t 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, ( 2 . 3 7 )

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

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 (Npl ) 'S I, (2.39)

which can be transformed to give

Irradiance is then given by

which can be shown to be equal to

(2.40)

(2.41)

(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

(2.43) 6

2 _ - - mr, m =integer,

since at those points,

sin'( y ) lim [

S / 2 - m r r (I") (2.44)

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

DIFFRACTION A N D SCATTERING

I I I I I

I I I

I A

I I A

m=O m = l m = 2 m =3

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

The phase difference 6 is given by:

2~rd(sin O1 -sin 0,) A

S = k O P D = (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 0 2 --

AA - d cos 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 AND 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

A ~ = - - c o s 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 AA _- - mN. (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 an 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 1 5

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 ka(sin O 1 -sin 0,) sin'( y )

sin (Ifi) (2.54) [ka(sin O 1 -sin O,)] '

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

(2.55)

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

2a(sin O1 -sin Oz) = nh, n =any integer excluding zero.

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 non- constant 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 2 4 6 m 0

(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 Ander- son,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.62 They are due to nonperiodic errors in the line spacing. These lines are usually observed at large distances from the parent lines.

DIF

FR

AC

TIO

N G

RA

TIN

GS

77

0

rn m

5 LI1

5 0

5


A ) TRANSMISSION 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.h4 and 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 ' a=-

and the total deflection angle is

y = o1 + 62.

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

mh Y _- - 2 cos - sin a. d 2

(2.57)

(2.58)

(2.59)

A case of special interest arises, y = 0; ( a = e l = &), reducing Eq. (2.59) to

mh = 2 d sin a = 2 h , (2.60)

RESOLVING POWER OF OPTICAL INSTRUMENTS 79

. \

B L A Z E D GRATING

GROOVE

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.

WoodG6 pointed 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,67768 S 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.


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 OF OPTICAL INSTRUMENTS

SUM OF TWO IMAGES

81

- 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.


Fic .26 . 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

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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:

2T L ’

w = - (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 w K 3

0 Lb

-

FIG. 28. Spatial frequency content of an image.

RESOLVING POWER OF OPTICAL INSTRUMENTS 85

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 xF and y F can be obtained from Eq. (2.23) if we first obtain the amplitude as follows:

dx 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(xF, y F ) is then given by:

T ( ~ , y ) e ~ ( k / F ) ( r ~ f i + i , i l ) I I-: A(XF, yF) =

and therefore, from Eq. (2.63):

F d x d y . (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.

1 . [I T*(x , y ) e - l ( k I ~ ) ( X X f i + Y v )

-‘m

(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 )


as follows:

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:

I I: F ( w , , w , ) =

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

k F

w , = - ( x - Z ) (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 w, =-.

F

2.6. The Abbe Theory of the Microscope

(2.72)

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

T H E A B B E T H E O R Y OF T H E M I C R O S C O P E

COMMON AREA: F(wx,wy)

87

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

\ \ INFOCUS-,

\ \

h

0 I I 1 I I I I dr

0 0.2 0.4 0.6 0.0 -.2 -

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


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.

THE ABBE THEORY OF THE MICROSCOPE 89

OBJECT FIRST ORDER IMAGE THIRD ORDER IMAGE

( A ) (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:

A

sin 8’ d = - (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


A r 2

A 6 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 phase- contrast 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. I t 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 i t 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

THE ABBE THEORY OF THE MICROSCOPE 91

CON DENSER OBJECTIVE

ANULAR OBJECT PHASE DIAPHRAGM 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.


FRONT FOCAL PLANE BACK FOCAL PLANE OF FOURIER LENS OF FOURIER LENS

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. I t 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


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.

S C A n E R I N G 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


of any friction or driving force, their movement equation would be

m - = - k d 2Y dt2 ’’

with the solution y = A e-‘wo‘

(2 .75)

(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

2q2 d 3 y 3 ~ x 1 0 ’ d t 3 ’

F , = - . -

Thus, the complete movement equation is

2q2 d 3 y dt2 3cx lo7 dt3

y + - - -qEo e-””. m - - - = - k d 2Y

(2 .80)

(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)

SCATTERING 97

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

(2.83) - 9Eo m [ w i - w 2 - i y w 3 / o i ] '

A =

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 >> wo, 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.

w 0 a

DRIVING ELECTRIC FIELD

W < < W o

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


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:

q 2 w 4 x 1 0 - ~ ( S ) = 8.rrcr2 - A: sin2 6, ( 2 . 8 5 )

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*,

w 4 sin2 6 (2.86)

q 4 E : x ( S ) = 8.rrm2cr2 .

SCAITERING 99

In the vacuum, using the values of the refractive index n = 1 and 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 an average distance between them that is larger than the wavelength of the light.

I f 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 or particle absorbs the luminous energy and 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,

and, using Eq. (2.88), we obtain by integrating

8 T q 4 w 4 ~ 10-'~(s~) W = 2,

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

(2.89)

(2.90)


4 -1 r sin t?

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

8Tq4w4 10-1~ U = (2.93)

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

3m'w:

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.


(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 - w O ) , thus:

2.rrq4w2 x 10-1~ (2.95)

We see that if u = wo, 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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80. G. Mie, Ann. Physik 25, 377 (1908). New York, 1969.

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.

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 boun- daries 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


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.

V x E + B = O

V X H - D = J

V - D = p

V . B = O

D = E E

B = p H

J = o E

Maxwell’s equations

material equations

(3.1)

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,

109 ELECTROMAGNETIC DESCRIPTION OF LIGHT

and the wave equation simplifies to

1 = - for a vacuum

C

1 6 = - for a dielectric V

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 dependen~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.


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 wt 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 = E H e-J(k r - w r ) + E~ e-J(k r - w r - 8 )

E = ( E H E H + e") e "k'r-" '

(3.4) S = O

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 ~ eJ*) e-J(k r - w r )

S = ~ / 2 , EH = Ev for circular polarization,

E, = E ~ ( E ~ f 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 EH by 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, &, = (2 ) l /z (EHf jEV) e - ~ l h . r - u f )

E(r, 1 ) = (E+&+eJd + E - E - ) e p J ' h ' r - w f ) (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- = E, indeterminate

E- < E, left handed

azimuth = A = 6/2

L/E, E, E_ Result.

= 1.0 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.


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 three- dimensional 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

R

+45.0’

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 E’ (o , 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

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.


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 ” ’ s i n 4i = ( ~ L ’ E ‘ / ~ ~ E ~ ) ’ ’ ~ sin 4, = n sin +i = n’sin 4t, Snell’s law

n = refractive index = c / o

P n cos 4 i - y n’cos cpl EsreAected - P - - rq = 9

b s i n c i d e n t P

P n cos 4i + y n’ cos

2n cos 4i sincident P n cos c$~ + y n‘ cos C#I~

P

= t , = , Fresnel equations (3.8) Estransmitted

6 n’ cos 4I - n cos 4t

6 n’cos 4 , + n cos 4t

Epretlected - P - rp = ,

pincident

P

Eptransmitted - 2n cos 4, - t p =

6 n‘cos + , + n cos E p incident

P

lrs12+lt,12= 1, Irp12+lfp12= 1,

R, = rsr?, 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 MEDIA 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’ n’- n rp = -

n + n” n + n” r, = - (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.’

ps = n cos c$~,

CL,-cL:

C L S + d ’

r, = -

The reflection coefficients can also incidence angle only.

p:= n‘cos 41,

n‘ CL; = -

cos cpl ’

CL; - CLP

CLP+CLb’ (3.10)

be written in terms of Snell’s law and

rp =

n’

n

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

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

v = -

r, =

(3.11)

rp =

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

4

- 10 n .om

4 - L -

OPTICAL POLARIZATION

- I - I - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - -

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

6, = 0, for 4i = 0, through zero for Brewster’s angle, 4B, to a negative value 8, = .rr, for 4i > c#J~. Figure 4 is a plot of r, and rp versus the incidence angle for Ivl< 1.

Brewster angle conditions are,

v 2 cos 4 B = ( v 2 -sinZ 4 B ) ’ ” ,

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.

j S p jSs r P = p P e , r s = p s e . (3.15)

WAVE PROPAGATION IN ISOTROPIC MEDIA 117

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 rp off 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

n

OPTICAL POLARIZATION

0 0.d 9O.U

Incidence angle, (pl

AV>I &<I - - - - - - -.

FIG. 6. A for u < 1 and u > 1 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 rp 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

WAVE PROPAGATION I N ISOTROPIC M E D I A 119

l.5

3

rP (s=45.0' rP , q=75.0'

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

normal incidence equation form is,

1 - v l + v

r = - r =-- - z,

1 - z l + z '

v=-

( 3 . 1 9 )

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

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

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

Z = 9

W = ( 1 + z)(z' - az + 1

a = 4 t a n 2 c p i + 2 ,

'

p = [ ' - " ] ' . I+w

(3.20)

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 2 . Where the two curves meet determines v and z.


...'..,

[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, k2 = 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.

WAVE PROPAGATION FOR METALS 121

(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 9O.U

Incidence angle, 9

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).


0 0.0‘ 90.0-

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 x , R, transmittance, and angle of incidence for ( R , / Rp)max for 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 a x stays 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 ec 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.


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 rp 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 equal to 52" at Lyman-a, the phase difference is ~ / 2 . Their experimental 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

THIN FILMS 125

3.5. Thin Films

3.5.1. Single-layer Films

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,

3.5.1 .l. Polarization Effects.

(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:

cos 4, phase thickness = /3 = W r = 2 m 2 d -. A

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.

r I 2 + r2, e-J2@ - ' S . P - 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 = pZ3 eJ'23,

(3 .25)


FIG. 13. Multiple reflections in a thin film.

The reflectance thus varies periodically with thickness with extrema given by dR/dd = 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 h . ~ ~ Figure 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 A for 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

THIN FILMS 127

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 -.-.-.- Irpl

0.u 90.0- Incidence angle,

FIG. 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.


- 1 4 p = 30'

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 de~reased .~ ' 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

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

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,

2 r n , z ,

A 0 A,=-. (3 .28)

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)


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 single- film 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.

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

3.5.2.3. MacNeille Polarizing Prisms.

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 = nH 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 ~ is ib le .~ '

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


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 IN ANISOTROPIC M E D I A 133

FIG. 18. Total internal reflection devices. (a) Fresnel rhomb, (b) Mooney rhomb, (c) achromatic device 1 (ADl) , 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


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.

E , , + E x , E y y + E y , E Z Z + E Z ,

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

= phase velocity = V, (;): =velocity in i direction = (kY2

(3.33)

(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 MEDIA 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, = v,, st = cos2 8,

( V ; - vf)[( V ; - ~ f ) sin’ 9 + ( 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.

(3.37) birefringence = nE - no

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 crystal- retarding 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.

nL- nR = *0.000071 for 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 Z Y

FIG. 21. Three cross sections

vx > vy > vz

of phase-velocity surfaces described by the Fresnel wave normal equation for a biaxial crystal.


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

0 E

_._.-.- ,-I

- - - - - - -. FIG. 23. Clan-Thompson prism drawn with normal and extreme rays. The optic axis is

perpendicular to the page, 0.


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 (90”-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,

( n h - n:)”,

2n2 tan S = (3.41)

The acceptance angle reaches a maximum for n2 = n,.

beam is refracted along the cut. For nE < n,, the maximum accepted incident angle for the transmitted

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, = nE. 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 1 A t a n s ’ (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.


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 Glan- Taylor 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. Marple- Hess 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


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


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 empiri- cally 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

1 4 4 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.

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.

3.6.3.3. Achromatic Wave Plates.

C O S ~ O = - T ~ ~

cos 6 +- = dl sin(26,), ( 3 (3.46)

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


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 four- element variable-retardation plate which rotates the polarization axis. Nine- element 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


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 0.001 YO 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, electro- optic, 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, = rzk Ek + 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 I3 = 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, ekli is 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

n i p s 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.

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.66-68 Spec- tral transmission, chemical stability, optical constants, and modulation frequency determine the material’s application. The natural birefringence

3.6.4.1. Electro-optic Effect.


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


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

dt’ 6nll - Sn, = n28 I:,, E 2 ( t ) exp[-(t - r’)] -,

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

I (Acne,) ‘

6 = 27rn,,P( t ) ~ (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.


3.6.4.2. Acousto-optic Effect. There are several ways to use the elasto- optic effect depending on incident beam diameter and frequency modula- t i ~ n . ~ ~ A 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,

3 PS- Anli = -no- - SC. 2 (3 .58)

C = stress-optic constant

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 2 TSCl A A ’

6=--- - (3 .59)

where 1 is the distance through the crystal.

5500 A is an order of magnitude less than needed to destroy the plate.

beam diameter, the phase modulation varies sinusoidally with the stress.

s = s,,, e - i (kr -RO - - s,,, COS(Rf)

for A > than the beam diameter

For quartz, the stress needed to produce quarter-wave retardation at

If the stress is a sound wave with a wavelength, A, much larger than the

(3.60)

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 type73 or 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


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,

NA A

sin+N-sin+,,=-, 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 Nth 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.


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 band~ id th .~ ’ 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 ‘ ’= -[ ( p v 3 h A 2 ) ] (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.


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.

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

3.6.4.3. Magneto-optic Effect.

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.

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 polycrystal- line aluminum oxide or thorium oxide, cold-pressed magnesium oxide, and cloudy fused quartz are examples. The powder-liquid mixtures can effectively depolarize 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

3.6.4.4. Depolarizers and Polarization Scramblers.

154 OPT1 C 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

S: = S, cos S - S3 sin 6

total intensity polarized + unpolarized

horizontal linear polarization

s; = s,

S ; = S, sin 6 + S3 cos S

+45" linear polarization

circular polarization,

where S' and S are the transformed and incident polarization vectors. The phase-modulated beam will always have one linear component unmodu- lated. 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.


S = S,,, sin wt,

sin 6 =sin(S,,, sin w t ) = 2 C J2n+l(Smax) sin[(2n + l)ot],

cos 6 =cos(&,,, sin wt)=Jo(6,,,)+2 C JZn(Smax) sin(2nwt).

(3.68) u)

n = O

m

f l= 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 time- average effects for rotating retardation plates.83

s = plate retardation

st, = s, total intensity polarized + unpolarized

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

S; = $( 1 + cos 6)S2

s; = s3 cos 6

horizontal linear polarization

+45" linear polarization

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


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, thin- display 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 plastic- sheet 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, AND 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.87 Grid 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 configur- atiom9*

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 demon- strate 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 AND 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. Pol a riza t ion Deter m i nation and M at he mat i ca I D escri pt io n

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 = Exex + EyeY = , (3 (3.70)


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

E ‘ = M * E

transparent isotropic medium in x - y

M = ( k x 0 ), kx E x ) 0 k, 1 Eb k, E,

partial linear polarizer in x - y

6 phase retardation in x - y

reflection in x - y

Ex cos a + E, sin a -s ina c o s a -Ex sin (Y + E, cos a

rotation of an angle a counterclockwise looking into the beam

1 0 cos2 a cos a sin a

general orientation of linear polarizer

cos2 a eirrI4 j ( 2 ) ” 2 cos a sin a sin2 a ePJrrI4 M = R ( 7 Y ) ( ‘6“ e P * , 4 ) R ( a ) = ( j ( 2 ) ’ / 2 cos a sin a

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 )

( S o s, s2 S J

( I M C S )

POLARIZATION DETERMINATION AND MATHEMATICAL DESCRIPTION 16 1

The following relations hold between the Stokes' vectors and the time- averaged electromagnetic vectors,

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

2 Q2+ 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.

( Q 2 + U 2 + V2)'/2 P =

IT 3

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


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

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65. E. K. Sittig, “Elastooptic Light Modulation and Deflection,” in Progress in Optics, Vol.

66. C. Forno and 0. C. Jones, “Hexamine Electro-Optic Light Modulators,” J. Phys. E 7,

67. B. P. Blumich and R. Germer, “Electro-Optic Shutters with PLZT Ceramics for Infrared

68. E. R. Kocher and R. P. Novak, “Fast Electro-Optical Shutter with Programable Trans-

69. T. G. Baur, D. E. Elmore, R. H. Lee, C. W. Querfeld, and S. R. Rogers, “Stokes 11. A

70. J. Etchepare, G. Grillon, R. Muller, and A. Orszag, “Kinetics of Optical Kerr Effect

71. M. A. Duguay, “The Ultrafast Optical Kerr Shutter,” in Progress in Optics, Vol. 14, Chap.

72. A. Korpel, Applied Optics and Optical Engineering, Vol. 6, p. 89 (R. Kingslake and B. J.

73. J. C. Cheng, L. A. Nafie, S. D. Allen, and A. I. Braunstein, “Photoelastic Modulator for

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79. J . B. Houston, Jr., M. Gottlieb, Shi-Kay Yao, I . C. Chang, J. Tracy, L. M. Smithline, and G. J. Wolga, “The Potential for Acousto-Optics in Instrumentation: An Overview for the 1980s.” Opt. Eng. 20, 712 (1981).

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4. HOLOGRAPHY

R. D. Bahuguna*

Rose-Hulman Institute of Technology 5500 Wabash Avenue

Terre Haute, Indiana 47803

D. Malacara

Centro de lnvestigaciones en Optica. A X Apartado Postal No. 948 37000 Leon, Gto. Mexico

4.1. Introduction

The invention of the laser has had a great impact on the science of optics. In particular, the high coherence of the laser light has completely revitalized those methods of optics which use interference as the basic phenomenon. Among these is the method of wave-front reconstruction. Also called holography, this method was invented by Denis Gabor in 1948'*2.3 to improve the resolution in the image obtained with the electron microscope. Although the initial goal did not prove successful, holography in itself became an important subject of investigation. However, due to the lack of a coherent light source, it remained in obscurity for more than a decade.

In the 1960s, with the availability of the laser, a major advance was made at the University of Michigan Institute of Science and Technology. Leith and Upatnieks in 1962495*6 applied the laser light to holography producing excellent three-dimensional images which astonished all those who saw them. In the same year, Denisyuk' made a remarkable contribution to holography by combining it with the volume-recording process, invented by Lippmann' in 1891, to make color photographs. Using a thick emulsion to record the interference fringes, a method was produced for color holography in which reconstruction could be carried out with a point source of white light. His hologram is, in effect, a hologram and a narrow-band spectral filter, combined into a single structure.

*Work was done while the author was at Centro de lnvestigaciones en Optica, A.C., Apartado Postal No. 948, 37000 Leon, Gto. Mexico.

167

METHODS OF t X P E R I M t N T A L PHYSICS Vol ?b

Copyright 0 1988 by Academic Press. Inc. All rights of reproduction in an) form reserred.

I S B N 0-12-475971-8

168 HOLOGRAPHY

Stephen Bentong of the Polaroid Corporation made another significant contribution in display holography. The method, known as rainbow holography or Benton holography, makes it possible to view the hologram with an extended white light source. Cross” combined the rainbow holograms with a technique known as composite or multiplex holography to produce cylindrical holograms.

Holography, although applicable to all waves, i.e., optical waves, micro- waves, X-rays, electon waves, seismic waves, and acoustic waves, has progressed exceptionally fast in the first category. This is essentially because of the availability of a highly coherent source: the laser. Research continued throughout the sixties and seventies. To date, hundreds of scientific papers and many books like those by Stroke,’’ Smith,” Collier et al.” Francon,14 Caulfield,” Abramson,I6 and Hariharan” have been published about holography and its applications to various other areas.

Holography is a relatively new process that is similar to photography in some respects, but is nonetheless fundamentally different. Some of the basic differences are the following:

(a) Photography records the image of an object formed by the lens of a camera. The image is recorded as two-dimensional and contains information about the object as viewed from a single particular angle. In holography, the object wave front is reconstructed, giving a three-dimensional view. It is as if one is viewing the object itself, possessing depth and parallax properties.

(b) The photograph is in itself a direct record of the scene and can be viewed by focussing one’s eyes onto its flat surface. The hologram is the interference record of the object wave with a reference phase-related wave. To reconstruct the image, the record is illuminated by the reference beam. The image is reconstructed at the same position as the object (or its image) during recording, and thus, not necessarily on the holographic plate itself.

(c) Photography can be done in ordinary light, whereas holography requires some amount of coherence.

(d) In photography the intensity is a quantity averaged over all the phases of the light wave, and so the phase information about the object is lost. On the other hand, in holography the amplitude and phase are encoded with the help of the reference wave and can be decoded in the reconstruction step.

4.2. Theory of Holography

The of-axis holography4 is the best-known form of holography and is, conceptually, the simplest to treat from the theoretical point of view. In this section, we describe the theory of this method for plane holograms and

THEORY OF HOLOGRAPHY 169

later see the effect of thick emulsions. Finally, we discuss the nonlinear effects and aberrations involved in the process.

4.2.1. Basic Theory of Plane Holograms

By a plane or thin hologram we mean one in which the recording material is thin compared to the spacing of the highest spatial frequency of the exposure variations, i.e., the recording medium is essentially two- dimensional.

Consider the arrangement shown in Fig. 1 , where 0 is the object wavefront and R is the reference field. They both interfere on the photographic emulsion H. The total field on H is O + R.

The irradiance I ( x ) on the photographic plate is given by:

I ( x ) = ( O + R ) ( O + R)*

= I 0l2 + I R 1' + OR * + O* R. (4.1)

Assuming that we are in the linear region of the amplitude transmittance f ( x ) versus the exposure E ( x ) curve, we have:

t ( x ) = t o + B E ( x ) , (4.2)

where to is a constant background transmittance and B is a parameter determined by the emulsion and the processing conditions. Since E ( x ) for a given exposure time is directly proportional to Z(x), we may write using eqns. (4.1) and (4.2):

t ( x ) = t ,+B'(1012+lR12+OR*+O*R) (4.3)

where B' = BT; t is the exposure time.

FIG. 1. Basic arrangement when recording a hologram.

170 HOLOGRAPHY

The developed plate which is now a hologram is illuminated by the reference wave R. The transmitted field X ( x ) at the hologram plane is then given by:

X ( x ) = R ( x ) t ( x )

= R ~ , , + B ’ ( R ~ O ~ * + R I R ~ ~ + I R ~ * o + R * o * ) . (4.4)

Now, if the reference beam is sufficiently uniform so that IRI2 can be treated as constant, then the fourth term of Eq. (4.4) is merely a constant times the object wave, showing that the object is reconstructed as such. The hologram is then like a window through which a virtual image of the object is seen. The last term represents the conjugate of the object wave times R’.

( C 1 ( d )

FIG. 2. Method to determine the conjugate image: Join the reference point source R and the object point 0, by a straight line. Draw a perpendicular R P to the plate H through R. Measure the angle RPO = a say. Draw a line passing through P at an angle ( - a ) and let it meet O , R at 0;. 0; is then the conjugate image. (a ) , (b) and (c) are different possible recording geometries; (d) is similar to (a ) except that one is viewing the image in reflection. Reflection images of 0, and 0: are seen as 0, and 0; respectively.


The conjugate image might be a real or a virtual image, depending on the particular geometry used to form the hologram. When the image is real, it is spatially located in front of the hologram. A simple method to determine the type of the conjugate image and its position is described in the book by AbramsonI6 and is shown in Fig. 2.

The paths of the light in a hologram are reversible. Thus, if the reconstructing beam reaches the hologram from the opposite side and in an antiparallel direction to the original reference beam, the two images are formed at the same places. The only difference is then that a virtual image becomes real, and vice versa.

If, when making a hologram, the object is real on the same side as the collimated reference beam, we have the following property. The real images produced by holograms, unlike those produced by lenses, are pseudoscopic, that is, they are “inside out.” Concave surfaces appear as convex, and vice versa. However, an orthoscopic image may be obtained if the pseudoscopic real image of another hologram is used as the object when taking the hologram. Figure 3 shows two alternative ways to reproduce these holographic images.

FIG. 3. Basic arrangement to reproduce the images of a hologram: (a) recording set up; (b) reconstruction with a beam identical to the reference beam; (c) reconstruction with a beam antiparallel to the reference beam.

172 HOLOGRAPHY

4.2.2. Nonlinear Effects and Aberrations

The effect of film nonlinearities on holograms has been studied by many authors, like Friezem and Zelenka,'' Goodman and Knight," Bryngdahl and Lohmann,20 and Kozma."

When the visibility of the fringes is high, the variations on the exposure E ( x ) are large, and hence the linear approximation in Eq. (4.2) might not be valid. Then, in general:

t ( x ) = t o + B l E ( x ) + B 2 E 2 ( x ) + B 3 E 3 ( x ) ... . (4.5)

The third- and higher-order terms produce higher-order images. The contrast of the object is reduced and some ghost images appear. If the object is a set of discrete points, false images may be produced.

This analysis explains nonlinear effects in flat transmission holograms, but not in volume or phase holograms. Besides film nonlinearities, intrinsic- phase nonlinear effects also appear in phase holograms.

The images of a hologram may be reproduced by using any of the two basic configurations described in the former section. However, the images may also be obtained after changing one or more of the following parameters: (a) position of the reconstructing light source, (b) wavelength, ( c ) size of the hologram, etc. The changing of these parameters in general affects the position of the images, the type of these images, and their size. However, it is very important to point out that these changes also affect the quality of the images, because some aberrations are also introduced, as pointed out by Meier.22

4.2.3. Effects of Thickness: The Bragg Effect

If the photographic emulsion is somewhat thicker than the period of the interference fringes, the hologram is called a volume hologram. This means that the interference pattern extends not only across the surface of the emulsion but, since the emulsion is quite transparent, throughout its depth as well. A volume hologram has a number of properties not shared by the plane hologram.

Volume holograms can be made to produce an image either by light transmitted through them (transmission holograms) or by light reflected from their surfaces (reflection holograms).

We can schematically represent the formation of a volume-transmission hologram, as in Fig. 4, where R and 0 are the reference and object plane waves. (The results of plane waves can be generalized for any object, as the object wave can always be decomposed into plane waves). The waves R and 0 interfere with an angle 4 between them and 4/2 with respect to the normal to produce a set of parallel fringes perpendicular to the emulsion.


R

3 Emulsion

FIG. 4. Arrangement to form a volume-transmission hologram.

The spacing is easily seen to be given by:

A d =

(2 s in(4/2)) ' (4.6)

Let us now illuminate the hologram at an angle 8, which is, in general, not equal to 4/2 at recording. In order to obtain an image reconstructed from the plane object wave, it is necessary that all the rays add in phase after reflection from the semireflecting planes. The following relation must then be satisfied:

+ A 2d

sin 0 =-, (4.7)

which is the familar Bragg condition. On comparing Eqs. (4.6) and (4.7), we have the following four solutions:

and e = * ( n - 4/21.

We will consider each case separately as follows:

(a) When the illuminating beam is identical to the reference beam ( 0 = 4/2), the object wave is reconstructed, and the virtual image formed. No conjugate image is observed.

(b) When the illuminating beam is in the opposite direction to the initial reference wave 6 = - (n - 4/2) , a real image is reconstructed. This real image is pseudoscopic, and no virtual image is formed.

174 HOLOGRAPHY

(c) When the illuminating beam is in the direction of the object beam ( 0 = -4 /2) , only an imperfect real image, with aberrations, is formed. The complete object cannot be seen, becasue the Bragg condition is not satisfied for all waves from the object.

(d) When the illuminating beam is in an opposite direction to the object wave ( 0 = T - 4/2) , a partial, imperfect virtual image, with aberrations, is reconstructed. The complete image is not observed for the same reason given before.

Summarizing, we can say that in thick holograms only one image is reconstructed, at,the same position where the object was when the hologram was formed. If the reconstructing beam is parallel to the reference beam, the image is orthoscopic. However, if the reconstructing beam is antiparallel to the reference beam, the image is pseudoscopic.

So far, we have discussed the transmission volume holograms. The reflection type can be made as shown in Fig. 5 . The object wave 0 and the reference wave R come from opposite directions. The interference fringes thus recorded are approximately planes parallel to the emulsion. The distance between the planes is h / 2 . If such a hologram is illuminated with a beam identical to the reference beam, it will reconstruct the virtual image of the object. The reconstruction is viewed in reflection rather than transmission.

The major feature of the refection holograms is that a single-color wave front may be reconstructed with white-light illumination. The Bragg planes act as an interference filter and have a high reflectivity only for those wavelengths that produced them. However, in practice, such a hologram recorded at one wavelength may reconstruct in a slightly shorter wavelength.

Object wave 0

i-

' Reference wave R

FIG. 5. Arrangement to form a volume-reflection hologram.

DIFFERENT TYPES OF HOLOGRAMS 175

For example, the recording in red color may reconstruct in green. This is because of emulsion shrinkage during photographic processing. The Bragg planes pull together so that the final spacing is smaller than the one at recording, leading to reconstruction at a different wavelength. It is worth pointing out that white-light illumination results in a loss of image resolution, since the hologram acts as a spectral filter with a bandwidth of about 100 A, which is too large for high-resolution imagery.

4.3. Different Types of Holograms

There are a number of hologram forming configurations depending upon the position and form of the object and the reference beams. The simplest one, invented by Gobor, is the in-line hologram. In this method, the object is a transparency illuminated by a reference beam. The scattered light contains information about the object, and the unscattered light acts as a reference wave. The source is the green line of the mercury arc lamp, Although successful in reconstructing the transparency, this hologram suffered from a serious drawback: the twin-image problem. The primary image and the conjugate image are in the same direction and so are inseparable. The problem was overcome successfully by Leith and Upatnieks4 using their off-axis configuration, and later by Bryngdahl and Lohmann” and by Meier24 using single-side band holography.

In the following section, we describe Fresnel, Fraunhofer, Fourier transform, image-plane, color, composite holographic stereogram, rainbow, and computer (synthetic) holograms.

4.3.1. Fresnel Holograms

When the recording plate is placed in the Fresnel diffraction region of the object and at an arbitrary distance from the reference source, we obtain a Fresnel hologram. A practical arrangement is shown in Fig. 6 . It is the simplest and most convenient configuration, requires no lenses either in formation or in reconstruction, and produces images of three-dimensional objects as shown in Fig. 7.

4.3.2. Fraunhofer Holograms

A Fraunhofer hologram can be defined as one which records interference of the far-field diffraction pattern of the object, with a reference light source not coplanar with the object.

176 HOLOGRAPHY

L S

FIG. 6. Recording a Fresnel Hologram.

FIG. 7. Reconstructed image from a Fresnel Hologram.


\

d

--

. $

-4

H

0

FIG. 8. Recording of a Fraunhofer hologram with a free space configuration.

The following are the two geometries with which to record such a hologram.

(1) This configuration is a free-space geometry (see Fig. 8). The subject (a small transparency) is illuminated by a broad plane-parallel beam. The recording plate is kept in the far field of the subject. The light scattered from the subject interferes with the uniform background to form the hologram.

This geometry was utilized by Thompson et aLZ5 for measuring the size and shape of dynamic aerosol particles. The twin-image problem in this geometry is apparently removed.

(2) In this configuration (see Fig. 9), the subject is placed close to the lens and is illuminated by a plane-parallel beam. The recording plate is

FIG. 9. Recording of a Fraunhofer hologram with a lens.

178 HOLOGRAPHY

kept at the back focal plane of the lens, and the reference wave is incident obliquely from the side.

4.3.3. Fourier Transform Holograms

A Fourier transform hologram records the interference pattern of the Fourier transform fields of both the object and the reference source. A practical arrangement is shown in Fig. 10.

Such holograms are useful as spatial filters for pattern recognition, where the properties of the Fourier transform provide the basis for the recognition process, as described by Vander Lugt.26

It is worth noting that when the Fourier transform hologram is formed with a plane-reference wave, the generated image remains stationary with respect to the translation of the hologram. For example, a moving reel of such holograms would project stationary images.

By a variation of the above geometry, one obtains the so-called quasi- Fourier transform h01ogram.l~ In this case, the subject is adjacent to the Fourier transforming lens with a coplanar reference source. As is well known, parallel illumination of the transparency produces its Fourier transform at the back focal plane together with a phase factor, as shown by G ~ o d m a n . ~ ’ The coplanar reference source removes the phase factor, and thus we obtain the same properties as obtained with the Fourier transform hologram.

Another variation of the geometry leads to the lensless Fourier transform hologram by virtue of the absence of a lens.13 Here the Fresnel diffraction field of the transparency is recorded together with the field from a coplanar reference source. The phase factor is again removed, and the hologram

L, H

FIG. 10. Recording of a Fourier transform hologram.


formed with this arrangement has a transmittance resembling that of the Fourier transform hologram.

4.3.4. Image-Plane Holograms

When the image of an object is formed on the plane of the recording plate and a hologram is formed with the help of a reference wave, the hologram is known as an image-plane hologram. A practical arrangement is shown in Fig. 11.

The image reconstructed with the reference beam in the original direction is similar to that obtained with a Fresnel hologram, except that the field of view is limited by the aperture of the lens. The field of view can be increased significantly by using the conjugate illumination. Then, the illuminating beam is on the opposite side and in opposite direction to the original reference beam, as described by Brandt28 and Ro~ens .*~

Since the conjugate illumination is preferred, the image in the reconstruction is pseudoscopic, or “inside out.” The image becomes orthoscopic if a conventional hologram is used instead of the lens to form a real image over the holographic plate.

In image-plane holograms, the requirements for the illuminating source are less stringent. The effect of the extension of the source on the resolution or image blur As of the reconstructed image can be seen from the following relations.

As = (2) Ar,

\

I I 1

Q2 L

FIG. 1 1 . Arrangement to form an image plane hologram.

p,

(4.8)

180 HOLOGRAPHY

where Ar is the extent of the source, Z, its distance from the hologram, and Z1 the distance of a point on the object from the hologram.

Further, the equation describing the effect of the finite bandwidth Ah on the resolution Au of the reconstructed image is given by the following relation:

AU = z1 sin 0, (y ) , (4.9)

where Or is the angle of the reference beam with the normal to the hologram plane (the subject and image are considered to be on-axis).

Equations (4.8) and (4.9) reveal that for small 2, , extension of the source and finite bandwidth have little effect. Thus reconstructions from image- plane holograms are sharp and achromatic even with white-light-extended sources. However, for image points which are far outside the plane of the hologram, the reconstruction shows color dispersion and blur.

In conclusion, image-plane holography is particularly suitable whenever display of holograms, double-exposure interferometric holograms, or vibration-analysis holograms are desired in white light.

