Progress in Optics, Volume 27

EDITORIAL ADVISORY BOARD G . S. AGARWAL,

Hyderabad, India

C. COHEN-TANNOUDJI, Paris, France

F. GORI,

Rome, Italy

A. KUJAWSKI,

Warsaw, Poland

A. LOHMANN,

Erlangen, F.R . G .

M. SCHUBERT,

Jena, G.D.R.

J. TSUJIUCHI,

Chiba. Japan

H. WALTHER,

Garching, F.R.G.

W. T. WELFORD,

London, England

B. ZEL’DOVICH,

Chelyabinsk. U.S .S.R .

P R O G R E S S IN OPTICS VOLUME XXVII

EDITED BY

E. WOLF University of Rochester, N.Y.. U.S.A.

Contributors K. PATORSKI, L.M. SOROKO, I.M. BASSETT, W.T. WELFORD, R. WINSTON, D. MIHALACHE, M. BERTOLOTTI, C. SIBILIA, R.P. PORTER

1989

NORTH-HOLLAND AMSTERDAM. OXFORD * NEW YORK .TOKYO

0 ELSEVIER SCIENCE PUBLISHERS B.V., 1989

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise. without the prior permission of the Publishers, North-Holland (Elsevier Science PublishersB. V.), P. 0. Box 103, 1000 AC Amsterdam, The Netherlands. Special regulationsfor readers in the U.S.A.: Thispublicationhas been registered with the Copyright Clearance Center Inc. (CCC). Salem. Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A.. should be referred to the Publisher, unless otherwise specified. No responsibility is assumed by the Publisherfor any injury anaYor damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. LIBRARY OF CONGRESS CATALOG CARD NUMBER 61-19297 ISBN: 0 444 87425 9 PUBLISHED BY

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PREFACE This volume consists of five articles which present reviews of various topics of modern optics. The first article, by K. Patorski, discusses the theory and applications of self-imaging,i.e. of the phenomena manifested by a periodic repetition of planar field distributions in certain types of wavefields. This phenomenon is finding applications not only in optics, but also in acoustics and in electron microscopy. In optics self-imaging is being explored particularly in image processing, in the production of spatial-frequency filters and in optical metrology. In the second article, contributed by L.M. Soroko, a class of somewhat unconventional optical elements and systems, known collectively as meso-optical devices, is discussed. It includes, for example, the so-called axicons, which image points into line segments. The article covers the theory and some uses of devices of this kind. In the article which follows, I.M. Bassett, W.T. Welford and R. Winston describe the principles and the developments that have been made mainly in the last 10 years in the field of nonimaging optics. This is an area of optics which deals with devices that provide highly efficient concentration of the energy flux without imaging. Their main applications are connected with concentration of solar energy. In the fourth article D. Mihalache, M. Bertolotti and C. Sibilia explain the principles of nonlinear guided waves that propagate in layered planar structures and they discuss some of their potential uses. These include, for example, the design of non-linear distributed couplers and optical switching devices. The concluding article, contributed by R.P. Porter, reviews the field of generalized holography and its application to various inverse wave-theoretical problems. Conventional holography utilizes planar recording surfaces. Recent researches have shown that when the recording surfaces are of more general shapes, holographic principles can be used to extract from such recordings solutions to various inverse source problems and inverse scattering problems. This development is also providing new insights into the important subject of diffraction tomography, especially in its acoustical analogue relating to ultrasonic imaging. This article presents a comprehensive review of generalized holography and of its main uses. I would like to use this opportunity to express my sincere thanks to Professors L. Allen, M. Francon, M. Movsessian, and Dr. G. Schulz, who have recently retired from the Editorial Advisory Board of Progress in Optics, for much valuable help and advice that they gave me over a period of many

years. Also, I welcome Professors G.S. Agarwal, C. Cohen-Tannoudji, M. Schubert, H. Walther and B. Zel'dovich as new members of the Board. As this volume went into production, I received the sad news of the death of Erik Ingelstam, Professor Emeritus at the Royal Institute of Technology, former Director of the Institute of Optical Research in Stockholm and a member of the Editorial Advisory Board of this series of books since its inception in 1961. Professor Ingelstam was greatly admired by the whole optics community, not only for his many valuable scientific contributions, but also because of the warmth ofhis personality and for his readiness to help his friends and colleagues when help was needed. He will be sadly missed. EMILWOLF Department of Physics and Astronomy University of Rochester Rochester, NY 14627, USA April 1989

E. WOLF, PROGRESS IN OPTICS XXVII

0 ELSEVIER SCIENCE PUBLISHERS B.V., 1989

I THE SELF-IMAGING PHENOMENON AND ITS APPLICATIONS BY

KRZYSZTOFPATORSKI Institute for Design of Precise and Optical Instruments Warsaw University of Technologv 02-525 Warsaw, Poland

CONTENTS

. . . . . . . . . . . . . . . . . . THEORETICAL CONSIDERATIONS . . . . . . . . .

PAGE

§ 1 . INTRODUCTION

.

3

§ 2*

.

6

. . . . . . ..

4 8

§ 3*

APPLICATIONS OF THE SELF-IMAGING PHENOMENON . . . . . . . . . . . .

§ 4. THEORETICAL CONSIDERATIONS : INCOHERENT

ILLUMINATION

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

§ 5 . APPLICATIONS OF SELF-IMAGING UNDER

INCOHERENT ILLUMINATION

. . . . . . . . . . .

7 5

. 9 1

. . . . . . . . .

99

. . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

101

§ 6 . PERIODICITIES OF OPTICAL FIELDS § 7 . CONCLUSIONS

101 101

1. Introduction

At the time of this writing, the optical effect described in this review article has had the 150th anniversary of its discovery. In 1836 H. F. TALBOT illuminated a diffraction grating and a rectangular array of tiny holes with a very small white light source. Behind the structures he observed colorful intensity patterns resembling the periodic objects themselves. LORD RAYLEIGH[ 18811 was the 6rst person to explain this experiment analytically, attributing its origin to the interference of diffracted beams. He showed that in the case of plane wave front illumination of a linear grating, characteristic intensity distributions repeat along the illumination direction with a longitudinal period equal to 2d2/1, where d stands for the grating period and 1denotes the wavelength of light. Chronologically the next studies that related to the phenomenon were by WINKELMANN [ 19081, WEISEL[ 19101 and WOLFKE[ 191l,1912a,b,c, 19131, who examined grating image formation in microscopy. Once more it was necessary to wait for many years until the problem was approached again. COWLEYand MOODIE[ 1957a,b,c, 19601 in their pioneering work made an intensive study of the properties of the Fresnel diffraction field behind periodic objects. They were motivated to find a high-magnification imaging process that was not using classical optical elements and investigated in detail the case of spherical wave front illumination, taking into account the source size influence. The authors called well-defined object images Fourier images, whereas the intermediate-intensity patterns appearing between the Fourier image planes were called Fresnel images. The latter images received much attention in the subsequent work of ROGERS [ 1962,1963,19641 and especially WINTHROP and WORTHINGTON [1965]. In his fundamental study MONTGOMERY [ 19671 developed general conditions that needed to be satisfied by an object to obtain repetition of its complex amplitude transmittance along the illumination direction. The term “self-imaging” was introduced by Montgomery and was used together with the term “Talbot effect” in the publications that appeared in the 1970s and 1980s. The simplicity and beauty of self-imaging still attract many researchers and result in interesting and original applications 3

4

SELF-IMAGING AND ITS APPLICATIONS

[I, 8 1

that represent competitive solutions to various scientific and technological problems. Although spatially periodic objects represent only a subgroup of all objects that can generate self-images, they are of primary interest. In optical studies the objects are usually in the form of a transparency, recorded, for example, on a photographic material. However, in other fields of science the origin of periodic structures can be different in examples such as crystallographic structures or ultrasonic waves. In the first case there is an analogy between light diffraction effects produced by a phase grating and scattering of electrons by atoms (COWLEYand MOODIE [1957d]). The role of self-imaging in electron microscopy was discussed, for example, by ROGERS[ 1969, 1970al. Similarly, the interaction of light with ultrasonic waves can be described by the theory of thin phase gratings (RAMANand NATH[ 19351). Studies of the self-imaging phenomenon or, in general, Fresnel diffraction field properties of periodic or quasi-periodic objects, were conducted more or less independently in optics, electron microscopy, or acousto-optics. The previously mentioned pioneering works of COWLEYand MOODIE[1957a,b,c, 19601 that examined optical self-imaging were stimulated, in fact, by the need to develop a lensless highmagnification imaging system for electron microscopy. On the other hand, the monograph of BERGMANN[ 19541 provides examples of the early uses of Fresnel zone diffraction to visualize progressive and stationary ultrasonic waves. The present review discusses “light optics” self-imaging, a process in which the object is inserted into a light beam. It can be of amplitude or phase type or, in general, have a complex amplitude transmittance. This approach, which requires an object at the initial plane, might seem to restrict the scope of these considerations, since some other phenomena exploiting or closely related to self-imaging cannot be treated directly in this way. They include the selfimaging phenomenon in waveguides (ULRICH [1975a,b], ULRICHand KAMIYA[ 19781, BLUME[ 1976, 19771) and the general problem of spatial [ 19831, periodicities of light fields (LOHMANNand OJEDA-CASTANEDA LOHMANN, OJEDA-CASTANEDAand STREIBL [ 19831, INDEBETOUW [ 1984a,b]). In the present article I have chosen the light optics self-imaging approach because of its applicational aspects. The problems of self-imaging in optical waveguides and spatial periodicities of a light field will be reviewed in separate sections. The self-imaging phenomenon requires a highly spatially coherent illumination. It disappears when the lateral dimensions of the light source are increased. On the other hand, when the source is made spatially periodic (an incoherently

1 3 0 11

INTRODUCTION

5

illuminated amplitude grating can serve as an example) and it is placed at the proper distance in front of the periodic structure, a fringe pattern is formed in the space behind the structure. The first experiment of this type was performed by LAU [ 19481, who used two amplitude gratings of the same spatial period illuminated incoherently. The observation was conducted in the back-focal plane of a converging lens. Fringe patterns were noted for some discrete separation distances between the gratings. Other studies extended the possible configurationsofthis experimentto gratings of unequal periods (ROBLIN[ 1971, 19731, EBBENI[ 1970a1, PEITIGREW [ 19771). After the work of JAHNSand LOHMANN [ 19791the name “Lau effect” became widely used in papers dealing with diffraction configurations of this type. Although their performance can be rigorously described by the coherence theory approach (GORI[ 19791, SUDOL and THOMPSON [ 1979, 19811, SUDOL[ 1980]), it is convenient to adopt the approach proposed by PATORSKI[ 1983bl. It is based on the idea of multiple incoherent superposition of the self-imaging phenomenon. This approach represents continuation and extension of the studies of the coherent illumination case and gives a good intuitive understanding of the phenomenon. It will be used throughout the section dealing with double diffraction systems under incoherent illumination. The theory of partial coherence, although elegant and rigorous in itself, sometimesobscures the physical insight into the phenomenon, making it less comprehensive to a wider group of readers. This work is divided into five main sections, the first of which are devoted to theoretical and applicational aspects of the self-imagingphenomenon, that is, the property of the Fresnel diffraction field of some objects illuminated by a spatially coherent light beam. The paraxial approximation is assumed throughout. Initially the group of self-imaging objects is defined (MONTGOMERY [ 1967]), followed by a discussion of aberration-free illumination modes with the Gaussian beam treated as a general topic (SZAPIELand PATORSKI[ 19791). The case of high-magnification self-imaging under highaperture illumination is addressed. The most important properties of intensity distributions in the self- and Fresnel-image planes of objects of various types, that is, amplitude-phase, amplitude, and phase objects are reviewed. For clarity and simplicity the discussion is restricted to one-dimensional periodic objects. Although the analyses of self-imaging usually concern the Fresnel diffraction field behind a single object structure, it is obvious that this phenomenon influences the performance of double or multiple grating systems. The literature that deals with this problem and those concerning the Fraunhofer and Fresnel field double diffraction systems is reviewed as well. Next the influenceof several factors relating to the degree of coherence of the illuminating

6

SELF-IMAGING A N D ITS APPLICATIONS

[I. § 2

beam, that is, the lateral extent of the source and its spectral band width, as well as finite object dimensions and periodicity are discussed. Following the theoretical analysis of the self-imaging phenomenon some specific techniques for its production are reviewed, including spatial filtering, holography, and self-imaging in optical waveguides. In 5 3 the applications of self-imaging are described. They are summarized in four main groups, namely, image processing and synthesis, technology of optical elements, optical testing, and optical metrology. Sections 4 and 5 are devoted to double diffraction systems using spatially incoherent illumination. The first periodic structure plays the role of a periodic source composed of a multiple of mutually incoherent slits. Depending on whether the periods of two periodic structures are equal or unequal, we speak about the Lau or the generalized Lau effect. Their interpretations are reviewed, and one of them, namely, the approach of multiple incoherent superposition of the self-imaging phenomenon, is discussed in considerable detail. The cases of finite and infinite separation distances between two periodic structures are discussed, followed by the special configuration that is required for spatial selection of diffraction order pairs in a double grating system. The method allows achromatic fringe formation under incoherent extended source illumination. Finally, double diffraction systems that use an incoherent source with limited lateral dimensions are reviewed. In 0 5 various applications of incoherent double-grating systems are described in the fields of optical testing, image processing, and optical metrology. Finally, after examining in detail some cases of coherent and incoherent illumination, the general issue of spatial periodicities of optical fields and its relevance to the replication of partially coherent fields in space is discussed. Throughout this review article there is an effort to present the physical aspects of the self-imaging phenomenon clearly. Mathematical formulations are reduced to the indispensable ones. Readers who prefer strict mathematical treatments should use the extensive list of references. New ideas and applications of self-imaging are still expected in the future.

0 2. Theoretical considerations 2.1. SELF-IMAGING OBJECTS

TALBOT [ 18361was first to observe the characteristic light intensity distributions behind a periodic object, that is, a coarse amplitude type of diffraction

1 5 8

21

THEORETICAL CONSIDERATIONS

1

grating. RAYLEIGH[ 18811 derived the mathematical formula expressing the distance between diffraction images of a linear grating in the case of quasimonochromatic plane wave front illumination. In the extensive studies of and WORTHINGTON COWLEY and MOODIE[ 1957a,b,c, 19601 and WINTHROP [ 19651 many theoretical and practical aspects of the effect were investigated. From the point of view of the object parameters Cowley and Moodie were the first to consider two-dimensional periodic objects composed of rectangular or square unit cells. Winthrop and Worthington extended the class of twodimensional structures, introducing objects with unit cells forming parallelograms and including hexagonal array (ROGERS[ 1962, 19691). The objects mentioned represent discrete combinations of periodic structures. [ 19671 was first to ask whether lateral periodicity of the MONTGOMERY object is a necessary or a sufficient condition for the Talbot effect to occur. The aim of his investigation was to establish the properties of the wavefield that repeats itself along the propagation direction (self-imaging phenomenon). We will now briefly review the analysis of Montgomery and discuss its results following the approach of LOHMANN [ 19781. Under the assumption of scalar diffraction theory (to be followed throughout this article) the reduced wave equation is V2u + k2U = 0 ,

(2.1) where k = 24A, 1 is the light wavelength, and u designates a three-dimensional wavefield function. If u(x, y, z) is to be periodic along the z-axis with period Az, it can be described by a Fourier series u ( x , y, z) =

c

um(x, y ) exp (i2 nrnz/Az)

m

with the boundary condition u(x, y, 0)=

m

u,(x,

u) = Ax, v)

9

wheref(x, y) is the amplitude transmittance of the object. When the wavefield u ( x , y, z) is introduced into eq. (2. I), the following two-dimensional differential equation should be satisfied

-+aZum ax2

a2um + k2[l -ay2

($>'I

urn = 0

.

(2.4)

Excluding two unimportant solutions, let us discuss here only the case corresponding to the condition 1-

(3 -

>o

8

SELF-IMAGING AND ITS APPLICATIONS

or Az/A > m 2 1. Introducing the Fourier transform notation urn(x,

Y)=

jj

Drn(5,

O0

-a

exp [i2n(xt + y q ) ~d t dq

(2.6)

where the tildes indicate Fourier transformation, we can represent the differential equation for urn(x,y ) in the spatial frequency domain

- w

x

{

-(2n)2(52

+ $2 + k2

[1

e r ] > d 5 d 4 = 0.

(2.7)

Equation (2.7) is satisfied if Dm = 0 or the term in braces is 0 or if both conditions are met simultaneously.The condition that the term in braces be 0 imposes

In other words the spatial frequenciesof self-imaging objects should be discrete and located on the rings of radii pm. Equation (2.8) describes the so-called Montgomery rings. The integer m is limited to 0 < m < mmax< -.

bZ

rz

The lower limit follows from disregarding the evanescent waves (weak selfimaging), whereas the upper one corresponds to the condition p: > 0. It can be readily proved that the equidistant rings in the frequency plane that are characteristic of a circular diffraction grating, or the equidistant spots that are characteristic of a linear diffraction grating, correspond to the Montgomery rings with the number n = 0, 1,4,9, ...,m2.Therefore laterally periodic objects represent only a subset of all objects capable of self-imaging,that is, reproducing their amplitude transmittance along the propagation direction. Lateral periodicity of an object is sufficient but not necessary for self-imaging to occur. In spite of the availability of a general theory of self-imaging specifying a very broad class of objects that can be used for experimentalpurposes, only periodic structures are widely used because of their relatively easy technology. It is hoped, however, that the future will extend the applications of self-imaging generated by more sophisticated structures. The work of SZWAYKOWSKI

1, t 21

THEORETICAL CONSIDERATIONS

9

[1988] might serve as a recent example. Starting from the results of Montgomery, he derived a general shape for the function describing the selfimaging object in polar coordinates. Szwaykowski showed that diffracting structures consisting of a set of evolutes of a circle meet the self-imaging demands and can be used in Talbot interferometryfor displaying the radial and azimuthal derivatives. The theory of self-imagingof doubly periodic objects, in addition to previously mentioned works of COWLEY and MOODIE[ 1957a,b,c, 19601, WINTHROP and WORTHINGTON [ 19651, and MONTGOMERY [ 19671, was extended in the subsequent works of SMrRNoV [ 1977, 1986a1, KONITZ[ 19791, SMIRNOV and GALPERN [ 19801, and IOSELIANI [ 19831. Various applications using this type of object structure will be discussed in Q 3. Several theoretical aspects of the Fresnel diffractionfield of self-imaging objects, common to all types of objects, will be explained in the next subsection, using the example of a one-dimensional periodic structure.

2.2. ABERRATION-FREE ILLUMINATION MODES

After defining the conditions that must be met by self-imaging objects, the properties of diffraction images under various illumination modes will be discussed. Intuitively we can say that an aberration-free self-image can be obtained using only a nonaberrated illuminating wave front. All theoretical works on self-imagingdeal with spherical wave front illumination as a general case, including plane wave front illumination as a special case. Usually a parabolic approximation is used when solving the diffraction integral. On the other hand, the departure from the Fresnel approximation is encountered in the case of objects with high spatial frequency, highly convergent or divergent illuminating beams, and large angles of incidence. In the following the Fresnel approximation will be used for a discussion of aberration-free illumination modes. As it is the most general mode, Gaussian beam illumination will be assumed. It includes spherical or plane wave fronts with uniform intensity distributions as special cases. The discussion will be restricted to one-dimensional objects, but extension of the analysis to twodimensional objects is straightforward. The case of departure from the parabolic approximation will be addressed in $2.3. To show the most important features of various aberration-free illumination modes, the intensity distributions in the self-image planes of periodic objects will be analyzed. For the same purpose it is plausible to assume infinite object

10

[I, 8 2

SELF-IMAGING AND ITS APPLICATIONS

dimensions and spatially coherent quasi-monochromatic illumination. Other cases differing from these assumptions will be discussed separately in § 2.6. The characteristics of self-imaging reviewed here can be directly applied to the explanation of coherent lens imaging of periodic objects in the presence of focus and ROBLIN[ 19741). defect (HOPKINS[ 19531, MALLICK 2.2.1. Gaussian beam illumination Propagation of Gaussian beams has received much attention because of the widespread use of laser radiation. On the other hand, spatially coherent beams with uniform amplitude or intensity distributions can be treated as a special case of the Gaussian beam. Therefore, by firstly performing the analysis for a general case (SZAPIELand PATORSKI [ 1979]), we will be able to compare the Fresnel diffraction field properties of periodic objects for different aberrationfree illumination modes. Let us assume a simple, single TEM,, mode Gaussian laser beam. Its propagation properties are schematically shown in fig. 2.1. The beam parameters at a distance zG from the waist plane (KOGELNIK and LI [ 19661) are given by (2.10a) (2. lob) (2.10c) where 2w, designates the waist diameter and I is the light wavelength. A

sx

Fig. 2.1. Gaussian beam parameters.

tx

1 . 8 21

11

THEORETICAL CONSIDERATIONS

one-dimensional periodic object will be represented as (2.11) where d is the spatial period and a, is the amplitude of the nth harmonic. Since we will not specify the form of the Fourier coefficient a, at this point, a n y type of periodic object can be assumed for the present analysis. Using the Fresnel-Kirchhoff diffraction integral (BORNand WOLF[ 1964]), the light field VG (the subscript denotes Gaussian beam illumination) at a distance z from the object is proportional to

x exp

where x, y and respectively; k

(

- I ’ k(x2 +

2R

”))

exp( - i k 9 ) dx dy , (2.12)

y’ are the coordinates in the object and observation planes, 2n/A and

XI,

=

exp( - i k 9 )

=

{ [+

exp - ik z

~

x2;y2

-1 (xx’ Z

+ yy‘) +

+

22

(2.13)

y’2]}.

Equation (2.13) expresses the distance between the points in the object and observation planes, using the parabolic approximation. Theresult of integration [ 19791) in eq. (2.12) can be written in the form (SZAPIELand PATORSKI 4n2n2 z2 4nn zx’ VG(x’,y’,z)ocCu,,exp - _ _ n k2d2 w , ” ) e x P ( - k b

(

exp

7)

(. I-

2nnx’ M , d

<) , w2

(2.14) where w, denotes the Gaussian beam radius at the observation plane R+z Mu=-.

R

XI,

y ‘ and

(2.15)

12

SELF-IMAGING AND ITS APPLICATIONS

[I, s 2

The form of eq. (2.14) is limited here only to the terms resulting from the presence of a periodic object in the x , y plane and describing its Fresnel diffraction pattern. Other terms describing the free-space propagation of the Gaussian beam were taken under the sign of proportionality. The first two exponential terms describe amplitude changes along the axial and lateral directions imposed by the object diffraction orders. Since these terms are not responsible for diffraction image formation, they will not be discussed. The next two phase terms are of basic imporpce to self-imaging. The term (2.16)

will be called the “localization” term, since it describes the phase changes of the diffraction orders with axial distance z. All orders are co-phasial and reinforce at distances z, satisfying the relation d2 w2 ~ R + zw 2v- = zMu- = z -

1

w,‘

R

(K)

’

(2.17)

where v is a positive integer refered to as the self-imagenumber. As mentioned in the previous section, the self-image planes correspond to the planes where the objeci light amplitude transmittance is repeated. This happens under the condition described by eq. (2.17); the diffraction orders are in phase. The second phase exponential term of eq. (2.14) carries information about the lateral magnification MG of diffraction patterns arising as a result of the Gaussian beam illumination, that is, (2.18)

It is more convenient to express eqs. (2.17) and (2.18) in a fong involving a smaller number of beam parameters. After simple but tedious calculations.we arrive at the following relationships: (2.19) M , = - , R, R, - z

(2.20)

1. $21

13

THEORETICAL CONSIDERATIONS

where R , is the Gaussian beam radius at the observation plane. In addition, and PATORSKI[ 19791 gives eqs. (2.19) and (2.20) as a the paper of SZAPIEL function of the object distance from the beam waist and the beam waist diameter. We recall that the factor of 2 in eqs. (2.17)and (2.19)can be omitted. In such a case, when v is an odd integer, we can speak about object images with half a period lateral shift with respect to the object. This results from the A phase shift of odd-number diffraction orders with respect to the zero and evennumber orders. As a special case, let us discuss the configuration with the periodic object placed at the waist plane, fig. 2.2. In this case R --t (plane wave front illumination at the object plane) and, correspondingly, (2.21) (2.22) In this case the lateral magnification and axial localization of self-images are proportional to the square of the ratio of the beam radii at the observation and beam waist planes. This determines the hyperbolic character of changes in the self-imaging phenomenon parameters that is inherent in Gaussian beam propagation. 2.2.2. Spherical wave front illumination Since eq. (2.14) is a general equation, it includes uniform illumination as a special case. This is obtained if we note that for uniform illumination, w 3 co

-

- t I - - - - t . + ,

2”

Fig. 2.2. Self-image formation with an object placed at the waist plane ofa Gaussian beam. (Atter SZAPIEL and PATORSKI[1979].)

14

[I, 0 2

SELF-IMAGING AND ITS APPLICATIONS

and (wlw,)’ = Mi2.Under these conditions we obtain from eq. (2.14)

(2.23)

This expression is in perfect agreement with the results obtained by several and authors using the Fresnel-Kirchhoff integral (see, for example, COWLEY MOODIE[ 1957al and WINTHROPand WORTHINGTON [ 1965]), geometrical (YOKOZEKI, PATORSKI and OHNISHII [ 19751) and Fourier plane (JOZWICKI [ 19831) approaches. The lateral magnification of self-images is now given by the ratio of the distances of the observation and object planes from the point source plane, fig. 2.3. The axial localization of self-images is governed by the relationship d2 R 2v- = z - ,

I

(2.24)

R+z

expressing the phase coincidence between the interfering diffraction orders. This occurs for the discrete axial distances z, given by z=

2vdtR I R - 2vd2 ‘

(2.25)

Comparing eqs. (2.15) and (2.24) with eqs. (2.20) and (2.19), respectively, the difference between the formulas corresponding to the self-imagingphenomenon produced by a uniform amplitude (intensity) and a nonuniform Gaussian beam can be readily understood. Relation (2.24) determining the self-image locations may be compared with the well-known lens equation -1 + - 1= - , 1 z R f

(2.26) 0

,

R

OP

I

z

!

I

Z‘

Fig. 2.3. Formation of self-imagesunder illumination by a point source S. The case of a diverging beam is shown. The distance z‘ of the observation plane OP measured from the point source was introduced by JOZWICKI [ 19831 for complete analogy analysis with the lens-imaging formulae.

1, I 21

THEORETICAL CONSIDERATIONS

15

where f denotes the focal length and z and R correspond to the image and object distances from the lens, respectively. Comparing this equation with relation (2.24), we have f,= 2d2/2. Therefore the self-imaging phenomenon can be considered as the equivalent effect of a sequence of apparent lenses of focal lengthy, situated in the plane of the periodic object. The point source now plays the role of the object and the self-images correspond to the images given by the sequence of lenses. However, this analogy is not complete because the magnification fulfills the condition of the geometrical shadow with a point source as the projection center (COWLEYand MOODIE [1957a]). This ambiguity was not avoided in a modified analogy proposed by ROBLIN[ 19731, in which a sequence of multiple object planes instead of the lens sequence was introduced. A new phenomenological interpretation that avoids this ambiguity [ 19831. The field distribution center was subsequently proposed by JOZWICKI has been attached to the point source from which all the distances are measured, for example, see distances R and z' in fig. 2.3. Therefore, if one has a periodical object at a distance R from the point source, then self-images arise at distances z' , fulfilling the relation 2v d 2 R2 A

1 R

1 z'

(2.27)

This can be interpreted as the equation of a lens situated at the point source plane. The magnification of the lens is given by z ' / R . This interpretation is in full agreement with the geometrical shadow principle. It should be noted that v can be a positive or negative integer. In these cases we are considering real ( z > 0) and virtual ( z < 0) self-images of the object, respectively. Moreover, the object can be located to the right or to the left of the point source localization plane; that is, it is illuminated by a convergent or a divergent spherical wave front beam. From a practical point of view the main interest in self-imaging under spherical wave front illumination seems to be dictated by the possibility of obtaining high lateral magnification imaging of periodic objects, without using conventional optics. This concept led to the papers of COWLEYand MOODIE [ 1957a,b,c, 19601, who proposed the method for crystal observation by electron microscopy. More detailed theoretical studies concerning high-magnification [ 19821 and configurations were reported by JOYEUXand COHEN-SABBAN COHEN-SABBAN and JOYEUX[1983]. Since they do not use a parabolic approximationfor expressing the distances between the points in the object and observation planes, a review of their work will be given separately in 3 2.3.

16

SELF-IMAGING AND ITS APPLICATIONS

2.2.3. Plane wave illumination

Plane wave illumination is the most common illumination mode used for various applications of self-imaging. It represents a special case of the preceding analyses of Gaussian and uniform-amplitude spherical wave front beams. It is obtained from eq. (2.23) by putting R = a,resulting in (2.28)

The diffraction orders are co-phasial at the distances z that fulfill the condition d2

z = 2v-

(2.29)

3,

and form self-images of a periodic object T(x). Again, by considering the self-image planes, the planes with exact replication of the object amplitude transmittance can be understood. Similarly, as in the previous case, if we omit the factor of 2 in the last equation and let v be an odd integer, we obtain the light field as in the object plane but laterally displaced by half a period with respect to it. Equation (2.28) can be obtained in a faster and simpler way by using the approach of the angular spectrum of plane waves. Because of its wide use in the following examples, we will shortly rederive the form of eq. (2.28) as presented by EDGAR[ 19691. For the sake of simplicity we will assume that the illuminating beam is impinging at a small angle with respect to the grating normal. In addition, the incidence plane is assumed to be perpendicular to the grating lines. The general case of non-normal incidence of the illuminating beam could have been assumed at an earlier stage, that is, for the Gaussian or spherical OP

Fig. 2.4. Geometry for calculating the Fresnel diffraction field of obliquely illuminated periodic object G. For simplicity only the orders + n, 0, and - n are indicated.

1 7 8

21

THEORETICAL CONSIDERATIONS

17

wave front beam illumination. However, to present the various factors influencing the parameters of self-images clearly, we prefer to introduce an oblique illumination at this point. Figure 2.4 shows the geometry for calculation. The field just behind the linear periodic structure G is given by E(x, z

=

0) = exp(ikx sina) C anexp

(2.30)

n

where a denotes the wave front incidence angle. At a distance z the light field becomes E ( x , z ) = Canexp{ik[xsinO; +zcose:,]},

where sine:,

(2.3 1)

n

=

I

nd

+ sina.

(2.32)

The origin of the coordinate system xyz, common to the object and observation planes, is placed at the object plane. Because of the small value of the incidence angle a and relatively small basic frequency of a periodic object, we can assume equal angular distances between the adjacent diffraction orders. The process of highly oblique illumination studied in detail by PATORSKI [ 1984bl and BIALOBRZESKIand PATORSKI[ 19851 will be commented on below. It is derived from the paraxial approximation assumed for calculating the optical path of object diffraction orders. In the configuration under study the parabolic approximation is expressed by the equality essential for the formation of self-images. By performing simple calculations in the exponential term we obtain

x exp[ - i 2 n ( z i s i n a

+

(2.34)

Only the terms that are responsible for the formation of the diffraction image are shown under the summation sign in eq. (2.34). The other terms are included

18

SELF-IMAGING AND ITS APPLICATIONS

[I. B 2

under the proportionality sign. After rewriting the above expression we obtain n

In comparing the last equation with eq. (2.28), we note their similar character, Z a/d), which expresses the lateral except for the additional term exp( - i2 R ~ sin shift of diffraction patterns proportional to z sina. They are centered (geometrically projected) along the chief ray, inclined to the optical axis (grating normal) at an angle equal to the angle of incidence of the illuminating beam. It must be emphasized that the results obtained are valid when the parabolic approximation (2.33) is satisfied for all diffraction orders. This implies that all diffraction orders are simultaneously co-phasial in the self-image planes. Because the angle 0, increases with n, we observe a departure from the parabolic approximation. This is also obviously true for the case of a normally incident illumination beam. Therefore, for higher diffraction orders the next term in the binomial expansion of the square root, that is, should be taken into account. This problem was addressed, for cxample, by SCIAMMARELLA and DAVIS[ 19681 and CHANG[ 19741. By setting a tolerance on the maximum allowable optical path difference introduced by the third term in the binomial expansion, we can determine the maximum number v of the self-image formed by a specified number of object diffraction orders. The general conclusion is that due to gradual violation of the parabolic approximation by higher diffraction orders, we cannot speak, in a strict sense, of selfimages at larger observation distances. In terms of the observed intensity [eq. (2.35) should be multiplied by its complex conjugate] in the case of a binary amplitude grating as an object, the higher the self-image number v the less sharp are the lines of the intensity distribution observed. As mentioned in the discussion following eq. (2.35), the lateral displacement of the point source from the optical axis introduces a displacement of the light field in the observed self-image plane. It is proportional to the observation distance z and the illumination angle. The same is true for the case of a nonparallel illumination beam, where the incidence angle can be replaced by the ratio of the distance of the point source from the optical axis to the distance R between the source and the periodic object. The shadowlike projection effect is readily understood for the self-imaging phenomenon. The preceding discussion leads us to another important property of the self-imagingphenomenon. Let us replace the point source by an extended one

1. I21

THEORETICAL CONSIDERATIONS

19

that can be thought as of being composed of a multitude of mutually incoherent point sources. Each one produces the self-imaging phenomenon. In a selected detection plane the mutually incoherent self-images are laterally displaced. In effect the overall contrast of the superposed intensity patterns is reduced, which provides an heuiistic explanation of the fact that the self-imaging phenomenon requires a highly spatially coherent point source illumination. This problem will be discussed in detail in 5 2.6. It can easily be proved that when the incidence plane of the illuminating beam is parallel to the grating lines and the incidence angle is small, there is no influence on the parameters of the Fresnel diffraction field. This finding means that a slit-type source with the slit direction parallel to the grating lines can be used for the generation of self-images. 2.3. NONPARAXIAL SELF-IMAGING

In all the foregoing considerations and referenced papers the paraxial approximation was assumed. The explanations using the Fresnel-Kirchhoff diffraction integral or the approach of an angular spectrum of plane waves involve the parabolic approximation of optical paths between the object and observation planes. For simplicity an infinite object size is usually assumtd as well. Although the last assumption results in an adequate description of the basic parameters of self-imaging (utilized in regions far enough from the edge shadow), it is contradictory to the paraxial assumption. This fact was pointed [ 19821 when studying high-magnification out by JOYEUX and COHEN-SABBAN diffraction configurations with spherical wave front illumination. Therefore, when dealing with high magnification self-imaging systems, other mathematical models must be used for an accurate description of the diffraction images. COHEN-SABBAN and JOYEUX [ 19831 proposed to compute the diffraction fields using a ray-tracing technique based on the Fermat principle. Accurate results are obtained for any numerical aperture (proportional to the ratio of the object diameter to the spherical wave front radius R of the illuminating wave at the object plane) and image field. The effects which arise as a result of finite object dimensions, that is, the limitation of the number of diffraction orders contributing to the image and the diffraction at the object edges (these effects are discussed in $2.6), can be readily taken into account. The aberrations pertinent to high magnification self-imaging were defined and a method for their reduction was proposed. The inability to obtain perfect self-images under high aperture illumination is primarily the result of the limited number of diffraction orders interfering in

20

SELF-IMAGING AND ITS APPLICATIONS

[I, § 2

the observation plane due to the “walk-off’ effect (see 8 2.6), the nonexistence of planes of fully constructive interference due to the dispersion of relative phases of diffraction orders interfering at each image point, and the nonharmonicity of imaged object Fourier series. The last feature can be attributed to a different lateral magnification encountered for different harmonics increasing with the distance from the optical axis in the observation plane. Using the calculation model proposed by COHEN-SABBAN and JOYEUX [ 19831, concrete high magnification applications of self-imaging can be solved with great accuracy. The ray tracing approach of COHEN-SABBAN and JOYEAUX [ 19831 to highmagnification nonparaxial self-imaging was supplemented by PATORSKI and KOZAK[ 19881. Using a simple analytical approach (in the series expansion of an optical path the terms up to the fourth power in the lateral coordinates were included) they found that in the range of relatively small magnificationsrealized at relatively short distances of the observation plane from the source and the self-imagingobject, a strong contrast modulation is encountered.Zero-contrast regions forming an elliptical fringe-like structure limit the useful area of the self-image to be utilized. Coma aberration in interfering diffraction orders introduces a strong departure of image lines from straightness. Experimental results were given. Additional investigationsbased on the paraxial approximation were reported [ 1984bl and BIALOBRZESKI and PATORSKI [ 19851, who studied by PATORSKI in detail a highly oblique plane wave front illumination of a linear periodic structure. The change in object spatial frequency distribution as a function of KORONKEVITCH, KRIVENKOV and MIKHLYAEV the object tilt angle (CHUGUI, [ 19811, GANCI[ 19811, PATORSKI [ 1983a1) had to be taken into account. Specifically,the diffraction orders are no longer equally separated, which is the basic difference between the normal and the slightly oblique illumination case. The cases where the incidence plane is parallel or perpendicular to the grating lines were treated separately. It was found in both cases that well-defined diffraction images are periodically detected in the observation planes parallel to the grating plane by changing the observation distance. They were called the “self-images under oblique illumination”. Analytical expressions were obtained for the case of a cosinusoidal type of diffraction grating. As with the studies and JOYEUX [ 19831, because of complex phase relationof COHEN-SABBAN ships between the diffraction orders, computer calculations are required to obtain general diffraction field expressions. Nevertheless, the experimental results obtained with the square-wavetype of transmission gratings agreed very well with the properties established analytically.

1 9

8 21

THEORETICAL CONSIDERATIONS

21

2.4. SELF-IMAGING OF LINEAR PERIODIC OBJECTS

As mentioned before, the self-imaging phenomenon relates to the effect of repeating the object amplitude transmittance in its Fresnel diffraction field along the illumination direction. Conditions for self-imaging objects and the parameters of the Fresnel field under various illumination modes have been discussed in previous sections. However, since the eye or recording materials are energy detectors and are not sensitive to the complex amplitude of a light field, only its intensity can be monitored. Therefore it is important to analyze the intensity patterns in the diffraction images of self-imaging objects being described by different transmittance functions. For the sake of a simple analysis a one-dimensional periodic object, illuminated by a plane wave front beam, will be assumed. 2.4.1. Objects with complex amplitude transmittance

A general object case is represented by an object influencing the amplitude and the phase of the light transmitted through it. As an example, consider a photographically produced diffraction grating, that is, a recorded two-beam interference pattern. Shrinkage of the emulsion after development introduces variations of the optical path length through the grating. In the most general case of complex amplitude transmittance, arbitrary profiles of amplitude and phase modulations can be assumed, together with arbitrary spatial periods. In addition, an arbitrary phase displacement between amplitude and phase modulation parts can be considered. The analysis of such a general case would be of rather academic interest and a formidable task, even when resorting to very fast computer calculations. Again, to simplify the analysis and its practical consequences we will limit our discussion to the case of a sinusoidal complex object model with a small phase modulation. Such an object model allows us to show, without any further approximations, interesting differences between the Fresnel intensity patterns of complex objects and pure amplitude or pure phase objects. The latter objects can be treated as special cases. The following [ 19811. discussion is based on the analysis of PATORSKI and PARFJANOWICZ The amplitude transmittance of a single frequency complex object under plane wave front illumination can be expressed in the form O(x)=

(2.36)

where A and Care the parameters of absorption modulation, B is the amplitude

22

[I, 0 2

SELF-IMAGING AND ITS APPLICATIONS

of the sinusoidal phase variation, d is the spatial period of the object, and $ denotes the relative phase angle along the x direction between the amplitude and phase gratings. Two cases of B are of practical importance; that is, $ = 0 (B > 0) and $ = K (B < 0); the case $ = 0 will be discussed in detail. To calculate the intensity distribution at a distance z from the object, the approach of the spectrum of plane waves will be used (EDGAR[ 19691). For this purpose it is convenient to rewrite eq. (2.36) in the form

a> 2

C + A cos 2 A -

m

(2.37)

imJm(B)exp

for $ = 0. The light field at a distance z is easily calculated by multiplying each spatial harmonic by exp { - ip2nAz/d2}, where p designates the relevant harmonic number (see 3 2.2.3). Correspondingly, the light amplitude at a distance z becomes (origin of the coordinate system placed in the object plane) E(x,z) = C

c imJm(B)exp m

+ f A c imJm(B)exp m

+ f A 2 imJm(B)exp m

I> I>

(m + 1)2n- - a(m + 1YAd2 d X

X

(m - 1)2n- - n(m - 1)"Ad

d2

(2.38)

The intensity distribution is calculated as im-"Jm(B)Jn(B)exp

I(x,z)= m

n

+ ACexp +A2cos

( - i d);

cos (27t

.

I>

X

(m-n)21r--(m2-n2)aAd d2

9-

2mnA -

.

(2.39)

The basic characteristics of the Fresnel field intensity patterns should now be established. From theoretical and practical points of view the most important

1, B 21

THEORETICAL CONSIDERATIONS

23

characteristics are the axial localization and the contrast value of the bestvisibility diffraction images. These parameters must be specified for various combinations of profile parameters A, C, and B of a sinusoidal complex object. The problem cannot be solved analytically, and a numerical analysis is required. The conclusions are as follows: (1) In the planes z = vd2/rZ, where v denotes an integer, the intensity distribution is given by =

C2

+ ;A2 + ~ A C C O S ( V ~ ) C O S

+ $A2 cos (4 n

i)

(2.40)

and matches the intensity in the object plane z = 0. This obvious fact, valid for any object amplitude transmittance and following immediately without numerical calculations, is due to the basic property of the self-imaging phenomenon. At the planes z = vd2/rZone obtains images of the absorbing part of the complex object. There are no planes, however, where images of phase modulation only appear. As explained before, one encounters a lateral shift of half a period in the case v = odd integer. (2) The localization of the best-visibility planes is a function of the amplitude contrast A/C and the amplitude B of the sinusoidal phase variations. The exception is the case of maximum amplitude contrast equal to unity, where localization is independent of the magnitude of the phase modulation. It coincides with the self-imageplanes. In general, the axial repetition period d2/I of the Fresnel diffraction field is preserved. (3) The shift of the best-visibility planes from the self-image planes is directly proportional to B and inversely proportional to A/C. (4) The departure of the best-visibility planes from the self-image planes in the cases B > 0 and B < 0 are equal but of opposite sign. For B > 0 the shift direction is the same as the propagation direction of the light beam. ( 5 ) The character of the intensity distribution in the best-visibility planes when B > f n is mainly determined by the phase modulation. For higher phase modulations the definition of diffraction image visibility is ambiguous. (6) Best-visibility planes do not coincide with the planes of maximum peak-to-trough difference. This coincidence is only encountered in the cases of pure amplitude and pure phase objects. (7) A simplified analytical method for studying self-imaging and employing only the three lowest harmonics of the object amplitude transmittance cannot

24

SELF-IMAGING AND

ITS APPLICATIONS

[I, 0 2

be used in the case of a complex object. The reason is that the phase difference between all adjacent object harmonics is no longer the same as in the case of pure amplitude or phase objects. The phase angles between generally complexvalued adjacent harmonics depend on the object profile parameters. Correspondingly, all harmonics have to be taken into account. Some of the properties just mentioned were previously noticed in the and DAVIS[ 19681. They can research by BURCH[ 19631 and SCIAMMARELLA be of practical importance for measuring the phase modulation introduced by photographic materials. The method proposed by PATORSKI[ 1980a,b] uses a sinusoidal diffraction grating with amplitude contrast A / C < 0.4. For weak modulations, as mentioned earlier, the localization of the best-visibility planes is a function of the phase modulation depth (fig. 2.5). If we know A/C and measure the distance of the first best-visibility plane from the grating, the parameter B can be determined. Diffraction images observed in planes different from the self-image planes are generally called Fresnel images, and the best-visibility planes, described earlier, can be treated as such. However, because of the definition of WINTHROPand WORTHINGTON [ 19651, the term “Fresnel images” refers only to diffraction images that show the interesting property of multiplication of the number of composite elements in a periodic structure. Since this phenomenon is of little importance in the case of complex transmittance objects, the discussion of Fresnel images is deferred to the next subsection, which examines amplitude objects.

tZ

..

0.9

.‘ I

.

\

‘.

‘. c

.

Fig. 2.5. Axial localization of the first maximum contrast plane in the Fresnel diffraction field of a sinusoidal complex grating as a function of the object profile parameters. (A/C) amplitude contrast; (B) phase modulation amplitude. Z = 1 corresponds to z = d 2 / I (Mer PATORSKI [198Oa].)

1,s 21

THEORETICAL CONSIDERATIONS

25

2.4.2. A mplicude objects The case of a sinusoidal amplitude object is readily obtained from eqs. (2.36) and (2.39) by putting B = 0. The intensity distribution in the self-image planes z = 2vd2/3, is given by =C2++A2+2ACcos (2.41) It matches perfectly the intensity in the object plane. On the other hand, at observation distances equal to an odd multiple of d2/21, double-frequency intensity patterns are observed. Their utilization in the precise measurement of small phase retarders was proposed by LOHMANN[ 19561. It should be remembered that in the case of an arbitrary (i.e., nonsinusoidal) transmittance, we require that all object spatial harmonics take part in the diffraction image formation. Moreover, the parabolic approximation should be satisfied by all of them. Various cases of the linear amplitude type of gratings with special transmittances over a single period were analyzed, for example, by MALACARA[ 19741, KONITZ[ 19841, and WRONKOWSKI [ 19871. The process under consideration is based on the free space propagation of light beams; no spatial filtering is allowed. Therefore for pure amplitude objects we cannot expect any contrast enhancement in the diffraction images as a function of the propagation distance z. However, for certain amplitude objects the Fresnel diffraction images, lying between the self-image planes, display very interesting properties. They were studied extensively by several authors, including COWLEYand MOODIE[ 1957a,b,c], ROGERS[ 1962, 1963, 1964, and WORTHINGTON [ 19651, GUIGAY [ 19711, BRYNGDAHL 19721, WINTHROP [ 19731, SMIRNOV[ 1977, 1979, 1986a], EBRALIDZE[ 19811, PATORSKI, WRONKOWSKI and DOBOSZ [ 19821, IOSELIANI [ 19831, PATORSKI[ 1983~1, SZWAYKOWSKI [ 19841, BAR-ZIV[ 19851 and WRONKOWSKI [ 19871. Other references will be given in $ 3 , which examines applications of the self-imaging phenomenon. Although the studies of Fresnel images represent a fascinating theoretical field, their mathematical treatment is extensive and cannot be presented here. Discussion will be limited to their most important properties. As mentioned earlier, the longitudinal period between the self-imagesis equal and MOODIE[ 1957a,b,c, 19601 and ROGERS[ 1962, 1963, to 2d2/1. COWLEY 19641 introduced the concept of suborders related to this basic distance. The suborder n was defined as the Fresnel image lying at a distance 2d2/nl along

26

SELF-IMAGING AND ITS APPLICATIONS

[I, 8 2

the illumination direction. Furthermore, WINTHROPand WORTHINGTON [ 19651 in their very important paper introduced a more general notation. The location of Fresnel images, with the interesting property of multiplication of the composite element of periodic structure, can be readily obtained by replacing v in the expression 2vd2/1 with v + N/n.Here N and n, with N < n, denote integers with no common factor. The period in the Fresnel image, defined as the distance between the elements forming the object periodic structure, is reduced M times. For a one-dimensional periodic object M = n when n is an odd integer, and M = in when n is even. Alternatively, M can be called the multiplication factor. To obtain the multiplication effect, the images of object component elements must not overlap. Moreover, to have the intensities of multiplied images equal, coherent noise must be avoided. In other words the object transmission between the composite elements must be equal to zero; otherwise some of the elements in the multiplied Fresnel image will be of [ 19771). Another parameter diminished intensity (ROGERS[ 19631, SMIRNOV of Fresnel images, namely, their lateral displacementwith respect to the optical axis, was determined by PATORSKI [ 1983~1 and SZWAYKOWSKI [ 19841. Cases with no lateral displacement and half a period shift are possible. In general, it follows from the theoretical investigations(COWLEYand MOODIE[ 1957a,b,c], WINTHROPand WORTHINGTON[ 19651, SMIRNOV [ 19771, EBRALIDZE [ 19811, IOSELIANI [ 19831, PACKROSS, ESCHBACHand BRYNGDAHL [ 19861) that Fresnel images can be represented as superpositions of displaced and phase-shifted object field distributions. The simplest object generating Fresnel images, with the aforementioned properties, has the form of a binary amplitude grating with transmittance levels of 0 and 1 and is composed of slits which are narrow when compared with the grating period. The ratio of slit width to grating period is called the duty cycle or the opening number. In the experiments of WINTHROP and WORTHINGTON [ 19651, ROGERS[ 19721, and BRYNGDAHL [ 19731, binary objects of this type were used. On the other hand, it follows that a square type of binary grating with an opening number equal to 0.5 (frequently used for experiments) cannot form its multiplied images. In the Fresnel image planes the images of composite elements (slits) mutually overlap. For example, in the case of the Fresnel image plane located at distances d2/21from the object or self-image planes (i.e., N = 1 and n = 4), the multiplication factor is equal to M = in = 2. This means that two slits will fill the image period, resulting in a uniform intensity distribution at those planes. This property is well known from various self-imaging experiments with square wave binary gratings. Another characteristic feature of Fresnel images of a grating of this type is that the intensity patterns in the planes

I, B 21

21

THEORETICAL CONSIDERATIONS

located at the distance d2/4A from the object or self-image planes are very similar to the object intensity pattern itself (PATORSKI,WRONKOWSKI and DOBOSZ[ 19821, BAR-ZIV[ 19851). The only difference is a smaller peak-totrough intensity modulation. When such a Fresnel image is visually observed, it can easily be mistaken for a grating self-image. 2.4.3. Pure phase objects The case of a pure phase sinusoidal object, chosen for the sake of simplicity and clarity of the analysis, will be considered. It is readily obtained from eq. (2.36), putting A = 0; that is (2.42) The intensity distribution in the Fresnel diffraction field as a function of the axial distance z is calculated as (2.43) I= 1

This expression was derived by COWLEYand MOODIE [ 19601, REICHELT, [ 19711. The intensity distribution STORCKand WOLFF [ 19711, and GUIGAY in the self-image planes, that is, z = 2vd2/A and at distances z = 2(v + i)d2/A, where v is an integer, matches the uniform intensity observed in the object plane. For small values of the amplitude of phase modulation the planes with deepest intensity variations are encountered for z = (v + @'/A. The intensity pattern detected at those planes is expressed by

=

1

+ 2( - 1)" C ( - 1)'-

J2/-

I= 1

x cos [(21- 1) (an;)]

*

l(2B) (2.44)

For higher values of B the definition of contrast (the ratio of the difference and sum of the maximum and minimum intensity values, respectively, over a single period) of the intensity pattern loses its practical meaning. Figure 2.6 shows the intensity curves at the best-visibility planes for different values of B. The

28

SELF-IMAGING AND ITS APPLICATIONS

2

I

B=x/4

1

1

B=x/2

1

X

0

d/2

I 0 d

B=W4

Fig. 2.6. Intensity distributionsin the Fresnel diffraction field of a sinusoidal phase grating in the plane z = d2/2d represented over a single lateral period. B indicates the amplitude of phase modulation.

character of intensity patterns for intermediate values of B can be deduced by a close examination of the curves shown. It can be seen that only in the range 0 < B < in the maxima and minima are unambiguously defined and found at x = 0 and x = i d , respectively. For higher values of the phase modulation, additional extrema appear. Moreover, they shift their position along the x axis with changes in B. A closer look at eq. (2.44)reveals that the intensity spectrum of the Fresnel field of a sinusoidal phase grating at distances z = (v + :)d’/A is composed of odd harmonics only. At the same time the harmonics attain their maximum value. The application of this property to the measurement of the amplitude of phase modulation, as well as other applications of the self-imaging of phase structures, will be described in 8 3.4.

1 . 8 21

THEORETICAL CONSIDERATIONS

29

A considerable interest in the self-imaging phenomenon of the phase type of periodic objects follows from two important facts. The first feature is the correspondence between the scattering of electrons by a crystal lattice and the diffraction of light by a phase object (COWLEY and MOODIE[ 1957dl). In fact, and MOODIE[ 1957a,b,c] were motivated by the pioneering works of COWLEY the development of a high magnification, lensless imaging system for crystal lattices. Light-optical analogs are frequently used by electron microscopists to verify various theoretical investigations. Because this review paper is restricted to optical self-imaging, further discussion of the self-imaging phenomenon encountered in electron microscopy is outside the scope of this work. The reader is referred for further details to Konitz [ 19791. The second area of interest is related to the phase grating theory of ultrasonic light diffraction (RAMAN and NATH [ 19351). The excellent monograph by BERGMANN[ 19641 provides references of early studies on visualization of ultrasonically produced phase gratings using the Fresnel ditfraction field. The observed intensity distributions were called “secondary interference” patterns. The works Of HIEDEMANN and BREAZEALE[ 19591 and COOK [ 19631 are the first ones to be reported in the optical literature. Phase gratings can be produced with standing or progressive ultrasonic waves. In the latter case, for intensity pattern visualization stroboscopic illumination or photodetector-oscilloscope detection is required. Further uses of the Fresnel diffraction field in acousto-optics were primarily concerned with the determination of the so-called Raman-Nath parameter describing the and amplitude of the phase modulation. The relevant works of COLBERT ZANKEL[ 19631, COOK[ 19761, RILEY[ 19801, and PATORSKI[ 1981a, 1984dl will be reviewed in 8 3. 2.5. DOUBLE DIFFRACTION SYSTEMS

In all preceding considerations we were concerned with the Fresnel field properties of a single object with self-imaging properties. The question might arise, however, what are the properties of systems using two or more diffracting structures of this type separated in space. By double diffraction systems we mean optical configurations composed of two periodic structures separated along the optical axis, that is, the illumination direction. The second structure is placed in the Fresnel diffraction field of the first one. In general, both gratings can be of amplitude or of phase type with equal or unequal spatial periods. We can distinguish two basically different

30

SELF-IMAGING AND ITS APPLICATIONS

[I, 8 2

localizations of the observation plane. It can be in the Fraunhofer or in the Fresnel diffraction region of the second structure. 2.5.1. Far-@Id configurations In the far-field configurations of double diffraction the intensity changes in the Fraunhofer diffraction “spots” are functions of the mutual lateral displacement between the two structures and are of primary interest. They can be utilized, for example, in the design of optical displacement transducers (KODATE,KAMIYA and KAMIYAMA [1971], KODATE,KAMIYAand TAKENADA [ 19751, PATORSKI [ 19781, HANEand HATTORI [ 19841, HANE, and HATTORI [ 19851);for alignment monitoring (FLANDERS, SMITH UCHIDA and AUSTIN[ 19771, TORIIand MIZUSHIMA [ 1977, 19791, BOWHUIS and WITTEKOEK[ 19791); and for acousto-optical light modulation (HARGROVE, and MERTENS[ 19621, CALLIGARIS, CIUTIand GABRIELLI HIEDEMANN [ 1977, 19801, DERIUGINand KOMOCKIJ [ 19791, PATORSKI[ 1981a,b, 1983d,e]). Figure 2.7 shows the far-field double diffraction system. Two diffraction gratings G1 and G2 are illuminated by a spatially coherent quasi-monochromatic plane wave front. The observation is conducted in the back-focal

c

L1

Fig. 2.7. Far-field double-diffraction system: S, point source; L1, collimator lens; GI and G2, periodic structures; L2, decollimating(transforming) lens; OP, observation plane.

I, 8 21

THEORETICAL CONSIDERATIONS

31

plane of the second lens. If the laser beam is unexpanded, both lenses can be omitted and the observation plane is at a distance far enough from the second grating to give clear separation of the double diffraction orders. The fundamental parameters of the intensity changes in the far-field orders that are a function of the mutual lateral displacement between the gratings G 1 and G2 include (i) light modulation frequency as a function of the ratio of the spatial periods of the gratings, (ii) phase relationships between modulation waveforms in the far-field orders of opposite and of the same sign, and (iii) modulation depth in the separate double diffraction orders. It was shown [ 19791 and PATORSKI [ 1983d,e] that the in detail by TORIIand MIZUSHIMA previously mentioned parameters depend on the type of gratings used and their axial separation distance. Because of the extent of the mathematical analysis describing the performance of far-field double diffraction systems in the literature, it will not be reproduced here. Only the most important conclusions will be given, as follows: (1) Two characteristic separation distances can be distinguished, namely, z = d2/I and z = (v + j ) d 2 / I . When comparing, for example, the intensity distribution in a system composed of two phase gratings, with that in a system composed of an amplitude and a phase grating, an interesting feature can be observed. Features of the changes in far-field light intensity for one double diffraction system (with one of the characteristic separation distances) are found in the second system for another characteristic axial distance. This conclusion does not only concern the situation where all the emerginglight from all far-field orders is detected. In this case, in the system using two phase gratings, there are no intensity changes, irrespective of the separation distance between the gratings. (2) The amplitudes of intensity changes, optimized at the aforementioned characteristic separation distances, are proportional to the profile parameters (amplitude transmittances) of both diffraction gratings. This property and the relative phase relationships between modulation waveforms can be applied to the problem of absolute position monitoring. Detection of intensity changes in two symmetrical far-field orders provides complete information on the amount and direction of the movement. (3) The light modulation frequency of double diffraction orders depends on the ratio of the spatial periods of the gratings. For example, if the spatial frequency of the stationary gratings is p times higher than the frequency of the moving one, the light modulation frequency is p times higher than the frequency obtained with two gratings of the same period. Other modulation parameters, that is, relative phase relations between modulation waveforms in various

32

SELF-IMAGING AND ITS APPLICATIONS

[I, 8 2

orders and the modulation depths, depend on the type of gratings used, their separation distance, and whether p is an odd or even integer. (4) The far-field light modulation features of the configuration using two antiparallel phase gratings, that is, acousto-optic light modulators using standing waves or two independent progressive sonic beams, are the same as the modulation features for a system composed of one stationary and one moving phase grating. The only difference is a twofold increase of the light modulation frequency. All the preceding properties, derived analflcally or using computer calculations, were verified experimentally (see references quoted at the beginning of this section). When studying the systems with ultrasonically generated structures, the Raman-Nath thin phase grating approximation was used.

2.5.2. Fresnel &Id double diffraction systems In a Fresnel field double diffraction system, the observation plane is located in the Fresnel diffraction region of the second and, at the same time, of the fist diffraction structure. Figure 2.8 depicts the general geometry of the Fresnel double diffraction system. A spherical or plane wave front beam from a point source S illuminates two linear amplitude gratings separated along the optical axis. Becausz the main interest in the analysis of such configurations (EBBENI [ 1970b1, SZWAYKOWSKI and PATORSKI [ 19851) is to determine the localization of the planes where moire fringes can be obtained, an inclination angle

Fig. 2.8. Geometry of the Fresnel field double-diffractionsystem composed of linear patings GI and G2. (Adapted from SZWAYKOWSKI and PATORSKI[1985].)

I , § 21

THEORETICAL CONSIDERATIONS

33

between the grating lines is allowed. Generally the grating periods can be different. Again, because of the extent of mathematical analyses, we will only present the most important properties of the system. The intensity distribution in the Fresnel region of grating G2 is influenced, simultaneously, by the self-imaging phenomenon of the two gratings and the spatial beat frequencies generated by them. The values of the beat frequencies depend on the angle between the grating lines. Therefore, the self-imaging phenomenon of moiri: fringes, the lowest beat frequencies, changes correspondingly. For example, in the case of diffraction gratings G1 and G2 of the same period and illuminated by a plane wave front beam, the best contrast of all the aforementioned three diffraction fields is obtained when the second grating G2 is placed in the self-image plane of the first grating G1 and the observation plane is chosen at the distance equal to a multiple of the self-imagingdistance of G2. At this plane, simultaneously, the best visibility of the intensity pattern formed by selected beat frequencies is encountered. Certainly, the moire pattern can be treated independently. Moiri: fringes under plane wave front illumination can be treated as a diffraction grating with period d/2 sin 6, where d is the period of G1 and G2 and 28 is the angle between the lines of these gratings. For any axial separation distance 1, we can find the observation plane behind G2, in which the contrast of the moire fringes attains a maximum value. This property of a double diffraction system under spatially coherent illumination cannot be found in the case of spatially incoherent illumination (EBBENI[ 1970a,b]), as will be shown in 8 4. In the latter case the grating separation distance is limited to some discrete values only. and PATORSKI [ 19851 the analyses of other In the paper of SZWAYKOWSKI systems were included as well. They concern the multiple beam illumination of the system shown in fig. 2.8 using an additional diffraction grating placed in front of G1 or between G1 and G2. Such systems were studied experimentally [ 1984~1and theoretically by SMIRNOV [ 1986~1.The properties by PATORSKI established find application as a method of producing binary amplitudegratings [ 1985]), as will be described with controllable parameters (SZWAYKOWSKI in $ 3. Another approach to study the Fresnel field of a multiple grating system was proposed by JOZWICKI [ 19871. It is based on the analysis of changes of the Fourier spectrum of the optical field when propagating through the component gratings. Specific features of this approach when compared with calculations conducted directly in the Fresnel field are the constant position of the Fourier plane, the discrete character of the spatial frequency distributions, and simple

34

SELF-IMAGING AND ITS APPLICATIONS

[I, 8 2

mathematical operations describing the transition between the spectra of and PATORSKI [ 19851). adjacent gratings (EBBENI[ 1970b1, SZWAYKOWSKI General relations describing cases with the point source at finite and infinite distance were derived.

2.6. PRACTICAL DEPARTURES FROM THE THEORETICAL MODEL

At the beginning of this chapter, to clearly show the basic properties of the self-imaging phenomenon, we introduced certain assumptions concerning the light source and the object. The case of a quasi-monochromatic and spatially coherent light beam illuminating a strictly periodic object of infinite lateral dimensions was assumed. Such a theoretical description is sufEcient for many studies. However, since it is important to realize the effects arising as a result of departures from the theoretical model assumed, this section examines the problem. 2.6.1. Influence of the source parameters Self-imagingis a diffraction-interference process. When defined in this way, it is dependent on the coherence of the light source. The two kinds of coherence that are distinguished, spatial and temporal, are produced with sources of small size and narrow spectral bandwidth, respectively. The requirements of hi@ spatial and temporal coherence are satisfied when low-power gas lasers are used. In fact, these are frequently used in various self-imagingexperiments. On the other hand, the use of other light sources is required to reduce the coherent noise at the observation plane or to increase the amount of information. As an example, various wavelength-dependent coding techniques can be mentioned. We will briefly review here the theoretical aspects of the self-imaging phenomenon related to the influence of the source coherence. Application aspects will be described in $ 3 . 2.6.1.1. Spatial coherence The influence of the source size on the intensity distribution in the Fresnel field of a periodic object was mentioned earlier in 8 2.2. The extended source can be treated as if it is composed of many mutually incoherent point sources, each of which generates the diffraction images. However, because of the lateral displacement between those sources, the corresponding self- or Fresnel images are also mutually displaced. The amount of displacement depends on the

1 9 8

21

THEORETICAL CONSIDERATIONS

35

experimental configuration, that is, the axial distances between the source, object, and observation planes, and on the object period. As a result, overlapping patterns form a blurred image, which disappears rapidly by increasing the source size. COWLEY and MOODIE[ 1957~1were the first investigators to discuss the light field in the self-imageplanes for some special cases of extended sources. Analytical and experimental studies of the phenomenon using the theory of partial coherence were presented by FUJIWARA [ 1974, 19771, DECKERS [ 19751, GROUSSON and MALLICK[ 19751, and SMIRNOV [ 1986bl. The case of periodic partial coherence was investigated by IMAIand OHTSUKA[ 19821. FUJIWARA analyzed the cases of a sinusoidal [ 19741 and binary [ 19771 type of amplitude grating. He found that spatial coherence greatly affects the contrast in self-image planes and the axial localization of images of maximum contrast. Figure 2.9 shows the normalized longitudinal intensity distributions along 3

-:* r

In

C

1

C-0.6

Fig. 2.9. Normalized axial change of the intensity value along the directions of x = 0 (upper diagram) and x = i d (lower diagram) under various coherence conditions: a , source radius; C, contrast factor of a sinusoidal amplitude grating; N,self-image number. (From FUJIWARA [1974].)

36

SELF-IMAGING AND ITS APPLICATIONS

[I, 8 2

directions of x = 0 and x = i d for various values of the radius of a circular source; Ccorresponds to the amplitude contrast of the grating and N describes the self-image number. A damped oscillating behavior is easily seen with the

b

.

OJC

d=0.08mm d O d-0.16mm d Fig. 2.10. Intensity distribution over a single period in the self-image planes of number N under various spatial coherence conditions; the case of a sinusoidal transmission grating. (From FUJIWARA [ 19741.)

1,s 21

THEORETICAL CONSIDERATIONS

31

decrease of spatial coherence. At the same time an increasing shift of the maximum contrast planes from the self-image planes can be noted. In the coherent case these characteristics are not encountered. Figure 2.10 shows the intensity curves over a single period in the self-imageplanes for different spatial coherence conditions. Contrast values decrease with the increase of both the self-imagenumber and the source diameter. Further investigations concerning Fresnel field properties of a sinusoidalobject under partially coherent illumina[ 1986bl. tion were reported by SMIRNOV The analysis of DECKERS[ 19751 was also based on the propagation laws of the mutual coherence function. It shows the existence of the so-called “critical coherence” field structures, reducing fringe visibility even in the self[ 1986b1). image planes (the case of Fresnel images was discussed by SMIRNOV Although the study was conducted for the case of self-imaging of moir6 fringes, the results clearly show the influence of the coherence of the light on diffraction image parameters of periodic objects. The characteristics of partially coherent imaging in the presence of a defect of focus can easily be explained by making use of the properties of self-imaging phenomena (MALLICKand ROBLIN[ 19741). GROUSSON and MALLICK [ 19751derived an expression for the three-dimensional intensity distribution in the neighborhood of the geometricalimage plane of a diffractiongrating, taking into consideration the coherence of the light. In the case of three diffraction orders passing through the optical system, the amplitude of the fundamental frequency term of the intensity distribution is modulated by factors including the source size parameters, namely, the spectral bandwidth and the lateral extension of the source. A modern interpretation of the influence of finite source dimensions based on the concept of the transfer function was given by SWANSON[ 19831. For point source illumination the transfer function is equal to unity for all frequency components forming the self-image. When the light source becomes extended, the higher spatial harmonics are attenuated. When the angular extension of the source, normalized with respect to the wavelength, becomes equal to the fundamental object frequency, the transfer function becomes zero for all intensity spatial frequencies except for the zero-order component. 2.6.1.2. Temporal coherence The dependence of the self-imagingphenomenon on the spectral content of the light source is clearly seen, for example, from eqs. (2.14), (2.23) and (2.28). In the case of a light source placed at a finite distance from the periodic structure, each spectral component produces self-images with a wavelength-

38

SELF-IMAGING AND ITS APPLICATIONS

[I. 8 2

dependent lateral magnification and located at different planes. Axial chromaticity is also encountered under plane wave front illumination, although the lateral magnification remains unity for all wavelengths. Successful techniques utilizing self-imagingchromaticity for real-time depth and range sensing as well as achromatization methods will be described in f 3. 2.6.2. Infuence of finite object dimensions The discussion presented up to this point has concerned periodic objects of infinite lateral extent. However, when using an object composed of only a few repeated elements or when conducting the observation in the region of a diffraction field close to the geometrical shadow of the object edge, we can expect that the Fresnel field properties are different from those in the case of an idealized infinite object. In all applications of self-imaging we can usually dispose of the effects of finite object dimensions. Nevertheless it is worthwhile to describe the properties of diffraction images, including the effects just mentioned. The first examination of this problem was by MENZELand MENZEL[ 19481. They stated that in order to have the same intensity distribution on axis, as in the case of an object with infinite extent, the object should contain at least 17 spatial periods. Further detailed investigations were carried out by COWLEY and MOODIE[ 1957b], ROGERS[ 19631, DECKERS [ 19761, KALESTY~~SKI and SMOLINSKA[1978], SMIRNOV[1978], and KEREN and KAFRI [1985]. Additionally, qualitative explanations using the concept of the walk-off effect RAMISHVILI and CHAVCHANIDZE [ 19711 and were presented by DENISYUK, SILVA[ 19721. First, let us briefly present the explanation of the walk-off effect and, second, the conclusions following from the more rigorous explanation based on the [ 19781. Figure 2.11 shows a schematic representation approach of SMIRNOV 0

OP

L t-------------

Fig. 2.1 1. Walk-off effect. Diffraction orders separate laterally in space behind an object 0 of finite dimensions. Observation plane OP is placed at the distance z where the orders + 1 and - I mutually “walk-off.