4.3.5. Color Holograms

Multicolor holography was first suggested by Leith and Upatnieks6 The basic idea of color holography is to use three-color object and reference beams to record a hologram on a single photographic plate. If the object and the reference beams contain the three primary colors, the hologram would, when illuminated with a beam identical to the reference beam, reconstruct the object wave in full color.

There are basically two main techniques to record color holograms (a) holograms recorded as planar diffraction gratings and (b) holograms recorded as volume diffraction gratings. The difference in response of the two grating types is made evident by considering the grating equations:

d(sin i + sin 0) = A (planar grating), (4.10)

2d sin i = A (volume grating). (4.11)

The planar grating equation reveals that for a given A, diffracted light would always be observed for any incidence. Thus the response is that broad-band and special techniques are required to separate or eliminate the spurious images which result from light of wavelength A l , diffracting from the component hologram recorded with the wavelength A2.

The volume grating equation, on the other hand, states that the incident and diffraction angles are equal, and thus only wavelengths satisfying Eq.


(4.11) would be significantly diffracted. Thus the spurious images are suppressed by the filtering action of the volume grating. Further, there is no loss of resolution, contrast, or spatial frequency bandwidth.

We now describe the various techniques used to record color holograms. 4.3.5.1. Plane Hologram Techniques. There are several basic tech-

niques to make color plane holograms, as will be described next. In the three-reference-beam method due to Leith and Upa tn iek~ ,~ the

object beam is a mixture of three primary colors. The reference beam is a combination of three separate beams in the three different primary colors. Each reference beam is inclined at a different angle to the recording plate. In this way, the spurious or the cross-talk images resulting from diffraction of light of wavelength A , and from the component hologram recorded at wavelength A 2 are spatially separated.

This method, however, is not very practical because the field of view is very much restricted. Furthermore, during reconstruction, the illuminating beam must be accurately aligned so as to be identical to the reference beam.

The spatial multiplexing method is essentially due to Collier and Penningt01-1.~' Here, we spatially multiplex, in a nonoverlapping manner, the several holograms corresponding to the several colors used, i.e., the various colored interference patterns are not allowed to overlap on the recording medium. During reconstruction, the colored illuminating light is incident in such a way that each component hologram is illuminated with only that color light used to form it.

There are several experimental arrangements which could be used to form such a hologram. One possible scheme is shown in Fig. 12. A mask F consisting of strips of red, green, and blue filters is imaged onto the hologram plane by a lens with the multiwavelength reference beam. The image, which consists of discrete areas of single-wavelength light, acts as

FIG. 12. Recording a color hologram by spatial multiplexing.

182 HOLOGRAPHY

the reference wave front. The light in each strip then interferes with the light of the same color coming from the object, giving nonoverlapping multiplicity of single-color holograms. The reconstruction step consists of placing the hologram in register and illuminating through the same filter mask.

Some resolution is sacrificed in this method, since each component hologram will be smaller than the total recording. Also, this scheme results in the production of some noise, because in a A ,-component hologram there will also be light present of A2 and A 3 from the object, thereby decreasing the diffraction efficiency of the hologram ( A , , A 2 , A 3 are the three primary colors).

The coded-reference-beam method, also due to Collier and Pennington,30 involves coding of the reference beam. The amplitude and phase of the reference wave are made to vary across the hologram plane in a significantly different manner for each wavelength. During reconstruction the developed plate has to be replaced precisely in the same position that it occupied during the formation of the hologram.

A possible arrangement for recording such a hologram is shown in Fig. 13. The ground-glass screen D serves as the reference-beam coding plate. Each colored component of the reference beam, when passing through the diffuser, produces a complicated speckle pattern on the hologram plane. The component speckle patterns are randomly different from each other with respect to phase and intensity at a point, meaning that each colored reference wave is uniquely coded. Collier and Pennington have noted that when only light of wavelength 0.514 km was used, an excellent image was obtained. However, when the wavelength was changed by only 0.0128 km and the coding plate was illuminated in exactly the same way, only a uniform noise was observed.

FIG. 13. Recording a color hologram using a coded reference beam.


Successful holograms have been recorded using this method, although the noise problem still remains. The geometry of the code plate plays a major part in determining the degree to which the noise affects the reconstruction. The noise, which is principally due to the interaction of the wavelengths A , and A 2 , with the hologram formed with A 3 , is more uniformly distributed over the image area by increasing the solid angle subtended by the diffuse screen at the hologram.

There are two kinds of volume hologram techniques, namely, transmission and reflection volume holograms, as will be described in this section.

The transmission volume hologram technique was used to record the first mulitcolor hologram by Pennington and Lin3' in Kodak 649F photographic emulsion. Light of wavelength 6328 8, from a He-Ne laser was mixed with light of wavelength 4880 8, from an argon laser, using a beam splitter. Of the two doubly colored beams so produced, one acts as a reference and the other illuminates a color transparency. The reference and the subject beams then combine together at an angle of 90" with respect to each other at the hologram plane. As noted before, the interference is throughout the depth of the thick emulsion.

An arrangement to record a three-color hologram devised by Friesem and Fedorowicz3* is shown in Fig. 14. The two three-colored wave fronts are generated in a similar manner as in the earlier re feren~e .~ ' A better color fidelity is, of course, obtained in the reconstruction than with two-color wave fronts.

Another technique uses the properties of the reflection holograms already discussed in Section 4.2.3. The hologram can be reconstructed with a multicolor wave front identical to the reference, or with white light as noted earlier.

4.3.5.2. Volume-Hologram Techniques.

L 2 0

FIG. 14. Recording a color volume-transmission hologram.

184 HOLOGRAPHY

4.3.6. Composite Holographic Stereograms

An interesting application of holography is the three-dimensional display of an object by the use of two-dimensional pictures or projections. This field, known as composite holography, started in the late sixties with McCrikerd and George,33 George et

Composite holography consists of producing a multiple hologram formed by many vertical strip holograms, each with a different paralax. The observer then obtains the stereo effect by looking at a collection of very thin vertical holographic windows. A different way to synthetize three-dimensional holograms from two-dimensional pictures has been described by Sopori and Chang3' and Haig.38 They multiplexed many images in a single plate by changing the angle of incidence of the object or the reference beams when making the hologram.

Kasahara et ~ l . , ~ ~ Groh and Koch? and Grossman4' applied composite holography to the three-dimensional display of internal human organs from ordinary X-ray projections.

Redman," and De B i t e t t ~ . ~ ~

4.3.7. Rainbow Holograms

White-light illumination of display holograms has proven to be so much more practical than laser or arc-lamp illumination, that most of the holograms that we see today are of some white-light type. One of these is the rainbow or Benton hologram. Rainbow holography was first reported by Benton' as a two-step process, but recent developments have simplified the procedure to one step with some limitations.

Another development is the cylindrical holographic stereogram derived from the rainbow holograms by Cross." Below, we describe these three methods one by one.

A conventional hologram H , is first recorded as shown in Fig. 15(a). The real image is then reconstructed by a beam that is antiparallel to the original reference beam. However, during reconstruction, instead of illuminating the whole plate, only a horizontal slot about 10mm wide is illuminated. A second hologram plate H 2 is placed near the real image, and a new hologram is recorded with a reference beam coming from the same side as the rays producing the real image [see Fig.

After development, the second hologram H2 is illuminated by a white-light beam in a direction antiparallel to the reference beam. The object rays are retraced in the direction they came from, back to the position of the slit. In other words, the light is diffracted only towards the real image of the thin horizontal slot of the master hologram. Further, because of dispersion, we obtain many colored slit images one over the other (see Fig. 1%).

4.3.7.1. Two-step Method.

15(a)l.


FIG. 15. Recording a rainbow hologram by the two-step method.

The presence of the slit makes a significant reduction in the information content in the second hologram, allowing a relaxation in the coherence requirements for the reconstructing light source. When the eye is placed at one of the real images of the slit, one sees the complete virtual image of the object in one color, depending on through which colored slit image one is watching. The holographic image of the object is almost monochromatic, and there is almost no blurring effect due to dispersion.

Instead of a point light source, an incoherent vertical line source may be used to produce a continuum of vertically displaced horizontal strip images. Another advantage is that the reconstruction efficiency is very high, since most of the light is diffracted towards the observer’s eyes. A disadvantage is that the vertical parallax is sacrificed. This is not a serious drawback, as most of the time the horizontal parallax is sufficient, and it is the one that is necessary for stereoscopic vision.

As noted earlier, in a Lippman-Denisyuk hologram, the colors are separated by an interference filter effect, and one single color is selected by the illumination angle. On the other hand, in rainbow holography, the colors are separated by diffraction, and only one single color is selected by the observation angle.

186 HOLOGRAPHY

4.3.7.2. One-Step Method. The one-step method was first introduced by Chen and Yu4* and was further developed by Chen.4'.44 Review papers on this subject have been presented by Chen et al.45 and by Benton et al.46

The orthoscopic recording scheme is shown in Fig. 16(a). A horizontal slit A is placed on the lens which images the object near the recording plate P2. The reference-wave point source SL is located in the plane of the lens and is centered above the slit.

The finished hologram, when reconstructed with the same reference wave, produces an orthoscopic image. But, the observer can only view a part of this image directly in front of the slit image; the rest is vignetted out and one must move about to sample it. Illumination with a white light source S,, however, produces a dispersed image of the slit, allowing the observer to see more of the image but in a spectrum of colors (Fig. 16(b)]. Chen and Yu42 have shown that by locating the slit between the object and the

FIG. 16. One-step orthoscopic recording scheme of a rainbow hologram (a) Recording; (b) Observation.


front focal plane of the lens, a monochromatic image can be obtained. This is because the image of the slit is formed on the same side of the plate as the observer, thus allowing the observer to place his eyes directly at the position of the slit image. But the size of the image becomes limited due to vignetting by the lens aperture.

The pseudoscopic configuration is shown in Fig. 17(a). The geometry is almost similar to the one used for orthoscopic configuration, with the only difference that the reference wave is convergent instead of divergent. In the reconstruction step [see Fig. 17(b)], the conjugate illumination with white

\ . \ '\

\ \ \

FIG. 17. One-step pseudoscopic recording scheme of a rainbow hologram. (a) Recording; (b) Observation.

188 HOLOGRAPHY

light is used to give real images of the slit through which the observer can see the image of the object. Unfortunately, the real image that is obtained positions the front of the object closest to the lens, farthest from the viewer. Thus the viewer is presented with the familiar “inside-out” or reversed-depth experience of a pseudoscopic image. However, there is an advantage that the image is bright and sharp and is undistorted and unvignetted by the lens aperture. Therefore, for applications where the depth perception is not important, the pseudoscopic configuration would be more desirable.

The astigmatic recording scheme is yet another method for producing rainbow holograms in one step and produces a good orthoscopic image with large field of view in the horizontal direction. The recording is illustrated in Fig. 18. The cylindrical lens L, images the object 0 in the vertical direction near the hologram plate H. The slit S is also imaged but beyond the recording plate. A divergent beam is used as the reference wave front. The image of the object can be reconstructed with a white-light point source in the same position as the reference source. Since in the vertical direction the hologram is an image plane hologram, the image in this direction is sharp (see Section

It is worth pointing out that in the one-step rainbow process using conventional spherical lenses, the field of view is narrow in the horizontal direction, which is a major drawback. This is primarily because of the difflculty in getting a large aberration-free spherical lens. The astigmatic recording has removed this obstacle, since long convex cylindrical lenses are easily available.

The lensless method of Bahuguna and Santoyo4’ is particularly suitable for transparencies. In this method, a master hologram of a diffuse slit is recorded. By using conjugate illumination a wave converging to a slit is reconstructed. This reconstructed wave is utilized to illuminate the object transparency for creating a rainbow hologram. Since the master hologram can be used for making a rainbow hologram of an unlimited number of transparencies, this method is essentially a one step method. A variation of the above method has been described by B a h ~ g u n a ~ ~ .

4.3.4).

FIG. 18. Astigmatic recording scheme of a rainbow hologram.


4.3.7.3. Rainbow Cylindrical Holograms. Lloyd Cross” invented the cylindrical holographic stereograms, utilizing the earlier work in rainbow holography invented by Benton,q and composite holography.

Composite holography consists of making a motion picture film of a subject, placed on a slowly rotating table, with a cine camera. The individual frames of the film are hologramed on strips of about one millimeter wide and twenty cemtimeters long. The holographic film is then processed and bent into a cylindrical shape. When the observer views the film, each eye sees a photograph of the object taken from a different angle (just as with a stereo viewer), and consequently a three-dimensional image is perceived at the center of the cylinder as the observer walks around the hologram. An ordinary tungsten lamp with a transparent glass envelope is used as a light source.

A machine to make this kind of hologram using anamorphic optics (plastic and holographic cylindrical lenses) was described by Fuesk and Huff .49

4.3.8. Computer Holograms

A computer-generated hologram (CGH), also called synthetic hologram, is produced with a graphical output from a digital computer. The hologram is calculated from the knowledge of a wave front or the object to be represented. The actual hologram is made by photoreducing the computer graphical output on a photographic film or glass plate.

The first CGHs were developed by Brown and Lohmann” with the purpose of making spatial filters for optical-data processors. Since then, the number of applications of CGHs have grown, and they may now be divided into the following five’areas:

a. three-dimensional image display b. optical-data processing c. interferometry d. optical memories e. laser-beam scanning

All of these applications have been described in a very complete review

Computer-generated holograms may be of three kinds:

1. Binary holograms. Most computer plotters may produce only high- contrast prints without gray tones. When the print is copied and minified by means of a high-contrast film, the resulting hologram is like a black screen with many small holes and is called a binary hologram, as described by Lohmann and Paris.” Efficient holograms have a total transparent area of about 50%.

by Lee.s’

190 HOLOGRAPHY

Binary holograms are analogous to normal holograms recorded on a very-high-contrast and nonlinear photographic film. Then, high-order images should be expected, but they do not overlap if the sampling theorem is obeyed.

There are many applications for binary holograms, but their most interesting applications are probably in the field of optical testing, as shown by W ~ a n t ~ ~ and loo mi^.'^ Computer-generated holograms for this application are used to produce reference aspheric wave fronts in order to test an aspheric optical surface or lens. This kind of hologram looks like an interferogram, as shown in Fig. 19.

2. Half-tone holograms. The computer hologram may also be obtained by displaying the results on a suitable half-tone printout or on a high- resolution television screen. This kind of representation approaches a real hologram.

3. Phase holograms (kinoforms). A kinoform, as described by Lesem et a ~ , ~ ’ is a phase hologram obtained by the following method. A variable- density display is calculated in such a way that after bleaching, the profile of the emulsion gives a phase hologram which reconstructs only one image.

FIG. 19. Binary computed holograms to test aspherical surfaces (from Ref. 53) .

SOME APPLICATIONS OF HOLOGRAPHY 191

Then, the picture is taken and bleached. Such a hologram is of interest, because the entire flux is concentrated in this image.

4.4. Some Applications of Holography

There are numerous applications for holography. Most of them utilize and exploit the fundamental difference between holography and photography, namely that holography records the whole wave front, while photography is a recording of the irradiance distribution in the image. In this section, we deal with the more important applications.

4.4.1. Holographic Interferometry

Interferometry is a laboratory technique by which extremely small movements of objects are detected and measured by exploiting the interference properties of light. It can detect changes as small as a fraction of the wavelength of light. Conventional interferometry is mainly useful for making measurements on highly polished surfaces of relatively simple shape. Holo- graphic interferometry extends the range to include three-dimensional surfaces of arbitray shape and surface condition. It is concerned with the formation and interpretation of the fringe pattern which appears when a wave generated at some earlier time and stored in a hologram is later reconstructed and made to interfere with a comparison wave.

Holographic interferometry has been applied with success to test and measure aspheric optical surfaces, for example, by Broder and Mala~ara , ’~ FouCrC and Mala~ara,~’ Malacara and M a l l i ~ k . ~ ~ The time-delay aspect makes the holographic method a very useful tool in many fields. For example, a machine part can be tested before and after stress. Further, due to the basic three-dimensional structure of the holographic reconstruction, a complex object can be examined interferometrically from many different perspectives: a single interferometric hologram is equivalent to many observations with a conventional optical interferometer. We shall discuss below the three basic forms of holographic interferometry.

The hologram of the test object is first recorded in the usual way. The hologram plate is processed and put back in exact register on the plate holder, with the object and the reference beam undisturbed. The illumination can be adjusted so as to equalize the absolute values of the original and reconstructed object-wave amplitudes. Under the ideal conditions, the two wave fronts must exactly cancel, and the object should not be visible. In practice, at least one broad

4.4.1 .l. Real-Time Holographic Interferometry.

192 HOLOGRAPHY

bright fringe is usually observed, because the processing of the hologram inevitably distorts the emulsion to some small degree.

In the next step, the object may be placed under stress and caused to deform, or it may be released from stress and allowed to flow and creep or expand. As a result, the actual object wave differs at several points with the reconstructed wave, and the field of view is now crossed with fringes.

Mathematically we can write the object wave in the initial and final state as Oo(x) exp(i+o(x)) and Oo(x) exp(+;(x)) respectively. The location of the fringes would then be defined by the locus of all points for which

&(x) - 40(x) = const. (4.12)

A dark fringe is produced whenever

4 & ( ~ ) - 4 ~ ( ~ ) = 2 ( m + l ) . r r ; m = O , 1,2 ,.... (4.13)

Many authors, like Aleksandrov and Bon~h-Bruevich,~~ Ennos,60 Froehly et a1.,6' Tsuruta et S0llid,6~ Molin and Stetson: and Hecht et ~ 1 1 . 6 ~ have worked on the problem of relating the observed fringes to the actual object deformation. According to Abramson,""-" the interpretation of the fringes is simple in most of the cases. His holodiagram allows measurement of deformation even for relatively complex and large objects. This technique and other methods, for example the method of Hecht et al.,6' have made holographic interferometry a very useful metrological tool for routine measurements.

A spherical form of the single-exposure interferometer is shown and described in Fig. 20. First a hologram of a diffuser is taken [Fig. 2O(a)]. The hologram is then put back in the plate holder in exact register so that both the actual and the reconstructed waves superpose. If a test flat T is inserted in the object beam [Fig. 20(b)], interference fringes appear which

D D T

P H

(a)

P H

(b)

FIG. 20. Instrument to perform single-exposure holographic interferometry


are related to the optical inhomogeneities, wedge and thickness variations present in the test plate.

4.4.1.2. Double-Exposure Holographic Interferometry. When it is sufficient to form a permanent record of the relative surface displacement occurring after a fixed interval of time, a simpler technique, the double- exposure method is employed. In this method, two exposures are taken, one for the initial state and the other for the final state of the object, on the same hologram plate. The interferogram so formed, yields the necessary fringes on illumination by the reference beam. Figure 21 shows an interferogram of an aluminum block tilted between the exposures.

This method is similar to the single-exposure method in most respects. Unlike the previous method, exact manual registration of the reconstructed wave with the original one is not required. Further, distortion due to emulsion shrinkage is identical for both reconstructed waves and therefore does not affect the spacing of the fringes formed by the interference of the two waves.

FIG. 21. Holographic interferogram of an aluminum block, tilted between two exposures.

194 HOLOGRAPHY

4.4.1 3. Time-Averaged Holographic Interferometry. The multiple exposure interferometry can be extended to the limiting case of continuum of exposures. The resulting technique, known as time-averaged holographic interferometry, is used to study how surfaces vibrate. It was first reported by Powell and Stetson.”

Suppose the object is vibrating in one of its normal modes. The time- averaged hologram of such an object gives what is effectively a double exposure of the vibrating surface seen at either extreme of its oscillation. Thus, the object would appear crossed with fringes.

Figure 22 shows some of the time-average interferograms of a 35-mm film can at its second resonance. Here the line across the middle of the can is a node of vibration of the can, and the contours to either side are loci of constant amplitude of vibration. Each bright area represents an amplitude of vibration of approximately a multiple of a half wavelength.

4.4.1.4. Sandwich Holography. The fringes in interferometric holography appear because the object either moved or changed its shape between the first and the second exposure. Many times, however, it is not easy to eliminate the effect of the movement of the object as a rigid body and to determine the object deformations due to loads or stresses.

Once the second exposure is taken, the effect of the movement of the object as a rigid body can only be compensated if the two holograms were recorded with two different and separated reference waves. However, an elegant solution is the sandwich hologram invented by Abram~on.’*-’~

In sandwich holography, a pair of holographic plates are simultaneously exposed by placing them in the same holder, with their emulsions towards the object and with the antihalation backing removed from the front plate. Two sandwiches at least must be exposed, one without any stress in the object and the second after the stress is introduced, as shown in Fig. 23. Upon reconstruction, both sandwich holograms will reconstruct the object

FIG. 22. Time-average interferograms of a 35-mm film can at its second resonance (from Ref. 12).


A, A2 OBJECT

OBJECT

FIG. 23. Sandwich holography.

without any fringes. However, if the sandwiches are now formed by the pairs of plates Al with B2 or by BI with A*, the reconstructed image of the object will show the fringes.

The tilting of a hologram laterally moves the image of the object when reconstructing the image of an object. However, if the sandwich pair is tilted, the images of both holograms move almost together, producing only a very small tilt of one image with respect to the other. This property of sandwich holograms permits compensation or simulation of small rigid movements of the body by means of small tiltings of the sandwich. In this manner, the fringe pattern due to the object stresses is easily isolated from any possible object rigid motion fringes.

4.4.2. Holographic Microscopy

The application of holography to microscopy dates back to the origin of holography itself. Gabor had intended to change the wavelength between formation of the hologram and reconstruction of the same in order to obtain a magnification in the ratio of the wavelengths. The magnification can also be obtained by scaling down the hologram, or by decreasing the curvature

196 HOLOGRAPHY

of the reconstructing wave as compared to the reference wave. The general formula for magnification is given by

mL

I*--- M =

m2Zo z,' PZC Z R

(4.14)

where the upper sign is for the primary image and the lower sign is for the conjugate image. Here m is the linear magnification of the hologram, is the ratio of the two wavelengths, and Z,, Z,, and 2, are the z-coordinates of the object-point, the reference-point source and the illuminating point source respectively.

The above principles have been effectively utilized to form holographic microscopes. For details of the theory, construction, and working, the reader is referred to the literature, specially the papers by E l - S t ~ m , ~ ~ El-Sum and K i r k p a t r i ~ k , ~ ~ B a e ~ , ~ ~ Van Ligten and O ~ t e r b e r g , ~ ~ and Thompson et al.25

4.4.3. Imagery through Diffusing Media

The recovery of the image of an object obscured by a distorting medium, for example the distorting atmosphere, has been a subject of investigation for several years. One possible use for holography is to solve such problems.

Let the object be obscured by a diffuser. To image the object faithfully, we proceed as described by Leith and UpatnieksBO A hologram of the object wave through the diffuser is taken with the help of a reference wave. The processed plate is put back in register, with the diffuser still in place. The hologram is now illuminated by a conjugate wave. This reconstructs the complex conjugate of the signal plus diffuser which, after passing through the diffuser, produces the conjugate real image, with the noisy part due to the diffuser having been filtered out.

The explanation of the phenomena is as follows. The diffuser can be mathematically represented by the function exp[ i e (x , y )]. During reconstruction, the conjugate real image of the diffuser, given by exp[-iO(x, y ) ] , is superposed on the diffuser, mutually cancelling the phase factors. It is as if the diffuser was not there, and we observe the real image of the object.

In case the disturbance is time varying, the above method does not help. The possibility of a solution exists if the reference and the object wave are made to traverse essentially through the same part of the intervening medium, as explained by Goodman." Then, at a first approximation, the phase difference between the two waves at the hologram will be largely independent of the intervening medium, implying that the hologram is practically unaffected by the perturbing medium.


4.4.4. Matched Filtering and Character Recognition

An optical matched filter is a photographic recording of the Fourier transform of the amplitude distribution of the object. Since the amplitude as well as the phase are recorded, this is essentially a Fourier transform hologram of the object. The primary use of such a filter is character recognition, as shown by Vander LugtZ6 and Vander Lugt er a/.'* The procedure is as follows. The filter, corresponding to the character to be read, is placed in the filter plane of a 4f-filtering system. The filtering process performs a cross-correlation between the object and hologram filter. In the image plane, we get a bright spot corresponding to the detected character. In case a set of characters are to be stored, the hologram filter consists of a number of holograms, corresponding to these characters, stored in a single master hologram.

4.4.5. Holographic Optical Elements and Aberration Corrections

Conventional optical elements like lenses and prisms may be replaced by holographic optical elements in many applications, with a lower cost, as described by Close." Computer-generated holograms may be produced to simulate and substitute aspheric lenses that produce complicated wave fronts.

A holographic optical element may even be analyzed by conventional lens methods, like ray tracing. It has been shown by Sweattx4 that the analogy is so close that a holographic lens may be considered as a normal lens with an infinite index of refraction.

A hologram may also be used as an extra optical element in an optical system in order to correct some aberrations, as shown by Upatnieks et a/.,*' and to improve its performance. An obvious problem, however, is that the light efficiency of a hologram is, in general, quite low. A second problem is that monochromatic light must be used due to the color dispersion of the hologram.

4.4.6. Acoustical Holography

Most of the principles pertaining to optical holography are also applicable to acoustical holography. However, the recording and the readout techniques so far applied have been rather complex and difficult, and the results have been of a poor quality. But, due to the large number of interesting applications, it is hoped that new and simpler techniques will evolve in the near future.

198 HOLOGRAPHY

There are principally four methods for recording an acoustic hologram:

a. recording by the surface of a liquid b. recording by a matrix of detectors c. hologram is obtained by scanning, pointwise d. hologram is obtained by direct interaction of light and acoustic waves

In this section, we only deal with the first method. For details and other methods, the reader is referred to the literature like the papers by Mueller and Sheridon,86 Young and Wolf,87 and Preston.88

Figure 24 illustrates the geometry for recording and reading such a hologram. S , and S2 are the two coherent ultrasonic sources. S , produces the reference wave and S2 illuminates the object A. The reference wave and the scattered wave from the object interfere on the surface of the liquid. The information about the object is in the form of ripples on the surface of the liquid.

In the reconstruction step, the distorted surface is illuminated by a parallel beam of monochromatic light, and the scattered light wave is collected by the lens, as shown in Fig. 24. The image of the light source is blocked by a small stop. The two reconstructed images A' and A" appear very near the focal plane of the lens.

The large difference between the recording ultrasonic wavelength and the illuminating light has two deteriorating effects: ( a ) the images are formed far from the surface, ( b ) the images are greatly aberrated. In order to

s, s,

FIG. 24. Recording a reconstruction of an acoustic hologram

EXPERIMENTAL PROCEDURES IN HOLOGRAPHY 199

eliminate these effects, the photographic record of the distorted liquid surface is taken with a lens of sufficiently small aperture and later reduced by the ratio of the two wavelength. This photograph can now serve as a hologram to obtain an optical reconstruction of the acoustically illuminated object.’“

Acoustical holography is finding several applications in various areas as described by Green,’g and by Hildebrand and Brenden.” Since many objects apaque to light waves are transparent to ultrasonic waves, they can be optically examined through ultrasonic holography. Because of this feature, it is also applicable to nondestructive testing, as shown by Byron and Collins,” and to medical imaging, as shown by Malacara.’’

4.5. Experimental Procedures in Holography

The basic requirement in holography is to record a stable fringe pattern due to the interference between two light beams. In order to achieve this, certain procedures are to be followed in general. In what follows, we discuss these together with the necessary equipment required to produce a good hologram.

4.5.1. Light Sources for Holography

The requirements placed on the light sources to be used in hologram formation depend on the subject and the experimental setup. Specifically, the light disturbances reaching any point on the hologram plane (via reflections from the subject surface or reference mirrors) must be mutually coherent. In other words, the maximum optical path difference for light rays between the source and the hologram via the above-mentioned path must be less than the coherence length. Besides, the object and the reference waves must be spatially coherent. Poor coherence would effect the fringe contrast and thus the diffraction efficiency. Fortunately, with the availability of the laser, the above requirements are easily achieved, and thus making a hologram has become a routine task in the laboratory. I t is, however, worth noting that Leith and Upatnieks,’3 by using an achromatic fringe system, have produced holograms with a high-pressure mercury arc, although of a lower quality than those made with a laser source.

During reconstruction, the restrictions on the source depend on the type of the hologram. For example, an image-plane hologram, a rainbow hologram, or a volume hologram can be reconstructed with an extended white-light source. For best results, however, the reconstructing wave must be identical to the one used during formation of the hologram. In general,

200 HOLOGRAPHY

deviation from this condition and poor coherence of the source affect the resolution in the image.

Normally, for a small object of a few centimeters in size, a 2-mW He-Ne laser is quite adequate in terms of the required power. For larger objects, one can use a higher-power He-Ne laser. If still higher power is needed, one may want to use an argon laser which gives output in the range of a few watts. To study transient phenomena, Brooks er have shown that a pulsed laser is a must.

4.5.2. Mechanical Stability

There are essentially two stability conditions that must be met to record a fringe pattern. We discuss them in the following two sections.

4.5.2.1. Stability of the Recording Plate. The recording plate must be stable relative to the fringe pattern in order to record the pattern faithfully. Its stability requirements depend on the highest frequency content in the fringe pattern. In most holograms, the minimum fringe spacing is approximately equal to the wavelength of the forming light. So the recording medium must not move more than a small fraction of a wavelength during exposure.

Special efforts have to be made to ensure the stability of the fringe pattern. The whole optical setup must be isolated from external vibrations. An inexpensive method is to mount the optical setup on a massive slab of iron or concrete (weighting about 0.5 tons) supported on partially inflated motorcycle tubes. The resonance frequency must be much lower (about 1 Hz) than the frequency of the building vibrations. Sophisticated ready-made vibration-free tables are available from the Newport Corporation.

Acoustic and thermal disturbances are more troublesome. The best way is to eliminate the sources of such disturbances themselves, for example switching off air-conditioning units, fans, and heaters, etc. Reducing the exposure time is also advantageous, but at the expense of having a more powerful laser. One must also reduce, as far as possible, the optical path lengths, especially those between the beam splitter and the recording medium.

4.5.2.2. Stability of the Fringe Pattern.

4.5.3. Beam Expansion

The laser-output cross-section is typically 1 mm in diameter to illuminate the subject. While making a hologram, one needs a larger cross-section, and hence the need of expanding the beam arises. The beam can be expanded by one or more lenses or spherical mirrors.


Because of the diffraction effects at the walls of the laser cavity, the TEMoo mode (the usual mode of operation) may be modulated, giving a speckled appearance to the expanded beam. This is especially the case when the laser resonant cavity is formed with two large-radius mirrors. One can get rid of this modulation by focusing the laser output through a microscopic objective and putting a spatial filter (a fine pinhole) of diameter between 10 to 40 pm, depending on the magnification of the microscope objective being used, at the focus of the beam. The output is a clean expanded beam.

4.5.4. Holographic Materials

The subject of holographic materials is so important that complete books have been written about it, for example, by Smith.95 Good chapters in books covering this topic have been written by Hariharan” and by Gladden and L e i g h t ~ . ~ ~ Additional general references are given in this last chapter, an interesting reference being that of Bartolini et aL9’

Silver halide photographic emulsions are by far the most popular of all recording media. They are available in film or glass plates, mainly from Kodak or from Agfa-Gevaert. Their spectral sensitivities are of different types, as shown in Fig. 25 and Fig. 26. With

4.5.4.1. Photographic Emulsion.

Y a

e m -

Moo-

swo 4000 4MK) 1100 eooo 0100 Too0 7100

5 1000 a; WAVELENGTH OF LIGHT

F i c . 25. Spectral sensitivity curves for some Kodak holographic silver halide emulsions.

202 HOLOGRAPHY

7

WAVELENGTH OF LIGHT

FIG. 26. Spectral sensitivity curves for some Agfa-Gevaert holographic silver halide emulsions.

them it is possible to match most laser wavelengths. Table I summarizes some of the most important characteristics of these photographic materials.

For ordinary practical purposes, the optical density versus log E (exposure) is the characteristic normally specified. For holographic purposes, the transmittance t versus the exposure E was originally suggested; however, from the point of view of diffraction efficiency, it is more important to plot the transmittance t versus the logarithm of the exposure. These curves for some holographic photographic emulsions are given in Fig. 27.

Glass photographic plates are available with and without antihalation backing. If desired, the antihalation backing may be removed with alcohol.

4.5.4.2. Dichromated Gelatin. The most important feature of hardened gelatin is that it is not dissolved in water, but it only swells three to four times its original thickness.

If a gelatin film contains a small amount of dichromate(( NH.&CrzO,), when exposed to blue light, the Crh' ion is reduced to Cr3+, making the gelatin even harder, as described by ChangY8 and Curran and Shankoff." If the plate is adequately processed, as shown for example by Hariharan," a modulation of the thickness as well as of the refractive index is obtained, producing in this manner a highly efficient phase hologram.

EX PER1 M E N T A L PROCEDURES I N HOLOGRAPHY 203

TABLE 1. Silver Halide Films and Plates for Holography

TY Pe Resolving Emulsion Spectral Power in Thickness

Plates Films Sensitivity I/mm. in pm.

Kodak

649 F 649 F Panchromatic 2000+ 17/16 120-01 /02 SO-I73 Blue and Red 2000+ 6 125-01 /02 SO-424 Blue-Green 1250 7/3 131-01/02 SO-453 Blue-Green 1250 9

Agfd-Gevaert

10E56 10E56 Blue-Green 1500f 7 10E75 10E75 Panchromatic 1500+ 7

8E56 H D 8E56 H D Blue-Green 2500+ 7 8E75 H D 8E75 H D Blue and Red 2500+ 7

EXPOSURE IN ERGS/cm'

FIG. 27. Curves of amplitude transmittance versus exposure, in a logarithmic scale, for some holographic silver halide emulsions, using a wavelength of 6328 A.

204 H 0 LOG RAPH Y

4.5.4.3. Photoresists. The photoresists are resins that, like dichromated gelatin, are hardened by the action of the light. Their use in holography has been described in several articles by Bartolini100-'02 and by Norman and Singh.Io3 Blue or green light produces chemical changes in the resin that make it more difficult to dissolve with a solvent. Phase-diffraction gratings or holograms may be produced with photoresists, but unlike those produced with dichromated gelatin, no modulation of the refractive index is obtained, but only of the thickness.

The developer is a suitable solvent that dissolves the unexposed region faster than the exposed ones, or vice versa, depending upon whether the photoresist is negative or positive, respectively.

A positive photoresist should be used for holographic purposes. Shipley AZ-1350 has been the most widely used photoresist, but recently it has been replaced by Shipley Microposit 1350 J photoresist. Since the sensitivity of this photoresist is greater for shorter wavelengths, blue light for a He-Cd laser or green light from an argon laser should be used.

If linearity is desired, the appropriate development procedure should be performed, using a developer different from that used for the normal uses of the photoresist, as described by Norman and Singh."'

Besides the materials just described, there are various other recording materials, for example: photochromics,'04~'09 photopolymers,' ' O J ' ~ thermoplastics,' l4 electro-optic crystaIs,"5-'20 magneto- optic materials,'2"'22 metal films,123 and ferroelectric mate~ia1s.I~~

4.5.4.4. Other Materials.

4.5.5. Recording, Processing, and Bleaching

These three basic steps in hologram-making using photographic emulsions have been described in detail by Benton'" and will now be considered.

4.5.5.1. Recording. There is no universal recording material that is suitable for all of the varied applications for holography. However, the photographic emulsion is still the principal recording medium, as it comes close to having ideal properties for a wide range of requirements. The most essential characteristic of the recording medium is that it should be capable of recording the generally high frequencies encountered in holography. For example, for an angle of 30" between the object and the reference beams, the frequency is about 1000 cycles/mm, which is beyond the resolution limit of most of the recording materials. Besides this, the medium must be reasonably sensitive to the particular radiation involved.

The following procedure, not very different from processing of the high-resolution photographic image, is recommended for the exposed holographic plate:

4.5.5.2. Processing.


1. Development: 5 minutes continuous agitation in Kodak D-19 developer at 20°C, followed immediately by a 1-minute rinse in water.

2. Fixation: 5 minutes continuous agitation in Kodak rapid fixer, followed by a 1-minute rinse in water.

3. Removal of residual fixer: I f minutes in Kodak hypoclearing agent. 4. Wash: 5 minutes in flowing water. 5 . Removal of dye-sensitizer: 5 minutes in methanol, followed by a

l-minute rinse in water. 6. Drying: Emulsion shrinkage is a real problem in holography. A one-

wavelength shrinkage can cause a 180O-phase reversal of the reconstructed wave front, a disadvantage for interferometric applications. Also, there is a color shift in reflection holograms when reconstructed in white light (see Section 11.2.2). Some special measures are to be taken in order to avoid emulsion ~hr inkage . '~ . '~

4.5.5.3. Bleaching. An absorption hologram recorded in a photographic emulsion is converted into a phase hologram by bleaching out the silver image, thus obtaining higher diffraction efficiencies. There are essentially two methods, as described in detail by Benton'" which are: ( a ) Direct bleaching and (6) Reversal bleaching.

In the direct bleaching method, the film or plate is developed and fixed as usual. It is then put in a bleach bath which converts the metallic silver into a transparent insoluble salt having a refractive index significantly higher than that of the gelatin. One of such agents is potassium ferricyanide, as described by Burckhardt and Doherty.'" The hologram efficiency of this and other similar processes is analyzed in a paper by Upatnieks and Leonard,'27 showing that it is between 25% and 50%. Higher efficiencies, up to 70%, may be obtained with a wet bromine process described by Lehmann er al.,'" or with a dry process using bromine vapor, as shown by G r a ~ b e . ' ~ ~ Unfortunately, holograms bleached by these methods are associated with flare light due to the enhancement of the low spatial frequencies. This is due to the fact that the resulting relief images augment the index images. The noise can be substantially reduced by substituting Kodak D-76 for the usual D-19 developer. It can be further reduced by immersing the hologram in an index-matching liquid gate or by coating the hologram surface with an index-matching film.

The reversal bleach method described by Smith'30 overcomes the noise problem associated with the previous method by producing relief images which tend to cancel the index variation at low spatial frequencies. Here, the plate is developed but is not fixed. It is then put in the R-9 bleach bath, which converts the metallic silver to a soluble salt and removes it entirely from the emulsion. The result is that the remaining silver halide appears

206 HOLOGRAPHY

reversed with respect to the case of the direct bleach method, the relief however remaining at regions where development took place. Thus the relief image tends to cancel the phase variations due to the silver halide, and consequently the speckle noise is reduced.

4.5.6. Embossing of Holograms

Phase photoresist holograms may be replicated by a process similar to the one used to make photographic records, as described by Bartolini ef C I ~ . , ’ ~ ’ , ” ~ Clay and Gore,’33 and by Burns.’34 The process consists first of coating the photoresist hologram with a thin layer of nickel or silver either by vacuum evaporation or by an electroless method in order to make it conductive. The second step is to deposit on top of this metallic film a thicker layer of nickel by electrolysis to form a metal stamper master.

The next step is to emboss this master into a transparent flexible thermoplastic like PVC or polycarbonate that softens at a temperature far from that of thermal decomposition, as shown by Clay,’35 to produce the desired replicas. Thousands of holograms can be produced with a single nickel master, at a very low cost for each copy.

Acknowledgments

The authors want to thank Prof. N. Abramson for his very valuable comments, Mr. Fernando Mendoza for his help with the laboratory work, and Mr. Krishna Morales for the drawings in this chapter.

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208 HOLOGRAPHY

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83. D. H. Close, “Holographic Optical Elements,” Opt. Eng. 14, 408 (1975). 84. W. C. Sweatt, “Describing Holographic Optical Elements as Lenses,” J. Opt. Sac. Am.

67, 803 (1977).

210 HOLOGRAPHY

85. J. Upatnieks, A. Vander Lugt, and E. N. Leith, “Correction of Lens Aberrations by Means

86. R. V. Mueller and N. K. Sheridon, “Sound Holograms and Optical Reconstruction,”

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103. S. L. Norman and M. P. Singh, “Spectral Sensitivity and Linearity of Shipley AZ-1350

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106. D. R. Bosomworth and H. J. Gerritzen, “Thick Holograms in Photochromic Materials,”

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108. A. S. Mackin, “Holographic Recording on Electron Beam Colored Sodium Chloride

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REFERENCES 21 1

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5. PHOTOMETRY AND RADIOMETRY

William L. Wolfe

Optical Sciences Center University of Arizona

Tucson, Arizona 85721

5.1. Introduction

The practice of radiometry deals with the calculation of radiation from objects of interest, in optical systems, and on detectors, and with the measurement of various radiometric quantities on both a relative and absolute basis. The calculation of radiative transfer alone has been the subject of several monographs (e.g., Chandrasekhar’), and the measurement of radiation has likewise been the subject of entire books (e.g., Bauer,’ and Smith et d3) . This can only serve as an introduction to the field in general and descriptions of several topics of special interest.

All equilibrium radiation problems can begin with a consideration of the famous blackbody distribution first described by P l a n ~ k . ~ The angular, spatial, and areal characteristics of real bodies can then be expressed in terms of an efficiency factor, the emissivity, that multiplies a blackbody function to obtain the true flux characteristics. Several important laws of radiation distribution and transfer apply to the blackbody idealization, and most experimental situations can be modeled with only minor variations from these laws. Accordingly, the first portion of the chapter is devoted to blackbody laws, partly because many of the variations and forms of this law are not widely known.

An area of radiation transfer that is becoming increasingly important is transfer in partially transparent media. The new techniques of sounding the atmosphere from satellites to infer temperature and of probing media like molten glass depend upon this theory. It also gives some insight concerning the nature of Lambertian radiators.

Temperature measurements made with multispectral radiometers have received new emphasis from the satellite techniques of reconnaissance with “crude spectrometers.” The techniques have application in areas other than those of assessing crops and drainage patterns and may prove to be a most powerful and practical radiometric technique. The section on multicolor

213


Copyright 0 1988 by Academic Press. Inc. All rights of reproduction in any form reserved

ISBN 0-12-475971-8

214 PHOTOMETRY A N D RADIOMETRY

techniques provides some new results or perhaps an enhanced understanding of old techniques applied to new problems.

Last and maybe least is the question of nomenclature. There has been confusion in the field of radiometry. Some of it is due to the many different fields in which radiometric measurements are made and discussed, and some of it is apparently due to the fact that photometry preceded radiometry just as the awareness of light preceded that of infrared and ultraviolet radiations (Herschel,’ and Ritter6).

5.2. Symbols, Units, and Nomenclature

Table I presents a list of all the symbols used in this chapter along with their definitions and units associated with them. The units are given not just to define the dimensionality, but they are intended to help clarify the meaning. SI units are generally used, although frequently used units like centimeters and micrometers are indicated rather than meters where it is appropriate. The symbols are listed alphabetically, but the ensuing discussion treats them topically.

Perhaps the simplest of the variables to be discussed is the independent spectral variable. Most of us think in terms of the optical wavelength A of the radiation to be measured. It is also useful to think in terms of the frequency which can be expressed in several ways. The wave number i j is just the reciprocal of the wavelength. The frequency v is the speed of light times that ( v = c/A), and the radian frequency w is 27rv. Similarly, k, the radian wavenumber is 27rl or 2 7 r l A . The nondimensional frequency x is c2/AT (where c2 is the second radiation constant defined below, and T is the temperature), a convenient independent variable for many calculations. The spectral distribution of blackbody radiation can be given in terms of any of these variables. In order to treat this somewhat generally, the symbol y is used to represent any of these spectral variables.

Radiation transfer is ultimately expressed in terms of a quantity like power, the time rate of energy. An equivalent concept is the time rate of the transfer of a number of photons or of the amount of power which evokes a visual response. Each of these is a flux and is denoted by the symbol P. To distinguish between them, the subscripts u, q, and u respectively, are recommended by the International Illumination Society.’ I have not used a subscript when the symbol can refer to any of these or when it refers specifically to energy transfer. I believe, no confusion will result. The rest of the symbols which are used to describe radiative transfer are L, M , E, I, and S. These are, respectively, the radiance or flux per unit normal area and solid angle; the irradiance or incidance or incident flux per unit area;

SYMBOLS, UNITS, A N D NOMENCLATURE 215

the emittance or exitance (emitted flux per unit area); the intensity or flux emitted per solid angle-generally from a subresolution source; and the radiance per unit path length in a medium which scatters or is a volume emitter. This is also called the volume radiance. In several sections I have dealt with irradiance and electric field strength in the same equation and have used U for the electric field. These equations have also involved m U U * for which I have used 9. I would prefer using I and E for these, and J and H for intensity and irradiance, but that would be contrary to the recommendations of the International Commission on Illumination (CIE). Several writers have proposed nomenclature schemes to make these definitions logical, consistent, and easy. Table I1 is a summary of these. The fluometry scheme of Jones has as its main feature the unifying concept that these flux terms are really all geometric concepts. He therefore used sterance, incidance, exitance, intensity, and sterisent as generic terms to be modified by “radiant,” “photon,” and ‘‘luminous.’’ Nicodemus,’ largely in an effort to overcome the ambiguity of “intensity” (it is variously used to mean E, M, I, and L by different scientists), has coined the geometric terms of areance, and pointance for E or M and I , respectively. Worthing,” years ago proposed that “-ancy” be used to mean “per unit area.” He went further and proposed that the transmission, absorption, and reflection properties be specified with “-ance” endings when a sample is specified, and “-ivity” endings when a substance is specified.” This, of course, makes emittance an ambiguous term if one uses the “conventional” terms. The nomenclature system has not yet been proposed. The concept of path radiance may be somewhat foreign to some readers. In a scattering medium or volume radiator the radiance is not constant. It can increase with path length, because each element of path contributes more radiation to the total. It thus has units of W cm-* sr-’ cm-I, where the final cm-’ is the (reciprocal of the) unit of path length. There is no recommended symbol for this. I have chosen S for this quantity because it agrees with “sterisent,” and many other appropriate letters have been preempted. I have also replaced with P (to stand for phlux) for ease of typing. Finally, to indicate any of these radiometric quantities, I have used the symbol R.

The atomic constants are listed in a conventional way: cI and c2 are the first and second radiation constants, equal to 27rc2h and hc/ k g , respectively. The speed of light in vacuo is c; Planck’s constant, h ; the Stefan-Boltzmann constant a; that for photon distributions, aq; and the Boltzmann constant k g (because k is the radian wave number).

Some of the common symbols are x, y, and z for general distances. These do not get confused with the spectral variables. Angles are denoted by 6 and 4; solid angles by R; areas by A; time by 1 ; temperature by T ; integers by m, P, 4.

216

Symbol

a, b, c A B BRDF b C

CI

c2

D D* E F f g h I K

k

L I M m. P, 9 N NEFD NEAL n P R 9 r S SNR T

U

V V(A)

C

Kt,

k ,

I

U

u W

X

Y I

PHOTOMETRY A N D RADIOMETRY

TABLE 1. Symbols, Units and Nomenclature

Definition Units

constants area effective bandwidth Bidirectional Reflectance Distribution Function base of a prism constant speed of light in a vacuum first radiation constant second radiation constant diameter specific detectivity irradiance or incidance focal ratio focal length or area factor constant (2 or 4) Planck's constant or height intensity luminous efficacy radiation interaction factor radian wavenumber Boltzmann's constant radiance length radiant emittance or exitance integer constants number of photons noise equivalent flux density noise equivalent change in radiance refractive index power or flux general radiometric variable or range responsivity radius volume radiance signal-to-noise ratio temperature time electromagnetic field quantity ( E or H) normalized nondimensional frequency volume or voltage or voltage ratio visibility curve velocity width nondimensional frequency or coordinate length general spectral variable or coordinate length coordinate length

- mZ Hz sr ' m

m s- ' w cm2 s-I cm K m cm Hz'/ 'W I

w cm-:

m or -

w s2 W sr-' ImW '

J K - '

Wcm 'sr- ' m Wcm '

-

-

-

-

-

- - W cm-' W cm-'sr-'

W various VW-' or AW-' m W cm-'sr-'

K

v m - ' or v

m' or v or -

m S C I

m - or m various or m m

-

-

S

-

-

SYMBOLS, UNITS, AND NOMENCLATURE 217

TABLE I. (coni.)

Symbol Definition Units

a

Y E

1)

1)"

8 A v

v

P

u ( A )

@ R

U

T

w

Subscripts

a, b a

d e h

q

C

I

S

U

V

1.2.i W

Superscripts

BB h m

P 0

-

absorptivity or absorptance absorption coefficient emissivity quantum efficiency luminous efficiency angle wavelength frequency wavenumber reflectance or reflectivity Stefan Boltzmann constant slit function transmittance or transmissivity angle solid angle radian frequency

indicating different spatial points atmosphere collector or optical element detector emitted hemispherical incident photon source energy visible first, second ith of a group infinite (number of reflection)

blackbody hemispherical maximum, molecular optics projected time rate weighted average

TABLE 11. Radiometric Nomenclature

Quantity

~~

Jones

~ ~

Field Nicodemus Worthing Theorists Meteorologists

Flux Density [Wm-*] Areance Intensity Flux Density Incident Incidance Incident Areance Radiancy - - Exiting Exitance Exitent Areance Irradiancy - -

- - - Flux per solid angle [w sr-'] Intensity Pointance Flux density per solid

Volume Radiance Sterisent angle [ w sr-' m-*] Sterance Sterance - - Specific Intensity

- - - -

> z 0

rn -I P <

FORMULAS FOR BLACKBODY RADIATION 219

5.3. Formulas for Blackbody Radiation

The derivation of the Planck equation can be based on the product of the function which gives the density of states available for photons and the average number of photons in a state. The expression is derived in terms of the radian wave number k ( = 2 r / h ) . The number of photons nk per unit volume with wave number between k and k + d k is given by

nkdk = NkdkV-' = k 2 K 2 ( e " - l ) - ' d k [m-3], ( 5 . 1 )

where V = volume x = c 2 / h T c2 = hc/ k , h = Planck's constant c =velocity of light

k , = Boltzmann's constant

Notice that the subscript notation here indicates a partial derivative. The quantity nk is the photons per unit volume per unir radian wave number. Other forms of the Planck equation are obtained in subsequent sections.

5.3.1. Geometric Distributions

It can be shown12 that the radiant exitance and radiance are related to nk by the following:

M q k = (f) nk

(5.3)

These are purely geometric relationships and apply to other spectral distributions. One important and very useful characteristic of the radiation is that the geometrical characteristics are independent of the spectral distributions. There are basically four spectral distributions: power vs frequency, power vs wavelength, photons vs frequency, and photons vs wavelength.

5.3.2. Generalized Spectral Distributions

The spectral distributions can be written in terms of the concentration of flux and the sterance. The different spectral forms of the basic four are

PA = 2 r ~ ~ h A A - ~ ( e " - 1)-' [W CLm-lI, (5.4)


(5.5)

(5.7)

Other variations involve different arrangements of the atomic constants, and different powers of a frequency or wavelength variable. They can be written in general as

R , = Cy*'"(ex - l)-', where

(5.8)

R is any radiometric quantity y is any spectral variable rn is the proper power of the spectral variable C is a constant made up of atomic constants.

In this formulation, when y represents a frequency variable, the positive sign applies, and when y represents a wavelength variable, the negative sign does. Then expressions for the maximum can be derived for the general case and applied to specific problems. The result is that the basis for finding the maximum of any spectral distribution is to find the value x, for which

(5.9)

The usual expression for the distribution of power vs wavelength can then be found from

c2 hc hu AT A k R T - kBT'

x=-=---

and

( 5 1 1 ) c 1.43883

A m T = L = - - - cm K =2898 [ p m K]. Xm Xrn

This is the common expression for the Wien displacement law." The constants for the other forms are given in Table 111.

The distribution can also be expressed usefully in terms of the dimensionless variable x by applying the technique described above:

R, = 2ch - x'(e" - 1)-' ( C 3

(5.12)

FORMULAS FOR BLACKBODY RADIATION 22 1

TABLE 111. Maxima of the Dependent and Independent Variables for Different Isothermal Planck Spectral Distributions

Function

Dependent Independent

Variable Variable m Xmax R m a x

Photons Y 2 1.593624260 0.6476 Power Y 3 2.82 1439372 1.4214 Photons A 4 3.920690395 4.7796 Power A 5 4.965 11423 21.2036 Power contrast A 6 5.96940917 115.9359

The relative value can be written as

(ex"' - 1) = u3

R, x3(err - l)- ' Rx,, , x ~ ( e x m - l ) - ' ( e ' - l ) '

-- - (5.13)

Any of the distributions will be of this form but with a different exponent on the relative dimensionless frequency u (=x/x,).

The radiance can change either as a result of a change in temperature or a change in the spectral variable.

Here it can be seen that the factor xex(e' - 1)-' is the factor for spectral changes. Each variation individually can be viewed as an exponential relation, as pointed out in part by

R,. = TP, (5.14)

where

p = xe'(e'- I ) - '

and

R, = y 4 ,

q = * rn + xe '( e* - 1 ) - I .

The temperature-variation portion of this development is described and discussed by DreyfusI4 and several applications are given.

The reader can find descriptions of the properties of blackbody radiation in several texts (Hudson"; Holter et al."; Bramson18; Kruse et al."; and Kingston").

A general treatment follows for the integrals of these functions in terms of the dimensionless variable x.

where

222 PHOTOMETRY AND RADIOMETRY

5.3.3. Generalized Integrals

The flux and change in flux over a finite spectral band are found by integrating the appropriate monochromatic function over the spectral band. The functions can be the radiance, photon sterance, or their respective changes with temperature:

(5.15)

(5 .16)

(5.18)

The finite integrals of the Planck function can be found by expanding (e" - 1)- ' or (e" - 2 + Kx)-' in a series. The results of these expansions are given below.

r n - 4 e e - m " [ ( r n ~ ) 3 + 3 ( r n x ) 2 + 6 r n x + 6 ] ,

e ~ m x [ ( r n ~ ) 2 + 2 m ~ + 2 ] ,

=-(-) 2c2h T 1 rn~4e-"A[(rn~)4+4(rn~)3+12(rnx)2

T c2 m ; l

+ 2 4 r n x + 2 4 ] ,

C rn-' e - m x [ ( m x ) 3 + 3 ( r n x ) 2 + 6 r n x + 6 ] .

5.3.4. Infinite Integrals

The value of the integral of a blackbody function is of course independent of the spectral variable, and the simplest variable to use is the nondimensional frequency x. Then the total radiance is

L = 2c2h ( 3' x 4 ( e x - I)- ' dx

(5 .19)

S I M P L E RADIATIVE T R A N S F E R

For photons, one has

223

L, = 2 c ( z )3 x3(ex - 1)-' dx

2kiT' T' c 2 h 3 ?l

=2.4041-- - 1.5202 x 10" -

u T' - --.

?l (5.20)

Both of these are solved by resourse to zeta functions. This latter expression can also be written as

L = & (5.21) ' 2.15 kBT'

This would indicate that the average value of the energy of a photon is 2.15 k B T .

5.3.5. Some Useful Blackbody Curves

Tabulations of various blackbody functions are presented in various texts.20 These generally include the distribution of power and photons with respect to wavelength and the integrals of these functions from A = 0 to the tabulated wavelength. Some also give tabulations of the change in the radiant emittance with respect to temperature. None of those I have found do the same for photon distributions. Computers make easy the calculation of the appropriate quantity of Sections 9.3.2 and 9.3.3, and they make tabulations of these values essentially unnecessary. In the spirit of providing approximate values for ready access, curves are given in Figs. 1-8. These include: the relative spectral distribution of photons and flux, their relative cumulative distributions, the relative distribution and relative cumulative distributions of their temperature, all as a function of the normalized nondimensional frequency.

5.4. Simple Radiative Transfer

This section deals with the radiative transfer in nonscattering media. The primary purpose is to describe the characteristics of the radiometric variables rather than the physical processes which give rise to absorption and emission. The first part of the description then applies to transfer in nonabsorbing, nonscattering, nonemitting media, and the second to absorbing and emitting but nonscattering media.


1

8.9

0.8

0.7

8.6

8.5

0.4

0.3

8.2

0.1

0 1

0 1 2 3 4 5 NORNALIZED NONDIMENSIONAL FREQUENCY

FIG. 1. Normalized blackbody energy distribution as a function of normalized nondimensional frequency of the radiation.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

NORMALIZED NONDIMENSIONQL FREQUENCY

FIG. 2. Normalized blackbody photon distribution as a function of normalized nondimensional frequency of the radiation.

SIMPLE RADIATIVE TRANSFER 225

1 -

0.9 - 0.8 - 0.7 - 0.6 -

0.5 - 0.4 - 0.3 - 0.2

0 .1

0

NORFlALIZED NONDIMENSIONAL FREQUENCY

FIG. 3. Normalized blackbody energy contrast distribution as a function of normalized nondimensional frequency o f the radiation.

1

0.9

0.8

0.7

0.6

e. 5

0.4

8 .3

0.2

0.1

0

NORNALIZED NONDIIIEHSIONAL FREQUENCY

FIG. 4. Normalized blackbody photon contrast distribution as a function o f normalized nondimensional frequency o f the radiation.

226 P H o r o M E r u Y A N D RADIOMETRY

1

8.9

9.8

0.7

0.6

0.5

0 .4

0.3

8.2

0.1

0

NOREIALIZED NONDIHENSIONAL FREQUENCY

FIG. 5. Normalized blackbody energy integral as a function of normalized nondimensional frequency of the radiation.

1

0.9

0 .8

0.7

8.6

0.5

0.4

0.3

8.2

8.1

8 0 1 2 3 4 5 6 7 8 9 18 NORMALIZED tiONDIMENSIONAL FREQUENCY

FIG. 6 . Normalized blackbody photon integral as a function of normalized nondimensional frequency of the radiation.

SIMPLE RADIATIVE TRANSFER

1

0.9

0 .8

0 .7

8.6

6.5

0.4

0 .3

1:: 0

227

/ * , . . . . . ~

1 -

0.9

0.8

8.7

0.6

0.5

0 .4

0 .3

. - - - - -

-

:::t/ . . . . . . . , 0

B 1 2 3 4 5 6 7 8 9 10 NORFIALIZED NONDIMENSIONAL FREQUENCY

FIG. 8. Normalized blackbody photon contrast integral as a function of normalized nondimensional frequency of the radiation.


5.4.1. Transfer in a Vacuum

The basic equation of radiation transfer involves the radiance of the source, the areas of the emitter and receiver, and the distance between them. Two general surfaces, A , and A 2 , shown in Fig. 9, have on them differential elements dA, and dA2. The line-of-centers distance between these elements is x, and the angles made between the normals and x are 8, and 02. Then the amount of flux dPI2 radiated from dA, and received by dA2 is given by

L dA, cos 0 , dA2 cos O2 X 2

dPI2 = = L dA, cos 0, d R 2 ,

The expression on the right was written four different ways to illustrate the different interpretations which have been applied to it. The first can be viewed as a variation of the definition of radiance: since radiance is power per projected area and solid angle, then power is radiance times projected area and solid angle. The second form shows that the solid angle can be written as a projected area divided by a distance squared. This form gives a physical and symmetrical expression for the transfer: the product of two projected areas divided by the square of the distance between them. The third form just indicates that in this chapter (and other places in the literature) the projected area is indicated with a subscript. The final form uses the “p” to indicate a “projected solid angle.” This is a nonphysical quantity, which when multiplied by an area and a radiance gives the flux transfer.

The expression can be used to obtain the relation between radiance and flux density for Lambertian bodies and the very useful configuration factor that is found in extensive tables; see Sparrow and Cess2’ for more on this.

The relationship between Land M (regardless of whether photons, power, or any other quantity is being transferred) is M = TL. One can imagine that a differential area of a radiator which has the same radiance in all directions

FIG. 9. The geometry of radiative transfer. Radiation travels from a differential element dA, of surface A , along R , , to dA, of A,. The angles between R , , and the surface normals are 0, and 0,.

S I M P L E RADIATIVE TRANSFER 229

is at the center of a sphere. The flux per unit area dA, of this source radiated into a hemisphere is given by

L dA, cos 8, dA2 cos e2 M = - = (5.23)

dA I x’dA,

The radiance is independent of angle, and O2 is zero, so that

M = L lo2= cos 8 sin e de d 4

= TL. (5.24)

It should be noted that the roles of emitter and receiver can be reversed either by use of Helmholtz’s reciprocity theorem or by direct integration.

This relationship can be used in connection with the flux transfer theorem to obtain a configuration factor (also called angle factor and view factor). On a differential basis, one has

L I 2 dA, cos 8, dA, cos O2 d R I 2 =

X-

For a Lambertian source MI2 = 7rLI2. Then

dA, cos 8, dA2 cos tI2

T X 2 d P l z = M

(5.25)

(5.26)

The configuration factor KI2 is the fraction of radiation emitted by the first surface that is received by the second. Thus,

dP12 ( MI2 dA, cos 8, dA2 cos O2)/x2 -

dP, MI2 dA, &Al - dA2 - 9

(5.27)

The configuration factor is a function of the “tilt” of the radiating surface and the solid angle subtended by the receiving surface.

View factors for surfaces of a finite size are more complicated. The definition is still the same, but integration over the source, the receiver, or both must be performed. The configuration factor for a differential element dA, and an extended area A 2 is obtained as follows:


Extended sources, of course, are not the only type of radiator one encounters in practice. The second major class is generally called “point sources,” although in theory there is no such.thing (at least one that radiates appreciably). What is usually meant is a subresolution source, one which is small compared to the image of the detector on the source. For such a source, radiance is not a good descriptor, but the integral of radiance over the area of source is appropriate. Such a quantity is usually called intensity and labeled I by radiometrists.

It can be defined in two ways:

Then the transfer is written as

I , ? dA, cos O2

X Z d P , , =

(5.29)

( 5 . 3 0 )

The radiance is a function of both the direction and the location on the surface, whereas intensity is a function only of direction.

Before considering transfer in media that absorb but have inappreciable emission, it is useful to consider two brief calculations involving flux on the detector of a radiometer. The first calculation involves an extended source, the second a subresolution one.

Assume that a radiometer views an extended source of area A , at a distance x with radiance L. The radiometer has a collector area A , and focal length f, detector area A , optical speed or Flnumber F, and transmission r. Then, if all solid angles can be approximated by areas divided by distances, the flux is given by

This can be reformulated in terms of solid angles and Flnumbers:

( 5 . 3 1 )

( 5 . 3 2 )

The total flux o n the detector must be obtained from one of the last two expressions. The last expression on the right is in terms of the collector area and the field of view of the detector. To keep a constant flux on the detector, these two quantities must be varied inversely. Usually one wants as small a detector field as one can get, but this must be accompanied by a concomitant increase in collector area (even without diffraction considerations). The penultimate expression shows that the flux concentration on

SIMPLE RADIATIVE TRANSFER 23 1

the detector is independent of the diameter of the optics and is a function of transmission and Flnumber . The total flux increases with detector size but only until the source becomes subresolution.

5.4.2. Transfer in Transparent, Nonscattering Media

The basis of the relationship that extends the transfer of flux from source to receiver and then to detector is that radiance through an optical system is constant (Nicodemus,” Rainwater”). The simplest extension of the transfer theory from a vacuum to real media is the consideration of a medium which absorbs, has a refractive index different from 1.0, and has no measurable emission. The first change is that the radiance is no longer conserved. Rather, it is L /n2 , the radiance divided by the refractive index squared, which is a constant. So the product of L K 2 and n2AR, which gives the flux, is still the same and still constant-except for any transmission losses. These transmission values can be applied to the optical elements, the overall system, and the intervening atmosphere T,,. In fact, the flux on the aperture of the radiometer in such a situation is given by

P, = T,LA,R,, = T,LA,R,, .

The flux on the detector can be written as

(5.33)

(5.34)

Careful practice results in accurate determination of T,,, A d , and a,,,, because these are all properties of the radiometer and are readily accessible (but sometimes hard to measure). In many cases, the atmospheric transmission is very hard to measure. Consequently, many investigators report not only their estimate of L but their estimate of T,L, which can be determined much more accurately. The quantity T,L is dubbed the apparent radiance.

5.4.3. Properties of Partially Transparent Materials

Partially transparent materials are bulk radiators. Every portion of the volume of the material contributes in a significant way to the radiation measured at a distance from the surface. Understanding of the phenomena is important for certain technical reasons as well a s for an improvement in the understanding of what are usually regarded as surface radiators. Some of the main contributors to this work have been G a r d ~ n , ~ ~ McMahon,2’ and Czerny and Genzel.26 This discussion is based on the treatment of Gardon.


FIG. 10. Geometry of radiative transfer in a volume radiator, shown as a side view. An annular volume element radiates a distance x/cosO to the surface.

A partially transparent, semi-infinite body at a uniform temperature T and with an absorption coefficient y, and a spectral path radiance S, sterisent [W m-3 sr-' pm-'] or just S is the subject of consideration. The radiance from the surface of a medium with a uniform absorption can be obtained from the path radiance by treating a differential element of volume as shown in Fig. 10. The result is that the surface radiance L of a slab of infinite lateral extent is given in terms of the volume or path radiance S as

S L = - .

Y (5.35)

The basic assumptions in this derivation are that the exponential law of absorption is valid and that the volume emission is the same in every direction (an isotropic radiator). Integration over the hemispherical projected solid angle yields

7TS M = - .

Y (5 .36)

From the earlier discussion of radiance in media of any refractive index n,


it is clear that (with a little algebraic manipulation)

(5.37)

This gives the volume radiance in terms of the absorption coefficient y and refractive index of the medium and the “standard” surface radiance. The external radiation will be in a solid angle that is larger by n 2 than the internal angle and is reduced by the surface reflection losses. Further, radiation cannot exit at angles which exceed the critical angle.

If the partially transparent medium is of a finite rather than semi-infinite thickness denoted by X , then the integration yields a radiance given by

L = n2~BB(1 - e - ~ . ~ / ~ ~ s @ ) . (5.38)

As Gardon points out, the radiance from the semi-infinite plate follows Lambert’s law, whereas that for the finite plate does not.

Several interesting characteristics of radiation in partially transparent media can be developed from this beginning: the expression for radiation from plane parallel plates, for directional and hemispherical emissivities, Bouguer’s law for finite radiators, and the technique of inversion applied to the atmosphere and to radiance solids like glass.

Integration of the radiance over the hemispherical projected solid angle yields the following:

M = n 2 ~ n B [ 2 - e - y ‘ ( 1 - y r ) - ( y r ) * ~ i ( - y t ) I , (5.39)

where

This exponential integral is to diffuse radiation what the simple exponential function is to directional radiation.

Gardon shows further that the directional and hemispherical emissivities are given respectively by

and

Eh = I:* &(ei ) sin 6, cos ee de,, (5.41)

where

ei = angle of incidence ee = angle of emission

234 P H O T O M E T R Y A N D R A D I O M E T R Y

The quantities r and p are the directional transmittances and reflectances of a single surface; the subscripts indicate the state of polarization (s perpendicular and p parallel); the derivation is (as they say) straightforward but tedious, making repetitive use of Fresnel’s equation at each surface for reflecting the volume radiance contribution. This result is then compared to the radiation from a blackbody.

The transmission, reflection, and absorption of a plane parallel plate can be done in various degrees of detail that can take into account polarization, self-emission, scattering, etc. Several of these situations are discussed below; any special situations can be obtained by extensions of these results.

The first situation encompasses plane radiation incident normally on a plane parallel plate of material which absorbs but whose reemission is negligible. Then the total reflectivity, taking into account an infinite number of multiple reflections, can be written

pe = p + ( l - p ) ’ p e-2yx + ( I - p ) ’ p 3 e - 4 Y X + . . . . (5.42)

This is an infinite geometric series that can be written as

p a ? = p + ( l -p) ’pe-*YX(1 - p 2 e-2y.r)-’ . (5.43)

The nomenclature is somewhat simpler if the single-pass internal transmittance e-”‘ is written as 7. Then

(5.44) p m = p + ( 1 - p ) 2 p ? 2 ( 1 - p 2 7 2 ) - 1 .

Notice that when r = 1 ,

(5.45)

Similarly the transmittance can be written

7,= (1 - p ) 2 7 + (1 -p)2r3p2+ ...