1. § 21

THEORETICAL CONSIDERATIONS

39

of the diffracted waves behind the linear grating. For clarity only the lowest diffraction orders are shown. Because of diffraction, the beams behind the grating are laterally displaced proportionally to a distance z. Because of the finite object dimensions, high harmonics gradually “walk-off’ the zero-order beam and do not take part in the image formation. The further the observation plane, the smaller is the number of diffraction orders that contribute to the image. A characteristic periodic change of the field contrast along the z axis is present if at least three diffraction orders overlap in space. Due to the limited number of harmonics, even in the self-image planes, a complete replication of the object amplitude transmittance cannot be achieved. Additional effects are introduced by diffraction at the object edges. [ 19781 presented a quantitative analysis of diffraction images of SMIRNOV periodic objects with finite dimensions using Linfoot criteria (LINFOOT[ 19641) for evaluating the quality of optical images. After establishing a spatial frequency analogy between a periodic transparency and a diffraction limited lens, he showed that it is possible to determine the isoplanatic region in which the quality of the image is approximately constant and has a predetermined value. Certainly an increase of object dimensions leads to larger dimensions in the isoplanatic region. It is worth emphasizingthat the proposed theory enables one to find the minimum number of object spatial periods giving a satisfactory [ 19791 also addressed the image quality over a required region. SMIRNOV problem of the influence of a h i t e object dimension on the depth of focus of Fresnel images and on the process of their formation in the case of an initially defocused transparency. 2.6.3. Eflects of periodicity errors Among the objects satisfying the Montgomery condition for self-imaging, periodic objects are most commonly used. Therefore these objects are of theoretical and practical interest when considering the problem of the influence of periodicity errors on the quality of the images. Errors can be of two types: large errors corresponding, for example, to the absence of several elements in a periodic structure, and small deviations of the period value. The first type was illustrated by COWLEYand MOODIE[ 1957al and DAMMAN, GROHand KOCK [ 19711 and was treated theoretically and S K I SMOLINSKA experimentally in considerable detail by K A L E S ~ ~ C ~and [ 1978al. Because only the periodically distributed elements are self-imaged,the nonperiodic elements are not reproduced in the self-image planes. They contribute, however, to background noise in the regions where they should have been

40

SELF-IMAGING AND ITS APPLICATIONS

[I, B 2

reproduced. The restoration of missing object elements improves with increasing number of self-images being detected. This is because any element of the diffraction image is formed by all object elements. A similar statement is true for the case of an object with some foreign elements replacing the basic ones. Figure 2.12 is an example illustratingthe latter case. When increasing the self-image number for better object restoration, the “walk-off” effect of high spatial frequencies should be taken into account. This results in the loss of fine details in the image. The case of quasi-periodicobjects with smalldeviationsfrom periodicity was and KALESTY~WKI studied analytically and experimentally by SMOLINSKA [ 19781 and SMIRNOV and GALPERN [ 19801. The intensity distribution in

Fig. 2.12. Restoration by self-imaging of a periodic object with foreign patterns included: (a, c) parts of the faulty object; (b, d) their respective self-images. (From KALESTY~SKI and SMOLINSKA [ 1978al.)

I,!3 21

THEORETICAL CONSIDERATIONS

41

self-image planes is formed mainly by the periodic object field disturbed by the noise from the nonperiodic light field. When using an object with a periodicity error, the contrast of a diffraction pattern decreases with increasing number of object elements (periods). On the other hand, it is well known that the image [ 19761, KALESTYASKI and quality increases with aperture size (DECKERS SMOLINSKA [ 1978a,b], SMIRNOV[ 19781). Therefore there is an optimum number of periods that give the best image quality. If the departure from periodicity is described by a function under investigation, for example, in the case of carrier frequency photography (BIEDERMANN [ 19701) or moire fringe technique (THEOCARIS [ 1969]), the self-imaging phenomenon can be used to find the spatial derivative of this function. This and SZWAYKOWSKI [ 19841 and PATORSKI problem was studied by PATORSKI and SALBUT[ 19851. The optical set-up suggested is very similar to that of a Talbot interferometer,which is explained in detail in Q 3.3. Two identical copies of the deformed grating, illuminated by a plane wave front beam, are axially separated along the illumination direction. The second copy is placed in the self-image plane of the fist one. The product-type moire fringes in the plane of the second copy display gradient information about the deformation function. Both direct and cross derivatives can be obtained optically by a one-step process. The second copy can be replaced by a nonlinear detection process or an undeformed grating slightly rotated about its normal. In these cases the derivative is displayed as a contrast modulation of the diffraction images or moire fringes, respectively. The method proposed represents an improved version of the shearing interferometry-holographic method as proand CHANG[ 19711. posed by SCIAMMARELLA

2.7. SELF-IMAGING BY SPATIAL FILTERING

Although the axial repetition of the complex amplitude transmittance of an object in its Fresnel diffraction field is characteristic of objects with spatial frequencies located on Montgomery rings (MONTGOMERY [ 1967]), the phenomenon can be “artificially” produced for other objects as well. In a strict sense we have to multiply in space the elementary object amplitude transmittance to obtain, for example, its periodic matrix. In the Fresnel region behind that matrix the light field will show the self-imaging phenomenon. This technique, which is based on spatial filtering in a coherent optical system, was proposed by KALESTYSISKI and SMOLINSKA[1977] and KALESTYASKI[ 19801. A periodic sampling filter in the form, for example, of

42

SELF-IMAGING AND ITS APPLICATIONS

[I, s 2

a periodic pinhole array is placed in the Fraunhofer region of the object (fig. 2.13). The product of the filter transmittance and spatial frequency spectrum in the image space results in a convolution operation between the object amplitude transmittance and the Fourier transform of the filter. This basic principle is utilized in various image multiplication techniques with focusing optics, as [ 19761. In the image plane the multiplied object image reviewed by THOMPSON is modulated by the Fourier transform of the single aperture of the sampling filter. The distance between sampling apertures must be properly chosen with respect to the Fourier spectrum of the object; it should satisfy SHANNON’S [ 19491 sampling theorem. It becomes increasingly obvious that, because of periodic modulation of the light field by the sampling filter, the object self-images are formed in front of and SMOLINSKA[1977] and and behind the image plane. KALESTY~~SKI KALESTYRSKI [ 19801 proved this fact even for a more general system that did not involve imaging optics. Their sampling filter was located in the far-field diffraction region of an object illuminated by a plane or spherical wave front. In the Fresnel diffraction zone of the filter, multiplied object images appear along the optical axis. The authors gave a detailed mathematical model of the system performance. Figure 2.14 shows the intensity distribution in the central zone of one of the multiplied images. A single object is indicated in the top left corner. A more general analysis of this technique was presented by KOLODZIEJCZYK [ 1986a1, who showed that the sampling filter can be described as a sum of exponential functions with square phase expressing the transmittances of convergent lenses. This approach allowed the author to explain the filter imaging properties irrespective of the type of illumination (spatially coherent or

Fig. 2.13. Technique of multiple image formation by filtering in the spatial frequency plane: 0, object; L1 and L2, coherent optical processor; OP, observation plane.

THEORETICAL CONSIDERATIONS

43

Fig. 2.14. Central part of the self-image of a multiplied singular object (top left corner) formed by the spatial filtering technique. (Courtesy K A L E S T Y ~ ~and S KSMOLINSKA I [ 19771.)

incoherent). A detailed theoretical analysis was conducted to determine the axial localization of the image planes, the lateral magnification of individual images and their periodicity lattice, and other imaging characteristics. The theoretical results were confirmed by experiments. In the methods just described the self-imaging phenomenon was generated by imposing a lateral periodicity onto the light field. Axial periodicity of a light field can also be produced, using a general nonperiodic filter in the form of Montgomery rings. This idea was first proposed by LOHMANN,OJEDACASTANEDAand STREIBL[ 19831. This type of filter can be produced by photographing the Airy transmittance function of a Fabry-Perot interferometer. It is interesting to note that the rings with numbers equal to square integers represent the periodic filter, giving both lateral and axial periodicity. The same principle of the “Montgomery rings-like” effective filter was independently discovered and implemented in a different arrangement by INDEBETOUW [ 1983, 1984al. Figure 2.15 shows the proposed experimental setup. Instead of using a spatial filter in the frequency plane, the Fabry-Perot interferometer is placed in the image space of the imaging optics. The multiple reflections are formed by projecting the object image between the reflecting

44

SELF-IMAGING AND ITS APPLICATIONS

Q

Fabry Perot

Fig. 2.15. Arrangement for self-imaging through a Fabry-Perot INDEBETOUW[1983].)

interferometer. (From

surfaces of the interferometer. They generate equally spaced images by a distance 2d along the optical axis, where d denotes the cavity length of the interferometer. The transmittance peaks of the interferometer (BORN and WOLF [ 19641) were found by INDEBETOUW [ 19831 to correspond to spatial frequencies located, approximately, on rings satisfying the Montgomery condition. The Limitations of the method imposed by the finite half-width of the Airy function peaks and the finite lateral extent of the field were presented. They influence the number of obtainable self-images and their quality, depending on the number of spatial frequencies taking part in the image formation. Lateral periodicity can be obtained by transmitting the rings described by square integers only, namely, 0, 1, 4, etc. Continuation of the studies on axial replication of object images by spatial filtering represents the work of OJEDA-CASTANEDA, ANDRESand TEPICHIN [ 19861. By relaxing the Montgomery conditions for longitudinal periodicity, they showed that any zone platelike spatial filter is adequate for self-replicating the object image at a specified axial distance. In other words, the filter should be laterally periodic in the square of the spatial frequency. The result can be used under either coherent or incoherent illumination. [ 19881treat the standard optical In the latter case DAVILA and LANDGRAVE processor as a unit magnification apodized system with a zone plate serving as an apodizer. The same system is capable of imaging periodic object planes into a single image plane. The authors give a simple explanation of the performance of both imaging configurations using geometrical optics and calculate the relative intensity of the images on the basis of wave optics.

1 . 8 21

45

THEORETICAL CONSIDERATIONS

2.8. EXPLANATION OF SELF-IMAGING USING A HOLOGRAPHIC APPROACH

After presenting some of the more usual approaches to the self-imaging phenomenon, we shall briefly present some studies in which the effect is treated as a holographic process, including those by ROGERS[ 197Ob1, KAKICHASHVILI [1972, 19741, and ARISTOVand IVANOVA [1975]. A hologram of a periodic structure can easily be formed in the Fresnel diffraction field of the structure. The zero-order diffraction beam can serve as a reference wave. Some similarities to Gabor in-line holography can be noted. If the holographic plate is placed in one of the self-image planes, the object and hologram happen to be identical (ROGERS[ 1970bl). Upon reconstruction, self-imaging occurs as described earlier. Generally, the holographic plate can be placed at an arbitrary distance from the object. The field just behind the hologram is replicated in the self-imageplanes, whereas object images are reconstructed at the planes shifted from the self-images. Figure 2.16 shows the recording and reconstruction stages. In the latter stage only the first real and virtual images of the object 0 and hologram H are indicated. A hologram is a record of the object Fresnel diffraction field. H,, and H , = are the first real and virtual self-images of the hologram H, and 0,-and 0,-- are the corresponding first self-images of 0. Further experiments dealing with generation of self-images by holographic methods were reported by KONITZ,BOSECKand LASH [ 19841 and SARMA, SHENOYand PAPPU[ 19851, who investigated the reconstruction properties of side-band Fresnel holograms of self-imaging objects.

,

-,

2.9. SELF-IMAGING IN OPTICAL WAVEGUIDES

According to the discussion presented in 8 2.1, only some objects with a specified spatial frequency spectrum possess the property of self-imaging;that is, their image can be formed under free-space diffraction field propagation. 03.4

I I

I

09.1

I

I

I

I

H3.1 I I I

--c I I

I

Fig. 2.16. Schematic representationof a holographicrecording of a periodic object without using a reference wave and hologram reconstruction. The hologram is a record of the object Fresnel diffraction field. H,, and H,, are the first real and virtual self-images of hologram H. 0,I and O,= - I designate the corresponding first self-images of an object 0. (Adapted from KAKICHASHVILI [ 19741.)

,

-,

46

SELF-IMAGING AND ITS APPLICATIONS

[I. ! I2

Nevertheless, there exists another type of self-imaging phenomenon, suggested by BRYNGDAHL[ 19731. The use of classical refractive elements is abandoned and, moreover, the formation of images of objects that cannot be self-imaged is possible. In spite of such a promising introduction, however, the self-imaging phenomenon in planar optical waveguides, which will be examined here, is a highly specialized technique with a much smaller number of applications. The explanation given by ULRICH [ 1975a,b] and ULRICH and KAMIYA [ 19781 will be followed, and for further details the works of BLUME[ 1976, 19771 are recommended. Figure 2.17 shows a cross section through a planar waveguide with the object schematically indicated. The guide has a uniform refractive index n, over its thickness. The treatment will be for a onedimensional slab waveguide, but it can be readily extended to square or rectangular cross section waveguides, producing two-dimensional self-imaging (VOGES and ULRICH [ 19761). If an object is placed at the entrance of the waveguide, under certain conditions the object intensity distribution is repeated at some distances along the guide. This effect results from interference between the waveguide modes being excited by the object. At an arbitrary distance from the object plane the modes are, in general, dephased because they propagate along the waveguide with different phase velocities. If the phase difference between the modes is equal or almost equal to integer multiples of 2n, a self-image of the object is formed. This explanation is analogous to that given in 5 2.4. The modes of a multimode planar optical waveguide can now replace the object diffraction orders. In the following short mathematical description, the object field is represented as a superposition of wave functions Fm(x) of the guide with complex amplitudes a, (2.45)

where m is a positive integer denoting the mode number. Taking into consideration only the guided modes, the field in the observation plane at a distance z

Fig. 2.17. Cross section of planar dielectric optical guide. (Adapted from ULRICH[ 1975al.)

1.8 21

THEORETICAL CONSIDERATIONS

41

(2.46) (2.47) where zm stands for the transfer coefficient of the mth mode. The exponential term exp(i&z) gives the phase shifts experienced by the mode$ along the guide with 8, describing the propagation constant of the mth mode. The form of the intensity distribution along the z axis depends on the phase differences between the modes. They can be referred to the fundamental mode m = 0 in the form

~ X[i@m(z)} P = ~ X [i(Bm P - 80)zI Z

+ 2m + 8,)

J,

(2.48)

where z, denotes the distance of the first self-imageand ,6 is a phase coefficient correspondingto the phase dispersion of the modes. To a good approximation (ULRICH [ 1975a1)we have z1 = 4n,w2/L, where w denotes here the waveguide or object width. The phase error expressed by Sm leads to a decrease in resolution of the images. Assuming an ideal waveguide with 6, = 0 and I ,z I = 1, and expressing z as z, = vzl with v = 1, 2, 3, .. ., we find that erect self-images are formed in the planes z2,z,, z6, . . .. On the other hand, inverted images are encountered at the distances zl, z3,z5, .. ..In real waveguides the coefficients 6, cannot be omitted and the higher-order modes degrade the image quality. A detailed discussion of this problem is given by ULRICH[ 1975a,b] and ULRICHand KAMIYA [ 19781. "Geometrical" parameters of self-imagingin planar optical waveguides (i.e., the magnification or demagnification of self-images), the multiplication and ANKELE[ 19751. effect and image inversion were analyzed by ULRICH The experimental results of ULRICH[1975a] are given in fig. 2.18. Figure 2.18a shows the first image v = 1 of a 3-pm-wide slit produced at a distance z = 24 mm by a 50-pm-wide guide with refraction indexes n, = 1.496 and n, = 1.457 (see fig. 2.17). Figure 2.18b shows the slit image as observed in the microscope, fig. 2.18~shows the intensity distribution at the same observation plane as in fig. 2.18a but for the slit with a wrongly adjusted width (the phase coincidence of a given number of modes at a specified observation distance depends on the object width), and fig. 2.18d shows the intensity distribution at a distance 150 pm further apart and the object slit width adjusted appropriately.

48

SELF-IMAGING AND ITS APPLICATIONS

Fig. 2.18. Experimental results of ULRICH[1975a] obtained with a planar waveguide. See text for explanation.

Applications of the phenomenon in the field of integrated optics and dealing POOLEand with the design of directional couplers were reported by NIEMEIER, ULRICH [ 19851 and NIEMEIER and ULRICH[ 19861.

8 3. Applications of the self-imaging phenomenon The beauty and simplicity of the self-imaging phenomenon have not only resulted in many theoretical works but have also led to many applications in science and technology. Irrespective of the variety of studies reported, it seems that the phenomenon has not been fully exploited up to the present time. For example, in the overwhelming majority of applications described, periodic objects are used, whereas other structures satisfying the general Montgomery conditions have not been exploited. The works reported in this section will be divided into the following primary groups: image processing and synthesis, production of optical elements, optical testing, and optical metrology. Some of the projects can belong to more than one group.

1.8 31

APPLICATIONS

49

3.1. IMAGE PROCESSING AND SYNTHESIS

The first proposal to use the Fresnel field properties of self-imaging objects in image processing and synthesis appears to have been reported by REICHELT, STORCKand WOLFF [1971]. They addressed the problem of redundant coherent imaging without granulation using a phase modulator based on the Fresnel diffraction of a sinusoidal grating. The diffraction field close to the grating satisfies the required properties of a wave front with a constant amplitude and spatially varying phase over the object. The axial range in which these properties are satisfied depends on the grating period and the amplitude of phase modulation, which was explained in 8 2.4. Although not in chronological order, it is thematically justified at this point to mention another application of the Fresnel diffraction field in the proximity and ZINKE [ 19831 for behind a phase grating. It was reported by DAMMANN the purpose of spatial frequency color encoding of blue, red, and green light with different spatial frequencies. The axial localization of the self- and Fresnel images of a periodic structure is wavelength dependent, and unless we are very close to the object (the authors quote an observation distance of the order of about 10 grating periods), the diffraction images can be significantly blurred. For the purpose of spatial frequency encoding of selected lines of the source spectrum, the authors described the use of Fresnel images in the proximity of specially designed twin gratings formed by superposition of two binary phase gratings with different spatial frequencies. Each composite grating encodes a different color in the Fresnel image plane. The influence of the size of the white light source on the encoded modulation depth was also studied. The use of a Fresnel image of a phase grating in an array illuminator was proposed by LOHMANN[1988]. Optical microcomponents are placed at a fractionalTalbot distance d2/21from a binary phase grating where the periodic intensity distribution is found. [ 19731described the Eu and LOHMANN[ 19731 and Eu, LIUand LOHMANN use of self-imaging for producing spatial filters for image differentiation. Although the authors proposed a solution for generating the first- and higherorder derivatives, this review will be limited to a description of the case of the first derivative only. It is well known that to obtain the derivative of the object amplitude transmittance in a coherent image-forming system it is necessary to attenuate and phase-shift the spatial frequency spectrum according to F(<)K (LOHMANN and PARIS[ 1968]), where F(<)denotes the filter transmittance as a function of the spatial frequency 5. The object spectrum should be multiplied

<

50

[I, 8 3

SELF-IMAGING AND ITS APPLICATIONS

by i2n& which can be achieved by a special copying process of the Fourier hologram in which the object information is recorded. Figure 3.1 shows the three stages required for derivative generation as proposed by Eu and LOHMANN [ 19731. In the first stage the Fourier hologram H, is recorded. It can be considered as a quasi-periodic grating if the object’s lateral dimension is small compared with the distance between the object and the reference source. The second step is simply to copy the hologram using the self-imaging phenomenon. However, the recording plate H, is slightly tilted with respect to the illuminating wave front. As explained in 8 2.2, the contrast across the plate varies due to changes in the axial distance between the copied hologram and the recording plate. For example, the contrast of the fundamental harmonic becomes zero (see eq. 2.28) when z is equal to d2/21, 3d2/21, etc. As previously, d denotes the hologram spatial period and 1 is the light wavelength. This contrast modulation can be used for the aforementioned spatial frequency attenuation, as required for derivative observation upon reconstruction of the hologram copy H,. In detailed theoretical studies the authors found the proper value of the tilt angle of the recording plate H, and established that plates H I and H, should touch each other at one edge. The experimental conditions for producing higher-order derivatives were established as well. Figure 3.2 shows the experimental results for an object in the form of the Chinese character meaning LIGHT and the letters COPY. [ 19731 uses the copy Another solution described by Eu, LIUand LOHMANN plate of a square-wave diffraction grating directly as a spatial filter in a coherent optical processor. Again the recording plate is tilted to introduce fringe contrast variation across the field. One of the diffraction orders of the filter, as seen in

H,

L

OP

Fig. 3.1. Optical differentiation method by hologram copying and reconstruction: (a) hologram recording; (b) hologram copying; (c) reconstruction. (Adapted from Eu and LOHMANN [1973].)

1, § 31

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51

Fig. 3.2. Experimental results obtained using hologram copying method: (a) object; (b) x-derivative pattern; (c) y-derivative pattern; (d) another result. (From EU and LOHMANN[1973].)

the processor output plane, carries the information proportional to the first derivative of an object located in the output plane. Higher-order derivatives can be obtained, for example, by bending the grating or film on which the filter is recorded. Another image processing operation that can be performed with the help of the self-imaging phenomenon is image subtraction or addition (EBERSOLE [ 19751). PATORSKI, YOKOZEKI and SUZUKI[ 1975al used the self-images of a periodic structure for the formation of complementary gratings. They are required by the grid coding method of image subtraction, as described by PENNINGTON, WILLand SHELTON [ 19701. The phase shift of A between the complementary gratings can be realized by self-imagingin double-exposure and real-time configurations. In double-exposure configurations the object transparency to be processed is placed in one of the self-image planes of a square-wave amplitude binary grating. Its image contains the sampled version of the object; the usual sampling condition (SHANNON [ 19491) must be fumed. To obtain a complementary sampling grating we can, for example, tilt the illuminating beam properly (fig. 3.3). As was shown in 2.2, the self-images shift laterally when the beam is tilted. The amount of shift is proportional to the tilt angle and the self-image number. The second exposure is done with the second object transparency (to be subtracted from the first one), which is introduced in the recorded complementary self-image. After development the plate is placed in

52

SELF-IMAGING AND ITS APPLICATIONS

[I. 8 3

Fig. 3.3. Schematic representation of the double-exposure self-imaging method for image subtraction: So and s,,point source at two lateral positions; L,collimatorlens; G,diffractiongrating; I, self-image plane coinciding with the input plane of an imaging system IL; OP, recording plane. YOKOZEKIand SUZUKI[ 1975al.) (Adapted from PATORSKI,

a conventional coherent optical processor, and the direction of the first orderdiffraction is observed. The contrast-modulated carrier is recorded in those areas of the double exposure plate where the different intensities of the two images are overlapping. The contrast is proportionalto the differencesbetween the objects. Modulated areas appear bright at the output plane. Figure 3.4 shows the experimental results obtained in the setup of fig. 3.3 using half a period lateral shift of the self-imaged grating between exposures. Real-time subtraction can be obtained using a Mach-Zehnder interferometer configuration with two object transparencies and two gratings placed in its separate arms. A more complicated method for image subtraction, using side-band Fresnel holography of self-imagingobjects to form complementarycarrier gratings, was reparted by SARMA,SHENOYand PAPPU[ 19851. The carriers were formed by making use of two real images generated during the reconstruction of the hologram of an amplitude diffraction grating.

Fig. 3.4. Experimentalresults obtainedby the double-exposuremethod using half a period lateral shift of the grating between exposures: (a) object image; (b) processed image. (From PATORSKI, YOKOZEKIand SUZUKI[1975a].)

1,s 31

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53

Another approach to image subtraction or addition was undertaken by [ 1978bl and PACKROSS, ESCHBACH and KALESTYRSKI and SMOLINSKA BRYNGDAHL[ 19861 for image synthesis. The technique does not require imaging optics or spatial filtering. It uses properties of the Fresnel images of a periodic object composed of various elements (substructures) to be synthesized. The elements are properly distributed over the object periodic lattice. In this way they form mutually displaced sublattices. Under spatially coherent illumination in the Fresnel image planes, where multiplication of the elements forming the periodic object occurs (see $2.4 and the references quoted), the object substructures overlap, being laterally displaced and phase shifted. The overlap conditions were calculated in detail for various Fresnel images by [ 19651, and IOSELIANI ROGERS[ 19631, WINTHROPand WORTHINGTON [ 19831. When the phase factors are equal to + 1 or - 1, addition or subtraction of periodic object substructures is encountered, respectively. The compound periodic object can be formed by a step-and-repeat photography or spatial frequency sampling technique. Figure 3.5 shows the result of image synthesis (addition operation) as reported by PACKROSS, ESCHBACH and BRYNGDAHL[ 19861. The object was composed of two two-dimensional lattices of the letters L and F. Each lattice has a period din the vertical direction and i d in the horizontal direction. If the

Fig. 3.5. (a) Object composed of sublattices of letters L and F; (b) synthesized image. (From PACKROSS, ESCHBACHand BRYNGDAHL[1986].)

54

SELF-IMAGING A N D ITS APPLICATIONS

[I, 0 3

observation is conducted at a distance z = d2/21, we face twofold “multiplication” in the Fresnel image in the vertical direction, and the self-image in the horizontal direction. In the former case the letters L and F add constructively, forming the letter E. As a result, a square lattice of E is observed. The intensity distribution in the plane z = d2/2il for the object transparency can be treated, equivalently, as the sum of the intensities of component elements. Particular design of the elementary cell of the object makes it possible to obtain real-time contrast reversal at the same Fresnel image distance, that is, z = d2/21. This technique was described by KOLODZIEJCZYK [ 1986bI. In the same paper the author proposed a complementary technique not requiring periodic replication of the elementary cell in the object. It can also be achieved by a spatial sampling technique in the object frequency plane as described in 8 2.7. In the plane defocused by the distance z = d2/21 with respect to the image plane of a coherent optical processor (in the focused image plane a sampling filter generates a multiplied version of the isolated object), a Fresnel image with contrast reversal can be obtained. It is interesting to note the use of a recording plane located at a distance d2/21from the periodic structure in a double diffraction system for producing [ 19851. This method (see special binary gratings as reported by SZWAYKOWSKI 3 3.2) can be treated as Fresnel image synthesis under multiple beam illumination. Another important application of Fresnel field properties of periodic objects [ 19731 used an object to is related to multiple image formation. BRYNGDAHL be multiplied as a diffuse light source illuminating a square array of small apertures (fig. 3.6). In such a case the intensity distribution in the Fresnel field is described by the convolution of the source intensity function with the intensity distribution generated by a point source. The latter one corresponds to the impulse response function of an imaging system. Therefore the self-image or Fresnel image intensity distribution obtained with highly spatially coherent illumination can represent the intensity impulse response function of the free space diffraction optical system, using incoherent or partially coherent illumination. As a result, square arrays of the object intensity distribution in the selfor Fresnel image planes are obtained if the lateral extent of the object is properly selected with respect to the spatial period of intensity distributions in those planes. BRYNGDAHL[ 19731 verified this principle experimentally and compared it with a pinhole-array camera approach. A generalized treatment of this method and the technique using a sampling I SMOLINSKA[ 19771, K A L E S ~ S K I filter approach ( K A L E S ~ S Kand [1980]) was given by KOLODZIEJCZYK [1986a]. The performance of the

1, I 31

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55

Fig. 3.6. Schematic representation of multiple image formation using self-imaging with extended light source: S , object source; 0,pinhole array. For simplicity only one of the diffraction images DI is shown. (Adapted from BRYNGDAHL [1983].)

sampling filter, that is, the square array of apertures pictured in fig. 3.6, was shown to be equivalent to the performance of an array of converging lenses introducing specified phase shift values. The focal length of these lenses was expressed as a rational value of the distande d2/2L,where d denotes the spatial period of a pinhole array (sampling filter). The axial localization of multiple images, their lateral magnification and their mutual separation depend on the system geometry, that is, on the axial distances between the object, sampling filter and observation planes, lateral object dimensions, filter period, etc. The theoretical model proposed is valid for spatially coherent and incoherent illumination. The chromaticity of the effect should be taken into account. It is worth noting that the sampling technique can be used in spectral regions where refractive optics cannot be used. In the paper mentioned earlier BRYNGDAHL[ 19731 proposed to generate multiple images by applying the condition for self-imaging to a single optical fiber used as an optical tunnel. The principle of the method can be seen in fig. 3.1.

, Fig. 3.7. Principle of image multiplication of an object 0 using the self-imaging effect generated in an optical tunnel. (Adapted from BRYNGDAHL [1973].)

56

SELF-IMAGING AND ITS APPLICATIONS

117

03

To obtain like the self-imagingmultiplication of an object, the length L of the optical tunnel should be equal to 2vb2/rZ,where b is the side length of the square cross section of the tunnel and v is a positive integer. The physical explanation of the technique was given in 5 2.9; detailed studies of the parameters of multiplied images have been given by ULRICHand ANKELE[ 19751. Another application of self-imaging to image-processing-likeoperations is related to the case of using quasi-periodic objects. One of the methods resulting in self-restoration of faulty objects was described in $ 2.6 where we reviewed and I the works of DAMMANN,GROHand KOCK [1971], K A L E S ~ S K SMOLINSKA [ 1978b], and SMOLINSKA and K A L E S ~ S K[ 19781. I Another application to obtain the derivative of a quasi-periodic function, developed by PATORSKI and SZWAYKOWSKI [ 19841 and PATORSKI and SALBUT[ 19851, was described in $ 2 . 6 in the discussion of the self-imagingphenomenon with quasi-periodic objects. The self-imaging phenomenon was proposed by VANYAN, GARIBASHVILI, KLIMCHUK, MUMLADZE, RAMISHVILI and CHAVCHANIDZE [ 19771 to improve the sharpness of blurred images. The defocused image of an object was replicated to form a periodic array and illuminated by a spatially coherent wave front. With the blurred elements carrying sutficient information about the object, a sigdicant improvement in image sharpness was obtained in axially periodic planes in the Fresnel diffraction field. No special deblurring filters are required, and information about defocus can be recovered. The possibility of using self-imaging for simple band-stop filtering of the selected class of binary objects composed of Rademacher functions (BEAUCHAMP [ 19751) was proposed by OJEDA-CASTANEDA and SICRE [ 19831 and LOHMANN, OJEDA-CASTANEDA and SICRE[ 19841. The filter is placed at the selected Fresnel image of the object, composed of sections of square-wave amplitude gratings. The rejection bandwidth selectivity can be improved by multiple interaction filtering. An achromatization technique of the self-imaging phenomenon can be of interest in optical information processing operations. It was described by PACKROSS, ESCHBACH and BRYNGDAHL [ 19841. Since localization of the self-image planes is wavelength dependent, polychromatic illumination results in a loss of contrast of the images. (On the other hand, applications making use of the longitudinal dispersion of diffraction images will be described later in $3.4.) To allow for white light illumination,the authors proposed an optical system compensating the chromaticity for a given self-image plane and preserving the image magnification. This was achieved using an imaging system with reversed dispersion. Figure 3.8 shows the optical system.

1, § 31

51

APPLICATIONS

0 SI

L3

L2 L1

I

Fig. 3.8. System for achromatization of the self-image: 0, object; SI, achromatized self-image; L1 and L3, achromatic lenses offocal lengthf, andf3, respectively;L2, zone plate offocal length b; I, image plane; z = 2vdZ/1.(Adapted from PACKROSS, ESCHBACH and BRYNGDAHL [1984].)

Elements L1 and L2 are placed in the back focal plane of the achromatic lens L3 to correct lateral magnification. The observation plane is at a distance b = fi from L1 and L2. Figure 3.9 shows the experimental results obtained under three-wavelength illumination without and with achromatization. Another theory concerning achromatizationof the self-imagingphenomenon was suggested by SICRE, BOLONGINIand GARAVAGLIA [ 19853. They predicted that partial achromatization of a selected self-image can be realized by introducing a spectral filter in the source. The filter should be designed in correspondence with the spatial frequency of the object being self-imaged. Most recently the necessary condition for self-imaging in incoherent polychromatic light was derived by INDEBETOUW [1988]. Synthesis of such incoherent self-imaging fields can be obtained by spatial frequency filtering in a space-invariant imaging system. Two types of self-imaging pupils were analyzed: a) a diffracting mask (of which the Fresnel zone plate is a special case) resulting in self-image planes depending on wavelength, (b) a resonant cavity, i.e., Fabry-Perot etalon, producing achromatic self-images. The etalon placed in the object (image) space selects longitudinal spatial frequencies coinciding with the resonant modes of a Fabry-Perot cavity.

3.2. TECHNOLOGY OF OPTICAL ELEMENTS

When making grating copies by contact photography, RAYLEIGH[ 18811 noticed that in some zones the contrast of lines might be appreciably reduced. He explained this as a consequence of imperfect contact between the grating and the recording material. Because of the finite gap, self-imaging determines the local contrast of the diffraction field recorded on the plate. For example, when the air gap has a thickness equal to d2/2A, the contrast of the diffraction image of a square-wave type binary amplitude grating becomes zero. The

58

SELF-IMAGING A N D ITS APPLICATIONS

Fig. 3.9. Left column: diffraction images in the first self-image plane for the light wavelength I = 514.5 nm recorded with (a) I = 514.5 nm, (b) 1 = 476.5 nm, and (c) I = 454.5 nm. Right column:recordings obtained in the image plane ofthe system shown in fig. 3.8. (From PACKROSS, ESCHBACHand BRYNGDAHL[1984].)

APPLICATIONS

59

undesired separation between the master grating and the recording plane might arise from their imperfect contact or unflatness. The diffraction effects become less important with a decrease in the spatial frequency of the copied grating. On the other hand, it is desirable in the copying process of a low-frequency (nonspectroscopic) grating to avoid direct contact between the master ruling and the recording medium, to prevent rapid damage of the master. For this purpose the use of the self-imaging phenomenon was proposed by BURCH [ 1960, 19631. The photosensitive plate is then located in the self-image plane where the contrast of the diffraction image of the amplitude master grating attains its maximum value. This technique was extended by PATORSKI[198Oc], PATORSKIand SZWAYKOWSKI [ 19831, and SZWAYKOWSKI [ 19851to produce and test binary amplitude gratings with an arbitrary opening ratio and a variable period. The use of gratings with a selected opening ratio (duty cycle), defined as the ratio of the transparent line width to the grating period, permits a sensitivityenhancement of the moirC f h g e method. Fringe sharpening (POST[ 19671, PATORSKI, YOKOZEKIand SUZUKI[ 1976b1) and fringe multiplication (POST [ 19671, LANGENBECK [ 19691, BRYNGDAHL [ 1975l)methods are used for this purpose. The required opening ratio can be obtained using a commercially available square-wavetype of binary amplitude grating with an opening ratio equal to 0.5. This grating is imaged onto the photographic recording plate using self-imaging. A controlled lateral displacement of the grating or the photographic plate between the two exposures results in a composite grating with the required opening ratio. For the same purpose the incidence angle of the illuminating beam can be changed, as proposed by JAROSZEWICZ and KOLODZIEJCZYK [ 19851. Since the displacement should be smaller than the half-period value, opening ratios smaller than 0,5 can be obtained. A single-exposure method requires two identical master rulings overlapping mechanically (where the rulings are in contact) or optically (where the second grating is located in one of the self-image planes of the first). The case of optical overlap (double diffraction system) is interesting from the and PATORSKI[ 19851) and practical points of theoretical (SZWAYKOWSKI [ 19851). It allows us to produce linear and cross-type view (SZWAYKOWSKI gratings with opening ratio values in the range between 0 and 1. Figure 3.10 shows the optical system proposed. A plane coherent wave front illuminates two linear gratings (cross-type gratings can be used as well) of the same spatial period d and separated by the first self-image distance z1 = 2d2/A. The recording plane is located at a distance z = d2/21from the second grating. This is the Fresnel image plane where, in the absence of the fist grating, a uniform

60

SELF-IMAGING A N D ITS APPLICATIONS

L

GI

G2 OP

Fig. 3.10. Double-diffraction system for producing binary amplitude gratings with arbitrary opening ratio. Plane wave front illuminates two identical binary gratings GI and G2 of period d; z, = 2d2/11; z = d2/2A. OP, recording plane.

intensity distribution is encountered. Nevertheless, a composite grating is observed in the double diffraction system. This results from the interference of all the diffraction orders from the second grating, arising from its mutually coherent multiple beam illumination. Figure 3.1 1 shows photographs (nega-

Fig. 3.11. Photographs (negatives) of linear diffraction gratings with opening ratios (a) 0.26, (b) 0.32, (c) 0.72, and cross-type gratings with ratios (d) [0.26,0.26], (e) [0.35,0.35], and (f) [0.21,0.70]. (From SZWAYKOWSKI [1985].)

APPLICATIONS

61

tives) of linear and cross-type diffraction gratings produced by the technique of SZWAYKOWSKI [ 19851. The period of composite intensity patterns can be varied by using spherical wave front illumination. In addition, the so-called “zoom effect” related to the formation of self-images at infinity (PATORSKI[198Oc], SUDOL[1980]) permits continuous variation of the grating period over an appreciable range. When the master grating is located at a distance equal to a multiple of d 2 / I , then one of the self-images is located at infinity. Therefore the image contrast behind the self-image preceding the one located at infinity varies very slowly, and gratings with a good contrast and different periods can be photographed. The application of self-imaging to the production of gratings with an asymmetrical groove profile was reported by JAROSZEWICZ and KOLODZIEJCZYK [ 19851. The recording plate is placed at the self-image plane of a binary amplitude grating with a small opening ratio. When the illuminating beam is tilted at a constant speed, the narrow bright lines in the self-image plane shift correspondingly. A proper change of the intensity of the illuminating beam during the lateral shift over one period allows us to obtain the desired exposure distribution. After a subsequent change of the density pattern to a phase pattern, the grating with the desired groove profile can be produced. From the practical point of view the line density generated with this technique is limited to approximately 300 lines/mm. An exciting application of the self-imaging phenomenon in the technology of periodic and quasi-periodic structures of submicrometer spatial period was reported by FLANDERS, HAWRYLUK and SMITH[ 19791. The technique, called spatial-period division, employs Fresnel images from a mask of a given period and a very small opening ratio to produce intensity patterns with a spatial period M times smaller than the object mask, where M denotes the multiplication factor discussed in § 2.2.4. For example, at the planes lying at a distance z = d 2 / M I ,the Fresnel images will have a spatial period d/M if the object mask has an opening smaller than iM.To obtain submicrometer period structures, radiation of a short wavelength must be used, since the condition A 4 d should be satisfied. Soft X-ray radiation was used in the experiments, reported to obtain gratings of period 0.1 pm. A high-coherence synchrotron source was suggested for obtaining Fresnel images with higher values of the multiplication factor. In principle the technique can challenge the limits of microlithography, namely spatial periods of about 10 nm. Subsequently,ARISTOV, AOKI,ERKO, KIKUTAand MARTYNOV [ 19851 reported experimental results obtained using ultra-soft X-ray radiation and two-dimensional periodic objects. Complex intensity patterns with very small elements were generated.

62

SELF-IMAGING AND ITS APPLICATIONS

3.3. OPTICAL TESTING

By optical testing we mean the analysis of phase distributions generated by optical elements (i.e., mirrors or lenses) and transparent and reflective objects. The self-imaging phenomenon was applied in this field to create a new type of shearing interferometer. It was developed independently by YOKOZEKIand SUZUKI[ 1971a,b] and LOHMANNand SILVA[ 1971, 19721. Ten years later a similar technique, known by the name of moire deflectometry, was described by KAFRI [ 19803. The first authors used the diffraction approach to describe the system performance, which led them to shearing interferometry characteristics. Kafri's explanation was based on a geometrical optics approach. The results for the two treatments are equivalent (PATORSKI [ 1986e]), although KERENand KAFRI[ 19861and their colleagues (BAR-ZIV,SGULIM, KAFRIand KEREN[ 19831) were fist to mention the difference between placing the phase object in front of and behind the self-imaged grating. The very origin of both Talbot interferometry and moire deflectometry methods can be traced to the [ 19641. However, because of a work of OSTER,WASSERMANand ZWERLING lack of close analysis, some ambiguities were left to future investigators. Taking into consideration historical aspects and the one-step character of their analyticalmodel, a descriptionthat follows the analyses of YoKoZEK1 and SUZUKI[ 1Y71a,b] and LOHMANNand SILVA[ 1971, 19721 will be presented here. It provides, simultaneously,information about the phase object and the influence of diffraction effects. A discussion of these effects in moirC deflectometry that supplements the ray tracing approach (KAFRI [ 19801) was given by KERENand KAFRI[ 19851. Figure 3.12 shows a schematic representation of the Talbot interferometer. A spatially coherent plane wave front beam illuminates a binary amplitude diffraction grating with a spatial frequency of a few lines per millimeter or higher. Its self-images are detected using the moire fringe effect by placing the second grating in one of the self-image planes. When the illuminating beam is L

Fig. 3.12. Optical set-up of the Talbot interferometer.

63

APPLICATIONS

deviated by an optical system under test or when a phase object is inserted, the fist grating diffraction orders become aberrant, leading to a distortion of the grating self-images. To find their description, we assume for simplicity a cosinusoidal amplitude transmittance T(x, y) of a linear diffraction grating. This assumption does not influence the generality of the approach, since the results can be directly extended to binary gratings. Let us write T(x,y)=A,+A,exp

):*( I-x

+A-,exp

( "d") , -1-x

where A,, A and A - are the amplitudes of the diffraction orders and d is the grating period. Let the function g(x, y ) describe the optical path introduced by a phase object placed at a distance z, from the fist grating plane. In the observation plane the light field formed by three diffraction orders can be represented as H x , y , 4 = A , exp [ikg(x, A1 x

"")I

+ g(x - A, y, z) - 2d2

where k = 2n/1 and I A1 = 1(z - z,)/d denotes the lateral displacement of the phase object function, carried by the nonzero diffraction orders, as measured from the optical axis in the observation plane. The value exp(ikA2z/2d2) expresses the phase shift of the f 1 orders with respect to the zero order due to free space propagation over a distance z; g(x, y , z) describes the object function in the observation plane. Under the approximation of slow variations of g(x,y) with the axial propagation distance, we could write g(x, y, z) z g(x, y). Retaining the general notation g(x, y , z), we assume, however, slow variations of the phase distribution with respect to the lateral coordinates x and y. It can be readily shown that for small values of A the intensity distribution formed by the three terms of eq. (3.2) can be written as Z(x, y , z ) = A;

+ 2 4 + 4AJl

+~A:cos-

(

cos

(2) -

cos $!

4n d a&, Y , 2 ) ) X--A d 1 ax

( - dI A

ag(xy" 'I) ax (3.3)

64

SELF-IMAGING AND ITS APPLICATIONS

3

Equation (3.3) was derived under the approximation

At the observation planes located at a distance z = 2vd2/IZ (the factor 2 may be omitted; in that case for v equal to an odd integer the grating dark lines become white in the diffraction image, and vice versa), the maximum contrast in the intensity distributions is encountered. It can be seen from eq. (3.3), that the grating self-image lines are no longer straight. Their deformation is proportional to the first derivative of the phase distribution under study. It is detected visually by a moirC fringe technique. The derivative information is averaged over the distance A, recognized as the lateral shear. If the detecting grating placed in the observation plane has the same spatial period as the self-imaged one and the lines of both gratings are set mutually parallel, the moirC fringes give a contour map of the derivative distribution. The uniform field display mode is encountered. By slightly rotating the detecting grating about its normal, horizontal reference fringes parallel to the shear direction are introduced. The derivative information is displayed as a departure from straightness of the moirb fringes. Another finite fringe detection mode with reference fringes perpendicular to the shear direction can be obtained using a detection grating with a slightly different period. The great simplicity of this interferometer must be emphasized. It only requires a well-collimated spatially coherent light beam and two relatively low-frequency gratings. The chromatic character of the self-imaging phenomenon requires a quasi-monochromatic source to assure good contrast in the detected self-images. The first grating plays the role of a beam splitter, whereas the second grating magnifies the information encoded in the Fresnel field diffraction patterns. The shear value or sensitivity of the method is easily changed by varying the distance between the object and the second grating (when the object is placed between the gratings) or by varying the separation between the gratings (when the object is in front of the beam splitter grating). For the same purpose a change in grating spatial period might be used. In the case of more significant changes of g(x, y) or larger values of lateral shear A, the approximation expressed in eq. (3.4) is not sufficient and the second term of the Taylor series expansion of g ( x , y ) must be taken into account. In this case eq. (3.3) transforms into

APPLICATIONS

65

It is seen that the ‘secondorder derivative term enters the contrast (amplitude modulation) term in the intensity distribution in the self-image planes. Therefore the form of the self-image lines carries information about the first derivative of the phase distribution and, at the same time, the contrast modulation gives the second derivative. Certainly, since it is mainly focused on the gradient information, the second factor should be considered as unwanted noise. It can be avoided by using two-beam lateral shear interference instead of the three (at least) diffraction orders required for the self-imagingto occur. This concept was proposed by SILVA[ 19721, KEREN,LIVNAT and GLATT[ 19851and PATORSKI [ 1985, 1986al. In the first two studies two-beam interference was obtained by using a spatial filtering technique and a specially prepared beam splitter grating, respectively. The self-imaging phenomenon is no longer present in those configurations. On the other hand, the configurations proposed by PATORSKI [ 1985,1986alfor improving the contrast of moirb fringes at higher shear values use the basic configuration of a Talbot interferometer, shown in fig. 3.12. To avoid the addition of two-beam interference patterns formed by the diffraction orders (0, + 1) and (0, - l), respectively, they were isolated by utilizing their different spatial frequencies or orientations. This technique was achieved by rotating the beam splitter grating about the axes parallel and perpendicular to its lines, respectively. However, when the object is placed in front of the first grating, the change in shear value over the observation plane must be taken into account when quantitatively analyzing the interferogram. It is worth observing that Talbot interferometrycan be treated as a modification of the Ronchi test (RONCHI[ 19641). In the latter configuration the beam splitter grating is placed near the focus of the beam under test, which results in high lateral magnification of the grating lines, and the detecting grating is not required. In the Ronchi test configuration the self-imaging effect is treated as noise, highly degrading the fringe contrast, and two-beam interference is preferred. Nevertheless, several attempts were made to use three- (or more) beam interference for optical testing under small (MALACARA and CORNETO [ 19741, PATORSKI [ 1986f1) and large shear values (KOMISSARUK [ 19641, LIN [ 19861). Similar principles were applied to the study of slopes and and COWLEY curvatures of flexed plates (ASSA,BETSERand POLITCH[ 19771). When using linear diffraction gratings, Talbot interferometry provides derivative information along the direction perpendicular to the lines of the beam splitter grating. The use of circular gratings (LOHMANN and SILVA[ 19721,

66

SELF-IMAGING AND ITS APPLICATIONS

[I. 0 3

SILVA[ 19721) provides the radial derivative. This type of derivative, together with the azimuthal one, can be obtained using evolute gratings as proposed by SZWAYKOWSKI [ 19881. Before listing the applications of Talbot interferometryand/or moirt deflectometry, it is worthwhile mentioning the modifications to the interferometer system itself that are directed toward improving its performance. To remove the influence of errors of the illuminating wave front and grating line straightness, YOKOZEKIand SUZUKI[ 1971bl and KAFRI and MARGALIT [ 19811 proposed the production of a detection grating by photographing the self-image in the observation plane before inserting the phase object to be tested. An electronic superimposition video technique was described by YOKOZEKI [ 19811. On the other hand, as proposed by LIMand SRINIVASAN [ 19831, the detection grating can be computer generated to produce a uniform intensity or a distinct moirt pattern that is characteristic of a specific object, thus obtaining a null test. The use of anisotropic gratings was proposed by RABAL, FURLANand SICRE[ 19861. The sensitivity can be improved in a [ 19851. On the telescopic arrangement, as proposed by KAFRIand KRASINSKI other hand, the spatial resolution can be enhanced by a technique involving [ 19851). Various simultaneous vibration of the gratings (GLA'ITand UFRI methods of modem interferogram analyses (SCHATZELand PARRY[ 19821, STRICKEK [ 1985a,b]) enable fast data analysis. Because of limited space, we will only list the applications of Talbot interferometry and moirt deflectometry. The following applications were reported for Talbot interferometry: beam collimation test (SILVA[ 19711, FOUREand MALACARA [ 19741, YOKOZEKI, PATORSKI and OHNISHII [ 19751, PATORSKI, and SUZUKI[ 1976a1, KOTHIYAL and SIROHI[ 1987]), testing and YOKOZEKI focal length measurement of optical elements (YOKOZEKIand SUZUKI [ 1971a,b], SILVA[ 19721, YOKOZEKI and OHNISHII[ 19751, MALACARA and CORNEJO [ 19741, BENTONand MERRILL [ 19761, WADAand SAKUMA[ 19831, TAKEDAand KOBAYASHI[ 19841, NISHIMURA, ISHIGUROand YOKOZEKI [ 19851, PATORSKI[ 1985, 1986a1, BERNARDOand SOARES[ 1988]), analysis of phase objects (YOKOZEKIand SUZUKI[ 1971a,b], LOHMANNand SILVA [ 1971,19721, SILVA[ 19721,DALLAS and SIGELMANN [ 19761, KORIAKOVSKIJ and MARCZENKO [ 19801, NAKANOand MURATA[ 19841, JUTAMULIA, LIN and Yu [ 19861, SMIRNOV [ 1986d]), optical differentiation of quasi-periodic [ 19841, PATORSKIand SALBUT structures (PATORSKIand SZWAYKOWSKI [ 1985]),measurement of small tilts (NAKANO and MURATA[ 19861, NAKANO [ 1987]),analysis of vibrating objects (KAIJUN,JAHNSand LOHMANN[ 1983]), and optical alignment (KINGand BERRY[ 19721, PATORSKI,YOKOZEKIand SUZUKI[ 1975b1).

APPLICATIONS

Fig. 3.13. MoirC patterns for the camera lens tested in the Talbot interferometer with (top) mutually rotated and (bottom) parallel grating lines for different grating separation distances. (From YOKOZEKIand SUZUKI[1971a].)

67

68

SELF-IMAGING AND ITS APPLICATIONS

Fig. 3.14. Moire fringes for a candle flame placed between diffraction gratings of the Talbot interferometer. Grating lines are slightly mutually rotated. (From LOHMANN and SILVA [ 19711.)

For example, figs. 3.13 and 3.14 show the fringe patterns obtained when testing the spherical aberration of a lens and temperature gradients around a candle flame, respectively. The photographs are taken from the original papers describing the Talbot interferometry method. Applications of the moire deflectometrytechnique were reviewed by KAFRI and GLATT[ 19851. Their paper contains an extensive list of references and photographic documentation of the experimental studies.

3.4. OPTICAL METROLOGY

Although some of the already-mentioned applications of self-imaging could fit under this section, they will be described separately, with emphasis on their primary features. Here we present the applications of self-imaging in spectrometry, range and depth monitoring, and measurement of periodic phase modulations generated optically and acousto-optically.

1,

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APPLICATIONS

69

The use of self-imaging to characterize the temporal spectrum of a light source was proposed by LOHMANN[ 19621 and studied by KLAGES[ 19671. The distance z of self-images from the object plane under plane wave front illumination is given by z = 2vd2/1, where, as previously, v is an integer, d denotes the object spatial period, and 1is the light wavelength. The longitudinal chromaticity of self-imagescan be exploited to study the spectral content of the source. The contrast of self-images as a function of the axial distance z corresponds to the Fourier transform of the source spectral content. When a second grating is inserted into the Fresnel field of the first one and displaced in a direction perpendicular to the grating lines, the uniform field intensity changes will be monitored just behind the second grating. The amplitude of intensity changes attains its maximum when the second grating coincides with a self-image for a particular wavelength. Moreover, the amplitude increases with the sharpness of the spectral line. An interesting application of the self-imagingphenomenon in real-time range and depth measurement was described by CHAVELand STRAND[ 19841 and LEGERand SNYDER[ 19841. The property of the periodic change of contrast of quasi-monochromaticdiffraction images with the propagation distance was used for this purpose. The contrast of the fundamental spatial frequency in the intensity distribution of diffraction images provides a measure of the distance z. To optimize the access to this information, Chavel and Strand studied quantitatively the change of the basic harmonic as a function of z for different grating transmittances and light source structures. It was shown that the synthesis of a desired contrast change as a function of these two parameters is possible. If the observation plane in the Fresnel field behind the grating is scanned by a video camera, the spatial intensity modulation is converted to a temporal signal. The latter one gives a real-time measure of the distance z from the object to the sensing plane. Figure 3.15 shows a schematic representation of a system based on the principle just explained. The video camera scans the image of the diffraction grating being self-imaged onto an object with variable depth. The grating frequency was about 5 lines/mm; for the illumination wavelength 1= 514.5nm, the distance z = 2d2/1 was equal to 155.2mm. Figure 3.16 shows experimental results obtained for depth measurement of a small solenoid. CHAVELand STRAND[ 19841 discussed in detail various aspects of the method, that is, real-time properties with a single view of the measured space, solutions to overcome the range ambiguity problem resulting from the periodicity of the contrast function, influence of the nonuniformity of the illuminating beam, the dynamic range of the video camera, and the requirements for spatial

70

SELF-IMAGING AND ITS APPLICATIONS

Range Measurement Using Talbot Effect

z-I

4

@ 3-D Object

Camera

Analog

Fig. 3.15. System for range measurement as described by CHAVEL and STRAND[1984].

and temporal coherence of the source. The depth of the field sensed is limited by the depth of focus of the television camera. LEGERand SNYDER[ 19841 used a similar approach, supplemented by a white light optical processor to display depth information in pseudocolor. To remove coupling between the unambiguous range and the object field, they proposed to use a modulated cosine grating as the self-imaging object. In this way many relf-imaging longitudinal periods within the available depth of field set by the camera optics can be included. The amplitude of the cosine grating proposed is also cosine modulated. Because the problem of self-imaging such an object was not treated in the theoretical part of this article, it will be presented here. The object amplitude transmittance can be described as

where d and d , denote the periods of the object grating and its amplitude modulation, respectively. It is interesting to note that this object can be treated as additive-typemoire fringes (see, for example, BRYNGDAHL [ 1975]), formed by the sum of two periodic modulations. Correspondingly, eq. (3.6) can be rewritten in the form

Using the approach previously presented, based on the spectrum of plane waves (EDGAR[ 1969]), the intensity distribution in the observation plane at

1, B 31

APPLICATIONS

71

Fig. 3.16. Experimental results of the depth encoding technique using the self-imaging phenomenon: (a) original object; (b, d) modulated images at 488 nm and 514.5 nm, respectively; (c, e) processed depth slice at 488 nm and 514.5 nm, respectively. (From CHAVEL and STRAND [1984].)

12

SELF-IMAGING AND ITS APPLICATIONS

[I, § 3

a distance z can be readily calculated. Since the observation system acts as a low-pass filter, only the lowest spatial frequency terms will be detected. They are in the form

It can be seen that the intensity distribution with a lateral period fd, is now amplitude modulated as a function of the propagation distance z. The axial period is equal to dd1/2L. By choosing an appropriate value of the amplitude modulation period dI, the value dd1/2Lcan be made smaller than the longitudinal contrast change period of an unmodulated grating and equal to 2d2/L. We cannot speak in this case about true self-image planes. LEGERand SNYDER [ 19841discuss in detail the relationship between object field width and the unambiguous range resulting in an optimization of both parameters and of the depth resolution. Next they describe a real-time processor for representing depth information in pseudocolor. Two separate channels are required for pseudocolor coding. They are realized by using as the self-imaging object a cross-type grating with two component linear gratings with unmodulated or modulated amplitudeof different spatial periods. A liquid crystal light valve is used as an intermediate device to send the depth signal from the monitor to the coherent white light processor. Further details of this processor are outside the scope of this review. Another application of the self-imaging phenomenon in the field of optical metrology concerns the measurement of the phase modulation amplitude of a sinusoidal phase grating. The assessment of the modulation depth is required in order to specify the performance of optically and acousto-opticallyproduced diffraction gratings. In the former case the modulation depth determines the grating diffraction efficiency, whereas in the latter case it is related to the ultrasonic signal, which must be precisely calibrated for medical ultrasonic transducers. and ZANKEL [ 19631. The fist qualitative studies were reported by COLBERT From the form of the intensity distributions at the best-visibility planes in the Fresnel diffraction field (see, for example, fig. 2.6), they estimated the modulation depth of ultrasonically produced phase gratings. As explained in § 2.4, the harmonics of the intensity distribution in the Fresnel diffraction field of a weakly modulated sinusoidal grating change their value with the observation distance z. The amplitude of the Zth harmonic is described by J, [2B sin (Idz/d2)], where J, is the Ith order Bessel function of

1 - 8 31

APPLICATIONS

73

the first kind, B is the phase modulation amplitude of the grating under investigation, and d denotes the grating period. COOK[ 19761 was first to exploit this process to determine quantitatively the value of B. In the case of ultrasonically produced gratings, B is known as the Raman-Nath parameter. Cook measured with a selective photodetector and oscilloscope the fist temporal harmonic of the intensity distribution in the Fresnel diffraction field of a progressive ultrasonic beam. By performing two measurements for two distances z, the value of B can be readily found from tables of Bessel functions. RILEY[ 19801 and PATORSKI [ 1981al independently described the modification to Cook‘s Fresnel diffraction field method. Their approach only requires one-plane measurement without axial movement of the photodetector (fig. 3.17). The amplitude of the first temporal harmonic of the intensity pattern is measured at a distance z = ( v + ;)d’/A, where it attains its maximum value. This property can be used for the precise localization of the measurement plane. After normalization with respect to the bias term, the value of B can be found from the calculated curve 2J,(2B). The technique is particularly suited to a determination of very small phase variations because the amplitude of the temporal intensity harmonics is described by a Bessel function with double argument (2B). Therefore the sensitivity in the range of very small values of B is higher than that of conventional diffraction efficiency far-field measurement methods. Detailed investigations concerning the application of the concept just described to the analysis of ultrasonically produced phase gratings were given by RILEY[ 19811. PATORSKIand SZWAYKOWSKI [ 19811 proposed to exploit the Fresnel field method for the detection of harmonic distortion that occurs at higher values [ 19851investigated theoretically of the modulation depth. SAIGAand ICHIOKA L

Fig. 3.17. Optical configuration of the Fresnel diffraction method for the evaluation of the Raman-Nath parameter of a progressive ultrasonic beam: S, point source; L, collimatinglens; UB, ultrasonic beam; PH, slit limited photodetector plane; z = ( v + i)d2/A.

14

SELF-IMAGING AND ITS APPLICATIONS

[I98

3

and experimentally a Fresnel imaging technique with fast-switched stroboscopic illumination for visualization of the strain wave of a progressive acoustic beam. The Fresnel field properties, under double-beam illumination, of a progressive ultrasonic beam were theoretically studied by PATORSKI[ 1984dl. Another metrological application of the Fresnel diffraction field of thin phase gratings was described by HARDINGand CARTWRIGHT [1984]. In surface contouring it is frequently convenient to project a two-beam interference pattern onto the object and detect it with a reference grating. Because of the sinusoidal intensity distribution of the projected fringes, it is desirable to use a reference grating that is also sinusoidal. The authors proposed to employ a sinusoidal phase grating placed in front of the image plane of the system that images the object with projected fringes. When the diffraction pattern of a phase reference grating with good intensity contrast coincides with the image plane of the projected fringes, moirC fringes with very good visibility can be observed. Figure 3.18 shows comparative experimental results obtained using an amplitude grating and a phase grating serving as the reference grating. The use of self-imaging for fringe formation in the differential laser Doppler technique was studied by CHANand BALLIK[ 19741. They found the phenomenon to be applicable in particle velocity measurements and suggested its use, especially in studies of the velocity of particles on a rough surface and in fluid flow.

Fig. 3.18. Video recorded real-time moire patterns obtained with (a) amplitude, and (b) phase grating detecting two-beam interference pattern projected onto the object. (From HARDING and CARTWRIGHT [1984].)

I , § 41

THEORETICAL CONSIDERATIONS: INCOHERENT ILLUMINATION

75

4 4. Theoretical considerations: Incoherent illumination 4.1. THE LAU EFFECT AND ITS INTERPRETATION

Spatially coherent illumination of a periodic structure results in self-imaging. Self-imaging disappears when the lateral source dimensions are increased. However, an interesting effect can be observed if a binary amplitude grating is placed in front of an extended incoherent source. Well-defined fringe patterns in the diffraction field of the second grating can be obtained when special relationships between the spatial periods and axial separation distances of the gratings are satisfied for a given light wavelength. Moreover, to observe fringes the lines of both periodic structures must be mutually parallel. LAU [ 19481 was first to observe this effect using two binary amplitude diffraction gratings of the same spatial period. Spatially incoherent illumination of the first grating was provided, and the observation was conducted at infinity, that is, at the back focal plane of a lens placed behind the second grating. For some discrete values of the separation between the gratings, fringelike intensity distributions were observed. Their period decreases with increasing grating separation. When a white light source is used, fringes appear in various color combinations. The first extensive theoretical study of this effect, including the general case of diffraction gratings of unequal periods, was provided by ROBLIN[ 1971, 19731. The formation of fringes was treated in terms of a multiple combination of grating harmonics in many different image planes. They included finite distance, virtual and real patterns formed by gratings of equal and unequal periods. EBBENI[ 1970a1, using the Kirchhoff diffraction integral approach, treated the special set of interference patterns that correspond to moire fringes. Analyses of these patterns which were based on geometrical optics were [ 19691. Moire fringes reprepresented by MCCURRY[ 19661 and THEOCARIS sent one family of fringes among many others investigated by ROBLIN[ 1971, 19731. PE-ITIGREW[ 19771, although not referring his investigation to the Lau effect, was first to represent the double grating system in terms of a general theory of imaging of a periodic structure by an optical system with a periodic pupil function. The achromatic imaging conditions established by Pettigrew are similar to those of Roblin. A few years later the Lau effect, as it was called by the investigators, received considerable attention. Different theoretical approaches were adopted for its explanation. JAHNSand LOHMANN[ 19791, who first introduced the name “Lau effect”, studied the field properties behind the second grating, using diffraction theory and the self-imaging (Talbot effect) approach. The observed

76

SELF-IMAGING AND ITS APPLICATIONS

[I, B 4

field intensity distributions were shown to be expressed by a correlation of the intensity distribution of the first grating with the modulus square of the Fresnel diffraction pattern of the amplitude transmittance of the second grating. Some specific similarities between the Lau effect and self-imaging were noted. GORI [ 19791, SUDOL[ 19801, and SUDOLand THOMPSON [ 1979,19811based their analyses on coherence theory. Fringe formation was shown to result from matching between the period of the second grating and the period of the complex degree of spatial coherence of the field illuminating this grating. [ 19831 and SWANSON and LEITH[ 1982, 19851 used the grating SWANSON imaging approach (independentlyof the one presented by PEITIGREW[ 19771). They showed in detail that the Lau fringes represent only one family of many interferencepatterns produced by a double-gratinginterferometer. The second grating was treated as an imaging element with the assigned transfer function [ 19851 and action similar to that of a Fresnel zone plate. LEITHand HERSCHEY studied a spatial filtering approach for grating interferometers to use them as imaging systems that were linear in amplitude or intensity. Studies of the generalized Lau effect for imaging of periodic structures in spatially incoherent radiation were reported by ARISTOV,ERKOand MARTYNOV [1985]. The and OJEDA-CASTANEDA [ 19831 and explanations by BRENNER,LOHMANN JAHNS,LOHMANN and OJEDA-CASTANEDA [1984] are based on optical transfer function theory and the concept of virtual transforms, respectively. It was found in the latter work that good visibility patterns are observed when the virtual spectra of the second grating, generated by each slit of the source grating, are in consonance. JUTAMULIA, ASAKURA and FUJI[ 19851 treated the two gratings in the original Lau set-up as channel selectors. Each channel comprises mutually parallel, doubly diffracted waves originating from a specific point of an extended incoherent source placed at an infinite distance from the two gratings. Therefore multiple beam interference fringes are formed at infinity for each channel. The interrelationship between the Lau and Talbot effects was studied by SMIRNOV[ 1986~1using the convolution formalism of Fresnel diffraction. The optical transfer function approach was used by HANEand GROVER [ 1987a,b]to analyze the imaging behaviour of the double grating system under spatially incoherent illumination. Due to the high magnification involved the application to precise displacement measurement was proposed. The incoherent superposition of multi-diffraction patterns was studied in a hybrid system consisting, effectively,of three diffractiongratings (transmission,reflec[ 19871. tive and transmission type gratings) by HANE,HAITORI and GROVER Well-defined fringes were found to appear for grating separations which were integer multiples of the self-imaging distance.

1, § 41

THEORETICAL CONSIDERATIONS: INCOHERENT ILLUMINATION

I1

An explanation of the Lau effect will be given following the theoretical approach proposed by PATORSKI[ 1983b, 1986bl. It is based on the concept of incoherent superposition of multiple self-imaging. It was chosen here because of the direct adoptability of the considerations presented for the coherent illumination case. In this way it unites the analyses of coherent and incoherent double grating systems. Moreover, this explanation gives a simple description of all important parameters of the effect. It is hoped that the physical insight and intuition presented here will make it understandable to a wider community of readers. Both cases of finite and infinite distances between the two gratings are examined.

4.2. MULTIPLE INCOHERENT SUPERPOSITION OF SELF-IMAGING

The optical field behind the second grating can be treated as superposition of mutually incoherent Fresnel diffraction fields. In the first approximation the first (source) grating is considered as an array of mutually incoherent line sources, after which the influence of the finite width of the slits is taken into account. To observe well-defined patterns behind the second grating, the self-imagesor Fresnel images resulting from each line source of the first grating must overlap in space. In other words, the in-registry condition must be fulfilled, which results in spatial localization of the patterns and discrete grating separation distances. 4.2.1. Finite separation of difraction gratings Figure 4.1 shows the geometry for a double-grating diffraction system. Let us take two binary amplitude diffraction gratings of spatial periods d , and d2 with lines mutually parallel and separated along the optical axis by a distance z,. The planes of both gratings and the observation plane are perpendicular to the optical axis. The first grating, denoted as GS, is illuminated by a spatially incoherent quasi-monochromatic light source. Let us suppose, in the first approximation, that the grating GS consists of narrow slits. An arbitrary cross section of GS perpendicular to its lines consists of a row of mutually incoherent point sources, which emanate spherical wave fronts illuminating grating G. Diffraction images behind G overlap in space and their intensities add up. To observe well-defined patterns, the images must coincide. This in-registry condition involves pattern overlap in the plane perpendicular to the optical axis and along the axis. Thereforethe fringes are spatially localized. In the case described

78

SELF-IMAGING AND ITS APPLICATIONS

Fig. 4.1. Incoherent-illumination double-grating diffraction system: S, point source within one of the slits of the source grating GS; Ax, lateral shift of the origin of diffraction images of the second grating G in the observation plane OP. (Adapted from PATORSKI[1983b].)

here a coincidence of the self- or Fresnel images is required. Since all spherical beams radiated by the point sources of GS have identical wave front curvature at the grating plane G, the axial localization and the transverse magnification of diffraction patterns of G are the same for all mutually incoherent self-imaging phenomena. For example, the self-images are specified by the well-known equations (see Q 2)

where the distances zo and z are indicated in fig. 4.1, d2 is the period of the grating G, v is an integer, and d' designates the period of the self-image of number v in the observation plane. The lateral displacement of the patterns, given by

is generally different for each pattern. Here x,, denotes the lateral position of the point source S as measured from the axis in a direction perpendicular to grating lines, and Ax depends on the distance of point sources (slits) from the optical axis. We can expect, however, a coincidence of patterns for specified values of zo and z. It can be easily shown that the point source displacement in the direction parallel to the grating lines has no effect on the diffraction field properties under discussion. The cases of gratings of equal and unequal periods will be discussed

1, B 41

THEORETICAL CONSIDERATIONS: INCOHERENT ILLUMINATION

19

separately. In the first case the coincidence conditions relating to both the self-images and Fresnel images will be derived, whereas in the second case only self-images will be discussed. The Fresnel image overlap will be discussed by analogy. 4.2.1.1. Gratings of equal period In examining the case of gratings of equal periods, the goal is to find values of zo and z for which the lateral shift of the diffraction images of G, due to each source point of GS, is equal to the diffraction image period or its multiple. In the case of self-images of G, their lateral shift for the elementary source displacement xo = d is derived from eqs. (4.1) and (4.2) to be

Since A x must equal a multiple m of the self-image period d ’ , then by putting A x = pd‘, we obtain from eqs. (4.1) and (4.2) zo=-

2v d 2 -, P A

P z=zo-

1-p’

(4.4)

where p designates a positive integer. These expressions determine the inregistry condition for the self-images of G produced by two adjacent slits of GS. The period of well-defined patterns is given by

It is seen that for a specified value of zo equal to an integer or noninteger multiple of the distance 2vd2/L,there exists only one localization plane z for which well-defined patterns can be observed. Putting p = 1, we require a shift of the diffraction images, generated by two adjacent slits of the source grating in the in-registry plane, to be equal to the diffraction image period. The fringes are located at infinity, and the separation between the gratings equals a multiple of 2vd2/1. This case corresponds to some patterns of the whole family of Lau effect fringes. It can be easily shown that for observation in the back focal plane of a lens with a focal length f, the image period is equal to df/zo. Other Lau patterns are expressed, as will be shown below, by the in-registry conditions of the Fresnel images. For other choices of v and p, virtual patterns are obtained that differ from the infinity-localized fringes studies in the literature. Recalling that v might

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SELF-IMAGING AND ITS APPLICATIONS

assume negative values, we have z = -z,-

P p + 1'

d' = d - . 1

(4.7)

P+ 1

The equation describing z, remains unchanged, since the separation between the gratings should be a positive value. For the special case of p = 1, we have d2 z0=2v-,

1

z=

-1,

0 ,

d ' = '2d .