(5.46)

Noticethatwhenr= 1, ~ ~ = ( l - p ) / ( l + p ) = 2 n / n ’ + l . Inthesimplestcase, this is a collimated beam both into and out of the sample. Therefore, power in the beam must be conserved. That which is not reflected or transmitted must be absorbed:

= ( l - p ) ’ T ( l - p 2 7 2 ) - I .

(5.47)

The proof that a, = E, for a partially transparent material requires the calculation of E ~ , the emittance of the partially transparent plane parallel plate divided by that of a blackbody.


One can apply the same techniques of integration, but for the sake of variety it will be done in terms of flux density rather than radiance. In a small solid angle dR, the flux density is (1/47r)SdxdR in a small element dx. This quantity multiplied by the transmission is integrated to the surface

SdR

47rY -- - (1-T).

This is the radiant exitance just inside the surface; outside, it is

(5.48)

(5.49)

If the body is opaque, then T = 0, and Kirchhoff's law provides the following result:

1dR M H H

4ff Y 7r M =- ( 1 - p ) =- ( 1 - p ) dR. (5.50)

From this, one can write that S / y = 47rM BB. Therefore, the nonreflected volume radiation gives rise to one component of radiant exitance:

M I = ( 1 - p ) ( 1 - T ) M B B r - ' dR. (5 .51)

That which is reflected from the top and the bottom and emerges again is

M2 = P ~ T ~ M , . (5.52)

After one more trip, one has

M3 = p27'M2 = p 4 r 4 M I . (5.53)

The sum of the infinite geometric series is 00

C M, = M,7r-'dR = (1 - p ) ( 1 - T ) M ' ~ T - ' dR/(1 - P ~ T ~ ) . (5.54)

The emissivity is the same as the absorptivity, and Kirchhoff's law applies to semitransparent substances.

There is one difference between this development and McMahon's: the radiant exitance is derived from a consideration of radiance flowing into a projected hemisphere. This changes the relationship between volume radiance and radiant exitance by a factor of 2.

i = O


5.4.4. Path Radiance

A special application of the radiometry of partially transparent media is the flux one obtains at the top of the atmosphere in a relatively well- collimated beam. If L, is the radiance of the ith element of the atmosphere, and L"" is the blackbody radiation of that element, then

L, = &,LY". ( 5 . 5 5 )

If T, is the transmission of the ith element, then the radiation at the top of the atmosphere is

BB &,L, T,-17,-2 ' ' * 71.

I f the atmosphere does not scatter, then

&, = (Y, = 1 - 7,. (5 .56)

The difference in transmission between the top and the bottom of the ith element is

d T , = ( l - ~ , ) 7 , - , ~ , - ~ . . . T ~ . (5 .57)

Therefore the contribution of the ith element is

LY" dr,.

The total beam radiation then is

Iho L"" dT.

This is the path integral, emphasizing the variation of transmission along the path but with a simplified geometry.

5.4.5. Atmospheric Sounding Techniques

There are two techniques by which a radiative inversion can be accomplished: nadir and slant path. The radiative transfer calculation in a partially transparent medium consists of determining the amount of radiation at some point for known temperatures and quantities of the constituents along the path. The inverse calculation is to determine the temperature and quantity of the constituents along the path from the radiance at the end of the path, usually in several spectral bands. These techniques have been reviewed by Wark and Fleming,27 by Hanel and Conrath,28 and by Gille.29 A short description of the techniques is given here to illustrate the radiometric principles involved.

The temperature profile can be determined by measuring the irradiance at the aperture of a satellite instrument in several narrow spectral bands in


the absorption region of a gas whose mixing ratio is known. Two atmospheric gases meet this requirement: CO, and 02. Kaplan3" proposed using the 667 cm-' (15 pm) band of C 0 2 for this purpose, and experiments have been performed based on that ~uggestion.'~

Once the temperature profile has been established, other spectral bands can be used for determining the profiles of other gases. The radiative transfer equation for monochromatic radiation can be written in terms of the radiance at a given frequency F and zenith angle 8 as'

dL(F, e)={-L(F, e ) + LBB[V, T(z)])yrn(V, z)p(z)secedz, (5.58)

where

T( z) = atmospheric temperature at z p( z) = atmospheric density at z

z =height at local vertical.

The equation can be recognized as the one derived earlier. In this case, however, the path differential is sece dz and the absorption is pym, because the product of the gas density and molecular absorption coefficient gives the overall absorption coefficient. The assumptions include a plane parallel atmosphere in which there is no scattering or refraction and which is in local thermodynamic equilibrium. The term L(V, 6 ) is the radiance at the lower boundary; the minus sign arises from the integration convention. If the instrument views a narrow cone, vertical to the surface of the earth, then sece = 1. The independent variable can be converted from altitude to the pressure at that altitude:

d(F' ') d(log P) . (5.59) I,,' LBB(F' T , d(1og P) L(F,O)= E(F)LBB(F, T) r (F ) -

The radiance at the top of the atmosphere L(F,O) is found from the contribution oftheground E ( P)r ( F)LBB( F,T), where r ( P) is the transmission of the entire path from the ground to the top of the atmosphere, and the path radiance expressed by the integral. The limits specify that the integration is from the top of the atmosphere down to a height at which the pressure is p. I t is worth writing this one more time and somewhat more compactly. For each wavelength or wave number, the transfer equation is

d r Lo=&rL- L d(1og P) . I d( logP)

(5.60)

An iterative inversion approach is then used. One tries a reasonable T(z) in the expression for each of the L( V ) until a best fit is found to the measured values of Lo( F). Various techniques have been used to obtain convergence."


A second way to accomplish atmospheric sounding is to use a long horizon path, and, in fact, a series of paths none of which intersect the earth itself. Inversions are done on each of these paths, and an entire distribution of temperatures and constituents can be generated. For this horizon case, there is no earth contribution, so the radiative-transfer equation is

T ( Y , P )

L( a ) = LRR( a, T ) dT (a, P ) . (5.61)

The integration proceeds from a no-atmosphere case (7 = 1) to the full- atmosphere transmission case. Instruments used for this technique need to have narrow fields of view to attain good vertical resolution. consequently, they need relatively wide spectral bands to establish sufficient signal-to-noise

Inversion R I d iorondr

Temperature (OK)

FIG. 1 1 . Temperature profile derived from IRIS data by inversion of the radiative transfer equation; data were recorded near Brownsville, Texas, 22 April 1969, at 1937 GMT. Data from a radiosonde ascent at 1800 GMT are shown for comparison.

R A D I O M E T R I C T E M P E R A T U R E M E A S U R E M E N T S 239

ratio. Accordingly, an integration using the radiometer spectral responsivity is required.

The inversion procedure most often discussed which uses this technique is the "onion peel" approach. Inversion is carried out for the highest layer; then successive layers are added, and each time a new inversion is accomplished. Thus, inversions for a series of almost horizontal lines are obtained. The data need only to be rearranged to get the vertical profiles.

The radiometric considerations involved in this technique include the choice of the right gas, a set of spectral lines that are narrow enough and far enough part for good inversion but wide enough to encompass enough photons-and, of course, the design of an instrument which provides plenty of throughput. One result of the vertical sounding techniques is shown in Fig. 11.

5.5. R ad i omet r ic Te m pera t u re Measure men t s

A radiometer can be used to measure the temperature of remote objects in several different ways: total radiation, radiation in a narrow spectral band, and by fitting the blackboy curve to two or more points. This section includes discussions of some of the accuracies and techniques by which these different measurements can be accomplished.

5.5.1. Radiation Temperature

One measures the radiation in as wide a spectral band as is instrumentally possible with a detector which is as spectrally flat as possible in order to obtain a radiation temperature. The output voltage is given by

V = lorn ~ ( A ) E ( A ) T ( A ) c , A - ' ( ~ " - I ) - ' dA. ( 5 . 6 2 )

We assume that 9, the responsivity, includes in it any necessary constant geometrical factors. The usual and simplest analysis includes assumptions that & ( A ) = T ( A ) = 1 and % ( A ) = 9. Then

V = aT4%. Therefore,

T=(&)"4. ( 5 . 6 3 )

The radiation temperature is the temperature of the blackbody which gives the same flux as the body being measured. We can now deal with successively more complicated situations: a constant emissivity and infinite spectral


band, constant emissivity but finite spectral band, etc. For constant emissivity and infinite spectral band, the expression is

V = 9 ~ a T

T=(&)"' (5.64)

One is usually in possession that E is 1. Then the relative

of the values for 9 and V but must assume error is

1 - &-1 /4

The error is different for different values of E and is negative for positive values of E. This means that the error is such that a higher temperature is inferred than is really the case.

The next complication is that 9 ( A ) = T ( A ) = & ( A ) = 1, but that the bandwidth is finite. Then the relative error is

IA, c,A-'(e" - l)- ' dA

aT4 1- (5.65)

A further complication is that % ( A ) and T are constant. One can usually measure 9 (A ) to sufficient accuracy. Then

V = J ~ ( A ) C , A - ~ ( ~ " - l)- ' dA. (5.66) A.4

In principle, the temperature can be found because this is an integral of the product of two functions which are known. The only errors will arise from inaccuracies in the determination of % ( A ) and the measurement of the voltage. If, in addition, & ( A ) and T( A ) are not constants and are unknown, then the error is

(5.67)

The effects of E and T can be combined into a single product spectral function for most purposes. Then the error is

1 - (E)-I (5.68)

where is the effective epsilon-tau product.

RADIOMETRIC TEMPERATURE MEASUREMENTS 24 1

5.5.2. Brightness (Radiance) Temperature

spectral band: This simple measure infers temperature from a radiance value in a narrow

LA dA = &(A)c I r -1A-5(eX - 1)-' dA. (5.69)

Such a measurement is made in a narrow spectral band but not in an infinitesimal or square one. Errors arise from the fact that the narrow band has a shape, that & ( A ) is not known, and that there is some uncertainty in picking dA. If the Wien approximation (x >> 1) can be used and & ( A ) = 1, then

L, = c ,T- 'A- ' e-x, (5.70)

(5.71)

If the line is chosen in a region in which the Planck law must be used, then

T = cz/ A In( c, r - ' h -').

The error is given by

ln(9&cl r - l / A V+ 1) ln(9c,.rr-'/ASV+ 1) . 1-

(5.72)

(5.73)

This expression is not in an easily used form and requires specific details for evaluation.

5.5.3. Ratio Temperature

The simplest form of a two-color or two-wavelength or multispectral technique is just a measurement of the ratio of the outputs in two narrow spectral bands. This voltage ratio V is given by (in the region where the Wien law holds):

then

(5.74)

(5.75)

In the region for which the Planck function must be used, the temperature cannot be found explicitly. The voltage ratio is

(5.76)

242 PHOTOMETRY A N D R A D I O M E T R Y

Generally one chooses the two spectral bands such that = E ? , but only a knowledge of the ratio is necessary. The measurements are usually made through an atmosphere of transmission T, so that an ET product in each band should be used instead of just E. Then, since xI = c2/A I T = x2 = c2/A2 T, there is only one value of T which satisfies the set of conditions.

The errors resulting from an inaccurate assessment of the ratio of the ET product in the two bands can be made by assuming the ratio is 1 and by calculating the T values for other values of the ratio. The error can be evaluated by calculating the value of T, using the proper ET ratio, then calculating T for an ET ratio of 1. The relative difference between the two temperatures is the error. As usual, the expression can be formulated in terms of the dimensionless variable x. The solution, however, is a function of both x1 and x2, not the ratio or the sum.

The analysis for finite spectral bands proceeds in almost the same way as for the monochromatic analysis. The radiation received in each of the spectral bands is a combination of the measured spot and any background in the field of view. Now, however, for the radiances, one must write integral expressions. Thus, for a spot of area A

A [ E T ( A ) A - ~ ( ~ " - l ) - ' dA + ( 1 - A )

A f' E T ( A ) A - ~ ( ~ " - 1)-' dA + ( 1 - A ) ' E T ( A ) A - ~ ( ~ ' - I ) - ' dA '

& ~ ( h ) A - ~ ( e ' - l ) - ' dA V = I,,

Ah2 I,,, (5.77)

where

V = voltage ratio Ah I = first spectral band Ah2 = second spectral band

XI = C 2 / A / TI XI = CI/A/TZ

It is assumed that a quantum detector is used so that photon flux is expressed. The emissivity & ( A ) and the transmittance can be removed from the integrals by the use of effective emissivities as follows:

[ ~ ( h ) A - ~ ( e " - l ) - ' dA

[ K4(e" - l ) - ' dh '

- &T = ( &T),R = (5.78)

In general, these effective or weighted average emissivities are functions of the spectral band, the temperature, and of course & ( A ) . In this case, we can

RADIOMETRIC TEMPERATURE MEASUREMENTS 243

include the transmission E T ( A ) . The rest of the procedure involves appropriate regions of the spectrum, in which the transmission is the same in both regions. the detailed calculations cannot be included here, but it can be shown as a representative example that temperatures of about 500°K can be measured to within 5% to 10% through the atmosphere by using the spectral region from 3 pm to 15 Figures 12 and 13 show some examples.

5.5.4. Ratio Temperature Difference

In some applications of scanning radiometers, the temperature difference from one point in space to another is the main concern. Medical thermography is one important application. The clinician who, for example, is screening women for the presence of breast cancer, is interested in regions of increased temperature, an almost sure sign of pathology. He is not necessarily interested in a variation in skin emissivity. In fact, medical thermography to date has used radiation patterns as a diagnostic tool usually with the tacit or explicit assumption that the emissivity is constant-or even

f V U .- > 0 3 -

2 - 0 -

0 2 -

01 -

Emissivity- transrni?lance ratio

FIG. 12. Error in temperature measurement by a two-color technique as a function of the ernissivity-transmittance ratio for different ratios of background area.


0 3

0

a 0 5 I

Background temperature 0 s percentape of cargef temperalure Mortmum error occurs when Dockground I S 7 5 % of torpet

FIG. 13. Error in temperature measurement by a two-color technique as a function of percent of background area for different emissivity-transmittance ratios.

1. The technique of using the ratio of radiation in two spectral bands can be used to measure temperature differences over the surface.33

To emphasize the main point and to avoid getting lost in the details, we shall use the calculations based on very narrow bands and shall ignore the contributions of the background. Then the voltage ratio from the two spectral measurements at any scanned point is given by (where responsivities have been cancelled)

(5 .79)

where

c2

A~ T’

c2

2 - ~ Z ~

x, =-

x --.

RADIOMETRIC TEMPERATURE MEASUREMENTS 245

The difference in this ratio when the system looks at two different spatial points a and b can be written

(5.80)

etc.

In this difference expression, A l and A 2 are both known. If the emissivity ratio E ~ / E ~ is the same for both spatial positions, then one has

(5.81)

Only T, and Tb are unknown. For the temperature and wavelengths peculiar to medical thermographic

applications, ex >> 1, so that the 1s can be dropped with only a few percent error. Then

The temperature difference can be found uniquely, provided that the ratio of emissivities is spatially invariant. Dereniak has described this technique in more detail.34

It seems important to emphasize the difference between two measurement situations. The ratio of radiances can be used to measure temperature if the body is gray-or at least if the emissivity is the same in two different spectral bands. However, the ratio of radiances can be used to measure the difference in temperature of two spatially separated points if the ratio of emissivity in the two spectral bands is spatially invariant-even if the emissivities are different in the two spectral bands.

How does one choose the spectral bands? They should be narrow enough that the theory applies, broad enough to get satisfactory signal-to-noise ratio, far enough apart to reduce the errors, and close enough together that the emissivities are highly correlated.


5.6. Radiometric Instruments

Devices used to measure the amount of radiant flux are generally called radiometers, but there are radiometers of various sorts and kinds, and even with different names. This section describes some of these different types and their salient characteristics. Devices called radiometers by most people measure the amount of energetic flux in a relatively wide spectral band and with a flat response. Photometers measure the flux in the visible spectral region with a response that matches that of the eye. Pyrometers are devices which measure the temperature of relatively hot bodies by virtue of some radiation property of the body. Pyrheliometers measure the radiation from the sun. Some measure incident flux, and others measure net flux. Radio- meters which have a very narrow spectral bandpass are generally called spectroradiometers, although some are also inconsistently called spec- trophotometers. In addition to these classifications, which are generally spectral in nature, radiometers can be classified as to whether they measure the spatial and/or temporal distribution of radiation. Spatial radiometers can be line scanners or can scan in two dimensions, creating the information necessary for constructing an image. Whether an image is created or not, these are classified as imaging radiometers. Another distinction which can be made is whether the instrument measures the average or dc value of the flux or whether it measures the high-frequency fluctuations. The latter is often called a temporal or fluctuation radiometer. The most recent types of radiometers are those which detect coherent radiation, or at least radiation which is in some way coherent with a specified source, and those which might be called power and energy meters, which are spectrally uniform and relatively slow.

5.6.1. "Standard" Radiometers

These devices, which collect all the flux over a designated and relatively wide spectral band, are designed with or without radiation choppers. The radiation is incident upon the entrance aperture and brought to a focus at the position of a chopper if there is one. Good design practice then calls for a lens or mirror to collect all the radiation and put it onto the detector by imaging the entrance pupil on the detector, a so-called field lens. This makes the detector an exit pupil and the chopper the field stop. The chopper can be either absorbing (black) or reflective. If the latter, then the surface should shine a reference source onto the detector; if the chopper is absorbing, then the radiation reference level of the radiometer is that of the chopper. The detector senses the radiation level on its surface, first that of the field and then that of the chopper or reference. Two main electronic

R A D I O M E T R I C I N S T R U M E N T S 247

processing schemes can be employed. One is to synchronize a rectifier with the chopper and record only the signals that occur during positive periods. Another is to record the difference between the reference signal and the signal from the field. The second scheme appears to have the advantage of an internal reference and no need for external calibration. In most cases, however, the internal reference can only serve as a transfer standard. These chopper radiometers are by far the most frequently occurring instruments for general laboratory measurements. They usually incorporate either a thermistor bolometer or thermopile as their sensing element. Their noise- equivalent flux density can be estimated as described below.

The signal-to-noise ratio (SNR) can be written in terms of the specific detectivity D* of the detector of area Ad and the power Pd on its surface:

D*Pd dA I,, m . S N R = (5.83)

Both D* and Pd are in general spectral-varying quantities, but for a good radiometer, the D* will be quite constant over the spectral band AA. The power on the detector is related to the irradiance E at the entrance pupil as described before:

Pd = TOEA,. (5.84)

where A,, is the area of the entrance pupil. Then the noise equivalent irradiance or noise equivalent flux density (NEFD) is given by

E m NEFD =---. s N R - D* T O ~ O

(5.85)

Values of D* for the thermal detectors used in most instruments range from lo8 to109 cm Hzl” W-’. , th ey are typically 0.1 cm on a side; and 7,) is about 50%. Therefore, the NEFD is about 5 x lo-“’ W cm-* for a 1-Hz bandwidth and 1-cm2 entrance pupil. The time constants are about 1 ms. A typical radiometer of this type is shown in Fig. 14. The operation is as follows.75 Light from a distant source is focused onto the detector through the chopper mirror, filter, and field stop. The detector views either the field of view when the open part of the butterfly-shaped chopping mirror is in position or the light from the blackbody which is brought to the detector in the same cone as the main radiation by the conformal mirror. The position of the chopper is monitored by a small detector which views the light from the lamp. In this instrument, the “back” side of the chopper is used. A sighting scope, a light stop, and rudimentary baffle are also shown.

OPTICAL DIAGRAM PRIMARY

FIELD STOP

CAVl T Y BLACKBODY

PREC I S ION APERTURE , n

CONFORMAL M I R R O R

F I L T E R

F I E L D STOP

DETECTOR

PELTIER COOLER OR HEATER

CHOPPER MOTOR

VIEWING

RETICLE

OPT1 CS

REFLECTIVE CHOPPER BLADE SYNCHRONOUS

PIC KOFF

L A M P

SECONDARY M I R R O R

c F C C E +

LLIGHT STOP

FIG. 14. The schematic of a typical chopper radiometer

RADIOMETRIC INSTRUMENTS 249

5.6.2. Astronomical Radiometers

Radiometers of this general type have also been used in infrared astronomy.36 Larger apertures, long observation times, and laboratory-like installations characterize their configurations and environments. The iadi- ation from the atmosphere has usually been the limiting noise and, as is so often the case, the problem was seeing the small source in a large background. The technique employed by most was the wobbling secondary. The secondary of a Cassegrain system, for example, is tilted back and forth so that the instrumental field of view included alternately the star or planet under observation plus some sky, and then only the sky. If the sky is sufficiently uniform, then this background-cancellation technique works well. Most of the detectors for these applications are described by their noise equivalent power (NEP) values, because they do not follow a square- root-of-area law. The ratio of the radiometer NEFD to that of the detector

FIG. 15. Optical train of the balloon-experiment: 1 m primary ( I ) ; wobbling secondary (2); field stop (3); off-axis mirrors (4,6); lamellar grating (5); bolometer (7); instrument box (8).


is Ad/rOAc. Since NEFD values are typically lo-' and detector areas are about 0.01 cm, it is easy to see how NEFD values of W cm-* are obtainable with reasonable instruments. Better (lower) detector values, larger entrance pupils, and smaller detectors can move these values down another decade or two with great care. Then the limitations are radiation fluctuations, emissions from the optics, and inhomogeneities in the atmosphere. Figure 15 shows one of these astronomical radiometers.

5.6.3. Space-defense Radiometers

Infrared radiometers have been used as part of missile detection systems. They have been at least proposed for detecting the radiation from rockets in their powered or boost phase, in their midcourse or coast phase, and when they reenter the atmosphere. The radiometric aspects of each of these applications are considered below.

When the rocket is in its boost phase, it emits very large amounts of radiation coming from a point (or subresolution) source. The problem is to design an instrument that can cover a large area all the time and still be quite sensitive. Because it keeps a selected area of the earth continuously in its field, a synchronous satellite is a very desirable platform. A field of view of about 5000 km2 is also desirable and probably not excessive or unreasonable. The signal-to-noise ratio can be calculated using the same equation for SNR. The problem is one of detection rather than measurement, and so the most sensitive detector will be used. Further, one can expect most of the radiation to be in the emission bands of water at 2.7 pm and of carbon dioxide at 4.3 pm (because these are the main combustion products of hydrocarbon fuels). Detectors for these regions can have D* values of l o i 2 to l oL3 andI2 are constant over the relatively narrow spectral bands. Thus,

SNR = l O I 3

Since this is a narrow band, the power can be written

Pd dA = Pd AA = roPA AA.

Then the NEP of the system is given by

(5.86)

(5.87)

NEP = low3 m / r o [ WI. (5.88)

The spectral NEP is

NEP= 1 0 - ' 3 m / A A 7 0 [ W pm-'I . (5.89)

RADIOMETRIC INSTRUMENTS 25 1

The power on the entrance pupil can be found in terms of the target radiance L,, even though this is a subresolution target. In the spectral band AA, it is

A P = L,A, 7 = L,A,R.

d (5.90)

Here L,A, represents the watts per steradian radiated on the average from the entire target, and L? is the solid angle that the entrance pupil subtends at the target. The range is approximately 33 x lo6 m, so that for the square meter entrance pupil, the power is

P = L,A, x 10-15. (5.91)

Thus the signal-to-noise ratio is

(5.92)

If the optical transmission is 50%, the detector area is (50 pm)', and the bandwidth is lo6 Hz, then a target intensity of lo3 W sr-' gives a signal-to- noise ratio of 1. The system must be designed so that photon noise is low enough for this D* of l O I 3 to be attainable. The problem now becomes one of scanning and numbers of detectors. The smaller the instantaneous field of view, the better the chance of seeing the difference between the target and background. The faster the scan rate, the more looks obtained but the larger the bandwidth. The smaller the detector, the better the signal-to-noise ratio, and (other things being equal) the smaller the instantaneous field and the bigger the bandwidth required. Suffice it to say that these radiometer designs use many detectors and various ingenious optical and scanning systems. They can also take on a variety of forms, because there are so many parameters that can be varied.

The midcourse detection problem is a different one. Here we can assume that there is no thruster power; the only signal is that from the ballistic projectile. Its radiant intensity is then calculable if certain assumptions are made. The radiant intensity in the forward hemisphere is &ALBB(A, T). Its emissivity is between 0 and 1. Thus an estimate of 0.5 is reasonable. A better estimate reiuires knowledge of the material, its state of aging, exposure, etc. For estimation purposes 0.5 is pretty good. The projected areas of most missiles and satellites range from 10m2 to 1OOOm' but are mostly about 100 m2. They sit on the launch pad, where they come into equilibrium with their surroundings at about 300" K. So a midcourse object has the intensity of a 300" K blackbody with an emissivity-area product of (cross section) 50 m2. It peaks at 9.6 pm, where its spectral radiant intensity is approximately 350 W sr-' pm-'. The SNR is again given by the same equation. The power


on the detector is obtained from

(5.93)

where

T, = optics transmission (0.5) EALT-’ =target radiant intensity (350 W sr-’ km-’)

A,/ R 2 = solid angle of sensor at the target

The noise equivalent range can be calculated by assuming Ad = 50 km, B = lo6 Hz, and by setting S N R equal to 1. Then the noise equivalent range is

NER = 6 x 105AA12.

The units are those of A;/’. If A;’’ is 10cm for instance, then the noise equivalent range for these conditions is 60 km. We can see that this is a more difficult problem that should be solved by the use of a larger aperture and more detectors.

The radiometric problems of reentry are trivial. The distances are hundreds of kilometers, and the bodies have become very hot from aerodynamic friction. Clearly the calculations above show that detection is easy. The problems relate to search rates and discrimination. The multicolor techniques described above may be appropriate.

5.6.4. Scanning and Imaging Radiometers

Instruments which belong to this category utilize a mirror or mechanical mechanism to cause the image of the detector to scan a larger field of view. The detector angular subtense is called the instantaneous field of view; the larger angular subtense is the total field of view. These devices come in single- and multiple-detector versions and have been used for medical investigations, nondestructive testing, industrial inspection, and for military reconnaissance and night-vision applications.

Imaging radiometers provide a spatial map of the flux distribution over an area. The usual procedure is to couple the detector to the electronics through a capacitor. In this way, only changes are recorded; these changes, as with the standard radiometer using synchronous rectification, are then coupled to an artificial average level or pedestal. Much of the technology of modern television is applicable to the design of these systems. The equations for signal-to-noise ratio and related measures are derived here. As usual, the starting place is the SNR equation. Radiation from extended


targets is measured; the detector power is therefore given by (for a distant scene)

(5.94)

where 70 , is the optics transmission, L is the radiance, Ad is the area of the detector, and A is the area of the optics.

If the optical aperture is circular, then

SNR = T ~ ~ ~ L D * dhrD~AB/4f2B”’, (5.95) I where

Do = optics diameter f/ Do = focal ratio

Estimates can be made for the performance of such a system, based on values chosen for the optics and detector. The instruments are AC-coupled; they measure difference in signal level from point40 point in the field. Thus the important parameter for assessing their performance is the change in radiance that is equal to detector noise. This is the change in signal-to-noise ratio, and it can be written as

D A9 4f B1/’ *

r,rOD* A L d h r -

If r,, T ~ , and D* are constant over the spectral band, then

D A0 ASNR= ToTaD* AL r-

4fB1l2‘

(5.96)

(5.97)

This is a relatively easy (simplified) equation with which the performance can be evaluated. Typically, To = 0.3, = 0.8, D* = 10”. Then

DAB h s N R - 2 ~ 10’OAL-

fB’ /*‘

The noise-equivalent radiance difference is ALISNR:

fB1” NEAL= 5~ 10-94 -

d AB’

(5.98)

(5.99)

Although there are many nuances in the design of these imaging radiometers, the main factors are displayed in this equation: make the Flnumber as small as possible and use a minimum bandwidth. The limits of performance are based on what the optics and the detector can do. The Rayleigh criterion


indicates that DAB is approximately equal to the wavelength of the center of the spectral band. The D* of the detector, when the detector is photon- limited, is given by’*

(5.100)

where

g = 2 or 4 for a photovoltaic or photoco,nductive detector 7 = detector quantum efficiency

When the system is limited by photon noise, the SNR is independent of the focal ratio of the optics. This condition is often met by systems which view the ambient background of the earth. The value is about lo”, as given above. Therefore, an approximate value of the performance of such systems is

NEAL = 5 x 10-6B1’2. (5.101)

The bandwidth can be calculated on the basis that there are about 500 lines in a TV frame and the frame rate is 30 sec-’. Thus there are 7 x 10 picture elements scanned every second. If only a single detector is used, it must have a time constant less than 1 ps, and the NEAL will be about lo-* W cm-’ sr-’. Such a radiance change corresponds to a change of about 10°K or 10% in emissivity, neither of which is very interesting for most applications. Most such instruments therefore use 100 or more detectors to reduce the bandwidth (or signal average in some other way). Since the bandwidth can be reduced in proportion to the number of detectors used, the NEAL is proportional to the square root of the number,

An important question to be asked about such scanning radiometers is how, and how accurately, can they be calibrated. The calibration schemes vary, based on the intended use. Almost all are ac-coupled instruments; in temporal frequency space, there is a low-frequency cuton determined generally by an RC pi network in series with an amplifier. This means that the average or dc value is lost. It must be returned as a pedestal level on which small variations ride, and this combined signal is recorded on a cathode ray tube (CRT), fi1m;magnetic tape, or other appropriate medium. If there is little concern about the true radiometric level, then averaging the signal level over some appropriate length of time (e.g., one line or one frame) provides an excellent reference. If the scene does not change much, then the small signals are displayed well and do not get lost in one end of the dynamic range. If, on the other hand, one is really interested in measuring radiometrically, then on-board calibration is required. Probably the simplest

RADlOM ETRIC INSTRUMENTS 255

way to do this is with a small source that periodically floods the detector (array). If this source is not collimated, it will not illuminate the detector in the same way as the distant objects, but it can be calibrated. If the source does not shine through all the optical elements, then its calibration gradually becomes invalid as the elements deteriorate. Obviously this source should be stable or measurable. The most accurate way seems to be to use a small heated cavity at the focus of a well-protected collimating system. The resultant beam shines at’the detector system through all the optical elements from just outside the active field of view. Ideally, two or more such sources are used, each at a different temperature and with their temperatures monitored. Diode emitters are a reasonable second choice. They are not as good, because they do not match the spectral input and because the stability of infrared diodes (8-14 pm) has not yet been established. In the visible, a stable silicon diode could be used, but any spectrally selective deterioration of optical elements would alter the calibration of the instrument.

There are several ways to calibrate these instruments in the laboratory and in the field. The usual specifications of performance are the NETD and the MRT-the noise-equivalent temperature difference and the minimum resolvable temperature. The first of these is defined as the difference in temperature between two blackbodies which both fill the field of view of the radiometric scanner that gives a peak-to-peak signal difference equal to the rms noise. The second is the minimum temperature difference that can be resolved by an observer at a specified spatial frequency. To make the first measurement, one needs two large-area blackbodies (almost a contradiction in terms) that can be carefully controlled in temperature. The usual sources are large blackened tanks of water with stirrers and thermometers of some type. They are set so that they are different in temperature by some 10°K or 20°K and so that a large SNR is obtained. The temperature difference is reduced until the signal and noise are about equal. A curve of ATISNR is plotted. Each point is a measure of the NETD-almost. Two types of errors occur: one because the sources are blackbodies, and one because they are not. Let us see how this comes about. The difference in signal-to-noise ratio is given by

ASNRKA 7(h)D*(A)L(A, T ) dA. (5.102)

The terms that would make the proportionality an equality are all geometric and temporal bandwidth factors. The ~ ( h ) factor is the product of the spectral transmission of the optics and the atmosphere. The scanning radiometer views first one body and then the other. It is assumed that they are close together so that there is no difference in atmospheric transmission,


optical transmission, or D* between the two measurements. Thus,

T(A)D*(A) AL(A, T ) dA. (5.103)

Only the radiance changes. Let us consider only that term. The radiance projected toward the sensor is the sum of the radiation emitted from the source and the background radiation reflected from the source to the sensor.

AL(A, T ) = A [ & ( A ) L R B ( A , T)+pbE(A, T ) ] , (5.104)

L - E + - - d T + A ( y ) L a T A L AE 1 a L

(5.105) -_ AE xex AT E PbAE - + - - + - A p b + - .

E ex-1 T L L

The first term is typically about 1 %, a change in emissivity from about 0.98 to 0.97 of these near-blackbody sources. The second term is about 5 AT/ T or 0.017 AT The next two terms represent background contributions. In fact, each would really be the sum of all the variegated components of the background. The first one is the contribution of the variation in reflectivity with a constant background. The second one represents the contributions of different background terms if the reflection geometry with the two samples is slightly different. These effects have been considered in more detail.I2 The analytical results there show that errors of several percent are easily attained. One message is clear: the radiometric differences of such test objects should be measured radiometrically.

When really large patterns for field work are to be used, the use of water baths and other actively heated structures gets cumbersome. Bartell” has described a test pattern design that can be used for NETD and MRTD measurements. As Lloyd38 shows, an MRTD test involves several important psychophysical factors. These include the temporal and spatial integration properties of the eye and its tendency to be a matched filter. Accordingly, the test pattern must have groups of four bars with a 7 to 1 aspect ratio. If one can accept that the effects of the psychophysical factors can be calculated, then simpler means, including edge gradients and line spread functions, are a~ailable.~’

5.6.5. Spectroradiometers

These devices differ from standard radiometers principally in that they use some technique for selecting narrow portion of the spectrum, and only radiation in that band is measured. The radiometric calculations of the


responsivity and SNR are in principle identical to those of the standard radiometers. The user must pay particular attention to the definition of the selected spectral band (the passband) and to the rejection of the rest.

The passband is usually characterized by the spectral width between the wavelengths at which the transmission is one half maximum. Older literature generally calls this the half-width, while most modern writers use the phrase “full width at half maximum.” The symbol is Ah. Sometimes this spectral band is normalized to the central wavelength Ao, and the reciprocal of the ratio is the resolving power RP.

- I RP=l--l AA - I .

There are several direct methods for filtering, and there are some that might be called exotic or indirect. Certainly the use of a prism, grating, and filter must be considered direct and straightforward-even if they are not always simple.

5.6.6. Filter Radiometers

The simplest filter in a conceptual sense is surely the absorption filter. One can define a passband by the transmission cuton of a material like germanium and the cutoff by the detector spectral response roll-off. Many combinations exist, especially with the advent of mixed crystal materials. For example, one could pick a detector made of either HgCdTe, PbSnTe, or InAsSb and obtain a sharp long-wavelength cutoff which occurs anywhere from 1 pm to 15 km. One could combine this with absorption filters made of germanium, silicon, zinc selenide, sodium chloride, fused silica, potassium bromide, and others. The combinations, although not endless, are certainly too large to enumerate here. The user should be aware of the possibilities. One should be aware, too, that the spectral response must be calibrated at the temperature of use. All of these materials have transmission characteristics that are strong functions of temperature.

In many instruments, interference filters are utilized either to define the passband or to provide an antireflective coating on other optical elements. These filters or coatings should usually be put in a collimated beam, because the essence of their operation is the interference of forward and backward light waves. This, in turn, depends upon the optical phase difference k n d cos8. For a given wavelength (and refractive index), the phase difference and therefore conditions for maximum and minimum of the filter transmission depend upon the cosine of the angle between the normal and the direction of the ray. If the beam is not collimated, the filter transmission curve is a weighted average over all these angles. For an F / 1 cone of light,


FIG. 16. Rays on a thin film illustrating the reflection from the first surface plus two from the second.

for instance, there is a phase shift variation with angle of 50% between central and marginal rays. This is equivalent to moving the central wavelength from 2 pm to 1 pm or from 10 pm to 5 pm. When the filter is used in a collimated beam, it can be tipped to change the passband position; such a procedure entails somewhat complicated polarization effects which must be calibrated. An interference filter in general has what might be called side lobes or higher-order passbands. For the simplest case of obtaining good transmission at A o , the total reflected radiation should destructively interfere. The first ray has no phase shift. The once-reflected ray undergoes a phase shift given by 2knd cos 0 + rr. The additive rr comes from the phase shift which arises from reflection when light goes from a dense to a rare material as it exits from the film. The third ray which undergoes two reflections is just twice this, etc., which is shown schematically in Fig. 16. For destructive interference, each of these rays should have a phase shift with the wave reflected from the front surface that is exactly rr, or an odd multiple of T. The design equation then is

mA 4

ndcos @=--, even m. (5.106)

Although more complicated designs exist, they generally have the characteristics that a variety of combinations of m and A satisfy the conditions for maximum t r ansmis~ ion .~~ For example, a filter designed for one-quarter wavelength thickness at 10 pm will have a peak at 10 pm, 5 pm, 2.5 pm, etc. The radiation in these higher-frequency passband regions is usually absorbed by the substrate. Caution should be exercised by the investigator when using such a filter in that this absorption can be highly temperature- dependent-the substrate is an absorption filter and has the properties described above.

Radiometers often use these filters in a fixed position or in a filter wheel, a set of different filters which can be introduced into an appropriate part

RADlOM ETRIC INSTRUMENTS 259

of the optical train one by one. The device is usually circular (a wheel) but can take on other forms, linear or cylindrical for example.

One very useful device for defining a passband may be thought of as a continuously varying interference filter; its most popular form is the circularly variable interference filter or CVF.4' Its center wavelength (or frequency) is determined by position on the filter, which consists of a circular substrate on which are evaporated interference layers whose thicknesses vary with circumferential position around the circle. The calibrations, cau- tions, and limitations which apply to interference filters apply also to CVFs. In addition, positional calibration must be considered. The bandpass is an average over the area of the filter that is viewed, and the center wavelength is a function of the position of that area-which is usually a slit. Half of a CVF typically covers one octave of the spectrum, from A,, to 2Ao. Thus, one generally has a bandpass of about A,/ 180". The RP is 180 per degree of the CVF that is viewed.

The concepts of throughput and the optical invariant are required for further useful discussions and comparisons of these instruments. The optical invariant when applied to pupil and image planes is just that ny sin 6 is a constant, where n is the refractive index of the medium, y is the height of the image or pupil, and 6 is the angle the next image or pupil subtends at the one being considered, as shown in Fig. 17. The throughput is the square of this optical invariant.

Suppose that an optical system with area A. and focal length f focuses radiation onto a CVF over a width x and length y (as defined by a mask or slit in front of the CVF). Then the throughput is Aoxy / f2 . If the slit extends from r / 2 to r (where r is the radius of the CVF) and is x wide at the circumference of the CVF, then the slit area is ry/4. The resolving power is 47r/x (using the average width of the slit). Therefore, one increases the resolving power at the expense of throughput, which is directly related to flux. There is a linear and inverse relationship between these two. The loss in throughput resulting from reduction of slit width can be compensated for by making the slit longer, thereby requiring the radius of the CVF to be larger.

FIG. 17. The Laprange invariant for pupils. The product of n , y , sin 0, is equal to nzy , sin 0 2 . In this example, n , = n z .


This problem can be avoided to a first approximation by using the individual absorption or interference filters just discussed. It has already been pointed out that an interference filter should be used in a collimated beam to obtain a narrower spectral band. If the filter is made the size of the entrance pupil and placed there, all is well, but economics drives one toward using smaller filters in smaller collimated beams. Then the optical invariant says that rays will diverge more or less as the size of the beam is decreased or increased. Over relatively small angles, and the operation region of most spectroradiometers, there is a linear proportionality between angle and diameter and a cosine relationship between resolving power and the linear dimensions of the aperture. If, however, an absorption filter is used, the relationship does not apply. Systems which use absorption filters are generally so crude spectrally that resolving power is hardly meaningful.

5.6.7. Prism Disperser Radiometer

Prism systems can have resolving powers considerably better than the filter systems discussed above. A collimated beam is directed toward a prism which has length 1, width w, and base b, as shown in Fig. 18. The prism refracts the light at different angles according to the value of the refractive index at each wavelength. The beam is focused to a plane where a single slit or many can be placed to select radiation of different frequencies.

Its resolving power is

The linear dispersion dx/dA

A dn AA dA

RP=-=b-.

of a prism is given by42

-g=f (&. b dn

(5.107)

(5.108)

The throughput Th can be calculated as the area of the beam on the prism times the area of the slit divided by the square of the focal length:

h dx Th=(hwcos 0)-.

f’ (5.109)

If the above expression is used for the linear dispersion dx/dA, then

dA Th = RP h2 cos 8 - = the throughput. (5.110) f

RADlOM ETRlC INSTRUMENTS 26 1

,

FIG. 18. Prism geometry.

5.6.8. Grating Disperser Radiometer

Grating systems make use of the interference of many diffracted waves arising from the n slits of the grating. The grating equation can be written

mh = d(sin 4 +sin 0), (5.111)

where

m = order of the maxima d = line spacing 4 = angle of incidence 0 = angle of diffraction

The resolving power is obtained by differentiating with respect to A :

m = d cos 0 (s) d0 dD m dh dh d cos 0

d cos d D dh = m

- -

A d(sin 4+s in 8 )

dh d cos 0 d D RP=---= (5.1 12)

5.6.9. lnterferometric Radiometer43

A third major type of spectroradiometer is the interferometer. This is an indirect device which first forms the Fourier transform of the radiation; the


spectrum is obtained by the use of a computer to transform the interferogram. It is appropriate to review the basic operation of an interferometer to obtain some of its radiometric characteristics. A simple Michelson interferometer is taken as the prototype. When quasi-monochromatic light illuminates the instrument, the center of the focal plane will have a maximum when there is no phase difference between the two arms. If one arm is moved at a constant rate, the phase difference changes at a constant rate, thereby causing a sinusoidal modulation on a detector placed at the center of the focal plane. If a second monochromatic beam of different frequency and different wavelength is also part of the input radiation, its frequency of modulation at the output will be different, and by superposition the output will be the sum of these two signals. The output flux density for light of a single wavelength (traversing paths a and b ) is given by

E = uu* = u, &‘‘ + u,, & ( “ “ + ~ ) 1 2 , (5.113)

E = E , + E , + 2 E R b c o s -ndcos8 . (5.1 14)

The optical path difference nd cos 8 is just the difference in the length of the two arms, assuming that a compensator plate has been inserted in the horizontal arm. The length of the horizontal arm then is

[: 1 l = t u ( t ) + I o .

The output flux density for a monochromatic input is given by

(5.115)

For a polychromatic input, the output intensity as a function of time (where it is assumed the beamsplitter is 50% so that E, = Eb) is

E ( t ) = 2 EA ( t ) 4- EA COS[ ( t u ( t ) -4- 1.3)- I (5.1 16)

If EA is a constant, then E ( t ) and EA are cosine transforms. This means that the spectrum of the radiation can be found this way as long as the intensity of any spectral component remains constant. The effects of changes in intensity with time are interesting, but their discussion will be postponed until the resolving power and resolution characteristics have been described. The resolution of an interferometer of this type is usually specified in wave numbers and is equal to the reciprocal of the total path difference. The shape of the bandpass is a sinc function, and the bandwidth is specified as the interval between the first zeroes. Many systems use electronic processing to alter this sinc-shaped spectral band in order to reduce the contributions

MEASUREMENTS 263

from the side lobes. The resolving power can be written as nd cos B / A ; it is the optical path difference measured in wavelengths.

The throughput of an interferometer like the Michelson is the area of the mirrors times the area of the detector divided by the square of the focal length of the final beam. If a Twyman-G~een~~ version is used, it is easy to see that this is (2.44A/D)*(.rrD2/4)=A2.

The use of an interferometer like the Michelson for radiometric purposes requires some special considerations. The most important of these are that there normally is no dc reference available, and that the source intensity must remain constant during the time of the scan. The lack of a dc reference, of course, means that the spectrum can be obtained, but only on a relative basis. A fairly obvious solution to this is to measure the total radiation with an auxiliary radiometer and use it to set the radiation scale. Variations of the source output intensity generate much more subtle results. It can be seen that the spectrum is obtained by a cosine transform:

co

Z cos(kvr) dr. /-. If the intensity varies, then the interferogram is affected according to the nature of that variation, and the effect ofthis variation is different in different parts of the radiation spectrum.

5.7. Measurements

Radiometric and photometric measurements are not difficult in principle. They are very hard to accomplish with a precision or accuracy better than about 1 YO. Most field measurements which are made with portable equipment and distant (in both space and time) from a primary calibration have errors of more like 20%. The measurement of the spectral radiant intensity of rocket plumes has been considered satisfactory when the error was less than an order of magnitude! These errors can usually be attributed to the fact that the measurement of the unknown was different from the calibration. Accordingly, rule 1 of radiometric measurement is to calibrate the radiometer in a configuration that is as similar to the measurement situation as possible.

The responsivity of a radiometer depends upon very many factors. It can be a function of wavelength position, angle, polarization, flux level, time, humidity, temperature, and many other things. Phase of the moon and the name of the operator have also been proposed as factors influencing the measurement. This great variability leads to rule 2, which is to think of everything.


These two rules are very general. The rest of this section is intended to make them a little more concrete by discussions of examples. The discussion starts with standards, the primordial element of every radiometric measurement.

5.7.1. Standards

Any radiometric measurement must be considered in the light of the calibration against known standards. It can be no better than the standard. The National Bureau of Standards (NBS) maintains and provides radiometric standards in the form of 1000 W DXW-type tungsten-halogen lamps. They are described in detail elsewhere.I2 The lamps are given a primary calibration and their irradiance is fit to the following curve:

E ( A , t ) = A - 5 e(o , -4 .7+~f+bf lA) (5.117)

The values of the constants ah, 6, and c are found from least squares fitting for different values of the wavelength A and time t . Four lamps are used and the proper 3a error found from among these, as shown in Table IV.45

Table IV summarizes the uncertainties assigned by NBS to these standards at different wavelengths.* Item l(a) is the error in primary calibrations; l (b) represents long-term reproducibility rather than error with respect to SI units. Item 2(a) represents the transfer to an irradiance scale, while 2(b) is the error in reproducibility of the four standard lamps. Item 2(c) represents the errors involved in the fitting procedure described above. Item 3 represents the error in transferring the value of the four standard lamps to the working standard being calibrated.

The total uncertainty with respect to SI units then is found from the rss of l(a), 2(b), 2(c), and 3. Reproducibility is found from l(b), 2(a), 2(b), 2(c), and 3. Note that these are all three standard deviation errors.

It should be observed that the errors increase rapidly in going toward 250nm largely because the flux falls off so rapidly. It is enlightening to calculate the ratio of the total 2800°K blackbody to the amount of radiation at each wavelength (the approximate temperature of these sources). These sources are limited in their spectral extent at the short wavelength end by lack of flux and at the longer wavelength end by the absorption of their glass or quartz. These are the certified standards available from NBS. Instructions for their use come with them and have been published elsewhere.'*

A very encouraging development in radiometric standards is that of the electrically calibrated radiometer ( ECR).46 The receiver is blackened and

* I am indebted to Karl Kessler for providing a detailed explanation of this table.

TABLE IV. NBS Radiometric Standard Uncertainties (%)

1. NBS Spectral Radiance Scale

(a) Absolute error (with respect to SI units) (b) NBS long term reproducibility

2. NBS Spectral lrradiance Scale (a) Systematic errors (b) Random errors (estimated reproducibility of the mean

(c) Model error of standards I , - I,, 3 - u)

3. Transfer Calibration of a Test Lamp (a) Random errors (3u precision)

4. Uncertainty of Reported Values (a) With respect to SI units (b) With respect to NBS (long term reproducibility)

250 350 450 555 nm nm nm nm 1.66 1.20 0.93 0.65 1.06 0.76 0.61 0.30

0.16 0.16 0.10 0.10 0.72 0.55 0.53 0.56

1.35 0.80 0.73 0.73

1.35 0.65 0.50 0.48

654.6 nm 0.65 0.26

0.08 0.54

0.78

0.41

800 1300 1600 nm nm nm 0.46 0.28 0.27 0.22 0.12 0.18

3 rn

0.08 0.07 0.07 c 0.51 0.80 0.80

I 0.76 0.77 0.82

R

4 v1

0.51 0.39 0.36

2.6 1.7 1.4 1.2 1.2 1.2 1.2 1.2 2.3 1.4 1.2 1.1 1.1 1.1 1.2 1.2


shaped to be as black as possible; its surface also contains windings of heating coils. This cavity interacts optically with a detector (a pyroelectric one in the current realization). The detector senses first the cavity which is heated by the radiation from the field of view and then from the heating coils. The development of such an ECR as a standard with accuracy of 1 Yo requires excruciating care and attention. It appears that this receiver will become the standard over a wide spectral range because it is more accurate than the current source standards and because it can cover a wider spectral band.

A more recently developed, more accurate standard has also been developed by G e i ~ t , ~ ' a device he calls a self-calibrating detector. Presently it is a silicon diode and is therefore useful only from about 0.3 pm to about 1 pm. He estimates inaccuracies of approximately 0.1%, more than an order of magnitude better than previous calibration capabilities. A silicon diode is used. First the reflectivity is measured by one of the methods described later. These are all basically ratio methods, so that inaccuracies of less than 0.1% can be attained. Then light illuminates the diode, which is used in a back-bias mode. The back-bias is gradually increased as the output is monitored. When there is no longer any increase in output with increase in back-bias, the internal quantum efficiency is 1. The external or total quantum efficiency is the product of 1 minus the reflectivity and the internal quantum efficiency. Careful attention to the design of the diode, especially its thickness and the geometry of the junction region, assures that there is very little error in the assumptions about the internal quantum efficiency reaching unity. This is a very exciting development in high-precision radiometry.

5.7.2. The Accuracy and Precision of Field Measurements

Experienced radiometrists will often state that a radiometric measurement made in the field (top of a mountain, floor of a desert, instrument bay of an airplane, or a satellite) is good if the error is 50% or less. This section is an attempt to explain why the errors are so large when the (present) NBS standards are about 2%.

Radiometric errors in the ultraviolet have been discussed in two articles in Optical Radiation N e ~ s . ~ ' , ~ ~ The discussion is given in the context of a spectroradiometric measurement in the 0.25-pm to 0.35-pm region. The errors in general are a function of the complexity of the radiation as well as the type, duration, and intervals of measurement. The uncertainties one should add to those of the standards described in Section 9.7.1 are discussed below.

MEASUREMENTS 267

If the radiant power is uniformly distributed with respect to angle, area, and polarization, if it does not vary much with wavelength, if it is in the middle of the dynamic range, and if the irradiance is about equal to the calibration irradiance, then the additional uncertainty is about 0.1 YO to 0.5%. If these conditions do not obtain, then the additional errors are as follows: position, direction, and responsivity variations add 1 ‘/o ; wavelength variation and wavelength calibration errors add 3% ; dynamic range, nonlinearities, 0.5% ; polarization responsivity, 0.5%. If the irradiance level is too low or the time of response of the instrument is inadequate to obtain a signal-to-noise ratio above 10, one should probably not make the measurement. Otherwise, important errors relating to random noise in the instrument will exist. The errors listed above, which apply to relatively good measurement conditions add in quadrature to values of 3.3%. Note that the angular and spectral contributions are dominant.

Among the contributions listed for unfavorable conditions of measurement are the following: uncorrected instrument temperature variations (0-lo%), old standards (2%), time interval since last calibration YO), variation in system alignment (0-5%).

The authors of the first article48 conclude, “thus under very favorable measurement conditions, the state of the art of spectral irradiance measurements in the 250-nm to 350-nm region varies from 3% to 9% . . . .” They emphasize that his can be obtained only with the best instruments and techniques. I believe one can make measurements with an uncertainty of 2% to 3% for well-behaved radiation and 3% to 5% for more complex radiation. This, of course, is not a major disagreement; my reasoning is as follows: the uncertainty in the irradiance standard is given as 2.4% at 0.25 pm. If the radiation is well behaved, an additional 0.1% to 0.5% is likely to be encountered with great care and excellent instrumentation. This gives a total (rms) error of 2% and 3%.

In the second article, the authors conclude that for performing terrestrial solar irradiance measurements in the UV-B region, an ultra low scatter, double monochromator with 1-nm spectral bandpass should be used, that the response should be deconvoluted, and that “it should be possible to limit the measurement uncertainty to about 15% at 295nm and to about 5% at 320 nm and longer wavelengths.” The errors they discuss are instructive. The first relates to the slit function or spectral response of the spectroradiometer. The output voltage (or current) V is given by

m

V = E, , (h)%!(h) dh. -m

(5.118)

If the overall responsivity takes into account the slit function as well as the


instrument responsivity by expressing % ( A ) as % r ( A ) a ( A - A o ) , where it is assumed that r is independent of A o , then

V = % I E,,(A)r(A)cT(A - A o ) dA. (5.119)

If the slit can be described accurately enough as a delta function, the integral unfolds immediately:

v = B E A ( A O ) r ( A O ) = % ( A O ) E A ( A O ) . (5.120)

The spectral response at A. can be found by the use of any sufficiently well-known standard source. In practice, of course, there are no delta functions. Slit functions peak and have long tails that are relatively wide. The long tails which reach longer wavelengths at values of to lo-' are the source of error in measurement-at least until they are corrected. The interesting features are the means by which such corrections can be made- and the limitations in making them. The slit function can be measured by the use of a device like the ECR and a source (like a laser) with a line width smaller than the spectral slit width. The spectral slit function can be taken as known at least to the resolution of the best laser in the spectral region of interest. But even though the function is known, the integral is not unfolded simply. The output voltage is given by

V = % I E , ( A ) r ( h ) a ( A - A o ) dh.

We can assume r ( A ) is constant over the band in order to arrive at the essence of the problem. Then

a-

--oo

a-

-m

m

V = % I E , ( A ) a ( A - A o ) dh. -m

(5.121)

It would appear that one cannot obtain E A ( A ) from this, because it is an unknown and arbitrary function. Many different distributions of EA (A ) for a given a ( A - Ao) can give the same value of V. But a spectrometer makes many measurements at different values of A o , obtaining different values of V. Thus, for known values of % ( A ) and o ( A - A i ) , the appropriate expression is

00

vi= I_, % ( A )EA (A - hi) dA. (5.122)

The more individual measurements that can be made, the more determined EA(A) will become. The problem is one of estimating E A ( A ) from a set of samples with known sampling shape. One can argue that a(A - A i ) can change as hi changes. This is inconsequential as long as o ( A - A i ) is known.

MEASUREMENTS 269

Another way to view this procedure is to recognize that the right side of the equation is a convolution. Therefore, the Fourier transform of the voltage is equal to the product of the transforms of the slit function and irradiance. The procedure for obtaining the irradiance is then straightforward but involves much calculation. It might be added in passing that the use of this approach to measuring spectra leads to the possibility that the smallest possible slit width is not the best. An interesting tradeoff between slit width, signal-to-noise, and processing procedures arises. Too wide a bandpass leads to processing errors which are too large; too small a slit width leads to a signal-to-noise ratio which is too low.

The authors of the NBS article then provide a figure which shows the true spectrum and the inferred one. The major portion of the error is at the short wavelengths where there is very little radiation and where the longer- wavelength scattered radiation predominates.

5.7.3. The Measurement of Emissivity

the properties of an ideal radiator. Now we can imagine a body in thermodynamic equilibrium at temperature T. The radiation it emits into a hemisphere is specified as M ( A ) and that of a blackbody MBB(A ). Then the hemispherical, spectral emissivity of a material & h ( A ) is given by

We have already

(5.123)

The spectral emissivity of the material may also be described in terms of photon radiation

(5.124)

The total hemispherical emissivities are given by

M aT4'

-- - Iorn MA

Eqh= [" Mz: dA u q 7-4

&h = Iom M :B dA

low M'-lA " M - --.

(5.126)

(5.127)

J o .


Any of these ratios is a function not only of the properties of the material but also of the properties of its surface. In general, a rougher surface causes the ratios to be higher and the emission to be more uniform.

In addition to spectral and total values of emissivity, there exist the values of emissivity averaged over some spectral band, i.e.,

& ( h ) M ” ” ( h , T ) dA/ k f ” ’ ( A , T ) dh. (5.128) I,, I t can be seen that E ~ , ~ ~ is the weighted average of the emissivity over the spectral band, where the weighting function is the radiant emittance of a blackbody at temperature T. Whereas the spectral emissivity is a function of temperature through the variation of material properties and the average emissivity is a function of temperature through the variation of material properties, the average emissivity also depends upon the temperature via the weighting function M ( h , T ) . These are the spectral categories of emissivity: spectral, total, weighted average, power, and quantum.

Emissivities can also be categorized in terms of the geometric relationships. We have already dealt with the hemispherical reflectivities, &h = M / M ~ ’ in their various spectral guises. Directional emissivities are important for samples which are not close to Lambertian. These emissivities are defined in terms of radiances. The spectral directional emissivity then is

It is true that a blackbody is Lambertian so that of 8, and 4.

(5.129)

LYB(A) is not a function

At one extreme of the measurements of emissivity is the total hemispherical emissivity. The essence of these measurements is the determination of the temperature of the sample and the total flux emitted by the sample. Three notable procedures are: the hot-filament method, the temperature- decay method, and the radiometric method.

In the first method, current is caused to flow through a long strip of the (metallic) material in question. Typical four-terminal measurements are made of the current and voltage in the strip. The temperature is measured by a thermocouple, a platinum resistance thermometer, or an optical pyrometer. This choice is made based on the temperature range and the personal preferences of the investigator. The inaccuracy of this method is

MEASUREMENTS 27 1

from 2% to 20%. The contributions to this total error include unaccounted flux losses by conduction along the wire, uncertainty in the temperature measurement (usually due to the fact that thermometer does not measure the temperature of the sample), uncertainty in the net flux transfer from the walls (because it is the net flux that counts), and any convection losses if the system is not evacuated.

In the temperature-decay method, a sample of small mass and large surface area is heated inside a cooled, evacuated chamber. It can be heated in any of the standard ways-resistively, inductively, optically-so that conducting or insulating samples can be measured. The temperature is measured as before, and the rate of cooling is noted. The inaccuracy of this method is also from 2% to 20% with most of the methods yielding about 5%. The errors are due to the same phenomena as before: temperature errors and nonradiative flux transfers.

The radiometric method utilizes a large specimen with a small detector next to it. Sometimes a reflector or light pipe is used to help define this hemisphere (or almost hemisphere). Calibration of the detector against a blackbody completes the technique. The uncertainty in this method is 5% to 10%.

Total normal emissivity, a special case of total directional emissivity, is also done in three different ways: emissometer, comparative radiometry, and by integrating spectral normal emissivity. The emissometer is an evacuated chamber which has cooled, blackened walls. The detector, a thermopile, views a cooled wall from the back and a sample through a KRS-5 window from the front. The field of view is a fairly large cone. The technique is comparison of the sample flux density to that from a reference-when the two are at the same temperature. This is more an inspection technique than a precise radiometric measurement. The method of comparative radiometry uses a black detector which views a blackbody at a known temperature and then views the sample. The sample temperature is measured as before. The sample can also be heated to any appropriate temperature, although there is no requirement for heating rate, temperature level, etc. The measurement is made at the temperature required. The measurement uncertainty depends upon the temperature required. The measurement uncertainty also depends upon the wavelength and temperature values involved. The radiation fields of the sample and the blackbody are being measured; all the caveats and techniques of field measurements apply. The final technique is integrating the spectral normal emissivity, although that is really the hard way. These measurements have an uncertainty ranging from 2% to 3% for high- temperature samples. The measurement of directional emissivity can be made by a geometrical adjustment of the comparative radiometric method for normal emissivity.


5.7.4. The Measurement of Reflectivity

The reflection of light from a sample is a function of the spectral interval and geometry and depends upon the nature of the source, the sample, and its surface. The spectral considerations are the same as for emissivity and will not be discussed here. The different geometric concepts apply for monochromatic, total, and band-limited radiation. These are the spectral, total, and weighted average values. The main difference between emission and reflection is that the latter is a function of the source geometry as well as the receiver geometry.

The well-known law of reflection states that an optically smooth surface reflects the light in the plane of incidence and at an angle equal to the angle of incidence. Thus, the ratio of the flux density in a collimated beam to the incident flux density also in a collimated beam is the specular reflectivity; it is a function of the angle of incidence for any material. If all the above conditions are met, then Fresnel's equations apply. Note that the specular reflectivities are functions of the state of polarization and the refractive index of the material.

If the reflected radiance is not a function of angle, i.e., if the reflected radiance is uniform, then the surface is said to be a uniform or Lambertian reflector. There is no such thing as a true Lambertian reflector which generates uniform reflected radiation, no matter what the angle of incidence. Some materials, like magnesium oxide, barium sulfate, and sodium chloride are Lambertian out to about 70" or 80" and for angles of incidence smaller than about 30". Many real objects like paint, sand, dust, and fabric are also Lambertian to the extent just described.

Any general surface consists of both specular and diffuse components. The reflectivity must be described in a more general way. J ~ d d ~ ~ has described nine different types of reflectivity. The beam can be collimated, conical, or hemispherical. The illuminating and collecting beams can be any of these. The combination comes to nine. The most general is conical- conical, and the others are limiting cases. In every case, the reflectivity is the ratio of two flux densities. A more useful concept is the bidirectional (or bistatic) reflectance distribution function (BRDF). This is a complete description of the radiance reflected in some direction 8, and 9, as a result of the flux density incident on the surface from some incident direction B i , 9i:

(5.130)

In most radiative transfer problems, the irradiance on a surface can be calculated from the source radiance and the geometry. The next step is to

MEASUREMENTS 273

use the reflected radiance as the source for the next transfer. The BRDF is just the right function for this.

Usually the spectral nature of the measurements is determined by the source and the detector. Only one system (Gier-Dunkle) uses a different method. The spectral region is determined by such difficulties as ambient fluxes, gradients, etc.

The main methods for the measurement of normal specular reflectivity are those of Strong” and Bennett and Koehler.’l These are illustrated in Figs. 19 and 20. In the Strong method, light shines into the sample area that has no sample. The beam reflects once from the folding mirror. Then the mirror is rotated to its sample position and the sample is inserted. Now the beam reflects once off the folding mirror and twice off the sample. The geometry of the apparatus prevents true normal-incidence measurements. They certainly are directional. Strong’s apparatus operated between 0.3 pm and 10 pm, with the spectral region limited primarily by the integrating sphere used for putting uniform flux on the detector. The inaccuracy is about 0.1%. The diagram of the Bennett-Koehler apparatus shows how it works. The beam with and without the sample travels along the following

FIG. 19. Sketch of the Strong-type reflectometer used in the visible. The sample reflectance R is obtained from the square-root of the ratio of two oscillograph galvanometer deflections G, and G, which correspond to SAMPLE-IN and SAMPLE-OUT.


FIG. 20. The Bennett-Koehler reflectometer.

sets of mirrors: MI to M6, and MI3 to Mzl. These are the input and output groups, respectively. When the sample is in, as shown, the light follows the path M 6 - M , - S - M 8 - S - M 9 - M 1 3 . There are two reflections off the sample as with the Strong reflectometer. When the sample is out, the path is M6-M,-Ml1-M9-Ml3. The reflection of the mirror MI, replaces that of two sample reflections and M 8 . The ratio is the square of the sample reflectance, except for the ratio of M, to MI . To correct for this, the sample as well as mirrors M6 and MI, are rotated 180". Then the light follows the dashed lines with the sample in, i.e., M13-Ml,-S-M,l-S-M12- MI3. With the sample out, the path is Ml3-MIO-M8-MI2-Ml3. This ratio is then S-MII-S to M 8 . Thus the ratio from the first two measurements would be S 2 M 8 / M , , , and that from the second would be S 2 M l , / M 8 . The ratio of these ratios gives the fourth power of the sample reflectance. The sample labeled T is a transparent one, the transmission of which can be measured with a sample-in sample-out technique. Although this technique does require the extra measurements for its ultimate accuracy (about 0.001), it does not require the use of an integrating sphere as does the Strong system (or it would be very sensitive to small misalignments of the sample).

5.7.5. The Measurement of Directional Reflectance

The bidirectional reflectance distribution function is, as the name implies, a distribution. This means that true measurement is that of an infinitesimal and can never be carried out perfectly. Differentials must replace derivatives,

M EASUREM ENTS 215

and the requirements for satisfactory signal-to-noise ratio dictate that these quantities are sometimes larger than one would really desire.

The geometry of the measurement is shown in Fig. 21. The angles of incidence are Oi and +i; those of reflection are Or and 4,. The sample is placed with its normal along the z axis and is assumed to be flat (on the average). Two geometries have been used in measurement^.^^'^^ The first emulates the diagram just discussed; the second places the beam and detector in a horizontal plane and tilts and rotates the sample. The first geometry requires no coordinate transformation to describe the defining geometry, but it is not capable of measuring with detectors that are cooled in dewar flasks with liquid refrigerants.

In both cases, the azimuthal angle of incidence is chosen to be zero. This is a free choice for the origin of the coordinate system. The incident flux is almost always a laser beam which has been expanded and clipped by an appropriate pinhole. It is then aimed at the sample as a slowly convergent beam that is focused at the detector position. The beam could be collimated, but this would eliminate one of the two methods of calibration described below. The function of the beam is to illuminate a portion of the sample with a uniform irradiance. The sample then scatters light to a detector- receiver assembly-a form of radiometer. The sizes of these can be determined to some extent by writing the definition of BRDF in its limiting form:

Ai A n d BRDF= lim -,

L\A,-O ai an-0 -

(5.131)

AAi

where AAi is the illuminated area and And is the solid angle of the detector as subtended at the sample (=Ad/d2). The flux on the detector is an integration over the sample area and the detector area. It is said that the BRDF is measured at angle Or, but it is really an average from O,-AO, to O,+AO, and over all points on the sample. Reasonable values are d =

1 m, = 1 mm. Then the averaging range is 1 mr. For most purposes, this angular resolution is finer than necessary. The illuminated area should be large enough to adequately represent the sample unless several measurements are to be made of several portions of the same sample. Rough samples typically have roughness zones as large as 500 pm. Thus a sample illumination of 5 mm is about right to characterize these. An F/100 beam will encompass a 1-cm sample and focus at a detector 1 m away from the sample. But the beam may not be uniform.

For specular reflection methods, a ratio technique may be used for calibration. Somewhat similar approaches have been tried for these direc-


X

FIG. 21. BRDF angles. The incidence angle is Bi. The angle of specular reflection is Bo. The arbitrary polar angle of reflection is 8,. The azimuthal angle of reflection is br.

tional methods. The two main calibration techniques are what we have called the “reference method” and the “no-sample method”. We can gain an understanding of them from an analysis of the detector output voltage. It is found to be

(5.132)

The reflected radiance L is by definition equal to the sample irradiance times its BRDF. Thus,

%LAd COS 8dA, COS 8, d 2

v=%!Pd=

(5.133)

The detector is always arranged to be perpendicular to the radiometer line of sight, so cos8d is 1 . If there is no sample, then all of the flux at the sample area gets to the detector, and

%E,PhAd COS 8dA, COS os d 2

v, =

VNs= %!E,Ai.

If we take as the sample area the illuminated portion only, then A, = Ai. Therefore,

(5.134)

The BRDF is given by the product of the voltage ratio and the reciprocal of the projected solid angle of the detector.

MEASUREMENTS 277

Alternately, one can use a reference whose BRDF is known. Then

(5.135)

where VK is the voltage from the detector when the known sample with BRDF pK is in place. This is simple in concept and calculation but requires a reference whose BRDF is known for all conditions and comparison measurements made at all positions. The obvious question then is: “How did you measure the reference?” The procedure which seems to devolve from this is to use a Lambertian reference. It has a BRDF which is Phr-’, where Ph is the hemispherical reflectivity. Then

(5.136)

But there is no such thing as .a true Lambertian surface. Some materials, as was discussed earlier, are very good approximations when they are illuminated normally. Then

(5.137)

where V(0) is the voltage resulting from the reference illuminated normally. This approach has the advantage that the instrument parameters are not involved; however, a separate measurement must be made of Ph.

Measurements of specular, hemispherical, and directional transmittance follow those of reflection with an obvious change in geometry.