The last equation corresponds to the self-image in-registry pattern mentioned [ 19701 in one by ROBLIN[ 19731 and used by BOURGEONand FORTUNATO of the grating interferometers. In the case of Fresnel images we need to find values of z, and z for which the lateral shift of these images equals their period or its multiple. The period denoted as d , is

where M corresponds to the period reduction factor discussed in Q 2.4 and introduced by WINTHROPand WORTHINGTON[ 19651. By using similar calculations to those just presented, the equations expressing the in-registry conditions of Fresnel images take the form

z=zo- P , M-P

dk=d-

1

M-P

, (4.10)

where N and n are integers with no common factor related to the formula describingthe Fresnel image localization z = 2( v + N/n)d2/A.Let us choose the Fresnel images located between the grating G and its fist self-image plane. For this purpose, we may put v = 0. The properties derived for this case will be repeated in an appropriate scale for the Fresnel patterns lying between the other self-image planes. Cases corresponding to various combinations of M,N , and n were extensively discussed by PATORSKI[ 1983b], and the reader is referred to this analysis. The general conclusion is that various combinations of N, n, and p cor-

I,! 41

THEORETICAL CONSIDERATIONS: INCOHERENT ILLUMINATION

81

respond to infinite- and finite-distance localized patterns. The latter patterns can be real or virtual. All cases discussed in the literature relating to the Lau fringes localized at infinity (JAHNSand LOHMANN[1979], GORI [1979], and SUDOL and THOMPSON[ 1979, 19811) are readily described by the in-registry Fresnel pattern approach. In addition, this approach makes it possible to predict the finite-distance localized fringes that can be obtained with gratings of the same period. These fringes were not mentioned in the earlier literature. The shape of the fringes is considered next. In the preceding discussion the transmission function of the source grating GS was approximated by an array of very narrow slits. Under this approximation the finite width of the slits of grating G influences the formation of Fresnel images. The product of the multiplication or period reduction factor (WINTHROPand WORTHINGTON [ 19651) and the grating opening ratio (defined as the ratio of the slit width to the grating period) must be less than unity. The influence of the finite width of the slits of GS should be considered at this point in the discussion. It determines the contrast of mutually incoherent overlappingpatterns and can be explained as follows.The point sources in each slit of GS form their own mutually incoherent diffraction patterns of G. These images are mutually displaced depending on the point source location in the direction perpendicular to the slits. For this reason the overlapping patterns become less defined with an increase in the width of the slits. Mathematically, in the case of spatially incoherent sources the diffraction field intensity is given by convolving the diffraction intensity distribution generated by a point source with the actual function of the intensity distribution of the source. In our superposition approach each of the mutually incoherent sources is described by a rectangular pulse intensity distributionwhen a binary type of source grating is used. When grating G is of the same type, the convolution operation (for the self-image or Fresnel image planes) results in a trapezoidal fringe shape for each of the overlapping patterns. The degree of overlap of mutually incoherent patterns defines the parameters of the trapezoid. The visibility of fringes is good when the width of the trapezoid is smaller than the period of the overlapping patterns. The base width is given by z zo + z z o + 22 (4.11) = 2a-, 2a- + 2 a ZO

ZO

ZO

where a width 2a is assumed for the slits of both gratings. The first term describes the width of the image of the source grating slit in the observation plane. Its magnification is due to the pinhole (slit) camera-like performance of

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SELF-IMAGING AND ITS APPLICATIONS

[I, 8 4

the slits of G. The second term expresses the width of the slit of grating G in the observation plane due to the self-imaging phenomenon. When observing at infinity, we obtain from eq. (4.11) the value 2a(2f/zo). The preceding explanation provides an easy tool for contrast interpretation of the in-registry patterns. It agrees well with the results discussed in the literature (JAHNSand LOHMANN[ 19791, SUDOLand THOMPSON[ 1979, 198 13). It is appropriate here to explain the origin of the necessary condition of mutual parallelism of lines of GS and G. In-plane rotation of GS about its normal causes a lateral shift of the diffraction images of G. The patterns generated by different cross sections of GS parallel to the x axis, fig. 4.1, are mutually displaced in a continuous way. Therefore, the fringe patterns observed become blurred with the rotation of one of the gratings and quickly disappear. The mathematical treatment of this phenomenon was presented by OJEDACASTANEDA and SICRE[ 19861. 4.2.1.2. Gratings of unequal period For the sake of brevity we will limit discussion to self-image in-registry conditions. As before, we start with the assumption of narrow slits of the source grating GS. The discussion of the influence of the finite width of the slits, given earlier, can be applied directly to the case here. The lateral shift A x of the self-images of G for the elementary source displacement x , = d , , where dl designates the spatial period of GS, is calculated as (4.12) where d2 is the spatial period of G. Setting the condition A x = pd;, where d; denotes the spatial period of the self-image of G and p is a positive integer, we obtain the following relations pertinent to the self-image in-registry condition

z,=--2~ did2 , P A

z = z o - Pd2 , dl - Pd2

dl d2 d;=------. dl - Pd2

(4.13)

It can be seen again that for a specified grating separation distance z,, only one in-registry plane can be obtained. Again various combinations of v and p result in fringe patterns located at finite and infinite distances. In particular it is interesting to note two cases. For p = 1 the fringe pattern period d; is equal to the period of moir6 fringes formed in the case of two gratings being superimposed. This configuration corresponding to the case of moir6 fringe forma-

41

THEORETICAL CONSIDERATIONS: INCOHERENT ILLUMINATION

83

tion in space with noncontact parallel gratings was analyzed by EBBENI [ 1970al. In addition, MACGOVERN [ 19741 derived eq. (4.12) for the case p = 1, using a slightly different argument. On the other hand, assuming d2 = fd,,we have zo = vd:/A, z = zo, and d; = d , . This configuration corresponds to the case of exact imaging of the source grating by the second one, as discussed by CHANG,ALFERNES and LEITH[ 19751. PETTIGREW [ 19771 arrived at the same configuration when discussing a general theory of grating transducers. It follows from the equations describing the configurations with gratings of equal and unequal periods that the distance z of the in-registry plane with well-defined fringes is wavelength dependent. This means that the Lau effect is chromatic irrespective of the localization of the observation plane. ROBLIN [ 19711, PETTIGREW [ 19771, CHANG,ALFERNES and LEITH[ 19751, and SWANSON and LEITH [ 1982, 19851 considered the possibility of obtaining achromatic fringes by spatially selecting two beam interferences. Because we are interested in the general case of all diffraction orders of both gratings taking part in the image formation process, we will not consider this issue in detail. A heuristic explanation of the performance of such systems will be given only. Figure 4.2 shows schematically one of the achromatic grating configurations. The first diffraction orders of the incoherently illuminated source grating GS impinge on the grating G of double the frequency of GS. The zero and higher diffraction orders of GS do not take part in further interference processes. The observation plane is located at the same distance zo from G as the source grating GS but on its opposite side. Two diffraction orders of G, one from each of the illuminating wave fronts, form the fringe pattern in this plane. The analyses of Leith and his co-workers (CHANG,ALFERNES and LEITH[ 19751, GS

G

Fig. 4.2. Incoherent double-grating configuration with achromatic fringe formation properties. Identical two-beam interferencesfor every wavelength are detected at the observation plane OP. GS and G, diffraction gratings with periods dandjd, respectively.B is the opaque screen blocking the zero order of GS.

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SELF-IMAGING AND ITS APPLICATIONS

[I, B 4

CHANG[ 19741, SWANSON[ 19831, and SWANSONand LEITH[ 1982, 19851) provide an elegant description of the unusual properties of the achromatic broad-source grating interferometer originally proposed by WEINBERGand WOOD [ 19591. The interferometer configuration using cross gratings was analyzed by CHENG[ 19861. Although in the preceding discussion both gratings were assumed to be of the binary amplitude type, this requirement is only essential for the source grating. The grating G can be of the phase type as well. In this case the diffraction images with best visibility (see Q 2.4) formed by each slit source are displaced along the optical axis by a quarter of the distance 2d2/I, as compared with the case of the amplitude grating. However, the intensity distributions in those planes are highly dependent on the depth of modulation of the phase grating and can be quite complicated in form. Therefore, for image contrast interpretation, the incoherent superposition model must be accompanied by a knowledge of those intensity distributions. Computer calculations are required for this purpose. Theoretical analysis of the Lau effect with a sinusoidal phase grating was presented by JINGHONG [ 19871. Obviously, when the source grating is replaced by a phase grating, the multisource effect is lost and fringe patterns cannot be observed.

4.2.2. Injinite separation of dfraction gratings To complete the studies of double-grating systems under incoherent illumination, we will now discuss the fringe formation process when the grating separation distance is infinitely large. This case corresponds to illuminating the second grating by a multiple of mutually incoherent quasi-plane wave fronts. In other words, the first grating serves as an extended periodic light source placed at the front focal plane of the collimator objective. Similarly to the case described in Q 4.2.1, the optical field behind the second grating will be treated as the superposition of mutually incoherent Fresnel diffraction fields of this [ 1986b1). grating (PATORSKI Let us treat the first grating GS as an array of mutually incoherent line sources. In this case (see fig. 4.3) the second grating G is illuminated by many mutually incoherent plane wave front beams. The angular separation angle y between the adjacent illuminating beams satisfies the relationship tan Y = di/f,

(4.14)

where d, is the spatial period of GS and f is the focal length of L. As in previous discussions, we assume the validity of the Fresnel parabolic approximation. We assume also that the choice of the lateral extent of GS and the collimator focal

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THEORETICAL CONSIDERATIONS: INCOHERENT ILLUMINATION

85

GS

Fig. 4.3. Geometry of the system with infinitely separated diffraction gratings. GS, incoherently illuminated grating placed at the front focal plane of the collimator lens L; G, second diffraction grating. Observation plane OP is located in the Fresnel field of G. (Adapted from PATORSKI [ 1986bl.)

length assures that, even under the maximum incidence angle, the properties of the Fresnel diffraction pattern of G are the same as those of the pattern generated by an axially propagating wave front, except for the lateral shift in the direction perpendicular to the grating lines. Again, to observe well-defined fringe patterns behind G, its self- or Fresnel images due to each line source must overlap in the observation plane. The lateral shift between the overlapping patterns must be equal to the pattern period or its multiple. In the following discussion we will only look at the in-registry conditions for the self-images. The overlapping of Fresnel images can be analyzed in a similar way. The self-images overlap at distances z equal to 2vd;/A. Each slit of GS generates self-images with adjacent separation distance Ax Ax = ztany, (4.15) which amounts in the self-image planes to 2Vd; tan y AX

=

1

(4.16)

The lateral period of overlapping self-images is equal to d2 (plane wave front illumination of G), so for the in-registry condition we require Ax to equal an integer multiple of d,. The following cases can be encountered: (1) tan y z sin y = y = n(1/d2).The angular separation angle y is equal to an integer multiple of the lirst order diffraction angle of G. The special case of n = 1 corresponds to the equality of the angular separation between the adjacent slits of GS as seen from the collimator lens and the diffraction angle of G. The in-registry condition is satisfied in every selfimage plane of G.

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[I, § 4

( 2 ) tan y w sin y w y = (r/s)(A/d2).Here r and s designate integers without a common factor. Since now we have A x = 2v(r/s)d2, the in-registry condition is found in a self-image plane of number v when the value 2v(r/s)becomes an integer. It follows from the preceding cases that to obtain well-defined fringe patterns, it is necessary to match the angular separation between the mutually incoherent illuminating beams with the first-order diffraction angle of the grating being self-imaged. This can be done by adjusting the parameters d , and f.Grating GS with the required period d , can be very easily produced using the self-imaging phenomenon of G under spherical wave front illumination (SUDOL[1980], PATORSKI[198Oc], and JAHNS, LOHMANNand OJEDACASTANEDA [ 19841) and by putting the recording plate in the back focal plane of a convergent lens. These results could be expressed using the partial coherence theory approach. According to previous works (GORI[ 19791, SUDOLand THOMPSON [ 1979, 19811, SUDOL[ 1980]), fringes are obtained when the spatial period of the modulus of the mutual intensity of the field illuminating the grating G is equal to its period or its multiple. For example, in the first case each slit of G is coherent with other slits. In our configuration (see fig. 4.3) this situation is encountered when the angular separation between adjacent plane beams illuminating G is equal to the first-order diffraction angle of G. The influence of the finite width of the slits of G has to be taken into account in the experiment. It determines the contrast of overlapping fringe patterns. The approach presented in the case of a finite grating separation distance can be used for the case under consideration as well. The intensity distribution in the Fresnel field of G is obtained by convolving the diffraction pattern intensity generated by a point source with the function describing the intensity distribution of each of the sources of grating GS. Subsequently,all overlapping patterns must be added. The trapezoidal fringe profile encountered in the self-image planes of binary amplitude grating G, illuminated by another binary amplitude grating GS, depends on the opening ratios of the two gratings. As explained previously, these two parameters are related to the intensity distribution of the self-image of G and the pinhole camera-like image of GS. Both intensity distributions should be evaluated in a given observation plane. The detailed analysis of various cases of the fringe profiles is given in the paper by PATORSKI [ 1986bl. The amplitude grating G can be replaced by a phase grating. As explained earlier, the most interesting planes to discuss the overlap in-registry condition are those which are axially shifted by d2/2Afrom the self-image planes. The

s

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THEORETICAL CONSIDERATIONS: INCOHERENT ILLUMINATION

87

intensity patterns in this case have to be calculated by computer because of their dependence on the phase modulation depth. 4.3. TEMPORALLY INCOHERENT SUPERPOSITION OF SELF-IMAGING

So far in this chapter the double-diffraction systems using spatially incoherent illumination have been discussed and their performance has been explained using the concept of multiple superposition of mutually incoherent Fresnel fields. PATORSKI [ 1983f,g, 1984el showed that the same approach can be used to characterize systems composed of a progressive ultrasonic beam and an optical grating. Although they employ a spatially coherent illuminating beam, the time dependent (visually detected) intensity distribution in the observation plane can again be treated as the s u m of mutually incoherent diffraction fields. For example, when a plane wave front light beam intercepts the ultrasonic beam first, a generator of mutually incoherent beams is formed. Similarly to the discussion given in Q 4.2.2, if the angular separation between the illuminating beams is properly matched to the diffraction angle of the self-imaging structure, one obtains spatial in-registry superposition of the Fresnel fields in the observation plane. It is necessary to note that in this case temporal incoherence between the overlapping fields is encountered in contrast to spatial incoherence found in the generalized Lau effect. Therefore, as the result of using a point source illumination, high-contrast self-images can be observed at larger observation distances. Their contrast depends on the amplitudes of the diffraction orders of the ultrasonic grating. Additional characteristics following from the point source illumination relate to the fact that the lines of both diffracting structures do not have to be restricted to be mutually parallel (PATORSKI[ 1984e1, SZWAYKOWSKI and PATORSKI[ 19851). In effect, the matching of the angular separation distance of the illuminating beams to the diffraction angle of the self-imaging structure can be realized in a continuous way even for a fixed acoustic wavelength. This fact represents an advantage over the Lau double-diffraction systems using optical gratings, in which only the step-wise tuning mode is possible. 4.4. DOUBLE-DIFFRACTION SYSTEMS UNDER PARTIALLY COHERENT

ILLUMINATION

In Q 4.1 and Q 4.2 we discussed systems of two gratings in series using spatially incoherent illumination. An extended source of unlimited lateral extent

88

SELF-IMAGING AND ITS APPLICATIONS

[I, 8 4

is placed next to the first grating or imaged onto it. Behind the second grating, in its Fresnel diffraction field, fringe patterns are formed for discrete separation distances between the gratings. The fringe period also depends on the gap between the gratings. In particular, when gratings with an identical period are used, Lau fringes are observed at infinity and their period decreases when the two gratings are separated. The variation of contrast of Lau fringes when changing their frequency is troublesome when it is desired to have variable period patterns formed with incoherent illumination.Lau and generalized Lau effects cannot be used for this purpose. This problem was solved by ROBLIN[ 1972, 19731, who proposed a double-diffraction system with a spatially incoherent source of a limited extent, which was removed, however, from the grating planes. (From the historical point of view it is necessary to note the pioneering studies reported by CROVA [ 18741, GARBE[ 18901, and LENOUVEL [ 19281.) In such a case partially coherent illumination of the first grating is encountered. The observation plane is set to be conjugate to the source plane. In particular, Roblin studied the case of infinite distances of both planes from the gratings. The most important characteristics of this system will be reviewed next. Figure 4.4shows the optical configuration. The lateral extent of the source was selected to avoid an overlap between the source images in adjacent double-

Fig. 4.4. Schematic representation of the double diffraction system (gratings G1 and G2 with lines mutually parallel) under partially coherent illumination. Observation plane is conjugate to the plane of the source S. (Adapted from ROBLIN [1972].)

1,s 41

THEORETICAL CONSIDERATIONS: INCOHERENT ILLUMINATION

a9

diffraction orders. The intensity distributions in each order were studied in detail as a function of the grating separation distance z, grating periods and profiles, eventual angular misalignment between grating planes and ruling direction and so on. The source images can be considered to be composed of mutually incoherent overlapping points. Intensity values of composite points depend on the interference effects between several doubly diffracted beams originating from the plane wave front generated by a particular point of the source S. For example, in the zero double-diffraction order under quasi-monochromatic illumination, one observes a fringe pattern of spatial period equal to f(d/z),where d denotes the period of both gratings, z is the grating separation distance, and fdenotes the focal length of the second lens. The contrast of the fringes changes periodically and attains maximum values for z = vd2/A. It should be noted that the same results were obtained in the case of the Lau effect (see 3 4.2.1). In addition, an achromatic term appears in the zero doublediffraction order; it corresponds to a fringe pattern of double spatial frequency as compared with the Lau fringes just mentioned. Its contrast is independent of the grating separation gap. In higher-order double-diffraction directions the frequency of fringes is a correspondingmultiple of the frequency of fringes in the axial order. Moreover, certain interesting properties are characteristic for low-order double-diffraction patterns. For example, in the first orders in the case of quasi-monochromatic illumination, there is no contrast modulation of the frequency changing fringe patterns as a function of z. The contrast value is close to unity. This arrangement has an obvious advantage over the totally incoherent double grating arrangements (Lau and generalized Lau experiments), when a continuous change of fringe patterns is required. This is a consequence of the introduction of a source with a limited spatial extent, resulting in spatial separation of different intensity patterns in the observation plane. In particular, when squarewave binary rulings are used the fringe profile is sinusoidal. ROBLIN[ 1972, 19731 presented extensive experimental verification of the theoretical studies. Various applications of the double-diffraction system with partially coherent illumination were reviewed by the same author (ROBLIN[ 19731). They include spectrometric configurations,holography with partially coherent light, variablefrequency sinusoidal test patterns for measurements of the optical transfer function, and several techniques using a light source or, in general, intensity patterns, modulated periodically in space. At the end of this section one of them, which deals with image subtraction will be reviewed (ROBLIN[ 19741). Figure 4.5 shows the double-diffractionconfiguration used for simultaneous

90

SELF-IMAGING AND ITS APPLICATIONS

[I, § 4

\

Fig. 4.5. Recording of the interlaced sampled images of objects 01 and 0 2 to be compared by optical subtraction. Simultaneous exposure coding method is realized in a double diffraction system performing under partially coherent illumination. (Adapted from ROBLIN [ 19741.)

recording of two carrier-modulated and interlaced images of objects 01 and 0 2 to be subtracted. Incoherently illuminated objects 01 and 0 2 serve as the sources for the two grating system. Their diffraction images appear at the observation plane OP. The objects 01 and 0 2 are properly located with respect to the optical axis to obtain overlap of their images. In the configuration shown in &. 4.5 the image of 01 created by the double diffraction beam ( - 1 , O ) coincides on the optical axis with the image of 0 2 generated by the beam ( + 1,O); the numbers in parentheses denote order numbers at G1 and G2, respectively. Each of the images is spatially modulated in intensity with period (d/z)f. When the separation distance z between the gratings is an odd multiple of d2/21, the carrier fringes in the two overlapping images are mutually displaced by half a period. According to the technique proposed by PENNINGTON, WILLand SHELTON[ 19701, after developing the photographic plate with registered coded images and filtering it in a coherent optical processor, information about the difference between 0 1 and 0 2 is obtained.

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6 5. Applications of self-imaging under incoherent illumination 5.1. GRATING SHEARING INTERFEROMETRY

Grating interferometers use diffraction gratings as beam splitters and recombiners. In the case of spatially coherent illumination the most popular version is the one described by RONCHI[ 19641. It uses a single diffraction grating placed near the focus of the beam under test. The interference fringes formed by the grating diffraction orders provide information about the phase gradient. As explained in 3 3, which examined applications of the self-imaging phenomenon, an interesting modification of the Ronchi set-up is the so-called Talbot interferometer. The distortions of self-images of the beam splitter grating are detected by a second grating. The resulting moire fringes provide information about the first derivative of the tested phase distribution. Incoherent versions of both the Ronchi and the Talbot interferometer exist. For a long period of time they were presented independently in the literature. The works of BARTELT and JAHNS [1979], PATORSKI[1984a], and BOLONGINI,OJEDA-CASTANEDA and SICRE [ 19851 provided a unified description. Chronologically, to the author’s knowledge, the incoherentillumination double-grating interferometers were described by ANDERSON and PORTER [ 19291, SCHARDIN[ 19421, BURTON [ 19491, BOURGEONand FORTUNATO [ 19701, MURTY and CORNEJO [ 19731, WYANT [ 19741, CORNEJO,ALTAMIRANO and MURTY [1978], and BARTELTand JAHNS [ 19791. Figure 5.1 shows two basic configurations of incoherent double-grating shearing interferometers, which are described in the cited references. They use two gratings of equal periods with their lines mutually parallel. The two gratings are optically conjugated or slightly defocused by an imaging system. This system can be optics under test or an aberration-free optics with a phase object to be tested inserted in its pupil. In both cases the imaging optics can be of refractive or reflective type. The observation is conducted in the image plane of the tested phase distribution, which lies in the far diffraction field with respect to the second grating. Although the aforementioned configurations were interpreted differently in the original articles, a careful examination proves their equivalence. This can be done using the approaches based on the Lau effect (BARTELTand JAHNS [ 19791, BOLONGINI,OJEDA-CASTANEDA and SICRE[ 1985]), the theory of partial coherence (WYANT[ 19741, GORI[ 19791, SUDOLand THOMPSON [ 1979, 19811, CARTWRIGHT and LIGHTMAN[ 1986]), multiple incoherent

92

SELF-IMAGING AND ITS APPLICATIONS

GS

0

L1

G

L2

OP

L3

I ,

f

,

f

,

f

,

f

,

f

,

f

b).

Fig. 5.1. Two basic configurations of double-grating shearing interferometers using spatially incoherent illumination: GS and G, diffraction gratings of the same spatial period; L1 and L2, grating imaging optics; 0,phase object; L3, object imaging lens; OP, interferogram observation plane. (Adapted from PATORSKI [1984a].)

superposition of the Ronchi test (PATORSKI [ 1984a]), and quasi-ray description (BOLONGINI, OJEDA-CASTANEDA and SICRE[ 19851). For example, the interferometer shown in fig. 5. l a can be considered to be based on the Lau effect (BARTELTand JAHNS[ 19791). If there is no object present, the Lau set-up is obtained with zero distance between the gratings. A uniform field intensity distribution is encountered in the observation plane. When a phase object is present, its multiple images, displaced by a distance equal to (A/d)f, appear in the observation plane. For example, in the case of a lenslike phase object placed exactly in the middle between the imaging lenses L1 and L2 (see fig. 5.la), an axial shift of the image of GS is encountered. Therefore gratings GS and G are no longer coincident, and finite-period Lau fringes form at the observation plane. The same explanation is valid for a configuration of cross-type gratings (BARTELT and LI [ 19831). BOLONGINI, OJEDA-CASTANEDA and SICRE[ 19851 expanded this explanation, using a ray-optics approach and the Wigner distribution function. The influence of the axial position of different phase objects on the image of GS was discussed. The local deformations of Lau fringes were found to be proportional to the first derivative of the object phase distribution. Considerations based on coherence theory, physically interpreted by FRANCON [ 19661, WYANT[ 19741, and CORNEJO-RODRIGUEZ [ 19781, lead to the statement that the lateral shear value in the object plane should be equal to the distance between the peaks of the complex degree of coherence of the light field generated by the incoherently illuminated periodic structure, namely, the source grating.

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Another interpretation is based on the concept of incoherent superposition [ 1984a1). Since this test is very well known of multiple Ronchi tests (PATORSKI [ 1978]), the principles of incoherent (RONCHI[ 19641, CORNEJO-RODRIGUEZ grating shearinginterferometry can easily be grasped. For example, in analyzing fig. 5.la, we can note that the phase object is illuminated by a multiple of mutually incoherent plane waves generated by the slits of the source grating. By considering a single beam, we note that the second grating performs the Ronchi test on this beam. Similar tests are encountered for each beam generated by the slits of the source grating. The mutually incoherent interferograms are identical (to a good approximation for the low-frequency gratings and slow phase changes in the object), and they overlap. This latter condition is fulfilled by imaging the phase distribution onto the observation plane and using two diffraction gratings of the same period. The contrast of overlap fringes, in addition to its dependence on the axial distance z between the image of the source grating and the grating G, is a function of the width of the slits of GS and G. As mentioned in 4, the fringe profile is defined by the convolution operation between the Fresnel pattern intensity distribution of G generated by a point source and the source grating function. The configuration shown in fig. 5.lb can be explained using the same [ 1986a,b]). The only difapproach (PATORSKIand CORNEJO-RODRIGUEZ ference is that the Fresnel diffraction patterns resulting from each slit of the source grating are now formed by quasi-spherical aberrated wave fronts. A remarkable property of the extended periodic source Ronchi test, in a folded version of the optical configuration, is worthy of mention here. It is shown in fig. 5.2 and represents the test of a concave mirror at its center of M

Fig. 5.2. Schematic representationof an extended source Ronchi test at the center of curvature of a concave mirror: GS, source grating; G, detecting grating; M, tested mirror with radius of curvature R.

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SELF-IMAGING AND ITS APPLICATIONS

[I, § 5

curvature. Its use was first described by ANDERSON and PORTER[ 19291. The source grating GS and the test grating G are parts of the same ruling. For this system the in-registry condition of the individual mutually incoherent Fresnel diffraction patterns of G generated by each slit of GS is automatically fulfilled. Therefore the fringe contrast is solely determined by the self-imaging phenomenon of G illuminated by the single slit of GS. The intensity distribution of diffraction patterns generated by individual slits of the source grating are highly dependent on wave front aberration introduced by the object under test. This is because, in general, the observation plane does not have to coincide with one of the self-image planes of G. In other extended-source Ronchi test configurations the fringe contrast is also primarily determined by the grating self-imaging phenomenon and the multiple superposition effect is of secondary importance. Figures 5.3 and 5.4 give experimental evidence of the foregoing statement. They represent the fringes observed in lens testing configurations, shown schematically in fig. 5. lb, and the corresponding point source illumination diffraction patterns. The barrel and pin-cushion type of fringes in fig. 5.3 were obtained with a test grating G placed to the right and left of the image of GS, which is given by the lens under test, respectively. It can be seen that the Fresnel diffraction pattern corresponding to the former axial localization of G has better bright-fringe definition than the one for G located to the left of the source image. Correspondingly, when convolved with the intensity distribution of the source grating GS (referred to the observation plane), an interferogram of better contrast is obtained. It is worth noting at this moment that the diffraction patterns of binary amplitude gratings with narrow slits (MURTYand CORNEJO[1973]) or

Fig. 5.3. Extended source Ronchi test fringes obtained with gratings GS and G of spatial frequency 2 lines/mm. G is placed to the (a) right and (b) left of the image of GS, respectively. (From PATORSKIand CORNUO-RODRIGUEZ [ 1986bl.)

o

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95

Fig. 5.4. Fresnel diffraction patterns obtained with point source illumination. They correspond to the fringe patterns shown in fig. 5.3. (From PATORSKI and CORNEIO-RODRIGUEZ [1986b].)

pseudorandom sequences (KATYL [ 19721, CORNEJO,ALTAMIRANOand MURTY[ 19781) contain narrow bright fringes in their intensity distribution. Therefore, if such gratings are used for a test grating G and the appropriate source grating is selected (for simplicity, square wave binary ruling can be used), fringes with well-defined boundaries can be obtained. Additional proof of the assumption that the incoherent-illumination doublegrating configurations shown in fig. 5.1 represent multiple superposition of the Ronchi test was given by PATORSKI[1982] in a simple experiment. The configuration of fig. 5. l b was used, with grating G replaced by a single slit. Such a system corresponds to the Ronchi test with a source slit and grating mutually replaced. It is interesting to note some similarities between this experiment and the SALET[1939] and IKEDA[1951] method for recording partial slopes (contour derivatives) in flexed plates. However, the order of magnitude of deformations measured in mechanical and optical testing experiments is very different, which is why, in the latter experiments, much denser diffraction gratings are required. Another type of the extended-source lateral shear interferometer is the so-called Lau-phase interferometer, described by BARTELTand JAHNS[ 19791 and BARTELTand LI [ 19831. It is shown schematically in fig. 5.5. Since it is based on a different principle, it deserves a separate explanation. A phase object is placed between two gratings. The narrow slit placed in the optically conjugate plane to the source plane enables us to observe fringes in the output plane. The slit transmits the double diffraction orders formed by the illuminating plane beams originating from certain parts of the source S. Thus the periodic

96

SELF-IMAGING AND ITS APPLICATIONS

+.+.#.++.++ S

L1

G I 0 G2

I

,

f

,

L2

SF

OP

L3

I

,

f

.

f

!

f

,

f

,

Fig. 5.5. Lau-phase interferometer. Spatially extended source S is coupled with a narrow slit SF in the frequency plane: L1, collimator lens; L2 and L3, imaging optics; 0,phase object; G1 and G2, diffraction gratings; OP, observation plane. (Adapted from BARTELT and JAHNS [1979].)

structure of the source is effectively used. The angular separation between adjacent illuminating beams corresponds to the first-order diffraction angle of [ 1984a], such a configuration can gratings G1 and G2. As noted by PATORSKI be considered as the multiple incoherent superposition of the Talbot interferometer configuration. From this explanationit seems logical to replace an extended source coupled with a narrow filtering slit by a periodic spatially incoherent light source. In this way a multiple mutually incoherent plane wave front beam illuminating system is formed. Its interaction with the second diffraction grating was described in the previous section. When a phase object 0 is placed behind or before the second grating, grating self-images are detected by using the moirk fringe technique (see fig. 3.12). The properties of the interferometer shown in fig. 5.6 and its various modifications have been discussed in detail by PATORSKI[ 1986bl. They are limited to small shear values because of the rapid decrease in fringe contrast with an increase in the observation distance. Contrast improvement can be obtained at the cost of considerable light loss by using gratings GS and GD

Fig. 5.6. Modified Talbot or Lau-phase interferometer using a periodic spatially extended light source. GS, source grating; G1 and G2, diffraction gratings; L, collimatorlens; 0,phase object. (Adapted from PATORSKI [1986b].)

o

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APPLICATIONS UNDER INCOHERENT ILLUMINATION

GI 0 G2

L

97

OP

Z A

Fig. 5.7. Extended source grating interferometer as proposed by BOURGEONand FORTUNATO [ 19701. GI, G2, and G3, diffraction gratings; 0,phase object; L, imaging optics; OP, observation plane.

with narrow slits. However, due to the extended source illumination, the coherent noise is greatly reduced. Finally, at the end of this section we would like to mention another version of the incoherent lateral shear interferometer. It can also be treated as a multiple incoherent version of the Talbot interferometer but under spherical-like beam [ 19701in order illumination. It was proposed by BOURGEONand FORTUNATO to study phase objects and is schematically shown in fig. 5.7. Its principle can be interpreted as follows. Grating G1 plays the role of the multiple slit source. Each of the slits illuminates the second grating G2, which has the same period as G1, and creates its real and virtual self-images. The images are distorted because of the presence of phase object 0. According to eq. (4.4) for the distance z, equal to an integer multiple of 2dZ/L(where d denotes, as before, the period of G1 and G2), one of the virtual self-images is located in the middle between G1 and G2. The phase object is also located in this plane. Since the self-image period equals fd,it is monitored in the observation plane by grating G3 with double frequency, in comparison with the frequency of gratings G1 and G2. The self-imagesof G2 generated by various slits of G1 satisfy the in-registry condition. Because of using the imaging lens L, the images of 0 for each mutually incoherent interferometer system coincide in the observation plane. A related configuration described in terms of geometrical optics and the moire technique and not using the detection grating G3 was described by THEOCARIS [ 19691. Again, because of the large deformations encountered in experimental mechanics studies, the gratings used are much coarser than those employed for optical testing. 5.2.

IMAGE PROCESSING

Although it is believed that applications of the Lau effect to image processing are still to appear, it should be noted that some configurations,described before

98

SELF-IMAGING A N D ITS APPLICATIONS

[I. 8 5

the term “Lau effect” was introduced, have much in common with image processing. For example, the method of incoherent Fourier transformation described by LEIFER,ROGERSand STEPHENS[ 19691 might be thought of as being based on the optical principle of the generalized Lau effect. Figure 5.8 shows the shadow-casting process used for spatial frequency analysis. The shadow of a binary grid is incoherently illuminated by a character whose spatial frequency content must be found. At a given observation distance the grid shadow becomes sharp if the object contains a corresponding spatial frequency. Its value depends on the axial distance between the character, the binary grating and the observation plane, as well as on the period of the projected grating. The same process works when the character is placed in the focal plane of a collimator lens. In this case the distance between two interacting objects (i.e., the character and projected grid) is infinitely large. Although the analysis of the shadowing process in the referenced paper was based on a geometrical argument, only the interrelationship obtained between the spatial periods in the three planes (object, grid, and observation planes) and their axial separation distances represents one of the cases described by eqs. (4.13) and (4.16). It agrees with the discussion by MACGOVERN[ 19741 as well. The convolution operation describes the intensity pattern detected in the observation plane. The work of LEIFER,ROGERSand STEPHENS[ 19691 might be treated as one of the first applications of the generalized Lau effect in the field of optical image processing and analysis. The authors refer to other works using similar techniques for other purposes. A spatially incoherent technique for decoding theta-modulated information (LOHMANN and ARMITAGE [ 19651) and based on the Lau effect was described by OJEDA-CASTANEDA and SICRE[ 19861. It is shown schematically in fig. 5.9.

Fig. 5.8. Incoherently illuminated letter E produces a well-defined periodic intensity pattern [1969].) behind a diffraction grating. (From LEIFER,ROGERSand STEPHENS

I , § 61

99

PERIODICITIES O F OPTICAL FIELDS

GS

G

-2

I

<+ + , z

L1

GF

L2

OP

I ,

f

---

f

-

~

f - f

4

Fig. 5.9. Theta-modulation decoder based on the Lau effect. (Adapted from OJEDA-CASTANEDA and SICRE[1986].)

A grating G, with many angularly separated gratings encoding the selected parts of a picture, is placed at a distance z from the source grating GS. The component gratings of G and the grating GS are of the same spatial frequency. When the lines of one of the information-encoding gratings are parallel to the lines of the source grating, Lau fringes are formed at infinity, that is, at the back focal plane of lens L1. The periodic binary spatial filter GF with the distances between the openings tuned to the period of Lau fringes is placed in this plane. This filter enables us to select a component of grating G that is aligned with respect to GS. The technique proposed has greatly reduced noise, compared with theta-modulation decoding performed with spatially coherent illumination. A lensless version of a theta-modulation decoder, based on the concept of incoherent superposition of the self-imaging phenomenon at finite distances from the coded grating G (with a different period from the source GS, see 5 4.2), was proposed by ANDRES,OJEDA-CASTANEDA and IBARRA[ 19861.

8 6. Periodicities of optical fields This chapter is primarily concerned with the longitudinal periodicity of the optical wavefield under coherent and incoherent illumination. In the first case we speak about the self-imaging phenomenon or Talbot effect, whereas the second case relates to the Lau or generalized Lau effect. Although the case of incoherent illumination can be rigorously represented by the coherence theory approach, it can be treated satisfactorily as a multiple incoherent superposition of the self-imaging phenomenon. On the other hand, the coherence theory treatment is indispensable for describing other interesting effects that show longitudinal periodicity, which also is exhibited by the mutual intensity function. In the first work on spatial periodicities in partially coherent fields, and OJEDA-CASTANEDA [ 19831 proved that the lateral periodicity LOHMANN

100

SELF-IMAGING AND ITS APPLICATIONS

[I, 0 6

of partially coherent illumination imposes periodicity in the longitudinal direction. In other words, if the mutual intensity function r(x, - x,, z = 0) in the initial plane z = 0 is periodic in the lateral coordinate difference (xl - x,), then the mutual intensity r(xl,x,, zl, z,) at the point pairs (xl, zl) and (x,, z,) is also periodic behind z = 0. This occurs for both the lateral (x, - x2) and the longitudinal (zl - z,) coordinate differences. Mathematically, this statement can be expressed as exp{ -ik(zl

- z,)}

r=C lun12exp n

where an designates the Fourier coefficients of the moving periodic object (to obtain mutual incoherence between the various diffraction orders), l/d is the fundamental spatial frequency, and zT = 2d2/rl.The expression is valid under the usual parabolic approximation. Compared with previously derived expressions for the Talbot effect, where the periodicities were related to the x, z coordinates themselves, in the case of partially coherent fields we are faced with the coordinate differences x1 - x2, z1 - z,. [ 19671 concerning a generalized group Like the hidings of MONTGOMERY of objects that can be self-imaged, LOHMANN,OJEDA-CASTANEDA and [ 19831 proved the existence of some laterally nonperiodic partially STREIBL coherent fields repeating themselves longitudinally. It was noted that this set of fields is very similar to Montgomery’s aperiodic self-imaging objects in spatially coherent illumination when the coordinate x is replaced by the difference or sum of the coordinates. In other words, for example, Montgomery rings in the Fourier plane now describe the form of an incoherent source, giving a longitudinal periodicity of mutual intensity after collimation. Such a source can be created, for example, by an incoherently illuminated Fabry-Perot interferometer. The intensity pattern, with peaks specified by the Airy formulas (BORNand WOLF[ 1964]),is similar to the spatial distribution of Montgomery rings. Therefore, after setting the collimator lens to the left of the output plane of the Fabry-Perot interferometer, an axially periodic mutual intensity function will be obtained. Similar problems were studied theoretically and were experimentallyverified [ 1984a,b]. In addition to analyzing the fact that the lateral by INDEBETOUW periodicity of mutual intensity leads to longitudinalperiodicity, he discussed the converse statement and found that only a subclass of axially periodic fields

1,s 71

CONCLUSIONS

101

exists which exhibit lateral periodicity. For example, in some cases a Fabry-Perot etalon placed in the image plane can produce lateral periodicity.

Q 7. Conclusions The self-imagingphenomenon was observed and analyzed long before highly coherent and intense laser radiation became widely used. However, the ease and simplicity of the experiment when conducted with a low-power gas laser created great interest in the phenomenon and led to extensive theoretical and experimental studies. At the present moment it seems that the principles of optical systems using one or more periodic structures performing under various coherence conditions of illuminating radiation are fairly well’established.On the other hand, in spite of the appearance of several papers proposing and implementing various applications of self-imaging, it is believed that many significant works are still to appear. For example, the broad class of self-imaging objects is still to be investigated. Rapid progress in the technology of periodic and quasi-periodic structures must be taken into account. At the same time nature provides us with such structures. Pioneering work on the self-imaging phenomenon was performed independently in the fields of optics, acousto-optics, and electron microscopy. This work was published in various journals over a long time period; several papers went unnoticed and some “rediscoveriesy’can be observed. The purpose of this work is to give as broad as possible a review of the studies presented in the literature and hopefully to stimulate further interdisciplinary or highly specialized studies.

Acknowledgement I wish to thank Dr. D. W. Robinson of National Physical Laboratory, the United Kingdom, for his help during the preparation of the manuscript.

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PATORSKI, K., 1983g, Acoust. Lett. 7, 39. PATORSKI, K., 1984a, Opt. Acta 31, 33. PATORSKI, K., 1984b, Optik 69, 30. PATORSKI, K., 1984c, Opt. Appl. 14, 375. PATORSKI, K., 1984d, Ultrasonics 22,253. PATORSKI, K., 1984e, Acoust. Lett. 7, 119. PATORSKI, K., 1985, Appl. Opt. 24,4448. PATORSKI, K., 1986a. Appl. Opt. 25, 1111. PATORSKI, K., 1986b, Appl. Opt. 25,2396. PATORSKI, K., 1986c, Appl. Opt. 25, 3009. PATORSKI, K., 1986d, Appl. Opt. 25, 3146. PATORSKI, K., 1986e, J. Opt. SOC.Am. A 3, 667. PATORSKI, K., 1986f, J. Opt. SOC.Am. A 3, 1862. PATORSKI, K., 1987, Acustica 62, 236. PATORSKI, K., and A. CORNWO-RODRIGUEZ, 1986a, Appl. Opt. 25,2031. PATORSKI, K.,and A. CORNEJO-RODRIGUEZ, 1986b, Appl. Opt. 25,2790. PATORSKI, K., and S. KOZAK,1988, J. Opt. SOC.Am. A 5, 1322. PATORSKI, K., and G. PARFJANOWICZ, 1981, Opt. Acta 28, 357. PATORSKI, K.,and L. SALBIJT,1985, Opt. Acta 32, 1323. PATORSKI, K., and P. SZWAYKOWSKI, 1981, Opt. Appl. 11, 627. PATORSKI, K., and P. SZWAYKOWSKI, 1983, Opt. & Laser Technol. 15, 316. PATORSKI, K., and P. SZWAYKOWSKI, 1984, Opt. Acta 31, 23. PATORSKI, K., S. YOKOZEKI and T. SUZUKI,1975a. Nouv. Rev. Opt. 6, 25. K.,S. YOKOZEKIand T. SUZUKI,1975b, Opt. & Laser Technol. 7, 81. PATORSKI, PATORSKI, K., S. YOKOZEKI and T. SUZUKI,1976a. Appl. Opt. 15, 1234. PATORSKI, K., S. YOKOZEKI and T. SUZUKI,1976b, Jpn. J. Appl. Phys. 15,443. PATORSKI, K., L. WRONKOWSKI and M. DOBOSZ,1982, Opt. Acta 29, 565. PENNINGTON, 1.: S.,P. M. WILLand G. L. SHELTON,1970, Opt. Commun. 2, 113. PEITIGREW,R. M., 1977, Proc. SOC.Photo-Opt. Instrum. Eng. 136, 325. POST,D., 1967, Exp. Mech. 7, 154. RABAL,H. J., W. D. FURLAN and E. E. SICRE,1986, Opt. Commun. 57, 81. RAMAN,C. V., and N. S.N. NATH, 1935, Proc. Indian Acad. Sci. 2,406. REICHELT, A,, E. STORCKand U. WOLFF,1971, Opt. Commun. 3, 169. RILEY,W. A., 1980, J. Acoust. SOC.Am. 67, 1386. RILEY,W. A., 1981, J. Acoust. SOC.Am. 70, 1089. ROBLIN,M. L., 1971, Opt. Acta 7, 539. ROBLIN,M. L., 1972, Nouv. Rev. Opt. Appl. 3, 253. ROBLIN, M. L., 1973,Modulation Spatiale &Intensite Produite par 1'Associationde Deux Resaux Paralleles (Ph.D. Thesis, Univ. Paris VI). ROBLIN,M.L., 1974, Opt. Commun. 10,43. ROGERS,G. L., 1962, Proc. R. Soc. London Ser. B 157, 83. ROGERS,G. L., 1963, Br. J. Appl. Phys. 14, 657. ROGERS,G. L., 1964, Br. J. Appl. Phys. 15, 594. ROGERS,G. L., 1969, J. Microsc. 89, 121. ROGERS,G. L., 1970a, J. Microsc. 91, 67. ROGERS,G. L., 1970b, Handbook of Gas Laser Experiments (Ihffe, London). ROGERS,G. L., 1972, J. Opt. SOC.Am. 62, 917. RONCHI,V., 1964, Appl. Opt. 3,437. SAIGA,N., and Y. ICHIOKA,1985, Appl. Opt. 24, 1459. SALET,G., 1939, Bull. Assoc. Tech. & Marit. Aeronaut. 43, 107.

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WINTHROP, J. T., and C. R. WORTHINGTON, 1965, J. Opt. SOC.Am. 55, 373. WOLFKE,M., 1911, Ann. Phys. (Germany) 34,277. WOLFKE,M., 1912a, Ann. Phys. (Germany) 37,96. WOLFKE,M., 1912b, Ann. Phys. (Germany) 38,385. WOLFKE,M., 1912~.Ann. Phys. (Germany) 39, 569. WOLFKE,M., 1913, Ann. Phys. (Germany) 40, 194. L.,1987, J. Modern Opt. 34, 1057. WRONKOWSKI, WYANT,J. C., 1974, Appl. Opt. 13, 200. YOKOZEKI, S.,1981, Opt. & Lasers Eng. 2, 13. YOKOZEKI, S., and K. OHNISHII,1975, Appl. Opt. 14, 623. YOKOZEKI, S., and K. PATORSKI,1978, Appl. Opt. 17, 2541. YOKOZEKI,S., and T. SUZUKI,1971a, Appl. Opt. 10, 1575. YOKOZEKI, S.,and T. SUZUKI,1971b, Appl. Opt. 10, 1690. YOKOZEKI, S.,K.PATORSKI and K. OHNISHII,1975, Opt. Commun. 14,401. YOKOZEKI, S.,Y.KUSAKAand K.PATORSKI,1976, Appl. Opt. 15.2223.

E. WOLF, PROGRESS IN OPTICS XXVII

0 ELSEVIER SCIENCE PUBLISHERS B.V., 1989

I1

AXICONS AND MESO-OPTICAL IMAGING DEVICES BY

LEVM. SOROKO Joint Institute for Nuclear Research Laboratory of Nuclear Problems Head Post O f i e , P.O. Box 79, Moscow. USSR

CONTENTS PAGE

§ 1*

INTRODUCTION

. . . . . . . . . . . . . . . . . . .1 1 1

. . . . . . . . . . . . . . . . . . . . . . . 112 § 3 . THE GRAVITATIONAL LENS . . . . . . . . . . . . . 120 § 4 OPTICAL TRANSLATORS AND FOCUSATORS . . . . . 122 126 § 5. MESO-OPTICS . . . . . . . . . . . . . . . . . . . . § 2 . AXICONS

*

§6

.

THE MESO-OPTICAL FOURIER TRANSFORM MICROSCOPE . . . . . . . . . . . . . . . . . . . . . . . . 128

§ 7. A

MESO-OPTICAL MICROSCOPE FOR VERTICAL STRAIGHT LINES . . . . . . . . . . . . . . . . . . . 147

§ 8 . THE CYLINDRICAL LENS AS A MESO-OPTICAL ELE-

............... § 9. CONCLUSIONS . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . LIST OF SYMBOLS . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . MENT

. . . . .

. . . . . . . . . 152 . . . . . . . . . 155 . . . . . . . . . . 155 . . . . . . . . . 156 . . . . . . . . . 158

1 1. Introduction An axicon is an optical element which focuses a point on its symmetry axis into a straight line segment of finite length. Typical examples of axicons are a conical lens and a conical mirror. A lens-plus-axicon combination, which works as an optical doublet, produces a ring-shaped image of a pointlike object. A configuration of two mutually complemented axicons, one of them divergent and the other convergent, can be used as a beam expander. This system converts a disk-shaped beam of light into a ring-shaped beam, or vice versa. An optical triplet made of one positive lens and two mutually complemented axicons can be used to change the ring diameter continuously. Another example of an axicon is a circular diffraction grating or its kinoform equivalent. This system converts a large part of incident monochromatic light into one diffraction order, giving rise to a finite-length image of the point. New optical elements named “focusators” have been developed recently. These computer-synthesized optical elements convert incident radiation into an arbitrary line with the required intensity distribution. These elements are manufactured by the photolithographic technique in the same manner as kinoform elements. Between 1973 and 1983 many papers in the field of axicons, kinoforms, focusators, and their various developments were published. The term “axicon” seemed to become insufficient to examine the new optical elements and devices, and thus a new term, “meso-optics”, was introduced. This term covers a wide class of nontraditional optical systems, which transform a pointlike object as manifolds a zero-dimensional(OD) manifold into various one-dimensional(1D) such as straight line segments, rings, ellipses, crosses, and more complicated lines on the plane or even in 3D space. In addition, systems may be conceived that convert a OD object into some prescribed 2D manifold. Thus the term meso-optics covers axicons, mirror axicons, kinoforms, and focusators, as well as various combined devices comprising traditional optical imaging elements, Fourier optics, and holography. The purpose of this review article is to acquaint readers with axicons and meso-optical imaging systems. A description of axicons is given in 5 2, which includes various types of axicons, W-axicons, circular diffraction gratings, 111

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AXICONS AND MESO-OPTICAL IMAGING DEVICES

[II, 8 2

kinoform axicons, and some multicomponent devices. The gravitational lens and its role in our universe are discussed in $ 3. Holographic translators as one-shot systems, computer-synthesized phase reflection holographic scanners that translate a traditional spot focus over a 2D pattern, and focusators, are described in $ 4 . The term “meso-optics” is introduced and explained in $ 5 , and the classification of the meso-optical elements is outlined. A new meso-optical microscope of high resolution is described in $ 6. This device permits one to detect and measure straight line objects with small vertical inclinations. The information about these objects is derived through the stereoscopic effect without any scanning along the axis of the microscope. All these and many other useful applications have been realized by the synthesis of 2D spatial Fourier transformationwith the meso-optical principle of imaging. The meso-opticalconfocal scanningmicroscope and meso-optical microscopes that see only vertical straight lines are described in $ 7. A microscope that operates in accordance with the principle of reconstructed tomography is explained. Finally, experimentswith a cylindrical lens as a meso-optical element of the meso-optical microscope for observation of the tracks of accelerated relativistic neon nuclei in nuclear research emulsion are discussed in $ 8.

2.1. CONICAL LENS AND MIRROR

The simplest axicon (MCLEOD[ 1954, 19601) is a conical lens (fig. 1). The pointlike object 0 on the symmetry axis is imaged by the conical lens into a straight line segment along the same symmetry axis. The depth of focus of this axicon is determined by the wedge angle a, the axicon diameter D,and the distance from the pointlike object 0 to the axicon, and it does not depend upon diffraction of light. The focal line length L or the depth of focus of this axicon for a collimated beam of light with uniform intensity distributionand with sharp

Fig. 1. A conical lens as the simplest axicon:4D is the radius, u the wedge angle and L the length of the focal line.

I L § 21

AXICONS

113

edges is equal to L z D/2(n - l)a, for a < lo", where n is the refraction index of the axicon material (RIOUX,TREMBLAY and BELANGER[ 19781). The depth of focus of the conical mirror is equal to L z D/2 tg2a. The properties of the conical lens can be explained if we consider the transformation of the wavefronts and their caustic surfaces (RAYCES[ 19581). The incident spherical wavefront is transformed by an ideal lens into a convergent spherical wavefront, which collapses into a one-point image without any caustic phenomena. Two real sheets of the caustic are produced in the image space by an ordinary lens with spherical aberrations (fig. 2). The primary sheet of the caustic C, has a complex conical surface with a generatic line in the form of a semicubic parabola. The secondary sheet degenerates into a line segment of spherical aberrations of the lens on its axis of symmetry. The conical lens forms a real image in the image space and a virtual image in the object space (RAYCES[ 19581). The virtual image of the pointlike object, in the form of a bright ring, sees the observer looking backward through the conical lens. The complete separation of these two images results from the conical nature of the emerging wavefront. Since the field of view of the conical lens is very small,it can only be used with coaxial collimated beams, in a scanning system, or for alignment purposes (EDMONDS [ 1974a,b]). To gain some insight into the optical properties of the conical lens and conical mirror, LIT and BRANNEN [1970] and LIT [1970] calculated the irradiance distributionin the transversal plane of a conical mirror for a pointlike source on the axis of symmetry. The wavefront reflected by the conical mirror goes from a ring-shaped virtual image; its radius and distance from the cone vertex are determined by the wedge angle of the cone cr, the distance from the pointlike sourceto the conical mirror I , , and the distance 1, from the observation point P(p, 0,12)to the cone vertex of the mirror. The radial irradiance distribution may be approximated by the expression

Fig. 2. The caustic sheets of the bispherical lens: W, divergent wavefront; W', convergent wavefront; C,, primary sheet of the caustic arc; C,, secondary sheet in the form of a straight line segment.

114

AXICONS AND MESO-OFTICAL IMAGING DEVICES

where PCM

1 +I, 2ka11

=1 ,

2 K

k=-

1'

I is the wavelength of the light, C, is a constant, and Jo is the zero-order Bessel function. For I, 9 1, we have

Here the factor 1/1: is due to the natural divergence of light. In the limit a -,0, we have p:" + CQ,and the conical surface becomes a plane mirror.

2.2. CIRCULAR DIFFRACTION GRATING

A circular diffraction grating consists of a number of circularly concentric grooves of constant radial spacing on a plane surface (DYSON[ 19581). A point S on the axis of symmetry is transformed by the circular diffraction grating into a system of secondary circular virtual sources, shown in fig. 3 in the meridional section as S,, S,, . ..,and S ; , S ; , .. . on both sides of the axis. The first-order diffracted wavefronts W,and W; which go toward the axis, intersect to give rise to a finite-length image on the axis of symmetry. The depth of focus L is given by the expression -=---

1

21

1

L

Da

It'

I...! I

---+---

-

Fig. 3. Diffracted wavefronts W, and W; and secondary virtual sources S, and S; of the point S produced by the circular diffraction grating CDG.

1 1 9 8 21

AXICONS

115

where I , is the distance from the point S to the circular diffraction grating with external diameter D. The pair of diffracted wavefronts W, and W; in the meridional section has both spherical and conical components. If the dimension of the pointlike sourced is such that dD < I:, then all wavelets are coherent and a sharp diffraction pattern is observed near the axis of symmetry over the depth of focus L. The intensity of this diffraction pattern is (DYSON [ 19581)

where I, is the distance from the circular diffraction grating to the observation point P(p, l,, (p) and

For 1, % I, we have

From eq. (2.7) we see that the intensity of the diffraction pattern grows proportionally to I, the scale of the radial diffraction pattern does not change with distance, and what is more important, the diffractionpattern, given by eqs. (2.5) and (2.7), is achromatic. By comparing eq. (2.1) with eq. (2.5), we may conclude that the Ji-dependence of the radial diffraction pattern is inherent in many optical systems: a conical lens or mirror, a circular diffraction grating, and a lens with an annular aperture of small width (STEWARD[ 19281). The radial envelope of the diffraction pattern fot all these systems is falling as l/p instead of l/p2, the dependence for an ordinary lens with an open aperture. A brief comment follows concerning gratings in the form of an Archimedean [ 19581). The normal to the groove of this grating in spiral of pitch a (DYSON its plane does not cross the optical axis, and the locus of the maximum intensity has the form of a ring-shaped region with a dark spot on the axis of symmetry. The radial distribution of the intensity in the diffraction pattern is described by

where J , is the fist-order Bessel function and ptRCH= pZDG.

116

AXICONS AND MESO-OPTICAL IMAGING DEVICES

2.3. KINOFORM AXICON

A kinoform (LESEM,HIRSCHand JORDAN [ 19691) is a diffraction element, the optical thickness of which is varied stepwise over its surface, in such a way that a wavefront of the light at its output is of a certain, desired form, but the maximum depth variations do not exceed the wavelength of the light. In contrast to holograms, a kinoform gives only one image and its diffraction efficiency is close to 100%. A kinoform axicon as an equivalent of the conical lens was made by MIKHALTSOVA, NALIVAIKO and SOLDATENKOV [ 19841. The intensity distribution over the diffraction pattern for the kinoform axicon is determined by

where a is the wedge angle of each zone of the kinoform axicon, and (2.10)

Equation (2.9) results from eq. (2.5) if we replace A/a by a(n - 1) or from eq. (2.1) if 201 is replaced by a(n - 1). The desired profile of the kinoform axicon was approximated (MIKHALTSOVA, NALIVAIKO and SOLDATENKOV [ 19841) by a six-step geometricalrelief of each zone of the kinoform with a linear law of phase delay. A computer-controlled laser photoplotter was used for rapid production of the optical relief with axial symmetry on glass or fused quartz (VEDERNIKOV, VIJUKHIN,KIRIJANOV,KORONKEVITCH, KOKOULIN,LOKHMATOV, NALIVAIKO, POLESHCHUK, TARASOV, KHANOV, SHCHERBACHENKO and YURLOV[ 198 11, KORONKEVITCH, KIRIJANOV, KOKOULIN,PALTCHIKOVA, POLESHCHUK,SEDUKHIN,CHURIN,SHCHERBACHENKO and YURLOV [ 19841). The problem of the light intensity distribution along the axis of symmetry was studied by PALTCHIKOVA [ 19861. The performances of the kinoform axicon were: (1) I=0,63pm, D = lOmm, L = 2m, p,KFA%20pm, atI, = 1.6m; (2) rZ = 0,63 pm, D = 100 mm, L = 50 m, hKFA % 35 pm, at I, = 19 m.

K 8 21

AXICONS

117

2.4. LENS PLUS AXICON

A positive lens combined with one axicon in the form of a conical lens, convergent or divergent, forms a ring-shaped image of a point (fig. 4) ( F b o u x , TREMBLAY and BELANGER [1978]); the radius of the ring is equal to Ro x (n - l)aF, a < lo", where F is the focal length of the positive lens. The diffraction pattern in the vicinity of this ring is determined by the amplitude of light (BELANGERand RIOUX[ 19761)

2 UL+A(P) =,Jl(:Pl), ZP1

(2.11)

where

(2.12) and D is the diameter of this optical doublet. By comparing eq. (2.11) with the amplitude of light in the A q disk of the positive lens alone,

P

(2.13)

where p = nDr/#IF,we see that the width of the diffraction pattern in the ring is twice the width of the Airy disk. If this doublet is illuminated by a Gaussian beam of light, the diffraction pattern on the focal ring is also a Gaussian (BELANGERand RIOUX [ 19781).

Fig. 4. Lens-plus-axicon system producing nng-shaped images: (a) with convergent axicon; (b) with divergent axicon.

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AXICONS AND MESO-OPTICAL IMAGING DEVICES

[II, 0 2

The effective width of this distribution is 1.65 times the effective width of the Gaussian crossover produced by an ordinary positive lens.

2.5. SYSTEMS WITH TWO AXICONS

Two mutually complemented axicons, one divergent and the other convergent, form a kind of Galilean telescope (BRYNGDAHL [ 19711) (fig. 5), which can be used for adjustable radial displacements in a wavefront and for converting a disk-shaped collimated beam into a ring-shaped one with variable magand BELANGER[ 19781): nification (RIOUX,TREMBLAY M % l + % (Dn - 11) ,- ~ r

0!<10",

(2.14)

where 5 is the distance between two mutually complemented axicons. This system serves as a beam expander without energy losses. The radial reversal of the wavefront is performed by two conical lenses [ 19701). placed vis-a-vis or by a conical lens and a plane mirror (BRYNGDAHL Two conical surfaces are present in a solid rod prism and in a rod prism with concave entrance and exit surfaces (LAVOIE[ 19751) (fig. 6). These optical elements, as well as folded and inverted axicons (PETERSand LEDGER[ 1970]), are used in laser machining for compression or expansion of the light beam and

$ 1 \\

Fig. 5. System of two mutually complemented axicons for two different positions of axicons: a, radius ofthe light beam; A,, convex conical lens; A,, concave conical lens; a, wedge angle;!Mu, radius of the ringform beam.

119

AXICONS

L

I

Fig. 6. Conical axicons producing:(a) polar reversion with 180" rotation of the field (rod prism); (b) polar reversion without field rotation (rod prism with axial hole); (c) cup prism with concave entrance and exit surfaces. D is the diameter of the prism rod, 2t the diameter of the axial hole.

for interfacing the laser output in the TEM,, mode with a centrally obscured aperture. We can also mention a radial collimator with a 45" conical surface and an axial hole (BICKEL[ 19671, STONER[ 1970]), which transforms a collimated laser beam into radially convergent rays collapsing into a bright line segment on the axis of symmetry, or conversely, a radially divergent beam of rays into a collimated one. The same performances are provided by reflaxicons with two conical surfaces (EDMONDS [ 19731, ROSIN [ 19741, EDMONDS [ 1974b1) and W-axicons (CHODZKO,MASON, TURNERand PLUMMER[ 19801, ZAVGORODNEVA, KUPRENYUK, SERGEEVand SHERSTOBITOV [ 19801,GAVRIKOV, PISKUNOV and SHCHEGLOV [ 19861).W-axicons proved to be the most popular. Chodzko and co-workers used a W-axicon as the end mirror in a laser with an annular gain medium. However, W-axicons have two principal disadvantages: They provide the output light beam with nonuniform radial distribution, and they do not preserve polarization. To obtain a uniform intensity distribution of the light at the output of the W-axicon, OGLAND[ 19781 proposed some new types of the W-axicon with a

120

AXICONS AND MESO-OPTICAL IMAGING

DEVICES

[II, § 3

curved generating line (slanting side) similar to the system of LIT and [ 19731 and a nonlinear W-axicon mirror system. TREMBLAY A uniformly linearly polarized beam is converted into a beam in which the direction of polarization depends on the azimuthal angle 8as 26(FINK [ 19791). This polarization scrambling effect can be seen in the interferogram as a reduction of the fringe contrast along the lines at 45 with the direction of initial polarization (FERGUSON [ 19821). The theory of polarization effects in W-axicons was developed by FINK[ 19791 and DENTE[ 19791. AZZAMand KHAN [ 19821 proposed a solution for overcoming the polarization scrambling effect, which consists of producing single-layer-coated polarization-preserving axicons. Dielectric layers of a specific thickness on metallic substrate induce equalization of the net complexp and s reflection coefficients, and thus uniform polarization reflectancethat does not depend on the input polarization.Another solution to the polarization problem in W-axicons without any coating layers was proposed by FINK [1979]. The polarization can be preserved if one reflection takes place at an angle less than Brewster’s angle and the other at a greater angle. Finally, it should be noted that a reflaxicon with conical mirrors does preserve polarization of the light beam. O

4 3. The gravitational lens A famous example of the axicon in nature is the gravitational lens (KLIMOV [ 19631,LIEBES[ 1964]),which bends the light rays by means of the gravitational field of the massive astronomical object of our universe. A light ray incident at the impact parameter p in the vicinity of a star with a mass Mg is deffected at an angle a(p) = k/p, where k = 4GMg/c2,with G the gravitational constant (LANDAU and LIFSHITZ[ 19731) and c the velocity of light. Let the gravitational radius of the nearest star rg = 2GMg/c2 be small relative to its geometrical and NOVIKOV[1967]). Unlike the traditional ideal radius R (ZELDOVITZ positive lens with the deflected a, as a growing function of p, namely, ~ ( p =) C tg p with C a constant, the gravitational lens bends the light ray inversely proportional to p. The depth of focus of the gravitational lens L = 00, and the smallest focal length is equal to

If the observer is on the straight line which connects the centers of the source star and the nearest star and r > fmin, eq. (3.1), then he or she sees the source star as a ring (LIEBES[19641).

11-8 31

THE GRAVITATIONAL LENS

121

The formation of the meso-optical images of the pointlike source star by the gravitational mass can be explained by the scheme shown in fig. 7. Since all the angles involved are very small, the following geometrical equations are true [ 19811): (MUCHANOV

The unknown angle 8 is determined by the quadratic equation

which has two solutions. In the general case the observer will see two virtual meso-optical images of the pointlike source star. Only in the case where the observer is on the straight line connecting the centers of the source star and the nearest star the gravitational lens produces a ring-shaped meso-optical such image ofthe pointlike source star. For a distributed gravitational mass Mg, as a galaxy, the number of solutions of a more complex equation than eq. (3.2), and the number of virtual meso-optical images, increases. In most cases the number of virtual meso-optical images must be odd, for example, 1, 3, etc. A gravitationallens induced by a distant galaxy, which is on the line of observation in front of a double quasar, has now been discovered (CHAFEE[ 19801). Experiments have been carried out with the laboratory equivalent of the gravitational lens, made of plastic (HIGBIE[ 19811, EGOROVand STEPANOV [ 19821). The profile of the plastic gravitational lens is pseudospherical with negative Gaussian curvature. For a small incident angle and a refracted angle a, the following differential equation couples the slope of the generating line

Fig. 7. Two meso-opticalimages of a star produced by a gravitational lens; SS, source star; NS, nearest star; I; and I;, two virtual meso-optical images of the source star; p , impact parameter of the ray; P, observer.

122

AXICONS AND MESO-OPTICAL IMAGING DEVICES

(profile) dp/dz with the refracted angle a: dz a(p) = (n - 1)(3.4) dP By substituting eq. (3.1) in eq. (3.4), we get the exponential profile of the plastic gravitational lens

In a real experiment some simplifications can be introduced. The profile p(z) can be made like an arc of the circle, such as in the pseudospherical surface of the stem of a wine glass. The real meso-optical image produced by a plastic gravitational lens is demonstrated with the optical system shown in fig. 8 (EGOROVand STEPANOV [ 19821).

6 4. Optical translators and focusators From a ray-optics viewpoint a traditional lens transforms a pointlike object into a pointlike image. To obtain the image in the form of some complex curve in a one-shot operation, we must construct an analogous curve in the object space. This technique is used in showing slides. We can achieve the same goal with a pointlike object by moving the @aging lens in a prescribed manner. This scanning procedure has an advantage: The energy of the source is concentrated at every instant in a small area, giving rise to a very high energy density. This technique is used by children who burn out a picture on a wooden plate by means of a sun image produced by a lens. The one-shot procedure can be used only for those experiments in which the energy problem is not of significance.

PG L

L,

L2

I

Fig. 8. Optical system used for observation ofthe real meso-optical image produced by the plastic equivalent of the gravitational lens; S, pointlike source; PGL, plastic gravitational lens; LI and L2, two positive lenses; Di, display.

I I , 41 ~

OPTICAL TRANSLATORS AND FOCUSATORS

123

However, there are a few simple problems that can be resolved in a one-shot procedure. As an example, let us consider an image in the form of a straight line segment. It may be produced by an ordinary cylindricallens and a pointlike source of light (fig. 9). The generatingLine of the cylindricallens must be parallel to the prescribed straight line segment. The problem becomes more complex if the desired image curve has a complex form or is off the plane. An extreme solution may be to use a very intricate cylindrical lens with prisms that have various focal lengths and inclination angles. Despite its seeming impossibility, this can be realized by means of the 2D holographic translator (LOHMANN, PARISand WERLICH [ 19671) as a computer synthesized hologram. The same problem can be solved by scanning, using a computer-generated phase reflection holographic scanner (CAMPBELL and SWEENEY [ 19781). This scanner is designed for generating a 2D pattern by 1D moving the corresponding hologram, which focuses a laser beam onto a single spot with high energy density. The successive scanningof the spot image over the required 2D pattern is controlled by the phase function $(x, y) recorded on the computer-generated hologram. At each instant only a small part of this hologram is illuminated by the laser beam. However, even with this simplified restriction, the problem of and SWEENEY finding the function $ ( x , y ) is not a simple one (CAMPBELL [ 19781). A new class of optical elements named focusators has been developed to generate a focal line with an arbitrary pattern in a one-shot fashion (GOLUB, KARNEEV,PROKHOROV, SISAKYAN and SOIFER[ 19811, DANILOV, POPOV,

Fig. 9. Straight line segment as an image of a point, produced by an ordinary cylindrical lens; CL, cylindrical lens; , straight line segment; S, pointlike light source.

124

AXICONS AND MESO-OPTICAL IMAGING DEVICES

[II, 4 4

Prokhorov, Sagateljan, Sisakyan and SOIFER[1982]). The first stage of its manufacture is to solve the following inverse problem: how to find a scalar wave function

U(u, 4 = A@, 4 exp [ik,cp(u,

41

in the plane of the focusator that goes into the arbitrary line in the plane z = f with prescribed intensity distribution. If amplitude A(u, u) and phase (eikonal) q(u, u) are slowly varying functions over the wavelength 1, then this inverse problem can be formulated in the frame of the geometricaloptics approximation (GONCHARSKII, DANILOV, POPOV,PROKHOROV, SISAKYAN, SOIFERand STEPANOV [ 19831). In general, the eikonal equation

(Vd2= n2(x, y, 4

(4.1)

9

where n(x, y , z), is the refraction index of the nonuniform medium, defines the phase function cp(x, y, z ) (eikonal) exactly. The behavior of the amplitude A(x, y, z ) is specified by the transfer equation AAcp + 2Vcp.VA = 0 .

(4.2)

In the early stage of the focusator development, amplitude-phase optical elements were searched for (SISAKYAN and SOIFER[ 1984]), but later only phase relief was studied (DANILOV, POPOV,PROKHOROV, SAGATELYAN, SISAKYAN and SOIFER[ 19821). The unknown function cp(u, u) in the plane z = 0 is determined by the following nonlinear operator equation (GONCHARSKII, DANILOV, POPOV, SISAKYAN and STEPANOV[ 1986a1):

; 1 J s, Jm

E(x, Y ) =

exp ((2 xi/a [ d u , 0) + R(% v; x , J41) du

N u , 0;x , v)

12,

(4.3) where E(x, y) is the intensity of the radiation in the plane z = f,G is the closed domain in the plane (u, u) which is mapped into a prescribed line, Z(u, u) is the intensity of radiation falling on the focusator, and R(u,u;x,y)=JfZ+(x-u)2+(y-

uy.