5.7.6. The Measurement of Laser Power

There are basically two ways to measure laser power: with an appropriate detector or with a calorimeter. The latter technique has come into consider- able use, because lasers usually have enough power to generate a useful signal and because a calorimeter is color-blind; it responds in the same way to laser radiation of any wavelength. One example of a calorimeter has been given by NBS. Figure 22 is a schematic diagram of the instrument. A laser beam enters the baffle section from the left (as usual) and is absorbed in the convoluted shaped cavity-a cone plus cornucopia shape. The water of the reservoir is pumped through the walls of this cavity and its temperature is raised. The increase in temperature is related to the input energy. The relationship is determined by heating the cavity electrically and monitoring


FIG. 22. Schematic diagram of an NBS laser calorimeter. Explanation in the text.

the current, voltage, and time. Nitrogen is flowed through the cavity to reduce scattering, and temperature sensors are used to record any stray absorption in the baffle structure. They claim 13% single-measurement uncertainty for power levels of about 100 kw in the wavelength range from 1 to 11 pm (13 kW). The unit is 1.2 m high, 1.5 m long and 0.65 m wide and weighs about 400 kg.

Although they do not discuss the sources of uncertainty, one can attribute them in general to differences between the measurement and the calibration. It is unlikely that the calibration source is 100 kW; it is probably considerably less. The errors then can arise from different amounts of conduction and convection of heat away from the cavity. Although the baffle structure is monitored, some errors will arise from spillover onto it. The relative contribution of heating from the pump will be different, and no cavity is a 100°/~ absorber.

Calorimeters exist for measuring laser powers from milliwatts to hundreds of kilowatts. The uncertainties are lowest, of course, for the middle of the range where the calibration can be very similar to the measurement and where there is sufficient energy that it causes an appreciable temperature rise. Of course, for fluxes of less than 100 kW, the instruments use smaller reservoirs, sometimes just the heating of an appropriate detector mass. At the lower end of the scale, direct detection by pyroelectric detectors is very competitive with these calorimetric techniques. When the measurements are made in the visible, silicon detectors can be used.

MEASUREMENTS 279

5.7.7. Synchronous Detection

When the investigator has control over the source of the radiation, he should use the technique of synchronous or lock-in detection. Some older radiometers of this type were said to have synchronous rectifier amplifiers (or SRAs). A small source of radiation is placed close to one side of the blades of the chopper, and a small detector is placed facing it on the other side. In this way, both the frequency and the phase of the chopper is measured. I f the optical frequency is labeled vo, the electrical frequency ver and the chopper frequency v,, then the electrical signal will be given by

V( t ) = % cos 2 ~ ~ , t P ( ~ 0 ) dvo dt. (5.138)

The optical frequencies are so much higher than the electrical frequencies that very many optical cycles are integrated, and P ( v o ) may therefore be considered a constant, Po. We are also assuming that the only temporal variations in the power are in the modulator-which is assumed to be cosinusoidal. .

I I'%%

I, V( t ) = %Po cos 27~v, t dt. (5.139)

One nonsynchronous way to handle this signal is to consider the frequency domain version of P ( t ) which is just a delta function at v,. Then a very narrow bandpass filter centered around v, may be used. This is not synchronous detection; it is simply narrow-band detection by which much noise is eliminated. Establishing a narrow bandpass at high frequencies is difficult (10% of v, is good!), and so a heterodyne approach is used in the synchronous system. An electronic bipolar switch is used. This can be represented as a rectangle function. Then the output is given by

V( t ) = %Po cos 27rv,t rect( v,t - 4) dt. (5.140)

The phase of the electronic switch is set relative to the zero phase of the chopper. If it is zero, then the output V ( t ) is just

I,

V( t ) = %P0lcos 2 ~ v , t l dt. (5.141)

Nonzero phase shifts will provide different values for V( t ) . The final output voltage is obtained by integrating over the time constant 7 which is set by a low-pass filter. It is worth considering all of this more generally and in the frequency domain.

I,


The output is given by the product of the Fourier transforms of the modulation function, the chopper function, and the low-pass filter. The latter is usually adjustable on the console of the amplifier to values of integration times ranging from 0.01 s to 1 s (f=0.16 to 0.60 Hz). This filter can be modeled reasonably well by a single time constant filter whose frequency characteristic is (1 + jwT) - ' . The switch can be modeled as a rectangle function with values of +1 and -1. The modeling of the temporal waveform of the flux is a much more complicated matter. If the flux is all concentrated in a spot that is small with respect to the opening of the chopper, it can be modeled as a rectangle function or square wave as well. The modulation and flux waveforms can then both be represented as square waves, and their Fourier spectra are sinc functions. Now we can see for these assumption that the phase relationship is indeed important. There is a direct and linear relationship between the percentage of error in period matching and in the flux that is sensed. Of course, the flux waveform will not be a perfect square wave as indicated, but it will be the convolution of the shape of the spot with the chopper. Detailed considerations of the various geometries are beyond the scope of this treatment, but the principles have been established. The investigator must consider the power that is converted to frequencies above the cutoff of the low pass as well as errors in phase or frequency adjustment, and as a result of any undue spreading of the size of the spot. Some lock-in systems now use quadrature addition; they eliminate errors due to phase shifts.

5.7.8. Integrating Spheres

In many radiometric measurements, one wishes to provide uniform radiation on a detector or a sample. The so-called integrating sphere is one of the methods for doing this. The basic concept is that light entering a port in the sphere is reflected diffusely many times until it becomes well random- ized. It then exits a second port as diffuse illumination. These results are derived in two steps. The first is the very simple theory described many years ago. The second is a more useful, more complete treatment.56

The theory is based on some of the geometric properties of the sphere. Figure 23 shows radiative transfer between two elements of the sphere. The general expression for radiative transfer is

@ = L dA , cos 0, dA2 cos 0,/ d '. (5.142)

In this case, the angles 0, and O2 are equal (0, = O2 = 0). Further, the center-to-center distance is d = 2R cos 0. Thus the flux on dA2 from dA, is

MEASUREMENTS 28 1

FIG. 23. Integrating sphere geometry.

given by

@ = L dA, dA2 COS’ 8/4R2 COS’ 8

= L dA, dA2/4R2. (5.143)

Since the points were chosen arbitrarily, this shows that dA, will irradiate all elements dA, uniformly. Now if Ed is the irradiance on element dA, caused by a source shining through the entrance port, then for a Lambertian surface the total diffuse irradiation is

Ed,-=pEd+p’Ed+p’Ed+. * .

When the direct irradiation is added to this, the total becomes

Ed E = ~ ~ ( i + ~ + ~ * + . .)=- ( 1 - P I ’

(5.144)

The resultant light is diffused and is an inverse function of the absorptivity of the wall. The hemispherical reflectivity p has a maximum value of T-’ - 0.3. Thus the ratio of total irradiance to direct irradiance is at most about 1.4.

A more detailed story described by Goebel and others is given be lo^.^',^^ Imagine a sphere divided up into n spherical caps, each of which rep-

resents an entrance port, exit port, or some other discontinuity in the wall of the sphere. It can be seen from Fig. 2 that the fractional area of any of these caps is given by

=” 1 - Jq] = f ( l - C 0 S 2 0) 2Rh h R - J R ’ - ~ ~

2R 2 f = - - - - 4.nR2- 2R-

= sin’ 8. (5.145)

The radiation entering is labeled O0 and reflects from area a, an amount p0@, uniformly in all directions. I f f ; = ai/A,, where a, is the ith special


area and A, is the spherical area, then the average wall reflectivity or Pw is n

Pw = c Pf; + P w ( 1 -EL). (5.146)

This is just the area-weighted reflectivity. The unabsorbed flux after the second reflection is pwpoQo. This is uniform and is distributed over the wall and the special areas so that the flux after the third reflection is just p ~ p o @ o . The total unabsorbed flux after an infinite number of reflections then is po - po/ ( 1 - pw). The flux on aperture a, then is given by

f; P O Q O

= [ 1 - P w ( 1 -@) + c 4 ’ (5.147)

and the ratio of the flux out of ai to the input flux Q,, is the cavity efficiency

E =&Po[ 1 - P w ( 1 -if;) (5.148)

A simple two-port integrating sphere where a; is the illuminated portion of the wall so that po= pw; the entrance port has area a , ( p , = 0); and the

Fractional Hole Area

FIG. 24. Sphere efficiency as a function of the fractional hole area for different numbers of holes.

MEASUREMENTS 283

8.6 - 0.5 -

0.4

8.3 - 0.2

0.1

-

- -

Fractional hole area

FIG. 25. Sphere efficiency as a function of the fractional hole area for different wall reflectivities.

exit port has area a&, = 0). Then - I

& =hw[ 1 - P w ( 1 -@)] . (5.149)

In this form, we can see that the simple model above is modified by J p , in the numerator. This represents the amount of flux getting out the ith hole. It is also modified by 1 - Z J ; to indicate that some of the wall surface is not available for reflection. Curves of sphere efficiency for different area ratios and coatings are shown in Figs. 24 and 25. The message is clear. Keep the reflectance high; make the ports as large as the uniformity considerations allow. Another message is apparent: there is a strong dependence on the value of pw. Any spectral variations in the coating are magnified in the sphere. The relative error in efficiency is

(5.150)


Notice that a 1% error in p for a p of 0.9 gives a sphere error of about 10%. Other uses of the sphere can be analyzed in a similar way.

Unfortunately, there is no such thing as a truly Lambertian source, so the theory described above is subject to question. Fortunately, the manifold reflections serve to help the diffuseness of the result.

Although it is beyond the scope of this treatment, we can mention the more general treatment and refer the reader to it.59

5.8. Photometry: Radiometry of Visible Light

The visible region has a very special significance in the entire electromagnetic spectrum. Only visible light can be detected in a useful way by humans, and the process of seeing is important to work, leisure, recreation, and everyday living. Accordingly, a special branch of radiometry, photometry, deals with flux levels of radiation as they generate a response in the eye.

A typical problem is the calculation of the proper light source to illuminate a work surface for a particular function. Psychologists have measured the appropriate light levels for performing many different tasks. Light sources with different spectral distributions but the same total radiation elicit different amounts of response from the eye according to how the spectrum of the source matches that of the eye. This can be expressed in terms of response of the eye at its maximum K, the spectral distribution of the eye’s response V ( A ) , and the spectral distribution of power:

P,= K V ( A ) P ( A ) dA [lm]. (5.151)

The visible flux P, can be thought of as “light watts” or “eyeball watts.” It is measured in lumens. The efficiency with which any source provides lumens can be written as

jom

K jOm V ( A ) P ( A ) dA (5.152) PV

pe j9mP(A)dA 0 v = - =

Therefore, for any source, the luminous output is

P,= qv P ( A ) dA = v v P . (5.153)

All the geometric relations which apply to flux in general apply to luminous flux in particular. The main differences and special features rest in the

jOW

PHOTOMETRY: RADIOMETRY OF VISIBLE LIGHT 285

handling of the spectral normalization and the many units and names for them which have grown like topsy with this area of application.

5.8.1. Normalization and its Use

In the previous section, it was shown that the luminous flux can be found from the product of the luminous efficiency and the flux of energy, the power. The response of the eye can then be obtained directly. We show that this is true by writing the responsivity of the eye as a function of wavelength as %(A). Then the output V is

a3

V = [ %(A)Pe(A) dA. -W

(5.154)

Each of the quantities in the integral can be written as a constant times a normalized distribution:

P ( A 1 = @(A ),

% ( A ) = % r ( A ) .

Then the output is m

V = BP, [ r(A)p(A) d. -m

(5.155)

But r ( A ) is the same as V(A), so that m

V = %P, j-, V(A)p(A) dA. (5.156)

The constant % is proportional to K, the maximum of the luminous efficiency curve, so that

V=(const) K [ V(A)@(A) dh W

-a

m

(5.157)

The constant depends upon one’s choice of what constitutes a barely perceptible constant level, a flicker detection, enough detection for the performance of detailed tasks, or some other criterion. This is a matter left for the applications and the psychophysicists.

The effects of spectral distribution are, however, a different matter. They arise from careful considerations of the response of the eye and calibration of sources. It is known that the scotopic response of the eye (under low levels of illumination) is redder than the photopic response (under normal


levels). This means, of course, that r ( h ) is changed. Another, more subtle effect relates to the distribution of the source used for calibration. Many photoelectric detectors are calibrated with a tungsten lamp source operated at a temperature of about 2800°K. These lamps are almost blackbodies and even closer to gray bodies. Accordingly, the output can be written

m

V = j p m s l ( A ) P ( A ) dh. (5.158)

The luminous efficiency is written m

vv= I K V ( A ) P ( A ) dh Im P ( h ) dh

= K I V ( h ) p ( A ) dh I* p ( h ) dh.

-m -m

m

(5.159)

Thus for a calibration source with spectral distribution of flux given by Pc, one has for the luminous responsivity

I (5.160)

-m -m

9 r ( A ) P c ( A ) dh

K V ( h ) P c ( A ) dh I slv= V I P c =

The flux distribution from some other source P, (h) evokes an output given by

V = s l ( h ) P , ( h ) dA

(5.161)

I

I V(A)pc (h) dA

I ( A ) p c ( A 1 dh

I V(A)pc(A) dh J r ( h ) p s ( A ) dh

= P, r ( h ) p , ( A ) dA.

By substitution for 3, one can obtain

V = % , K - ps I r (A ) p , ( h ) dh.

Divide both sides by Pv[ = P,K V(A)p , (A) d h ] . Then the output is given by

v = slvPv (5.162)

REFERENCES 287

This result deserves discussion. The output is not always the luminous flux on the receiver times the luminous response of that receiver. It is true if the integral factor is equal to 1. For that to be true, either r ( A ) must equal V(A), or p c ( A ) must equal p , ( A ) . If one calibrates an eye with p c and then uses the eye, such is the case. If one calibrates a TV tube with a tungsten lamp and then uses the same kind of tungsten lamp to illuminate the televised scene, it is also true. It is not generally true, no matter how logical it sounds. This should be borne in mind in the photometric calibration of TV and other kinds of tubes, which d o not usually have reponses that match the eye.

5.8.2. Photometric Units

In radiometry, in which watts are the unit of flux, the units of sterance and concentration are W m-’sr-* and W m-’ (or some appropriate variation). In photometry these units become, respectively, Im m-’sr-’ and Im m-’. These are surely true, but history and usage have provided other units. Other references are available for a discussion of these complicated units.60

References

1. S. Chandresekhar, Radiatiue Transfer, Dover, New York, 1960. 2. G. Bauer, Measurements of Opfical Radiation, Focal Press, London, 1965. 3. R. A. Smith, F. E. Jones, and R. P. Chasmar, The Defection and Measurement oflnfrared

4. M. Planck, “Zur Theorie des Gesetzes der Energieverteilung im Normalspectrum,” Ver-

5. W. Herschel, “Experiments on the Refrangibility of the Invisible Rays of the Sun,” Phil.

6. 1. W. Ritter, as described in Optical Anecdotes, p. 55, by D. J. Lovell, SPIE, Bellingham,

7. USA Standard: Nomenclature and Dejinitions for Illuminating Engineering RP- 16. Unfted

8. R. C. Jones, “Terminology in Photometry and Radiometry,” J. Opt. SOC. Am. 53,1314 (1963). 9. F. E. Nicodemus, as reported by 1. Spiro in Opt. Eng. 13, G183, (1974), and C. L. Wyatt,

Radiation, Clarendon Press, Oxford, 1968.

handlungen der Deutschen Physikalischen Gesellschafr 2 , 237 ( 1901).

Trans. Roy. SOC. 90, 255 (1800).

MA, 1981.

States of America Standards Institute.

Radiometric Calibration: Theory and Methods, Academic Press, New York, 1978. 10. A. G. Worthing and D. Halliday, Heat, Wiley, New York, 1948. 11. A. G. Worthing, in Temperature: Its Measurement and Control in Science and Industry,

12. W. L. Wolfe and G. J. Zissis, The InJrared Handbook, Office of Naval Research, Washington,

13. J. Strong, Concepts of Classical Optics, Freeman, San Francisco, 1958. 14. M. G. Dreyfus, “Spectral Variation of Blackbody Radiation,” Appl. Op t 2 , 1113 (1963). 15. R. D. Hudson, Infrared System Engineering, Wiley, New York, 1969. 16. M. R. Holter, S. Nudelman, G. H. Suits, W. L. Wolfe, and G. J. Zissis, Fundamentals of

Reinhold, New York, 1941.

D. C., 1978.

InJrared Technology, Macmillan, New York, 1962.


17. P. Kruse, McLaughlin, and McQuistan, Elements of Infrared Technology, Wiley, New York,

18. M. A. Bramson, Infrared: A Handbook for Applications, Plenum, New York, 1968. 19. R. H. Kingston, Detection of Optical and Infrared Radiation, Springer-Verlag, Berlin, 1978. 20. J. N. Howard, Book review of Seven Place Tables of the Planck Function for the Visible

Spectrum, by D. Hahn, J. Metzdorf, B. Schleef and J. Seich, Academic Press, 1964, in Appl. Opt. 4, 808 (1965); other volumes are mentioned.

21. E. M. Sparrow and R. D. Cess, Radiation Heat Transfer, Brooke-Cole, Belmont, Calif. 1966. 22. F. E. Nicodemus, “Radiance,” Am. 1. Phys. 31, 368 (1963). 23. J. Rainwater, “Generalization of the Abbe Sine Law in Geometric Optics,” Am. J. Phys.

24. R. Gardon, “The Emissivity of Transparent Materials,” J. Am. Ceram. SOC. 39,278 (1956). 25. H. 0. McMahon, “Thermal Radiation from Partially Transparent Reflecting Bodies,”

26. M. Czerny and L. Genzel, “Uber die Eindringtiefe raumlich diffuser Strahlung in Glas,”

27. D. Q. Wark and H. E. Fleming, Monthly Weather Reo. 94, 351 (1966). 28. Hand and Conrath, Science 165, 1258 (1969). 29. J. C. Gille, “Inversion of Radiometric Measurements,” Bull. Am. Mer. SOC. 49, 903 (1968). 30. L. D. Kaplan, “Inference of Atmospheric Structure from Remote Radiation Measure-

31. S. Twomey, J. Geo. Res. 66, 2153 (1961); 1. Assoc. Comp. Mach. 10, 97 (1963); J. Frank

32. W. L. Wolfe and H. P. Stahl, “Some Calculational Results Using Multicolor Radiation

33. W. L. Wolfe and E. L. Dereniak, in Imagingfor Medicine, Vol. I Chap. 13 (S. Nudelman

34. E. L. Dereniak, Ratio Temperature Thermography, Ph. D. Thesis, University of Arizona,

35. Electro-Optical Industries Brochure. 36. R. Hofmann, S. Drapitz, and K. W. Michel, “Lamellar Grating Fourier Spectrometer for

37. F. 0. Bartell and W. L. Wolfe, “Cavity Radiators: an Ecumenical Theory,” Appl. Opt. 15,

38. J. M. Lloyd, Thermal Imaging Systems, Plenum, New York, 1975. 39. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics, Wiley, New York, 1978. 40. W. G. Driscoll and W. Vaughan, Handbook of Optics, McGraw-Hill, New York, 1978. 41. R. A. Sawyer, Practical Spectroscopy, Dover, New York, 1963. 42. R. J. Bell, Introductory Fourier Transform Spectroscopy, Academic Press, New York, 1972. 43. A. C. Candler, Modern Interferometers, Hilger and Watts, London, 1951. 44. R. D. Saunders and J. B. Shumaker, Optical Radiation Measurements: The 1973 Scale of

Spectral Irradiance, U. S . Government Printing Office, 1977. 45. J. Geist and W. R. Blevin, “Chopper Stabilized Null Radiometer Based upon an Electrically

Calibrated Polyelectric Detector,” Appl. Opt. 12, 2532 (1973); W. M. Doyle, B. McIntosh, and J. Geist, Roc. SOC. Phot. Inst. Eng. 62, 166 (1975); and R. J. Phelan and A. R. Cook, “Electrically Calibrated Pyroelectric Optical Radiation Detector,” Appl. Opt. 12, 2494 (1973).

46. J. Geist, E. F. Zalewski, and A. ‘R. Schafer, “Spectral Response Self-Calibration and Interpolation of Silicon Photodiodes,” Appl. Opt. 19, 3795 (1980).

47. National Bureau of Standards, Optical Radiation News, 20 April (1977) and 24 April (1978).

1962.

32, 626 (1964).

1. Opt. SOC. Am. 40, 376 (1950).

Glastech. Ber. 25, 134 (1952).

ments,” J. Opt. SOC. Am. 49, 1004 (1959).

Inst. 279, 95 (1965); Monthly Weather Reo. 91, 659 (1963).

Inversion,” IR Phys. 20, 293 (1980).

and D. D. Patton, eds.), Plenum, New York, 1980.

Tucson, 1976.

a Balloon-borne Telescope,” IR Phys. 17, 451 (1977).

84 (1976).

REFERENCES 289

48. F. E. Nicodemus, J. C. Richmond, 1. W. Ginsberg, and T. Limpens, Geometrical Consider- ations and Nomenclature for ReJlectance, NBS Monograph 160, U.S. Dept. of Commerce, 1977.

49. D. B. Judd, “Terms, Definitions and Symbols in Reflectometry,” J. Opt. SOC. Am. 57, 445 (1967).

50. J. Strong, Procedures in Experimental Physics, Prentice-Hall, New Jersey, 1938. 51. H. E. Bennett and W. F. Koehler, “Precision Measurements of Absolute Specular Reflect-

ance with Minimized Systematic Errors,’’ J. Opt. SOC. Am. 50, 1 (1960). 52. J. W. Harvey, Light-scattering Characteristics of Optical Surfnces, Ph. D. Thesis, Optical

Sciences Center, University of Arizona, Tucson, 1976. 53. F. 0. Bartell, E. L. Dereniak, and W. L. Wolfe, “The Theory and Measurement of

Bidirectional Reflectance Distribution Function (BRDF) and Bidirectional Transmission Distribution Function (BTDF),” Proc. SOC. Phot. Instr. Eng. 257, 157 (1980).

54. NBS Dimensions, April, 1977. 55. J. A. Jacques and H. F. Kuppenheim, J. Opt. SOC. Am. 45, 460 (1955). 56. D. G. Goebel, “Generalized Integrating-Sphere Theory,” Appl. Opt. 6, 125 (1967). 57. B. J. Hisdal, “Perfect Samples in Integrating Sphere,” J. Opt. SOC. Am. 55, 1122 (1965)

and “Non Perfect Surfaces in Integrating Sphere,” J. Opt. SOC. Am. 55, 1255 (1965). 58. F. 0. Bartell, Blackbody Simulator, Cavity Radiation Theory, Ph. D. Thesis, University of

Arizona, Tucson, 1978. 59. J. Meyer-Arendt, “Radiometry and Photometry: Units and Conversion Factors,” Appl.

Opt. 7 , 2081 (1968).

6. DETECTORS

T. 0. Poehler Applied Physics Laboratory

Johns Hopkins Road Laurel, Maryland 20707

6.1. Introduction

Quantitative measurements of optical radiation are usually accomplished by converting the radiant energy into electrical signals. The electrical output consists of the response of the detector to the radiation, which is designated the signal and random fluctuations known as noise. Detectors may be characterized by the physical mechanisms by which they convert optical to electrical energy, the spectral range over which they respond, the rate at which they respond, the minimum radiative power that they can detect, and the amplitude of the electrical signal generated for a unit of incident radiation.

Optical radiation may be detected by either thermal detectors where an increase in temperature of the sensing element exposed to radiation is observed, or by quantum detectors where photons are directly converted into the electrons or mobile charge carriers in the detector. Many physical effects have been used to detect optical radiation. Devices based on either a photoexcitation or photoemission of electrons have produced detectors with the highest performance. Devices in this general category, which on the one hand include photomultiplier tubes and semiconductor photodetectors constructed from such elements as silicon or germanium primarily in a photodiode configuration, have been used to detect most radiation in the shorter wavelength regimes. On the other hand, certain simple photoconductive detectors have found a widespread application in the long-wavelength portion of the spectrum. Detectors based on the conversion of photons due to a thermal effect within the detector medium have also found wide application. The primary practical modern device of the thermal variety currently being utilized is the pyroelectric detector, which is used for both energy and power measurements over a range of the spectrum extending from visible to far-infrared wavelengths.

29 1


Copyright 0 1988 by Academic Press Inc. All rights of reproduction in any form reserved

ISBN 0-12-475971-8

292 DETECTORS

6.2. Figures of Merit

Since there are a wide range of physical mechanisms employed in radiation detectors, several figures of merit are used to quantitatively characterize performance, independent of the physical basis of their operation.' A simple criterion that can be applied to typify the response is the responsivity R ", generally defined as the ratio of the rms value of the signal voltage, V, to the rms power, P, incident on the detector and specified in units of volts/watt. The total power incident on a detector of area A is associated with an irradiance H (watts/cm2), so that the responsivity is

vs vs R - - = - (volts/ watt). " - P H A

The responsivity is usually a valid description of the detector performance over the three to six orders of magnitude of radiant power that is typical of the range over which detectors respond with a linear transfer function. The responsivity may be specified either as R( Tf), which represents the output voltage resulting from input radiation from a blackbody at a temperature T with a signal derived from a source modulated at a frequency

or R(A,f), which is the responsivity signal associated with a monochromatic source at a wavelength A with modulation frequency f:

The limiting sensitivity of any detector is determined by random fluctuations which exist in its output, and hence it limits detection to a certain minimum detectable power for any system. The power required to generate an output signal equal to the noise is frequently referred to as noise- equivalent power (NEP). To determine the noise-equivalent power requires measurement of the amount of radiative power from a blackbody source which must fall on a detector to yield an rms signal V, equal to that generated by the detector noise VN when it is shielded from the source. The noise-equivalent power must be specified for a particular source temperature, signal modulation frequency, and amplifier-system bandwidth. The NEP for detection under a particular set of operating conditions including a 500°K blackbody radiation source modulated at 900 Hz with a 1-Hz bandwidth is written as

P H A HA NEP( 500"K, 900 Hz, 1 Hz) = - = - - - (watts), (6 .2)

S / N S / N - v.1 V N

where P is the detected power, H the irradiance, A the detector area, S and N are the signal and noise measured, respectively. This latter ratio may be expressed as either a voltage or current ratio. Often the reciprocal of the NEP designated as the detectivity is used to characterize some detectors and is specified in units of watts-'. To make comparison between

T H E R M A L DETECTORS 293

detectors, their respective performance must be reduced to representative conditions, so the detectivity is often normalized to an amplifier bandwidth of 1 Hz and a detector area of 1 cm2. This figure of merit’ called D* is

D * = L N ( A ) l V ‘I2 =- VS/vN(AfA)’”(cm(Hz)’~’/W) (6.3) H P

where Af is the electrical bandwidth in Hz. A detector whose performance is specified at a particular wavelength A from a monochromatic source is designated as D,*(A,f; Af).

The performance of low-noise detectors may also be limited by radiative noise arriving at the detector from the background environment. When operating under these conditions, a detector is said to be background-limited in performance or in the BLIP mode. Since this measure of the detectivity is dependent upon the half angle 6 from which the detector can see background radiation, a normalization including this dependence on field of view is used and is designated D**, where

D** = D* sin 6, (6.4)

where the half angle of the field of view is 6.

6.3.. Thermal Detectors

6.3.1. Thermocouple Calorimeters

Thermocouples have been employed to detect radiation since the early nineteenth century3 subsequent to the discovery of the Seebeck effect. The thermocouple radiation detector is simply composed of a circuit of two metal conductors and the junctions between them, where one of the two junctions is exposed to radiation, thus raising its temperature above the unilluminated cold junction. A potential difference will be generated by the Seebeck effect4 between the two junctions, which can then be observed by a simple voltage measurement. .A number of thermocouples connected in series are called a thermopile andhave the advantage of a larger amplitude output signal than that generated by a single thermocouple under ordinary conditions.

Thermocouple detectors are often made from fine wires attached to a thin-film radiation absorber whose temperature rise in sensed by the thermocouple.’ The pairs of metals used in these devices include bismuth-silver and copper-constantan. The sensitivity of such a device is enhanced if it is placed in a vacuum enclosure with a window that is transparent in the spectral range of interest; commercially available units are supplied in evacuated mounts. Devices such as these are usually used for detection of

294 DETECTORS

0.75 mm

4 b -

I Radiation

Silver contacts

5 0 2 Silicon

Bismuth

FIG. 1. Thin-film thermocouple, fabricated by vacuum evaporation. (After Contreras and Caddy.”)

infrared radiation in the 1- to 20-pm range where the “black” film is highly absorbing, but are characterized by low-power-handling capability and long response times. Thin-film thermocouples fabricated by vacuum evaporation5 of the components often exhibit improved properties in comparison to bulk devices. These detectors, made by precision photolithographic techniques with small dimensions and low mass, can yield lower time constants and impedances better matched to electronic circuitry (Fig. 1 ) . Evaporated arrays of thermocouples composed of antimony and bismuth are used in commercial thermopiles.

While thermopiles are satisfactory for many spectroscopic applications, their low-power-handling capability and long time constant make them inappropriate for many laser measurements. In particular, measurements of CW laser power in the multiwatt range or pulsed lasers with pulse energies of several joules require calorimeters capable of withstanding these inputs without damage. These requirements have stimulated the construction of a multitude of calorimeters for high-laser-power measurements. Typically, these instruments use a disc6 or cone’ coated with an absorbing layer to absorb the laser radiation, while the temperature of the absorber can be monitored with a thermocouple to obtain an estimate of the energy deposited. While absorbing discs are common, radiation trapping can be increased by directing the radiation into a cone which is coated with absorbing material. Multiple reflections of the incident beam will cause nearly complete absorption and will also tend to increase the damage threshold of the detector. A diagram of a typical absorbing cone calorimeter typical of those used for power and energy measurements is illustrated in Fig. 2. Commercial calorimeters are available which are capable of measuring power of up to several kilowatts and short-time-duration pulses of greater than 1000 joules, using these principles. With the absorbing coatings generally used on these calorimeters, the spectral range which can be

T H E R M A L DETECTORS 295

Black surface

I Thermal sensing element

FIG. 2. Absorbing cone calorimeter, used for power and energy measurements.

measured is typically from the visible spectrum through approximately 30-km wavelength in the infrared (IR).

6.3.2. Bolometer

The heating effect of absorbed radiation can be used to cause a change in the resistance of a metal or semiconductor from its normal resistance at room temperature in devices which are classified as bolometer detectors. This change in resistance can be measured when the resistive bolometer element is incorporated in a bridge circuit in combination with a matching bolometer element not exposed to radiation to compensate for temperature changes in the environment. Initially, bolometers were constructed of evaporated metal films, but these devices have been noisy and less sensitive than thermocouples. Thermistor bolometers composed of semiconductor alloys have been used in modern infrared work.899

A number of specialized bolometer detectors have been applied to measurements of incident laser pulses at intensities much greater than those of interest in other applications. Early work on laser-power measurements utilized a tangled mass of enamel-insulated wire contained in an enclosure." These "rats-nest" calorimeters effectively absorbed all the incident optical energy after a sufficient number of reflections from the wire, and the change in resistance of the wire was used directly as a measure of incident power. The system could be calibrated by passing a known current pulse through the conductor. The upper limit on the optical energy before the thin insulating coating on the wires was damaged was typically several joules per square centimeter. Several thin-film bolometers have also been used as room- temperature detectors of high-power-pulsed 10.6-km radiation.'' The low

296 DETECTORS

mass of these devices permits response times of less than 1 psec., and at least one device has been reported capable of withstanding 100-5 input energy.I2

Two important quantities necessary in characterizing the performance of a sensitive bolometer are the specific heat capacity C and the normalized temperature coefficient of resistance a, where

1 dR R dT'

a = - -

Here R is the bolometer resistance (ohms) and T is the temperature of the element (OK). The responsivity of R of an element with a bias voltage V applied is

R=aV/G, (6.6)

where G is the thermal conductance to a heat sink or bath. A high- responsivity bolometer thus requires a large temperature coefficient of resistance, as is observed in most semiconductors at cryogenic temperatures. Hence, extremely sensitive bolometers can be obtained by using materials such as carbon,I3 ger rnani~m, '~ or ~ i l i c o n ' ~ at temperatures below 20°K. Operation at low temperature also improves the response time T,

C G '

75-

At low temperatures, the specific heat capacity is, of course, proportional to T3 so that the speed of response can be made faster according to (T/ Td)3, far below the Debye temperature, Td. Since both the responsivity and time constant of a bolometer depend inversely on the thermal conductance, bolometer performance can be tailored to a particular application by the degree of thermal contact with the bath. This parameter is usually varied by altering either the length or the composition of the lead wires to the element. Carbon bolometers with R L- 1 x lo4 V/W at 2°K have been constructed, while Ge and Si bolometers with R = 1 x lo5 V/ W and response times of 10 psec have been demonstrated. Since the specific heat capacity of silicon at low temperatures is eight times smaller than that of germanium, silicon has a faster response and is used in commercial bolometers. Since these are thermal detectors, the spectral response is uniform throughout much of the infrared spectrum and usually limited by the transmission of windows or radiation shields used to admit radiation to the vacuum-enclosed bolometer element. Doped-germanium bolometer elements have been used effectively by Lowl4 for infrared astronomical measurements. Metal bolometers cooled to near their superconducting transition temperature where very large temperature coefficients of resistance occur have also been

THERMAL DETECTORS 297

FIG. 3. Cooled Si semiconductor balometer arrangement.

suggested. Devices of this type using thin tin films are sensitive infrared detectors,16 but require a high degree of temperature stability compared to cooled semiconductor bolometers.

A typical cooled Si semiconductor bolometer arrangement is shown in Fig. 3.” The bolometer is mounted in a vacuum rather than immersed in the cryogenic fluid bath to allow the sample temperature to change in response to incident radiation. The bolometer is connected to wires in good contact with the temperature bath. Room-temperature radiation is prevented from heating the bolometer by cooled filters, usually quartz and high-density black polyethylene. The bolometer is often mounted at the focal point of a spherical or conical cavity to enhance the absorption of radiation. A simple dc bias circuit is often used in combination with a low-noise pream- plifier and a signal-averaging system if the signal is repetitive. The maximum bias current is limited by dc ohmic heating which will raise the equilibrium detector temperature, or by excess noise generation in the bolometer as the bias field is increased. This excess noise occurs at relatively low applied electric fields preliminary to breakdown effects associated with impact ionization phenomena, which typically occur at field strengths of a few volts per centimeter in Ge and Si.