Effective algorithmsfor solvingthe inverse problems that arise in the producDANILOV, tion of the focusators have been developed by GONCHARSKII, POPOV, PROKHOROV, SISAKYAN,SOIFER and STEPANOV[ 19831, GONCHARSKII, SISAKYANand STEPANOV[ 1984a1, and GONCHARSKII,

11,s41

OPTICAL TRANSLATORS AND FOCUSATORS

125

DANILOV,POPOV,PROKHOROV, SISAKYAN,SOIFER and STEPANOV [ 1984bl. Some features of the methods described in the cited papers should be noted. Given the function cp(u, u), the family of rays reflected from the mirror focusator can be found (GONCHARSKII, DANILOV,POPOV,SISAKYAN, [ 1986b1). The ray that emerges from the point (u, u) SOIFER and STEPANOV at z = 0 plane intersects the plane z = f at point P(x, y ) with coordinates

where (p: and (p: are the spatial derivatives of cp along the u and u coordinates, respectively. Equations (4.4) define the mapping of the domain G into a line segment with small dimensions relative to f.Only smooth unambiguous mappings with nonzero Jacobian have been studied by GONCHARSKII, DANILOV, POPOV,SISAKYAN, SOIFER and STEPANOV[ 1986bl. Each point M(I) on the line segment is generated by light rays going from the smooth curve r(l)inside the domain G. When the curves r(I)for different indexes I do not cover the domain G completely or compactly, or when these curves intersect each other, piecewise smooth functions cp,(u, u) rather than the unique function cp(u, u) must be studied. As a result, the focusator is made of several pieces. In rare cases the phase function q(u, u) can be represented analytically. For example, the axicon focusator for an input spherical wave and the straight line segment on its axis with constant linear density energy has a phase relief 1

q(u, u) = -ln[2bJr2

b

d m ,

+ (f- br2), + 2b2r2 + 1 - 2fb],

(4.5)

b = 4L/D2,D is the diameter of the focusator, and L is where r = the length of the straight line segment (GOLUB,KARNEEV,PROKHOROV, SISAKYAN and SOIFER[ 19811). The focusators were manufactured both for CO, laser radiation and for visible light with I = 0.6 pm (GONCHARSKII, DANILOV, POPOV, PROKHOROV, SOIFERand STEPANOV [ 19831, GONCHARSKII, DANILOV, POPOV, SISAKYAN, SISAKYAN, SOIFERand STEPANOV[ 1984b], GONCHARSKII, PROKHOROV, POPOV,SISAKYAN, SOIFER and STEPANOV [ 1986b1). Plane focusaDANILOV, tors were used in ophthalmological operations involving laser treatment of myopia (laser keratotomy) (AKOPYAN, DANILENKO, DANILOV, NAUMIDI, [ 19851). POPOVand SISAKYAN

126

AXICONS AND MESO-OPTICAL IMAGING DEVICES

Q 5. Meso-optics The growing number of nontraditional optical imaging devices such as axicons, conical kinoforms, and systems with prescribed aberrations, which transform a pointlike object into a complex image, necessitates a classification scheme for these optical devices. One possible approach has been suggested by SOROKO [ 1982,1983a, 19841 and BENCZE and SOROKO [ 19841. To explain this scheme, let us consider two possible transformations of spherical wavefronts. A divergent spherical wavefront issued from a pointlike source is reversed by a perfect lens into a convergent spherical wavefront, which gives rise to a pointlike image (fig. 10). In a meso-optical system, such as an axicon, the divergent spherical wavefront from the pointlike source is transformed into a conical wavefront, which forms a meso-optical image of the point as a straight line segment (fig. 11). An optical imaging device is referred to as meso-optical if it produces desired conical wavefronts. The unwanted conical wavefronts in traditional optical imaging devices induce the aberrations. The term “meso-optics” covers a wide class of non-traditional optical imaging devices that transform a pointlike object as a zero-dimensional (OD) manifold into various one-dimensional (1D) manifolds, such as straight line segments, circles, ellipses, crosses, and more complex lines on the plane or even in 3D space. The geometrical transformations performed by meso-optical imaging devices

Fig. 10. A perfect lens reverses a divergent spherical wavefront into a convergent spherical wavefront.

MESO-OPTICS

,

/

I

127

I

\

\

\

I I

\

\

'.

Fig. 11. An axicon, which transforms a divergent spherical wavefront into a conical convergent wavefront.

can be represented by the following scheme: (OD) + (1D) ,

(1D) + (2D),

(2D) + (3D) .

(5.1)

A point goes into a line segment, a line segment goes into a restricted 2D

surface, and a surface is transformed into a 3D manifold. In general, an n-dimensional object goes into an (n + 1)- or (n + 2)-dimensional images. On the contrary, in a classical optical imaging system a unique transformation (OD) @ (OD)

(5.2)

takes place in the frame of the geometrical optics approximation. There are two main types of meso-optical imaging systems: with longitudinal or transversal properties of the meso-optical images (BENCZEand SOROKO [ 19841). An example of the longitudinal meso-optical devices is an axicon with one conical surface. The kinoform with ring response (KORONKEVITCH, LEUKOVA,MIKHALTSOVA,PALTCHIKOVA, POLESHCHUK,SEDUKHIN, CHURIN and YURLOV [ 1985a1, KORONKEVITCH,PALTCHIKOVA, POLESHCHUK and YURLOV[ 1985b1) is a transversal meso-optical device. A positive lens combined with a convergent conical lens produces two mesooptical images of a point: one longitudinal, on the symmetry axis, and one transversal,in the plane perpendicular to the symmetry axis. A positive lens and a divergent conical lens give only one, transversal, meso-optical image of a point. A system may also be built that converts a OD object into a 2D manifold

128

AXICONS AND MESO-OPTICAL IMAGING DEVICES

lK86

in the form of a truncated conical surface (SOROKO[ 1987e1).This device can be described as double-meso-optical. Meso-optics offers many advantages over the classical imaging optics, some of which will be explained in the following section.

8 6. The Meso-optical Fourier Transform Microscope 6.1. GENERAL DESCRIPTION

The new Meso-optical Fourier Transform Microscope permits the detection and measurement of straight-line objects with small vertical inclinations. The geometrical information about these objects is extracted through the stereoscopic effect without any scanning operations along the optical axis of the microscope or otherwise. These and many other applications were realized by the synthesis of 2D spatial Fourier transformations (FT) with the meso-optical principle of imaging. This new optical apparatus is called the Meso-optical Fourier Transform Microscope (MFTM) (SOROKO[ 1981a1, ASTAKHOV, KOMOV,SIDOROVA,SCRYLand SOROKO[1983], BENCZEand SOROKO [ 1985a,b,c; 1987a,b,c], ASTAKHOV,BENCZE, YANVEG, KISHVARADI, NITRAIand SOROKO[ 19871).

Fig. 12. Optical part of the Meso-optical Fourier-Transform Microscope (MFTM), involving a convergent beam of light, meso-optical mirror with ring response, nuclear emulsion layer and system of observation.

I I , § 61

THE MFTM

129

The optical part of the MFTM is shown principally in fig. 12. Straight-line objects with small vertical inclinations randomly distributed over a restricted volume of depth h are illuminated by a convergent beam of light, the crossover of which is near the surface of the meso-optical mirror. Because of the properties of a convergent beam of light, the rays diffracted on a straight-line object produce the 2D spatial FT of the object. Its FT is a long bright strip crossing the optical axis of the MFTM. The width of the straight-line objects is of the order of 0.5-1 pm, the numerical aperture of the meso-optical mirror is of the order of 1, and the width of the bright strip amounts to 30 pm. The meso-optical mirror of an MFTM with ring response produces two meso-optical images of a straight-line object: one on the left (L)and one on the right (R). Thus a straight-line object is mapped by the MFTM into itself and is twice multiplexed. The meso-optical images of the straight-line object lie on opposite sides of the focal ring on the line going through the optical axis of the MFTM and remain tangential to the corresponding focal circle (fig. 13).

Fig. 13. Two meso-optical images (the left, L, and right, R) of the straight line object A: D, diameter of the field of view; OAL, the apparent angle left; OAR,the apparent angle right; OA, the true orientation angle of the object A.

130

AXICONS AND MESO-OPTICAL IMAGING DEVICES

PI, 8 6

When the original straight-line object changes its orientation in the field of view of the MFTM, the two meso-optical images rotate in the same direction as the original object, and all three remain parallel. A cross-section of the restricted volume, taken perpendicular to the straightline object, is shown in fig. 14. The location of each straight-line object in 3D space is specified exactly by its angle of orientation in the plane perpendicular to the optical axis of the MFTM, its distance po from the optical axis of the MFTM and its z-coordinate, with the z = 0 plane chosen as the reference plane. The apparent distances from the point 0 of the left (pl) and right (p2) meso-optical images are related to the true coordinates po and zo by zo =

P I + P2

2 sina,/,

,

Po =

P I - P2 2 cos a1/2’

where a1,2is the angle between the optical axis of the MFTM and the central ray of the diffracted light spanned by one half of the meso-optical mirror. The

Z,=pl(sind)-.1

z,=p,(sinocT’,

z,=L(zp z,)= (p,+p2)(2si nd1-1 2

-1

p,=$pl1 92)( 2 cos o( 1 * Fig. 14. Cross section of the volume with straight-line particle tracks by the plane which is perpendicular to a given straight-line particle track: p1 and p2 are the apparent distances of the particle track relative to the point 0, t l and z2 are the apparent z-coordinates of the particle track, po is the true distance from the optical axis, and z, is the true zcoordinate of the particle track.

11, i3 61

THE MFTM

131

observation and registration of two meso-optical images of straight-line objects of given orientations $ f A$ with A$ PZ 2" have been performed with a 2D matrix of charge-coupled devices (CCD) equipped with an anamorphic optics (BENCZE,KISHVARADI, NITRAI and SOROKO[ 1986b1).

6.2. MERIDIONAL SECTION OF THE MFTM

The properties of the meso-optical mirror of the MFTM in the meridional section of the MFTM were tested by means of the experimental setup shown in fig. 15 (BENCZE, KISHVARADI, NITRAIand SOROKO[ 1986b1). A laser beam with wavelength I = 0.63 pm traverses the microscopic objective (90 x with a numerical aperture 1.2), forms a pointlike source with diameter 0.48 pm on the axis of the MFTM, and exposes the meso-optical mirror, the surface of which is a figure of rotation with an arc of an ellipse as the generating line. When the pointlike source coincides with the first focus of this ellipse on the optical axis of the MFTM, the observer sees through the microscope a sharp focal circle as a locus of the second focus of the ellipse. The total width of the profile at half-maximum is equal to AxexpPZ 1.5 pm. The diffraction-limitedresolution n

Fig. 15. Optical system used for observation of a straight-line object in the meridional section of the MFTM: (1) collimated beam of light; (2) objective of the microscope; (3) meso-optical mirror; (4) objective of the microscope; (5) photographic film.

132

AXICONS AND MESO-OPTICAL IMAGING DEVICES

[II, § 6

of one half of the meso-optical mirror amounts to Ax,, = 1.4 pm (BENCZE, KISHVARADI,NITRAI and SOROKO [ 1986b1). These authors believe that they are the 6rst to succeed in obtaining such a high spatial resolution in a mesooptical system with a large numerical aperture. The purpose of the MFTM is to observe and measure the straight-line particle tracks in nuclear research emulsion. Observation of the tracks of relativistic protons with minimal ionization in the nuclear research emulsion has been accomplished by means of an MFTM (BENCZE, KISHVARADI, NITRAI and SOROKO [ 1986bl). The meso-optical image of the track produced by protons with energy 250GeV in nuclear research emulsion is shown in fig. 16. The total width of the profile at half-maximum is equal to Ax x 1.8 pm.

Fig. 16. The meso-opticalimage of a track produced by a proton with energy 250 GeV in nuclear research emulsion.

11,s 61

THE MFTM

133

6.3. SAGITTAL SECTION OF THE MFTM

The MFTM in the sagittal section can be considered as a one-dimensional “pin-hole” chamber with a transmitted slit oriented perpendicularly to the given particle track (SOROKO[ 1987f1). The rays of the diffracted light in the sagittal section of the MFTM are shown in fig. 17 for a straight-line particle track of length I uniformly illuminated over its length. The diameter of the crossover of the convergent beam of light can be estimated as A w = AH/l, where H is the distance from the nuclear emulsion layer to the meso-optical element with ring response. The side lobes of the diffracted pattern can be neglected, since we use a Gaussian laser beam of light. The light intensity distribution I ( x ’ ) in the sagittal section of the meso-optical image can be approximated for 1 > A w by a trapezium with small sloping regions on both sides. When the length of the straight-line particle track 1 decreases, the sloping region increases until it takes the value of lo, which is determined by the relation 1: = 21H. For 1 < lo the function I ( x ’ ) spreads out over the length 2 A 0 , and the intensity of light drops to the value 1/2 A w (fig. 18). Experiments which support the validity of this scheme have been done. The scheme of the first experiment,

Fig. 17. The meso-optical image of thc straight-lime object in the sadttal section of the MFTM for 1 > I,; A m is the diameter of the crossover of the convergent beam of light.

134

AXICONS AND MESO-OPTICAL IMAGING DEVICES

X'

Fig. 18. Same as in fig. 17, but for 1 < I,.

Fig. 19. Optical system used for observation of the meso-optical image of a straight-line object in the sagittal section ofthe MFTM for 1 > I , and the results of the experiment:(1,2) lenses which form the convergent beam of light; (3) object; (4) kinofom with ring response; (5) objective of the camera, (6) photographic film.

11, 8 61

THE MFTM

135

Fig. 20. Same as in fig. 19, but for 1 < 1,: (1,2) lenses which form the collimated (Fresnel) beam of light; (3) object; (4) kinoform with ring response; (5) objective of the camera;(6) photographic film.

with 1 > I,, and its results are shown in fig. 19. In the second experiment, with 1 < lo (fig. 20), the meso-optical image of the straight-line object ceases to be straight and has the form of an arc of the focal circle, with an angle AO= 6/1, where 6 is the width of the straight line object. These experiments support the suggestion that the MFTM in the sagittal section can be considered as a one-dimensional pinhole chamber with a virtual transmitted slit which is always directed perpendicularly to the observed particle track and has a width equal to the diameter of the crossover of the convergent beam of light. The angular resolution of the MFTM for A = 0.63 pm,D = 3 mm, H = 150 mm, and R = 80 mm is equal to

6.4. VERTICAL INCLINATION

If the angle of the vertical inclination of the particle track is to be estimated, the sets of z-coordinates of this particle track in several neighboring fields of view of the MFTM are measured and then processed properly (fig. 21).

136

AXICONS AND MESO-OPTICAL IMAGING DEVICES

1

E!

PI, 0 6

H

Fig. 21. A particle track with vertical inclination and its meso-optical images observed in neighboring fields of view; z , , z2 and z 3 are the distances between the two meso-optical images of the particle track in three different fields of view. 6.5. MESO-OPTICAL IMAGES OF CIRCLES AND RECTANGLES

The imaging properties of the MFTM for open circles, rectangles, and semiplane shutters were investigated by BENCZE, PALTCHIKOVA, POLESHCHUK and SOROKO[ 1986al. The meso-optical element of the MFTM was a kinoform with ring response (KORONKEVITCH, KIRIJANOV, KOKOULIN, PALTCHIKOVA, POLESHCHUK, SEDUKHIN,CHURIN,SHCHERBACHENKO and YURLOV[ 19841,KORONKEVITCH, LEUKOVA, MIKHALTSOVA, PALTCHIKOVA, POLESHCHUK, SEDUKHIN,CHURINand YURLOV[ 1985a1, KORONKEVITCH, PALTCHIKOVA, POLESHCHUK and YURLOV[ 1985b1). The diameter of this kinoform was 40.8 mm, the focal length was f = 202 mm, and the radial resolution was of the order of 10 pm. The experimental set-up is shown in fig. 22. The phenomena observed in this experiment can be described by the Hilbert transform and by the Dirac delta-plus function (SOROKO[ 1987~1).The convergent beam of light goes through the quasi-one-dimensional object in the plane x1 with amplitude transparency f ( x , ) . In the plane of the spatial frequencies %, the Fourier transform F(w,) of the function f(xl) is displayed. The function F(w,) is subjected in the plane 0, to two operations that are typical for meso-optical systems: (1) to the multiplication by the unit step function Y ( y ) (Heaviside function); and (2) to the multiplication by the function i 0,.

I t § 61

THE MFTM

137

Fig.22. Optical system used for observation of the meso-optical images of a circle and a rectangle: (1) source of the collimated beam of light; (2) Fourier Transform lens; (3) object; (4) kinoform with ring response; (5) objective of the camera; (6) photographic film.

Each meso-optical image (left or right) of the straight-line object contains only positive (left) or negative (right) spatial frequencies. In contrast to onedimensional Hilbert optics (SOROKO[ 1981b]),each straight-line particle track has its own virtual knife-edge filter. The second operation is the result of the phenomenon that is observed in the W-axicon as a nonuniform radial intensity distribution of the light at the output [ 19781). Because of this phenomenon, the light of the W-axicon (OGLAND producing each meso-optical image of the straight-line segment is gathered over the part of the meso-optical element with ring response in the form of a narrow sector with its center on the optical axis of the MFTM. The original quasi-one-dimensional optical field fo(x) is subjected in the MFTM to the following chain of transformations:

where ~ ( x is) the Hilbert transform of the function fo(x):

138

AXICONS AND MESO-OPTICAL IMAGING DEVICES

[II, § 6

with the integral taken in the sense of the Cauchy principal value. From eq. (6.3) we may conclude that, in general, the convolution kernel of the MFTM is described by the first spatial derivative of the Dirac delta-plus function, d/dx (6, (x)) (SOROKO [ 1987~1). POLESHCHUK and The experimental results (BENCZE,PALTCHIKOVA, SOROKO [ 1986a1) are in close agreement with these phenomenological predictions. A fragment of the meso-optical image of a spectral slit of width 6 = 40 pm is shown in fig. 23. The effect of spatial differentiation of the object is seen clearly. The general view of the meso-optical image of a rectangle of dimensions 0.283 x 0.378 mm2 is shown in fig. 24. Two independent spatial differentiations along two mutually orthogonal axes (x and y ) are clearly demonstrated.

Fig. 23. Fragment of the meso-optical image of a spectral slit of width 40 pm.

I I , § 61

THE MFTM

139

Fig. 24. Meso-optical image of a rectangle: only one quadrant is shown.

An object in the form of an open circle is transformed by the MFTM into two concentric circles. These properties of the MFTM can be explained as a natural apodization in meso-optics (BENCZEand SOROKO[ 1987dl).

6.6. METRIC PERFORMANCE OF THE MFTM

6.6.1. Longitudinal resolution

When the length of the particle track is smaller than the diameter of the field of view, its meso-optical images change their initial form (fig. 25). If the observer sees only one of two meso-optical images, he or she may interpret the reduction in length as a change of the orientation angle 8. The apparent orientation angles 8, and OR are coupled with the true orientation angle 0, of the particle track by the equation 8, + 8 R = 28,. Thus for an estimation of 8,, the two apparent angles 8, and 8R should be measured. Their difference OL = (D - L)/R, where D is the diameter of the field of view and L is the length of the particle track. Since all the angles can assume both positive and negative values, the length of the particle track L can be positive or negative. Thus we may estimate the coordinate of the end of the particle track (SOROKO [ 1987a1).

140

AXICONS AND MESO-OPTICAL IMAGING DEVICES

Fig. 25. Comparison of the meso-optical images of particle track AB of full length D and particle track OB of length 4D.

When the field of view is moving along a long particle track with p1 - pz = 0, the apparent orientation angles eLand change in the following way (fig. 26). The intensity of the meso-optical images approaches zero value for apparent orientation angles 0, or OR close to the values 0, f Aewith A0 x: D/R.The total length of the particle track is equal to L , = X + D, where Xis the length of the translation segment over which the apparent orientation angles OL and 8, are constant. As the intensity of the meso-optical images is proportional to (L/D)2, we cannot measure the length of a particle track with L ? L o , where Lo = T H = 0.3 mm. The error of length measurements by means of this algorithm is of the order of 30 pm. 6.6.2. Meso-optical analogue of the moire efect When two regular gratings are superimposed, the moire effect is observed. It comprises the difference and sum fringes, which are oriented perpendicular and correspondingly parallel to the bisector of the angle a between the two

141

THE M F I U

X

Fig. 26. Behaviour of the apparent orientation angles 4 and 4 along a particle track with length greater than the field-of-view diameter.

gratings. If this angle is very small, the period of the s u m fringes is equal to the period of the grating and the s u m moird effect cannot be seen on the background of the difference moire effect (DURELLI and PARKS[ 19701). When one of the gratings moves relative to the other in a direction perpendicular to the grooves of the grating, the fringes of the differencemoird move in a direction perpendicular to the relative velocity vector. The fringe period of the difference moird is l/a times the period of the grating, and the velocity of the linear movement of these fringes is l/ct times the relative velocity movement of the gratings. A meso-optical analogue of the classical moird effect is observed in the MFTM (SOROKO[ 1987b1). When the external tip of the right half of the particle track is rotated around the center of the field of view in a direction perpendicular to the orientation of the particle track, the meso-optical images of this half-track are moved as a whole along the meso-optical images that are perpendicular to the velocity vector with which the external tip of the right half of the particle track is moving. The linear velocity of this movement is RID times the true velocity of the external tip of the right half-track. The same relation holds for absolute displacement values. Let us consider the nuclear interaction of a charged particle, the “event”, in

142

AXICONS AND MESO-OPTICAL IMAGING DEVICES

[II, § 6

the form of the “kink” shown in the center of fig. 27. The left and right meso-optical images of this event are different. The left observer sees the event ABC as a convex obtuse angle, whereas the right observer sees the same event ABC as a concave obtuse angle. This meso-optical effect can be used for an estimation of a very small scattering angle. The left (L) and right (R) mesooptical images of the “kink” event with A1 = R AOare shown in fig. 28. The error of this measurement technique is given by the diameter of the crossover of the convergent beam of light and amounts to 1’ for D = 3 mm, H = 150 mm and R =80m.

6.6.3. Curved particle tracks

A new technique of the particle track curvature measurement by means of the MFTM was developed by SOROKO[ 1987dl. To evaluate the curvature radius of the particle track, it is sufficient to measure two linear quantities: the lengths of the left and right meso-optical images of the same particle track. The

Fig. 27. The meso-optical images of the kink-event;two particle tracks AB and EC with relative angle 8.The left meso-optical image is shown at the top, the right one is at the bottom.

THE MFTM

143

R

Fig. 28. The structure of the meso-optical image of a kink-event with very small scattering angle.

principle of this technique is explained in fig. 29. The lengths of the two meso-optical images, I, and I, are coupled by the equation

and the radius of curvature is equal to

with = I, - I,. The ratio of the difference length curvature radius p is equal to

4

-=-

S

16R L

< to the sagittal S = L2/8p for the same (6.7)

144

AXICONS AND MESO-OWICAL IMAGING DEVICES

Fig. 29. Measurement of particle track curvature.

For R = 80 mm and L = 1 mm we have r / S = 1.28 x lo’: the difference length 5 is about lo’ times the sagittal S (!). For example, if S = 0.1 pm, the difference length 5 amounts to 128 pm or about 0.13 ZAv with lAv = i(lL+ ZR). Experiments that support the validity of this approach have been performed. Two particle tracks have been measured, with small and large curvature radii. The meso-optical images of these particle tracks are shown in fig. 30. The distinguishingfeature of this technique of curvature radius estimation is that the ratio 5/S has a very large value. Meanwhile, in the simulated experiments mentioned earlier, the diameter of the field of view is 4 times and the radius of the focal circle is (4.37)- times the corresponding values in the MFTM. Therefore we have 51s = 73.

6.7. EVENT-SEARCHING ALGORITHMS

The purpose of the MFTM is to search for nuclear interactions (“events”), consisting of several particle tracks in the nuclear research emulsion with a common vertex. The coordinates of this vertex can be found directly from the meso-optical images of the particle tracks without first performing the reconstruction of all particle tracks in 3D space. It was proved (BENCZEand SOROKO [ 1985133)that the meso-optical images of the particle tracks which form an event are lying on the sinogram both for the left and the right observers. This conclusion is valid even when the vertex

145

THE MFTM

Fig. 30. Meso-optical images of two particle tracks with different curvature radii.

of the event is outside the field of view or is beyond the nuclear emulsion layer. For small scattering angles the meso-optical images lie on a straight line. As an illustration of these properties of the MFTM, a pattern of seven straight-line particle tracks is presented in fig. 3 1. Only six of them come from a common vertex; the other particle track does not belong to this event. The tangent-event-searching algorithm was developed as well (BENCZE and SOROKO[1985d]). Only those particle tracks are observed and registered which cross the center of the field of view in a given instant. The event and the coordinate system used in the tangent-event-searching algorithm are shown in fig. 32. A particle track crosses the center of the field of view if the conditions y, = (x, - xo) tan + + y o ,

x, = (yo -yc) tan ++ x o ,

e= + -

:.,

(6.8)

146

AXICONS AND MESO-ORICAL IMAGING DEVICES

7 P.3

-100

-3

Fig. 31. Pattern of seven straight-tineparticle tracks (left) and meso-optical images shown right in the format: radial and angular coordinates. Only one of the meso-optical images is shown.

are fulfilled. If the particle tracks have a common vertex with coordinates ( x o , yo), the meso-optical images lie in the coordinate system ( x , tan 0) on a straight line with the angle equals f l = tan - ( y o - yJ. A computer simulation

of the tangent-event-searching algorithm was accomplished by BENCZE and SOROKO [ 1985dl). It should be apparent that if the vertex of the event is on the optical axis of the MFTM,the meso-optical images of the corresponding particle tracks will lie on a focal circle with its center on the optical axis of the MFTM.The radius of this focal circle is specified by the z-coordinate of the event vertex. The number of independent events that can be recognized by this technique is

h h Sin&,,, N,=-=

Az

Ap

Fig. 32. The tangent-event-searchingalgorithm: xo and yo are the coordinates of the vertex; x, and y , are the coordinates of the centre of the field of View: 9,.rpz, Q,, rp, are the orientation angles of four particle tracks which form a vertex.

11, B 71

MESO-OPTICAL MICROSCOPE FOR VERTICAL STRAIGHT LINES

147

For A p = 5 pm, a,,2 = 30" and h = 200 pm, we have N, = 20. This technique of direct event searching was tested in simulated experiments by BENCZE, PALTCHIKOVA, POLESHCHUK and SOROKO[ 19871. A kinoform with ring response and a plate with concentric transmitting rings were used.

8 7.

A meso-optical microscope for vertical straight lines

7.1. MESO-OPTICAL SCANNING MICROSCOPE

In physical experiments with neutrinos from an accelerator, the nuclear emulsion layer is sometimes aligned perpendicularly to the trajectory of the charged particles. The particle tracks in that layer are directed along the optical axis of the microscope and are identified as vertical. At high linear magnification with the microscope these vertical particle tracks can only partially be seen. Therefore the observer continuously varies the position of the sharp focus along the z-axis. The productivity of this operation is very low. Meanwhile the scanning along the z-axis can be withdrawn completely if we use an optical axicon rather than the traditional objective in the microscope. The required minimal value of the meso-optical depth of focus is Lmin= h, where h is the depth of the nuclear emulsion layer. A pointlike photoelectric pick-up element is on the axis of this meso-optical scanning microscope. Moving the object along the x and y axes is accomplished sequentially in the usual way. Scanning along the z-axis is completely absent. To decrease the intensity of the side lobes in the spread point function of a single axicon, a new meso-optical element in the form of a double axicon or a double circular diffraction grating should be inserted into the scanning mesooptical microscope. The radial light amplitude distribution on the axis of this device is

where po is the same as shown in eq. (2.2) and y < 1 . However, adeep z-modulation of the light intensity arises on the axis of the device with period

where F is the focal length of the double axicon and R is its external radius. For 1 = 0.63 pm, F = 150 mm, R = 60 rnm and y = 0.7, we have A z 5.5 pm.

148

AXICONS AND MESO-OPTICAL IMAGING DEVICES

W,8 1

This z-modulation is no barrier to the application of the double axicon for observation of the vertical particle tracks in the nuclear emulsion of depth 0.2 mm. This z-modulation can be suppressed meanwhile with the aid of a phase plate (SOROKO[ 1987g]), which covers one of the axicons and involves N sectors with N 2. The phase shift of the light over each sector is equal to 36O0/N(k- l), where k is the index of the sector, k = 1, 2, ..., N - 1, N. Because of this structure, each z-modulation pattern assumes a lateral shift A/N(k - l), k = 1,2, . ..,N - 1,N,and the resulting patterns have no variations along the z-axis. The multichannel meso-optical scanning microscope (SOROKO[ 1986a1) involves a 2D array of prisms, a 2D array of photoelectric pick-up elements, [ 19821) (fig. 33). The vertical and a spherical meso-optical objective (SOROKO particle tracks are seen in the parallel light rays, whereas the observation is accomplished by light rays that are going through the center of the spherical meso-optical objective. The conversion from one system of rays to the other is made by a (2D) array of N 2 prisms, where N is an odd number. The 2D arrayed holes in the plate in front of the photoelectric pick-up system are on

n

Fig. 33. Multichannel meso-optical scanning microscope: [1) light source; (2) spherical mesooptical objective; (3) prisms; (4) shutter with 2D array of pin-holes; (5) photoelectrical pick-up system; (6) display.

II,§ 71

MESO-OPTICAL MICROSCOPE

FOR VERTICAL STRAIGHT LINES

149

the straight lines connecting the center of the spherical meso-optical objective with the center of each prism. The scanning, which is accomplished within one prism over an area a’, covers an area N’a’. The coordinates of the observed vertical particle track are specified by the coordinates of the fork with the nuclear emulsion layer and by the indexes m, and myof the hole through which the rays of the diffracted light passed. The discrete positions of the nuclear emulsion layer are a distance Nu apart along the x and y axes.

7.2. MESO-OPTICAL CONFOCAL MICROSCOPE

A traditional optical confocal microscope has both an illuminating and an imaging objective and a photoelectric pick-up element with a pinhole (SHEPPARDand CHOUDHURY[ 19771, WILSON [ 19811, SHEPPARD, HAMILTONand Cox [ 19831,WILSONand SHEPPARD [ 19841) (fig. 34). The axial scanning of a 3D object is carried out at audio frequencies, and the lateral scannings along the x and y axes are performed with an ordinary mechanical system. The image of an object, such as the curved wing of an insect, is stored on the screen of the display. The coherent transfer function of a confocal

Fig. 34. Traditional optical confocal microscope:( 1 ) pointlike light source;(2) illuminating objective; (3) object; (4) imaging objective; (5) photoelectric pinhole pick-up element; (6) axial scanning system; (7) x-y scanning system.

150

AXICONS AND MESO-OPTICAL IMAGING DEVICES

PI, 8 7

microscope (SHEPPARD,HAMILTONand Cox [ 19831) is the product of the corresponding transfer functions of two objectives. Therefore the side lobes of the point spread function are so low that a superposition of many partial images may be done without any deteriorationof the contrast and of the signal-to-noise ratio. In the meso-optical confocal microscope (SOROKO[ 1985, 198783) the axial scanning along the z-axis is completely removed (fig. 35). Each objective involves the traditional lens-collimator and a multiaxicon. To hold the confocal topology of the rays in the meso-optical confocal microscope, we must meet the following condition: the light rays in the space between two multiaxicons should be parallel in the meridional section of this device. To cancel the longitudinalmodulation of light induced by the multidirectional propagation of light, the geometrical dimensions of the meso-optical elements must be large and the width of each ring axicon must be made small. The fringe period of the longitudinal interference pattern in the region of the focus is equal to

El

Fig. 35. Meso-optical confocal microscope: (1) pointlike light source; (2) illuminating multiaxicon objective;(3) nuclear research emulsion;(4) imaging multiaxicon objective;(5) photoelectric pick-up element; (6) x-y scanning system.

11, o 71

MESO-OPTICAL MICROSCOPE FOR VERTICAL STRAIGHT LINES

151

For h = 0.2 mm, sin f a x 0.5 and L = 100 mm, we have A x 0.63 mm. Therefore the efficiency variation induced by the longitudinal interference is very small over the depth h. In general, the suflicient condition for this process is LA > h2 sin

..4

7.3. MESO-OPTICAL CONDENSER

To increase the productivity of the observation of vertical particle tracks further, the movement of the nuclear emulsion layer along the y axis must be withdrawn completely. This was achieved by introducing a meso-optical condenser into the traditional optical microscope (fig. 36) (SOROKO[ 1987g1). This element comprises a cylindrical lens with a generating line in the form of a polygon (SOROKO [ 1986b1). It can be interpreted as a one-dimensional analogue of the multiaxicon with many conical surfaces. In the region of superposition of two plane wavelets that issue from two elementary prisms at the angles f 46 on both sides of the symmetry plane, an interference pattern with fringe period d = A16 is produced. The patterns with different 6 are added

L

Fig. 36. Microscope equipped with a meso-optical condenser: (1) collimated beam of light; (2) stop; (3) cylindricallens with generating line in the form of a polygon; (4) cylindrical immersion bath; (5) objective of the microscope; (6) ocular of the microscope.

152

AXICONS AND MESO-OPTICAL IMAGING DEVICES

PI, 5 8

constructively only on the central fringe. The width A of the elementary prism must be chosen proportional to the distance x from this p r i m to the symmetry plane: A = xh/L, where L is the distance from the nuclear emulsion layer to the meso-optical condenser. The width of the central fringe is equal to A/a,,, where clef is the effective aperture of the meso-optical condenser. For L = 100 mm and a lateral dimension of the meso-optical condenser B = 60 111111, the total aperture is a, = 0.58 radian, and a,, = 0.33 radian. The difference between a, and a,, is explained by the fact that light which comes from the external prisms is less efficiently focused than that from the central prisms. The spatial resolution of the meso-optical condenser is equal to Ape, = 2 pm. When a vertical particle track is in the illuminated region in the form of a long “fence” of width 2 pm, the diffracted light is gathered by the objective of the = 0.08 radian from the total microscope with numerical aperture a1 = length h of the vertical particle track. Despite this fact, the spatial resolution of the system as a whole amounts to 2 pm. The estimation of the x and y coordinates of the vertical particle track can be accomplished by algorithms taken from reconstructed tomography (SOROKO[ 1983b1). For this purpose the nuclear emulsion layer with vertical particle tracks is scanned twice by means of the meso-optical condenser, both times along the x-axis but with different orientations of the “fence” relative to the x-axis. For angles of orientation equal to + 45 O and - 45 O, the coordinate x and y coordinates of the vertical track can be found from the equations

a

2x=x, +x2,

2y=x,

-x2,

(7.4)

where x , and x2 are the x-coordinates of the nuclear emulsion layer at the instant when the vertical particle track is in the illuminated region. If the density of vertical particle tracks is high, the number of different orientations of the fence must be increased to 2N, where N is the number of vertical tracks inside the field of view. The productivity of this meso-optical device can be equal to 10-30 mm2/s.

8 8. The cylindrical lens as a meso-optical element In accordance with the definition of the term meso-optics, the cylindrical lens can be considered as a meso-optical element, since it transforms a pointlike source into a straight-line segment with a length that does not depend on diffraction of the light. In addition to this minor application of the cylindrical lens, it can be used in the Meso-optical Fourier Transform Microscope

11, I 81

CYLINDRICAL LENS AS MESO-OPTICAL ELEMENT

153

(MFTM) as a true meso-optical element. The purpose of this MFTM is to find the end of the particle track and to estimate its coordinates. In the experiments by BENCZEand SOROKO[1984] and ASTAKHOV, BENCZE,KISHVARADI, NIEDERMAYER, NITRAI and SOROKO[ 19851 on semi-automatic registration of relativistic accelerated neon nuclei in the nuclear emulsion, it was proved that these particle tracks can be seen through the MFTM with a radial resolution of 60 pm, that is, with a numerical aperture of the order of radian. The main feature of these particle tracks is that they are almost parallel within an angle spread of the order of 0.5-1 ", and in addition, the average multiple scattering angle over the length of the nuclear interaction is small. As a result, the meso-optical images of the parallel particle tracks are formed by a very small part of the meso-optical element. The small arc of the focal circle can be approximated by the tangent line, and the meso-optical element with a complex toroidal surface can be replaced by an ordinary cylindrical lens (SOROKO[ 1987hl). To retain the stereoscopic properties of the MFTM, the central rays of light diffracted by the particle track, with I x 30" relative to the a width of the order of 1.5 pm, must be at an angle I optical axis of the MFTM. The experimental set-up is shown in fig. 37. The meso-optical images of the straight particle tracks are focused in the sagittal section by the Fourier transform lens, and in the meridional section with a cylindrical lens. The radial resolution Ap = A/Afi, where AB is the aperture spanned by the cylindrical lens. The minimal angular resolution is equal to A P i n z 2A/D, where D is the diameter of the field of view. The radial resolution Ap = 5 pm was sufficient to obtain the required depth of focus of the order of h = 0.2 mm. A value of Afi < 0.12 radian was chosen. The monocular stereo-effect has been observed by varying the angle fl in the range 13" < fi < 43". The binocular stereo-effect can be observed as well by means of two cylindrical lenses on both sides of the optical axis. When the particle track orientation angle + 8 increases, the left meso-optical image goes up and the right one goes down. Both horizontal and vertical parallaxes are observed. The number of information degrees of freedom of the MFTM with a cylindrical lens along the radial coordinate is N, =

&-

2h sin p , nA cos fi

where n is the index of refraction of the nuclear emulsion. The corresponding

154

AXICONS AND MESO-OPTICAL IMAGING DEVICES

Fig. 37. Meso-optical system with a cylindrical lens used for the observation of meso-optical images of quasi-parallel straight-particle tracks, and the results of the experiment.

number along the angular coordinate is D N, = - h e , 23,

(8.2)

where 68 is the tolerant spread of the particle tracks orientation. For /3 = 50", n = 1.51, h = 0.2 mm, 3, = 0.6 pm, 68 = 0.02 radian, and D = 2 mm, we have N t = N , * N , = 6 , 6 x lo4.

(8.3)

The total number of information degrees of freedom of the MFTM is 160 times the corresponding factor of the traditional optical microscope, N;TRAD= D/Ap = 4 x lo2.This difference can be explained by the fact that the central angle of observation has a very large value: /3 z 50". As a result, the particle tracks are displayed along both radial and angular coordinates. Therefore the MFTM allows a higher density of particle tracks than the traditional optical microscope with the same resolution along the radial coordinate. The MFTM with two cylindrical lenses has additional information degrees

11, 91

ACKNOWLEDGEMENTS

155

of freedom along the z-axis. The resolution along the z-coordinate is of the order of

and N , x 18 (SOROKO[ 1987hl).

0 9. Conclusions The term “meso-optics’’ is very new. For researchers in this field the term has become familiar in their day-to-day practice. However, the words “mesooptics” and “meso-optical” are not used widely in the optical literature, either in English or in Russian. Naturally, any new scientific term has its starting point and its history. We hope that this review article will favor the dissemination of this very useful term, and that the popularity of meso-optics as a new branch of imaging optics will increase significantly. Acknowledgements I wish to express many thanks to V. P. Dzhelepov, corresponding member of the Academy of Sciences of the USSR, and Director of the Laboratory of Nuclear Problems, Joint Institute for Nuclear Research (Dubna), for support of the investigations in meso-optics for many years. Gratitude is also extended to Professor S. A. Bunyatov, Chief of the project Neutrino Detector, for leadership in the work on the Meso-optical Fourier Transform Microscope for nuclear research emulsion and to Professor Yu. A. Batusov for many discussions and his deep understanding of the scientific and technological problems of meso-optics. The development of the MFTM has been accomplished as a part of the USSR-Hungary scientific collaboration, and I am grateful to Professor D. Kiss (KFKI Budapest), chief of the Hungarian scientific group. Warm appreciation is expressed to Gy. L. Bencze, A. Kishvaradi, L. Molner and G. Nitrai, as well to A. Ya. Astakhov from the Laboratory of Computing Techniques and Automation of JINR. For their fruitful collaboration I wish to acknowledge V. P. Koronkevich and I. G. Paltchikova from the Institute of Automation and Electrometry (Novosibirsk).

156

AXICONS A N D MESO-OPTICAL IMAGING DEVICES

List of symbols 1. GREEK

wedge angle of conical axicon effective aperture total aperture angle between the optical axis and the central ray of the diffracted light angle between the optical axis and the ray of the diffracted light going to the cylindrical lens ratio of the effective radii of the two axicons forming a double axicon width of a straight-line object delta-plus Dirac function tolerant spread of particle track orientations Laplacian operator spatial resolutions; the total width of the profile at half-maximum aperture spanned by the cylindrical lens distance between two mutually complemented axicons, the difference length t = I , - 1, wavelength of the light the period of z-modulation radial coordinate, curvature radius of particle track apparent distances between a particle track and the point 0 for the left and right observers true distance between a particle track and the optical axis eikonal, the phase function of the light field apparent orientation angles of a particle track for the left and right observers true orientation angle of a particle track Hilbert transform of the input function f ( x ) spatial frequency coordinate

111

LIST OF SYMBOLS

157

11. LATIN

radial spacing of the circular diffraction grating; radius of an axicon; the radius of a light beam amplitude of the light field diameter (width) of a meso-optical element dimension of the source; the fringe period diameter of the field of view; diameter of a meso-optical element intensity of radiation in the image plane Fourier transform of the input function f ( x ) gravitational constant depth of a nuclear emulsion layer point spread functions along x- or pcoordinates distance from nuclear emulsion to the meso-optical element of the MFTM intensity of radiation in the plane of the focusator wave number of the light, k = 2 a / l length of a particle track characteristic length: 1; = 21H distance between object and imaging element distance between imaging element and image lengths of the left and right meso-optical images focal line length total length of a particle track linear magnification of an optical system gravitational mass of a star refraction index of an optical medium information degrees of freedom: along z-axis, pcoordinate, &angular coordinate, and total point of observation radius of the focal circle pointlike object or source coordinates of a point in the plane of the focusator spatial coordinates coordinates of an event vertex coordinates of the center of the field of view relative to the nuclear emulsion layer unit step function (Heaviside function) true z-coordinate of the particle track

158

AXICONS A N D MESO-OPTICAL IMAGING DEVICES

111. SCRIPT

4 29-

'

operator of the direct Fourier transformation operator of the inverse Fourier transformation

IV. SPECIAL

v

8

nabla; gradient operator convolution operation

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DANILOV, V. A,, V. V. POPOV,A.M. PROKHOROV, D. M. SAGATELYAN, I. N. SISAKYAN and V. A. SOIFER,1982, Sov. Phys. Tech. Lett. 8, 810. DANILOV, V. A., V. V. POPOV,A. M. PROKHOROV, D. M. SAGATELYAN, E. V. SISAKYAN, I. N. SISAKYAN and V. A. SOIFER,1983, Preprint Phys. Inst. Acad. Sci. USSR, No. 69 (in Russian). DENTE,G. C., 1979, Appl. Opt. 18, 291 1. DURELLI, A. J., and V. J. PARKS,1970, Moire Analysis of Strain (Prentice-Hall, Englewood Cliffs, NJ). DYSON,J., 1958, Proc. R. SOC.London Ser. A 248, 93. EDMONDS, W. P.,1973, Appl. Opt. 12, 1940. EDMONDS, W. P., 1974a, Appl. Opt. 13, 1762. EDMONDS, W. P., 1974b, Appl. Opt. 13, 1764. EGOROV,G. S.,and N. S. STEPANOV, 1982, Usp. Fiz. Nauk 138, 147 (in Russian). FERGUSON, J. R., 1982, Appl. Opt. 21, 514. FINK,D., 1979, Appl. Opt. 18, 581. GAVRIKOV, V. F., A. K. PISKUNOV and V. A. SHCHEGLOV, 1986, Sov.J. Quantum Electron. 13, 2135 (in Russian). A.M. PROKHOROV, I. N. SISAKYAN and V. A. SOIFER,1981, GOLUB,M. A,, S.V. KARNEEV, Sov. Tech. Phys. Lett. 7, 618 (in Russian). GOLUB,M. A., V. P. DEGTJARJOVA, A. N. KLIMOV,V. V. POPOV,A.M. PROKHOROV, E. V. SISAKYAN, I. N. SISAKYAN and V. A. SOIFER,1982a, Sov. Tech. Phys. Lett. 8,449 (in Russian). GOLUB,M.A., A. M. PROKHOROV, I. N. SISAKYAN and V. A. SOIFER,1982b, Sov. J. Quantum Electron. 9, 1866 (in Russian). V. V. POPOV,A. M.PROKHOROV, I. N. SISAKYAN, V. A. GONCHARSKII, A. V., V. A. DANILOV, SOIFERand V. V. STEPANOV, 1983, Dokl. Akad. Nauk USSR 273,605 (in Russian). GONCHARSKII, A. V., I. N. SISAKYAN and V. V. STEPANOV, 1984a, Dokl. Akad. Nauk USSR 279, 68 (in Russian). GONCHARSKII, A. V., V. A. DANILOV, V. V. POPOV,A.M. PROKHOROV, I. N. SISAKYAN, V. A. SOIFERand V. V. STEPANOV, 1984b, Sov. J. Quantum Electron. 11, 166 (in Russian). GONCHARSKII, A. V., V. A. DANILOV, V. V. POPOV,I. N. SISAKYAN and V. V. STEPANOV, 1986a, Dokl. Akad. Nauk USSR 291, 591 (in Russian). V. A. SOIFERand V. V. GONCHARSKII, A. V., V. A. DANILOV,V. V. POPOV,I. N. SISAKYAN, STEPANOV, 1986b, Sov. J. Quantum Electron. 13, 660 (in Russian). GONCHARSKII, A, V., V. A. DANILOV, V. V. POPOV,A. M. PROKHOROV, I. N. SISAKYAN, V. A. SOIFERand V. V. STEPANOV, 1986c, Sov. Tech. Phys. Lett. 11, 1428. HIGBIE,J., 1981, Am. J. Phys. 49, 652. KLIMOV, Yu. G., 1963, Dokl. Akad. Nauk USSR 148, 789 (in Russian). KORONKEVITCH, V. P., V. P. KIRIJANOV,F. I. KOKOULIN,I. G. PALTCHIKOVA, A. G. POLESHCHUK, A. G. SEDUKHIN,E. G. CHURIN,A. M. SHCHERBACHENKO and Yu. I. YURLOV,1984, Optik 67, 257. KORONKEVITCH, V.P., G.A. LEUKOVA,LA. MIKHALTSOVA, I.G. PALTCHIKOVA. A.G. POLESHCHUK, A. G. SEDUKHIN, E. G. CHURINand Yu. I. YURLOV,1985a, Automatriya 1 , 4 (in Russian). A. G. POLESHCHUK and Yu. I. YURLOV,1985b, KORONKEVITCH, V. P., I. G. PALTCHIKOVA, Preprint Inst. Autom. & Electrom., No. 265 (in Russian). LANDAU, L. D., and E. M. LIFSHITZ,1973, Theory of Field (Nauka, Moscow) p. 19. LAVOIE,L., 1975, Appl. Opt. 14, 1482. LESEM,L. B., P. M. HIRSCHand J. A. JORDAN,1969, IBM J. Res. & Dev. 13, 150. LIEBES,S., 1964, Phys. Rev. 133, B835. LIT, J. W. Y., 1970, J. Opt. SOC.Am. 60, 1001. LIT, J. W. Y., and E. BRANNEN, 1970, J. Opt. SOC.Am. 60,370. LIT, J. W. Y.,and R. TREMBLAY, 1973, J. Opt. SOC.Am. 63, 445.

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LQHMANN, A. W., D. P. PARISand H. W. WERLICH,1967, Appl. Opt. 6, 1139. MCLEOD,J. H., 1954, J. Opt. SOC.Am. 44, 592. MCLEOD,J. H., 1960, J. Opt. SOC.Am. 50, 166. I. A., V. I. NALIVAIKO and I. S. SOLDATENKOV, 1984, Optik 67,267. MIKHALTSOVA, MUCHANOV, B. F., 1981, Usp. Fiz. Nauk 133, 729 (in Russian). MUMOLA, P. B., H. J. ROBERTSON, G. N. STEINBERG, J. H. KREUZER and A. W. MCCULLOUGH, 1978, Appl. Opt. 17, 936. OGLAND, J. W., 1978, Appl. Opt. 17, 2917. PALTCHIKOVA, I. G., 1986, Preprint Inst. Autom. & Electrom., No. 328 (in Russian). PETERS,W. N., and A.M. LEDGER,1970, Appl. Opt. 9, 1435. RAYCES,J. L., 1958, J. Opt. SOC.Am. 48, 576. Rioux, M., R. TREMBLAY and P.-A. BELANGER, 1978, Appl. Opt. 17, 1532. ROSIN,S., 1974, Appl. Opt. 13, 1764. SHEPPARD, C. J. R., and A. CHOUDHURY, 1977, Opt. Acta 24, 1051. SHEPPARD, C. J. R., D. K. HAMILTON and I. J. COX,1983, Proc. R. SOC.London Ser. A 387,171. SISAKYAN, I. N., and V. A. SOIFER,1984, Thin optics synthesized by computer, in: Fizicheskie Osnovi i Prikladnie Voprosi Holografii (LIJAPH, Leningrad) p. 142 (in Russian). SOROKO,L. M., 1981a, USSR Patent No. 743424, Bull. No. 21, p. 262 (in Russian). SOROKO,L. M., 1981b, Hilbert-Optics (Nauka, Moscow) (in Russian). SOROKQ,L. M., 1982, Commun. JINR, DI-82-642, Dubna. SOROKO, L. M., 1983a, Mesooptics, holography and optical processor, in: Metody i Ustroystva Opticheskoi Holografii (LIJAPH, Leningrad) p. 189 (in Russian). SOROKO,L. M., 1983b, Introscopie (Energoatomizdat, Moscow) (in Russian). SOROKO, L. M., 1985, USSR Patent No. 1183934, Bull. No. 37, p. 191 (in Russian). SOROKO,L. M., 1986a, USSR Patent No. 1234796, Bull. No. 20, p. 203 (in Russian). SOROKO,L. M., 1986b, USSR Patent No. 1273861, Bull. No. 44,p. 180 (in Russian). SOROKO,L. M., 1987a, Commun. JINR, P13-87-169, Dubna (in Russian). SOROKO,L. M., 1987b, Commun. JINR, P13-87-170, Dubna (in Russian). SOROKO,L. M., 1987c, JINR Preprint No. E13-87-292, Dubna. SOROKO,L. M., 19874 Commun. JINR, P13-87-358, Dubna (in Russian). SOROKO,L. M., 1987e, USSR Patent No. 1283699, Bull. No. 2, p. 193 (in Russian). SOROKO,L. M., 19875 Commun. JINR, P13-87-527, Dubna (in Russian). SOROKO,L. M., 19878, Commun. JINR, P13-87-576, Dubna (in Russian). SOROKO, L. M., 3987h, Commun. JINR, P13-87-468, Dubna (in Russian). STEWARD,G. C., 1928, The Symmetrical Optical System (Cambridge University Press, Cambridge) p. 89. STONER,J. A., 1970, Appl. Opt. 9, 53. VASIN,A. G., M. A. G O L U B , A. ~ . DANILOV, N. L. KASANSKII, S.V. KARNEEV, I. N. SISAKYAN, V. A. SOIFERand G. V. UVAROV,1983, Preprint Phys. Inst. Acad. Nauk, USSR No. 304, M. (in Russian). P. KIRIJANOV, V. P. KORONKEVITCH, F. I. KOKOULIN, VEDERNIKOV, V. M., V. N. VIJUKHIN,~. A. I. LOKHMATOV, V. I. NALIVAIKO, A. G.POLESHCHUK, G. G. TARASOV, V. A. KHANOV, A. M. SHCHERBACHENKO and Yu. I. YURLOV,1981, Automatriya 3, 3 (in Russian). WILSON,T., 1981, Appl. Opt. 20, 3238. WILSON,T.,and C. J. R. SHEPPARD, 1984, Theory and Practice of Scanning Optical Microscope (Academic Press, London). ZAVGORODNEVA, S. I., V. I. KUPRENYUK, V. V. SERGEEV and V. E. SHERSTOBITOV, 1980, Sov. J. Quantum Electron. 7, 142 (in Russian). ZELDOVITZ, YA. B., and I. D. NOVIKOV, 1967, Relativistical Astrophysics (Nauka, Moscow) (in Russian).

E. WOLF, PROGRESS IN OPTICS XXVII

0 ELSEVIER SCIENCE PUBLISHERS B.V., 1989

NONIMAGING OPTICS FOR FLUX CONCENTRATION BY

I. M.BASSETT School of Physics University of Sydney Sydney, NS W 2006, Australia

W. T. WELFORD Blacken Laboratory Imperial College London SW7 282,England

R. WINSTON Enrico Fermi Institute University of Chicago 5640 Ellb Avenue, Chicago, IL 60637, USA

CONTENTS PAGE

. . . . . . . . . . . . . . . . . . . GEOMETRICAL OPTICS OF NONIMAGING SYSTEMS . 2D CONCENTRATOR DESIGNS WITH MIRRORS . . . . 3D MIRROR SYSTEMS . . . . . . . . . . . . . . . . HYBRID REFLECTING-REFRACTING SYSTEMS . . . . SECOND-STAGEOPTICSANDTRUNCATION . . . . . NONSPECULAR REFLECTORS . . . . . . . . . . . . THE GEOMETRICAL VECTOR FLUX FORMALISM . . . SOME PHYSICAL ASPECTS OF NONIMAGING OPTICS . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . INTRODUCTION

163 167 171 180 182 187 188 204 214 225

6 1. Introduction 1.1. HISTORICAL ASPECTS

The ideas underlying the topic of this review originated in the mid-1960s in a paper by HINTERBERGER and WINSTON[ 19661, in which deliberate nonimaging was used to enhance the efficiency of a flux-collectingdevice. At about the same time BARANOV[ 19651 described what is now known as a threedimensional compound parabolic concentrator (CPC) applied to solar energy collection, and PLOKE[ 19671 described a three-dimensional CPC used as a condenser for microscopes. However, the physical principle underlying these [ 18931. The ideas goes back much further in time, at least to BOEGEHOLD principle can be variously described as ageneralization of the Smith-Helmholtz invariant, as a form of Liouville’s theorem, or as a statement of the conservation (in the geometrical optics approximation) of radiance along rays. Neither Baranov nor Ploke seems to have published much after about 1970, but Winston went on with several collaborators to develop the original idea into a well-defined discipline, which acquired the name of nonimaging optics. Developments up to the mid-1970s were described in a book by WELFORD and WINSTON[ 19781. Since then, many developments in the geometrical optics and physical optics of concentrators have occurred, and these will be presented in this review. The collection and concentration of solar energy is one of the main applications of nonimaging optics ; however, this review is restricted to a discussion of the geometrical and physical optics of nonimaging systems, and solar energy and other applications will only be mentioned as they affect design details.

1.2. THERMODYNAMIC LIMIT ON CONCENTRATION

Let a hot body, for example the sun, transfer heat by radiation to a blackbody through an optical system such as a concentrator; then at equilibrium the temperature of the blackbody cannot exceed that of the sun whatever the design 163

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[I14 8 1

of the concentrator.* This fact tells us that there must be a limit to the concentration which can be achieved by an optical concentrator. To fmd this limit, let the concentrator have an entry aperture of area A, let the sun subtend the (small) solid angle 62 steradians at this aperture, and let the absorber be flat with area A’. (Nonflat absorbers are discussed in 5 3.) The absorber will reach its highest temperature if the concentrator irradiates it fully over the solid angle 2 R. Under that condition we should have 62A

=

RA‘ ,

(1.1)

so that the maximum concentration ratio, defined as the ratio of input to output areas, would be

Here we have assumed that the refractive index inside the concentrator is unity, but otherwise this result is completely general; that is, there are no assumptions about the optical system inside the concentrator; in particular, we do not need to assume that it is some kind of image-forming system. In the foregoing procedure we have stressed that the absorber is to be a blackbody. Paradoxically, if it is not a blackbody, it might reach a still higher temperature than that of a blackbody at a concentration ratio less than C,,,,,. This happens, for example, with selective absorbers used for solar energy, which absorb strongly in the visible and near infrared, where the spectrum of sunlight is strongest, but they have low emittance at longer wavelengths where the absorber would tend to reradiate.

1.3. PHYSICAL OPTICS LIMIT OF CONCENTRATION

The action of a concentrator, or any other optical system containing lenses, mirrors, diaphragms, and other elements, can be analyzed by geometrical optics; this means, in effect, tracing “rays” according to hell’s law, and it is a relatively elementary procedure, albeit in many cases very lengthy. We obtain more detailed information by an analysis on the basis of physical optics, but this can mean using any of several different models of wave propagation or perhaps a quantum optics model. We defer a detailed discussion to 5 9, but we

*

The heat transfer is to be purely by radiation; that is, we exclude, for example, intermediate stages of conversion to stored electrical energy.

111, § 11

INTRODUCTION

165

note here that for extended incoherent sources (but not, for example, for single mode lasers) the limit on concentration is never greater than that given by geometrical optics.

1.4. IMAGE-FORMING SYSTEMS AS CONCENTRATORS

The familiar example of a concentrator for radiation from the sun is a burning glass. Curiously enough elementarygeometricaloptics does not yield a theoretical limit on concentration for image-forming systems of this kind. According to the usual paraxial or Gaussian optics theory, when an object of linear dimension q is imaged at size q’, the convergence angles are in the inverse ratio of these dimensions; more explicitly, fig. 1.1 shows this process with media of different refractive indices in the source and absorber spaces, and we then have the Lagrange invariant: naq

=

n‘a’q’ .

(1.3)

For a source at infinity, as in fig. 1.2, the Lagrange invariant takes the form -nhj?= n’a’q’ .

(1.4)

If in fig. 1.2 we take the concentration ratio as the area of the lens divided by the area of the absorber, it is found to be

and a similar expression holds for the case in fig. 1.1 with the object at a finite distance. These expressions are based on paraxial optics, that is, the small-angle approximation, and there is nothing in them to say what would happen if we try to increase the concentration by increasing the angle a’. However, we know from aberration theory that if the lens is corrected for spherical aberration and chromatic aberration on axis, this does not guarantee that a good image will

Fig. 1.1. The Lagrange invariant.

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Fig. 1.2. The Lagrange invariant with one infinite conjugate.

be formed of an extended object: We have also at least to satisfy the Abbe sine condition. Thus in fig. 1.2, which represents perhaps the more important case, we need to ensure that h =fsina' ,

(1.6)

where f is the paraxial focal length, for all rays. However, it is well known to optical designers that ordinary imaging systems with a finite field of view and good aberration correction cannot be designed (or so far have not been designed) with the image side convergence angle greater than about 60°, so that the concentration is only about one half of what it could be if the convergence angle were 90". Such systems are typically microscope objectives, and they have to be built on a small scale (e.g., focal length 2 mm for the highest powers) in order to be able to keep the aberrations low. We make an exception here of certain nonordinary systems such as the Luneburg lens and GOMEZ-REINO [ 1985]), since these and generalizations of it (SOCHACKI cannot yet be manufactured for use in the optical region of the spectrum. Another example of an imaging system which is less than ideal as a concentrator is the concave paraboloidal mirror, often used for collecting solar energy; the paraboloid used with one infinite conjugate has large negative coma, which spreads the light flux away from the axis, thus decreasing the concentration below that for a notional aberration-free system. The Lagrange invariant can be regarded as a form of what is sometimes called the radiance theorem; radiance is power per unit projected area per unit solid angle integrated over an appropriate wavelength or frequency interval. (It should be distinguished here from specific intensity, which is power per unit projected area per unit solid angle per unit frequency interval (WOLF[ 19761); this quantity is used in radiative transport theory). The radiance theorem asserts that no optical system can produce an image of a source with greater radiance than the source itself, apart from a trivial factor involving the refractive

111, § 21

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167

indices in the source and output spaces. The theorem is usually given as a consequence of the Lagrange invariant and then it has only paraxial validity, but in fact, as we shall see in 3 2, a form of the radiance theorem can be given which is valid for all optical systems of which the performance can be predicted by ray-tracing when used in a light transmitting or transporting mode. Thus the system may contain elements other than ordinary lenses and mirrors (e.g., dsractive or holographic elements), it can have aberrations, and it need not even be an image-forming system at all. This theorem will form the basis for the development of the ideas of nonimaging optics.

8 2.

Geometrical optics of nonimaging systems

We may first ask, why nonimaging, anyway?If the object is simply to achieve concentration, whether for solar energy or for other applications which will be mentioned later, then some of the requirements for image-forming systems can be relaxed: We do not need to form images, we merely need to ensure that all the radiant flux entering a certain aperture emerges from another aperture of which the size is given by the theoretical limit of concentration, whatever that may be. It turns out that nonimaging concentrators can do much better than image-forming systems because of this relaxation in the specification. We fist have to recapitulate some definitions and frame them in terms suitable for the subject. Radiance is flux or power flow per unit area per unit solid angle, the area being measured in a plane perpendicular to the direction of the flux. A Lambertian radiator has the same radiance at a point for all directions (usually also the radiance is constant over the Lambertian source). We shall need to refer to numerical calculations in terms of rays and ray densities; in this sense a plane Lambertian radiator is assumed to emit a constant number of rays per unit solid angle per unit projected area. We set up a coordinate system with origin in the radiating surface and z-axis perpendicular to the surface, as in fig. 2.1 ; then the element of surface area is dx dy, and if directions are specified by the direction cosines (L, M, N), the element of solid angle in the direction (L, M, N) is dL dM/N. The element of area perpendicular to the flux direction is N dx dy. Thus a Lambertian radiator would have nominally the same number of rays in each element of solid angle dL d M / N and projected area N dx dy. There is also the concept of a limited or restricted Lambertian radiator, which is usually a defined surface in an optical system rather than an actual source and is Lambertian over a solid angle less than 211.

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Fig. 2.1. Coordinate system illustrating the concept of a Lambertian radiator.

If we start at the source with a uniform density of rays per unit solid angle and per unit projected area, this represents constant radiance, that is, a Lambertian radiator. At any other point in the system the radiance as a function of position and direction is given in terms of the ray density at that point.

2.1. RADIANCE THEOREM IN GEOMETRICAL OPTICS (LIOWILLFS THEOREM)

We can now state the generalization of the Lagrange invariant mentioned in 8 1.4. Consider any optical system, image-forming or not, containing lenses, mirrors, gratings, and holographic optical elements, as in fig. 2.2; the system has plane entry and exit apertures as indicated, but there are no symmetry restrictions. We assume that a ray going into the entry aperture emerges as a

0

Fig. 2.2. Notation for the generalized Lagrange invariant, or Liouville's theorem in geometrical optics.

111.5 21

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169

well-defined ray (or perhaps a finite number of diffracted rays) from the exit aperture; that is, it is not lost through scattering. Alternatively, this ray may be turned back inside the system so that it returns through the entry aperture or it may be lost by absorption inside the system. If the ray does get through we can, in principle, find its position and direction by ray-tracing inside the system. To parametrize the ray, we set up coordinate systems in the entry and exit spaces, respectively Oxyz and O’x’y’z’ as in the figure, and the directions of the ray segments can then be defined by direction cosines (L,M,N) and (L’, M’, N ) . The two coordinate systems do not need to have their respective axes parallel to each other; it is only necessary that the ray segments should not lie in the x-y planes. Consider a small displacement in position and direction of the incoming ray specified by dx, dy, dL, and dM; there will be a corresponding displacement of the emerging ray specified by primed symbols. The generalization of the Lagrange invariant asserts that the following relation holds between the two displacements: nz dx dy dL d M = n” dx’ dy’ dL’ dM’ .

(2.1)

If the system is lossless for the ray in question, this result asserts that radiance is conserved or not increased along the ray; since there are no restrictions on the form of the optical system, it enables us to obtain, in the geometrical optics model, limits on concentration which do not involve the paraxial approximation or assumptions about symmetry. Fortunately these limits agree with the predictions of thermodynamics. The statement “radiance is conserved along a ray” must be qualified to exclude certain devices such as dichroic or polarizing beam-combiners, since by combining beams of different frequencies or polarizations, radiance could be increased, and strictly it is specific intensity that is conserved or not increased; however, it is usual in the topic of this review to speak of radiance, and beam-combiners are not used as components of the nonimaging systems under discussion. The differential quantities on either side of eq. (2.1) have been given various names, such as &endue and throughput. An analogy with mechanics has also been pointed out in which dx, dy, dL, and d M are components in a fourdimensionalphase space; ndL and ndMare written as dp and dq and regarded as components of momentum, and eq. (2.1) then becomes an analog of Liouville’s theorem of conservation of phase space volume. Equation (2.1) as an optical theorem has been known for many years (e.g., BOEGEHOLD[ 18931, STRAUBEL[ 1902]), although it does not appear to have had many practical applications in optics until recently; we may note the work

170

NONIMAGING OPTICS FOR FLUX CONCENTRATION

[I119

s2

of LICHTENBERG [ 19691 on spectrometer design, BODEM[ 19781 on the problems of launching power into lightguides and fibers, and SCHUMACHER [ 19761 on relations between electron and light optics; in addition, occasional misguided attempts to circumvent the theorem are demolished in the literature (WELFORD and WINSTON[ 19821). An accessible proof of the optical form of [ 19861. A formulation of Liouville’s Liouville’s theorem is given by WELFORD theorem for optics in generalized coordinates was given by JANNSONand WINSTON[ 19861.

2.2. LIMITS ON CONCENTRATION

We can now obtain concentration limits similar to that suggested in 1.2 but not restricted to small angles. Let a concentrator have plane entry and exit apertures of areas A and A‘, respectively, and let the entry aperture be filled with rays incident at all angles from 0 to 0 degrees to the normal. Let all these rays get through the concentrator to emerge from the exit aperture at, say, angles up to 8’ to the normal. Then if we integrate both sides of eq. (2.1) over area and solid angle, we obtain n2nA sin20

=

nI2nA’ sin2@ ,

(2.2)

so that the concentration ratio is

The maximum possible concentration is obtained when the exit aperture is filled by rays up to 4 A to the normal, so the maximum possible concentration ratio is (WELFORDand WINSTON[ 19781)

c,

=

nr2 n2 sin20 *

~

Equation (2.4) is the fundamentaltheorem of nonimaging optics. It translates into the language of geometrical optics the limits on concentration imposed by thermodynamics, and thus it sets limits on the design performance of concentrators. The so-calledtwo-dimensional(2D) systems form an important special case; these are systems in which all the optical components are cylindrical with parallel generators and the entry and exit apertures are parallel-sided slots. It

I I L § 31

2D CONCENTRATOR DESIGNS WITH MIRRORS

171

then follows that if end effects are neglected or, as in practice, are cancelled optically, the maximum possible concentration ratio is n' n sine * Equations (2.4) and (2.5), obtained by integrating etendue, set limits to the designs of concentrators. It is possible to reach these limits by suitable designs of 2D nonimaging systems and to approach them indefinitely closely with 3D systems. Concentrators which fulfil eq. (2.4) or eq. (2.5) are called ideal concentrators. The same term is applied to concentrators which are designed to fill completely an exit aperture solid angle less than 27t (see Q 3).

6 3. 2D concentrator designs with mirrors Two-dimensional systems using mirroos were historically the fist to be designed as theoreticallyperfect concentrators, and as the designs are explained it will become clear why mirrors rather than refracting elements are so important; in fact, refracting elements have so far played more of an auxiliary role in nonimaging concentrators, as conveniences for making a system more compact or for facilitating the use of total internal reflection; this will be seen in Q 5.

3.1. 2D COMPOUND PARABOLIC CONCENTRATOR (CPC)

(WINSTON [ 19741)

Figure 3.1 shows the CPC in section; it consists of two concave reflectors, each a section of a parabola, but the two are not parts of the same parabola. Let the concentration ratio be C = l/sin 0, for example; starting from the exit aperture A'B', the parabola AA' has its focus at B' and its axis at an angle 0 to the concentrator axis, as shown. The other side is drawn similarly. The parabolas end at A and A', where their tangents are parallel to the concentrator axis. The mode of action of this system is described in detail by WELFORDand WINSTON[ 19781. The concentration ratio AB/A'B' is precisely @in 8; all rays entering at angles up to Bget through the exit aperture and no rays at angles greater than 6' get through (as must be true by Liouville's theorem, since the maximum phase space volume is transmitted for concentration C). Thus if the

172

NONIMAGING OPTICS FOR FLUX CONCENTRATION

Fig. 3.1. The compound parabolic concentrator, or CPC.

entry aperture is filled by rays up to the angle f 8, the exit aperture must be filled by rays up to f II in the plane of the diagram; that is, it appears to be a full Lambertian radiator in this section, and a plane absorber placed across the exit aperture would receive the maximum possible flux density that could be collected with the entry aperture AB. The rays shown in the figure can be regarded as the projections of skew rays, since the reflectors are parabolic cylinders with the generators normal to the plane of the diagram; from this it follows that a length of such a CPC can be finished with plane mirrors perpendicular to the generators, and it will then have the maximum possible concentration ratio for its entry angle for all rays of which the projections on the plane of the diagram fall within f 8. The 2D CPC has been applied on a large scale for collecting and concentrating solar energy.