6.3.3. Golay Cell

The Golay cell is a thermal detector with a nearly uniform energy response from the ultraviolet to the microwave region that has been widely used in

298 DETECTORS

infrared work." The detector consists of a gas-filled cell enclosed by two thin membranes. The incident radiation is absorbed by an aluminum layer deposited on one membrane, with the metal film thickness chosen for a surface impedance yielding maximum absorption of radiation virtually independent of spectrum. The absorbed radiation heats the gas in the cell with an attendant pressure increase which distorts the second flexible membrane. The distortion is sensed by a photodetector cell, which detects a light beam reflected from the membrane. The intensity of the light incident on the photodetector depends on the deflection of the beam caused by the membrane distortion. This effect is amplified by using two fine grids arranged to have the image of the lines of one grid incident on the spaces of the second grid, permitting no light transmission at equilibrium. When the membrane is distorted, the image of the first grid shifts, allowing significantly more light on the photodetector. This grid system is, in effect, a mechanical means of enhancing the effect of small displacements of the membrane (Fig. 4).

The Golay cell has a linear response for continuous power levels of 1 x to 5 x lo-' Wand can be used with caution up to the milliwatt level. While it has an NEP of 5 x lo-" W, the limiting sensitivity is usually fixed by the bias and amplifier electronics. In particular, a small light source which illuminates the optical system in the cell must be energized by a carefully isolated dc supply or batteries to attain low-noise performance. Since the Golay cell is operated at room temperature and has a wide spectral response, it has been used widely in many applications, particularly in infrared spectral measurement^.'^ The spectral response depends primarily on the input window, usually quartz, diamond, or some infrared-transmitting crystal. For many applications, however, the cell is undesirable because of its rather long time constant. The response time is determined by the rate

Cell Line grid

Incident radiation

Window

Absorber Flexible mirror

FIG. 4. Golay cell and optical system to measure small displacements of the membrane.


of gas expansion and physical motion of the membrane, so that a minimum time constant of about 1 msec is observed. The number of commercial suppliers for this historically important detector has been sharply curtailed.

6.3.4. Pyroelectric Detector

The pyroelectric detector is a thermal detector which responds to a temperature rise induced by absorbed radiation through a net change in surface The pyroelectric effect is observed in noncentrosym- metric crystals having a unique axis along which there is exhibited spon- taneous electric polarization. In insulating crystals of this type, net charge will be induced on the surfaces by the internal dipole moment until it is neutralized by extrinsic surface charge. The extrinsic surface charge is not mobile, so that if there is an abrupt change in the internal dipole moment resulting from a temperature change, the induced charge will produce a net transient electrical field for a substantial time period. This effect has been widely used as a room-temperature method of detecting infrared radiation by detecting the charge or voltage on a capacitor constructed using a pyroelectric material and a pair of metallic electrodes. The simple geometry of a pyroelectric detector is shown in Fig. 5(a); the radiation can be detected

,Electrodes,

Pyroelectric detector

Low noise amDlifier

FIG. 5 . Geometry of a pyroelectric detector and electrical equivalent, considering the detector as a voltage source.

300 DETECTORS

using partially transparent electrodes on the faces of the pyroelectric element, or electrodes may be attached to the edges of the sensor.

The pyroelectric effect can occur in crystals whose polarization is either reversible or nonreversible in an applied electric field. The former class, known as ferroelectric crystals, contains most of the important pyroelectric materials. Application of an electric field significantly greater than the coercive field in the direction of the polarization axis to convert a multi- domain crystal to a single domain is called poling. The pyroelectric effect occurs only below the Curie temperature in poled ferroelectric materials. If the temperature is allowed to exceed the Curie point, the pyroelectric crystal must be repoled to reestablish the single domain structure.

Pyroelectric detectors can be used to detect radiation modulated at a constant frequency w ; to observe transient radiation when operated in conjunction with an appropriate matching amplifier; or as an energy- measuring device by storing the total charge generated by an input signal. A pyroelectric detector responding to radiation modulated at a frequency w can be considered as a parallel plate capacitor in parallel with a resistance while the alternating charge generated by the radiation is equivalent to a current generator in parallel with the passive components. If the pyroelectric element is connected to an amplifier, the effective input impedance of the amplifier can be considered the resulting combined amplifier and detector impedances. Radiation modulated at a frequency w will generate a varying nonequilibrium temperature T, in the pyroelectric element, giving rise to an alternating charge PAT,,, ; where p is the pyroelectric coefficient and A is the detector area. If the detector is considered as a voltage source [Fig. 5(b)] operating into a high-input impedance voltage amplifier, it will be characterized by a voltage responsivity R v (volts/watt) of2’

Rv = 7 (wpAR/ G ) ( 1 + w27; ) 1 + O ~ T ~ ) - ” * , (6 .8 )

where 7 is the emissivity, R is the equivalent circuit resistance, and G is the thermal conductance, while T~ and T are the electrical and thermal time constants of the system, respectively. Since the T€ and T are on the order of approximately 1 sec in most cases, the high-frequency limit of this equation is usually applicable, yielding

(6.9)

where c’ is the volume specific heat and E is the dielectric constant of the material.

Alternately, if the detector is a current source operating into a low-input impedance current amplifier, it will have a current responsivity R,


(amperes/watt) at high frequency,

77P R. =- ' c'h'

(6.10)

where h is the detector thickness. The operating regime is usually in the high-frequency region where the current responsivity is independent of frequency, while the voltage responsivity decreases at 6 db/octave. If the detector is used as a voltage source, the ratio of the pyroelectric coefficient to the dielectric constant is the important parameter in achieving a high responsivity, while as a current source, the pyroelectric coefficient alone is the significant factor.

A good pyroelectric material is typified by a large pyroelectric coefficient, a small dielectric constant, a low thermal conductivity and heat capacity, and a high absorption coefficient in the spectral region of interest.22 Table I shows properties of some of the pyroelectric materials of greatest interest. In addition to a number of inorganic insulating crystals, such as strontium barrium niobate (SBN)23 with pyroelectric coefficients as high as 1 . 2 ~ lo-' C c K 2 K-', useful pyroelectric effects are observed in polymers such as polyvinylidene fluoride (PVF2).24*25 From Eqs. (6.8) and (6.9), it is apparent that the proper choice of a pyroelectric detector material depends on the required frequency response and mode of operation. In low frequency applications, SBN with its large pyroelectric coefficient can yield responsivities as high as 2.5 x lo3 V/ W, but its high-frequency performance

TABLE I . Properties of Pyroelectric Materials

Pyroelectric Curie Specific Coefficient Dielectric Temperature Heat

P Constant T, C Material (C cm-'K-') E ( W ( J gm-'K-')

Ba(Ti0,) 2 x lo-"

Triglycine Sulphate (TGS)

Li,SO,. H,O LiNbO,

LiTaO, NaNO, Strontium Barrium

Niobate (SBN) Polyvinylidene

Fluoride (PVF,)

4 x

1.ox 4 x 1 0 - ~

6 x lo-' 1.2 x lo-" 1.2 x lo-'

4 x

610 (ell) 399 0.5 4100(e,)

50 322 0.97

10 0.4 30 (el,) 1573

58 75 ( E L )

8 427 0.96 1400 333

15

302 DETECTORS

as a voltage source is strongly reduced by the high dielectric constant. At high frequencies, TGS'" and LiTa0" 27 with much smaller dielectric coefficients are preferred in the voltage source mode. While TGS has been widely used as a pyroelectric element, its low Curie temperature makes frequent repolarization by application of a poling electric field a necessity, and other more stable materials, such as lithium tantalate, have been widely adopted for general use.

The fundamental pyroelectric detector assembly consists of the pyroelectric element, a load resistor, and a field effect transistor (FET), all contained in an evacuated housing with a transmitting window. The pyroelectric charge developed in the thin crystal wafer is collected on metal electrodes in either face-electrode or edge-electrode configurations, with the face electrodes being the more common. Since the spectral response of pyroelectric detectors is dependent on the absorption of the radiant energy by the sensing element, the response is inherently large only when the fundamental absorption coefficient of the pyroelectric material is large, usually in the infrared region. For detection in the visible and near infrared, the front surface of the element must be coated with an absorbing layer to obtain adequate response. Because the detector elements are commonly mounted i n a hermetically sealed or evacuated package to minimize heat transfer, the spectral response is also restricted by the window transmission spectrum. Materials commonly adopted for windows include quartz, silicon, germanium, and irtran. Exclus- ive of the window properties, pyroelectric detectors with transparent electrodes have flat response from 2 to 30 microns, with substantial sensitivity throughout the far infrared.

6.3.5. Liquid-Crystal and Luminescent Detectors

Where the radiation intensity is sufficiently great ( > 5 mW/cm2), several methods are available for direct visualization of the radiation pattern in spectral regions where photographic film is not sensitive. Liquid crystals are cholesteric compounds whose structure varies with temperature, often in a small temperature range. Absorption of radiant energy by a thin liquid-crystal layer will raise its temperature and cause a change in the wavelength of light reflected from the crystal surface. Observation of this change provides a means of monitoring the spatial and temporal behavior of a radiant source, often an infrared laser. Liquid-crystal displays have been used to monitor laser mode patterns and infrared interferograms at wavelengths where conventional photographic plates fail to respond.28

Another widely used method of observing the spatial variation of infrared radiation is through thermal quenching of visible lumine~cence .~~ If a ZnCdS phosphor is excited by an ultraviolet lamp, visible yellow fluorescence is

PHOTON DETECTORS 303

TABLE 11. Thermal Detectors

Operating Temperature R" D* T

Detector ( O K ) NEP(Wj (V/W) (cm Hz'"/W) (secj

Thermocouple B o I o m e t e r Carbon Bolometer Ge Bolometer Si Bolometer TGS Pyroelectric SBN Pyroelectric Golay Cell

3 00 3 00

L

2 2

3 00 3 00 3 00

2 . 5 ~ lo-"' 1 3x10' I x 10' 2 x IOU

I x 1 0 - l ' 2 x 104 4.5 x io10

5 x 10-I) 1 x 10' IX 1 x 1 0 ~ 1 x loy

5 x lo-'' 1 x 10' 8 x 10"

5x10" 5x10-I ' 2x10' 1 . 6 ~ 1 0 ~

~

3 x 1 x lo-' 1 x

1.ox lo-' 2 x 1 x lo-) 1 x lo-' 1 x

induced which can in turn be quenched wherever the phosphor is locally heated by radiant energy. Thin plastic films can be coated by a lacquer bearing the phosphor, after which the film is mounted on an aluminum heat sink to provide units with sensitivities of approximately 100 mW/cm* and a response time of 0.1 sec.3"

6.3.6. Summary

The performance of a number of different thermal detectors is summarized by the characteristics shown in Table 11. The room-temperature thermal detectors, such as thermocouples and bolometers, are relatively slow and exhibit moderately good detection sensitivity as measured by NEP and D*. The cooled bolometers are significantly more sensitive, have better response times, but are vastly more difficut to use. The pyroelectric detectors exhibit a sensitivity similar to the other uncooled thermal detectors and are becoming the standard room-temperature sensor for energy and power measurements.

6.4. Photon Detectors

Optical radiation in the visible portion of the spectrum is most frequently detected by a photon detector where the incident photons directly interact with electrons in the detector medium to produce an electrical signal proportional to the intensity of the radiation. There are three basic processes which may be used to categorize the majority of practical photon detectors- the photoemissive effect, the photovoltaic effect, and the photoconductive effect. Since these processes directly convert radiation into electrical signals by a quantum (rather than thermal) process, detectors of this class are

304 DETECTORS

typically characterized by high sensitivity, short response times, and strong spectral variations in the response to radiation.

6.4.1. Photomultiplier

One of the most important direct photon effects used to detect radiation is the photoemissive effect, also known sometimes as the external photoeffect. Photoemissive devices are particularly well suited to detection of low-intensity radiation and offer major advantages in detection of fast low-level signals. In a photoemissive detector, the action' of incident radiation is to induce emission of an electron from the surface of a photocathode into space, where it is subsequently collected by an anode. There are a number of devices in which the photoemissive effect is applied for sensing in a simple photomultiplier or for imaging radiation in image intensifiers or image tubes, primarily in the visible spectrum. The most commonly employed device is the photomultiplier, where the photoemitted electrons impinge upon a sequence of electrodes called dynodes, which act as secondary electron Each electron which is incident on a dynode induces emission of one or more additional electrons, so that an amplification or gain mechanism exists for the production of large numbers of electrons.

The spectral properties of the photomultiplier are primarily determined by the photocathode. Photocathodes are generally thought to be of two types, conventional photocathode^^^ and so-called negative electron affinity cathodes.33734 The two types of cathodes differ from each other in the magnitude of the photoelectric work function associated with the energy required to remove an electron from its equilibrium thermal level inside of a solid to the energy level above the potential barrier at the solid surface which allows the electron to escape from the solid. The minimum energy which a photon must possess to induce photoemission equals the work function, 4, of the metal [Fig. 6(a)]. Within the category of conventional photocathodes, both metals and semiconductors are used as electrode materials. The energy band diagrams associated with the photoemissive properties in these types of materials are illustrated in Fig. 6(a) and (b), respectively. In the conventional semiconductor electrode, the minimum energy required for a photon to induce photoemission is still that quantity which will raise the electron to a level higher than the potential barrier at the surface of the material [Fig. 6(b)]. Semiconductor photocathodes with a positive electron affinity generally require less energy than metallic photocathodes and hence have spectral responses which extend to somewhat longer wavelengths than conventional metallic photocathodes. In either case, these photocathodes can respond to radiation of wavelengths extend-


Metal Vacuum

(a)

Semiconductor Vacuum

Photoexcitation -! \ Conduction

band Fermi level

Valence band

FIG. 6. Energy band diagrams associated with the photoemissive properties of photocathodes. (a) For metals, and (b) for semiconductors.

ing only from the visible spectrum into the near infrared. These material limitations, which have been encountered over most of the history of the use of photomultiplier tubes, have permitted only very marginal response at wavelengths as long as 1 p. In the high-energy ultraviolet part of the spectrum, the tubes often do not respond to wavelengths shorter than 0.2 p as a result of the transmission spectra of typical window materials used in their construction. There have been some extensions of the response to near 0.1 p by using rather delicate materials, such as lithium fluoride, and even further extension of their response into the X-ray spectrum by using very thin wafers and material such as gallium arsenide. An alternative procedure that is used to produce some ultraviolet response in photomultiplier tubes is to coat the entrance window of the tube with an ultraviolet-sensitive material which is capable of fluorescing in a region where the tube does respond sensitively. Materials such as sodium silicate have been used to produce relatively short wavelength response in conventional tubes by using this approach.

In recent years, many advances have been made in increasing the sensitivity in the long-wavelength portion of the spectrum and extending that response to the wavelengths as long as 1.5 p.33-36 Prior to this work, photo-

306

Semiconductor Vacuum

Conduction band

Photoexcitation +

Valence band Fermi leve

FIG. 7. Energy-band diagram associated with the photoemissive properties of photocathodes with sensitivity in the infrared. (After K r ~ s e . ~ ’ )

emissive detectors were not available for this portion of the spectrum. The extension of spectral response of photomultiplier tubes into the infrared has resulted from the discovery of the negative electronic affinity photocathode. It was discovered that by overcoating the surface of certain p-type semiconductors with evaporated layers of low-work-function materials, the resulting structure has a negative electronic affinity. This is illustrated in Fig. 7. Here, in contrast to the conventional photocathodes previously described, the incident photon must have an energy which only equals or exceeds the energy gap, E, , of a semiconductor to allow photoemission. Hence, by judicious choice of the semiconductor and associated coating, it is feasible to construct photocathodes where the spectral response does extend well into the near infrared. Even here, however, the quantum efficiency does diminish at increasing wavelengths, so that there is yet a relatively severe limitation on use of photomultiplier systems at wavelengths exceeding 1.2 p.

The conventional photomultiplier tubes are usually based on a photoemissive structure where a thin evaporated layer composed of a compound including some alkali metal (with the predominate example being that of cesium), metallic elements from Group V of the periodic table, such as antimony and other species, are combined in a ternary or similar alloy. Examples of the spectral response of representative conventional photocathode?’ are shown in Fig. 8. These show some of the conventional materials designated by the typical nomenclature referring to the photocathodes as S1, S17, and S20. For example, the S1, which is silver (Ag) on cesium oxide (CsO), has some response to wavelengths as long as 1.1 p. The conventional photocathode with the most sensitivity in the red portion of the spectrum is the so-called S 2 0 , composed of cesium-sodium- potassium-antimony [ (Cs)Na2KSb]. This system is able to respond, at least weakly, to almost 0.8 p. Some extended response versions of the S20

307 PHOTON DETECTORS

s

1 1 1 1 I 1 I 0.4 0.6 0.8 1

Wavelength (rm)

FIG. 8. Spectral response of representative conventional photocathodes. (After K r ~ s e . ~ ’ )

photocathode are able to respond sensitively to wavelengths as long as 0.9 p and are therefore useful in sensing red radiation from sources such as the helium-neon and ruby lasers and even the gallium arsenide semiconductor laser at 0.83 micron.

The so-called negative electron a f f i n i t ~ ~ ~ , ~ ~ class of photoemissive devices utilizes photoconductive semiconductors, such as gallium arsenide, with a very thin surface coating that is most commonly cesium oxide. The production of this surface layer on the semiconductor causes bending of the energy bands of the semiconductor at its surface, as was shown in Fig. 7. The effect of this band-bending is to produce a condition where the energy level of the electron at the conduction-band minimum in the bulk of the semiconductor crystal is higher than the energy level of an electron in the vacuum. Therefore, an electron in the conduction band in the bulk of the material will energetically fall out of the photocathode semiconductor into the vacuum if it is able to reach the activated surface region without recombin- ing. Thus the conduction-band electrons require no energy above that of

3 08 DETECTORS

I I 1 I I I 1

0.4 0.6 0.8 1 Wavelength (pm)

FIG. 9. Spectral response of a typical GaAs and InAs, P,-, photocathode.

E, to escape from the photocathode, while in conventional photocathodes substantially higher energy is required for the electrons to escape into the vacuum [see Fig. 6(b)]. The existence of a region in the negative electron affinity photocathode, where the energy at the bottom of the conduction band exceeds the vacuum potential, is peculiar to this class of photocathodes and does not occur in the conventional photoemitters. The spectral response of a typical GaAs and an InAs,P,-, photocathode is illustrated in Fig. 9.

While the photocathode is the most crucial part of the photomultiplier, since it converts the incident radiation to electronic current and hence determines the spectral characteristics of the detector and the ultimate sensitivity, the remainder of the structure is important in achieving the high amplification and noise figure as well as the fast time response associated with the photomultiplier. A simple schematic of a photomultiplier-type structure is illustrated in Fig. 10. The electrons that are emitted from the photocathode are electrostatically focused and accelerated toward a


Dynodes I I I I

Photocathode I

Focusing electrode (semitransparent)

FIG. 10. Schematic structure of a photomultiplier.

sequence of electrodes termed dynodes. The electrons from the photocathode arrive at the first dynode with a kinetic energy of from 1 to 400 eV. Secondary emission from the surface of the dynode causes a multiplication of the initial current. Using dynode materials which are chosen primarily for high secondary emission ratios, the secondary emission ratio can be from 5 to 10 per electrode, depending on the primary electron energy. Conventional materials such as beryllium oxide (BeO), magnesium oxide (MgO), and cesium antimonide (Cs,Sb) are used as dynode materials in most photomultiplier wavelength systems. The gallium arsenide or gallium phosphide negative electronic affinity secondary emitters are also used in multipliers, since they have high secondary emission ratios. Secondary emission ratios of 5 to 50 are obtained over a primary electron energy range of from 100 to 600 eV. The process of secondary emission from a succession of dynode surfaces causes a large multiplication of the initial current. The process is repeated at a sufficient number of dynodes, so that the initial current emitted from the photocathode will be multiplied by a large factor. For a secondary emission ratio of 5 at each dynode, the multiplication between cathode and anode is approximately lo7 for a series of 10 dynodes. The last electrode in this chain is a collector anode without gain.

The structure of the electron multiplier chain can take on a variety of forms, each with special characteristics as illustrated in Fig. 11. A circular arrangement provides a structure which is both simple and compact with relatively high-bandwidth and electron-collection effi~iency.~’ A variety of other ladder or venetian-blind structures have been constructed which are compact, exhibit stable gain characteristics, and allow a relatively large number of stages, but with rather poor time response. The relationship between response and electron-collection efficiency is an inverse one, since the time response is largely determined by a spread in the time of flight of

310 DETECTORS

FIG. 11. Circular arrangement of an electron-multiplier chain. (After Sommer.”)

high-velocity and low-velocity electrons. The avoidance of the large spread in time of flight implies a focusing structure where electrons with unequal paths are either forced into compensating long and short pathlengths, respectively, or rejected as they progress between electrodes. High collection efficiency requires that all electrons be collected independent of their velocity or path.

The necessity of minimizing time-of-flight variations between the photoemissive surface and successive electrodes has given rise to photomultiplier designs such as the crossed-field photomultiplier. Here the magnetic field is used together with an electrostatic field to produce varying pathlengths depending on the electron energy, that is, high-energy electrons are forced into long pathlengths while low-energy electrons are induced into correspondingly short paths. This design reduces time-of-flight variations and produces rise times at the anode of about 0.1 ns as compared to several nanoseconds for various other photomultiplier designs. Another rather different photomultiplier is the so-called channel phot~mul t ip l ie r .~~ In this device, large numbers of parallel channels, each channel being a small diameter cylinder of material exhibiting electron gain, are used to form continuous dynode surfaces. Voltage is applied along the channel length, and multiple reflections of electrons traveling down the channel produce extremely high gain. Devices of this type, where channels with density as high as a million channels per square inch exist in parallel, are used in imaging, and with a high channel density, they can retain a high degree of image resolution.


Because of the high current amplification and the noise, the photomultiplier is one of the most sensitive instruments for detection of visible and near infrared radiation. The photomultipliers have been used to detect optical power levels as low as lo-'' W. As a detector of an optical signal modulated at some low frequency, the limiting factor in sensitivity is the so-called dark current of the photomultiplier. The random fluctuations observed in a photomultiplier output are due to either cathode or dynode shot noise, Johnson and 1/ f noise. The cathode shot noise is composed of a fluctuating current emitted by the photocathode due to incident signal power and another component observed in the absence of radiation, which is the so-called dark current arising from random thermal excitation of electrons. All electrons in the cathode which are thermally excited to energies greater than the work function or electron affinity are emitted, and each emitted electron is then effectively replaced by another electron from the cathode. The minimum detectable optical power is determined by the magnitude of this thermally generated dark current and typically leads to a minimum detectable power on the order of W for simple video detection. An obvious means of reducing the thermal dark current is to cool the photo~athode.~'*~" If the temperature of a conventional photoemitter is lowered to at least -2O"C, the thermal dark current becomes negligible. For systems with lower thermal activation energy, such as the negative electron affinity emitters, correspondingly lower temperatures are required. Cooling of this type has traditionally been achieved by using coolants such as dry ice to reduce the temperatures to a point where thermal carrier generation is reduced to negligible levels. More recently, thermoelectric cooling has been found to be more convenient and able to achieve temperatures which are precisely those required for optimum cooling, without reducing the tube temperature to a point where factors such as moisture and condensation become troublesome.

6.4.2. Photoconductive Detectors

For radiation with wavelength in both the visible4' and near infrared42 spectrum, photoconductive and photovoltaic semiconductor detectors are generally employed. A semiconductor photodetector is a simple device with high detectivity. Incident photons excite charge carriers from bound states into free states, increasing the number of mobile charge carriers, and these photoexcited carriers change the electrical properties of the material. The physical realization of this effect depends on the form and properties of the material. In a photoconductive sample, radiation is detected through an increase in electrical conductivity, while in a photovoltaic device it is detected as a photon-generated voltage. Excitation of carriers in a semicon-

312 D E r ECrO RS

hv > Eg

hu 2 Ed

\G Donor level

FIG. 12. Energy-band diagram for a photoconductive detector.

ductor may take place from states near the top of the valence band across the energy gap into states near the bottom of the conduction band, producing excess electron-hole pairs, or alternately from impurity levels creating either excess electrons or holes (Fig. 12). The former are designated intrinsic photoconductors, while the latter, in which the photoexcitation is from impurity levels, are termed extrinsic. Extrinsic detectors may incorporate either donor or acceptor impurities, leading to sensors where the radiatively induced carriers are either electrons or holes, respectively. Extremely long wavelength detection may be obtained by electron intraband excitations in materials such as InSb, yielding response at wavelengths in the 100-km to 1-mm range.

Intrinsic photoconductors respond to photons of sufficient energy to cause an interband transition on being absorbed, thus creating excess electron- hole pairs in the semiconductor. The semiconductor will absorb only photons with energy greater than the energy gap, EB, so that the maximum wavelength A which can be detected is

(6.11)

where h is the Planck’s constant and c is the speed of light. The photodetector can respond to all radiation with wavelengths shorter than A,,,, with maximum sensitivity for photon energies slightly greater than the gap energy. The photoresponse of several common infrared photoconductors is shown in Fig. 13. The variation in spectral sensitivity for photon energies greater


FIG. 13. Spectral response of several common infrared photoconductors.

than the gap arises from the strong absorption of the radiation which produces excitation only near the surface. The location of the maximum in the spectral response is dependent on the thickness of the crystal and the carrier recombination parameters.

The spectral regions in which intrinsic semiconductor photoconductors are capable of responding depend on the energy band gaps in elemental and compound semiconductors which have been developed. As shown in Table 111, the range from the visible spectrum to approximately 8 pm is well covered by a number of common semiconductors. While the points shown are for a fixed temperature, the band gaps of most semiconductors vary strongly with temperature, so that the maximum sensitivity for a given material will change if the detector is cooled. For example, when InSb is cooled from 300°K to 77"K, the energy gap increases from 0.18 eV to 0.23 eV causing a reduction in A,,, from 7 pm to 5.5 pm. Materials such as PbS

314 DETECTORS

TABLE I 1 I . Energy Band Gaps for Photoconductive Semiconductors

Band Gap Cutoff Wavelength Material ( e v ) (wm)

~

Si 1.10 1.12 Ge 0.68 1.82 lnAs 0.33 3.75 InSb 0.18 6.88 PbS 0.4 3.1 PbSe 0.25 4.95 PbTe 0.31 4.0 GaAs 1.4 0.88 CdSe 1.74 0.71 CdS 2.4 0.52

Hg,-,Cd,Te( x = 0.2) 0.09 14 Pb,-,Sn .Te(x = 0.2) 0.083 15

and PbSe have a positive temperature coefficient describing the band-gap variation and respond to longer wavelengths at lower temperatures. The spectral variation in sensitivity of PbS and PbSe detectors as a function of temperature is illustrated in Fig. 13. Cooling is usually required in infrared photodetectors to reduce competition between the carriers generated by photons and thermally generated carriers.

Beyond about 8 pm, there are no simple intrinsic elemental or compound detectors available. The range of intrinsic detectors has been extended beyond this point by alloying two compounds of different bandgaps to yield materials of a desired energy gap. When a compound such as CdTe with a band gap of 1.5 eV (300°K) is combined with HgTe, which is a semimetal in which the valence band and conduction band effectively overlap by 0.15 eV, the resultant band gap is dependent on the fractional composition of the compounds (Fig. 14). In the alloy Hg,-,Cd,Te, the gap varies nearly linearly with x between the two extreme values, so that it goes through zero at x = 0.10 at 300°K.42*43 For x = 0.2 at 77”K, the gap is approximately 0.1 eV, yielding an intrinsic detector in the 8- to 14-pm range where there is an important “window” in atmospheric transmission. Development efforts in this material have centered on this region, although narrow-gap Hg, -,Cd,Te detectors have been made with response beyond 40 pm.44

The binary, semiconductors Pb,_,Sn,Te and Pb, -,Sn,Se also have composition-dependent energy gaps which can be made small corresponding to detection at wavelengths beyond 10 p.m.45.46 Development of the Pb, - .Sn,Te alloy system has received significant emphasis in recent years particularly as photovoltaic detectors. The limiting wavelength response of


I I I I

1.6 - 1.4 -

1.2 -

m - - -

- -0.4 -

0 0.2 0.4 0.6 0.8 1.0 HgTe CdTe

FIG. 14. Dependence of the resulting band gap on the fractional composition of the compounds in Hg,-,Cd,Te.

a number of the elemental, compound, and alloy semiconductor photoconductors is listed in Table 111. It illustrates that the intrinsic semiconductors cover the visible and infrared spectrum to about 15 pm with a readily available set of photodetectors.

In practical use, the PbS and PbSe detectors are widely used in the infrared from 1 to 5 pm, with a reasonable sensitivity at room temperature, although an increase in responsivity and decrease in noise is obtained by operating at low temperature. The intrinsic photoconductors InSb and InAs can be operated at room temperature, but the large number of thermally excited carriers at room temperature limit their response, so that they are customarily cooled. The narrow-gap alloys Hg, -,Cd,Te and Pb, -,Sn,Te, used even further in the infrared, must always be cooled to about 77°K to adequately reduce the number of thermally generated carriers.

As shown in Fig. 13, a number of detectors have enhanced sensitivity and extended cutoff wavelengths when operated below 300°K. In addition, detectors used in the wavelength range greater than 10 pm must be cooled to prevent a large number of thermally excited carriers from competing with the photoexcited carriers. These factors have led to the widespread use of cooled photon detectors.

When a detector is to be cooled, it is frequently mounted in an insulated dewar flask, illustrated in Fig. 15. The walls of the dewar can be made of glass, Pyrex, or metal and may either be permanently sealed on manufacture

316 DETECTORS

Pump out PO rt

Electrical leads Electrical leads

FIG. 15. Cooling of a detector mounted in an insulated dewar flask.

or provide for periodic evacuation. Most infrared detector systems are usually cooled to liquid-nitrogen temperature (-77°K). The semiconductor sensor is usually cemented to the wall of the coolant container and connected by small-diameter wires to an external electrical connector. A window admitting radiation to the detector is selected for a spectral band pass matching the useful range of the detector and may be either removable or permanently mounted to the dewar.

Several other methods are also used to cool detectors, in addition to the common cryogenic fluids such as nitrogen. Several open- and closed-cycle refrigeration systems have been used, including Joule-Thomson and Sterling refrigerators. These systems are generally capable of achieving operating temperatures of 70-80°K, with a refrigeration capacity of 10 to 20 W. Units of this type have been used to cool detectors in remote-sensing satellite systems where other methods were not feasible.

Finally, there has been a resurgence of interest in solid-state refrigerators using the thermoelectric effect. Following the discovery of the thermoelectric effect by Seebeck, Peltier discovered that when current flows in a circuit of two dissimilar metals, heat is absorbed at one junction and released at the other. The practical use of materials for thermoelectric devices can be evaluated by the thermoelectric figures of merit:

(6.12)

where a is the thermoelectric coefficient (V/"K), p the resistivity (a-cm), and k is the thermal conductivity (W/"K cm). Semiconductors generally have the highest thermoelectric figure of merit, with the commonly used bismuth telluride having 2 = 3 x OK-'. With a value of Z such as


described, a temperature differential of 60-70°K can be obtained with a semiconductor detector as a heat load.

6.3.3. Extrinsic Photoconductors

The requirement for quantum detectors beyond wavelengths of approximately 10 pm frequently makes it necessary to use extrinsic semiconductors. Impurity or defect states in a semiconductor create discrete energy levels in the band gap of a semiconductor with relatively low activation energies. Carriers can be photoexcited from these levels near the conduction or valence band edges, creating a decrease in the detector’s electrical resistance which can then be measured, as was previously shown in Fig. 12. Impurity levels near the conduction band edge are called donor levels, since photoexcitation of carriers will cause extra electrons to be added to the number already in the band at thermal equilibrium. Levels adjacent to the valence band edge are called acceptor levels, since excitation of carriers into these levels will leave excess holes in the band. The nature of the impurity level depends on the valence of the impurity relative to the host semiconductor. Since these are states with very low activation energies, detectors based on photoexcitation from these levels must be cooled to prevent thermal excitation from masking the photoeffect. The technique is limited to semiconductors of a high degree of purity and crystal perfection where the level of incidental impurities and defects is small.

In materials which do meet these requirements, the ionization energy of isolated impurities can be approximately treated by a description similar to an isolated hydrogen atom. The ionization energy in a semiconductor with a dielectric constant K and an effective carrier mass m* is given by

m* E = 13.6 - m K 2 eV9 (6.13)

where m is the free electron mass. The ionization energy of an impurity state is much smaller than that of

a hydrogen atom, due to the difference in the dielectric constant and the effective carrier mass in the solid. For example, the donor ionization energy in GaAs with K = 13.5 and m* = 0.072 m is approximately, 0.005 eV, equivalent to an excitation wavelength of 200 pm. This far-infrared transition is observed in pure GaAs epitaxial films and is used as the basis for a far-infrared photoconductive detector.

In materials such as the elemental Group IV semiconductors, Ge and Si, with complicated energy band structure, application of the simple expression for ionization energy yields only an approximate result. Accurate calcula-

318

Conduction band

- Au

0 2

025

cu 0.33 - 0.04

Zn -

0.02

0.03 -

Egap = 0.75 eV

Ga

Valence band

FIG. 16. Typical acceptor impurity levels in germanium.

tions using correct anisotropic masses predict donor ionization energies of about 0.01 eV for germanium and 0.03 eV for silicon. The ionization energies predicted are valid only for elements removed by the valence group of the host element by 1. For example, in Ge, elements of Group I l l are impurities with a single acceptor level, and elements of Group V are impurities with a single donor level with an ionization energy of approximately 0.01 eV. The typical acceptor impurity levels in germanium are illustrated in Fig. 16. Elements further removed from Group IV have more than a single level with substantially different ionization energies, as shown in Fig. 16.47*48*49 Impurity ionization energies in Ge that lead to extrinsic detectors, which are of practical importance, yield photoconductive response as shown in Fig. 17. The small values of most donor and acceptor ionization energies are comparable with thermal energy (0.025 eV) at room temperatures, so that the extrinsic photoconductors must be cooled to reduce thermal ionization.

For wavelengths shorter than the maximum, the radiation is strongly absorbed and produces excitation only near the surface. Since it is common that the surface imperfections are characterized by rapid nonradiative recombination, the excitation of photoconductivity and luminescence is less when limited to the surface. The location of the maximum in the spectral response is dependent on the thickness of the crystal and the recombination parameters. For wavelengths longer than the maximum, the excitation decreases simply because the absorption producing free carriers is decreasing.

1 .o

PHOTON DETECTORS

\ Ge:Au

Solay cell

I I 1 1 1 1 l 1 I I I I I 1 I l l I I 1 1 1 1 1 1

10 100 Wavelength ( rm)

1000

319

FIG. 17. Spectral response of some extrinsic photoconductors.

6.4.4. Photoconductive Response

A photoconductor is operated using a simple electrical circuit consisting of the detector, an external bias supply, and a load resistance. The photosignal generated when radiation is incident on the detector is usually measured across the load resistance, as shown in Fig. 18.

When a photoconductor with a bias voltage V, responds to radiation by a change in conductivity us, a signal voltage V, is produced, where

Vs=(: )V", (6.14)

with (T being the total detector conductivity. For an absorbed power per unit volume P,/Ax, the signal photoconductivity induced by the radiation

320

/

- / 1 1 1

DETECTORS

/ /-

/ Semiconductor /

Photo voltage

Bias SUPPlY

Load resistor

Radiation

contacts

FIG. 18. Measurement of the photosignal generated when radiation is incident on a photoconductive detector.

is

(6.15)

where q is the quantum efficiency, A is the wavelength, and p is the carrier mobility. The detector is assumed to have a thickness x and an illuminated area A. The spectral responsivity RA will then be given by the expression

RA =- qA "0' (volts/watt), hcnxA

(6.16)

where n is the carrier concentration. The response of a photoconductive detector depends on having a material with a low thermal equilibrium carrier concentration, small volume, high quantum efficiency, and a long carrier lifetime. Since the excess carrier lifetime is the parameter which determines the response time of the detector, the requirement for a long lifetime for high responsivity conflicts with a short photoconductive response time. Typical response times in intrinsic photoconductors range from lO-'sec to sec, making these detectors very superior in this respect to thermal detectors with equivalent sensitivity. The responsivity also has a linear bias field dependence which is, of course, limited in practice by either Joule heating or other high field effects. Operation of these detectors at low temperatures will typically reduce the equilibrium carrier concentration and

PHOTON DETECTORS 3 2 1

increase the excess carrier lifetime, leading to a significant increase in R,. The spectral detectivity 0: can be written as

( 6 . 1 7 )

where Af is the noise bandwidth and vT is the total noise voltage. Reducing the detector temperature will reduce the thermal noise component, so that the noise will be dominated by generation-recombination noise. The total effect of cooling is a significant increase in detectivity, as was shown for several intrinsic photoconductors in Fig. 13.

In addition to specification of responsivity and detectivity, the performance of either extrinsic or intrinsic photoconductors under steady-state incident radiation can be described in terms of the short-circuit photocurrent or the open-circuit photovoltage modes. The short-circuit photocurrent Z,, is typically given by Z,, = TqNg, where 7 is the quantum efficiency, q is the electronic charge, N is the number of photons absorbed in the sample per unit time, and g is the gain of the photodetector. If the photoconductive gain and photon-absorption rate are expressed in terms of fundamental materials parameters, the short-circuit photocurrent can then be written as

( 6 . 1 8 )

Here P is the absorbed power at a particular wavelength, A, V, is the bias voltage applied to the sample, 1 is the distance between electrodes on the sample, p is the majority carrier mobility, and T is the majority carrier lifetime. From this expression it can be seen that the low-frequency photocurrent is directly proportional to the absorbed power and also depends directly on the wavelength for values of the wavelength below the long-wave cutoff limit associated with each individual semiconductor. An important observation is that the photoconductivity depends on the lifetime and mobility of the majority carriers, since photoconductivity is in fact a majority-carrier phenomenon. In a photoconductive device, minority carriers, which typically have lifetimes significantly shorter than the majority carriers, contribute little to the change in photocurrent and hence can be ignored to a first approximation.

If a detector is inserted into an electronic circuit with some effective load resistance, the photocurrent in the total circuit is modified. The photovoltage which appears across such a load resistance can be simply expressed in terms of the product of the short-circuit current and the equivalent load resistance. If the load resistance, R, , is large with respect to the detector resistance, Rdr over its range of operation, the photovoltage is essentially

322 DETECTORS

the open-circuit photovoltage, Voc. When this is the case, the photovoltage V,, can then be written as follows

(6.19)

Here the detector resistance was expressed in terms of its majority carrier concentration n, mobility p, and respective length, width, and thickness 1, w, and x. As in the case of the short-circuit current, the open-circuit voltage is then dependent on the incident power, the wavelength, and bias and is inversely proportional to the majority-carrier concentration in the device. These expressions for short-circuit current and open-circuit voltage, are low-frequency or steady-state descriptions, and do not express information about the response time of the device which is crucial in many applications.

Since a photoconductor is, in fact, a majority-carrier device, the frequency response of the open-circuit voltage V, is then given by the expression

where w is the angular frequenccy of the modulation of wavelength of radiation of wavelength A. At low frequencies where the product W T is much less than 1, the device response is essentially independent of frequency, while at high frequencies where OT becomes much larger than 1, the response is inversely proportional to frequency and depends explicitly on the majority-carrier lifetime, which is thus the primary limit on high-frequency performance of photoconductive devices.

6.4.5. Photovoltaic Devices

A mode of detection of great practical significance is that using the photovoltaic eff e ~ t . ~ ~ While this effect relies on direct conversion of photons to carriers in the material, as does photoconductivity, it places additional requirements on the nature of the detector in that some form of potential barrier with an internal electric field is required to cause a separation of the photoexcited electron-hole pair. Photovoltaic detectors are constructed from materials where the intrinsic photovoltaic effect arising from band-to- band transitions generates electron-hole pairs which are subsequently separated by an internal field and separately collected at opposite electrodes of the device. The photovoltaic effect is essentially limited to materials where intrinsic photodetection occurs, although there have been a few limited reports of extrinsic photovoltaic detectors. The simplest and classical structure for separation of the photoexcited electron-hole pairs occurs in the


junction

simple p-n junction; however, a number of other structures have recently been vigorously investigated.

By far the most common form of photovoltaic effect is that which occurs in a p-n junction fabricated in a common semiconductor, such as silicon, by doping or formation of epitaxial layers at one surface. A simple p-n junction photodiode is illustrated in Figure 19, where two electrical contacts are formed and a thin region at the surface has been prepared to allow the incident radiation to be absorbed and to generate electron-hole pairs. The photoexcited electrons and holes are separated by the high field in the space

n-type P-n junction

Incident radiation

n-type

Top surface Top contact

I 1 Contact

N-region P-region

Conduction band

Valence band

FIG. 19. (a) A simple p-n junction photodiode. (b) Associated energy-band diagram.

3 24 DETECTORS

Current

Saturation current

current Dark \ Ill Voltage

Photovoltage (open

Photocurrent Photocurrent (short

circuit)

circuit)

IT-$ Radiation +~ Photodiode

Bias - source+ '6

I + R L I Load resistor Photovoltage

FIG. 20. (a) Observation of photovoltage and photocurrent in a photovoltaic device. (b) Circuit to observe the photovoltage.

charge region between the p and n regions. This charge separation creates an open-circuit voltage across the junction and leads to a voltage which can be detected that is proportional to the absorbed photon density. The direct observation of this photovoltage is the simplest and most straightforward mode of operation of a photovoltaic detector [Fig. 20(a)]. Junction detectors are often operated as well under some form of reverse bias voltage which leads to observation of a photocurrent [Fig. 20(a)] rather than a photovoltage and are said to be operated in a photoconductive mode, despite the fact that it is not equivalent to the photoconductive mode previously described.

The short-circuit current associated with the photovoltaic detector is identical in form to that for a photoconductive detector, with the exception


that the gain is unity, that is, the short-circuit current is given by

nqPA hc

I,=-, (6.20)

The open-circuit voltage, which is the commonly observed photovoltaic signal, is obtained as the product of the short-circuit current with the dynamic resistance of the junction in the absence of bias. Hence the open-circuit photovoltage V,, for a photovoltaic detector is approximately

(6.21)

Here I,,, is the saturation current of the junction in the absence of a photo signal, k is Boltzmann’s constant, and T is the absolute temperature. Just as in the case of the photoconductive detector, the photovoltaic signal depends on the incident power and wavelength, but it does not depend on the majority carrier lifetime. The photovoltaic effect, in fact, is essentially independent of the majority-carrier lifetime, since in creating an electron- hole pair, the determining factor is the minority-carrier lifetime, which is significantly shorter. Since the frequency response is determined by the product of the modulation frequency of the radiation w and the minority- carrier lifetime 7, the response is essentially independent of frequency for values of W T < 1. The point at which WT becomes unity is at a significantly higher frequency for a photovoltaic device; therefore, photovoltaic detectors generally display a higher frequency response or a faster time response than photoconductive detectors of exactly the same material.

The most widely used photovoltaic detectors for the visible spectrum rely on the common elemental semiconductors Ge and Si with cutoff wavelengths of 1.5 pm and 1.1 pm, respec t i~e ly .~~ For applications at longer wavelengths, photovoltaic structures have been developed using 111-V compounds or the alloy systems Pb, -,Sn,Te and Hg, -,Cd,Te. For optical communications systems, detectors utilizing Ge”,52 or 111-V alloys, such as InxGal-xAsyP1-y,53-55 have been emphasized with peak response in the 1.1- 1.7-pm range. For longer-wavelength infrared extending to wavelengths as long as 14 pm, the Hg,-,Cd,Te and Pb,-,Sn,Te alloy systems have been vigorously de~eloped.’~-’~

While the discussion of photovoltaic detection has centered on the classical p-n junction, there are a number of structures which have been employed for photovoltaic detection. The most direct extension of the simple p-n junction is the p-i-n photodiode which incorporates an intrinsic region between the p and n portions of the traditional junction. If either the p or n side of the junction is made very thin compared to the optical absorption

length in the medium, the incident optical radiation will readily penetrate into the intrinsic region. In the intrinsic region, the absorption will produce a large number of electron-hole pairs. As these electron-hole pairs are produced in a region of high electric field, the carriers will rapidly be drifted out of that region into the adjacent p and n sides of the junction. This rapid carrier separation leads to an improved frequency response and efficiency in the p-i-n diode as compared to that of a pn diode of a comparable material.

In addition to being used to enhance detection, p-i-n configurations have also been employed in so-called avalanche photodiodes.""."' An avalanche photodiode is a structure with an internal gain achieved through the so-called avalanche breakdown effect, which occurs in the high field regions of a p-n or p-i-n junction under high reverse bias. The carriers generated by photoexcitation are accelerated to velocities which are sufficiently high so that upon collision with atoms in the lattice, additional free electrons are generated by the impact ionization process. These free electrons are also accelerated and undergo more collisions and hence will also lead to the generation of more free electrons. The so-called avalanche effect, which occurs within a high field region in a reverse bias p-n or p-i-n junction, can lead to a significant internal gain."2 Hence, a larger photosignal is produced in a material with the same incident radiation power than would be produced under conventional diode operation. While the photogenerated current for a p-i-n diode is as previously described by Eq. (6.20), for conventional photovoltaic devices the value of current, I,", in an avalanche diode is

nqPA I,, = - M, hc

(6.22)

where M is the avalanche gain. The gain depends on device design and bias conditions and may be as high as 100. The avalanche process is a statistical one, where M fluctuates for each chain of events. Hence the amplification process in these devices is subject to an additional noise- generation process.

In many semiconductors, particularly those of significance for infrared detection, production of both n- and p-type materials of high quality is not always possible within the current state of the art. A method for forming photovoltaic structures in materials where p-n junctions cannot be formed is through creation of a Schottky barrier ph~todiode." ' .~~ A Schottky barrier is a junction formed at the interface of a metal semiconductor structure. In a manner analogous to that described for the p-n junction, a potential barrier is created at the metal-semiconductor interface, which will effectively separate electron-hole pairs generated by incident optical radiation. A diagram of a typical Schottky barrier junction is shown in Fig. 21. In most


EF -

Metal Semiconductor

FIG. 21. Diagram of a typical Schottky barrier function.

cases, the Schottky barrier is formed by producing a thin film of metal on the surface of the semiconductor, where the thickness of the metal is sufficiently small so that it is partially transparent to incident radiation at the wavelengths of interest. Incident photons can generate electron-hole pairs within the semiconductor or in the vicinity of the potential barrier at the metal-semiconductor interface. Depending on the specific structure, the Schottky barrier photodiode may be illuminated either from the metal side of the junction or it may, in fact, be illuminated through the semiconductor if it is made sufficiently thin so that carriers can reach the interfacial region.

6.4.6. Photon-Drag Detector

While most semiconductor quantum detectors respond to radiation by a photo-induced change in free carrier concentration, the photon-drag detector operates through a change in carrier momentum as photon energy is absorbed.65 The simplest description of photon drag attributes the phenomenon to radiation pressure which transfers momentum to the mobile carriers in the semiconductor. As the radiation propagates through the semiconductor, it acts upon mobile carriers, affecting carrier transport in the direction of light propagation. A voltage called the “photon-drag voltage” is generated between a contact at the face of the sample where the radiation enters and another contact where it leaves the sample. The photon- drag effect has been exploited for use as a high-speed room-temperature infrared detector, with wide use for high-power CO, lasers at 10.6 pm. While the simple description of the effect is accurate for C 0 2 lasers operating

328 DETECTORS

around 10.6 pm with the photon-drag detector at room temperature, observations of the frequency and temperature dependence indicate that for wider use, this description is inadequate. Gibson ef al.66.67 have shown that in germanium, a more accurate representation must take into account the differences in energies associated with interband transitions of electrons originating in quantum states of the same energy but different wavevectors when the transitions also involve momentum transfer from the photon filed to the electrons. Based on this model using an accurate band structure for p-type germanium, the frequency and temperature dependence of the effect can be explained.

While photon drag has been observed in compounds such as InSb and GaAs. Ge has been found to be the most efficient detector material and is used in commercially produced detectors. The responsivities in typical Ge detectors are on the order of 1 x lop6 V/W. While having a very low responsivity, the photon-drag detector does have a linear response at extremely high incident intensities and a very high frequency response. Most detectors are linear up to incident power density of greater than 10 MW/cm2 with response times of less than 1 nsec. It has been proposed that the response time is limited only by the transit time of photons through the detector.

6.5. Noise

A fundamental limit exists in the minimum radiation power that can be detected by all forms of detectors, arising from the noise that is present in addition to the Noise may arise from within the detector, may be associated with the source of radiation, or may appear in the electronic system used to enhance the electrical output signal of the detector. The latter element can be largely eliminated through attention to the proper electronic design and may, in fact, include certain low-noise amplifiers that can reduce electronic contribution to the noise to a very small level. The noise which actually arises due to mechanisms internal to the detector generally falls into several distinct categories.

Many sources of noise are common to or appear in most forms of radiation detectors. In particular, almost all forms of noise are present, to at least some degree, in the solid state detectors used to detect most forms of visible and infrared radiation. In the absence of electrical bias, there is a low-level source of noise usually described as Johnson noise. This type of noise is generated by the random motion of charge carriers in any resistive medium. The Johnson noise is present at a power level which is determined by the temperature of the material and the bandwidth of the detection system. In addition to these factors which determine the noise power for Johnson

NOISE 329

I I I I 1 1 I I 1 I

- -

-

Generation-recombination noise -

Johnson noise

-

I I 1 I I I I I I I 2 4 6 8 10

noise, the noise voltage and current also depend on the resistance value of the detector. The mean square Johnson noise power, which then appears even in the absence of electrical bias as a fluctuating current or voltage, is given by P, = kTB, where k is the Boltzmann's constant, T is the absolute temperature of the detector, and B is the measurement of bandwidth. The open-circuit noise voltage V, is then given by

V, = (4kTRB)' /2 , (6.23)

where R is now the resistance of the detector. In most solid state photodetectors, the Johnson noise is present at all frequencies, but it is not the dominant noise, except at high frequencies. At lower frequencies, several other sources of noise dominate. This is illustrated in Figure 22, which illustrates an ideal noise spectrum for a typical solid-state photodetector in the absence of radiation. At low frequencies, another noise source, called l/f noise, dominates, while at intermediate frequencies generation-recombination noise dominates. The specific ranges over which each of these noise sources is predominant varies according to the details of the composition and structure of the detector, but typically below 1 kHz is the range for l/f; which is the principal noise source. Between 1 kHz and as high as 1 MHz, generation- recombination noise would be the dominant noise, while above that frequency, the Johnson noise would be the predominant noise source.

At the lowest frequencies, the source of noise found in most solid-state detectors is l/f noise characterized by a spectrum where the noise power

FIG. 22. Different types of noise of a solid state photodetector.

330 DETECTORS

varies approximately inversely with frequency. Most solid-state detectors usually exhibit l/f noise at low frequencies. Although at higher frequencies, the amplitude drops below that one or below the other types of frequency- independent noise. The mechanism for l/f noise is not well understood. It has been attributed to the presence of potential barriers at contacts or at the surface of the detector. The understanding and reduction of l/f noise is then a process which depends on the procedures used in preparation of the contacts and surfaces of the detector and is very much an art. The noise current associated with l/f noise is often expressed as

I , = ( c1;- , (6.24)

where C is a constant, Ib is the bias current, B is the bandwidth, and f is the frequency. There are often exponents associated with the frequency given here by a, where a is approximately, although not always, precisely equal to unity.

A third and very important form of noise found in most solid-state detectors is the so-called generation-recombination noise. Generation- recombination noise is caused by fluctuation in the rate of thermal generation and recombination of charge carriers in a solid, giving rise to a fluctuation in the average charge-carrier density. Because of this variation in carrier density, the resistivity will also fluctuate, and this will, of course, give rise to a fluctuating voltage at the terminals of the detector when a fixed bias current is flowing through the solid. In many solid-state detectors, the fluctuations in charge-carrier concentration will arise either from intrinsic carrier generation, that is, the creation of free electron-hole pairs by the thermal energy of the lattice, or by extrinsic carrier generation where fluctuation in the number of carriers can arise from the generation and recombination rates associated with an impurity level. For a simple intrinsic photoconductor with a single type of thermally generated carrier, usually electrons, the noise current will be given as

I =(%)([*]”” 1 + w 2 r 2 (6.25)

where p is the total number of free holes in the sample, r is the free-carrier lifetime, B is the measurement bandwidth, and n is the total number of electrons in the material. It is clear that when the frequency becomes large with respect to the free-carrier lifetime, the noise power would diminish, which will limit the upper frequency limit over which generation and recombination noise is an important factor.

OPTICAL WINDOW MATERlAL 33 1

A final form of noise which is found in photo-emissive devices is that of shot noise. The shot noise in a photoemissive detector is equivalent to the generation-recombination noise in a semiconductor in that it arises from randomness in the emission of electrons. This noise is generated even at low temperature where thermal carrier generation can be neglected. The shot noise in a photomultiplier includes both the cathode shot noise due to random current fluctuations in the average current emitted by the photocathode due to the incident signal power, and the dynode shot noise due to random secondary emission at the dynodes. Since the current originating at a dynode does not see the full gain contribution of the tube, the dynode shot noise is a small fraction of the cathode shot noise.

6.6. Optical Window Material

One of the limitations in the spectral response in many detector systems is associated with the transmission spectrum of optical window material used to protect the detector element. There are approximately one hundred optical materials with a useful transmission band for application as windows. A variety of physical properties can be used to select a fraction of these as technologically significant. The most significant physical properties in window selections are spectral transmittance and the index of refraction of the material. In addition, a number of mechanical and thermal properties are often of importance, including: hardness, thermal conductivity and

TABLE IV. Optical Window Materials

Material Cutoff Wavelengths (pm)* Index of Lower Upper Refraction

Borosilicate Glass 0.3 Fused Silica 0.18 Sapphire 0.15 Magnesium Fluoride (IRTRAN-1) 0.45 Magnesium Oxide (IRTRAN-5) 0.40 Calcium Fluoride (IRTRAN-3) 0.15 Zinc Sulfide ( I RTRAN-2) 0.5 Sodium Chloride 0.2 Zinc Selenide ( I RTRAN-4) 0.5 Cadmium Telluride (IRTRAN-6) 0.9 Silicon*t 1.2 Germaniumt 1.8 KRS-5 0.5

2.8 4.3 6.5 9.5 9.5

12. 14.5 2.0

22. 30. 55. 5 5 . 55.

1.49 1.44 1.73 1.38 1.68 1.40 2.23 1.51 2.41 2.65 3.46 4.01 2.35

* Transmission of 10% through 2-mm specimen thickness. t Lattice bands cause discontinuous transmission spectrum.

332 DETECTORS

expansion coefficients, specific heat, elastic moduli, and variations of all of these with temperature.

There are about 10 to 12 materials that experienced optical-system design- ers incorporate in most practical systems. These are described in Table IV, which indicates the useful transmission range of each material and its index at the center of this range. Those materials useful as windows in the near ultraviolet and visible part of the spectrum include borosilicate glass, fused silica, sapphire, and calcium fluoride. In the infrared region, there are a number of elemental and compound semiconductors with useful passbands including Ge, Si, CdTe, ZnSe, ZnS, and MgO. As can be seen from Table IV, most of these semiconductors have a high index of refraction so that antireflection coatings are necessary for efficient transmission. A number of the latter group are available commercially prepared by a powder hot pressing process under the trade name IRTRAN.

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334 DETECTORS

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55. M. A. Washington, R. E. Nahory, M. A. Pollack, and E. D. Beebe, “High Efficiency In,~.,Ga,As,.P,_,/lnP Photodetectors with Selective Wavelength Response Between 0.9 and 1.7 km,” Appl. Phys. Lett. 33, 854 (1978).

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Noise, McGraw-Hill, New York, 1958.

A Brewster angle, 116, 121, 122 brightness temperature, 241 Burch interferometer, 20

Abbe theory of the microscope, 86 aberration correction by holography, 197 aberrations, 83

in holography, 172

devices 1 and 2, 133 wave plates, 144

Achromatic

acoustical holography, 197 acousto-optic effect, 150 Airy disc, 67, 80 angle measurements, 41 angular resolution of telescopes, 81 anisotropic media, 133 antireflection coatings, 33 astronomical radiometers, 249 atmospheric sounding techniques, 236

B

Babinet compensator, 143 principle, 68

band-pass filters, 33 beam expansion in holography, 200 beam splitters, 33 Bennett-Koehler reflectometer, 274 Bessel function, first order, 66 binary hologram, 189 birefringence

in crystals, 133 induced, 146

curves, 223 radiation, 219

blackbody

blaze angle in gratings, 78 blazed gratings, 78, 158 bleaching of holograms, 204 bolometers, 295 Bragg

effect in holograms, 172 regime, 151

C

channeled spectrum, 28 character recognition, 197 chromatic dispersion in gratings, 72 circular

aperture, diffracting, 58, 66 polarization, 110 polarizers, 124

coded reference beam for color holograms,

coherence length, 6, 41 color holograms, 180 common-path interferometers, 19 compensated interferometer, 11 compensating

plate, 11 retardation plate, 143

182

compensators, 143 complex plane representation of

polarization, 118 composite holography, 184 computer hologram, 189 conjugate image in holography, 170 Cornu spiral, 56 Cotton-Mouton effect, 153 crystal

polarizers, 138 retardation plates, 142 types, 135

cubic crystals, 135

0

depolarizers, 153 detection, synchronous, 279 detector polarization, 158

335

336 INDEX

detector Golay cell, 298 photon-drag, 327 pyroelectric, 299 quantum, 303

figures of merit, 292 liquid crystal, 302 luminescent, 302 noise, 328 photoconductive, 3 I 1 photon, 303 thermal, 293, 303

mirrors, 33 Fabry-Perot polarizers, 156 etalon, 31

detectors, 291

dichroic

dichroism, 33 interferometer, 25 dichromated gelatin for holography, 202 dielectric tensor and energy distribution,

Faraday effect, 153 field

133 equations, 108

electromagnetic description of light, 108 elliptical polarization, 113 embossing of holograms, 206 energy

distribution in gratings, 75 theorem, 70

equal-thickness fringes, 8 etalon, 41 experimental procedures in holography, 199 extraordinary beam, 136 extrinsic photoconductors, 3 17

F

diffracting circular aperture, 58, 66 rectangular aperture, 65 slit, 57, 65 straight edge, 57 two-dimensional array of holes, 80

images with aberrations, 83 diffraction, 49

dipole radiation pattern, 98 directional reflectance, measurement of, 274 division

of amplitude, 4 of wave front, 1

effect, 42 interferometry, 42

Doppler

double exposure holographic interferometry, 193

double-pass interferometer, 37

E

echelles, 79 echellettes, 79 echelon, 31 edge waves, 52 Edser-Butler fringes, 28 electro-optic

effect, 147 modulators, 147

measurements, 266 figures of merit for detectors, 292 filter radiometers, 257 finesse of interference fringes, 27 first order plate, 143 Fizeau interferometer, 8 four mirror polarizer, 124 Fourier transform, 62, 69

Fraunhofer holograms, 178

diffraction, 62 hologram, 175

biprism, 3 diffraction, 53 equations, I13 hologram, 175 integrals, 56 rhomb, 133 zone plate, 59

of equal chromatic order, 28 of equal inclination, 5 of equal thickness, 4

Fresnel

fringes

G

Gabor plate, 61 ghost lines in gratings, 75 Clan-Foucault prism, 141

INDEX 337

Glan-Taylor prism, 141 Clan-Thompson prism, 139 Golay cell, 297 grating

angular distribution of light, 70 disperser radiometer, 261 polarizers, 157

blazed, 158 chromatic dispersion, 72 resolving power, 72

gyroscope, optical, 43

gratings

H

Haidinger fringes, 11 half-tone hologram, 190 Helmholtz-Kirchhoff theorem, 50 Herpin matrix, 129 hertzian polarization, 157 hologram embossing, 206 hologram, Fourier transform, 178

Fraunhofer, 175 Fresnel, 175 Lippman-Denisyuk, 185 binary, 189 color, 180 computer, 189 half tone, 190 image plane, 179 phase, 190 rainbow cylindrical, 189 volume reflection, 174

rainbow, 184 reflection, 172 transmission, 172 volume, 172

bleaching, 204 interferometry, 191 materials, 201 microscopy, 195 optical elements, 197 processing, 204 recording, 204 stereograms, 184

holography, 167 applications, 191 composite, 184

holograms, effects of thickness in, 172

holographic

Huygens wavelet, 49 Huygens- Fresnel

principle, 49, 54 theory, 5 1

I

image-plane hologram, 179 imagery through diffusing media, 196 imaging radiometers, 252 impermeability tensor, 146 inclination factor, 52, 56 induced birefringence, 146 integrating spheres, 280 interference, 1

filters, 31 fringes, 1

interferometric radiometer, 261 interferometry applications, 38 irradiance interferometer, 24 isotropic

crystals, 135 media, 113

J

Jamin interferometer, 12 Jones’ matrices, 159

K

Kerr effect, 147, 153 kinoforms, 190 Kirchhoffs diffraction theory, 50, 5 1 knife-edge test, 39

L

lambertian sphere, 153 laser power, measurement of, 277 lateral shearing interferometer, 17 length measurements, 41 light source polarization, 158 light sources

for holography, 199 for interferometers, 6

linear polarization, 110 linear polarizers, 126, 130 Lippman-Denisyuk hologram, 185 liquid crystals, 155 liquid-crystal detectors, 302

338 INDEX

Littrow configuration, 158 Lloyd’s mirror, 1 localized fringes, I2 luminescent detectors, 302 Lummer-Gehrke plate, 30 Lyman ghosts, 76

M

MacNeille polarizing prisms, 130 Mach-Zehnder interferometer, 12, 17 magneto-optic effect, 153 matched filtering. 197 materials for optical windows, 331 Maxwell equations, 108, 134 measurement

of directional reflectance, 274 of emissivity, 269 of laser power, 277 of reflectivity, 272

measuring wave plates, 145 mechanical stability in holography, 200 metal grid polarizers, 157 metal mirrors protection, 33 metals, wave propagation for, 120 meter definition, 41 Michelson

fringes, 12 interferometer, 4, 9 stellar interferometer, 22

Michelson- Morley experiment, 38 microscopy, 42 microtopography. 29 Miyamoto- Wolf theory, 53 modulation transfer function, 85 Mooney rhomb, 133 Mueller’s matrices, 159 multilayer films, 35, 128 multiple reflection interferometer, 30 multiple reflections in a thin film, 126 multiple-beam interferometer, 25 multiple-pass

Fizeau interferometer, 37 Twyman-Green interferometer, 37 interferometers, 37

Fizeau interferometer, 28 interferometers, 25

multiple-reflection

N

Newton interferometer, 7 rings, 4, 5

Nicol prism, 139 noise in detectors, 328 nonlinear effect in holography, I72

0

off-axis holography, 168 one-step method for rainbow holograms.

optic axis of a crystal, 135 optical

186

activity, 138 elements, holographic, 197 polarization, 107 transfer function, 82, 84 window material, 331

ordinary beam, 136 orthoscopic recording in holography, 186

P

P-polarization, 114 Parseval’s theorem, 69 partial coherence, 89 partially transparent materials, 231 path radiance, 236 phase

contrast microscope, 90 difference in gratings, 72 holograms, 190 modulation, 146 plate, 91 retarders, 126, 130 shift upon reflection, 115 velocities in a crystal, 136

photocathodes, spectral response of, 307 photoconductive

detectors, 3 1 1 response, 3 19

photoconductors, extrinsic, 317 photographic emulsions for holography, 201 photometers, 246

lNDEX 339

photometric and radiometric

nomenclature, 214 symbols, 214 units, 214

measurements, 263 standards, 264 units, 286

photometry, 213 photomultiplier, 304, 309 photon detectors, 303 photon-drag detector, 327 photoresists for holography, 204 photovoltaic devices, 322 pinhole camera, 61 plane hologram techniques, 181 plane-holograms, 169 Pockels electro-optic effect, 147 PoincarC sphere, 112, 142 point spread function, 85 polarization

circular, 110 determination, 159 effects, 128

in metals, 120 in thin films, 125

elliptical, 113 linear, 110 mathematical description, 159 optical, 107 scramblers, 153

made of four mirrors, 124 made of three mirrors, 123

grating, 157 linear, 126, 130 metal grid, 157 slit, 157

polarizer

polarizers,

polarizing prisms, MacNeille, 130 principal angle, 122 prism

beam splitters, 138 disperser radiometer, 260 polarizers, 139

processing of holograms, 204 pseudoscopic configuration in holography,

pupil function, 68 187

pyroelectric detector, 299 materials, 301

pyroheliometers, 246

Q

quantum detector, 303 quarter-wave stack, 35

R

radial shear interferometer, 18 radiance temperature, 241 radiation temperature, 239 radiative transfer, 223

in a vacuum, 228 in transparent media, 231

interferometric, 261 prism disperser, 260, 261

astronomical, 249 filter, 257 imaging, 252 scanning, 252 space defense, 250 standard, 246

instruments, 246 measurements, 263 standards, 264 temperature measurements, 239

of visible light, 284

cylindrical hologram, 189 holograms, 184

radiometer,

radiometers, 246

radiometric

radiometry, 213

rainbow

Raman-Nath regime, 151 ratio temperature, 241

Rayleigh difference, 243

interferometer, 22 scattering, 99

real-time holographic interferometry, 191 recording of holograms, 204 rectangular aperture, 65 reference beam in holography, 170 reflectance, directional measurement of. 274

340 INDEX

reflection circular polarizer, 124 diffraction gratings, 78 holograms, 172 polarizers, 122

reflectivity, measurement of, 272 reflectometer, Bennett-Koehler, 274 refractometry, 39 relativity measurements, 38 resolving power

in gratings, 72 of optical instruments, 79

resonance anomalies, 157 retarding waveplates, 142 reversal shear interferometers, 19 ring interferometer, 42 Rochon prism, 138 Ronchi-test, 39 rotational shear interferometers, 19 Rowland ghosts, 76

S

S-polarization, 114 Sagnac interferometer, 42 saltus problem, 52 sandwich holography, 194 scalar theory region, 157 scanning radiometers, 252 scatter-plate interferometer, 20 scattering, 49, 92

by large objects, 94 by small particles, 95 cross section, 100

schlieren techniques, 39 second Brewster angle, 121 senarmount compensator, 144 shearing interferometers, 17 signal-to-noise ratio, 247 single layer films, 125 single-layer coatings, 33 single-slit diffraction, 53 slit polarizers, 157 slit, diffracting, 57, 65 solei1 compensator, 143 space defense radiometers, 250 spatial

filtering of images, 91 frequencies, 84 multiplexing method for color holograms,

181

spectral response of photocathodes, 307 spectroradiometers, 246, 256 spectroscopy, 43 spherical Fabry-Perot interferometer, 29 standard

lamp, 6 radiometers, 246

standards, radiometric and photometric, 264 star diameter measurements, 39 stereograms, holographic, 184 Stokes parameters, 159 straight edge, diffracting, 57 synchronous detection, 279

T

temperature measurements, 213 radiometric, 239

temporal coherence, 6 thermal detectors, 293, 303 thermocouple calorimeters, 293 thermography, medical, 243 thin films, 32, 125

matrix, 129 polarization effects, 125

Thompson scattering, 101 three mirror polarizer, 123 three-reference beam method for color

time averaged holographic interferometry,

total internal reflection devices, 131 transfer function

of lenses, 40 optical, 82, 84

diffraction gratings, 78 holograms, 172 polarizers, 122 volume color holograms, 183

holograms, 181

194

transmission

two dimensional array of holes, diffracting,

two-beam interferometers, 7, 22 two-step method for rainbow holograms,

Twyman-Green interferometers, 12

80

184

U

uniaxial crystals, 135

INDEX 34 1

V wave-front topography, 40 white-light compensation, 6 Wien displacement law, 220 Wollaston prism, 138 Wood anomalies, 157

visible light, radiometry of, 284 volume hologram, 172

volume-reflection hologram, 174 techniques for color holograms, 183

W

wave equations, 108 plates

achromatic, 144 measuring, 145

for metals, 120 in anisotropic media, 133 in isotropic media, 113

propagation

Y

Young's experiment, 1

2

zone plate, Fresnel, 59

physical optics and light measurements

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