4

3.2. THE EDGE-RAY PRINCIPLE

The 2D CPC shown in fig. 3.1 was first described several years after the 3D version (see 8 4), but it is convenient to consider it first because it illustrates the methods of design available for nonimaging systems, and concentrators in particular. First, it is clear from fig. 3.1 why the term nonimaging is used: The radiation

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2D CONCENTRATOR DESIGNS WITH MIRRORS

173

is supposedly coming from a Lambertian source at infinity subtendingthe angle f 8,but there is nowhere in the exit aperture space where an image, no matter how aberrated, could be said to be formed. Some rays go through the concentrator after one reflection, some after no reflections, and a sizeable proportion of the transmitted phase space volume is involved in two or more reflections, as can easily be seen with a few trial sketches. On the other hand, in an image-forming optical system all the rays which form the image meet each reflecting or refracting surface the same number of times, usually only once, and in refracting systems at least there is generally a well-defmed paraxial region in which the image formation is substantially perfect. Second, it can also be seen from trial sketches that the rays which enter at angles less than 8 follow a variety of different kinds of path, and we are not very interested in the details of these paths provided from the rays do get through. The rays at the extreme angle 8are more important, and they emerge just grazing the edge of the exit aperture. Thus it seems that what matters is that extreme rays (i.e., in this case rays at the maximum entry angle) shouldjust get through an exit aperture of the right size for the planned concentration ratio. This idea led to the formulation of the edge ray principle (WELFORDand WINSTON[ 19781); in its more general form it applies also to sources at a finite distance (see § 3.3) and it can be stated concisely in the following form: Extreme rays at the entry aperture should also be extreme rays at the exit aperture. No rigorous proof has been proposed for this principle, but it is undoubtedly helpful in initiating designs both in 2D and 3D, as will be seen in later sections. It could, with hindsight, be applied directly to evolve the CPC of fig. 3.1 : The extreme entry rays are those at 8and the extreme exit rays are those emerging at A' and B', so that an application of the principle yields the two parabolic profiles directly. Other design principles will emerge in later discussion of nonimaghg concentrators.

3.3. CONCENTRATOR FOR A SOURCE AT A FINITE DISTANCE

To design a concentrator for a source at a finite distance, the natural way to proceed for a conventional optical designer would be to use a collimating lens to produce a virtual source at infmity and follow this with a CPC as in Q 3.1,

174

NONIMAQING OPTICS FOR FLUX CONCENTRATION

Fig. 3.2. Concentrator for a source not at infinity (compound elliptical concentrator, or

CEC).

but this method introduces chromatic and other effects from the lens. It is possible to avoid this problem and to have an ideal, all mirror system as follows. Suppose that, still in 2D, we have a source PQ at a finite distance, as in fig. 3.2, and we wish to design a concentrator to produce a full Lambertian output at an exit aperture BB' at the position shown. This system will comprise two mirror surfaces AB and A'B' of shapes to be determined. To apply the edge-ray principle, we take the extreme entry rays to be those from P and Q, respectively, and according to the principle all rays collected from Q should emerge from B'. To achieve this, AB must be a portion of an ellipse with foci at Q and B', and similarly, A'B' must be a portion of an ellipse with foci at P and A'.Clearly all rays collected from PQ do emerge from A'B' ; however, in order to show that the concentration achieved is the maximum possible for given PQ, A'B', and the distance between them, we have to show that the &endue which is collected at AA' is precisely equal to that which emerges from B'B' .To do this, we use the following theorem. In fig. 3.3 rays from the Lambertian source PQ are collected by the aperture AB (not necessarily arranged symmetrically with respect to PQ); then the

Fig. 3.3. Hottel's theorem on &endue in 2D.

111, § 31

2D CONCENTRATOR DESIGNS WITH MIRRORS

175

collected etendue is (PB

+ QA) - (PA + QB) .

(3.1)

This result is due to HOTTEL[ 19541, and it is reproduced by WELFORDand WINSTON [ 19781 ;it can be proved by direct integration of the Ctendue collected by the aperture AB from PQ. From the property of the ellipse that the sum of the distances from the two foci of any point is a constant, it is then easily shown that, in fig. 3.2, PA

-

PB

=

A'B' ,

so that the output Ctendue is 2A'B'; since A'B' is perpendicular to the axis of the concentrator, this must mean that rays emerge from all points of the exit aperture with their direction cosines uniformly distributed over f 1, so that the output is fully Lambertian. The system of fig. 3.2 is usually called a compound elliptical concentrator (CEC); the design procedure as described is different from the procedure for a CPC in that the extreme rays are defined by the edge of the source, not by a given angular subtense; the concentration ratio is defined as AA'/BB', corresponding to what it would be if a collimator were used as suggested at the beginning of the section. 3.4. CONCENTRATOR WITH EXIT ANGLE LESS THAN n/2

A plane absorber placed at the exit aperture of a CPC or a CEC gets the maximum possible concentration, but the radiation arrives at angles of incidence ranging up to 4 n. This may not always be desirable or efficient, and thus we are led to consider concentrators that transform radiation which is Lambertian over, say, 8, to radiation which is Lambertian over e,, the so-called 4/0, concentrator (RABLand WINSTON[ 19761). To construct this concentrator, we again make use of the edge-ray principle. In fig. 3.4 let AB and A'B' be the required entry and exit apertures. We want to make the profile such that all rays emerging from A'B' at the extreme angle 0, must have entered at 8,. We start with a straight taper RA' at the exit end at an angle to the axis i(8, - 8,), the point R being such that the ray from it passes through B ' . Then from the edge-ray principle we want all rays leaving B' at angles less than 13,to have entered at 8,, and we do this as for the CPC with a parabola with its focus at B' and ending at A where its tangent is parallel to the axis. It then follows (WELFORDand WINSTON[ 19781) that this is an ideal concentrator with concentration ratio AB/A' B' = sin 8,/sin 8,.

176

[III, 8 3

NONIMAGING OPTICS FOR FLUX CONCENTRATION

P

Fig. 3.4. Concentrator with exit angle less than

4% (O,/O,

concentrator).

3.5. MAXIMUM SLOPE PRINCIPLE

It can be seen from the concentrator designs in $5 3.1-3.4 why particular importance is attached to mirrors in this subject: The experience of optical designers shows that very complicatedlens systems would be needed to achieve anything like compliance with the edge ray principle even for systems with an exit angle much less than $ II, whereas the required result can be obtained with almost absurdly simple mirror systems. We shall see in later sections that it is sometimes advantageous to make hybrid systems, that is, combininglenses and mirrors, particularly for 3D concentrators; the design of the mirrors in such systems is facilitated by another design principle, very similar in its effect to the edge ray principle but applicable specifically to mirrors either alone or in combination with refracting optics. The maximum slope principlk (WELFORD and WINSTON[ 19781) may be stated as follows: A mirror surface in a concentrator should have the maximum slope consistent with allowing extreme rays to emerge from the exit aperture.

“Maximum slope” means maximum angle of inclination to the axis of symmetry of the concentrator, or alternatively it can mean that extreme rays should all have the minimum possible angle of incidence on the mirror. As stated, the principle seems to be almost equivalent to the edge ray principle, and just as with the latter there is no formal proof, but it will be seen to be more readily applicable to, for example, designs where the mirror effect is obtained by total internal reflection (see $ 5).

111.8 31

2D CONCENTRATOR DESIGNS WITH MIRRORS

111

3.6. NONPLANE ABSORBERS

So far in discussing the form of concentrators we have spoken of entry apertures and exit apertures, with the tacit assumption that the radiation would be collected by a plane absorber across the exit aperture. This is a good geometry for many applications in which the energy of the radiation is to be transformed to electrical energy by means of, for example, photovoltaic cells, but if it is to be used as heat either directly or perhaps to drive a heat engine, it is more likely that a tubular absorber would be used to heat the working fluid. Thus we are led to consider how concentrators may be designed to achieve the maximum concentration to nonplane absorbing surfaces. Consider now a tubular absorber as in fig. 3.5, where the cross-section is assumed to be convex everywhere and symmetrical about the axis. We try a generalization of the edge ray principle, namely, that extreme entry rays shall be tangent to the absorber surface, and on this basis we try to design the mirror profile. For given concentration ratio C the width AA' of the entry aperture is given as the product of C and the perimeter of the absorber cross-section; if the entry angle 0 is given by sin 0 = 1/C and if we assume the profile is parallel to the axis at the entry aperture, this determines the position of the entry aperture relative to the absorber by the extreme ray shown with one reflection, and the tangent condition will determine the shape of the reflector from A' to Po, the point opposite the extreme ray from A. We then assume that the rest of the mirror profile is the involute of the absorber cross-section which passes

Fig. 3.5. Concentrator for a tubular absorber.

178

NONIMAGING OPTICS FOR FLUX CONCENTRATION

[III, I 3

through Po. It has been shown (WINSTONand HINTERBERGER [1975], WELFORDand WINSTON[ 19781) that a concentrator designed in this way delivers all rays entering inside 0 to the absorber, and since we have made the concentration ratio equal to l/sinO, it is an ideal concentrator. The same method may be applied, for example, to a two-sided flat absorber or "fin" (WELFORDand WINSTON[ 19781). WINSTON[ 1978bl and RABL, GOODMANand WINSTON[ 19791 also discussed nonplane absorbers, and they considered the problems arising when, for example, an absorbing tube is placed inside a transparent vacuum jacket, so that there are gaps between the reflector profile and the absorber surface. So far we have shown how to design the concentrator profile in principle, but we have not obtained an explicit expression for it; the expression depends on the form of the absorber cross-section, and a differential equation can be obtained for it; the equation is (RABL [ 1976a1, WELFORDand WINSTON [ 197811, isine+rcose- isin(8+$)--tcos(e+$)(1+ - i c o s e + rsine+ i c o s ( e + $1- tsin(e+ $)(I =

-tan;(&

4) -

+ 4)

+ e + $),

(3.3)

where the symbols are as shown in fig. 3.6. The independent and dependent variables are, respectively, 8 and t, and since r and $ are both known functions of 0, this is a first-order linear differential equation, soluble by quadratures. This

Fig. 3.6. Notation for the differential equation for a concentrator profile.

IK§31

ZD CONCENTRATOR DESIGNS WITH MIRRORS

179

equation holds for the portion of the profile from A’ to Po in fig. 3.5; the involute portion is obtained by the ordinary equation for involutes,

3.7. MECHANICAL CONSTRUCTION FOR CONCENTRATOR PROFILES

If the absorber profile is tangent to the extreme rays, then it is the evolute or caustic of surfaces orthogonal to the pencil of extreme rays; that is, it is the evolute of geometrical wavefronts from an extreme source point. We can make use of this fact by considering as an example the problem of a concentrator for a source at a finite distance and a tubular absorber, as in fig. 3.7. The extreme source point produces a circular wavefront (still in 2D), and the absorber must be the caustic after reflection at the mirror. This is achieved by applying Fermat’s principle of extreme paths: We tie a string of the right length between the source point and the point 1 at the rear of the absorber and pull it taut with a pencil, as in the gardener’s method for drawing an ellipse; the pencil then describes the mirror profile. It is easily shown (WELFORDand WINSTON [ 19781) that the profile so generated with the taut string as a ray does obey the law of reflection. This gives the profile up to the “shadow point” Po, and the involute can also be constructed by string in the usual way. This construction and that described in the preceding section can both be specialized to recover the original CPC of 5 3.1.

PO

Source point

Fig. 3.7. Mechanical construction for a 2D concentrator profile.

180

NONIMAGING OPTICS FOR FLUX CONCENTRATION

[IK8 4

The string construction can also be used to write down the solution of the differential equation (3.3), giving for the profile corresponding to 8> e,, t=

(s + s(eo))- rcos(e - e,) + r(e,) cOs(e, 1 - cos(e + - ei)

+

e,) 3

(3.5)

where 0, is the entry or collection angle.

0 4. 3D mirror systems By 3D systems we mean principally concentrators with axial symmetry, although forms with other symmetries or no symmetry at all could be envisaged. If we take the profile of the CPC (5 3.1) and rotate it about the axis of symmetry, we obtain the simplest form of 3D nonimaging concentrator, in fact the form which historically came before the 2D form (BARANOV[ 19651, HINTERBERGER and WINSTON[ 19661, PLOKE[ 19671). It is found that this leads to very good but not ideal concentrators; that is, some skew rays incident on the entry aperture at angles less than the design angle get turned back after several reflections and re-emerge from the entry aperture, and conversely some skew rays incident outside the design angle are transmitted. This result is not entirely unexpected, since in producing a 3D concentrator in this way we have not introduced any more free parameters into the design but we are trying to deal with an extra double infinity of rays, that is, the skew rays. (As noted in 5 3.1, the skew rays in a reflecting 2D system do not involve any more parameters than the in-plane rays.) It seems at present that no design exists for a 3D reflecting system with real entry aperture which has ideal behavior, although, as will be seen in 5 9, such systems exist with virtual entry aperture. The problem is akin to that which arises in the design of axisymmetricimaging optical systems where, except in certain cases of spherical symmetry, the aberrations cannot be reduced to zero over a finite field of view when skew rays are included. This point is discussed in some detail in Appendix B of WELFORD and WINSTON[ 19781.

4.1. THE 3D COMPOUND PARABOLIC CONCENTRATOR

The performance of the 3D CPC,a shape obtained simply by revolving the 2D profile about its axis, cannot be predicted analytically: It is necessary to

111, I 41

3D MIRROR SYSTEMS

181

carry out an extensive programme of ray-tracing to determine the departure from ideality. Rays are traced at and near the design acceptance angle in a grid covering the whole of the entry aperture, and they are followed until after a predetermined number of reflections they are either transmitted or turned back; that is, the z-direction cosine changes sign. The results can be displayed as a transmission curve (WINSTON[ 1970]), showing the proportion of rays transmitted as a function of input angle. A curve such as that shown in fig. 4.1 would have a sharp cutoff, as in the broken line, if the concentrator were ideal; in practice the curves are as the full line, with a transition typically taking place over an angular range of one or two degrees. Further transmission curves of this kind are given by WELFORDand WINSTON[ 19781, and this publication also shows the very complex patterns in the entry aperture formed by the boundaries of areas of rays transmitted and rejected at angles near the theoretical acceptance angle. It can be seen from such graphs as those in fig. 4.1 that the efficiency of a CPC as a concentrator far exceeds that obtainable from any conventional imaging-type system.

4.2. OTHER 3D REFLECTING SYSTEMS

The variants of the 2D CPC described in J 3 can, of course, be turned into 3D systems, and comments similar to those in J 4.1 apply, mutatis mutandis, to the transmission characteristics of these systems. For example, WINSTON [ 1978al described the 3D form of the CEC (J 3.3) and gave an example of a transmission curve for this type of system. It will be seen in 8 6 that a major application of such systems is as so-called “second-stage” nonimaging systems to follow a fist stage imaging system, which is a useful practical compromise for very large-scale concentrating systems.

Fig. 4.1. Transmission curve of a 16” 3D CPC; the graph shows the proportion of flux transmitted for the angle of incidence on the entry aperture indicated on the abscissa scale.

182

NONIMAGING OPTICS FOR FLUX CONCENTRATION

[I14 8 5

Since the 3D CPC is not ideal in its performance, as shown by its transmission curve, it might be thought that a better idea for a 3D system would be to combine two 2D CPCs at right angles, forming a 3D system with a rectangular or square aperture (WELFORDand WINSTON[ 19781); however, it is found by ray-tracing that this system is not ideal; that is, some rays inside the design entry angle are turned back. A more detailed study was made by MOLLEDO and LUQUE[ 19841, giving numerical data for both empty and dielectric filled systems and discussing the applications to solar energy.

8 5.

Hybrid reflecting-refracting systems

By now it will be clear that reflecting surfaces are essential for achieving a near approach to ideal concentrator performance; nevertheless, refracting components play a useful part, minor in principle but technically very important, in making concentrators more compact and more convenient to combine with other optical elements. In addition, total internal reflection, a completely lossless process, has its uses in concentrator design and this necessarily involves refraction at some stage.

5.1. SOLID 3D CONCENTRATOR

A 3D concentrator may be made from solid glass or plastic, as in fig. 5.1 (WINSTON[ 19761); on account of refraction according to Snell’s law at the entry and exit faces, the profile needs to be that of a 8,/8, system (Q 3.4) with angles given by

n sin 8,

=

sin Oi ,

n sin 0,

=

1,

(5.1)

Fig. 5.1. A solid 3D concentrator; the profile is that of a O,& concentrator, so that the rays externally have the required angles according to Snell’s law.

111,s 51

183

HYBRID REFLECTING-REFRACTING SYSTEMS

where Oi is the entry angle in air and n is the refractive index of the material. It can easily be shown that all rays are totally internally reflected at the profile if ncos$(e, +

e2)> 1.

(5.2)

5.2. CONCENTRATORS WITH LENS COMPONENTS

A practical disadvantage of the CPC geometry in both 2 D and 3 D is the length of the concentrator for high concentration ratios: For input angle 8 the length is proportional to cotan 8. This disadvantage can be partly overcome by using lenslike components. Systems using ordinary lenses and Fresnel lenses on both plane and convex base curves at the entry aperture of 2D reflecting concentrators were described by COLLARES-PEREIRA, RABLand WINSTON [ 19771; thus in fig. 5.2 the cylindrical lens is assumed to be aberration free for rays at the entry angle 8, and the mirror profile has to refocus these rays at the opposite edge of the exit aperture. Clearly the profile is an off-axis hyperbola in this approximation. A better approximation is obtained by ray-tracing through the lens and constructing the profile accordingly. Such systems achieve reduction of the overall length but at the expense of a nonideal performance, in that the transmission curve no longer has a sharp cut-off at the design entry angle. This is a good example of the application of the edge-ray principle. A different example involves the application of the principle of maximum slope (5 3.5). Figure 5.3 shows a 3 D concentrator of solid dielectric with a

0 0

Fig. 5.2. A concentrator with a lens at the entry aperture; the hyperbola profile has its foci as indicated.

184

NONIMAGING OPTICS FOR FLUX CONCENTRATION

Fig. 5.3. A solid concentrator with lenticular profile at the entry aperture.

lenticular entry aperture; the profile of the reflecting surface can be designed accordingto the condition that the slope must be as large as possible, consistent with total internal reflection for rays in the plane of the diagram (skew rays will then also have total internal reflection). Then (WELFORDand WINSTON [ 19781) it follows that the profile is a hyperbola from P, to P,, neglecting aberrations at the entry surface, and it is an equiangular spiral from P, to P,. NING,WINSTONand O'GALLAGHER [ 19871 carried this study further, showing diagrammatically the decrease in overall length obtained with increasing power of the lens entry aperture; they propose also a different method of designing the second portion of the profile (P, to P, in fig. 5.3), according to which the rays incident on this portion are made to emerge parallel to each other. There is little differencein the resulting performance, but this so-called "phase-conserving method" has the advantage that it yields an analytical solution for the profile. A different approach to the design of 2D-only concentrators was suggested by WELFORD and WINSTON[ 19791, who showed how an apparently conventional imaging system could be designed with lenses only as an ideal concentrator: By invoking the edge-ray principle and aiming at aberration-free imagery at the edge of the exit aperture (i.e., at the edge of the field of view in conventional terms), a system could be obtained with large aberrations except at the edge of the field. It is questionable whether such a system should be designated image forming or nonimaging. Other aspects of designs for 2D systems with tubular absorbers in media of O , and LUQUE refractive index not equal to unity are discussed by M I ~ ~ A NRUIZ [ 19831 and by WELFORDand WINSTON[ 19781.

111, S 51

HYBRID REFLECTING-REFRACTING SYSTEMS

185

5.3. INHOMOGENEOUS MEDIA

As for image-forming systems, it may be asked whether inhomogeneous media offer any gains for nonimaging optical systems, in spite of the fact that at present there is little hope of realizing such media except on a small scale for special purposes, for example, optical fibres and SELFOC lenses. Such ideas for imaging systems go back at least to JAMES CLERKMAXWELL [ 18541; LUNEBURG[ 19641 gave several examples in both 2D and 3D of imaging systems ideal in certain defined senses. WINSTONand WELFORD[1978] examined the general problem of 2D nonimaging systems with inhomogeneous media and obtained a generalization of Hottel's result (eq. 3.1) for the &endue of a beam. This is illustrated in fig. 5.4, where PapEand PSPF are wavefronts of extreme pencils in an inhomogeneous medium and AB is the entry aperture which they just fill; then the result for the dtendue is

H

=

[PJ] + [PpA] - [P,Al- [PpBl,

(5.1)

where the square brackets denote opticalpath lengths. The similarity to eq. (3.1) is obvious. From this it was shown that mirror surfaces could be constructed to transform the two incoming extreme pencils into extreme pencils at a specified exit aperture, that is, a generalization of the edge-ray principle. In a series of four papers MIRANO [ 1985a-c, 19861 examined two- and three-dimensional refractive index distributions in relation to nonimaging systems. MIRANO [ 19861 proposed a reformulation of the edge-ray principle and gave a proof of it in 3D for inhomogeneous refractive index without axial symmetry, but under the important restriction that all input rays within the

P

Fig. 5.4. The generalization of Hottel's theorem for inhomogeneous media.

I86

NONIMAGING OPTICS FOR FLUX CONCENTRATION

[IKB 5

design ktendue emerge from the exit aperture; that is, none is turned back. The rays in an incoming beam are described as usual by two spacelike coordinates and two direction cosine coordinates, forming a four-parameter manifold; a lifth coordinate, z for example, is introduced to allow for propagation of the rays. The incoming four-parameter manifold occupies a certain region of phase space, say R,,, and the emerging rays occupy R, . According to MiAano, the boundaries of the entering and emerging beams are specified as three-parameter manifolds, denoted respectively by a (R,) and a(R,),and he states the edge-ray principle as follows. If a(%) and a(R,)are the same three-parameter manifold of rays, then R,, and R, are the same four-parameter manifold. The proof depends on the fact that in the five-dimensional space the trajectories representing two different rays cannot intersect, because if they did, this would imply two different rays at a point with the same direction cosines. It then follows that the manifold a(&) completely cuts off R,, from all other rays and the theorem is proved by noting that no gaps can appear in the ktendue because it has been assumed that no rays are turned back. In the same paper Miflano uses this formulation of the edge-ray principle to show how axisymmetric concentrators with nonuniform refractive index can be designed. The rectangular coordinates ( x , y , z) are transformed to an orthogonal curvilinear set and the direction cosines to a conjugate set of variables. The surfaces of constant j , where j is one of the curvilinear coordinates, form the flow lines (see 9) of the system, and they also give the mirror profile. Unfortunately the numerical values of refractive index are such that it is unlikely that this type of concentrator could be made. The earlier papers by MIRANO [1985a-c] deal with two-dimensional aspects of similar problems. Two of these papers [ 1985b,c] give an ingenious design for a 2D concentrator consisting of straight-sided triangular sections of different refractive indices, to be known as the CTC or compound triangular concentrator, but again the refractive index values are rather unrealistic. Finally, it was noted in 5 3 that in purely reflecting 2D systems the paths of the rays shown in a diagram are also projections on the plane of the diagram of skew rays, so that no extra account need be taken of skew rays in the design process. However, this is not true for 2D systems involving dielectric or refracting materials, as can easily be seen from a few numerical experiments using Snell's law to calculate the paths and projected paths of rays refracted into glass. Thus it could be said that systems involving reflections but no refractions are the only truly 2D systems; nevertheless, in practice it is useful to consider as a separate class from 3D systems those systems in which the geometry of the surfaces and refractive index distributions is two-dimensional.

111, § 61

SECOND-STAGE OPTICS AND TRUNCATION

187

6. Second-stage optics and truncation

For large-scale concentration applications, that is, solar energy, nonimaging concentrators of CPC type get very long for high concentration ratios, since the overall length is approximately proportional to the product of the width or diameter of the entry aperture and the concentration ratio. Two ways of mitigating this disadvantage have been tried: (1) by simply truncating the CPC or other shape and (2) by using a nonimaging concentrator as a second stage to follow a conventional imaging system, such as a concave paraboloidal mirror. Both these approaches result in a loss of ideality; that is, the transmission curve (fig. 4.1) would be further degraded from the ideal rectangular shape, but the gain in compactness is often well worthwhile.

6.1. TRUNCATION

Truncation of a CPC at the entry end has been discussed by several authors (WINSTONand HINTERBERGER [ 19751, RABL [ 1976b], MIRANO[ 19831). The losses in concentration are moderate for appreciable gains in compactness, as shown by curves for 2D CPCs reproduced by WELFORDand WINSTON [ 19781; for example, for a CPC with 5 entry angle a truncation to 50% of the original length produces less than 20% loss in concentration, and the saving in material is correspondingly large. MIRANO[ 19831gives transmission curves for truncated CPCs (in that paper they are called directional intercept factors), and he also discusses concentrators designed for double-sided fin-like absorbers (bifacial absorbers). The disadvantage for some purposes of truncation is that the transmission curve is flattened; that is, there is not a sharp transition at the design entry angle. O

6.2. SECOND-STAGE SYSTEMS

The idea of second-stage concentrators was adumbrated by RABL and WINSTON[ 19761 in a discussion of the CEC (Q 3.3); it was taken further by WINSTONand WELFORD[ 19801in connectionwith solar energy concentration. A large concave paraboloidalmirror is the simplestway to achieve an immediate great concentration of direct sunlight, yet as pointed out in 1.4 this cannot approach the theoretical maximum concentration for a source subtending 0.5" on account of the large coma of such a mirror. However, it is possible to use

188

NONlMAGlNG OPTICS FOR FLUX CONCENTRATION

[IIL § 1

a paraboloid as a first stage of concentration and then follow it with a nonimaging concentrator. In the latter reference an example of a 10 m diameter f/0.5 paraboloidal mirror was taken, and it was shown that although even with a CEC second stage there was still a necessary lower concentration than the theoretical maximum (because of the coma in the paraboloid), yet the use of the second stage had recovered a considerable proportion of the loss of concentration. The question of losses of concentration due to aberrations in the first stage [ 1982a,b, 19831 in a was taken further in a series of papers by KRITCHMAN discussion of mainly 2D systems. KRITCHMAN [ 19831 concluded that the theoretical limit could be attained in a 2D two-stage system only if the imaging stage was without aberrations; in that case the pupil of the imaging stage could be regarded as an incoherent source, and the entry aperture of the second stage would coincide with the image of the primary source. For moderate scale systems a Fresnel lens as the first (imaging) stage has some attractions because of low mass and cheapness. NING, OGALLAGHER and WINSTON[ 1987al described such systems with a solid total internal reflector system as the nonimaging second stage. An experimental test with a photovoltaic cell as receiver was set up for a 3D system with nominal concentration x 200, and the results agreed well with ray-tracing predictions.

0 7. Nonspecular reflectors 7.1. INTRODUCTION

At a real surface only part of the incident light is reflected specularly;the rest is distributed over angles differing from the angle of incidence. An effectively random element is thus introduced, information is lost, and an ideal system cannot be realized. Consequently, if the reflectors used in a collector are appreciably nonspecular, the “ideal” property, and the associated edge ray principle, cannot be used effectively as design criteria. An alternative design criterion, however, is available - one which is broadly equivalent to ideality when the reflectors are perfect and specular. This is the criterion of maximum power transfer from entrance aperture to absorber. The simplest nonspecular form of reflection is Lambertian, in which the angular distribution of the reflected light is that of black-body radiation for any angle of incidence. Methods adapted from the theory of thermal radiation transfer can be employed. A useful notion is that of the “codguration factor”

189

NONSPECULAR REFLECTORS

Fig. 7.1. Hottel’s string construction for the view factor between cylindrical surfaces.

or “view factor” from one surface to another (SIEGELand HOWELL[ 19721, SPARROW and CESS[ 19781). Although this is a purely geometrical notion, the following physical definition makes its meaning clear. Let a surface 1 emit light uniformly and with Lambertian angular distribution. The “view factor” P,,of a surface 2 viewed from surface 1 is by definition the fraction of the light from 1 which reaches 2 directly; the view may be partly blocked by other bodies. If the geometry of the surfaces is “two-dimensional”, that is, if they are actually surfaces of parallel cylinders, the value of the view factor is given by the beautiful string construction of HOTTEL [1954]. Hottel’s strings are illustrated in fig. 7.1, which includes a third body partly blocking the view. When the strings are drawn taut, their lengths are related to the view factors PI2 and p2, by lip12

=

1 2 ~ 2 1= f(lAC

+

- /AD - IBC)

3

(7.1)

where I, is the length of the profile of surface 1, I, is the length of the profile of surface 2, IAc is the length of the string passing from A to C, and so on. The combination of string lengths on the right-hand side of eq. (7.1) is independent of the positions on the surfaces of the four ends A, B, C, and D, provided A and B are “out of sight” from any point on surface 2 and C and D are similarly out of sight from surface 1. In the general, or “three-dimensional” case a generalized Hottel string construction (DERRICK[ 19851) is available to simplify the calculation of view factors. In nonimaging optics, strings substitute for integration, either of a differential equation or, as in this case, of the integrand of a definite integral. Derrick’s method reduces a fourfold to a double integral, the integrand of which is determined by the lengths of certain geodesic strings. The 2D view factors may be derived from potentials, defined at any point by pairs of string lengths. For example, as illustrated in fig. 7.2, at any point r a potential V2(r)may be defined by V2(r)

= f(ID

- IC)

9

(7.2)

190

NONIMAGING OPTICS FOR FLUX CONCENTRATION

Fig. 7.2. String construction for potential from which view factors may be derived.

where 1, is the length of the string passing tautly from the point r around surface 2 to the left and 1, is the length of the similar string passing to the right. The view factor P I , may then be expressed as the difference between the values of this potential at two suitably chosen points on surface 1, divided by the profile length of surface 1. Thus f 1 p 1 2 = v2(r13)

-

v2(rA)

9

(7.3)

where rA and r, are the points A and B illustrated in fig. 7.1. Consider a collector system (fig. 7.3) consisting of a Lambertian source y, a Lambertian reflector ACA', and an absorber a. The reflector surface may be regarded as consisting of a large number of pieces, labelled i = 1,2, .. ., each piece being small enough to be effectively flat and uniformly illuminated. The power reaching the absorber depends on the various view factors: Pij between elements i, j of the reflector, Piybetween reflector element and source, and Pa, between source and absorber. The simplest case is considered in 5 7.2, in which 2D geometry is assumed, the Lambertian source y is taken to be the entrance aperture AA' of the reflector bowl, and the reflector is assumed to be perfect, that is, nonabsorbing. In this case there is a prescription for the optimal reflector, which rivals in simplicity the string or edge ray prescriptions for specular reflectors. The optimal reflector

Fig. 7.3. Collector system with cylindrical geometry consisting of light source 7, absorber a, and reflector ACA'.

111, § 71

NONSPECULAR REFLECTORS

191

is determined by the condition Pi, = U P i y ,

(7 * 4)

where u is a constant determined by the (fixed) geometry of absorber and entrance aperture. If the collector system has symmetry of rotation about an axis, and the reflector elements i, j are taken to be annuli, the optimal reflector profile may be shown to satisfy the same simple condition (7.4). If less than the whole incident hemisphere is illuminated at the entrance aperture, or if the reflector absorbs part of the light which falls on it, the simple method of f 7.2 fails. A method of handling this more general problem in 2D geometry is described in f 7.3. The most general problem is considered in f 7.4, namely, the problem of maximizing power transfer from entrance aperture to absorber with an arbitrary reflection law at both reflector and absorber, any incident light distribution, and without restriction to 2D geometry (DERRICKand BASSETT[ 1985a1). In place of the view factors PV, which characterize a Lambertian reflector of given geometry, there appear three-index quantities PVk. By definition, PVk is the fraction of the power reaching segment j direct from segment i, which is reflected to segment k. This formulation permits a reflection law which is partly or even, in particular, wholly specular. It constitutes in principle a general alternative to ray-tracing methods for evaluating the performance of an optical system, within the domain of geometrical optics. Experience with this alternative method suggests that it is faster than ray-tracing methods for some problems. In f 7.5, using methods similar to those of f 7.2 and f 7.3, an upper bound on collector performance is derived. Its main relevance is to the design of a totally (or nearly totally) absorbing cavity (in general with 3 D geometry). For a given area of absorber (per unit area of aperture) and for a given absorptivity a of the absorbing surface, the performance is shown to be limited by the dzfluse component of the light reflected from the absorber; better performance can be obtained if the light not absorbed is reflected specularly.

7.2. 2D SYSTEM WITH PERFECT LAMBERTIAN REFLECTOR AND FULL LAMBERTIAN SOURCE

Let us imagine that the reflector R (fig. 7.4) is divided into N small segments of length I,, labelled 1,2, ...,N. A ray of light that leaves segment i may strike another segment j , the absorber

192

WI,I 7

NONIMAGING OPTICS FOR FLUX CONCENTRATION

(b)

Fig. 7.4. Collector system with cylindrical geometry and full Lambertian light at entrance aperture.

a, or pass out across the aperture y. We define corresponding view factors P,,, Pia and Pi,; P,, is the fraction of the power leaving i which strikes j, Pia is the fraction which strikes a, and PtY is the fraction which passes out across y. Accordingly,

c Pu + Pi, + P I , N

i= 1

=1

.

(7.5)

Since the aperture y is, by assumption, uniformly illuminated in the whole inward hemisphere at each point, we may similarly define the probability P , that a ray passing inwards across y strikes the segment i, and the probability P,, that it strikes the absorber a. It follows that Pya +

c P,, N

i= 1

=

1.

(7.6)

If the entrance aperture y, of length L,, is illuminated at radiance B, the power entering is (per unit length parallel to the cylinder axis) nBL,. The fraction of this power which strikes the absorber directly is P,,, and the fraction which strikes the reflector once and then the absorber is C , P,,Pi,. The fraction which strikes the reflector exactly twice before striking the absorber is Xu Py,PuPja, and so on. The power reaching the absorber (per unit length perpendicular to the plane of fig. 7.4) is therefore

Let us write P for the N x N matrix Pu, P , for the N-rowed vector P,a, and P , for the N-rowed vector Pi,. Let us also assume for simplicity that the 1, have a common value, 1. Expression (7.7) for the power J reaching the absorber (per

111, I 71

NONSPECULAR REFLEaORS

193

unit length perpendicular to the diagram) is then J = nB(LYPya+ lP;(1

LYPya+ 1P;-

+ P + P2 +

l-P

*

*)Pa)

pa).

In eq. (7.8)PT is the row vector transpose to the column vector P,, and use has been made of the third of the following symmetry relations (which are all instances of the symmetry relation appearing in eq. (7.1)). Lapmi= l i p i a ,

lipu = lipji, LyPyi = lipjy

3

(7.9)

Lapay = Lypya

Equation (7.8) expresses the absorbed power J per unit incident radiance as a function of the system's geometry. If the absorber and the rim positions A, A' are held constant, J depends on the position vectors of the remaining N - 1 ends of the reflector segments, and these may be varied so as to maximize the [ 1980al have shown, there is a expression for J. As BASSETTand DERRICK very simple solution to this maximization problem, namely, eq. (7.4), or Pa

=

KP,.

(7.10)

The constant of proportionality

K

is given by (7.11)

and the fraction, q, of the incident power which then reaches the absorber is given by (7.12) Condition (7.10)is equivalent to the statement that the light leaving the reflector is of uniform radiance B', related to the incident radiance B by (7.13) An optimal reflector geometry may be obtained from eq. (7.10)with the help of potentials expressed in terms of string lengths, as in eq. (7.3);

1Pia = va(ri) - va(rj- 1)

3

(7.14)

194

NONIMAGING OPTICS FOR FLUX CONCENTRATION

1111,s 7

where ri- ri are the endpoints of the ith reflector segment. A potential V , may be similarly defined, for which (7.15)

/Pi, = V,(ri) - V , @ -

The reflector profile condition (7.10) may be rewritten in terms of these potentials: (7.16)

V,(r) - IcV,(r) = const.

The potentials V,(r) and V,(r) are given in terms of the string lengths of fig. 7.5: V,

=

$(I4 - 1 3 ) ,

V , = ;(I*

- I])

t

$(I6 - I,)

.

(7.17)

It should be noted that the solution of eq. (7.10) or the equivalent (7.16) is not unique; in particular, there may be solutions with and without cusps. All such solutions, however, must give the same efficiency (7.12). Some examples are shown in fig. 7.6, together with performance figures q and F. If 5 is the ratio of aperture area to absorber area, F = 5 q is the ratio of the power reaching the absorber to the largest value of this quantity consistent with radiance conservation (viz. nB x absorber area, if B is the radiance of the incident light). Figures 7.6a and b show two different solutions to the same maximum power transfer problem and necessarily achieve the same power transfer, that is, the same efficiency, q ; figs. 7 . 6 ~and d, with wider apertures, show a higher

A A’

Fig. 7.5. Strings defining the potential of absorber and aperture for the collector system ofFig. 1.4.

NONSPECULAR REFLECTORS

(a) 17 = 0.701,f =

1, F = 0.701

(c) q = 0.496,5 = 5/n,F = 0.790

(e) 17 = 0.516,f= 4.95/n,F = 0.814

195

(b)q = 0.701,f = 1, F = 0.701

(d) q = 0.512,f = 5/n,F = 0.816

( f ) q = 0.487,f = 4/r,F = 0.620

Fig. 7.6. Some optimum reflector profiles for a circular cylindrical perfect absorber. The reflector is assumed to be perfect and Lambertian, and the entrance aperture acts as a full Lambertian source.

illumination of the absorber, that is, higher F, and fig. 7.6e illustrates the possibility of unsymmetrical solutions; in fig. 7.6f the absorber is placed well behind the aperture.

7.3. 2D SYSTEM WITH IMPERFECT REFLECTOR A N D RESTRICTED

LAMBERTIAN SOURCE

When the source y of fig. 7.1 does not fill the whole sky (as seen from the reflectoraperture AA’ ), the optimization problem has no short-cut solution like

196

NONIMAOING OPTICS FOR FLUX CONCENTRATION

[III, B 7

eq. (7.10). However, there is an expression, analogous to that of eq. (7.8), for the power J reaching a (perfect) absorber. It is easy to make allowance, in this expression, for imperfect reflectivityof the reflector: We assume that a fraction p of the light incident at any angle on the reflector is reflected, with Lambertian angular distribution, the remainder being absorbed. We also allow the reflector segments to have different lengths 1,. Let A denote the N x N diagonal matrix whose (i, j ) element is liSu. Then in the notation of eq. (7.8)the power absorbed (per unit length perpendicular to the diagram), J, is given by

J = 7tB(L,Q,,

+ P:A(p-'Z

- P)-'P,).

(7.18)

The symmetry relations (7.9)show that equally

J = 7tB(LaQa,+ P;FA(p-'I - P ) - ' P , ) .

(7.19)

The symmetry between source and absorber exhibited in eqs. (7.18)and (7.19) is required by the second law of thermodynamics and is analogous to Kirchhoff s theorem. As a consequence, any reflector which serves to maximize the power transfer from y to o! will still be optimal if the roles of source and absorber are interchanged. Denoting by r,, rl, ....rN the vector positions of the N + 1 ends of the N reflector segments, ro and rN are fixed, and the problem is to maximize expression(7.18or7.19)forJwithrespecttothevariablesr1,r,, .... rN- ];Jdepends

.:

-4 -6-4

-4

-8

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

-2

0 2

4

6

(b)

..:---..-- - -,! .................

-12-8 -4

(C)

0 4 8 12 Fig. 7.7. Various optimum reflector profiles for a circular cylindrical perfect absorber for three values of the reflectivity, p = 0.8, 0.9 and 1.0 (respectively, full, dashed and dotted lines). The incident light is "restricted Lambertian"with a half angle of 30".The efficiencies q and fractions F of maximum illumination are shown in table 7.1 (F = [ q sin30").

111,s71

197

NONSPECULAR REFLECTORS

on these variables through the quantities P,,P, and P.The elements of P, and P, may be expressed in terms of potentials resembling those of 5 8.2, and the elements of the matrix P may also be expressed in terms of potentials U(r,r'), which are sums of string lengths connecting the points r and r' and passing over, under, or around the absorber. From these, convenient expressions for the derivatives aJ/ar, can be obtained. These expressions, like the expression for J itself, involve the inverse matrix ( p - ' I - P)-',which is evaluated in practice by truncating a geometrical series. A trial set of position vectors r , , . . .,r,-, can be adjusted by a rapidly convergent "steepest-descents'' method, using the string expressions for the derivatives aJ/ar,. Details of the and BASSETT[1985b]. Some examples of method are given by DERRICK optimal reflector profiles are shown in fig. 7.7 for three values of the reflectivity p and for light incident from a Lambertian source at infinity, symmetrically placed with respect to the aperture and subtending a half angle of 30" there. As would be expected, the higher the absorptivity of the reflector, the shorter is the optimal profile. The performance figures q and F a r e shown in table 7.1. In fig. 7.7b, 5 = 2 = l/sin30", the same value as that for an ideal reflector. The fraction F = 0.391 of maximum illumination attained, even with reflectivity p = 1, is appreciably less than that ( F = 1) attained by the ideal reflector.

7.4. GENERAL COLLECTOR SYSTEM

Consider a general three-dimensional system, such as illustrated in fig. 7.8, composed of a Lambertian source, y, a reflector, R, and an absorber, a. There is no loss of generality in supposing both R and a to be finite in extent - the nominally infinite periodic array of fig. 7.8b can be simulated by taking a finite unit cell bounded by perfect specular plane mirrors, which are included as part of R. TABLE7.1 Aperturelabsorberratio and performance figures q, F for the collector systems offig. 7.7 at three values of the reflectivity p. Figure

7.7a 7.7b 7.7c

5

1.O 2.0 4.0

F

rl p = 0.8

p = 0.9

p = 1.0

p = 0.8

p = 0.9

p = 1.0

0.590 0.335 0.176

0.632 0.362 0.140

0.679 0.391 0.204

0.295 0.335 0.352

0.316 0.362 0.379

0.340 0.391 0.409

198

NONIMAGING OPTICS FOR FLUX CONCENTRATION

Fig. 7.8. Two typical 3D collector systems, showing reflector R, absorber a, and light source 7.

Let us conceptually combine the absorber and the reflector into a single entity, the “sink,” comprising in general several disjoint parts. Divide the sink surface into a large number, N, of small-area elements, A,, j = 1,2, ...N. For each element Aj let there be defined a reflection function R,( p , p’ ) of two unit vectors p , p ’ with the following significance. If light is incident on A, in a direction - p , that is, coming from a source which A, sees in direction p , then Rj( p, p’) d’p’ is the fraction which is reflected onto an element of solid angle d’p’ ,centred about the direction p’ . The fraction a,( p) = 1 - 1R,( p , p’ ) d’p‘ is absorbed. Let p = (sin $ cos $I, sin 4 sin q,, cos $), p‘ = (sin e; cow#$, sin e,l sin $$ ,cos e; ), referred to a local set of axes with the normal to A, as polar (z) axis. The normal is to be chosen on the reflecting side of R and the absorbing side of ci, so that 0 6 4, t$ 6 A and 0 6 6,$$ 6 211. We may then write Rj(P, p ’ ) = R,<$, 6,e; Y q; 1 and

4

f 271

As examples, we may cite the special cases

R,(p, p ‘ ) = pi(4.)G(COS 4 - cos q) 6(& -

#

f

A)

,

(7.21)

which belong to an imperfect reflector with angle-dependent absorption coefficient, a,($) = 1 - pi($), which is Lambertian for eq. (7.20) and specular for eq. (7.21).

111,s71

199

NONSPECULAR REFLECTORS

Consider light of radiance B, which emanates from y and falls upon the sink, suffering multiple reflections between the various area elements Aj, with partial absorption. We assume that these elements may be taken sufficiently small for the following definitions of transfer coefficients to have a meaning: (7.22) B WYj power absorbed by Aj which comes direct from y, BQ,,

power reaching A , from y after reflection from A j , j # k,

(7.23)

Wij

fraction of power reaching Aj direct from A , which is absorbed by A j , i # j

(7.24)

Pijk

fraction of power reaching Aj direct from A , which is reflected to A,, i # j , k # j .

(7.25)

Wu is taken to be zero for i = j , and Puk is taken to be zero if j The power absorbed by a sink element A , may be written as

Jk = J:

+ Ji + J: + J,'+

*

a

*

=

i or j

=

k.

,

the various contributions coming from light suffering 0, 1,2,3, . .. intermediate reflections from other sink elements. The power absorbed by A , coming direct from y is 1:

=

BWyk ,

and after 1,2, 3, . .. reflections is, respectively, Jkl

=

Jk2

=

J: =

B

C

Tk

Qyj,

i

9

C QyljPrjk y il

B

k

9

C QymiPmijPijk y

jim

k

9

and so on. Adding all these contributions yields Jk=B[Wyk+

where

1 i

EyjkW/kl,

00

Eyjk

=

C

n = l

Qnjk

(7.26) (7.27)

9

with the recurrence relations (7.28) and n

=

1,2, 3,

.. . with Q:,

=

Qyij.

200

[III, 8 7

NONIMAGING OPTICS FOR FLUX CONCENTRATION

Finally we sum Jk over the subset of values of k,which belong to the absorber. Introducing eak = 1 if,& on a, and zero otherwise, we have for the total power absorbed by a, (7.29) In eq. (7.29) the summations are, at least formally, over all N sink elements. An analogous summation of eq. (7.26) over the reflector elements yields the power dissipated by the reflector. Equation (7.29) can, of course, be exact only in the limit of an infinite number of infinitesimal area elements. In practice the numerical value of the expression on the right-hand side depends on the choice of area elements, and on any approximations employed in the evaluation of the Pijk and other transfer coefficients (7.22)-(7.25). If the absorbing and reflecting surfaces are parallel cylinders, expression (7.29) for the power transfer may simplify. For example, for wholly specular reflection there is no coupling between rays with different inclination to the cylinder axis, and the power transfer is a sum of independent, effectively 2D, expressions. Physically, the decoupling arises from conservation of the component of linear momentum parallel to the cylinder axis. An analogous reduction occurs if the system has symmetry of rotation about an axis; the reduction in this case is a result of skew invariance, or the conservation of the axial component of angular momentum. With cylindrical geometry, purely Lambertian reflection, and absorptivity independent of the angle of incidence, the power transfer reduces to a single, effectively 2D, expression. A simpletest calculation is exhibited in table 7.2 and fig. 7.9. An infinite array of parallel circular absorbing cylinders, backed by a perfect plane specular mirror, is illuminated with full Lambertian light. Analytical solutions are obtainable with the help of Hottel strings and the method of images. For the geometry chosen, an incident ray undergoes at most three reflections before reaching the absorber. The calculated contributions, ql, q2, q3, and q4 of 1,2, 3, and 4 reflections, to the efficiency q, are shown separately. TABLE7.2 Efficiency‘1and contributions to q from 1,2, reflections,for the plane reflector test problem of fig. 7.9. For the results shown, BC was divided into 12 (unequal) segments, CC’ into 36, and A‘C’ into 60. The semicircular absorber profile AB may be treated as a single segment.

...

‘11

‘12

v3

v4

rl

0.1852 0.1850

0.0993 0.0984

0.0033 0.0034

0.0000 0.0000

0.9465 0.9455

~

calculated exact

NONSPECULAR REFLECTORS

C

20 1

C’

Fig. 7.9. The equivalentunit cell for the plane reflector test problem. The semicircle of unit radius AB is a portion of absorber surface, CC’ is a portion of plane reflector, and BC and A’C’ are notional plane reflectors. BC = 0.3030 and CC’ = AA‘ = 0.4726.

The method has been used to calculate the performance of tubular solar collector arrays with realistic absorption and reflection laws (DERRICK, BASSETT and MILLS [ 19861). Conventional ray-tracking methods (MILLS, BASSETTand DERRICK [ 19861)took about five times longer to give comparable accuracy. On the other hand, storage of the transfer coefficients PIjkuses much more high-speed memory than is required for ray-tracing. The problem of determining optimal reflector profiles, with partly specular reflectors, has so far proved intractable.

7.5. AN UPPER BOUND ON COLLECTOR PERFORMANCE

Let the light incident on the entrance aperture of a collector be uniform and fully Lambertian. At any absorbing surface let the fraction ps of the light falling on it be reflected specularly,the fraction pd diffusely, and the remaining fraction a = 1 - ps - pd being absorbed; ps and pd are assumed uniform over the absorbing surface and independent of the angle of incidence. The fraction aeff of the incident light absorbed satisfies the inequality

(7.30) where C is the ratio of absorber area to aperture area.

202

NONIMAGING OPTICS FOR FLUX CONCENTRATION

WI,8 7

We first obtain inequality (7.30) for ps = 0 (OGALLAGHER, RABL,WINSTON and MCINTIRE[ 19801, BASSETTand DERRICK [ 1980bl). The surface of the absorbing body is divided into N portions of equal area s, N being taken sufficiently large so that the individual portions may be assumed to be uniformly illuminated. If S , is the area of the entrance aperture (assumed plane), and it is uniformly illuminated in the whole inward hemisphere with light of radiance B, then the power entering is xB S,. Let the fraction of the incident power which strikes the ith portion of the absorber without first being reflected from the absorber be FA,.Let the fraction of power leaving portion i which strikes portion j without intervening reflection from the absorber be Fu. Let FA be the N-rowed vector whose elements are the FAI,and let F be the N x N matrix whose elements are F;, then the fraction a,* of the power entering the system which is absorbed is given by ae, =

aFz

1

-1 , 1 - pF

(7.3 1)

where 1 is the N-rowed column vector all of whose elements are 1, and p = 1 - a. Let FiAbe the fraction of the power leaving portion i which strikes the aperture A without any intervening reflection from the absorber; then S F ~ A= S A F A i

.

(7.32)

Furthermore, F1.J . = F.J i

(7.33)

and F1+ -SA FA

= 1.

S

(7.34)

Using the symmetry relations (7.32) and (7.33) and the sum relation (7.34), aea may be re-expressed in the form ae, = - '1.

S, N

1-F 1, a + p(1- F)

(7.35)

where S is the area of the absorber. We now introduce a matrix inequality (BASSETT and DERRICK [ 19811) O'$(X) u < $(uTXu),

(7.36)

111, § 71

NONSPECULAR REFLECTORS

203

whereX is any real symmetricN x N matrix, u is any N-rowed vector normalized so that uTu = 1 , and the second derivative $"(x) satisfies $"(x) < 0 in the domain of x in which the eigenvalues of X lie. In inequality (7.36) we set X = 1 - F, u = N - ' / * l , $ ( x ) = x / ( a + px). This gives 1-F 1-f 1< a+p(l-F) a+p(l-f)'

1

- lT

N

(7.37)

where f

=

1 lTFl N -

If in eq. (7.35) we set p

=

0, we obtain

Since aeff< 1 , 1 - f < - SA S Since x/(a + p x ) is a monotonic nondecreasing function of x ,

(7.38) From inequalities (7.37) and (7.38), -11 N

1-F -~ 1 a+p(l- F)

<

sA/s

a+ps,/s'

Substitution into expression (7.35) for aeRyields the inequality aetT 4

a

a

+ ( 1 - a)C

(7.39)

If ps # 0, the absorber surface may be modelled as a collection of small patches of two sorts, distributed uniformly on a macroscopic scale, some being diffusely reflecting absorbers and the remainder being perfect specular reflectors. Inequality (7.30) is then easily obtained from the special form (7.39) (BASSETT.and DERRICK [ 19811).

204

WI,8 8

NONlMAGlNG OPTICS FOR FLUX CONCENTRATION

+

The equality sign in the matrix inequality (7.36) is attainable if is linear and in some other special cases. An example of a reflector system which realizes the equality, that is, the upper bound, is provided by a perfect specular involute reflector formed for a cylindrical absorber. Light not absorbed is assumed to be reflected with Lambertian angular distribution. For such a system, C = 1 and ps = 0 and the fraction of incident light absorbed is simply or, the absorptivity of the absorber surface. An effective light trap, with a specularly reflecting absorbing surface, is described by BRENEMAN[ 19811.

8 8. The geometrical vector flux formalism The edge-ray principle and the associated idea of maximum slope have led to many useful concentrator designs, but they do not show how the jumble of rays from multiple reflections inside a concentrator eventually yields a Lambertian or near-Lambertian output. The geometrical vector flux formalism was developed in an attempt to understand this process (WINSTONand WELFORD[ 1979a,b]).

8.1. DEFINITION OF THE VECTOR FLUX

Consider any lossless optical system in which the radiation has originated in a Lambertian source. The Ctendue (I 2.1) is conserved along any ray in the system. Thus considering any one ray, we can write the differential element of Ctendue attached to this ray at P as d U = dpx dp, dx dy ,

(8.1)

where as usual the coordinate axes are chosen in arbitrary directions. This element of etendue can be written where dJ,

=

dp, dp,

.

Consider the quantity J, defined by J, =

jj

dPx dP,

3

W8 81

GEOMETRICAL VECTOR FLUX FORMALISM

205

where the integration is over the whole direction cosine space; it can be seen that J, is proportional to the flux per unit area crossing the x y plane at P. Similarly we can define Jx and J,,:

These equations give the flux densities across the yz and zx planes at P. The proportionality factor involved could be chosen to make the flux densities have the dimensions of power per unit area, but for our purposes it will turn out to be more convenient to have no dimensions attached to these quantities. If we take J,, J,, and J, together, they look like components of a vector J : J = ( J x , J,,. J,)

*

(8.6)

The fact that etendue is independent of the rectangular coordinate system used to d e h e it shows that J is indeed a vector quantity, and with the scaling to be explained next we define it as the geometrical vectorflux. In photometry J, is proportional to the illumination on a surface lying in the xy plane, and quantities like J have been defined in photometry (GERSHUN [ 19391, ZIJL[ 19511, BORN and WOLF [ 19751); in addition, a similar quantity is defined and its vector property pointed out in radiative transfer theory [ 1950]), where it is called the netflux. (CHANDRASEKHAR The scaling of the vector flux is defined by carrying out the integration in eqs. (8.4) and (8.5) at the surface of a Lambertian radiator in a space of unit refractive index. Taking the z-axis normal to the surface, we find

J

=

(O,O, 2 ~ ) .

(8.7)

Similarly for two-dimensional geometry, we obtain

J

=

(0, 0, 2 ) .

The vector flux has zero divergence in a region free from sources or attenuators; this follows from the constancy of etendue along rays and the fact that rays have no ending in such a region. However, the vector flux does not have zero curl, as follows from certain examples given in later sections.

8.2. SOME TWO-DIMENSIONAL EXAMPLES

Consider first a two-dimensional strip as in fig. 8.1 ;that is, the strip extends infinitely in the direction perpendicular to the paper. The direct calculation of

206

NONIMAGING OPTICS FOR FLUX CONCENTRATION

Fig. 8. I Calculating the geometrical vector flux from a strip source.

J according to the definition would be somewhat laborious, but we make use of the fact that all Lambertian radiators which subtend the same solid angle at a point (in the strict sense that the solid angle is bounded by the same cone) give the same vector flux at that point. Then to find the vector flux at P in fig. 8.1, we may replace the strip AA' by the strip AA", which is perpendicular to the bisector of the angle 0 subtended by AA' at P. It then follows immediately that

IJI

=

2 sinf0,

- /

(8.9)

and the direction of J is by symmetry along the bisector, as indicated in the figure. Unit of y&r

7 -

Fig. 8.2. Flow lines and levels of IJI from a strip source.

I K 8 81

GEOMETRICAL VECTOR FLUX FORMALISM

207

Then from elementary geometry J is constant along circles with AA' as a chord, and the lines of flow of J are confocal hyperbolas with A and A' as foci. Figure 8.2 shows the flow lines and levels of J for the strip. It can be seen (and it can be confirmed analytically) that the flow lines are not orthogonal to the levels, so that J is not the gradient of a scalar potential and curl J is not zero. The equation of the hyperbolas in fig. 8.2 is y=ka

(

l + -c 2

Ta2)"'

,

(8.10)

where 2c is the width of the strip, that is, the distance between the foci, and

Fig. 8.3. Flow lines and levels of IJI from a half-plane.

208

NONIMAGING OPTICS FOR FLUX CONCENTRATION

[IKI 8

a is the distance from the center at which a given hyperbola starts. The radii of the level circles are given by (8.11)

Figure 8.3 shows a semi-infinite Lambertian strip radiating to the right; by similar reasoning to the preceding or by treating it as a limiting case of the strip in fig. 8.2, it can be shown that the flow lines are confocal parabolas with the focus at the end A of the strip, and the levels of J are straight lines radiating from A. A final two-dimensional example is a Lambertian wedge. Figure 8.4 shows a 60” wedge, and in this case there are different regions of the flux field to consider. For points from which only one face of the wedge is seen, the effect is the same as for the semi-infinite strip of fig. 8.3, but where both faces are visible, it is equivalent to an infinite plane Lambertian radiator and J i s constant in this region, as shown.

1.8

Fig. 8.4. Flow lines and levels of IJI from a 60” wedge.

111,s 81

GEOMETRICAL VECTOR FLUX FORMALISM

209

8.3. THREE-DIMENSIONAL EXAMPLE: THE UNIFORM DISC

We consider a circular aperture in a black-body cavity, that is, a circular disc, as a Lambertian radiator. The components of J can be written down by means of eqs. (8.4) and (8.5), but the integrations are very laborious. It is more convenient to use a transformation of the surface integrals to line integrals, which was given by SPARROWand CESS[ 19781; this applies to Lambertian radiators of any shape. In fig. 8.5 dso is a line element on the boundary of the radiator with position vector r,; the point P at which the vector flux is to be calculated has position vector r. Then the component J,, of flux in direction n at P is given by (8.12)

For a disc of radius c with z-axisperpendicular to the surface the components of J are found to be J = - -

c’

+ r’ + ’z

+ r’ + z’)’ - 4c2r2}’/’ c’ + r’ - z2 + I}, {(c’ + r’ + z * ) ~ 4~’r’}’/~ {(c’

(8.13)

where r is the radial coordinate. The angle to the z-axis of the flow lines is arctan (Jr/Jz);it turns out that the flow lines are the same hyperbolas as for the strip in fig. 8.2. Figure 8.6 shows the flow lines and the surfaces of constant

.

Fig. 8.5. Line integral for calculating J from a Lambertian source of any shape.

210

NONIMAGING OPTICS FOR FLUX CONCENTRATION

Fig. 8.6. Flow lines and levels of I J I from a Lambertian disc of diameter 2c.

IJI; the latter are not the same as for the two-dimensional strip, as might be expected, since there are contributions to J from skew rays to be considered.

8.4. USES OF THE VECTOR FLUX FORMALISM IN CONCENTRATOR DESIGN

Since the vector flux field is solenoidal, there is a double infinity of surfaces which are not crossed by flow lines. Thus a small mirror element lying in one of these surfaces does not disturb the flux field locally, since there is no normal component of J . It may disturb the field at some distance, but even this may not happen in certain conditions. The general requirement is that every ray striking the mirror from one side in a certain direction should be paired by a ray striking the other side in a direction corresponding to reflection, as in fig. 8.7. This could happen in the simplest case if Lambertian sources were arranged symmetrically on either side of the plane in which the mirror element lies, but this is not essential. Let cylindrical mirror surfaces be placed along a pair of hyperbolic flow lines of the two-dimensional strip, as in fig. 8.8; then by an elementary property of conic sections any ray directed at one of the foci, that is, the edge of the strip, will be reflected towards the other focus, which continues until in the limit the ray emerges at B or B’. Clearly any ray directed at a point between the foci must

GEOMETRICAL VECTOR FLUX FORMALISM

21 1

Fig. 8.7. Placing a mirror element in a vector flux field.

emerge after a finite number of reflections somewhere between B and B‘. This arrangement fulfils the condition of pairing of rays proposed earlier; thus we have a 2D concentrator with a virtual entry aperture and a real exit aperture, and it is ideal in the sense that all rays aimed at the entry aperture (and at the insides of the mirror surfaces) must emerge from the exit aperture. The concentration ratio is AA’/BB‘, which can be shown to be equal to l/sin 0, where 8 is the semi-anglebetween the asymptotes ofthe hyperbolas. Figure 8.8 is drawn with symmetrical hyperbolas, but this is not essential: Any pair of confocal hyperbolas would form such a concentrator. Thus if we take the hyperbola from B and that from any other point B” between A and B, we obtain a concentrator with concentration ratio AA’IBB‘’, which is equal to l/(sine - sine,). In the same way we can consider the three-dimensionalcase of the field from a Lambertian disc radiator. The inside surface of a hyperboloid of revolution (fig. 8.6) could be used as a concentrator with the disc surface defining the

Fig. 8.8. The flow line concentrator, with virtual entry aperture AA‘ and real exit aperture BB’.

212

NONIMAGING OPTICS FOR FLUX CONCENTRATION

[IIL I 8

virtual entry aperture and the part of the disc inside the hyperboloid as the exit aperture. It is not obvious that the condition of detailed balance of rays is fulfilled, but this has been proved (WINSTONand WELFORD[ 1979a1). Thus again all rays directed at the original disc emerge after reflections inside the portion of the disc cut out by the hyperboloid. The concentration ratio is l/(sin 8)2, where 8 is the asymptote semi-angle. Next, we consider a two-dimensional case, the truncated wedge as in fig. 8.9. To work out the flow lines, we divide the space into four regions, A, B, C and D as shown in the top half of the diagram, and we find (WINSTON and WELFORD[ 1979b1) that putting (JI = 2 as usual on the surface the flow lines in region C are parabolas with focus at the opposite wedge comer; in region B they are hyperbolas with foci at the two corners, in region A they are parabolas with focus at the nearer comer, and in region D they are straight lines showing that J i s constant in that region. The diagram shows a 60" wedge angle and for this case IJJin region A is 1. Detailed balance of the rays holds everywhere in this two-dimensionalfield so that any pair of flow lines could be used to define a concentrator. In particular, if we take the pair starting from

\

A

I

I

D

'1.0 1.95 1.6

1.4

Fig. 8.9. Flow lines and levels of IJ I from a truncated wedge; the CPC profile is generated by the Bow lines from the comers of the wedge.

111, § 81

GEOMETRICAL VECTOR FLUX FORMALISM

213

the two corners of the wedge, we see that the original compound parabolic concentrator is obtained again, but from entirely different reasoning from that in 8 3. Other cases treated in more detail by WINSTONand WELFORD[ 1979bl are the e,/O, concentrator and the light cone with straight generators.

8.5. OTHER APPROACHES TO THE VECTOR FLUX

BARNETT[ 1979401 calculated the vector flux flow lines and contours of constant intensity inside a 2D CEC (compound elliptical concentrator, see 8 3), by a different route from that described in § 8.4. BARNETT[ 1980al has pointed out that in ideal concentrators like the 2D CEC and CPC the extreme rays anywhere inside the concentrator are rays from the edge of the source, this being a consequence of the edge-ray principle (5 3). From this information the general form of the phase-space volume anywhere along the concentrator can be deduced. Furthermore, in an ideal concentrator no rays are lost, and so the phase-space volume remains continuous as it propagates; thus at any point the

Fig. 8.10. Flow lines and contours in a 2D compound elliptical concentrator.

214

NONIMAGING OPTICS FOR FLUX CONCENTRATION

[III, 8 9

direction of the flow line is the bisector of the angle between the two extreme rays, and the magnitude of the flux is twice the sine of half the angle between the extreme rays. In this way fig. 8.10 was obtained, showing flow lines and contours of constant JJI.In a further paper BARNETT[ 1980bl carried out a similar calculation for a 2D e,/O, concentrator, which is a slightly more complicated case. WINSTONand NING[ 19861 described a derivation of the geometrical vector flux starting from Maxwell's equations; they showed that the quantity so derived was conserved and had all the other properties associated with the vector flux as defined in 8.1.

0 9. Some physical aspects of nonimaging optics In this section we discuss the attempts that have been made to extend the ideas of nonimaging optics beyond the geometrical optics model. It must be admitted that the progress here has been patchy and the main results, as will be seen in 3 9.2, amount to replacing equalities concerning radiance, etendue and specific intensity by inequalities. This can be seen from the fact that any wave motion is scattered in all directions by a variation in potential, that is, an opaque obstruction or a variation in refractive index, and although the back scatter is very small in many practical cases, it must exist; thus not all the flux impinging on the entry aperture of a concentrator can possibly get through to the exit aperture in any physical optics model. In 5 9.1 we note some discussion of physical optics effects with nonimaging systems of dimensions comparable to the wavelength of the radiation, and in § 9.2 some general results applicable to systems of any size are given.

9.1. VERY SMALL SYSTEMS

In infrared radiometry it is desirable to use as small a detector as possible to minimize noise. This is particularly important in infrared astronomical observations. HARPER,HILDEBRAND,STIENINGand WINSTON[ 19761, working at millimetric wavelengths and long focal ratios, had point spread functions (diffraction discs) several millimetres across, and they used a CPC to reduce the size of the light patch corresponding to a single point spread function, thus permitting the use of a smaller detector to respond to the radiation from the point spread function. The detector was actually placed in

111, I 91

SOME PHYSICAL ASPECTS OF NONIMAGING OPTICS

215

a cooled reflecting cavity fed from the exit aperture of the CPC; this system was called a heat trap by the authors. Thus with the 5m Hale telescope at wavelength 1.4 mm, HARPERand colleagues used a x 64 concentrator at the prime focus (f/4) with the entry aperture slightly larger than the central maximum of the point spread function; the advantage was that the small exit aperture made it possible to use a very small cavity and detector and still collect nearly all the flux from the point spread function. This provided a gain of a factor of 2 over the conventional method using an f/l field lens. With such a system, where the entry aperture is only about 10 wavelengths across, it would be unreasonable to expect the concentration to be predicted exactly by geometrical optics. In fact, for the examplejust quoted the theoretical gain would have been about 4, and although part of the difference can be ascribed to reflectivity losses in the cavity and the concentrator, some must also result from being able to use a cavity because of the concentration achieved. Systems for collecting and concentrating the light from (approximately) the area of one point spread function are called field optics in astronomy. HILDEBRAND and WINSTON[ 19821 calculated the losses involved with field optics due to the finite diameter of the field lens or of the entry aperture of the CPC or other concentrator; their results suggest that little flux is lost when the field optics aperture has a diameter equal to that of the first dark ring of the point spread function. Thus some of the loss in the heat trap is probably caused by effects that have not yet been evaluated involving the small dimensions relative to the wavelength of the light.

9.2. RIGOROUS LIMITS TO CONCENTRATION IN PHYSICAL OPTICS

The limits to concentration implied by the conservation of Ctendue, radiance and specific intensity provide absolute limits on the performance of passive optical devices, in the domain of geometrical optics. In 5 9.1 we described some devices which, although they are used at wavelengths comparable to their own dimensions, approach these limits. The limits established in geometrical optics cannot be beaten by using wave effects. As discussed in $ 1, the conservation of Ctendue may be applied in the presence of diffraction, provided rays may be accurately traced through the optical system. However, in general, the conservation laws (of Ctendue, radiance, or specific intensity) give way, in the wave domain, to inequalities. We consider here a number of inequalities implying limits to concentration that hold rigorously in electromagnetic theory.

c! m

TABLE9.1

Summary of various limits on the degree to which light can be concentrated by a passive optical system, in the domain of physical optics. upper bound on

In terms of incident (from vacuum)

conditions

inequality"

ReEb

1

energy density Uv at point 0

power W

(a)kR%- 1 (b) light passes only once through neighbourhood of 0

Uv < w k 2 / 3 x c

PI

2

energy density U, at point 0

specific intensity I (assumed uniform in range bk)

bkR 4 1, bk/k 6 1 (b) incident light describable classically as incoherent superposition of plane waves (a)

energy density U, at point 0

degeneracy parameter or mean occupation number ( N ) per mode (assumed uniform in frequency range)

6kR4 1,

4

specific intensity I '

specific intensity I (assumed uniform in frequency range)

I , I' well defined at points of comparison

5

radiance B'

radiance B

B, B' well defined, quasimonochromatic light

3

bklk-4 1

16n31 V\-

1 1 9

21

WkR)

s9 0

a

n

Ex 111

11.21

M

W

etendue U'

&endue U

U,U' well defined

total electromagnetic energy UR in resonant cavity

degeneracy parameter or mean occupation number ( N ) per mode (assumed uniform in frequency range)

frequency range covers just g modes with the same frequency

power per unit-frequency interval

power per unit-frequency interval

ingoing channels mutually incoherent

per channel,

d W' dw

~

dW per channel, dw

power per channel W'

power per channel W

ingoing channels mutually incoherent, light quasimonochromatic

number of outgoing channels C'

number of ingoing channels C

ingoing channels mutually incoherent

degeneracy parameter or mean occupation number (N')per mode

degeneracy parameter or mean occupation number ( N ) per mode

differs from 1 by less than 3% for v > 271 and F tends to 1 as v tends to infinity. Explicitly: + (vjo(v))2 - (j,(v)Y - 2vj,(v)j,(v), where the j, are standard spherical Bessel functions. For the other symbols, k is the wavenumber, w the circular frequency, and R the distance from point 0 to the nearest point at which the refractive index differs from 1. References [ l ] BASSETT[1985b,c]; [2] BASSETT[1984]; [3] WINSTONand WELFORD[1982]; [4] B A S S E[1987]. ~

a F(v)

F ( v ) = (vj,(v)y

218

NONIMAGING OPTICS FOR FLUX CONCENTRATION

[III, B 9

In ordinary (i.e., imaging) optics “concentration” suggests “focussingyy.It may be shown (BASSETT[ 19861) that if light of power W and wave number k is concentrated by a passive device (such as a lens or mirror), the stationary energy density U , attainable at any point 0 satisfies Wkz 3 ‘IIC

uv<-,

provided (1) the distance R from 0 to the nearest object is many wavelengths, kR B 1, and (2)after the light has passed through the neighbourhood of 0 any light which is subsequently reflected or scattered back to 0 may be neglected. Inequality (9.1) can be derived with the help of a multipole expansion of the stationary electromagnetic field about the point 0. The derivation depends on three propositions: first, contributions to U , and to the incoming power W are additive, that is, they are expressible as sums of contributions from the individual multipole terms; second, only the dipole terms contribute to U v ;and third, for each dipole term the ratio Uv/W is k2/3nc. In order to apply inequality (9.1) to a given physical system, it is necessary to make a clear distinction between incoming and outgoing light, at least near the boundary of the region of radius R about 0. (This is the reason for proviso (1) earlier, namely, the distance R from 0 to the nearest object is many wavelengths, kR % 1.) The distinction can be made in a stationary situation, in which light approaches 0 from a single converging lens, since the incident light then fills appreciably less than a hemisphere and is sufficiently separated in space from the outgoing light. In the same circumstances pure dipole radiation cannot be realized and the upper bound implies by the equality (9.1) cannot be attained. It seems fair to conjecture that half the upper bound (i.e., k ZW/6nc) might be nearly attainable. On the other hand, if light approaches from all directions, then after passing through the focus it must encounter some obstacle and must be at least partly reflected or scattered back to the focus, violating premise (2) of inequality (9.1) that light passes only once through the focus. However, premise (1) implies the possibility of forming a wave packet (with 6 k / k d l), which passes entirely through the focal region before any secondary reflected light reaches that region. Then a clear distinction can again be made between incoming and outgoing light, and inequality (9.1) may be applied at the initial passage of the pulse through the focal region. Pure dipole radiation is possible, and the upper bound Wk2/3nc is attainable in principle. These results are expressed briefly in the first row of table 9.1. The form of the dipole field near the origin shows that the energy density falls away from its peak in a distance roughly equal to the wavelength. With pure

111, § 91

SOME PHYSICAL ASPECTS OF NONIMAGING OPTICS

219

dipole radiation the equality sign in relation (9.1) is realized and large energy densities could result. For example, at wavelength 500 nm a pulse of 50 MW gives a peak energy density of about 3 x 10l2J/m3, which corresponds to that of blackbody radiation at a temperature a little below lo7 K. If premise (2) is removed, the nature of the limit to concentration changes completely. Concentration in a resonant cavity is then permitted, with many passages of the radiation through the point of interest. A periodic disturbance can eventually build up a large response in a resonant system, even though interacting weakly with it. Paradoxically, the weaker the interaction is (between the resonant system and the outside world, of which the periodic disturbance is a part), the greater the ultimate steady-state response. The reason for this is that a large response requires a high Q-value, that is, a narrow resonance, which in turn requires the interaction to be weak. In the present case the maximum concentration is achieved if the incident light is tuned to the resonance, so that it is not surprising to find an upper bound on the energy density, for given incident power, which is inversely proportional to the incident frequency range. If the incident light can be described as an incoherent superposition of plane waves, and if it is quasimonochromatic, the property determining this limit to concentration is, in fact, the radiance per unit frequency interval or specific intensity, I. For kR > 211 the result is

A more precise expression of this inequality is given in the second row of table 9.1. The incident light may not have a well-defined specific intensity, a quantity associated, by definition, with plane waves (and geometrical optics). More generally, it may be possible to characterize it by a mean occupation number (or “degeneracy parameter”) (N), in appropriately chosen modes (stationary solutions of Maxwell’s equations in vacuo). Apart from plane waves these might be spherical waves (with specified radial wave number and angular dependence) or the field from a single mode laser. In the last case

where W is the power, z is the coherence time, and o is the angular frequency (MANDEL[ 19611). When I i s well defined, it may be expressed in terms of ( N ) by (BASSETT[ 1985b1) I= (N)

h o3 ~

(2 11)4c2 ’

(9.4)

220

NONlMAGlNG OITICS FOR FLUX CONCENTRATION

[III, 8 9

In terms of ( N ) a more general version of inequality (9.1) may be written

U, 5 ( N ) hck3/nR

(9.5)

as in the third row of table 9.1. A basic inequality, in terms of which inequalities (9.1) and (9.5) can be derived, is expressed by the statement that no occupation number can be raised (by a passive system) above the maxirnum occupation number characterizing the incident ensemble, or ( N ' ) is the ensemble average occupation number of any normalized onephoton state, not necessarily stationary. If there are two or more incident streams or channels (in the sense of scattering theory), these are assumed to be mutually incoherent. Whereas inequality (9.5) is expressed in the language of quantum optics, the conclusionsdrawn from it here, such as inequalities (9.7) and (9.14), lie within the domain of classical electromagnetic theory. The photons to which inequality (9.5) refers are implicitly associated with a decomposition of the classical electromagneticfield, in the presence of a transparent medium, into harmonic oscillators (BASSETT [ 1985b, p. 15871). As discussed [ 1970, pp. 58 and 108-1 lo], classical and quantum optics give by GLAUBER the same results for the averagevalues of quantities linear in the field amplitudes. Thus the quantum methods used here do little more than provide a convenient short-cut to classical results. The simplest optical application which goes beyond geometrical optics is obtained by assuming that geometrical optics prevails at the two points of comparison, but interference and diffraction effects may occur as the light passes from one to the other channel. In such a case the occupation numbers ( N ) and ( N ' ) may be replaced by the corresponding specific intensities, by eq. (9.4), and the inequality becomes

I'

< I,,

,

(9.7)

which can be briefly expressed by the statement that the specific intensity cannot be increased by a passive optical system (see fourth row of table 9.1). In any sufficiently narrow range of frequencies Av, a specific intensity I entails a radiance B = IAv. Hence, in any sufficiently narrow frequency range inequality (9.7) implies a corresponding inequality in the radiance, B'

< B,,,

(9.8)

(see fifth row of table 9.1). If the input radiance B is uniform and the input &endue is U,the input power

111, I 91

SOME PHYSICAL ASPECTS OF NONIMAGING OPTICS

22 1

is UB. If U' is the output etendue, inequality (9.8) implies that the output power cannot exceed U'B. Thus if no power is turned back, U'

u

(9.9)

(see sixth row of table 9.1). In other words, the whole input power cannot be accommodated at the exit aperture in less than the input Ctendue: The geometrical optics limit to concentration cannot be exceeded. This result was first obtained, with the scalar wave approximation, from the linear unitary property of loss-free propagation through an optical system (WINSTONand WELFORD[ 19821). This property has its origin in the linear unitary character of time evolution, which is exact and general, and which also underlies inequality (9.6). The conclusion embodied in inequalities (9.7)-(9.9) can also be obtained from the second law of thermodynamics (RIES [ 19821). Another simple application of the basic inequality (9.6)gives an upper bound on the energy in a cavity resonator exposed to quasimonochromatic radiation. In this case the state to which ( N ' ) refers is a cavity mode. If the range of frequencies of the incident light is assumed to cover just one g-fold degenerate resonant mode of angular frequency o,the energy UR in the cavity satisfies uR

g( N ,

muho

(9.10)

(see the seventh row of table 9.1). A plane Fabry-Perot resonator at (nearly) normal incidence provides an effectively one-dimensional soluble problem in classical electromagnetic theory. It may be shown that in this case the left and right sides of relation (9.8) can approach equality as the width of the resonance tends to zero. Thus at least in some cases the equality sign in (9.6) and in the derived inequalities is attainable in principle, which suggests the possibility of obtaining very high energy densities by the resonance mechanism. If we set ( N ) = 10" (LOUDON [1983]), 2n/k = 10-6m, R = 2n/k, the highest energy density allowed by inequality (9.5) is about lOI9J/m3, an energy density attained by blackbody radiation at a temperature of about 10' K, a typical central stellar temperature. Equation (9.3) provides an approximate value of ( N ) in terms of classical quantities. An exact classical expression for (N) (eq. 9.13) allows the three inequalities (9.7)-(9.9) to be generalized to include those cases in which the light requires a physical-optics description before or after passing through the optical system - for example, if the incident light comes from a single mode laser, or if the light enters or leaves by a single-mode fibre.

222

NONIMAGING OPTICS FOR FLUX CONCENTRATION

[III, 8 9

We borrow the concept “channel” from scattering theory, and we also use the closely related notion of “number of degrees of freedom” of light (GABOR [ 19611). The light propagating at a single angular frequency o in a single-mode, single-polarization fibre is completely defined classically by two parameters, conveniently taken as amplitude and phase. Accordingly, it is said to have two degrees of freedom, F = 2. It corresponds to a single channel in the sense of scattering theory, C = 1. Similarly, the light from a single-mode, singlepolarization laser has C = 1, F = 2. An ordinary single-mode fibre with two propagating polarizations (at a given frequency) has C = 2, F = 4. A multimode fibre has correspondingly larger C and F ( = 2C). As the diameter of the fibre increases, the geometrical optics limit is approached. In this limit there is a simple relation between the number of channels C and the &endue U: C = 2- u (9.11) A2 ’ where A is the free space wavelength, a result obtained essentially by GABOR [ 19611. For example, for limited Lambertian light, occupying a cone of directions with half angle t( and axis normal to a plane aperture of area A, 2U 211An2 sin’u c=-= 3 (9.12) A2 A2 where n is the refractive index of the medium in which the experiment takes place. A steady flux of light characterized by a degeneracy parameter ( N ) may also be characterized by the power per unit angular frequency interval per channel d Wldo, which is related to ( N ) by dW 1 (N)hU. (9.13) d o 211 Consider, for example, a straight, uniform waveguide and a particular polarization channel for which the value of the propagation constant is fl = fl(o).The modes are made discrete by the device of a cyclic boundary condition. In a cycle of length L the number of modes in dfl is -=-

dS -Ld f l = - L -dm 211 211 d o Hence, if the mean number of photons per mode is (N(o)),the energy propagating per unit length in d o is

1 dfl o ( N ( w ) ) h a . 211 d o

- -d

111, § 91

SOME PHYSICAL ASPECTS OF NONIMAGING OPTICS

223

The power d W in d w is obtained by multiplying by the group velocity dw/d/l (SNYDERand LOVE[ 1983]), which gives eq. (9.13). The same equation may be shown to hold for light not confined to a waveguide, for example, that from a single-transverse-mode, single-polarization laser, again with the help of a device which makes the modes discrete: The light is imagined to lie between and LI the mirrors of a suitably constructed open resonator (KOGELNIK [ 19661). By means of eq. (9.3), inequality (9.6) may be rewritten in terms of the classical quantity d W/dw, the power per unit angular frequency interval per channel: (9.14) Hence, in any infinitesimal interval d a we have d W‘ ingly, for quasimonochromatic light,

W’ G

w,,

;

< d W,,

and, accord(9.15)

that is, the power in any outgoing channel is bounded by the largest power in any input channel, these input channels being assumed to be mutually incoherent. The last inequality also implies that the light in C mutually incoherent ingoing channels cannot be accommodated in less than C outgoing channels; that is,

C‘ 3 c .

(9.16)

Inequality (9.14) is the required generalization of (9.7), and similarly (9.15) is the counterpart of (9.8). Thus the power per unit angular frequency interval per channel, d W/dw, is a natural extension of the specific intensity I into the wave domain. In the geometrical optics limit, by eqs. (9.4) and (9.13), we have

do

(9.17)

where l is the free space wavelength. The power per channel, W, similarly extends the notion of radiance into the wave domain. In the geometrical optics limit W is related to the radiance B by (9.18) The corresponding generalization into the wave domain of the geometrical

224

NONIMAGING OPTICS FOR FLUX CONCENTRATlON

u

",f-L:::

WB

Fig. 9.1. Six-port coupler formed from three single-mode fibres.

optics concept of Ctendue is simply the number of channels C, and this is related to the Ctendue U in the geometrical optics limit by eq. (9.11). Inequalities (9.14)-(9.16) imply constraints on the steady power flows into and out of a passive optical network (GOODMAN[ 19851, BASSETT [ 19871). Consider, for example, a six-port coupler formed from single-mode fibres (fig. 9.1). If the light is quasimonochromatic and the three inputs with powers W,, W, and W, are mutually incoherent, then by (9.15) no output power W,! can exceed the largest of W,, W, and W,: W,! < max(Wl, W,, W,) Again, inequality (9.15) implies a precise limit on the fraction of the power in a multimode fibre (fully illuminated by an incoherent source), which can be accommodated in a single-mode fibre. For example, for a step index fibre of core diameter 50pm and numerical aperture 0.25, and light of wavelength 0.85 pm, the number of channels is, by eq. (9.12) C=

27r x ~ ( 2 5x 10-6)2 (0.85 x 10-6)2

z 1000.

In a single-mode fibre propagating two polarization states the number of channels is C' = 2. Thus the fraction of power which can propagate into the single-modefibre from the step index fibre is at most C ' / C = 1/500. In contrast to this, all the light from a single-modelaser can, in principle, be accommodated in a single mode fibre. Constraints equivalent to those described here have been derived in microwave network theory, from the unitarity of the scattering matrix (MONTGOMERY, DICKEand PURCELL[ 19481).

1111

REFERENCES

225

References BARANOV, V. K., 1965, Opt.-Mekh. Prom.-st. 6, 1-5. BARNETT, M. E., 1979-1980, Optik 54,429-432. BARNETT, M. E., 1980a, Optik 55, 343-350. BARNETT, M. E., 1980b. Optik 57, 391-400. BASSETT,I. M., 1984, Opt. Acta 31, 1211-1232. BASSETT,1. M., 1985a, Phys. Rev. Lett. 54, 2014-2017. BASSETT,I. M., 1985b, Opt. Acta 32, 1577-1592. BASSETT,I. M., 1986, Opt. Acta 33, 279-286. BASSETT,I. M., 1987, in: Proc. 14th Congr. Int. Commission for Optics, Quebec, SPIE 813, 543-544. BASSETT,I. M., and G.H. DERRICK,1980a, Opt. Acta 27. 215-230. BASSETT,I. M., and G.H. DERRICK,1980b. Opt. Acta 27, 1361-1366. BASSETT,I. M., and G.H. DERRICK,1981, Opt. Acta 28, 57-63. BASSETT,I. M., and R. WINSTON,1984, Opt. Acta 31,499-505. BODEM,F., 1978, Opt. & Laser Techno]. 10, 89-96. BOEGEHOLD, H., 1893, in: Grundzuge der Theorie der optischen Instrumente, eds S. Czapski and D. Eppenstein, 4th Ed. (Leipzig) ch. IV, p. 1924. BORN,M., and E. WOLF, 1975, Principles of Optics, 4th Ed. (Pergamon Press, New York). BRENEMAN, E. J., 1981, Appl. Opt. 20, 11 18-1 119. CHANDRASEKHAR. S., 1950, Radiative Transfer (Oxford Univ. Press, Oxford). COLLARES-PEREIRA, M., A. RABLand R. WINSTON,1977. Appl. Opt. 16,2677-2683. G. H., 1985, Opt. Acta 32, 39-60. DERRICK, DERRICK, G.H., and I. M. BASSETT,1985a, Opt. Acta 32,937-960. DERRICK, G. H., and I. M. BASSETT,1985b, Opt. Acta 32, 313-328. DERRICK, G.H., I. M. BASSETT and D. R. MILLS,1983, Proc. Solar World Energy Congr., Perth, ed. S.V.Szokolay (Pergamon Press, New York) p. 847. G. H., I. M. BASSETTand D. R. MILLS, 1986, Sol. Energy 37, 195-203. DERRICK, GABOR,D., 1961, in: Progress in Optics, Vol. 1, ed. E. Wolf (North-Holland, Amsterdam) pp. 111-153. GERSHUN, A., 1939, J. Math. & Phys. Sci. XVIII, 51-151. GLAUBER, R. J., 1970, in: Quantum Optics, eds S. M. Kay and A. Maitland (Academic Press, New York). J. W., 1985, Opt. Acta 32, 1489-1496. GOODMAN, HARPER,D. A., R. H. HILDEBRAND, R. STIENING and R. WINSTON,1976, Appl. Opt. 15,53-60. HILDEBRAND, R. H., and R. WINSTON,1982, Appl. Opt. 21, 1844-1846. HINTERBERGER, H., and R. WINSTON,1966, Rev. Sci. Instrum. 37, 1094-1095. HOTTEL,H., 1954, in: Heat Transmission, 3rd Ed., ed. W.H. McAdams (McGraw-Hill, New York). JANNSON, I., and R. WINSTON,1986, J. Opt. SOC.Am. A 3, 1-8. KOGELNIK, H., and T. LI, 1966, Proc. IEEE 54, 1312-1329. KRITCHMAN, E. M., 1982a, J. Opt. SOC.Am. 72, 399401. KRITCHMAN, E. M., 1982b, Appl. Opt. 21, 870-873. KRITCHMAN, E. M., 1983, J. Opt. SOC.Am. 73, 508-511. LICHTENBERG, A. J., 1969, Phase-Space Dynamics of Particles (Wiley, New York). LOUDON,R., 1983, The Quantum Theory of Light, 2nd Ed. (Clarendon Press, Oxford). LUNEBURG, R. K., 1964, Mathematical Theory of Optics (University of California Press, Berkeley). MANDEL,L., 1961, J. Opt. SOC.Am. 51, 797-798.

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[III

MAXWELL, J. C., 1854, Cambridge & Dublin Math. J. 8, 188 (see also M. Born and E. Wolf, Principles of Optics, for a more accessible description of Maxwell’s fisheye). MILLS,D. R., I. M. BASSEVand G. H. DERRICK,1986, Sol. Energy 36, 199-206. MIRANO,J. C., 1983, Appl. Opt. 22, 2747-2750. MIRANO,J. C., 1985a, J. Opt. Soc. Am. A 2, 1821-1825. MIRANO,J.C., 1985b. J. Opt. SOC.Am. A 2, 1826-1831. MIRANO,J. C., 1985c, Appl. Opt. 24, 3872-3876. MIRANO,J. C., 1986, J. Opt. Soc. Am. A 3, 1345-1353. MIRANO,J. C., J. M. Ruiz and A. LUQUE,1983, Appl. Opt. 22, 3960-3965. MOLLEDO, A. G., and A. LUQUE,1984, Appl. Opt. 231,2007-2020. MONTGOMERY, C. G., R. H. DICKEand E. M. PURCELL,1948, The Principles of Microwave Circuits (McGraw-Hill, New York). MOON,P., and G. TIMOSHENKO, 1939, The Light Field (Translation of: Svetovoe Pole, by A. Gershun; the book is about illumination calculations; a vector quantity similar to J is defined and used to calculate illumination from simple sources). NING,X., R. WINSTONand J. J. GAL LAG HER, 1987, Appl. Opt. 26, 300-305. and R. WINSTON,1987a, Appl. Opt. 26, 1207-1212. NING,X., J. J. OGALLAGHER OGALLAGHER, J. J., A. RABL,R. WINSTONand W. MCINTIRE,1980, Sol. Energy 24,323-326. PLOKE,M., 1967, Optik 25, 31-43. RABL,A., 1976a, Appll Opt. 15, 1871-1873. RABL,A., 1976b, Sol. Energy 18,497-511. RABL,A., and R. WINSTON,1976, Appl. Opt. 15, 2880-2883. RABL, A., N. B. GOODMAN and R. WINSTON,1979, Sol. Energy 22, 373-381. RIES,H., 1982, J. Opt. SOC.Am. 72, 380-385. SCHUMACHER, B. W., 1976, Optik 45,355-380. SIEGEL,H., and J. R. HOWELL,1972, Thermal Radiation Heat Transfer (McGraw-Hill, New York). SNYDER, A. W., atld J. D. LOVE,1983, Optical Waveguide Theory (Chapman and Hall, London). SOCHACKI, J., and C. GOMEZ-REINO, 1985, J. Opt. Soc. Am. A 2, 1297-1300. SPARROW, E. M., and R. C. CESS, 1978, Radiation Heat Transfer (McGraw-Hill, New York). STRAUBEL, R.,1902, Ann. Phys. (Germany) 4, 114-1 17. WELFORD, W. T., 1986, Aberrations of Optical Systems (Hilger, London). WELFORD, W. T., and R. WINSTON,1978, The Optics of Nonimaging Concentrators (Academic Press, New York). WELFORD, W.T., and R. WINSTON,1979, J. Opt. Soc. Am. 69, 917-919. WELFORD, W.T., and R. WINSTON,1982, Appl. Opt. 21, 1531-1533. WINSTON,R., 1970, J. Opt. SOC.Am. 60,245-247. WINSTON,R., 1974, Sol. Energy 16, 89-95. WINSTON,R., 1976, Appl. Opt. IS, 291-292. WINSTON,R., 1978a, Appl. Opt. 17, 688-689. WINSTON,R., 1978b, Appl. Opt. 17, 1668-1669. WINSTON,R., and H. HINTERBERGER, 1975, Sol. Energy 17,255-258. WINSTON,R., and X. NING, 1986, J. Opt. SOC.Am. A 3, 1629-1631. WINSTON,R., and W. T. WELFORD,1978. J. Opt. Soc. Am. 68, 289-291. WINSTON,R., and W. T. WELFORD,1979a, J. Opt. SOC.Am. 69,532-536. WINSTON,R., and W.T. WELFORD,1979b, J. Opt. Soc. Am. 69,536-539. WINSTON,R., and W. T. WELFORD,1980, Appl. Opt. 19, 347-351. WINSTON,R., and W. T. WELFORD,1982, J. Opt. Soc. Am. 72, 1564-1566. WOLF,E., 1976, Phys. Rev. D 13, 869-886. ZIJL,H., 1951, Manual for the Illuminating Engineer (Philips, Eindhoven).

E. WOLF, PROGRESS IN OPTICS XXVII

0 ELSEVIER SCIENCE PUBLISHERS B.V., 1989

IV

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES BY

D. MIHALACHE Deparimeni of Fundamental Physics Ceniral Insiiiuie of Physics PO Box MG-6. Bucharest. Romania

M. BERTOLOTTI and C. SIBILIA Dipariimento di Energetica Universiih di Roma 00161 Roma, Iialy

CONTENTS PAGE

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

5 1. 0 2.

INTRODUCTION

5 3.

TRANSVERSE ELECTRIC POLARIZED NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS 252

NONLINEAR ELECTROMAGNETIC WAVES GUIDED BY

A SINGLE INTERFACE

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

229 232

0 4. TRANSVERSE ELECTRIC POLARIZED NONLINEAR

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

SURFACE PLASMON POLARITONS . . .

5 5. EXPERIMENTAL STATUS . . . . . . . 5 6 . CONCLUSIONS . . . . . . . . . . . .

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

ACKNOWLEDGEMENTS REFERENCES

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

289 297 308 309 309

6 1. Introduction The discovery of optical bistability in semiconductors such as GaAs (GIBBS, MCCALL,VENKATESAN,GOSSARD, PASSNERand WIEGMANN [ 19791) and InSb (MILLER, SMITHand JOHNSTON [ 19791) with its multiplicity of applications to all-optical signal processing systems for optical communications and optical computing stimulated a great deal of theoretical and experimental activity in the last few years (see, for example, LUGIATO [1984], GIBBS, MANDEL,PEYGHAMBARIAN and SMITH[ 19861). Nonlinear optical devices such as bistable switches (GIBBS,MCCALL,VENKATESAN, GOSSARD, [ 19791, MILLER,SMITHand JOHNSTON [ 1979]), PASSNER and WIEGMANN [ 19831, logic gates (SEATON,SMITH,TOOLEY,PRISEand TAGHIZADEH JEWELL, RUSHFORDand GIBBS[ 1984]), and others have already been demonstrated in a plane wave context. Planar optical waveguides with their inherent confinement of the light in one dimension of the order of the wavelength of the light provide the optimum geometry for performing efficient nonlinear interaction in general and nonlinear optical signal processing in particular. The key concept on which all nonlinear guided wave optical devices are based is that the local intensity of the guided wave controls the propagation wavevector; that is, the field profile and propagation constant can become power dependent when one or more of the layered media is characterized by an intensity-dependent refractive index. Two categories of integrated all-optical devices can be anticipated on the basis of these nonlinear optical phenomena. The first class of optical devices are those in which the’nonlinear change in the refractive index is small in comparison with the refractive index difference between the guiding media. In this case the dependence of the propagation wavevector on the power flow can be evaluated from coupled-mode theory [ 19751, SIPEand STEGEMAN [ 19791, STEGEMAN [ 1982]), and (KOGELNIK the guided wave field distribution (i.e., field profile) can be approximated by linear guided modes. Devices that have been proposed and operate in this regime include nonlinear distributed couplers, such as gratings and prisms SHOEMAKER, (CHENand CARTER[ 19821, VALERA,SEATON,STEGEMAN, Xu MAI and LIAO [1984]), and the nonlinear coherent coupler (JENSEN [ 19821, CADA,GAUTHIER, PATONand CHROSTOWSKI [ 19861). 229

230

NONLINEAR W A V E PROPAGATION IN PLANAR STRUCTURES

[IV, § 1

The second category of nonlinear optical devices are those in which the optically induced change in refractive index is comparable with, or larger than, the index differences between the guiding media. In this case both the propagation wavevector and the field distribution are power dependent, and this dependence can be evaluated from a more exact approach in which the nonlinear wave equation is solved subject to continuity of tangential electric and magnetic fields across all the interfaces. Devices that have been proposed in this regime of operation include nonlinear guided wave optical limiters, lowpower threshold devices, and optical switching devices (SEATON,XU MAI, STEGEMAN and WINFUL[ 19851). Analytical solutions for the optical fields in Kerr-law media were found by LITVAKand MIRONOV[ 19681, MIYAGIand NISHIDA[ 19721, and KAPLAN[ 19761 and have been seminal to much of the progress in the field of nonlinear guided waves. The unique features of nonlinear guided waves in planar layered structures displaying both self-focusing and self-defocusingKerr-type nonlinearities have [ 19801, AGRANOVICH, been studied intensively in recent years (TOMLINSON BABICHENKO and CHERNYAK [ 19811, MARADUDIN [ 19811, LOMTEV[ 19811, AKHMEDIEV[ 19821, FEDYANINand MIHALACHE[ 19821, LEDERER, LANGBEINand PONATH[1983a,b], Yu [1983], BOARDMANand EGAN [ 1984a,b],BOARDMAN, EGANand SHIVAROVA [ 19841, STEGEMAN, SEATON, CHILWELL and SMITH[ 19841, MIHALACHE, NAZMITDINOV and FEDYANIN [ 19841, SIBILIA,BERTOLOTTIand SETTE [ 19841, LEUNG [ 1985a,b], STEGEMAN and SEATON[ 19851, MIHALACHE and MAZILU[ 1985, 1986a], MIHALACHE, STEGEMAN,SEATON,WRIGHT,ZANONI,BOARDMANand TWARDOWSKI [ 19871, KUMARand SODHA[ 19871). LANGBEIN, LEDERER and PONATH[ 19851 have developed a formalism that is capable of dealing with arbitrary lossless optical nonlinearities (non-Kerr-like ARIYASU,SEATON,SHENand MOLONEY[ 19851 nonlinearities). STEGEMAN, and MIHALACHE and MAZILU[ 1986bl have applied this technique, which does not require analytical field solutions to the nonlinear wave equation in order to evaluate the power dependence of the propagation wavevector (see and EGAN[ 1984bl) to a variety of planar layered structures. also BOARDMAN The question of stability or instability to propagation of various TE, nonlinear stationary wave solutions has been studied by numerical techniques (AKHMEDIEV, KORNEEVand KUZMENKO[ 19851, JONES and MOLONEY [ 19861, ARIYASU,SEATON,STEGEMANand MOLONEY[ 19861, MOLONEY, [ 1986a,b], LEINE,WACHTER, LANGBEIN ARIYASU, SEATONand STEGEMAN and LEDERER[ 19861, MIHALACHE and MAZILU[ 1987a,b]). WRIGHT,STEGEMAN,SEATONand MOLONEY[ 19861 studied Gaussian

IV, J 11

INTRODUCTION

23 1

beam excitation of TE, nonlinear guided waves using the beam propagation method. These authors have shown that for a thin dielectric film bounded by two self-focusing media, three different field distributions corresponding to the same guided wave flux level can be excited independently by suitable Gaussian input beams. The problem of multisoliton emission from a nonlinear waveguide has been considered by WRIGHT, STEGEMAN,SEATON,MOLONEYand BOARDMAN[ 19861. These authors have demonstrated numerically that external excitation of a nonlinear waveguide where the film and substrate are linear but the cladding displays a nonlinear refractive index (optical Kerr effect) can produce sequential threshold behavior by means of multisoliton emission from the waveguide. This behavior is similar to that predicted to occur at a GORDON,SMITHand nonlinear interface (ROZANOV[ 19791, TOMLINSON, KAPLAN[ 19821). The effects of linear absorption on the propagation of TE, nonlinear guided waves in an optical waveguide with a nonlinear Kerr-law cladding have also been investigated using the beam propagation method (GUBBELS,WRIGHT, [ 19871). STEGEMAN, SEATONand MOLONEY Only a few experiments dealing with nonlinear guided waves have been reported to date (VACH,SEATON,STEGEMAN and KHOO[ 19841, BENNION, GOODWINand STEWART[1985]). These authors used a single nonlinear self-focusing medium (liquid crystal MBBA or CS,) bounding a deposited dielectric film. These experiments can be interpreted in terms of nonlinear guided waves with flux-dependent field distributions. The aim of this article is not to give a review of all the papers on nonlinear guided waves but to select a few topics that illustrate in the simplest way the main physical principles of nonlinear guided waves in layered planar structures and related phenomena, Our selection is dictated by the general criterion of maximum simplicity, and for issues that are not discussed in this paper we advise the reader to consult the recent review by STEGEMAN,SEATON, HETHERINGTON, BOARDMAN and EGAN [ 19861. In the present review we analyze in detail nonlinear transverse electric (TE) polarized waves guided by layered planar structures for which the exact stationary solutions of nonlinear wave equations are available. In this chapter 5 2 is devoted to the study of electromagnetic waves guided by nonlinear interfaces, discussion of the basic concepts and method used to analyze nonlinear guided wave phenomena. The nonlinear TE polarized waves guided by thin dielectric films are studied in detail in 0 3. In 5 4 we show that nonlinear TE polarized waves can also be guided by very thin metal films (nonlinear surface plasmon polaritons). The experiments reported to date on

232

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

nonlinear guided wave phenomena are reviewed in present our conclusions.

[IV, $ 2

5 5. In the last section we

6 2. Nonlinear electromagnetic waves guided by a single interface 2.1. INTENSITY-DEPENDENT REFRACTIVE INDEX AND DIELECTRIC TENSOR

In the last few years new developments in nonlinear optics have been centered on third-order nonlinear guided-wave interactions that involve the mixing of three optical fields. The nonlinear polarization vector of an optically nonlinear medium is P,""(w) = %x$@,(w) EZ(4 E , ( d

(2.1)

9

where i = x, y, z, x(3) is the third-order susceptibility and E is the total electric field. Note that it is necessary to take the conjugate of one of the mixing optical fields so that the output signal is at the same frequency as the input signal, a prime requisite for all optical signal processing systems operating at a single frequency w. If the optical field associated with a plane or a guided wave is large enough, it can change the refractive index of the medium. For a plane wave in an isotropic material the Fourier component of the polarization field at the frequency w is =

h[XY + 3&

IE,(w)121Ei(4

(2.2)

Y

where xhl) = nf - 1 and no comprises the linear part of the refractive index. Expressing IE,(w)12 in terms of the local intensity Z = i c k n , IE,(w)l 2, the intensity-dependent refractive index can be expressed in the form n

=

no + n,,Z,

3x9

n,, = -,

ChnO

where n,, > 0 for self-focusing Kerr-like nonlinearities and n,, < 0 for selfdefocusing Ken-like nonlinearities. For guided waves propagating along the x-axis with z normal to the surfaces, the electric field is written as E~= .LE~(~)ei(8xox - 0 1 )

+ C.C.,

where is the effective guided wave index and K~

(2.4) = w/c is

the vacuum wave-

IV, I21

NONLINEAR EM-WAVES GUIDED BY SINGLE INTERFACE

233

vector. The expression for the nonlinear polarization vector of an isotropic [1964], STEGEMAN medium is (see, for example, MAKERand TERHUNE

For TE polarized waves propagating along the x-axis the electric and magnetic fields are E = (0, E,, 0) and H = (Hx, 0, H,) so that the only nonzero component of the nonlinear polarization vector is

y,""(z) = cgnOZn21 I EY(z)I2E,,(~).

(2.6)

In this case only the cYy component of the nonlinear dielectric tensor enters into Maxwell's equations, and this will be written as cJr = no'+

alEy12,

a = c&,,n,2n2,.

(2.7)

The Maxwell equations are dE dz

=

~dHx

dz

-iopoHx,

/IK~E, = opoHz,

i/Iq,H, = ioi+,,EY,

where cyy is given by eq. (2.7). Equations (2.8) and (2.9) give the following nonlinear wave equation for the amplitude function EJz): d2Ey dz2

~-

K;(/I'

-

nZ)E,

+ aKi1Ey(2Ey= 0 .

(2.10)

For real electric fields eq. (2.10) has an analytical solution (LITVAKand MIRONOV [ 19681, MIYAGIand NISHIDA[ 19721, KAPLAN[ 19761). This exact solution will be analyzed in detail in the next section.

2.2. TRANSVERSE ELECTRIC (TE) POLARIZED NONLINEAR SURFACE-GUIDED WAVES

Transverse electric (TE) polarized electromagnetic waves cannot be guided by the interface between two dielectric media whose refractive indices do not

234

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, 8 2

depend on the intensity. However, when at least one of the two dielectric media exhibits a power-dependent refractive index, a surface-guided wave can exist (LITVAKand MIRONOV[ 19681, TOMLINSON [ 19801, MARADUDIN[ 1981, 19831). In this case the self-focusing optical nonlinearity is not regarded as being small, and this gives rise to new types of waves that have no counterpart in the linear optics of surface and guided waves. (A certain critical power level must be reached before a nonlinear surface guided wave and, hence, a selffocused channel are created.) We consider a nonlinear interface between an optically linear semi-infinite dielectric medium (called the substrate) with dielectric constant E , in the region I (z < 0) and a semi-infinite Kerr-law nonlinear medium (called the cladding) with dielectric function E = E, + a, I E I * in the region I1 (z > 0). The TE polarized wave propagates along the x-axis with the z-axis normal to the surfaces. The only nonvanishing component of the electric field is written as EJx, z, t ) = 4 Ey(x, z) e i(Bmx - wr) + C.C. , (2.1 1) where K = Blc0 is the guided-wave wavevector, x0 = o / c is the vacuum wavevector, and p is the effective index. The nonlinear Maxwell equations for the x-independent guided wave fields (stationary field distribution) are z
(2.12) (2.13)

where q: = B2 - E,

,

qc2 -- 2 -

E,

,

a, = c&,,n,2n2,.

For waves guided by a single interface that are characterized by EJz) -,0 as I z I -, 03, that is, the fields decay exponentially away from the boundary. the solutions of eqs. (2.12) and (2.13) are well known (see, for example, LITVAK and MIRONOV [ 19681, MIYAGI and NISHIDA [ 19721, KAPLAN[ 1976,19771): E:(z) EJ'(z)

=

Eo eKoqaz ,

=

(~)1'2qc{cosh[~oqc(z - zc)]}-' ,

z
(2.14)

z> 0,

(2.15)

IV, § 21

NONLINEAR EM-WAVES GUIDED BY SINGLE INTERFACE

235

where q,

=

(/?' - E , ) ' / ' ,

a, > 0

qc = (/?' - E,)'/',

(self-focusing nonlinearity). For TE polarized waves both the field E,, and its derivative dE,,/dz are continuous across the interface z = 0 between nonlinear and linear medium. This leads directly to the eigenvalue equation E, = E,

+ $a,E;,

(2.16)

where Eo is the surface field. We see from eq. (2.16) that the field amplitude is fixed at the boundary because E, and E, are constants, and if the limit a, + 0 is taken in eq. (2.16), then Eo + + 00 ; that is, TE polarized electromagnetic waves do not exist in the linear limit for a single interface. The guided wave power in watts per meter of wavefront is given in terms of the Poynting vector as p='

.s, OD

Re@ x H * ) , dz = l3 2cpo

r

E,Z(z) dz .

(2.17)

--a0

For Kerr-law media this expression can be evaluated analytically (see, for example, TOMLINSON [ 19801): (2.18)

'.

where Po = (q,/p0)'/' (2aCx0)The /?-power formula (2.18) can be viewed as the nonlinear dispersion equation w = a(K, P),that is, the frequency-wave number ( o - K ) relationship for a given power level. It is possible to evaluate the nonlinear guided wave attenuation coefficient approximately from the imaginary component of the dielectric constant by assuming that the field distribution obtained in the lossless case will still be valid if the loss per wavelength is small (MARADUDIN [ 19831, ARIYASU,SEATON and STEGEMAN[ 19851, STEGEMAN,SEATON, ARIYASU,WALLIS and MARADUDIN [ 19851). A new formalism that is capable of dealing with arbitrary local lossless nonlinearities has been developed by LANGBEIN,LEDERERand PONATH [ 19851. This technique does not require analytical field solutions to the nonlinear wave equation in order to evaluate the flux dependence of the propagation wavevector.

236

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, 8 2

It is well known that the form of the dielectric function is determined by the physical processes which lead to the nonlinearity. The Kerr nonlinearity, which is quadratic in the local optical fields: eNL = a I E I 2, arises from electronic nonlinearities, thermal effects, and other factors. The dependence of the dielectric function on the electric field is not quadratic in semiconductor materials, where the nonlinearities are caused by absorption leading to the creation of excitons, plasmas, etc. In this case the optical nonlinearity takes the form cNL = ar 1 E 1 ', where 1 < r < 2 (JAINand LIND[ 19831, CHEMLA, MILLERand SMITH[ 19851, MATHEW, KAR,HECKENBERG and GALBRAITH [ 19851, YAO, KARAGULEFF, GABEL,FORTENBERRY, SEATONand STEGEMAN[ 19851). Furthermore, in realistic materials it is not possible optically to change the refractive index indefinitely, and a saturation effect sets in. The values of the For a saturated change An,,, of the refractive index vary from l o - ' to nonlinear interface the saturation effect is important because the interesting flux-dependent, surface-guided wave properties occur when the optically induced change in the refractive index Ansat is comparable with, or larger than, the refractive index difference n, - n, which exists at low powers between the substrate and the cladding. We model the dielectric function of the nonlinear self-focusing (a, > 0) cladding as (MARADUDIN[ 19831, LANGBEIN,LEDERER,PESCHELand PONATH[ 19851, STEGEMAN,ARIYASU,SEATON, SHEN and MOLONEY [ 19851, MIHALACHE and MAZILU[ 1986b1) E ~ ,=

cVv =

E,,

= E,

+ E,,,

[ - (-1

(2.19)

exp

a

3

1

9

(2.20)

E ~ ,= tYy= E,,

= E,

+ u,,~E;,

r2 1 .

(2.21)

Note that for both dielectric tensors (2.19) and (2.20) the maximum change in the dielectric function is that is, for I E I -,co,E -,E, + E,,,. For small fields E - + E, t acE,?, that is, the usual Kerr-law medium case. The dielectric tensors (2.19), (2.20), and (2.21) can be written in general form as follows:

'

tx, =

cYy =

E,,

= E~

+ &,NL(E;).

(2.22)

The nonlinear wave equation for the real quantity (in the absence of loss)

IV,I 21

231

NONLINEAR EM-WAVES GUIDED BY SINGLE INTERFACE

d2E, __ dz2

+ K;[E, + &,NLa(E:) - /?’]E,

=

0.

(2.23)

For surface-guided waves characterized by E,(z) integral of eq. (2.23) can be written as

(2r

=

Qc(E,,, 8) =

KO’

[

qfE,’ -

JOE’

+0

as /zI -+ co, the first

&,NL(E;) d(Et)]

.

(2.24)

The continuity of E,, and dE,ldz across the nonlinear boundary z = 0 between the linear substrate and the nonlinear cladding leads to the dispersion relation

(2.25) (2.26) (2.27) where E, is the surface field, ccNLis an averaged dielectric function of the nonlinear medium, M, = 1 if a self-focused peak occurs in the nonlinear cladding, and M , = 0 if there is no field maximum in the nonlinear medium. In our case of a self-focusing nonlinear cladding (ac > 0), a self-focused field maximum occurs in that medium (M, = 1) and the dispersion relation (eigenvalue equation) (2.25) becomes ES = &,

+

- JOE’

Eo2

&FL(E;) d(E,’) ,

(2.28)

from which the surface field E , can be determined. For a Kerr-law nonlinear cladding the eigenvalue equation (2.28) reduges to eq. (2.16). From eq. (2.28) we obtain the important result that TE polarized surface-guided waves can be supported by a single nonlinear interface if, and only if, cS > eC. The guided wave power flow parallel to the interface is P = P, + P,, where P, is expressed as (LANGBEIN,LEDERERand PONATH[ 19851)

(2.29) Here M ,

=

1 (there is a field maximum in the nonlinear cladding),

238

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, I 2

dEy/dz, and Eyis the field maximum evaluated from a$?,,) = 0. Finally we obtain the following expression for the total power flow P = P, + P,: @’/:

=

P,

U = ;pop-,

(2.30)

4s

(2.31) where M, = 1. Here we have q(x) = p2 - cC - E , , ~

+

(2.32)

X

(+

q ( x ) = p 2 - E, - Esat t - In 1 X

):6

,

(2.33) (2.34)

which correspond to the dielectric tensors (2.19), (2.20), and (2.21), respectively; u = cl,E,Z is obtained from the eigenvalue equation (2.28) and ii is determined from q(ii) = 0. For Kerr-law media we have q(x) = P2 - E, - i x , and the integrals in expression (2.31) can be evaluated analytically:

P, = 2P0p[(p2 - &,)1/2

+ (/3’

- EC - f U ) ’ / 2 ] ,

(2.35)

where u = u, E,Z = 2 ( ~ , E,) is obtained from the eigenvalue equation (2.16). In the case of a saturable nonlinear cladding characterized by the dielectric functions (2.19) and (2.20), there is a maximum in the change AE, of the dielectric constant which can be induced optically, that is, E + E, + esat for large fields ( 1 E I + 00). Thus the effective index /3 approaches its limiting value of (E, t ~ , , , ) l / ~asymptotically with increasing power. A necessary condition for the existence of a solution of the eigenvalue (2.28) in the unknown E, is E, < E~ + E,,,. From the condition that a,@,,)= 0 we thus obtain that p2 < E, + therefore the permitted /3-region for a nonlinear surface-guided wave is E:/’ < p < (8, t ~ , , ~ ) l /For ~ . a power-law cladding (r # 2) we have

r + 2) (E,

& , ) p,

(2.36)

ii = [;(r t 2) ( p - & , ) p ,

(2.37)

u = [g

-

‘P 16L 163 1 62 -

1 61 160 159 1 58 -

157156 -

PlmWlmm

I

30

20

40

50

60

70

80

90

100

110

120

-

I

Fig. 2. I. The dependence of the effective index B on the power flow for TE waves guided by the interface between a self-focusing cladding (n, = 1.55,nZC= 10-’mZ/W) and a linear substrate (n, = 1.56).The dashed and dotted lines correspond to the dielectric functions (2.19)and (2.20). respectively. (After STEGEMAN, SEATON,ARIYASU,SHEN and MOLONEY[ 19861.)

and the power flows P,,P, are given by eqs. (2.30) and (2.31) with Po replaced by P o , , = (%/Po)”2 (2w:/;)The numerical results for TE polarized surface-guided waves are shown in fig. 2.1 for Kerr-law claddings (r = 2), power-low claddings (r # 2), and saturable claddings. The dependence of the power flux P on the effective index /3 was evaluated for a nonlinear cladding characterized by n, = 1.55, n,, = m2/W (liquid crystal MBBA) at A = 0.515 pm (argon ion laser) in contact with a linear substrate with n, = 1.56. The values of the nonlinear coefficients a,, were chosen to produce equal minimum values of the power flux : a G , ]= 4.7 x 10W7m/V, a,,

2.5

=

2.3 x 10-

l4

ac,1.5= 1.75 x 10-9(m/V)1.5,

(m/V)’.5

SEATON,ARIYASU,SHENand MOLONEY[ 19861). We see (see STEGEMAN,

240

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, 0 2

from fig. 2.1 that the minimum power required for the excitation of nonlinear TE polarized surface-guided waves increases with decreasing E,, . Finally we note that the dispersion relation and the power flow for TE waves guided by a linear exponential-like graded-index medium in contact with a Kerr-like nonlinear cladding were also obtained (VARSHNEY, NEHME, SRIVASTAVA and RAMASWAMY [ 19861, WACHTER, LANGBEINand LEDERER [ 19871).

2.3. STABILITY TO PROPAGATION OF NONLINEAR SINGLE-INTERFACE GUIDED WAVES

The reflection of a plane wave from an interface between a linear medium and a nonlinear medium was apparently studied first by KAPLAN[ 19761. Following this pioneering work there have been several theoretical (ROZANOV [ 19791, MARCUSE[ 19801, TOMLINSON, GORDON, SMITHand KAPLAN [ 19821, MONTEMAYOR and DECK[ 19861) and experimental (SMITHand TOMLINSON [ 19841) studies of the interaction of Gaussian light beams with a nonlinear interface. Numerical techniques have been used by AKHMEDIEV, KORNEEVand KUZMENKO [ 19851 to study the excitation of nonlinear surface waves by Gaussian light beams. The question of stability to propagation of nonlinear surface-guided waves is crucial to the problem of the excitation of KORNEEVand these waves by external sources. It was found by AKHMEDIEV, KUZMENKO[ 19851 that both stable and unstable nonlinear surface waves can be excited by Gaussian light beams incident on the nonlinear interface at grazing angles. Consider the nonlinear interface between a linear substrate with dielectric constant E, in the region I ( z < 0) and a nonlinear Kerr-law cladding with dielectric function E = E~ + ac 1 E I in the region I1 ( z > 0). The TE polarized wave of frequency o propagates along the x-axis, and the only nonvanishing component of the electric field, E,, is homogeneous in the y direction ( z being the transverse coordinate). The parabolic equation for the slowly varying z ) is then amplitude A(x, z ) = LX;’~E,(X,

’

aA

a2A

+

- 2ifl~,,- = -- y’(z) K ~ AO(z) I C IA ~ I ’A

ax

az2

.

(2.38)

Here e(z) = 0 for z < 0 and e(z) = 1 for z > 0, y’(z) = p2 - n.’ for z < 0 and y2(z) = fl’ - n,‘ for z > 0. Note that the usual stationary solution of eq. (2.38), that is, A(0, z) = A,(z), can be obtained analytically (see eqs. 2.14, 2.15).

IV, § 21

NONLINEAR EM-WAVES GUIDED BY SINGLE INTERFACE

24 1

Equation (2.38) has two integrals of motion as follows: W

(2.39) -w

and for arbitrary solutions of eq. (2.38) we thus have dI/dx = 0 and dH/dx = 0. Note that eq. (2.38) is a mixed-type linear/nonlinear Schrbdingerlike equation with coefficients which depend on the transverse coordinate z. The absence of translational symmetry along the z-axis means that we cannot GREENE, use the elegant apparatus of the inverse scattering method (GARDNER, KRUSKALand MIURA[ 19671, ZAKHAROV and SHABAT [ 19711) to solve the problem analytically. The stability to propagation of the nonlinear stationary surface wave was investigated numerically on an IBM 370/135 computer. To avoid the numerical I

r 3.06

Z [pml

Fig. 2.2. Evolution of nonlinear surface wave field distributionwith propagation distance. Here n, = 1.55, nzc = 10-9m2/W, n, = 1.56, and the initial field pattern A&) corresponds to f?= 1.5607.

242

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV,

Ii 2

stability question, we selected for the difference approximation of eq. (2.38) the Crank-Nicolson scheme (see, for example, AMES [ 19651, GREENSPAN [ 19741). We chose the grid sizes equal in magnitude, K,AX = K,AZ = 0.4.The correspondingsystem of nonlinear equations was solved by the Newton-Picard method (see AMES [1965]). We have found that two iterations for the Newton-Picard method are sufficient in practice. This difference scheme makes it possible to conserve the integrals of motion (2.39) and (2.40) on the grid. The conservation of the total power flow was always better than 99%. For a Kerr-law nonlinear cladding and for j? = 1.5607 on the negative-sloped branch (dlldfi < 0) of the nonlinear dispersion curve (see fig. 2. l), the nonlinear stationary wave is unstable on propagation. In this case the nonlinear surfaceguided wave is ejected into the linear substrate as the result of the propagation (see fig. 2.2). The evolution of the nonlinear surface wave field distribution with propagation distance for fi = 1.574 on the positive-sloped branch of the nonlinear dispersion curve (dl/dS > 0) is shown in fig. 2.3. For this value of the propagation constant the nonlinear stationary wave A,(z) is stable on propagation over at least 300 wavelengths (see fig. 2.3). To conclude this section we point out that for the related problem of self-focusing of plane waves in infinite media, [ 19731 has shown that the solutions are stable for dZ/db > 0. The KOLOKOLOV 0.15 0 10

0.05 160

0.0

140 120 100

-5. 80 I

X

60 LO

20 0 -1 0

00

Z[,uml

1.5

Fig. 2.3. The same as in fig. 2.2, but for /3 = 1.574.

IV, § 21

NONLINEAR EM-WAVES GUIDED BY SINGLE INTERFACE

243

numerical results presented here for TE polarized waves guided by a nonlinear interface yield the same stability criterion; that is, dlldfi > 0.

2.4.

TRANSVERSE MAGNETIC (TM) POLARIZED GUIDED WAVES AT A NONLINEAR INTERFACE

Surface polaritons are electromagnetic waves guided by a single interface between two semi-infinite media or by single or multiple films bounded by two semi-infinite media (see, for example, MILLS and SUBBASWAMY [ 19811, AGRANOVICH [1982]). In all cases the electromagnetic fields decay with distance from the boundary into the semi-infinite media in an exponential-like fashion, resulting in fields localized near or at the surface. In the case of transverse magnetic (TM) polarization the magnetic vector is oriented perpendicularly to the plane of incidence, defined by the direction of propagation and the normal to the surface. Consider one of the simplest cases, namely, that of electromagnetic waves guided by a single interface between two semi-infinitelinear media. The dielectric constants are E~ and E,, where the subscripts c and s refer to the cladding and the substrate. In the linear case only TM polarized surface polaritons can be supported by a single interface and only if E~ > 0, E, < 0, and E~ < I E, I . The effectiveindex fi = K / K,, where K is the propagation wavevector and K, = w/c, is given by (see, for example, AGRANOVICH [ 19821) (2.h) In the following paragraphs we consider the effects of optical nonlinearities on surface and guided electromagnetic waves, in which these nonlinearities are not regarded as small and give rise to new types of waves, which in some cases have no counterpart in the linear optics of surface and guided waves. Note that the propagation of nonlinear TM polarized surfacewaves at a plasma boundary [ 19671. For the TM polarization was apparently studied first by ALANAKYAN and Kerr-like media there were two approximations frequently used in the literature: (1) the uniaxial E,( IEx I ’) approximation, in which the component of the dielectric tensor parallel to the surface E, depends on the field component parallel to the surface Ex (see AGRANOVICH, BABICHENKOand CHERNYAK [ 198 l]), and (2) the uniaxial E,,( I E, I 2, approximation, in which the component of the dielectric tensor perpendicular to the surface E,, depends

244

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV,8 2

on the normal field component E, (see SEATON,VALERA, SVENSONand [ 19851). STEGEMAN The dispersion relation of TM polarized nonlinear surface polaritons guided by the interface between a linear dielectric and a nonlinear Kerr-law dielectric medium in the uniaxial E ~ I Ex ~ I (') approximation, E~~ =

ex + axxlExl2

,

E,,

= E,,

(2.42)

was studied in detail by AGRANOVICH, BABICHENKO and CHERNYAK [ 19811. In this approximation, and in the case E~ > 0, E, > 0, dispersion curves of TM polarized nonlinear surface polaritons at a quartz-vacuum interface were [ 19831). An exact dispersion relation of nonlinear plotted (see MARADUDIN TM polarized waves at the boundary between two semi-infinite nonlinear uniaxial media characterized by diagonal dielectric tensors of the type (2.42) has been derived by LOMTEV[ 19811 and Yu [ 19831. The influence of oscillations in the transition layer on TM polarized nonlinear surface polariton [ 19821 in the case spectra was fist discussed by AGRANOVICH and CHERNYAK of the uniaxial E,( 1 Ex/2, approximation and subsequently by MIHALACHE [ 1986al for the uniaxial E,=( I E, I 2, approximation: Exx = Ex

Y

E,,

= 8,

+ ~zzIEzI'.

(2.43)

For TM waves in Kerr-law media and in the case of the uniaxial E,( I E,l approximation, the differential equation for the E J z ) field component is

2,

(2.44)

where y = s, f, c, refers to the substrate, film, and cladding, respectively. This equation has analytical solutions (see AGRANOVICH, BABICHENKO and CHERNYAK [ 19811). For example, if ac < 0 and a, c 0, we have (2.45) (2.46)

If a, > 0 and as > 0, then cosh is replaced by sinh. Note that, because the sign of the term proportional to a,, in eq. (2.44) is now negative, whereas it was positive for the TE polarized case, the field distributions differ between TE and TM polarized E,( 1 E,I 2, cases.

IV, § 21

NONLINEAR EM-WAVES GUIDED BY SINGLE INTERFACE

245

An alternative approach is to eliminate EJz) and EJz) from Maxwell's equations and to obtain a single equation for the remaining field Hy(z). In the uniaxial eZz( I E, I 2 , approximation we have

(2.47) This differential equation is not exactly solvable because of the E:, term in the denominator. We note that for most materials the quantity A E = 1 a,E; I is smaller than 0.01, and in some exceptional cases, for example, InSb, it can reach 0.1. The approximation E,, x E, in the denominator of the third term in eq. (2.47) implies only a small error in an already small term. In this limit the solutions for Hy(z) now have exactly the same form as those for the exactly a,. Because of these solvable TE case with a, replaced by a; = P4(c2.5:~:)similarities, the uniaxial eZz( I E, I 2, approximation leads to similar powerdependent behavior for both T E and TM polarized guided waves. TM polarized electromagnetic waves guided by a single interface were investigated in detail by STEGEMAN,SEATON, ARIYASU, WALLIS and [ 19851. Both cXx( I E,I 2, and E,,( 1 E, 1 2, approximations for TM MARADUDIN waves were analyzed, and the guided wave wavevector and the attenuation coefficient versus guided wave power were evaluated for a variety of material conditions. The nonlinear guided wave attenuation coefficient was calculated approximately from the imaginary components of the dielectric constants E, I and eC I . Assuming small losses, it can be easily shown that

(2.48) where PI and PR are the imaginary and the real part of the effective index /3 (see STEGEMAN,SEATON,ARIYASU,WALLIS and MARADUDIN[ 19851). AKHMEDIEV [ 19831 has given a theory of nonlinear surface TM waves, but the analysis was restricted to isotropic nonlinear media in which the two electric field components have equal weight in the dielectric constant. BOARDMAN, MARADUDIN, STEGEMAN, TWARDOWSKI and WRIGHT[ 19871 presented a numerical method for solving Maxwell equations for TM waves at a nonlinear interface that is applicable to arbitrary forms of dispersive nonlinearity. They did this by transforming the infinite transverse plane into a finite interval and making use of asymptotic boundary conditions. In the following paragraphs we derive an exact dispersion relation of TM polarized guided waves at an interface between either a linear dielectric or metal

246

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, 8 2

and a nonlinear Kerr-law dielectric (MIHALACHE,STEGEMAN,SEATON, [ 19871). This dispersion WRIGHT,ZANONI,BOARDMANand TWARDOWSKI relation is a polynomial equation involving the boundary values of the electric field components, the medium parameters, and the guided wave effective index p. Note that surface electromagnetic waves guided by the boundary between a nonlinear dielectric and a metal are of particular interest, since they correspond to the only type of nonlinear single-interface wave that does not have a power threshold. As is well known, TM polarized waves exhibit two electric field components, one parallel ( E x )to the wave vector and one perpendicular (E,) to the surfaces. To define the effects of an intensity-dependent refractive index, it is necessary first to examine the nonlinear polarization field. The electric field vector is

E(r, t ) = ~ [ E , ( zx) + E,(z) z ] exp [i(fiuOx- ot)]+ C.C. ,

(2.49)

'

where Ex(z)and E,(z) are 4 n out of phase with one another; that is, I E, I = E t and I ExI ' = - E,'. The nonzero components of the nonlinear polarization vector are (STEGEMAN [ 19821)

' + ax, I ~ , ( z I)')E,(z) ,

(2.50)

P,""(z) = %(azx~Ex(z)12 + az,IE,(z)12)E,(z).

(2.51)

P,""(z> = %(axxI E X ( 4I

Thus for a Kerr-law medium the components of the dielectric tensor are (2.52) (2.53)

where the values of the Kerr-type optical nonlinearities a. depend on the particular nonlinear mechanism under consideration. For electronic nonlinearities obtained from a power law expansion of the polarization in terms of field products, we have axx= a,, = 3ax2= 3aZx = c ~ g , 2 n , ,,

whereas for electrostrictive nonlinearities axx= a,, = ax, = azx= ce,,n,2n2, ,

where n = no + n,,l, no is the linear part of the refractive index, and n2, is the intensity-dependent refractive index coefficient. Starting from the Maxwell equations for TM polarized guided waves, the electric field components Ex(z)and E,(z) obey KO (c,, dEx - - - 1. - B2)E,,

dz

B

(2.54)

IV,§ 21

NONLINEAR EM-WAVES GUIDED BY SINGLE INTERFACE

241

(2.55) H,,= --&,,E,. c%l

(2.56)

B

The key point of our analysis is that for guided TM waves, that is, E + 0 and dE/dz + 0 as z + k 00, eqs. (2.54) and (2.55) have a first integral, which can be written as 2

I(“) + U(E,, E,) 2

=

dz

0,

(2.57)

where U(E,, E,)

=

f K:E,E~ + 4 IC;(B’ - e,)E; + f lc:axz E,’E,’ - a K,’axx(E: + E): ,

(2.58)

as was first shown by BERKHOER and ZAKHAROV [ 19701. The solution of Maxwell equations (2.54) and (2.55) in the semi-infinite linear substrate characterized by a dielectric constant E, (for a dielectric E, > 0, for a metal E, < 0), which occupies the lower half-space z < 0, can be written as E,(z)

=

E,, ercoqsz, z < 0 ,

(2.59)

where 4s

=

(B’ - E,)”’

9

E,,

=

E m

9

B’ > E,

for a dielectric. Furthermore, in the nonlinear medium eq. (2.54) can be rearranged as

(2.60) which gives

D, =

iBE,,

dEx

~ o ( & , ,- 8’) dz

’

(2.61)

where D,is the z component of the displacement vector D. Equation (2.61) also holds in the linear medium with E,, replaced by 6,. From standard electromagnetic theory D, and Exmust be continuous across the interface z = 0. Now we define E,, = E,(O) and the value q,, of the z component of the dielectric

248

[IV, I 2

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

tensor at the interface z enl = 6, - aZ,E&

=

0, which depends on the boundary fields:

+ azzE&.

(2.62)

From the continuity of D, at the interface z = 0 we obtain the following relation between the boundary values of the fields: (2.63)

In the limit of weak fields eq. (2.63) reproduces the usual relation between the boundary values of the fields. Furthermore, by using the first integral (2.57), we STEGEMAN, finally obtain the followingeigenvalue equation for /? (MIHALACHE, [ 19871): SEATON,WRIGHT,ZANONI,BOARDMAN and TWARDOWSKI fi4[2$kl - E,E,~C~&A' -

2 2 2 fa,,^,',(^," + 4 1 ) - ~,,E~,E, ~ 1 2

1

3 2

+ B Z ( ~ x ~ , 3+d~(,zE&~s41 + ~ l , , E o zEn1 ~ s- ~,4&)

- ~azzE&e,2E;fi =0.

(2.64)

Note that the particular case of an isotropic nonlinear cladding, that is, a, = a,, = a, = a, = a,, was studied in detail by AKHMEDIEV [ 19831. In this case the eigenvalue equation for /? has a simpler form:

8, = 8, = E,,

(2.65)

where G,= 6, t a,(E& - E;,) is the value of the dielectric constant of the nonlinear Kerr-law cladding at the interface z = 0. Given the material parameters, eqs. (2.63) and (2.64) allow the boundary values of the electric field components inside the nonlinear medium to be determined as functions of /?(at least numerically). Equations (2.54) and (2.55) can then be integrated using the boundary values to give the field distribution in the nonlinear medium. The field distribution in the linear medium is simple exponential (see eq. 2.59). Once the fields are obtained, the guided wave flux can be calculated by integrating the time-averaged Poynting vector over the depth z. Finally we have P = P, + P,, where (2.66) (2.67)

IV,8 21

NONLINEAR EM-WAVES GUIDED BY SINGLE INTERFACE

249

A first survey of the field patterns and the permitted /?regions can be obtained

from the inspection of the so-called “phase trajectories” of the nonlinear surface waves (see AKHMEDIEV [ 19831, LANGBEIN, LEDERERand PONATH [ 19851). Let us consider various particular cases for an isotropic nonlinear cladding with E~ =

eZ =

E,

and

=-

axx = azz = axz = a, = ac ,

( I ) E, > 0, a, 0, cS < 0. The dependence of the dimensionless power flow PIP,,on the effective index /?for E, = 2.25 and several values of E, is shown the guided in fig. 2.4a. It can be seen that by increasing the effective index /?, wave power increases up to some limiting value and then decreases to zero. This is related to the fact that, in a medium with negative dielectric constant, the power flow and the wavevector have opposite directions and in some region of /? values the power flow is decreasing with increasing effective index /?.We note that the magnetic field for this solution attains its maximum at the interface between two media. In this case the effective index for the nonlinear surface = [ E, I E, I ( 1 E, I - 6,)corresponding to wave is greater than the value /?, a TM polarized linear surface polariton.

0.08

OOL 0 02

P/po

I

06 04 0.2

18

22

76

30

17

21

25

29

-B Fig. 2.4. Dependence of the dimensionless power flow PIP, of the surface wave on the effective index B for four different cases. For the medium with positive dielectric constant (6, or 6,) we choose the value e = 2.25. The values of the dielectric constant of the adjoining medium are indicated above the curves. (After AKHMEDIEV[1983].)

250

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, $ 2

(2) E, < 0, a, > 0, E, > 0. Figure 2.4b shows the dependence of the dimensionless power flow PIP, on the propagation constant Pfor E, = 2.25 and several values of E,. In the present case the surface wave exists in a bounded range of variation of 1, and the effective index is greater than the value P, = [ I E, I E,( I E, I - e S ) Let us note that as in case (1) there is a maximum power which can be transmitted. (3) E, < 0, a, < 0, E, > 0. For this case fig. 2 . 4 ~shows the Pdependence of the dimensionless power flow PIP,. We see that the power flow is equal to zero for p = P,, and increases to infinity as /?approaches n, = ~,1/’. It should be noted that in the case where E, > I E, I TM polarized linear surface polaritons ~ polarized nonlinear surface do not exist. As we can see from fig. 2 . 4 ~TM polaritons can exist in the case E, > I eCl when the power flow exceeds some threshold value (the curves for E~ = - 2.1 and E, = - 1.45). (4) E, > 0, a, < 0, E, < 0. Figure 2.4d shows the /3 dependence of the dimensionless power flow PIPo for E, = 2.25 and several values of 8,. Thus for negative nonlinearity, that is, cases (3) and (4), as the power flow increases, the effective index fl decreases from the value P = j?,, corresponding to the TM polarized linear surface polaritons. ( 5 ) E, > 0, a, > 0, E, > 0. In this case the “phase-path diagram” and the field pattern show that the magnetic field attains its maximum in the selffocusing nonlinear cladding (a, > 0) and not at the interface z = 0. This non-

’

Fig. 2.5. Dependence of the energy flux PIP, of a surface wave on the effective refractive index /I in the case when E= = 2.25, E. = 2.5, and a, =- 0. (ARer AKHMEDIEV [1983].)

IV, 8 21

NONLINEAR

EM-WAVES GUIDED

B Y SINGLE INTERFACE

25 1

linear wave can be guided by the interface between a self-focusing dielectric cladding and a linear dielectric substrate, provided that the power threshold is exceeded and in this sense is similar to TE polarized surface polaritons at a nonlinear interface (see TOMLINSON [ 19801, MARADUDIN [ 1981, 19831). For the present case fig. 2.5 shows the /? dependence of the dimensionless power flow P/Po for tC= 2.25 and E, = 2.5. This TM polarized wave does not have [ 19831). an analog in the linear optics of surface-guided waves (AKHMEDIEV In fig. 2.6 we present the /?dependence of the power flow P for the nonlinear self-focusingdielectric-metal interface for both electronic, a, = 3 ax=(curve a), and electrostrictive, axx= axz (curve b), nonlinearities. Figure 2.7 shows the transverse distribution of the electric field component Ex for the electronic nonlinearity and for several values of the propagation constant p. Note that for both electronic and electrostrictive self-focusing nonlinearities there is a maximum power which can be transmitted, and the effective index /? is greater than corresponding to a TM polarized linear surface wave. the value /?, We finally mention that nonlinear TM polarized surface polaritons guided by thin metal films were studied in detail by using the uniaxial cZz( I EzI 2, approximation (see ARIYASU,SEATON, STEGEMAN,MARADUDINand WALLIS[ 19851, KUSHWAHA [ 19871, LANGBEIN, LEDERER, MIHALACHE and MAZILU[ 19871).

Fig. 2.6. Dependence of the power flow P on the effective index B for parameter values o = 3.66 x 101’rad s - ’ , cc = 8, = 8, = 2.405, cs = - 2.5, and a, = a,, = 6.4 x m2 V 2 : (a) arx = 3a,,; (b) ax, = ax=. (AAer MIHALACHE, STEGEMAN,SEATON,WRIGHT,ZANONI, BOARDMAN and TWARDOWSKI [1987].)

252

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

-0.2

0

[IV, B 3

02

Fig. 2.7. Ex versus transverse coordinate z for case (a) in fig. 2.6: (a) ,!I= 9, (b) j= 13, (c) ,!I= 25. (ARer MIHALACHE, STEGEMAN, SEATON,WRIGHT,ZANONI,BOARDMAN and TWARDOWSKI [1987].)

4 3. Transverse clectric (TE) polarized nonlinear optical waves guided by thin dielectric films 3.1. NONLINEAR GUIDED WAVE PROPAGATION IN THREE-LAYER

STRUCTURES WITH KERR-LAW MEDIA

A guided wave is an electromagnetic field that is guided by media with a high refractive index. A dielectric slab is the simplest example of an optical waveguide; it is actually employed for light guidance in integrated optics circuits (see, for example MARCUSE[ 19741, TIEN[ 19771, NOLTINGand ULRICH[ 19851). A slab waveguide is a thin dielectric film of thickness d and refractive index nf surrounded by media of lower refractive indices: the substrate with refractive index ns and the cladding with refractive index n,. For thin-film waveguides and TE waves (polarized along the y-axis) the only nonvanishing component of the electric field is

Iv, 8 31

253

NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

From the continuity of Ey and dEJdz at the interfaces z = 0 and z obtain the dispersion relation tan (~c,q,d)= 4 f h S + 4 c ) (4;

=

d we

(3.4)

- 4s4J

Equation (3.4) can be rewritten in the form (constructive interference condition): xOqfd=

Qsf

+ Qcf + m n ,

m

=

0, 1 , 2 , .. . ,

(3.5)

where tan Qsf

4

= 2,

4f

tanGCf= 42 . 9f

Solutions of eq. (3.5) can exist for a discrete number of values of m and are labelled TE, (m = 0, 1, 2, .. .). Note also that we have the following relations for the field amplitudes: E:(nf - n f )

=

Ef(n:

- p')

=

E,?(nf - nz) .

(3.6)

It remains to relate the amplitude coefficient E, of the electromagnetic field to the power carried by the mode. The guided wave power flow is obtained by integrating the x component of the Poynting vector

where d,, = d + ( ~ ~- 4+ (~1 ~)0 4 , ) -is the effective thickness of the thin-film waveguide. Therefore, from eq. (3.7) we can find the field amplitude E, as a function of the power flow P and the effective index fl. There are also transverse magnetic (TM) polarized modes with the magnetic field polarized along they-axis, that is, H,, # 0 and with Ex # 0 and E, # 0 (see, for example, MARCUSE[ 19741). Thus TM polarized waves exhibit two electric field components, one parallel ( E x )to the wave vector and one perpendicular ( E z )to the surface (a complication for nonlinear optics). Note that the dispersion relation for TM modes of a linear asymmetrical waveguide is given by eq. (3.4), with qYreplaced by 4,,/ey, where y = s, f, c. A guided-wave version of the slowly varying phase and amplitude approximation has been developed for guided waves and is known as coupled mode theory (KOGELNIK[ 19751, SIPE and STEGEMAN [ 19791). This method is useful for analyzing the generation of

254

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV,§ 3

new waves and for intensity-dependent refractive index phenomena. If the optical nonlinearities do not significantly alter the field distribution of the guided waves, the coupled mode theory can be used to calculate the intensitydependent wavevector or phase shift (STEGEMAN[ 19821). In the case where the optically induced refractive index change is comparable with or larger than the index differences n, - n,, n, - n, which exist at low powers between the dielectric film and the bounding media, both the field profiles and propagation constants are power dependent, and coupled mode theory which is essentially a form of first-order perturbation theory is inadequate to obtain even qualitative results. In this case an exact theory must be used. For Kerr-law media and TE surface-guided waves the nonlinear wave equation can be solved analytically. 3.1.1. Nonlinear cladding The asymmetrical dielectric layered structure we consider consists of an optically linear medium with refractive index n, (the substrate) occupying the half-space z < 0 (region I), a dielectric film of thickness d with refractive index n, in region I1 (0 < z < d), and a nonlinear Ken-law self-focusing cladding described by the dielectric function E = E, + a, I E I 2, a, > 0 in the region I11 (z > d). The Maxwell equations for the x-independent guided wave fields are z
--

Ic,'(fi'

- Ef)E;'

=0,

(3.10) The exact solutions of eqs. (3.8), (3.9), and (3.10) for ac > 0 (self-focusing optical nonlinearities) and fi > n, can be written as

E;Ir(z)=

(t)

4 cos(Ic,q,z)+"sin(Ic,q,z)

112

4,

1

qc{cosh[Icoq,(z-z,)l}~',

,

z
(3.11)

O
(3.12)

z>d,

(3.13)

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NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

255

where (3.14) u = tanh [ Koqc(z,

- d ) ].

(3.15)

The dispersion relations are obtained from matching the tangential electric and magnetic fields at the interfaces (STEGEMAN, SEATON,CHILWELLand SMITH[ 19841, MIHALACHE, MAZILUand TOTIA[ 19841): (3.16) This result is very similar to the linear case with the exception that qc is replaced by ( - uq,). When a, -,0, then z, -, - co, u -, - 1 and we obtain the dispersion relation of TE polarized modes of the linear asymmetrical thin-film waveguide (see eq. 3.4). For fl> n, the exact solutions of the Maxwell equations (3.8), (3.9), and (3.10) are ~ ; ( z=) a;

1/2E exp (Ic,q,z) ,

+-(4f - 4 s ) exp(24,

where qf = (8' - n:)'/' =

r0ilfz)],

z
(3.17)

0
(3.18)

and

4 sinh(rc,&d) [2(1 - ~ ' ) ] ' / ~ cosh(Koqfd) q, +2 4f

I-'

.

(3.19)

The dispersion relation obtained by ensuring the continuity of E,, and dE,,/dz across the interfaces z = 0 and z = d is (3.20) where u is given by eq. (3.15).

256

[IV,B 3

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

The guided wave power per unit length along the y-axis is obtained in the usual way by integrating the Poynting vector over the depth dimension z:

P P = __

“

2CPO

--oo

E;(z) dz = P, + P, + P, .

(3.21)

For /?< n, we have (see MIHALACHE[ 19851, MIHALACHEand TOTIA [ 19851)

ps

= ‘p 2 0

p-A”2 4,

(3.22)

9

Pf=iPoP/P{Kod(l+ $ ) + L n4f( K o 4 f d ) x

P, = 2P0Bq,(1

[( - 5) 1

11

cos(Icoqfd) + 2 42 sin(Koqfd) 4f

+ u),

,

(3.23) (3.24)

For /3> n, we obtain for P, and P,:

4s

Pf= :Po@’

(3.25)

{ ~ , d ( - 5)+ 1gf 1

sinh(IcoBfd)

x [(l +$)cosh(n,gd)+

11

2 ”4 s i n h ( ~ ~ q , d ) . (3.26) Br

We consider the liquid crystal MBBA (n, = 1.55, n,, = m2/W) as the nonlinear cladding medium deposited on a glass waveguide (nf = 1.61, n, = 1.52).The existence of the nonlinear cladding affects the cut-off conditions for an asymmetrical film waveguide. As is well known, a linear asymmetrical optical waveguide (n, # n,) cannot support guided waves below a critical thickness d,, (see, for example, MARCUSE[ 19741, TIEN[ 19771). In the case of an asymmetrical nonlinear optical waveguide there is a power threshold for TE, wave propagation for film thicknesses d < d,, (see fig. 3.1 for d / l = 0.1). This phenomenon can be used as a low-power threshold device, that is, one which begins to transmit above a certain minimum power (MIHALACHE

IV, 5 31

NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

251

I

60 50

-

40 30 -

2o

t

B 156

157

158

159

160

161

162

163

-

Fig. 3.1. TEo guided wave power versus effective index B. Here n, = 1.55, n,, = n, = 1.52, n, = 1.61, and 1 = 0.515 pm. (AAer MIHALACHE[1986b].)

m2/W,

[ 1986b1). A low-power threshold device can also be achieved with a saturable self-focusing cladding, provided that the saturation value nsat is not too large. The dependence of the guided wave power on the effective index /3 is shown in fig. 3.2 ford = 2 pm, n, = n, = 1.55, n2c = l o p 9m2/W, and n, = 1.57. The unique features of the TE, solution are the existence of wave propagation for /3 > n, and the local maximum in the guided wave power. For the TE, wave shown in fig. 3.2 the value of /3 never exceeds n, and there is an absolute maximum in the guided wave power. Moreover, the TE, branch terminates at some value of /3 < n, (see fig. 3.2). The evolution of the TE, and TE, field distribution with increasing /3 is shown in fig. 3.3, which illustrates one of the characteristic features of nonlinear guided waves, namely, power-dependent field distributions. As shown in fig. 3.3, as /3 increases, the TE, field maximum gradually narrows and moves into the nonlinear self-focusing cladding. With increasing /lthe TE, field maximum nearest the nonlinear cladding shifts into that medium (see fig. 3.3). The variation in the guided wave power with propagation constant /?for d/A = 6, shown in fig. 3.4, illustrates that the higher order TE, (m3 1) branches all terminate at some values of /3< np For a self-focusing cladding there is an absolute maximum in the power that can be propagated in any TE, (rn 2 1) guided wave. For all film thicknesses the lowest order TE, branch degenerates at high powers into a self-focused surface wave guided by the nonlinear interface between the film and the cladding media. Clearly, these

258

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

120

t

-

n 1

-

90 80

83

P'rnW'rnrn'

110 -

100

[IV,

d I X = 3.883

-

70 60 50

40 30

-

-

20 -

10 -

B I

1.552

1.556 1.560

1.564

1.[j68

1.572

1.576

1.580

1.584

1.500

*

Fig. 3.2. Guided wave power versus effective index j9. Here n, = n, = 1.55, n2= = lo-' mz/W, I = 0.515 pm. (After SEATON,VALERA, SHOEMAKER, STEGEMAN, CHILWELL and SMITH [1985].)

n, = 1.57, d = 2 pm and

TEi

TEO

Fig. 3.3. Field distributions associated with TE, and TE, nonlinear guided waves. The field evolution with increasing j9 is shown. (After SEATON.XU MAI, STEGEMANand WINFUL [1985].)

Iv, $ 31

280

-

240

-

NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

259

200-

dlX=6 160 120 -

80-

T Eo

40A1 @

B

Fig. 3.4. G u i d e d wave p o w e r versus effective i n d e x 8. Here n, = n, = 1.55, n20 = nr = 1.57, d/1 = 6, and 1 = 0.515 pn.

*

m2/W,

nonlinear guided waves can be used for optical limiters in a variety of applications. The limiting action for the case of a self-focusingnonlinear cladding has been demonstrated for TE, waves experimentally (see VACH,SEATON,STEGEMAN and KHOO[ 19841). 3.1.2. Nonlinear substrate and cladding In the following paragraphs we consider a symmetrical thin-film waveguide, which consists of a dielectric thin film of thickness d and refractive index n , bounded on both sides by identical self-focusing Kerr-law media (n, = n, and cr, = cl, > 0). The dielectric film fills the region I1 ( - f d < z < f d ) , and the two nonlinear dielectrics fill the half-spaces I ( z < - i d ) and I11 ( z > i d ) , respectively. The Maxwell equations for TE polarized waves are

_d2EJ __ dz2

d2EJ1 dz2

___

Ki(b2 K,'(P'

-

&,)E,!

+ i~;a,(E:)~

- Ef)E;'

=

0,

=

0,

Z<

-id,

(3.27)

- 2Id < z < { d , (3.28)

260

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, 8 3

The exact solutions of differentialequations (3.27), (3.28), and (3.29) in thc case a, > 0 (self-focusing optical nonlinearity) are E:(z)

=

($)1'2qc{cosh[~oqc(z

($>

1I2

EjlI(z)

=

- zJ]}-',

q,{cosh [ KoqC(z- z,)]} - ,

z<

-id,

z >i d .

(3.30)

(3.32)

From the boundary conditions we are left with an equation for the unknown zfas follows:

{ 1 - bf tanh2[ KOaf(id- ~ f ) ]cosh2[ }

- { 1 - b: tmh2[ x&(id

id + zf)]

+ ~ f ) ] cosh2[ } Ko&(id - zf)] = 0 ,

(3.33)

where b, = gigc. Equation (3.33) has a unique solution z, = 0 for all B > n,. The solution (3.31) for which zf= 0 corresponds to the symmetrical wave (S) of the symmetrical three-layer planar structure. In this case the field distribution is symmetrical to the center of the waveguiding thin film and z, = - z,. The eigenvalue equations for the symmetrical branch are

+ z , ) ] = b2 tan(lcoqfid), tanh[ ~ o 4 ~ (+i zdc ) ] = -b, tanh(K,qf;d),

tanh[ K,q,(;d

B < nf

1

(3.34)

/3> n,,

(3.35)

where b, = qJqc. Note that if aC--t 0, then z, + + 00 and eq. (3.34) reduces to the well-known dispersion relation for the symmetrical (even) modes of the symmetrical dielectric waveguide: 1 9, tan(lc,q,,d) =4f

(3.36)

For the symmetrical solution (S) the amplitudes of the electric field inside the dielectric thin film are given by

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NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

26 1

Next we obtain the dispersion relations for the antisymmetrical wave (AS) of the symmetrical thin-film waveguide. For this purpose we write down the second solution of the Maxwell equation (3.28) inside the film layer: (3.39) In this case the equations for the unknown zf have the form

{ 1 - b: cotZ[ icoqf(:d - z,)]} sin2[ KOqf(:d + zf)]

- { 1 - b2' cotz [ Koqf(;d

+ Zf)]} sinz [ Koqf(:d

- Zf)] = 0

(3.40)

for /?>nf and ,J

1 - b: cothZ[Ko&(;d - zf)]} sinh' [ K O & ( i d

+ zf)]

- { 1 - b: coth2[rcoQr(~d + zf)]} sinh2[ lcoqf(id- zf)] = 0

(3.41)

for /?>n,. It is easily verified that eqs. (3.40) and (3.41) have the solution zf = 0. This solution corresponds with the antisymmetrical wave (AS) of the symmetrical layered structure. We notice that eqs. (3.40) and (3.41) also have solutions z, # 0, that is, the asymmetrical wave (A) propagating in a nonlinear symmetrical layered structure. The asymmetrical wave only exists above a definite power threshold (AKHMEDIEV[ 19821). From the boundary conditions we obtain the following dispersion relations for the antisymmetrical wave (AS): (3.42) + z,)] = - b, cot (KOqfid), /9 < n f , tanh[Koq,(id + z,)] = -6, coth(K&d), /9> n f , (3.43) If a, + 0, then z, + + co and from eq. (3.42) we obtain the dispersion relation tanh [ Koq,(id

for the antisymmetrical (odd) modes of the linear symmetrical waveguide: cot (KOqf2I d) = - 4, -. 4f

The amplitudes B, and B, of the electric field inside the linear medium in the case of the antisymmetrical solution are given by B:

=

2 -q4,2[sin(~,q,;d)]-~[1 - b,ZcotZ(~,qf$d)], a,

/?
(3.45)

262

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

2

B; = - q ~ [ s i n h ( ~ , 4 , $ d ) ] -[ ~l - b : c ~ t h ~ ( ~ ~ Q ~ ffl> d ) ]n,, , aC

[IV, 8 3

(3.46)

In the case of the symmetrical wave (S)the time-averaged total guided wave power in W/M of wavefront is expressed as

P = 4PoBqc(l - r2) x

I

1

I

x 1

+ iqc(l + 1 2 ) [cos(icoqf~d)]-2 Kod + sin (K04f d ) ] }

[

(3.4,)

4f

+ fq,(l + r1) [ c o s h ( ~ , i j ~ f d d ) ] - ~ (3.48)

for B > nf, where

Equations (3.47) and (3.48)give us the dependence w = w(B, P), that is, the dispersion relation for the nonlinear symmetrical wave (S).We observe that for P = 0, eq. (3.47)gives 1 - r, = 0, that is, the dispersion relation of TE polarized symmetrical (even) modes of the symmetrical linear waveguide. It is also easy to verify that as d - t 00, eq. (3.48) yields double the energy flux of the surface waves at a nonlinear interface (see 2). In a similar manner, making use of eqs. (3.45)and (3.46)we obtain the power flow in the nonlinear antisymmetrical solution (AS):

P = 4P0B4,(l - t 2 )

for p< n, and

1

+ iq,(l + t , ) [sinh(~,q,$d)]-~ (3.51)

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NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

263

for > n,, where t , = - b , coth(K,g,$),

t , = -b,cot(~,q,$d).

(3.52)

Here, too, it is easily seen that at P = 0, eq. (3.50) reduces to 1 - t, = 0, that is, the dispersion relation for antisymmetrical (odd) modes of the symmetrical linear waveguide, whereas as d + 00, eq. (3.51) gives double the power flow of the nonlinear surface waves. For a material system with complete symmetry, that is, n, = n, and nZc = nZs> 0 (self-focusing nonlinearities), one expects that at high power self-focused fields can occur in one or both bounding media. The dependence of the dimensionless flux PIP, on the propagation constant /?is shown in fig. 3.5 for a symmetrical layered structure with parameters n, = 2.0, n, = n, = 1.5, and d/A = 0.6 (see AKHMEDIEV [ 19821). Branch S exhibits field distributions symmetrical with respect to the film center (symmetrical TE, branch). In this case with increasing power flow a field minimum develops in the center of the film, and two symmetrical field maxima move into the cladding and substrate media. Branch A only exists above a power threshold, and the associated fields are self-focused in either the cladding or the substrate (asymmetrical TE, branch). For curve AS the field distributions retain symmetry with respect to the center

Fig. 3.5. Normalized power Row P/P,versus effective index Bfor a symmetrical layered structure. Here n f = 2.0, ne = n, = 1.5, and d / t = 0.6.The curves are marked as follows: S, symmetrical TEo branch; A, asymmetricalTE, branch; AS, symmetrical TE,branch; B, asymmetrical TE,branch. (After AKHMEDIEV [1982].)

264

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, g 3

of the thin film (symmetrical TE, branch). This branch evolves from the usual low-powerTE, mode with field extrema in the film, to a high-power TE, branch with symmetricalfield maxima localized in the two nonlinear bounding media. Curve B has a power threshold and is similar to that labelled A, since the field distributions are asymmetrical with respect to the film center (asymmetrical TE, branch). In the following paragraphs we will derive the exact dispersion relations for TE polarized guided waves in a planar structure consisting of an optically linear dielectric film embedded in dissimilar optically nonlinear unbounded media. The three-layer waveguiding structure consists of a nonlinear substrate characterized by the Kerr dielectric function E = e, + a, I E I in region I (z < 0), a thin dielectric film of thickness d with dielectricconstant e, in region I1 (0 < z < d), and a nonlinear cladding characterized by the Kerr dielectric function E = eC + ac I E 1 in region I11 (z > d). The Maxwell equations for TE polarized waves propagating along the x-axis with effective index fi are

dZEF ~dz2

K:(p

- Ef)E;I

=

0,

z
(3.53)

0
(3.54) (3.55)

For the nonlinear substrate medium the field solutions are Ej(z) =

(:)"'

q,{cosh[ Icoq,(z, - z)]}-' ,

z <0

(3.56)

for a, > 0 (self-focusing nonlinearity) and

(3.57) for a, < 0 (self-defocusing nonlinearity), where qs = (/I2 The fields inside the optically linear dielectric film are written as cos(lcoqfz)+ 4 3[ t a n h ( ~ ~ q ~ z ~ sin(Koqfz)}, )]*' 4f

(3.58)

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265

NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

where 0 < z < d, qf = ( 6 , - Pz)l/' and the f 1 refer to the cases a, > 0 and a, < 0, respectively. For the nonlinear cladding medium the nonlinear wave solutions are E.Y(z) =

(:)"'

for ac > 0 and EJ1'(z) =

(1

~

q,{cosh [ Koqc(Z, - z)]} - ,

:

IJ1"

z >d

q,{sinh [ Koqc(zc - z ) ] }- ,

z >d

(3.59)

(3.60)

for a, < 0, where 4, = (B2 - E , ) ~ ' ~ . The dispersion relations are obtained from matching the tangential electric and magnetic fields at the film-cladding interface ( z = d ) :

(3.61) where

and 1 refer to self-focusing and self-defocusing nonlinearities, respectively. This result is very similar to the linear case, with the exception that 4, and 4, are replaced by u i ' q , and u : l q C , respectively. When as+O, ac-rO, then z, -+ + co,z, 4 - co,v, + + 1, and u, --+ + 1, and from eq. (3.61) we obtain the dispersion relation for TE polarized modes of the linear asymmetrical waveguide (see eq. 3.4). For B > n, the analytical solution of the Maxwell equation (3.54)is E;'(z)

=

I

EJ'(0) cosh ( KOgfz)

+2 4 [tanh (rc0q,z,)] * 4f

I

sinh(rcOQfz) ,

(3.62)

where 0 < z < d, & = (B' - E ~ ) " ~ and , the 1 refer to the cases a, > 0 and a, < 0, respectively. Continuity of the magnetic fields gives the dispersion relation for /?>n,:

(3.63)

266

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV,§ 3

The guided wave power per unit length along the y-axis is P = P, + Pf + P,, where (3.64) (3.65)

for fl< n,, and

x

[( + y) 1

cosh(K,q,d)

+ 2 480: ‘ sinh(K0qfd) ~

4f

for fl > n , (see SEATON,VALERA,SHOEMAKER, STEGEMAN, CHILWELL and SMITH[ 19851). Since the constants of integration z, and z, in eqs. (3.56), (3.57), (3.59), and (3.60) are related by way of the boundary conditions and are dependent on the power carried by the nonlinear guided wave, the propagation constant fl obtained by solving the dispersion relations (3.61) and (3.63) becomes power dependent. Next we show that a knowledge of the field shapes is not necessary, however, to determine the dispersion relations of the problem (see BOARDMANand EGAN[1984a,b, 19851). The Maxwell equations (3.53), (3.54), and (3.55) integrate to (3.68) (3.69) (3.70)

Iv, $31

261

NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

because Ey(z) and dEy/dz are required to vanish as z + + 00 ; here cf is an integration constant. Suppose that the electric fields at the boundaries z = 0 and z = d of the waveguiding film are Eo and Ed, respectively. It is also useful, at this stage, to define the quantities y = (fl' -

,

E -

1, ,E2)1/2 0 ,

so that the gradient of the field at z

y, = (fl'=

0 and z

E,

=

- '2 a c E2)'12 d

3

d can be written as (3.71)

Then by using eq. (3.71) the integration constant cf can be expressed as cf = &qf

+ y,Z)E; = K;(qf + y : ) E j .

(3.72)

After some manipulation it can be shown that the electric fields E; and E i at the boundaries of the waveguiding layer are related to each other through the following equation for a conic section (a hyperbola or an ellipse):

where qs = ef - E,, q, = E~ - cC. Depending on the specific material parameters of the three-layer planar structure, for a particular value of E; there may exist two, one, or no values of Ed'. It may be noted that for a purely symmetrical waveguide, that is, E, = E,, a, = a,, the conic section reduces to straight lines which are perpendicular to each other: (E; - E j ) [(Ef - E , ) - 4as(E; + E j ) ] = 0 .

(3.74)

This implies that both symmetrical (E, = E d )and antisymmetrical (E, = - E d ) waves exist in the symmetrical layered structure. The third option is the assymmetrical wave, for which E; # EZ: (3.75) This asymmetrical wave cannot exist in the linear limit (see AKHMEDIEV [ 1982]), BOARDMAN and EGAN[ 1984bI). The four eigenvalue equations for nonlinear guided waves (fl < n,) in the asymmetrical layered structure are (3.76)

268

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, 8 3

for even parity solutions, for which E, and Ed have the same sign (E, > 0, Ed > 0, or E, < 0, Ed < 0 ) and

(3.77) for odd parity solutions, for which E, and Ed have opposite signs (E, > 0, Ed < 0 or E, < 0, Ed > 0), where qf = (ef - f i 2 ) ' I 2 . For nonlinear surface waves (fi> nf) the eigenvalue equations are determined from eqs. (3.76) and (3.77); that is,

(3.78) for even parity waves and

(3.79) for odd parity waves. Next we follow the elegant analysis of BOARDMANand EGAN[ 1984bl to calculate the guided wave power flow without knowing the optical fields in the outside nonlinear media. The starting point of the analysis is again the first integrals (3.68), (3.69), and (3.70) of Maxwell equations. After differentiation with respect to z, eqs. (3.68) and (3.70) become

(3.80) (3.81) Taking into account eqs. (3.80) and (3.81) and using

(3.82) (3.83) after some manipulation we can calculate the power flows P, and P, in the

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NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

269

nonlinear bounding media: P S

= 2 ( ~ ) 1 / 2 ( 2 a s K o ) - ~ B (f4 Ys s ) ,

(3.84) (3.85)

In the linear waveguiding film we have (3.86) so that by integrating with respect to the depth variable z we obtain the following contribution to the total power flow:

p -1

d

Furthermore, by using eq. (3.71) the power flow in the linear film can be expressed as

The total power flow in the planar structure is P = P, + Pf + P,,where P, and P, are given by eqs. (3.84) and (3.85), respectively, and Pf is given by eq. (3.88). Since E: can be expressed in terms ofEZ and, through the eigenvalue equation, EZ is a function of the effective index 8, then the power flow P can be varied with fi for a particular value of the frequency w. As a specific example, consider the case n,, > 0 and nZc> 0; that is, both bounding media exhibit self-focusing nonlinearities. This case contains by far the richest set of new phenomena (see SEATON,VALERA, SHOEMAKER, STEGEMAN, CHILWELL and SMITH[ 19851). We see from fig. 3.6 that when n, = n, but n2c # n,,, that is, the optical nonlinearities are dissimilar, the nonlinear wave solutions evolve into two unconnected branches A and B. If, furthermore, n, # n,, the curves shift and distort with respect to the power axis but no new features emerge. Branch B in fig. 3.6 exists only above a certain power level. Branch A, which evolves from the linear case, exhibits field localization in the more nonlinear medium of the two (the cladding); that is,

270

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, $ 3

1.585

1.580

1.575

1.570

-

P(mWhnrn1

I

25

50

I

75

100

Fig. 3.6. TE, guided wave power versus effective index Bfor d = 2 pm,nc = ns = 1.55, n, = 1.57, nzc = 2 x m2/W, and n, = m2/W. (After SEATON,VALERA,SHOEMAKER, STEGEMAN, CHILWELL and SMITH[ 19851.)

it degenerates at high powers into the corresponding single interface surface wave. The second branch in fig. 3.6 (branch B) starts with a field extremum in the medium with the smallest nonlinearity (the substrate) and terminates with field maxima in both nonlinear bounding media. Nonlinear guided waves in the lower branch A are excited until the maximum is reached. A further increase in guided wave power can only be achieved by switching to the upper branch B, on which the field distribution and hence the attenuation are different from those corresponding to lower branch A. Therefore, switching between the two branches should be accompanied by a change in the transmitted intensity. A subsequent decrease in guided wave power on upper branch B leads to switching back to the lower branch A at a much lower power than for the switch-up. Therefore, a hysteresis loop or bistability could occur (BOARDMANand EGAN [ 19851).

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NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

27 1

3.1.3. Nonlinearflm In the case of a Kerr-law waveguiding film, the field solutions of the Maxwell equations are expressed in terms of Jacobian elliptical functions (MIYAGIand NISHIDA [ 19721,AKHMEDIEV, BOLTARand ELEONSKII [ 1982a,b], FEDYANIN and MIHALACHE [ 19821, BOARDMAN and EGAN [ 1984a, 19861, HOLLAND [ 19861). In the following paragraphs we consider an asymmetrical planar layered structure consisting of an optically linear medium (the substrate) with dielectric constant cSin region I (z < 0), a Kerr-law dielectric film of thickness ddescribed by the dielectric function E = 8, + afI E I in region I1 (0 < z < d), and a linear medium (the cladding) with dielectric constant E~ in region 111 (z > d). The Maxwell equations for TE polarized waves propagating along the x-axis with propagation constant B are d'E; dz2

~-

d2E:' dz'

~-

K;(/?'

-

+ ~;a,(l?;')~= 0 ,

Ef)E;I

z
(3.89)

0
(3.90)

z>d.

(3.91)

We look for solutions localized near the surfaces of thin film with fields that fall to zero as IzI + co. From eqs. (3.89) and (3.91) we then find E:(z) E;"(z)

=

Eo exP(Koq,Z)

=

9

Ed exp [ - Koqc(z - d ) ] ,

z
z

>d ,

(3.92) (3.93)

where qs = (p' - &,)I/' and qc = (p' - E,)'/'. Consider first the case of a self-defocusing Kerr-like nonlinearity (af = - I af I < 0). In this case we have fi < n,, where n, = $ / 2 and the exact solution of eq. (3.90) can be expressed in terms of Jacobi elliptic function (BYRDand FRIEDMAN [ 19541): (3.94) for 0 < z < d, where t = (1 + m ) - '/'qf, qf = (nf - f12)1/2, and 8 is a constant of integration. Here sn is a specific Jacobian elliptical function that is a type of sine function and m is the modulus of this Jacobian function (0 < m < 1).

212

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, I 3

From the boundary conditions we are left with the eigenvalue equations (3.95) (3.96) where cn2(8/m) = 1 - sn2(O/m) and dn2(O/m) = 1 - m sn2(O/m). Here cn is another specific Jacobian elliptic function that is a type of cosine function. It may be mentioned that for m = 0 one derives from eqs. (3.95) and (3.96) the dispersion relation of TE polarized modes of the linear asymmetrical waveguide. Making use of eqs. (3.92), (3.93), and (3.94), we find the total power flow P = P, + Pf + P, carried by the nonlinear guided wave (AKHMEDIEV, BOLTARand ELEONSKII[ 1982a,b], FEDYANIN and MIHALACHE[ 19821): P,

sn2(elm)

= Po/?mt2

9

(3.97)

Pf = 2P0Bt[ x0td - E(xotd + O/m) + E(O/m)] ,

(3.98)

P,

=

4s

sn2(xotd + O/m) PoBmt2 4,

9

(3.99)

where Po = (2 1 ufl ice)- (&,-JpO)l/' and E(O/m) is an elliptic integral of the second kind (see BYRD and FRIEDMAN [ 19541). In the approach of AKHMEDIEV, BOLTARand ELEONSKII[ 1982a,b] the propagation constant Pis treated as an implicit function of the modulus of the Jacobian elliptic function. Thus solving the dispersion equations (3.95) and (3.96), one obtains the values of the effective index /?for every 0 < m < 1. By means of eqs. (3.97), (3.98) and (3.99) we then find the corresponding values of the power flows P,, Pf and P, and therefore the dependence P = P(B). Another very useful technique developed by BOARDMAN and EGAN [ 19861 makes use of the boundary field amplitudes Eo and Ed to calculate the dependence of the propagation constant on the power flow down the guide. In the case of a self-focusing dielectric film (af> 0) the exact solution of nonlinear wave equation (3.90) is expressed as: (3.100)

Iv, 8 31

NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

213

for 0 < z < d, where t = (1 - 2m)- ”’q,

for /3 < n,,

t = (2m

- 1)- ”’9, for p > n,,

and qf = (p’ - nf2)”’. The dispersion relations for these nonlinear guided waves are (3.101) (3.102) from which one derives the dependence /3 = /3(m), where m is the modulus of the Jacobian elliptic functions. By means of eqs. (3.92), (3.93) and (3.100) we obtain the following expressions for the power flows P,, Pf and P, in the linear substrate, nonlinear waveguiding film, and linear cladding, respectively : P,

=

Popmt2

cn2(elm)

4s

+ elm) - E(O/m) - (1 - m)rc,td], cn2(ic,rd + elm) = Po/3mt2

Pf = 2P,/3t[E(rcotd P,

,

(3.103) (3.104) (3.105)

4c

In order to find the magnitude of the power flow P for each propagation constant /3, we compute numerically, using eqs. (3.101) and (3.102), the dependence of p on the modulus m of the Jacobian elliptic functions. For the material parameters used by BOARDMAN and EGAN[ 19861, that is, n, < n, < n,, all the branches TE,,, (m > 0) are induced by the self-focusing nonlinearity so that nothing is propagating until a certain power threshold is reached (see fig. 3.7). BOLTAR This threshold behavior was first shown numerically by AKHMEDIEV, and ELEONSKII [ 1982a,b] for higher-order TE, (m 3 1) branches. They used a different set of material parameters that allow a linear limit for the TE, branch. Since two different values of the propagation constants can exist for the same power level, and power thresholds exist, this nonlinear layered structure is of potential interest as a low-threshold device and, eventually, as an optical switch.

274

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCWRES

[IV, 8 3

1I

1

3.2. N O N L I N E A R T H I N - F I L M G U I D E D WAVES IN SATURABLE M E D I A

We now investigate the effect of saturation on the power-dependent wavevector and field distributions of nonlinear thin-film guided waves. We consider first a dielectric planar layered structure consisting of a linear substrate with dielectric constant E, in region I (z < 0), a linear dielectric film of thickness d and dielectric constant Ef in region I1 (0 < z < d), and a nonlinear self-focusing cladding in region I11 (z > d) characterized by the dielectric tensor (2.19) where the parameter esat is the maximum change in the dielectric function; that is, E + E, + E,, for large field intensities. For TE polarized waves we have E = (0, E,, 0), and Eyis a real quantity in the absence of loss. The Maxwell equations are d2EI dz2

2~ : ( / 9 ~-

E,)E:

=

0,

z
(3.106)

O
(3.107)

IV, f 31

215

NONLINEAR OPTICAL WAVES GUIDED BY T H I N DIELECTRIC FILMS

The first integrals of eqs. (3.106), (3.107), and (3.108) are z
(3.109) O
(3.110)

(3.1 11) where cf is the integration constant. For TE polarized waves both the fields E,, and their derivatives dE/dz are continuous across the interfaces z = 0 and z = d. Next we derive the following relations between the integration constant cf and the fields E, and Ed at the interfaces z = 0 and z = d, respectively: cf = KO'

( ~ -f E,)E; = (Ef - E , ) E -~

@,'!

Since dE:'/dz = film can be written as

(3.112)

and taking into account that the integration over the

(3.113) finally we obtain the dispersion relation for TE, waves (LANGBEIN, LEDERER and PONATH[ 19851): (3.114)

E,,,,~ = E,

+

1

-

E,:

IOE2

&FL(E;) d(E;).

(3.115)

216

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV,5 3

We have M, = 1 if a self-focused peak (real field maximum) occurs in the nonlinear cladding medium; otherwise Mc= 0 (virtual field maximum). It should be noted that when a peak occurs in the nonlinear cladding medium, the field amplitude Ey is evaluated from @,(EyY /3) = 0. For fl > nf the dispersion relation is (3.116)

This dispersion relation has solution /3 > nf if, and only if, there is a real field maximum (self-focused peak) in the nonlinear cladding; that is, Mc= 1 and ijc = - (8' - cCNL)'/' < 0. If there is no self-focused peak in the cladding medium (M,= 0), we have 4, = 4, = [q(u)]'/', where q(u) is given by eqs. (2.32) or (2.33). Thus by using eqs. (3.112) and (3.113) we obtain (3.117)

pf

- ZPOSU -1.

[

Kod (4: + 4 3 4:

+ -4,+ 4;

I

q* (4: + 4 3

4; (482 + 4 3

dx

(3.118) (3.119)

where u = a, E:. In the case of a Kerr-law nonlinear cladding the integral (3.119) can be evaluated analytically, and we obtain

P, = 2P0/l[(#?2- &,

+ $u)"2

- (8' - I'/),&

.

(3.120)

For a self-focusing cladding (ac> 0) a field maximum can occur in that medium at sufficiently high powers; that is, M, = 1 and 4, = - q c , where 4, = [ (P(U)]'/~, with q(u) given by eqs. (2.32) and (2.33). In this case the results can be summarized in the following form: (3.121) (3.122)

and P, is given by eq. (3.117).

Iv, 31

NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

211

For a Kerr-law self-focusing cladding the integrals in eq. (3.122) can be evaluated analytically, and we have P,

=

2P0p[(p’

- E,

+ $4)”’

+ (8’ - E,)”2]

.

(3.123)

For p > n, we finally obtain

(3.124) (3.125) and P, is given by eq. (3.122). We consider the case of liquid crystal MBBA (n, = 1.55, n,, = m2/W) on a glass waveguide (nf = 1.57, d = 2 pm) with a substrate chosen such that n, = n, = 1.55. We have performed numerical calculations of the effective index /3 as a function of the total power flow P. The numerical results for TE polarized waves guided by a planar layered structure with a nonlinear cladding described by the dielectric function (2.19) are shown in fig. 3.8 for several values of the dimensionless parameter esat. It should be noted that, compared with the the characteristic behavior of TE, and TE, nonlinear Kerr-law case (E,,, = a), wave solutions is preserved provided that the saturable value nsat = ( E , + E,,,)~’’ - n, is not too small (i.e., for E , , ~ = 0.1256 or nsat = 0.04 in fig. 3.8). The absolute maximum in the TE, guided wave power depends strongly on the saturation value, and for below a certain value of n,, (i.e. for E,, = 0.0155 or nsat = 0.005) no absolute limiting action is predicted (see approaches the value fig. 3.8). For the TE, branch the propagation constant /I asymptotically with power (see the curve corresponding to = (E, + E,,, = 0.1256 in fig. 3.8). Thus the characteristic features of the TE, branches are retained provided that nsat is larger than the low power refractive index differences ( n f - n,) or ( n f - n,). The important conclusion to be drawn from these calculations is that the saturation effects, if too large, can alter and in some cases eliminate the more interesting power-dependent features of nonlinear guided waves. For two self-focusing saturable bounding media (n2=# n2s) the effects of saturation on the two separate (unconnected) TE, branches is dramatic (see STEGEMAN, WRIGHT,SEATON,MOLONEY,SHEN,MARADUDIN and WALLIS [ 19861). In this case the switching characteristics are quite different when saturation is included in both the substrate and the cladding. This feature

278

",

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

I3

B €*at

1584

=-

1580

1576

1.572 1568

€sat

=0.0624

= 0.0624 1.564

= 0.0155

1.560 1.556 1552

t

I

I

,

P (mW/mml I

I

I

I

should be important for applications of nonlinear guided waves to all-optical switching devices. We consider now a three-layer asymmetrical structure consisting of a linear substrate with dielectric constant E,, a thin dielectric film of thickness d with dielectric constant q,and a nonlinear self-defocusing (ac < 0) cladding characterized by one of the two types of saturated nonlinearities:

(3.126) or

(3.127)

Iv, 8 31

NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

279

We observe that both dielectric tensors (3.126) and (3.127) are Kerr like, that is, E -+ E, - I a, 1 E,’ for small field intensities, and reveal a common saturable level ( E , - E , , ~ ) . In the case of a self-defocusingcladding the field pattern reveals a virtual field maximum in the nonlinear cladding (Mc = 0) and the dispersion relation is (3.128) where

and CP(U) =

B2 - ~c +

q(u) = B2 - E,

&sat

- exp(

-&[I

+ E,,~-

U

,I);-

(3.129)

(3.130) U

corresponding to the dielectric tensors (3.126) and (3.127), respectively. The total power flow is P = P, + Pf + P,,where Ps,Pf and P, are given by eqs. (3.117), (3.118) and (3.119). We have performed numerical calculations of the effective index B as a function of the total power flow P for several values of the dimensionless parameter E , , ~ . The numerical results for TE waves guided by a multiple quantum well (MQW) GaAl,As, -, structure are shown in fig. 3.9 for both dielectric functions (3.126) and (3.127). For Kerr-law ( E , , ~ = 0 0 ) self-defocusing cladding with n, n, there is a maximum in the power that can be transmitted. For a realistic saturable nonlinear cladding, optical limiting ) ~enough / ~ (see fig. 3.9 for occurs, provided that nsat = n, - ( E , - E ~ ~ is ~large E , , ~ = 0.0676, i.e., nsat = 0.01). We see from fig. 3.9 that for a self-defocusing cladding medium the effective index B decreases monotonically with guided wave power, and for n, > n, cut-off occurs at a finite power. This phenomenon can be used for upper-threshold optical devices in which the cut-off power can be tuned, for example, by using a thin film of bulk GaAl,As, --x with variable refractive index n, (x) (see STEGEMAN, SEATON, HETHERINGTON, BOARDMAN and EGAN [ 19861).

=-

280

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

3.3860

h

[IV, 5 3

TE

3 3854 ,Eso, = 0.0676

-.-. -. 12

16

PImWlmm I

20

24

Fig. 3.9. Effective index /3 versus power P for a self-defocusing cladding. Here nr= 3.390, nC=3.385,n,= - 2 x 10-9m2/W,n,=3.380,d= 1.07pm,andI=0.82pm.Solidanddashed lines correspond to dielectric tensors (2.19) and (2.20), respectively. (After MIHALACHE and MAZILU[ 1986bl.)

3.3. STABILITY TO PROPAGATION OF NONLINEAR SLAB-GUIDED WAVES

The question of stability to propagation of various TE, nonlinear guided wave solutions has only recently been studied by numerical techniques (JONES and MOLONEY[1986], MOLONEY,ARIYASU, SEATONand STEGEMAN [ 1986a,b], LEINE, WACHTER,LANGBEIN and LEDERER[ 19861, MIHALACHE and MAZILU[ 1987a,b]). Analytical stability analysis is complicated by the fact that the system under study is of Hamiltonian form (see JONES and MOLONEY [ 19863). Since the eigenvalues of the linearized system all lie on the imaginary axis, the usual stability arguments associated with dissipative systems do not apply (unless one deliberately introduces losses into the system). We treat first the specific case of a TE polarized guided wave in a symmetrical nonlinear planar waveguide consisting of a linear guiding film with refractive index n,bounded on both sides by identical self-focusing Kerr-law cladding and substrate layers; that is, n, = n, and n, = n,, (see fig. 3.10a). The dependence of the power flow P on the propagation constant /?is shown in fig. 3.10b for a symmetrical layered structure with the following parameters: n, = n, = 1.5, nr = 2, and d / l = 0.4 (see AKHMEDIEV [ 19821, JONES and MOLONEY [ 19861). In fig. 3.11 we present the results of the evolution of the field distribution with propagation distance, taking as initial data the electric field distribution

IV, 5 31

NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

I

28 1

I

2.4

2.0

1.6

(b)

(0)

Fig. 3.10. (a) Sketch of planar waveguide geometry; (b) nonlinear dispersion curves for a symmetrical planar waveguide. Here n, = 2.0, n, = n, = 1.5, and d/A = 0.4. The curves are marked as follows: S, symmetricalwave; A, asymmetricalwave. Beyond bifurcationpoint 8 = 8, m 1.89 both branches (S and A) are unstable (dashed line). The doubly degenerate A-wave becomes stable on the positively sloped region. (After JONESand MOLONEY [1986].)

immediately before and after the bifurcation point on the symmetrical TEo branch (S). At b = 1.89 < 8, the symmetrical wave is stable to propagation over at least 180 wavelengths (see fig. 3.1 la), whereas at @ = 1.90 > 8, the S-wave breaks

-063

0 la1

0632-

-063

0

0 6 3 Z-

(b)

Fig. 3.1 1. Evolution of the field distribution with propagation distance for the symmetrical wave (S) just below (a) /I= 1.89 and beyond (b) B = 1.90 bifurcation point B = 8,. (After JONES and MOLONEY [1986].)

282

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, § 3

symmetry after only 18 wavelengths (see fig. 3.1 lb). In this case the wave starts to drive either into the substrate or into the cladding layers. Therefore the symmetrical TE, branch (S) loses stability on the positive-sloped branch of the dispersion curve (see fig. 3.1 lb) at the bifurcation point (critical propagation constant fl = fl, = 1.89), where the double degenerate asymmetrical wave A appears (JONES and MOLONEY[ 19861). Next we consider an asymmetrical three-layer planar structure consisting of a linear substrate with dielectric constant E, = n: in region I (z < 0), a thin dielectric film of thickness d with dielectric constant E~ = nf in region I1 ( O < z < d ) , and a nonlinear self-focusing cladding characterized by the diagonal dielectric tensor (2.19) or (2.20) in region I11 (z > d). The TE polarized wave of frequency o propagates along the x-axis, and the electric field is homogeneous in the y direction (z being the transverse coordinate). The only nonvanishing component of the electric field EJr, t ) is given by eq. (2.1 1). Then in the usual slowly varying envelope approximation we obtain the following equation for the amplitude A(x, z) = oc,“’E,,(x, z): (3.131) HereB(z) = Ofor - 00 < z < dand O(z) = 1 forz > d, y’(z) = fl’ - n;forz < 0, y’(z) = fl’ - r$ for 0 < z < d, y2(z) = fl’ - n,” for z > d, and (3.132) (3.133)

corresponding to the dielectric functions (2.19) and (2.20), respectively. Note that for a Kerr-law medium we have f( IA 1)’ = IA I ’. The parabolic equation (3.131) is a complicated mixed type of linear/ nonlinear Schrodinger-like equation. The x-independent solution of eq. (3.131), that is, A(0, z) = A,(z), represents stationary nonlinear guided waves whose effective index fi is subject to a nonlinear dispersion relation fl = fl(P), where P is the power flow. Equation (3.131) has two integrals of motion: I ( 8 ) given by eq. (2.39) and

IV, 8 31

NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRICFILMS

283

Here we have g( IA I 2, =

’ * f( IA 12) d( IA , 2 ) ,

[,IA

(3.135)

when f(IA 1)’ is given by eq. (3.132) or (3.133). For Kerr-law media we obtain g( / AI 2, = $ ( A I and for saturable media g( ( A I ’) has the expressions

g( I A

I2

=

+

&sat

(

pn 1 +

z)]

IA I2

(3.136)

9

(3.137)

corresponding to the dielectric functions (2.19) and (2.20), respectively. Note that for arbitrary solutions of eq. (3.131) we thus have dZ/dx = 0 and dH/dx = 0. The stability of the stationary nonlinear thin-film guided waves was investigated numerically on an IBM 370/135 computer. For the difference approximation of eq. (3.131) we used the Crank-Nicolson scheme (see, for [ 19741). The corresponding system of example, AMES[ 19651, GREENSPAN nonlinear equations was solved on the successive steps in x by the Newton method combined with a matrix tri-diagonal inversion along z (see AMES [1965]). We have chosen the grid sizes equal in magnitude, K,AX = K,AZ = 0.4. This difference scheme makes it possible to conserve the integrals of motion l(8)and H ( b ) on the grid. The conservation of the total power flow P(B) was always better than 99%. Unstable stationary waves are defined as waves whose field distribution along the depth coordinate z changes with propagation distance x; otherwise, the stationary solution is called stable. For a Kerr-law nonlinear cladding the stationary wave is unstable on the negative-sloped region of the nonlinear dispersion curve P = b(P) and starts to drift into the nonlinear medium. In this case the emission of a single soliton, that is, a self-focused channel (see fig. 3.12) was found to be a route by which unstable nonlinear guided waves decay (WRIGHT,STEGEMAN,SEATON, MOLONEYand BOARDMAN[ 19861, MOLONEY,ARIYASU,SEATON and STEGEMAN [ 1986b], MIHALACHE and MAZILU[ 1987a1). In the case of a nonlinear saturable cladding described by the dielectric = 0.1256, the TE, wave on the negative-sloped branch tensor (2.19) with of the nonlinear dispersion curve (see fig. 3.8) is unstable (MIHALACHE and MAZILU[ 1987b1). Numerical propagation over the first 400 wavelengths is shown in fig. 3.13 for B = 1.5685. In this case the instability is weak, the field

284

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, 8 3

r 0.4

80

70 60

- 50 E

a40

I

X

30 20 10

0

-1.0

0.0

2.0

4.0

5.0

Fig. 3.12. Evolution ofthe TE, nonlinear guided wave field distributionwith propagation distance for a Ken-law cladding. The initial field pattern A&) corresponds to /3= 1.5685. (After MIHALACHE and MAZILIJ[1987a].)

remains confined to the waveguiding film, and the field maximum oscillates aperiodicallyback and forth between the 6lm boundaries. The evolution of the TE, nonlinear guided wave field distribution with propagation distance for the case of a saturable cladding described by the dielectric tensor (2.19) with = 0.1256 is shown in fig. 3.14 for fi = 1.58 on the positive-sloped branch of the nonlinear dispersion curve f l = fi(P). For this value of the effective index the nonlinear stationary wave A,(z) is stable to propagation (see fig. 3.14). Numerical propagation of the stationary wave confirms that the TE, nonlinear guided wave in an asymmetrical layered structure in which self-focusingoccurs in only one of the two bounding media is stable on the positive-sloped branch of the nonlinear dispersion curve (WRIGHT, STEGEMAN, SEATON, MOLONEY and BOARDMAN[ 19861). A TE, wave in an asymmetrical layered structure with a nonlinear selffocusing cladding is unstable on the negative-sloped branch of the nonlinear and LEDERER [ 19881). In this dispersion curve (LEINE,WACHTER,LANGBEIN case much of the TE, optical field energy is radiated by means of a soliton-like self-focused channel (see fig. 3.15).

IV,0 31

NONLINEAR OPTICAL WAVES GUIDED BY THIN DIELECTRIC FILMS

285

02 0 .I 00

200 180

160 140 120

x

80

60 40

20 0

-1.0

00

Z lpml

2.0

30

Fig. 3.1 3. Evolution ofthe TE,nonlinear guided wave field distributionwith propagationdistance for a saturable cladding described by the dielectric function (2.19) with = 0.1256. (After MIHALACHE and MAZILU[1987b].)

In the presence of dissipation the Kramers-Kronig relation dictates that at least linear absorption must accompany nonlinear refraction. Therefore we must include in eq. (3.13 1) an absorptive term: aA

- 2ipicO- =

aZ

a2A ~

azz

- y2(z) iciA + O(z)f( IA I ' ) A + iflic:r(z)A

, (3.138)

where T(z) is the absorption profile. The effect of linear absorption on TE, nonlinear guided waves in an asymmetrical waveguide with a Kerr-law cladding has been investigated by GUBBELS, WRIGHT,STEGEMAN, SEATONand MOLONEY[ 19871 by using the beam propagation method (see FLECK,MORRISand FEIT[ 19761, FLECK, MORRISand BLISS[ 19781). It was shown that it is preferable to fabricate nonlinear asymmetrical optical waveguides with lower absorption in the waveguiding film. It is interesting to note that the beam propagation method (incorporating a split-step fast Fourier transform) provides a unified treatment of

286

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

0.0

2 .o

Z [,urn1

[IV, 8 3

4.0

Fig. 3.14. The same as fig. 3.13, but for B = 1.58. (After MIHALACHE and MAZILU[1987b].]

various guided and radiation field problems subjected to the paraxial approximation. To investigate the effects of linear absorption on TE, nonlinear guided waves, eq. (3.138) was solved numerically with initial data corresponding to a nonlinear stationary TE, solution. Two cases are of interest, namely, lower absorption in the film than in the bounding media and vice versa. Consider first the case rs = r, > r,. In fig. 3.16 we show the evolution of the flux versus propagation distance for three points on the dispersion curve. Note that curve (b) in fig. 3.16 displays the least absorption and this is because the field is concentrated mainly in the film, which has the lowest absorption. When the absorption coefficients are reversed (rs= r, < rf),the results change drastically (see fig. 3.17). In fig. 3.18a we show the input wave used to generate curves (b) in fig. 3.16 and 3.17. The optical field after 60 wavelengths is shown in fig. 3.18b in the case where the film has the lower absorption and in fig. 3 . 1 8 ~in the case where the film has the higher absorption. We see from fig. 3.18b that the self-focused peak is attracted toward the region of lower absorption, that is, the film.

IV,9 3)

N O N L I N E A ROPTICAL WAVES G U I D E D BY T H I N DIELECTRIC

FILMS

287

10

05 00

0

2

L

6

8

ZIpl

Fig. 3.15. Evolution of the nonlinear stationary TE, input field for B = 1.5643. Here n , = n , = 1 . 5 5 , n,=1.57, n2,=10-9m2/W, d = 2 p m , and rl=0.515pm. (After LEINE, WACHTER,LANGBEINand LEDERER[1988].)

In figs. 3.19 and 3.20 we show the effects of linear absorption on one-soliton emission from a nonlinear optical waveguide. Here we have chosen the input flux as Pi, = 112 W/m, which gives one-soliton emission in the ideal case. For r, = r, = 0 and r, = the effects of linear absorption are pronounced. Although the emission of a localized wave-packet still occurs, the optical wave now loses a substantial amount of energy and broadens as it propagates away from the cladding-film interface (see fig. 3.19). For r, = r, = 0 and r, = l o - * there is no remaining remnant of the soliton emission (see fig. 3.20).

288

[IV,§ 3

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

P(W/rnl

120

Xlrml

I

10

20

30

LO

50

70

60

80

-b

Fig. 3.16. Power flux P versus propagation distance X for n, = nc = 1.55, n, = 1.57, m2/W, d = 2 pm, d = 0.515 pm, r, = r, = and r,= Curves (a) and (b) n,, = correspond to positive-sloped TE, branches of the nonlinear dispersion curve P = P(B)for the initial power flux P = 45 W/m. Curve (c) corresponds to the nonlinear surface polariton solution of initial flux P = 112.5 W/m. (After GUBBELS, WRIGHT,STEGEMAN, SEATONand MOLONEY [1987].)

t

120 -

P(W/rnl

~

I

10

20

30

40

50

60

70

~~

80

Fig. 3.17. The same as in fig. 3.16 but for r, = r, = lo-’ and r, = lo-’. (Atter GUBBELS, WRIGHT,STEGEMAN, SEATONand MOLONEY[ 19871.)

IV, § 41

w

T E NONLINEAR SURFACE PLASMON POLARITONS

289

0.030 0 015

00

(

-4 - 0.4

(a1

IN2

- 02

-2

0

2

4

00

2 [pml

Fig. 3.18. (a) Input wave used to generate curves (b) in figs. 3.16 and 3.17. (b) Field after 60 wavelengths (x = 30 pm) for r, = r, = 10- and rf= lo-’. (c) Field after 6 0 wavelengths (x = 30 pm)for r, = r, = and rf= 10- z. (After GUBBELS, WRIGHT,STEGEMAN, SEATON and MOLONEY[1987].)

Finally, in fig. 3.21 we show the two-soliton emission from an ideal nonlinear optical waveguide (i.e., r, = r, = r, = 0) for an input flux Pi, = 150 W/m exceeding the critical flux P, z 112 W/m. The input optical field was chosen as the linear TE, waveguide mode (see WRIGHT, STEGEMAN,SEATON, MOLONEYand BOARDMAN[ 19861).

8 4. Transverse electric polarized nonlinear surface plasmon polaritons 4.1. KERR-LAW BOUNDING MEDIA

In recent years there has been considerable interest in the theoretical (MILLS and SUBBASWAMY [ 19811, AGRANOVICH[ 19821)and experimental (USHIODA [ 19811, ABELESand LOPEZ-RIOS[ 19821) study of surface polaritons. As is well known, the boundary between two linear dielectric media cannot support a TE polarized surface polariton. However, if one or both of the media exhibit

290

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

10

0

[IV, 8 4

A

0

2

Z bml

4

6

Fig. 3.19. Soliton emission in the presence of absorption. The input flux in Pin= 112 W/m, r, = r, = 0,and r, = lo-'. The other parameters are like those in fig. 3.16.

an intensity-dependent refractive index, it has been predicted (see LITVAKand MIRONOV [ 19681, TOMLINSON [ 19801, MARADUDIN [ 19811, MARADUDIN [ 19831) that a TE polarized surface polariton should exist at powers exceeding a threshold value. In addition to the well-investigated nonlinear guided waves, which can be supported by a linear dielectric film surrounded by at least one medium

IV, 8 41

TE NONLINEAR SURFACE PLASMON POLARITONS

29 1

06

0.3 00

100

80

60

E

350 X

40

30

20

10

0

0

L

2

6

2 [pml

Fig. 3.20. The same as in fig. 3.19 but for

r, = rf=0 and r. = lo-’.

(cladding or substrate) with an intensity-dependent refractive index, new waves have also been predicted for metal films bounded on one of both sides by nonlinear media (STEGEMANand SEATON[ 19841, STEGEMAN, VALERA, SEATON,SIPEand MARADUDIN [ 19841, LEDERERand MIHALACHE [ 19861, MIHALACHE, MAZILUand LEDERER[ 19861). Nonlinear TE polarized waves guided by very thin metal films surrounded on both sides by nonlinear media

292

[IV,I 4

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

0.6

0.3 100

0.0

90

80

70

60

LO

30

0

i / \ l 0

2

L

6

Z [pml

8

10

Fig. 3.21. Two-soliton emission from a nonlinear optical waveguide. The input flux in r, = r,=r, = 0. The other parameters are like those in fig. 3.16. (After WRIGHT,STEGEMAN, SEATON,MOLONEYand BOARDMAN [ 19861.)

Pi,= 150 W/m and

exist only for power levels above a threshold that depends on the material parameters (see STEGEMAN, VALERA, SEATON,SIPE and MARADUDIN [ 19841). LEDERERand MIHALACHE [ 19861 demonstrated that nonlinear TE polarized surface plasmon polaritons also exist in planar configurations with either a nonlinear cladding or a nonlinear substrate. For this asymmetrical

IV, c 41

TE NONLINEAR SURFACE PLASMON POLARITONS

293

layered structure the nonlinear dispersion curves exhibit a definite power threshold and a limited range for the permitted propagation constants as well. We will now investigate the characteristics of nonlinear TE polarized waves guided by an asymmetrical three-layer configuration consisting of a linear substrate with dielectric constant E, in region I (z < 0), a very thin metal film with dielectric constant ef = - j < 0 in region I1 (0 < z c d), and anonlinear self-focusing Kerr-law cladding in region I11 (z > d). The field distribution in the substrate, film, and cladding regions, respectively, is given by

Here the surface field Eo is given by

a,E;

=

2{ 1 - tanh2[ rcoq,(z, - d)]} 4 cosh (KOBf d) + 2 sinh (x0Bfd)

(4.4)

Bf

As a result of the continuity requirements of E,, and dEJdz at the interfaces

z = 0 and z = d, the effective index 8 is subject to a dispersion relation

This dispersion relation has a solution if, and only if, n, > n, and z, > d; that is, a field maximum must always be situated within the cladding region. The power per unit distance along the wavefront carried by the wave is given by

294

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

x [(I

+$)cosh(rcOg,d)+ 25sinh(~,gd)]], 4f

[IV, 8 4

(4.7)

where Po = (2a, x0)- ' (q,/po)'/2 . Numerical calculations were performed with the following parameters : n, = 1.55, n,, = mZ/W, Ef = - 10, and I = 0.515 pm (argon ion laser). The dependence of the propagation constant /3 on the guided wave power P is shown in fig. 4.1 for different refractive indices of the substrate (see LEDERER and MIHALACHE [ 19861). As in the symmetrical configuration studied by STEGEMAN, VALERA, SEATON,SIPEand MARADUDIN[ 19841, we found a definite power threshold where the nonlinear TE polarized surface plasmon polaritons start. This threshold increases with the refractive index of the substrate. The dashed line in fig. 4.1 corresponds with the symmetrical planar structure with the following material parameters: n, = n, = 1.55, n,, = nZc = mZ/W.We see from fig. 4.1 that in the case of an asymmetrical configuration with n, > n, nd n,, = 0, two different values of the propaga-

Fig. 4.1. Propagation constant B versus guided wave power for n, = 1.55, n2c = m2/W, ef = - 10, I = 0.515 pm, and d = 0.005 pm. Solid curves: n, indicated at the curves and n,, = 0. Dashed curve: n, = n, = 1.55 and n2s = n,, = m2/W. (After LEDERERand MIHALACHE [ 19861.)

IV,8 41

TE NONLINEAR SURFACE PLASMON POLARITONS

295

tion constant correspond with the same value of the control parameter P, and an upper b-limit occurs in addition to the lower one that is determined by n,. The upper 8-limit results from the impossibility of field matching at the metal film-dielectric substrate interface. The field distribution for 8 near its lower limit n, shows that the electric field penetrates deep into the substrate region, and for 8 approaching the upper limit the field energy is mainly concentrated within the nonlinear cladding, which is favorable for diminishing absorption losses in realistic metal films. Thus, a possible switching between the upper and the lower branches of the dispersion curve is accompanied by a transition from a high transmission state to a low one. NON-KERR-LIKE BOUNDING MEDIA

4.2.

We consider an asymmetrical configuration consisting of a linear dielectric substrate with dielectric constant E,, a thin metal film of thickness d with dielectric constant Ef = < 0, and a nonlinear self-focusing cladding characterized by one of the dielectric tensors (2.19), (2.20) and (2.21). We use the formalism developed by LANGBEIN, LEDERERand PONATH[ 19851 to investigate the power dependence of the effective index for both saturable and power-law dielectric tensors (non-Kerr-like media). The dispersion relation of nonlinear TE polarized surface plasmon polaritons is given by tanh(KOSfd)= 4 f k C - 4.) where

a;

- 4s4c ’

(4.9)

and u = a,E:, where Ed is the electric field at the interface z = d and ~ ( uis) given by eqs. (2.32), (2.33) and (2.34). The dispersion relation (4.9) has a solution 8=-n, if, and only if, n, > n,; in this case a field maximum (self-focused peak) must be situated within the self-focusing cladding medium. The total power flow per unit length is P = Ps + Pf+ P,,where P, is given by eq. (3.122) and P, and Pfare given by eqs. (3.124) and (3.129, respectively. The numerical calculations were performed with the following parameters : n, = 1.55, n,, = 10-9m2/W (liquid crystal MBBA), ~f = - 10, n, = 1.6, d = 10W3pm, and A = 0.515 pm. The /3-power plots for the case of a selffocusing cladding characterized by the dielectric tensors (2.19), (2.20), and (2.21) are shown in fig. 4.2. As in the symmetrical configuration studied by

296

[IV,8 4

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

A

1.66 1.65

~

TE

1.64 1.63 1.62 -

i

Ii

&,1=0.25

I

I

‘.

0.25

8

1.60 -

P(mWhml

I

1 50

100

150

200

250

300

350

400

Fig. 4.2. The dependence of the effective index fl on the guided wave power P for n, = 1.55, n2= = lo-’ m 2 P , na = 1.6, E~ = - 10, d = lo-’ pm, 1 = 0.515 pm, a, = 1.75 x (m/V)1.5 (r = 1.5). and a, = 6.4 x 10- l 2 (m/V)2 (r = 2.0). Solid and dotted lines correspond to dielectric

tensors (2.19)and (2.20). respectively. Dashed lines correspond to a power-law cladding with the exponent r = 1.5 and r = 2.0. (After MIHALACHE, MAZILUand LEDERER[1986].)

STEGEMAN, VALERA,SEATON,SIPE and MARADUDIN[ 19841, we found a definite power threshold where nonlinear TE polarized surface plasmon polaritons start. The upper 8-limit that occurs in addition to the lower one (8 = n,) is determined from u = a,Ei = 0; that is, P, = Pf = 0 and P, # 0 (see termination points A and B in fig. 4.2). For the effective index 8 approaching the upper value, the optical field will mainly be concentrated within the nonlinear self-focusing cladding. We see from fig. 4.2 that the minimum power required for the excitation of nonlinear TE polarized surface plasmon polaritons increases with decreasing Furthermore, the effective index 8 approaches asymptotically with increasing power. The its limiting value of (8, + conclusion which can be drawn from these calculations is that the saturation effects can alter the specific power-dependent features of nonlinear TE polarized surface plasmon polaritons in configurations with Kerr-law media. Finally, it should be noted that the combination of the very thin metal film thickness required, the losses usually associated with surface plasmons for small I Ef I, and the large changes in the refractive index required, will probably make such nonlinear waves difficult to observe experimentally.

IV, § 51

EXPERIMENTAL STATUS

291

8 5. Experimental status In the last few years many experiments on third-order planar waveguide interactions have been reported, namely, nonlinear distributed couplers (CHEN and CARTER[ 19821, VALERA,SEATON,STEGEMAN, SHOEMAKER, Xu MAI and L ~ A O[1984], CHEN, CARTER,SONEK and BALLANTYNE [1986], FORTENBERRY, MOSHREFZADEH, ASSANTO,Xu MAI, WRIGHT,SEATON and STEGEMAN [ 19861, ASSANTO,SVENSON,KUCHIBHATLA, GIBSON, SEATON and STEGEMAN [ 19861); nonlinear coherent directional couplers (LATTES, HAUS,LEONBERGER and IPPEN [1983], LI KAM WA, SITCH, IRONSIDE,SEATONand MASON,ROBERTSand ROBSON[ 19851, CUL.LEN, STEGEMAN [ 19861); nonlinear guided wave optical limiters (VACH,SEATON, and KHOO[ 19841, BENNION, GOODWIN and STEWART[ 19851, STEGEMAN BERTOLOTTI, FERRARI,SIBILIA,ALIPPI and NESRULLAYEV [ 19871); and optical bistability and switching in nonlinear thin-film waveguides (MARTINOT, KOSTERand LAVAL[ 19851, LUKOSZ,PIRANIand BRIGUET [ 19861, PARDO, KOSTER,CHELLI,PARAIRE and LAVAL[ 19861, VALERA,SVENSON, SEATON and STEGEMAN [1986], VITRANT and ARLOT [1987], KIM, GARMIRE, SHIBATA and ZEMBUTSU [ 19871, CUSH,TRUNDLE, KIRKBYand BENNION [ 19871).

5.1. NONLINEAR DISTRIBUTED COUPLERS

The nonlinear coupling of a laser beam into a planar waveguiding structure containing an intensity-dependent refractive index medium is usually achieved [ 19841, LIAO, by distributed couplers such as prisms (LIAOand STEGEMAN VALERAand WINFUL[ 19851, ARLOTand STEGEMAN, SEATON, SHOEMAKER, VITRANT [1987]) and gratings (CARTERand CHEN [1983], VINCENT, PARAIRE, NEVIERE, KOSTERand REINISCH[ 19851). Basically, a high-refractive-index prism placed over a nonlinear optical waveguide constitutes a nonlinear prism coupler. When the direction of the incident laser beam is such that the parallel component of the plane wave wavevector of the incident light in the prism is equal to the guided wave wavevector, coupling becomes effective and the optical energy can be transferred from the prism to the waveguiding film and vice versa. However, the guided wave wavevector changes with power if one of the layered media has an intensity-dependent refractive index. Therefore, for a fixed angle of incidence of the input laser beam, the coupling efficiency becomes power dependent and decreases as the incident power increases.

298

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, B 5

CHEN and CARTER[1982] demonstrated the phenomenon of nonlinear coupling for grating coupling to surface plasmons guided by the interface between a metal (Ag) and a nonlinear medium (GaAs or Si). In this experiment the reflected intensity was measured by means of a standard attenuated total reflection geometry. Figure 5.1 shows the results of angular scans at both high (70 pJ/pulse) and low (10 pJ/pulse) input laser energy near the reflection dip (i.e., around the surface plasmon excitation angle) for the Si-Ag case at the incident wavelength A = 1.11 pm. Note that for high input laser energy the reflection minimum is shifted toward the larger angle, which indicates a selfdefocusing nonlinearity ( f 3 ) is negative). This power-dependent shift in the optimum coupling angle was used to measure both the sign and the magnitude of the intensity-dependent refractive index coefficient nz1 in semiconductors such as GaAs and Si. CARTER,CHENand TRIPATHY [ 19831 measured the intensity-dependent refractive index coefficient n,, of polydiacetylene films by using an attenuated total reflection method. They have grown multilayer samples of a polydiacetylene upon an Ag-overcoated grating forming a planar waveguiding structure. By measuring the change with optical intensity in the coupling angle between an input laser beam (A = 0.755 pm) and a planar waveguide mode in

X X

& 1.05

X X X X

0 90

'

i t

*

1 ' " ' I ' ' ' ~ 1 ' ~ ~ ' 1 ~ ~ ' ~ 1 ~ ~ ~ ' 1

16.4 17.2 180 18.8 19.6 2 0 4 INCIDENT ANGLE

Fig. 5.1. Angular scan of the reflected intensities around the surface plasmon excitation angles for the Si-Ag case. The curve marked high (0) was taken at an input energy 70 @/pulse, whereas the one marked low ( x ) was taken at 10 @/pulse. The arrows mark the excitation angles (in degrees) for the two cases. (After CHENand CARTER [1982].)

EXPERIMENTAL STATUS

299

Fig. 5.2. The experimental set-up for investigating nonlinear prism coupling. Symbols: P, coupling prism; M, mirror; LC, liquid crystal; WG, waveguide; S, substrate; BS, beam splitter; VBA, variable beam attenuator; DET, detector; L, lens. (AAer VALERA,SEATON,STEGEMAN, SHOEMAKER, X u MAI and LIAO [1984].)

the structure, both the magnitude and the sign of the intensity-dependent refractive index coefficient nZ1for the polydiacetylene films were estimated. The experimental set-up used by VALERA, SEATON, STEGEMAN, SHOEMAKER, Xu MAI and L ~ A O[ 19841 for investigating nonlinear prism coupling is shown in fig. 5.2. The sample geometry consisted of two strontium titanate coupling prisms and an optical waveguide (1.7 pm thick Corning 7059 glass film deposited onto a Pyrex substrate). The gap between one of the coupling prisms and the glass film was filled with the nematic liquid crystal MBBA, characterized by the intensity-dependent refractive index n = no + n,,l, where I is the local intensity. A few milliwatts of light from an

Fig. 5.3. Experimental results for the power coupled into the nonlinear waveguide versus incident power for TE, wave. (AAer VALERA,SEATON,STEGEMAN, SHOEMAKER, XU MAI and LIAO [1984].)

300

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

argon ion laser (A

=

[IV, 5 5

0.515 pm) was transmitted through the sample

(Po,,x 5 mW). As the guided wave power increases, the coupling efficiency decreases because the synchronous coupling condition is no longer valid. As shown in fig. 5.3, the transmitted intensity is not linear with the input intensity and the device operates as a power limiter. CHEN,CARTER,SONEKand BALLANTYNE[ 19861 observed nonlinear optical coupling to planar GaAs/AlGaAs waveguides at A = 1.06 pm from a Q-switched mode-locked Nd : YAG laser by using a grating coupling technique. The planar waveguides used in this experiment (1.7 pm GaAs on 3 pm A1,-,Ga,As) were epitaxially grown by a metal organic chemical vapor deposition technique, and at an input light intensity of approximately 100 MW/cm2 nonlinear switching to the waveguide mode has been observed. PATELA,JEROMINEK,DELISLE and TREMBLAY[ 19861 demonstrated experimentally the power-dependent prism coupling into the planar waveguiding structure consisting of a thin film of Coming 7059 glass (optical linear medium) deposited by radio frequency planar magnetron sputtering on the surface of CdS, Se, - doped glass (optical nonlinear medium). The limiting action in prism coupling of light into single and multimode nonlinear waveguiding structures was described. The nonlinear prism coupling was used to evaluate both the sign and the steady-state value of the intensity-dependent refractive index coefficient n,, of the nonlinear semiconductor doped glass. FORTENBERRY, MOSHREFZADEH, ASSANTO,Xu MAI, WRIGHT,SEATON and STEGEMAN[ 19861 reported a very large (twenty-fold) decrease in the efficiency of prism and grating coupling into low-loss ZnO thin-film waveguides with increasing incident power at I = 0.53 pm. The low-loss waveguide ( 1 dB/cm at A = 0.633 pm) consisted of a ZnO thin film (d x 0.6 pm) with a refractive index n f x 2 sputtered onto a glass substrate (n, z 1.47). Both strontium titanate prisms and gratings were used for in- and outcoupling into the nonlinear waveguide. The ZnO thin film waveguide was excited by highpower, single longitudinal mode, 10 ns long laser pulses with I = 0.53 pm. ASSANTO,SVENSON, KUCHIBHATLA, GIBSON,SEATONand STEGEMAN [ 19861 reported a detailed study of nonlinear prism coupling of argon ion laser light at A = 0.515 pm into a nonlinear ZnS waveguide under continuous wave equilibriumconditions. The nonlinear optical waveguide consisted of a thin film of ZnS (d x 1.3 pm, n, x 2.36) deposited onto a microscopic slide (ns= 1.51). The power-dependent change in the refractive index of ZnS has been identified as thermal (through absorption), and the intensity-dependent refractive index coefficient was found to be n,, = 10- l2 m2/W. GOODWIN,GLENNand BENNION[ 19861 measured the nonlinear response

301

EXPERIMENTAL STATUS

t

A 'Oo0

/

f'

,'

realignment

input. mW

Fig. 5.4. The response of a prism-coupled nonlinear organic waveguide at 1 = 0.515 pm. (After GOODWIN, GLENNand BENNION[1986].)

of a prism-coupled nonlinear organic waveguide at a wavelength of 0 . 5 1 5 pm from an argon laser. Figure 5.4 shows a significant reduction in coupling efficiency as the input light intensity was increased. Complete recovery of coupling efficiency at the highest intensity level was possible, as shown in fig. 5.4, by realigning the input beam. SVENSON,Xu Recently chirped gratings were used by MOSHREFZADEH, MAI, SEATONand STEGEMAN [ 19871 to obtain high coupling efficiencies into Xu MAI, SEATONand nonlinear polystyrene waveguides. MOSHREFZADEH, STEGEMAN[ 19871 described the fabrication by reactive ion beam etching of deep gratings into glass substrates for coupling into polymeric waveguides. Coupling efficiencies of up to 45 % have been realized for nonlinear polystyrene waveguides.

5.2. NONLINEAR COHERENT DIRECTIONAL COUPLERS

In recent years much attention has been given to nonlinear directional couplers that transfer optical power between two coupled channel waveguides (JENSEN [ 19821, SARIDand SARGENT [ 19821, MAIER[ 19821, KITAYAMA and WANG [ 19831, GIBBONS and SARID[ 19871, CAGLIOTTI, TRILLO, WABNITZ, DAINO and STEGEMAN [ 19871, THYLEN, WRIGHT,STEGEMAN, SEATONand

302

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, B 5

NONLINEAR MATERIAL

OUT, OUT,

Fig. 5.5. Schematic of an integrated optic nonlinear coherent coupler.

MOLONEY[ 19861, STEGEMAN, SEATON,IRONSIDE,CULLENand WALKER [ 19871). If the device is set to produce a complete transfer from one waveguide to the second waveguide at low powers, the transfer can be minimized at high powers and this corresponds to a power-dependent optical switch (see fig. 5.5). Intensity-dependent power transfer has been demonstrated using Ti in-diffused LiNbO, channel waveguides (LATTES,HAUS,LEONBERGER and IPPEN[ 19831 and in GaAs-G%,,Al,,,As multiple quantum well channel waveguides (LI KAM WA, SITCH,MASON,ROBERTSand ROBSON[1985]). Although the optical nonlinearity n21 in LiNbO, is small (the required power levels are high), the picoseconds response time of the device observed by LATTES,HAUS, LEONBERGER and IPPEN[ 19831makes it very attractive for an all-optical logic gate. LI KAMWA, SITCH,MASON,ROBERTSand ROBSON[ 19851 reported the nonlinear optical switching of laser diode radiation in multiple quantum well structures at a temperature of 180 K at low waveguide power levels (approximately 1 mW). These devices consisted of two parallel channel waveguides, and when light is initially inserted into one channel, the transfer efficiency into the second channel depends upon the initial power in the incidence channel. Recently, CULLEN,IRONSIDE,SEATONand STEGEMAN [ 19861 described the waveguiding characteristics of both planar waveguides and single-mode directional couplers fabricated by potassium/sodium ion exchange in CdS,Se, --x doped glasses.

5.3. NONLINEAR GUIDED WAVE OPTICAL LIMITERS

The existence of nonlinear guided waves was first verified experimentally by VACH,SEATON,STEGEMAN and KHOO[ 19841 and subsequently by BENNION, [ 19851. In both experiments nonlinear self-focusing GOODWINand STEWART cladding media were used (nematic liquid crystal MBBA and CS, respectively). The nonlinear planar waveguide used by VACH, SEATON,STEGEMANand KHOO[ 19841 consisted of the nonlinear liquid crystal MBBA as the cladding medium (n, = 1.55, n,, = m2/W, at A = 0.515 pm), a thin film of borosili-

IV, I 51

303

EXPERIMENTAL STATUS

cate glass (d = 1 pm, n, = 1.61), and a soda lime glass substrate (n, = 1.52). The light from an argon ion laser was coupled with a strontium titanate prism into a linear waveguide, propagated into, through, and out of the region with the liquid crystal on top, and then coupled out of the linear region with a second strontium titanate prism. As can be seen from fig. 5.6, for the TE, wave there is a pronounced saturation effect (limiting action) as well as hysteresis with respect to increasing as opposed to decreasing incident power. This result can be interpreted as follows. The TE, wave shows a maximum in the power which can be transmitted (see fig. 3.2) and as the propagation constant fl increases, one of TE, field maxima moves toward and into the nonlinear cladding (see fig. 3.3). The nematic liquid crystal MBBA is a lossy medium with an attenuation coefficient of approximately 20 cm - I . As the optical field becomes more 2 5

I

I

T E 1 MODE

7

-+--+ /+

20

15

-3E c

c

a" 10

P

0 5

+

increasing intensity

o decreasing intensity

I

0

0

0

100

200

3 00

yn (mW) Fig. 5.6. The power transmitted in a TE, wave for both increasing and decreasing incident powers. (After VACH, SEATON,STEGEMAN and KHOO [1984].)

304

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, $ 5

localized in the liquid crystal cladding, the transmission decreases because of absorptive loss in the MBBA (see the upper branch in fig. 5.6). For the subsequent decrease in power, both branches of the TE, curve are excited and hysteresis is observed since the high fl branch of the TE, curve in fig. 3.2 corresponds to higher losses (see the lower branch in fig. 5.6). Recently, BERTOLOTTI,FERRARI, SIBILIA,ALIPPIand NESRULLAJEV [ 19871 studied the optical nonlinearities in a lyotropic liquid crystal (potassium caprylate) in a waveguidingconfiguration similar to that used by VACH,SEATON,STEGEMAN and KHOO[ 19841. In this experiment a large hysteresis effect was found in the optical response in all mesophases of potassium caprylate.

5.4. OPTICAL SWITCHING AND BISTABILITY IN NONLINEAR PLANAR CONFIGURATIONS

A nonlinear interface (i.e., an interface between a linear dielectric and a nonlinear dielectric medium characterized by an intensity-dependentrefractive index) is potentially useful as a very fast (sub-picosecond) optical switch and and PETROV[ 19751, KAPLAN [ 1976, logic element (BOIKO,DZHILAVDARI 19771, SMITH,HERMANN,TOMLINSON and MALONEY[ 19791, ROZANOV [ 19791, MARCUSE[ 19801, SMITH,TOMLINSON, MALONEYand HERMANN [1981], TOMLINSON, GORDON, SMITH and KAPLAN [1982], SMITHand TOMLINSON [ 19841). The nonlinear interface configuration is illustrated schematicallyin fig. 5.7. In the negative x half-space we have a linear dielectric medium with an index of refraction no and in the positive x half-space we have a nonlinear self-focusing dielectric material (n2 > 0) with the intensitydependent refractive index n = no - A + n2Z(A > 0). We assume that the zerointensity refractive index of the nonlinear medium is less than that of the linear dielectric by a small amount A. For low intensities the incident beam will undergo total internal reflection at the nonlinear interface if the angle of incidence $is less than the critical angle $c = (2A/n0)'I2. There is, however, an

Fig. 5.7. Light incident on an interface between linear and nonlinear media. (After SMITH, TOMLINSON, MALONEY and HERMANN[1981].)

IV, § 51

305

EXPERIMENTAL STATUS

evanescent field in the nonlinear medium, which as a result of n2 > 0, increases the refractive index of the nonlinear medium and reduces the refractive index difference across the interface, thus reducing the critical angle. This in turn increases the evanescent field and a positive feedback is established. Therefore we expect that there will be some threshold input intensity at which there will be a sudden switch from total internal reflection to a state where both transmitted and reflected fields can occur. [ 19841 reported a detailed study of the switching SMITHand TOMLINSON behavior of a nonlinear optical interface. The nonlinear dielectric used in this experiment was an artificial Kerr medium with a very slow response time (- 100 ms) but with a very large nonlinear coefficient n , = 4.5 x 10- l 3 m2/W, so that it was possible to perform the experiment with the continuous wave output of an argon ion laser ( A = 0.515 pm). The linear dielectric was an optically polished single crystal of LiF (no = 1.391), and the nonlinear dielectric was an aqueous suspension of 800 8, quartz particles with no - A = 1.373 (on the use of a liquid suspension of dielectric spheres as an artificial Kerr medium see, for example, SMITH, ASHKIN and TOMLINSON[1981], ASHKIN, DZIEDZICand SMITH [1982], SMITH, MALONEYand ASHKIN [1982], BERTOLOTTIand SIBILIA[ 19841). Figure 5.8 shows the experimental measurement of reflectivity as a function of normalized incident intensity for the case I+$ = 6 " . The experimental results clearly display a second reflectivity jump corresponding to a second transmitted beam - a feature that was not predicted by the plane wave theory developed by KAPLAN[ 19761. Note that the experimental results (see fig. 5.8) are in excellent agreement with the predictions of

1.o

4

0.7 I

10.0 01

I

I

I

I

I

0.2 0.3 0.4 0.5 0.6 INCIDENT INTENSITY (n,I/Al

l

c

0.7

Fig. 5.8. Experimental measurements of interface reflectivity as a function of normalized incident intensity for the case $ = 6". (After SMITHand TOMLINSON [1984].)

306

NONLINEAR WAVE PROPAGATION IN PLANAR STRUCTURES

[IV, I 5

Surface plasmon wave

Fig. 5.9. Excitation of surface plasmons by the attenuated total reflection method. (Atter MARTINOT,LAVALand KOSTER[1984].)

numerical calculations based on a two-dimensional Gaussian input beam (see TOMLINSON, GORDON,SMITHand KAPLAN[ 19821). The excitation of surface plasmons by the attenuated total reflection method can also be used to obtain a bistable optical device (see MARTINOT, LAVALand KOSTERand LAVAL[ 19851). A standard way to KOSTER[ 19841, MARTINOT, excite surface plasmons is to use the Kretschmann configuration (see fig. 5.9). A thin metal film is sandwiched between two dielectric media I and 11, whose refractive indices are n, and n,, respectively. Assuming n, > n,, when a TM polarized light beam is incident in medium I1 with an incidence angle 0 larger than the critical one, it is totally reflected. A surface plasmon wave can then be excited across the metal layer through the evanescent wave. Resonant excitation is obtained when the incident beam wavevector component parallel to the interface (k II = konc sin 0) fulfils the plasmon dispersion relation. By tuning the incidence angle, a deep minimum in the reflected light power is then observed. Let us now consider that the material in which the incident beam propagates has nonlinear optical properties; that is, its refractive index n, depends on the light intensity. In this case the minima in the reflected power P can be observed at different values of the incidence angles 8 as the incident power is increasing. As the incident light propagates through the nonlinear medium, any variation of the incident intensity tunes the surface plasmon excitation through resonance, and the superposition of the incident and reflected beams induces a feedback in the system and allows bistable operation of the device. MARTINOT,KOSTERand LAVAL[ 19851 observed optical bistability from surface plasmon waves propagating along a silver-CS, interface by using thermally induced optical nonlinearities in CS, (see fig. 5.10). In the last few years several groups have reported optical bistability and switching in planar waveguiding configurations. Theoretical calculations by VINCENT, PARAIRE, NEVIERE, KOSTERand REINISCH[ 19851have shown that

EXPERIMENTAL STATUS

307

Fig. 5.10. Reflected power versus incident beam power for different values of the incidence angle. (ARer MARTINOT, KOSTERand LAVAL[1985].)

a planar waveguiding structure consisting of a nonlinear Kerr-like silicon layer sandwiched between a sapphire substrate and a silver layer, light being coupled in it by a diffraction grating, can exhibit optical bistability. Subsequently, PARDO, KOSTER,CHELLI,PARAIRE and LAVAL[ 19861 realized a bistable optical device using guided wave excitation in a nonlinear silicon epitaxial layer grown on a sapphire substrate. LUKOSZ,PIRANIand BRIGUET[ 19861 reported optical bistability and self-pulsingin ZnS and ZnSe planar waveguides by using [ 19861 reported input couplers. VALERA,SVENSON,SEATONand STEGEMAN intrinsic bistability and switching in a thin-6lm waveguidein which the oriented liquid crystal K18 was used as the cladding region. The bistable behavior was interpreted in terms of the nematic-isotropic phase transition. VITRANTand ARLOT[ 19871 demonstrated optical bistability with a nonlinear prism coupler. A guided wave was launched through a nonlinear prism coupler whose prism-waveguide gap was filled with a MBBA liquid crystal. It was observed in this experiment that the intensity of the excited guided wave exhibits optical bistability when the incident intensity is modulated. This result was shown to arise from a thermal averaging process in the liquid crystal. KIM, GARMIRE, SHIBATAand ZEMBUTSU[ 19871 reported optical bistability and nonlinear optical switching due to increasing absorption in single-crystal ZnSe waveguides with switching times of 10 ps and threshold switching power of 15 mW. We finally mention that KIRKBY,CUSHand BENNION[ 19851 studied optical nonlinearities of organic photochromic thin-films and CUSH, TRUNDLE, KIRKBYand BENNION[ 19871 investigated bistable optical switching and logic elements in photochromic fulgides.

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[IV,$ 6

0 6. Conclusions We have shown in this review that the use of media with nonlinear refractive indices enriches considerably the phenomenon of guided wave propagation in planar structures. Thus if one or more of the media bounding a dielectric or metal film exhibits an intensity-dependent refractive index, the number of nonlinear guided wave solutions, the propagation wavevector, the field distributions, the attenuation coefficient, and the waveguide cut-off and switch-on conditions all become power dependent. We have discussed a number of potential applications to all-optical signal processing of nonlinear guided wave phenomena in planar structures. Planar optical waveguides are primarily of interest for serial, rather than parallel processing systems. The power-dependent field patterns can be used for switching and thresholding operations in a waveguide context, whereas the power-dependent wavevector used in conjunction with a distributed coupler can lead to devices such as optical limiters. We have shown that low-threshold devices which only pass optical pulses above a threshold power, are possible based on self-focusing nonlinearities. Thus a planar optical waveguide with a nonlinear self-focusing cladding and a linear waveguiding film whose thickness is less than the TE, cut-off thickness at low power can become transmitting at high power levels. Nonlinear guided wave limiters can be obtained if one or both of the media boupding the waveguiding film are characterized by a self-defocusing nonlinearity. Optical switching action can potentially be achieved if both the cladding and substrate media exhibit self-focusing nonlinearities. Moreover, a combination of controllable thresholding and limiting actions can be used to fix a range of power levels that are transmitted down the nonlinear optical waveguide. Furthermore, the same type of phenomena should occur in a channel waveguide, which would reduce the waveguide volume and power operating levels considerably. For example, since most materials exhibit the saturated change in the refractive index Ansat in the range of to power limiting action in the range of a few milliwatts would be expected for a 1 mm wide light beam propagating in a planar waveguide. If one can extrapolate linearly with waveguide width to channel waveguides, then by using nonlinear guided waves, optical limiting action might be obtained at a few microwatts power levels. Materials in general limit the experimental realizations of all the nonlinear guided wave phenomena predicted to date. The fact that the refractive index differences n, - n, and n, - n,, which exist at low powers between the film and the bounding media, must be less than the saturated change in the refractive

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index Ansat,puts severe limitations on the material combinations which can be used for making a nonlinear planar optical waveguide. There is a need for new materials with optical nonlinearities nz1 greater than 10- l 3 m2/W and with attenuation coefficients less than 1 cm-I in waveguide formats. The optical nonlinearities should be sufficiently large so that the various nonlinear devices can be implemented at milliwatt power levels. Since the nonlinear waveguide phenomena will be used p r i m d y for serial signal processing, it is also necessary that, once the optical signal is turned off, the nonlinearly induced polarization should relax in the picoseconds range (the “turn-on” of the nonlinearities is usually instantaneous). Some candidate materials for nonlinear third-order integrated optics devices such as GaAs/GaAIAs multiple quantum well structures, semiconductor doped glasses, and nonlinear organic media (e.g., polydiacetylene films) satisfy the preceding criteria, although not all optimally. We look forward over the next few years to both the demonstration of more all-optical signal processing operations using currently available materials and to the development and utilization of new highly nonlinear materials with response time in the picoseconds range that will lead to many nonlinear guided wave devices for optical logic and signal processing.

Acknowledgements It is a pleasure to thank our friends Allan Boardman, Uwe Langbein, Falk Lederer, Hans Ponath, and George Stegeman for many stimulating discussions. One of the authors (D. Mihalache) also wishes to express his gratitude to Professors M. Ivascu, I. I. Popescu, I. M. Popescu, A. Corciovei, and I. Ursu for continuous support and valuable encouragement.

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E. WOLF, PROGRESS IN OPTICS XXVII

0 ELSEVIER SCIENCE PUBLISHERS B.V.. 1989

V GENERALIZED HOLOGRAPHY WITH APPLICATION TO INVERSE SCATTERING AND INVERSE SOURCE PROBLEMS BY

ROBERTP. PORTER Elecm'cal Engineering Department University of Washington Seattle. WA 98195, USA

CONTENTS PAGE

. . . . . . . . . . . . . . . . . . . 317 § 2. GENERALIZED BACKPROPAGATION . . . . . . . . . 319 § 3 . HOLOGRAPHIC IMAGING . . . . . . . . . . . . . . 337 § 4 . IMAGING SOLUTIONS TO INVERSE SOURCE PROBLEMS . . . . . . . . . . . . . . . . . . . . . . 351 § 5 . DETERMINATION OF STRUCTURE OF WEAKLY SCATTERING OBJECTS . . . . . . . . . . . . . . . . . . . 374 392 § 6 . CONCLUSION . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . 392 APPENDIX A. FIELD REPRESENTATIONS FOR SURFACE SOURCES . . . . . . . . . . . . . . . . . . . . . . . . . 393 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . 395 § 1*

INTRODUCTION

0 1. Introduction Holographic imaging is a fundamental imaging method that relies on general properties of diffracting wave fields to store information about threedimensional objects on two-dimensional recording surfaces. Most developments of holographic theory are restricted to planar recording surfaces and, consequently, do not exploit the full power of the holographic imaging concept. By generalizing the theory to surfaces of arbitrary shape, we can relate holographic imaging to the solution of inverse source and inverse scattering problems. Specifically we can show that generalized holography can accurately reconstruct the set of radiating sources. In addition, if a scatterer is illuminated by many incident waves and if the scattered field for each realization is measured by a hologram, we can reconstruct the scatterer by a technique known as holographic tomography. BREMMER[1951] attempted to develop a general theory of diffraction imaging by defining a generalized imaging system as a device that converts a diverging spherical wave into a converging spherical wave. Such a system would focus a radiating wave to an infinitesimally small region and, therefore, could perfectly reconstruct an arbitrary object. Unfortunately such an image field requires a sink at an unknown location. In the absence of this information we must assume that the image field is source free, satisfying the homogeneous wave equation. A perfectly converging spherical wave is not a solution of the source-free wave equation; perfect reconstruction for an arbitrary object in a source-free space is not possible. Other efforts aimed at developing a theory of diffraction maging that can be related to solutions of inverse problems include MITTRA and RANSOM[ 19671 and SHEWELLand WOLF [ 19681. The development of generalized holography was motivated by the need for a general theory of diffraction imaging, where the image field is source free, that could explain the limitations on reconstructions of arbitrary objects. Such an imaging system is physically realizable for planar holograms; our goal is the extension to nonplanar recording geometries. We seek to understand the capability of generalized holography to reconstruct the detailed structure of a radiator or scatterer. 317

318

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V, 8 1

Generalized holography is applicable to a wide range of inverse problems (LANGENBERG [ 19871). It can be used in ultrasonic medical imaging to design arrays contoured to body shape to optimize the focusing of energy into specific organs. It has been studied (ESMERSOY and LEVY[ 19861) to integrate surface seismic profiling with vertical seismic profding in a borehole for resolution of deep earth structure. It has been proposed as a method for improving resolution [ 19811). in microwave holography (TRICOLES,ROPEand HAYWARD Focusing in generalized holography is accomplished by complex conjugation of the recorded monochromatic field. Consequently this theory has significant utility in understanding phenomena arising in the optics of phase conjugate waves. In particular, this chapter provides a foundation for understanding phase conjugate optics with nonplanar boundaries. Holographic imaging theory was extended to recording surfaces of arbitrary shape for both scalar and electromagnetic waves by PORTER[ 1969a,b, 1970, 19711. He showed that all closed holograms surrounding any object embedded in a homogeneous medium have precisely the same image field. The image field is independent of the shape of the recording surface. He also developed image fields for open holograms defined on surfaces extending to infinity and showed that certain restricted types of sources could be accurately imaged by holographic systems (PORTERand SCHWAB[1971], PORTERand DEVANEY [ 1982b1). PORTER[ 19701 established that these sources satisfy a certain integral equation which relates the sources to the radiated field and that this equation arises directly from the definition of the generalized hologram. BLEISTEINand COHEN[ 19771 investigated a similar integral equation in connection with the inverse source problem of acoustics and electromagnetics. They attribute this formulation to a report by Bleistein and Bojarski (see BLEISTEIN and COHEN[ 19771). Also see BOJARSKI[ 1973,1981,19821. Their equation has the same solution as the Porter equation and can be derived from it (see PORTERand DEVANEY[1982a] and WOLF and PORTER[1986]). BLEISTEIN and COHEN[ 19771 show that purely radiating sources satisfy the integral equation and that the nonradiating sources, whose fields are identically zero outside a finite region, define the null space of the Porter equation. (See DEVANEY and WOLF[ 19731 for a discussion of radiating and nonradiating sources.) [ 1982al showed that the inverse source problem has PORTERand DEVANEY a unique solution when the source energy is minimized for a given radiated field. These authors (PORTERand DEVANEY[1982b]; see also PORTERand SCHWAB [ 19711) showed that these solutions are the set of radiating sources and presented a method for findingthem. They also investigated the inversion

v, 0 21

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319

process in the presence of noise and showed that the Rayleigh resolution criterion places a practical limit on the performance of any reconstruction and PORTER[ 19851extended the solution of the inverse technique. DEVANEY source problem to inhomogeneous media. WOLF[ 19691 laid the foundation for the holographic approach to the solution of the inverse scattering problem for weak scatterers. He showed that many holograms are necessary, correspondingto plane waves incident from different directions. This paper led to the principle of diffraction tomography, which has found extensive applications in the fields of ultrasonic imaging (MUELLER, KAVEKand WADE [ 19791) and seismic profiling (FITZPATRICK[ 19791). IWATAand NAGATA [ 19751proposed a reconstruction method that was tested by KENUEand GREENLEAF [ 19821. A scanning method for backscattered data and WADE[ 19851. DEVANEY [ 1982bl has been developed by LAN, FLESHER developed a filtered,backpropagation algorithm for invertingthe scattered field in an efficient manner. In this chapter we present the basic theory of generalized holography for scalar and electromagnetic waves. We show that the holographic images produced by films can approximate closely the images produced by generalized holograms of arbitrary shape. We also review the solutionsto the inverse source problem, paying close attention to the role played by radiating and nonradiating source contributions. The inverse scattering problem can also be solved with generalized holograms, which can produce a band-limited reconstruction of a weak scatterer. The duality between scanning the scatterer with plane waves from all directions or at all frequencies is shown. Solutions to the inverse scattering problem with wide-band plane waves and point sources are discussed.

0 2. Generalized backpropagation Backpropagation unifies two-step imaging systems in diverse fields such as holography (PORTER[ 1970]), acoustic tomography (DEVANEY[ 1982b]), and ORISTAGLIO and LEVY[ 19851). In holoseismic wave migration (ESMERSOY, graphy, film recording of the scattered field combined with a reference wavefront effectively measures the complex conjugate of the scattered wave, a term proportional to the scattered wave, and higher-order combinations. Illumination of the hologram launches the conjugate wave, which converges to the real image. Section 3 will review real and virtual imaging for holograms. Early attempts to reconstruct tomographically ultrasonic images of tissue were plagued by poor image quality resulting from literal application of back-

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GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V,s 2

projection ideas from X-ray tomography. DEVANEY [ 1982bl realized that the degradation resulted from diffraction effects which could be compensated by focusing. He achieved this by recording amplitude and phase and then performed the reconstruction by backpropagation, which can achieve the diffraction limit. Section 5, which describes inverse methods, will show how backpropagation tomography implements the generalized backpropagator to solve the inverse scattering problem. Wave migration attempts to extrapolate a wide-band seismic signal back to the scattering horizons, which can be readily identified with geological events. Classical wave migration has been restricted to reflection seismology, which is approximately understood through the use of the paraxial approximation to the wave equation or by using ray optics (CLAERBOUT [ 19851). DEVANEY [ 19841 introduced his backpropagation algorithm to permit wide-angle reconstruction in the new technique of large offset, seismic profiling using borehole receivers and well-to-well tomography. ESMERSOY, ORISTAGLIO and LEVY[ 19851 used the time domain extension of the spherical wave backpropagator of PORTER [ 19701 to develop a finite-difference tomographic algorithm that has been successful in reconstructing complex structures within the diffraction limit. We shall also discuss their method in $ 5.

2.1. TWO-STEP IMAGING SYSTEMS

In two-step imaging systems an illumination or radiation step produces a field expanding outward from the object. The field is then sensed photographically, electronically, or mechanically by a distributed continuous or discrete sensor. Ideally, we are able to sense the field from every direction with sensors distributed on an arbitrary surface. Figure 1 illustrates both the scattering problem whose source distribution (scalar waves) p( !P) is a function of the incident field and the radiation problem where pis arbitrary. Important imaging results are readily obtained when the field is sensed on a closed surface surrounding the object. Of course, most practical situations permit only limited angular coverage of the field. Important geometries include planar screens giving 180-degree coverage and screens asymptotic to the infinite wedge (see fig. 2). For all of these cases we shall see that the backpropagated image fields are spatially invariant or isoplanatic, permitting spatial deconvolution. For h i t e , open recording surfaces many of the properties of the backpropagator hold approximately.

GENERALIZED BACKPROPAGATION

321

Fig. 1. Scattering problem for which the source is dependent on the incident field. For the radiation problem there is no external incident field.

Fig. 2. Examples of hologram geometries.

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GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

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It is important to form images using both the scattered field and its normal gradient at each sensor location. Even though we can derive one field quantity from the other, it is usually easier to measure them independently. We shall show that the backpropagator based on both the field and its normal gradient produces an image field independent of the geometry of the recording surface. This backpropagator was defined by PORTER[ 1969bl as the Basic Image System (BIS). The image field produced by the closed BIS contains only propagating waves. Letting the recording surface be two infinite planes and using the angular spectra representation for the field, WOLF and PORTER [ 19861 have shown that the BIS image field contains no evanescent waves. The imaging or reconstruction step consists of establishing appropriate surface source distributions along the recording surface contour (fig. 3). In holography the film is illuminated by a field related to the reference wave of the recording step. Since a film is described by a transmittance as a function of

Fig. 3. Generalized holographic imaging. During the recording step, the radiated or scattered field and its normal gradient are recorded by the hologram. Reconstruction or imaging is accomplished by establishing surface sources proportional to the complex conjugate of the measured field and its normal gradient.

v, § 21

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position, we cannot establish both conjugate field and normal gradient surface sources. In 0 3 we shall devise bounds permitting an estimation of the errors present in holographic imaging. For electromagnetic waves we must, in general, record polarizations. We shall present the appropriate backpropagator. For backpropagation tomography and backpropagation seismic-wavemigration, reconstructions are computed. In principle, we can use both the field and its normal gradient to achieve nearly ideal backpropagation. Limitations arise because recorded data are sparsely sampled. Many interesting issues arise such as the sufficiency of sampling and the efficiency of algorithms; however, these are beyond the scope of this article. Seismic waves are generally elastic with both shear and compressional waves being important. It is possible to extend backpropagation methods to this type of polarized wave, and indeed, work in this direction is being pursued. The present article is confined to acoustic and electromagnetic waves.

2.2. GENERALIZED BACKPROPAGATOR FOR SCALAR WAVES

- CLOSED

SURFACES

The generalized backpropagator images a point source by launching a converging spherical wave from an arbitrary, closed recording surface. The backpropagator cannot image to a point because the real image space is source free and consequently the image field must satisfy the homogeneous Helmholtz equation. In this section we determine the image field for any source (radiating or scattering) for both open and closed surfaces. We set the stage by considering the fist step with a radiating source. Suppose we have a monochromatic source distribution p,(r) exp ( - iot), which produces the field satisfying the Helmholtz equation (with the periodic time-dependent factor exp( - iot) omitted from now on)

(V’ + k 2 ) Ul,(r)= with k

= w/c, c

ul,(r) =

- p,(r) ,

(2.1)

being the speed of light in vacuo. The field Y,(r)is given by

1”

d

P h ’ ) G&, r ’ )

Y

(2.2)

where the free space Green (MORSEand FESHBACK [ 19531) function satisfies the equation

(v2+ k 2 ) Gf(r, r ’ ) =

- 6,(r -

r’),

(2.3)

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GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V,0 2

with So the three-dimensional Dirac delta function with volume integral J d3r’ 6,(r’) = 1. The source p, can result from scattering of an incident wave and can be a combination of surface sources and point sources with compact support. We measure both the field and its normal gradient on S, (see fig. 3). The backpropagator is obtained by assigning to monopole and dipole surface sources the complex conjugate of the field and its normal gradient,

a,(r”) = ii V” Y : ( r ” ) ,

(2.4)

a , ( r “ )= Y : ( r ” ) .

(2.5)

Using the field representation for surface sources given by eq. (A.6) of Appendix A, we obtain for the backpropagated field the expression d S “ i i * [ G f ( r ,r ” )V” Y $ ( r ” )- Y $ ( r ” )V” G f ( r ,r ” ) ] .

Y,,(r) = Sh

(2.6) We now show that the image field is given by

where

Kb(r, r ’ ) =

e

;(r, r‘ - G f ( r ,r ’ )

- G f @r, ‘ )

r space, v space,

(2.8)

where the r space is inside S, and the v space is outside (see fig. 3). To establish eq. (2.8), substitute eq. (2.2) into eq. (2.6) and interchange the order of integration, yielding eq. (2.7) with

Kb(r, r ’ ) =

1 dS“ A

J

Sh

[Gf(r, r ” )V“ G f ( r ’ ,r “ )

- G t ( r ’ ,r ” )V “ G f ( r ,r ” ) ] .

(2.9)

Using Green’s theorem to convert eq. (2.9) to a volume integral and substituting eq. (2.3) (defining the Green function) into eq. (2.9), gives for &,the expression

Kb(r, r ‘ ) =

jvh [ G t ( r ’ ,r ” ) 6,(r - r ” ) - G f ( r ’ ,r ” )a&’, r ” ) ]. dV“

(2.10)

v 9

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325

Since we always have the source in the r or interior space but may have the image observation coordinate r either inside or outside, we find eq. (2.8). The imaging kernel has some simple physical interpretations. Substituting eq. (2.8) for the exterior field (v space) into the image field (eq. 2.7), we see that

yb(r) = - Ys(r),

v space

(2.11)

if ps is red. 'Yb is interpreted as the virtual image field. To evaluate the real image, we need the free space Green function Gf(r, r')

=

exp(ik1r - r'r I) 44r-r ' ~ '

(2.12)

yielding the real image kernel (2.13) where j, is the spherical Bessel function (COURANTand HILBERT[ 19621). It is evident that the image of a point object at r' is sharply peaked. In the limit of vanishingly short wavelength A, the image field is concentrated at r' because

where So is the one-dimensional Dirac delta function

s, 00

d x S0(x)

=

1.

(2.15)

It is tempting to infer from this result that we have perfectly imaged a point object (STONE[ 19811); however, it is easy to see a flaw in this argument (FISCHER and LANGENBERG [ 19841). For a vanishingly short wavelength and an arbitrary object the image field can be represented as (2.16) This integral is not well defined, so we consider the limit directly from eqs. (2.7) and (2.13), yb(r)

=

-1

2K

lim Jv dp dOd@p(sinO)(sinkp) p,*(r - I ( ) ,

(2.17)

2-40

where we have written the integral in the spherical coordinate system,

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GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

P,B 2

= r - r‘ . It is evident that the integrand is not localized at r and, therefore, Yb does not replicate ps. In 8 4 we shall find conditions on ps such that the source can be extracted from the image field. The image field is spatially invariant, as can be seen by displacing the source to some location u. Then the image field is identical but displaced by u, so that

p

Kb(r, r ’ ) = Kb(r - u, r’

- u)

.

(2.18)

The proof follows by direct substitution from eq. (2.8) into eq. (2.18). The basic image system is a perfect, isoplanatic instrument in the limit of a vanishingly short wavelength (BORNand WOLF[ 19751). The resolution, or width of the peak of Kbr,is and is independent of the shape of the recording surface s h . BLEISTEINand COHEN[ 19771 have derived an identity (see also BOJARSKI [ 1973, 1981, 19821) that is closely related to eqs. (2.6) and (2.7). To derive their equation, we note that -

s,.

d S ” n^ * [G,(r,r ” ) V” Ys(r”)- Y s ( r ” V” ) Gf(r, r”)] =

{ ys(r) 0

outside s h , inside s h .

64.7)

(see Appendix A). Taking the complex conjugate of eqs. (2.6) and (2.7) and adding the result to eq. (A.7) yields

j s dS”

n^* [Kbr(r,r ” )V” Ys(r”)- Ys(r”)V “ Kbr(r,r”)]

h

=

- Jv d V’ ps(r’)Kbr(r,r ’ ) .

(2.19)

On substituting from eq. (2.13) for Kbr, we obtain the Bleistein and Cohen identity for all r. In spite of the similarity of eq. (2.19) to eq. (2.6), Bleistein and Cohen’s identity cannot be interpreted as a surface source distribution on Sh, and it does not launch converging waves. It is easy to prove that the image kernel (eq. 2.8) is independent of the shape of s h . Consider another surface Sk enclosing S, and, of course, containing no sources during the recording step. Now evaluating the image kernel over the volume between s h and S,!, by Green’s theorems (see eqs. 2.9 and 2. lo), we find that (2.20)

v, § 21 2.3.

327

GENERALIZED BACKPROPAGATION

GENERALIZED BACKPROPAGATOR FOR SCALAR WAVES SURFACES

- OPEN

In many situations of importance in practice, one cannot obtain scattered or radiated information from all directions. For an arbitrary surface eqs. (2.6) and (2.7) are valid, with the image kernel or image field for a point object being given by eq. (2.9). However, this equation does not usually simplify to the sum of incoming and outgoing Green functions given by eq. (2.8). When the hologram surface S, is finite, a little thought will show that the kernel is spatially varying, and we thus lose one of the most convenient properties of the image field with closed surfaces. There is a class of surfaces - open surfaces - that are asymptotic to a wedge [ 19701). These surfaces have image fields that or a cone at infinity (PORTER are spatially invariant but depend on the wedge or cone angle. Furthermore, all hologram surfaces that are asymptotic to a given wedge or cone have identical image fields. Consider the wedge shown in fig. 4 with angle j3. Since we have been in $2.2 that the image kernel is independent of the shape of the hologram surface, we can evaluate the surface integral over S,, Kb(r,r’) =

jsa

d S “ n”*[G,(r,r”)V’G:(r’,r”)

- Gf (r’, r ” ) V ” Gf(r, r ” ) ] ,

(2.21)

OPEN HOLOGRAM

Fig.4. Geometry and coordinate system for an open hologram. S , infinity.

closes the hologram at

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GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

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which we take as a cylinder at infinity. We note immediately the important reciprocity property that

Kb(r, r ’ ) =

- K z (r ’, r ) .

(2.22)

The real image field can be evaluated by using the far-field limits of G f ( r yr ’ ) = $i Ho(k Ir - r’ I), viz, G f hr ” ) = f i

(&)

112

exp(i{k[r” - d cos(0” - O)] - an}), (2.23)

(2.24) Here r” is the distance to the screen, d = I r - r’ I ,and 8is shown in fig. 4. The

Wedge

0 = 45O

2

25

1

15

0.

05

Y

Y/X

-0

-05

-1.

-1 5

- 2 5.

-2.! 1 X/A

Fig. 5. Contour plots of the image field for two-dimensional wedge holograms with /?= 4 A. Contours are in decibels.

4 A and

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329

GENERALIZED BACKPROPAGATION

integral S, yields the real image field d S “ Gf(r,r ” )G:(r”, r ’ )

(2.25)

or, after substituting eqs. (2.23) and (2.24), Kbr(r,r ’ ) = -

4K -i

J@ -b

d0” exp[ -ikd cos(0” - 0)] .

(2.26)

In figs. 5a,b we present contour plots of the image field for /3 = IT and, the important case of the planar screen, /3 = in. In three dimensions we consider open surfaces asymptotic to a cone with angle /3, as shown in fig. 4, and the Green function given in eq. (2.12). Using

a

Cone B = 90°

Cone@ = 4 5 O

2:

1:

05

zlh

-0 5

-15

-25 2

1

rlh

0

i

1

rlX

Fig. 6 . Contour plots of the image field for conical holograms with j= a n and in decibels.

4 II.Contours are

330

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

the far-field limit as before yields the image field d8“ sin 8” exp [ - ikd cos 8” cos 81

4x

x J,(kd sin 8“ sin 8) ,

(2.27)

with contour plots shown in fig. 6 for the cases b = { A, ; A . It should be noted that for b = A the image kernel becomes the closed surface kernel given by eq. (2.13).

2.4. GENERALIZED BACKPROPAGATOR FOR INHOMOGENEOUS MEDIA

In this section we will discuss generalization of the backpropagator for scalar waves to inhomogeneous media (DEVANEY and PORTER[ 1985]), where the source distribution and field are related by the reduced scalar wave equation

[V’

+ k’n2(r)]

Ys(r)= -ps(r).

(2.28)

Here the index of refraction n(r) is assumed to be real (no absorption) and piecewise continuous. Furthermore, we assume that Ys and VYs are continuous and that the index of refraction tends to unity for large r, lim00 n(r) = 1 .

(2.29)

r-

This section is motivated by applications where the unknown source is embedded in a medium with known properties. Such applications arise, for example, in geophysical inversion, medical imaging, and nondestructive evaluation. The point impulse response for an inhomogeneous medium satisfies the equation [V’

+ k’n’(r)]

G(r,r ‘ ) = - h0(r,r ’ ).

(2.30)

Since we restrict ourselves to lossless media [n(r)is real], the Green function satisfies the symmetry condition G(r,r ’ ) = G(r’,r) .

(2.31)

Also, since n(r) is real, the imaginary part of G(r, r ’ )satisfies the homogeneous equation

(V’

+ k’n’)

ImG(r, r ’ ) = 0 .

(2.32)

v, § 21

33 1

GENERALIZED BACKPROPAGATION

The image field for conjugate surface sources is found by applying eq. (A.6), which holds for nonhomogeneous media, d S ” n^. [G(r, r ” ) V ” Y:(r”) - Y:(r”) V”G(r, r”)] .

YJr) =

(2.33) In terms of the scatterer source distribution the image field is Yb,(r)= 2i

jv

d V’ p s ( r ‘ ) ImG(r, r’)

inside S, ,

d V’ p:(r‘) G(r,r ’ )

outside S , ,

(2.34) (2.35)

where G is chosen to satisfy the Sommerfeld radiation condition. The derivation of eqs. (2.34) and (2.35) may be given by a straightforward application of the methods developed in previous sections for free space. Using Green’s theorem we find

Yb(r)=

-

jv

d V’ p:(r‘)

I,.

d V” [G(r, r ” )bO(r”- r ’ ) -

G*(r”,r ’ )b0(r - r”)] ,

(2.36)

which gives eqs. (2.34) and (2.35) after integrating over v h (see fig. 3). Equations (2.34) and (2.35) have the same form as the free space holographic imaging equation derived earlier. We shall show in 3 3 that the free-space image field is intimately related to the real image produced by a point reference hologram taken over the recording surface s h . Viewed in this way, the preceding equation for Y,,, (eq. 2.34) is an integral equation for the source given the holographic real image field that it generates. We shall refer to eqs. (2.33)-(2.35) as the generalized holographic imaging equations.

2.5. GENERALIZED BACKPROPAGATOR FOR ELECTROMAGNETIC WAVES

The backpropagator can be developed for electromagnetic wave radiators and scatterers embedded in free space (PORTER [ 19711) and a restricted class of inhomogeneous media with either linear polarization P or linear magnetization M. In the Gaussian system of units we have (BORNand WOLF[ 1975]), (2.37)

332

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

1 a D 4R VxH=--+-Je, at c

+~ R M D = E + 4nP, B

=

H

[V,fl2

(2.38) (2.39)

,

(2.40)

where c is the speed of light in free space, E and Hare the electric and magnetic vectors, and P and M are the polarization and magnetization. The treatment that follows may be generalized by including a magnetic current in addition to the magnetization M. For simplicity we will assume that M = 0, that the waves are monochromatic with wave number k = o / c , and that the dielectric is linear with

D=n2E.

(2.41)

In this case we have the wave equation i4nk 4n V x Q x E - k2n2E = -J, - -V x J , , C

C

(2.42)

where the inhomogeneity is characterized by the positional dependence of the index of refraction. Also we assume that the medium is finite. The radiation from an arbitrary point current is characterized by the dyadic Green function r ( r , r ’ ) , which satisfies the equation

V x V x r ( r , r ‘ ) - k2nz(r)r ( r , r ’ ) = 6,(r - r’)I ,

(2.43)

where I is the unit dyadic. The electric field radiated by the currents is given by (TAI[ 1971]),

E,(r) = 411 C

jv

dV r ( r , r ’ ) . [ikJ,(r’) - V x Jm(r’)].

For a homogeneous medium (n constant) the dyadic Green function is given by

(

r ( r , r ’ ) = I + -VV :z

) G(r, r ’ ) ,

(2.45)

where G is the scalar Green function given by eq. (2.12) with K = nk. The electromagnetic backpropagator can be obtained in a similar manner as the scalar wave backpropagator. We measure both the scattered electric and magnetic fields on S,, assuming, of course, that there are no free magnetic

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333

GENERALIZED BACKPROPAGATION

currents near S , such that the magnetic vector of the scattered field is 1

H , = T V x E,.

(2.46)

ik

The imaging step consists of establishing discontinuities in tangential components of E and H,equivalent to electric and magnetic surface currents, i h x (H, - H,)= - h x (V x EZ),

(2.47)

k

n^ x (E, - E,)

=

- h x E:(r”),

(2.48)

and backpropagating the image field according to eq. (A. 11). Substituting from eqs. (2.47) and (2.48) into that equation, using eqs. (2.44) and (2.46) for the scattered fields, and interchanging the order of integration yields the image field

C

dVq,(r,r’).[ -ikJ,*(r’) - V x J;(r’)],

(2.49)

where (after using suitable vector identities) the dyadic image kernel can be written as n&, r’) =

jsh

dS“

h

a

{r(r, r ” ) x [V” x r*(r”, r’)]

+ [V” x

r(r, r”)] x r*(rf’,r’)} .

(2.50)

If the hologram is closed, this equation can be analyzed by using Green theorem and the wave equations (2.42) and (2.43), resulting in the incoming and outgoing Green dyadic d r , r‘) =

r*(r, r f ) - r(r, r f )

inside S,,

- r(r, r’)

outside S ,

.

(2.51)

In free space the dyadic kernel is simply related to the scalar-wave free space kernel : (2.52)

where Kb is given by eq. (2.8). If the recording surface is open and extends to infinity, as shown in fig. 4, we may evaluate the kernel by performing the surface integral over the infinite

3 34

[V,8 2

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

cylinder or sphere. We assume constant E, p here. To evaluate the kernel over S,, we note that nb = 0 =

5%

.

d S ” n^ { - * } + 9

s,_

d S ” n^ { *

* *

as long as r and r’ are outside the closed surface S , S , goes to infinity, the free space Green dyadic

(+

rf(r, r ” ) = I

-V V 2:

}

+ S,.

(2.53) As the radius of

) Gf(r, r n )

(2.54)

and its curl become

rf(r,r ” ) = (I - PP’)Gf(r, r N ), V X rf(r,r”) = - ik(P” x I) Gf(r, r ” ) .

(2.55) (2.56)

Then eq. (2.50) evaluated over S , reduces to z,(r, r’) = 2ik

js_

d S “ Gf(r, r”) G:(r”,r ’ ) (I - 3 ” 3 ” ) ,

(2.57)

which also results if the dyadic operator operates on the scalar kernel:

( + -VV )

nb(r,r’) = I

Kb(r, r ’ )

(2.58)

2:

Here K, is given by eqs. (2.26) and (2.27) for cylindrical and conical open surfaces.

2.6. PHYSICAL CONTENT OF THE BACKPROPAGATOR

Most of the interest in the image field !Pbcenters on its use for solving inverse source and inverse scattering problems. Understanding the limitations on these solutions helps to relate holographic imaging to solutions of inverse problems. We shall see in this section that the information about the source distribution is limited to radiating sources that produce propagating waves; nonpropagating or evanescent waves are not present in the image field. In addition, it is important to distinguish between radiating sources, which can be imaged, and nonradiating sources, which cannot be imaged. We show in a later section, based on results developed here, that the backpropagated image field contains

V, 5 21

GENERALIZED BACKPROPAGATION

335

enough information to identify the radiating sources but contains no information about nonradiating sources. The most obvious property of the image field follows from the definition of two-step imaging processes, namely, that the image region must be source free and, consequently, the image field must satisfy the homogeneous wave equation (V2

+ k2)Yb = 0.

(2.59)

In any analysis of the image field (eq. 2.7) we must solve the eigenvalue problem

(2.60) showing that only sources satisfying the conditions (V2

+ k2)pn = 0 ,

(2.61)

where A,, # 0, can be found. These are the radiating sources. All propagating fields outside the source region can be generated by this set { p,}. The solutions in the null space A,, = 0 are the nonradiating sources pNR (DEVANEY and WOLF [ 19731) that can be generated from the equation

(V2+ k2)Q(r) = - p N R ,

(2.62)

where Q is identically zero outside the volume containing the source distribution. These results have been generalized to nonhomogeneous media by DEVANEY and PORTER[ 19851. (See 3 4.1.) The angular spectrum of any radiating source can be found from its image field, given by eqs. (2.6)-(2.13) (WOLF and PORTER[ 19861). Since the image field Ybr is independent of the shape of sh, we can take S h to be a sphere of large radius R + co, centered on some point in the source region. We can formally write

(2.63) where the integration extends all space and j, is again the spherical Bessel function (COURANT and HILBERT[ 19621) with representation daexp[iks-(r-r')].

(2.64)

Here s are real unit vectors and the integration is over the unit sphere. Substituting from eq. (2.64) into eq. (2.63), interchanging the order of integration,

336

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

and introducing the Fourier transform

p(K)=

Im

d3rrp(r’)exp(-iK-r’)

(2.65)

of the source distribution, we can express the image as an integral over the unit s sphere Ydr)= - ik 14nd62p*(ks)exp(-iks*r).

8 n2

(2.66)

The image field contains information only about those Fourier components ;(K) for which 1 K ( = k, which are precisely the Fourier components associated with the radiating part of the source. Other spectral components of p(r) produce no field outside the source volume. The image field produced by the Basic Image System, with surface sources Y:, n^.VY$*,contains only propagating waves; the evanescent waves are eliminated during backpropagation even though they are present in the scattered field. To show this, let Ys- and Y: be the radiated fields in the two half-spaces R- and R shown in fig. 7. These two fields may be expanded in the form of angular spectra: +

J J-m

(2.67)

CANCELLATION OF EVANESCENTWAVES

Fig. 7. Geometry to illustrate cancellation of evanescent waves.

VA 31

331

HOLOGRAPHIC IMAGING

+ qY - m d l

z
(2.68) where (2.69) The values of k and q for which p2 = q2 < 1 are associated with homogeneous plane waves that propagate away from the source region, whereas those for which p 2 + q2 > 1 are associated with evanescent waves decaying away from the source region. It has been shown (SHERMAN[ 19671) that the homogeneous plane wave portions of 'pa and Ys-,referred to here as and Y z , can be analytically continued into the opposite half-space. One can also show that the image field eq. (2.66) can be written in the form %r(r) = [ K m + Kil(41

Y

(2.70)

which contains analytical continuations of the propagating portions of the scattered field but no decaying waves. Evidently back propagation from both surfaces S,+, S , has cancelled the evanescent waves. WOLF and PORTER [ 19861 have shown that these waves are eliminated by using information contained in both the field and its normal derivative recorded on a surface S, intercepting all of the source radiation.

4 3. Holographic imaging Ideally, the generalized backpropagator is realized by establishing dipole and monopole surface sources proportional to the complex conjugate of the recorded field and its normal gradient. Holograms are recording films whose transmission or reflection coefficients are functions of the intensity of the total field incident on the film. The scattered field is reconstructed by illuminating the film with a reference beam. In this section we describe equivalent source models for holograms, which we then analyze and compare with the generalized backpropagator. We show that the real image formed by a hologram approximates the image formed by the generalized backpropagator. Planar or single-layer holograms are thin sheets of recording film whose and Lu [ 19701). Absorption thickness is less than a wavelength (CAULFIELD

338

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V.8 3

holograms have a transmission coefficient that is a linear or nearly linear function of the intensity of the total field during the recording or exposure step. Phase holograms, made with a recording material such as thermoplastic or photopolymers, produce a phase shift that is proportional to the intensity of the total incident field. Reflection holograms can be made by evaporating a highly reflective film on the surface of a phase hologram. Physically, these holograms have many of the properties of reflective gratings. Volume holograms are emulsions whose thickness is large compared with the wavelength. These holograms are of four types : transmitting absorptive, transmitting reflective, phase transmitting, and phase reflecting. Many single-layer holograms can be recorded in a single volume hologram, permitting color imaging and highly compact information storage. They can be modeled by equivalent surface sources, but the complexity of the analysis is beyond the scope of this article. The models developed in this chapter are restricted to single-layer transmission and reflection holograms.

3.1. SINGLE-LAYER TRANSMISSION HOLOGRAMS

COLLIER,BURCKHARDT and LIN [ 19711 have given a convenient definition of a linear, single-layer hologram with transmission coefficient T=AZ,

(3.1)

I = I Yi + !PSI*

(3.2)

where and A is a constant depending on the exposure time and sensitivity of the film. We also assume that the reflection coefficient of the developed film is zero. Consider the planar hologram shown in fig. 8. The field to the right of the film is Y-,.

=

TYi,

(3.3)

in agreement with conventional treatments. In these theories of holography the image field is obtained by substituting the transmitted field into the Huygens-Kirchhoff integral (GOODMAN[ 19681). Although this approach leads to correct results for a single, planar hologram, it is inappropriate for modeling imaging with multiple or nonplanar holograms. Suppose we have two planar holograms that have recorded the scattered fields from the front and back of the object as shown in fig. 9a. We assume that

HOLOGRAPHIC IMAGING

339

Fig. 8. Planar transmission hologram showing incident and transmitted fields.

Fig. 9. Hologram geometry with 2 planar screens. (a) Each screen is illuminated by an incident and a scattered wave; (b) the image reconstruction. Note that the reconstructed field !P2 also illuminates hologram (1).

the incident fields Yi are plane waves of unit amplitude. One can reconstruct the real image by illuminating each hologram with the complex conjugate of the incident field shown in fig. 9b. To simplify the discussion, we take the film constant A = 1 and neglect second-order terms in YS.Then the transmitted field through a single hologram is given by

T Y : = Y: + A ,

(3.4)

where A

=

Y,* -t (Yy)2!Ps.

(3.5)

340

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V,8 3

Let us also assume that we have arranged the holograms so that the incident field from hologram (2) does not illuminate hologram (1). Then, to fist order, the image field on the real image side ( + ) of hologram 1 is YT+ =

"5 + A I + Y2,

(3.6)

where Y2 is the image field from hologram (2) that illuminates hologram 1 as shown in fig. 9b. On the virtual image side ( - ) of hologram (1) the field is given by YT- = Y; + T Y 2 , (3.7) since the field from hologram (2) will pass freely through the film. As we stated earlier, Y2contains only the image field from hologram (2) so that Y2is of the order Y2 = O( Y*).

(3.8)

To first order, eq. (3.7) is YT- = y$

+ yz.

(3.9)

Comparing eq. (3.9) with eq. (3.6), we see that the field on the real image side of a hologram is the sum of the transmitted field plus contributions from other holograms. Consequently, the field on the surface of a transmission hologram is not uniquely determined from the incident field and the film transmittance. Fortunately, the difference in the field across the hologram, an equivalent dipole source layer,

(3.10)

a, = YT+ - YT- = A , ,

is uniquely specified by the incident field and the film transmittance. A complete equivalent source model requires specification of a monopole source layer, which can be found by noting that the normal gradient of the field on both sides of the hologram (1) is

d * V Y T += ( d * V Y Z ) T +d * V Y 2 , d'vyT-

=

d'vy$

+d'vyz

*

(3.11) (3.12)

For a plane wave incident at angle 8, as shown in fig. 9, the monopole source layer is . a

=

d*V(YT+ - YT-)

= ikAcos8.

(3.13)

Equations (3.5), (3.10) and (3.13) constitute an equivalent source model of a transmission hologram illuminated by a plane wave.

v, o 31

HOLOGRAPHIC IMAGING

341

The performance of the transmission hologram, when compared with the generalized backpropagator, depends on the accuracy of the approximation a,

=

(3.14)

ikAcosOX B-VY:,

(3.15)

a, = A X Y:,

where Yz, i - V Y z are the equivalent sources for the generalized backpropagator. The hologram equivalent sources contain the term (eq. 3.5) ( Y:

IZ ys

3

which is unimportant in this comparison because it produces the virtual image field which has a small effect on the real image. The real image sources ( Y z , ik cos OY:) will accurately approximate the sources for the backpropagator if Iri.VY$ - ikcosOY:I

x 0.

In the following section we consider the point reference hologram and show that its real image can closely approximate the image field of the generalized backpropagator.

3.2. POINT REFERENCE HOLOGRAM

The image field of a hologram whose recording arrangement consists of a point reference source, the object, and the recording film,can accurately approximate the image field produced by the generalized backpropagator (PORTER [ 19701). The recording arrangement is shown in fig. 10. When illuminated by a converging spherical wave, the point reference hologram produces

Fig. 10. Geometry and coordinate system for a point reference hologram. Y O L F

2 7

C H Q F

10

342

GENERALIZED HOLOGRAPHYllWERSE PROBLEMS

[V, 8 3

two images of a point object - an image coincident with the scatterer and a distorted image at a symmetrical location about the reference point. The image field and recording arrangement are similar to those of a lensless Fouriertransform hologram described by STROKE [ 19661. The derivation of the appropriate surface source model for a single-layer transmission hologram follows the procedure used in 5 3.1. Consider the point reference source S,(r - R,),which radiates a spherical wave. The total radiated field on the film is Y,(r") = Ys+

exp(ikr") 4nr"

(3.16)

The intensity recorded by the film is 1 I=(4 nr

+ Y:

exp(ikr") Ysexp( - ikr") + 4nr" 4nr"

9

(3.17)

where we have neglected the second-order term, assuming that (3.18)

As before, the film has transmittance T = AZ, where A is the film exposure coefficient. We can produce a real image by illuminating the hologram by a spherical wave converging on the source point R,. To prevent the converging wave from diverging from its focus and reilluminating the film from inside, we place an absorber at R, as shown in fig. 11. The image field converges on the image point

Fig. 1 1 . Image reconstructionwith a point reference hologram. An absorber is placed at R,, to absorb the reference wave.

v, 3 31

343

HOLOGRAPHIC IMAGING

r ” and then diverges outward, lighting the film from the inside. If Y’ denotes the diverging field on the inside surface, we find that the field just on the inside of S, is given by

YT,(r”) =

T exp( - ikr”)

4nr”

+

Y’,

(3.19)

which is the sum of the transmitted wave and the image field from other portions of the hologram. The field on the outside surface consists of the incident field plus the diverging image field transmitted through the hologram: YTv(r”) =

exp ( - ikr ”) 4nr“

+ TY’.

(3.20)

It should be noted that the subscript r in eq. (3.19) denotes the real image space which is inside the closed hologram. Similarly v denotes the virtual image space. As in the previous section we choose A to ensure a nearly transparent film. Letting A = (4nr”)’, the transmittance is

T = 1 + 4nr”[ Y .exp(ikr”) + Ysexp( -ikr”)]

.

(3.21)

Then, to first order, the equivalent dipole layer is =

YTr - YTV= Y$

+

Ysexp( -2ikr”).

(3.22)

Since Y‘ is the order of Yson the film, we have neglected products Y ‘ Y$ and Y’ Ys in the transmitted field. The equivalent monopole layer a, is found by taking the normal gradient of eqs. (3.19) and (3.20), which yields a, = ii

-

(V” YTr- V” YTv) = - ikii Vr”[Y. + Ysexp( - 2ikr”)l . (3.23)

We have assumed here the far-field condition kr” % 1, which allows us to ignore the term ii V( l/r”). We shall analyze the image field due to the Y$ and Ys source terms in eqs. (3.22) and (3.23) separately. The partial field Ya is due to the sources a,,

= - ik(ii

- V r ” )Y2 ,

(3.24)

and a ,, = Y:

.

(3.25)

The partial field Yc is due to the sources ,a,

=

ik(ii- V r ” )Ysexp( - 2ikr”),

(3.26)

344

and

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

cCl= Ysexp( - 2ikr”)

[V,s 3

(3.27)

The total reconstruction field is YT =

exp ( - ikr”) + Y, + Yc. 4nr”

(3.28)

3.2.1. Case I. Reconstructionjkld due to Y: term The partial field Y, is found by substituting eqs. (3.24) and (3.25) into the field representation Eq. (A.6). We find that the image field is given by

Ya(r)= -

js

h

dS” Y:(r”)[ik(i. V r ” )Gf(r,r ” ) + i * V ” Gf(r,r ” ) ], (3.29)

which, after substituting eq. (2.2) for Ys,can be written in the form

ya(r)=

s,

d Y’ p:(r’) Ka(r, r ’ ) ,

(3.30)

where the kernel, or point source image field, is

Ka(r, r ’ ) = -

s,,

d S ” G f ( r ” ,r ’ ) [ik(i*V r ” )Gf(r,r ” )

+i

V ” Gf(r,r ” ) ].

(3.31)

The difference between this kernel and the backpropagator is the error kernel K , = K , - K,,, where Kb is given by eq. (2.9), K,

=

-

jsh

d S “ G f ( r ”r, ’ ) [ik(A V r ” )Gf(r,r ” )+ i .V ” Gf(r,r ” ) ]. (3.32)

The error kernel vanishes as r“, the distance from R, to the hologram, tends to infinity because, in this limit,

v ” G : = - ik(i V’’r’‘)GY

(3.33)

1 l G f ( r , r ” ) G f ( r t ’ , rd’ )S/ = - d n , 4 7r2

(3.34)

and

where d62 is the differential solid angle.

v, 8 31

345

HOLOGRAPHIC IMAGING

An error-bound, when r f ris large, has been found for the two-dimensional problem. If r” ,the distance from a line reference source to a cylindrical surface s h with the geometry indicated in fig. 10, is large and varies slowly along s h , the magnitude of the error term is bounded for r and r’ in a circle of radius E, centered am,, by rf)l <

ELh

8n(r;)’

~

1

+ -,

(3.35)

8krk

where rk is the smallest distance between s h and R,, r; % &, and Lh is the length of the curve defining the cylindrical hologram S,. Some of the image-field error results from the displacement of the object from the reference point. The error vanishes as the hologram is placed further away from the object and the point reference source. (See PORTER[ 19701, Appendix for more details.) 3.2.2. Case II. ReconstnrctedJielddue to the Ysterm The sources proportional to Ys(eqs. 3.26, 3.27) yield a second image symmetrically located, relative to the reference point, at 7‘ = ( r f, $ + n) when the point object is nearby R,. This image is also found in the field produced by the planar Fourier-transform hologram (STROKE [ 19661). Substitution of the expressions for the sources acoand oc, into eq. (A.6) leads to the kernel &(r, r’)

=

-

s,,

d S “ exp( -2ikr”) Gf(rf’,r)

x [ikd.Vr”G,(r,r”)

+ d * V ”Gf(r,rff)].

(3.36)

We evaluate the image field in two dimensions using the approximation of the Green function for large rf’: expi(k1r - rs I -:in)]

Gf(r, r ) = ’I

(3.37)

After approximation I r - r ” I to second order for large r ”,the kernel becomes KJr,r’) =ajo2‘d$”exp[ -ik(rf cos(@- @ f f ) + r c o s ( $ -9”)

- r 2 sin2($” 2r”

- $) - (r’)’ sin2($” - #) 2r“

(3.38)

GENERALIZED HOLOGRAPHYjlNVERSE PROBLEMS

[V, I 3

in the real image space near Ro. At the image point (r’, +’

+ A) the kernel is

346

sin2(+” r”

+ I )

(3.39)

The field for the symmetrical image varies spatially because its magnitude and phase depend on the point object location r ’ ; we cannot write Kc(r,r ’ ) = f ( r - r ’ ) .

For an observation point r sucn that the exponent of eq. (3.38) is much larger than a wavelength, the kernel can be evaluated by the method of stationary phase. The stationary phase points, assuming r % r ’ , of the first two terms of eq. (3.38) occur at

r‘ sin(+’ - +) r

+ + A +

3

++

r’ sin(+’ r

- +)

The point response at r is Kc(r,r ’ ) = !.

(”>’” [exp

ik ( r

+ r’ cos(+’ - $1 - (r’)2sin2(+ -

4 lrkr

r’‘(+)

r+r‘cos(+‘ -+)+

+‘))I

(r‘)’ sin2(+ - +‘) r “ ( + + R)

(3.40) An observer at the point ( r ’ , +‘ + R) sees an image of the point object, but the field is not isoplanatic. An observer at some point r inside S, sees both a converging and a diverging wave whose focal point is at (r’, +‘ + A).The image field for some object ps is distorted unless the hologram is in the far field of ps where k(r’)2/r’’ 1.

+

3.3. SINGLE-LAYER REFLECTION HOLOGRAMS

A single-layer reflection hologram, shown in fig. 12, has the reflection coefficient

R

=AI,

(3.41)

where the intensity is given by eq. (3.2). A planar hologram illuminated by a plane wave !Pi will reflect a plane wave whose value on the hologram surface

V, I31

341

HOLOGRAPHIC IMAGING

Fig. 12. Planar reflection hologram showing incident and reflected fields.

is R = Yi,

(3.42)

if the reflection coefficient varies slowly along the film. It can readily be seen by setting A = - 1 that a reflection hologram specifies the field on the hologram surface. For an incident unit-amplitude plane wave the intensity is given by

I = 1 + YiY$+ Y?YS.

(3.43)

Illuminating the hologram by the conjugate of the incident field, the reflected field on the surface side ( - ) is

R Y i = -(1

+ !F,Y$+ Y ~ Y s ) Y ~ .

The total field on the hologram surface is given by YR = Y:

Y R = -[Y,*+(Y:)*Ys].

(3.44)

+ R !Pi,that is, (3.45)

A hologram of arbitrary shape (or multiple ho1ograms)can also be illuminated by some of the image field spreading out from other portions of the hologram. (We assume that the incident field is blocked so that its reflection cannot reilluminate the hologram.) To first order this spreading image field will be cancelled on the hologram surface, leaving eq. (3.45) unchanged. By comparison with the previous discussion we readily deduce that the Y$ term represents the real image. We define the boundary conditions for the real image as

YR =

- Y$

on S , .

(3.46)

348

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V,§ 3

If it were possible to have a film with reflection of + 1, we would find that the normal gradient of the reflected field is constrained to

fi.VYR = ii.VY$

on

sh.

(3.47)

Equations (3.46) and (3.47) are the boundary conditions for the real image field produced by a reflection hologram.

3.4. CLOSED REFLECTION HOLOGRAM

The image field inside a closed reflection hologram can readily be evaluated and has a simple interpretation. 3.4.1. Case I. Reconstructedjield due to the 4 ’: term As a consequence of illuminating the hologram from inside, the field on the

inner surface of the hologram is Ya(r) = Y$

(3.48)

and produces the image field !Pa(r) =

s,

d V’ p * ( r ’ ) Ka(r, r ’ ) ,

(3.49)

where the kernel is given by the integral Ka(r, r ’ ) =

1%

dS” G f ( r ” ,r ’ ) r i a V”G(r,r ” ).

(3.50)

The Green function satisfies the condition G ( r , r ” ) = 0 on S ,

.

(3.51)

We can evaluate eq. (3.50) by the method of Appendix A, yielding Ka(r, r ” ) = G f ( r , r ’ ) - G ( r , r ‘ ) .

(3.52)

Since the Green function must be a standing wave inside a closed reflector, we have G ( r , r ’ ) = i G f ( r , r ’ ) + f G f ( r , r ’ ) - Gh(r, r ’ ) ,

(3.53)

where Gh is the homogeneous portion of the solution required to match

HOLOGRAPHIC IMAGING

boundary conditions on

sh.

349

The image field for a point source is

Ka(r, r ’ ) = iKbr(r, r ’ ) + G,(r, r ’ ) ,

(3.54)

where Kbr is the image field produced by the generalized backpropagator. The homogeneous field can only be evaluated for specific geometries. More generally, there exists a countably infinite set of frequencies at which the structure is resonant. At these frequencies the Green function, specifically G,, diverges. At the same frequencies the image field is dominated by the resonant modes. Between resonant frequencies the homogeneous field will be minimized, permitting observation of the image. In certain geometries, such as an ellipsoid, a point object located at a focus will produce an image at the other focus. The image field corresponding to the geometrical optics focus is contained in the homogeneous field G,. Thus we can minimize, in certain structures, the effect of G, on the backpropagated image. 3.4.2. Case II. Reconstmctedfild due to the Ys term In this case the image field is constrained to YCW

=

(3.55)

ys

on the inner surface of the hologram. Following the analysis of case I, we find that the image kernel is KJr, r ’ ) = G&, r’) - G(r, r ’ ) ,

(3.56)

where G is identical to the Green function for case I and is specified by eq. (3.51). Substituting eq. (3.53) yields L

Kc(r, r’) = - iKbr(r, r’)

+ Gh(r, r’) .

(3.57)

It should be noted that case I1 also produces a real backpropagated image. Suppose we can illuminate the hologram with an incident field that has zero phase everywhere along the inner surface of the hologram. From eq. (3.45) the field on the hologram surface is

YR = - Y: - Ys.

(3.58)

Combining the results from cases I and I1 yields the total image field YR(r) = -

jv

d V’ [p*(r’) KJr, r’) + p(r’) Kc(r, r’)] .

(3.59)

350

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V,§ 3

Substituting expressions for the kernels yields YR = 4

+

dV’ [p*(r’) - p(r’)] Kbr(r,r’)

S,

(3.60)

d ~ [p*(r’) ’ + p(r’11 Gh(r, r’)

If the source p(r) is purely imaginary, the homogeneous image field is eliminated, leaving only the backpropagated image.

3.5.

OPEN REFLECTION HOLOGRAMS - THE PLANAR CASE

We shall consider only the case with Y: specified on the surface of the hologram. The reader is referred to $ 5.5 for more details on the Fourier expansion method used here. A related problem was analyzed by SHEWELL and WOLF[ 19681. The scattered field on the screen at z = zh is exp(ik1r - r’ I)

(3.61)

)r-r’)

The angular spectrum d x d y Ysexp[ -ik(px +qy)]

(3.62)

is given by (3.63)

(3.64) The Green function that has zero value on the hologram is

x {exp[ -ikm(z - z ” ) ] - exp[ikm(z

+ z ” - 2zh)]} . (3.65)

v, 9 41

IMAGING SOLUTIONS TO INVERSE SOURCE PROBLEMS

35 I

Substituting the Green function into eq. (3.63) yields Ka(r,rf)=* 8 7c2

k -

e e x p [ i k s - -(r-r')] m dp dq

exp[ikp(x - x ' ) + ikq(y - y ' )

k , / m ( 2 Z h - z - z')] ,

(3.66)

where s _ = ( p , q, - m), P is the region p 2 + q 2 < 1, and P is the region p 2 + q 2 > 1. The sign of m is chosen to guarantee decaying evanescent waves. The first term in the kernel is the image field due to the generalized backpropagator from the planar surface. The second term is a decaying evanescent field that degrades the image and arises because the image was produced with a reflection hologram.

8 4. Imaging solutions to inverse source problems An object consisting of widely separated point radiators can be imaged using the focusing property of the backpropagator, but the image is not a precise replica because any aperture has limited resolving power. For radiators so close to each other that their images overlap, we cannot determine the source simply by focusing the field. We pose a more general problem: Under what conditions can we extract the source distribution from the image field and what common characteristics have such source distributions? DEVANEY and WOLF [ 19731 have shown that any source distribution consists of radiating sources as well as nonradiating sources that produce no external field. BLEISTEINand COHEN[ 19771 have given examples of nonradiating sources and have shown that a procedure exists for decomposing any source into a distribution of radiating sources. They have also shown that radiating sources can be found from the backpropagated image field by eigenfunction decomposition for both electromagnetic and scalar waves. PORTERand DEVANEY [ 1982a,b] have demonstrated that a unique solution to the inverse source problem can be obtained by imposing the constraint that the source energy dVI p(r)l be minimized; radiating sources satisfy this constraint. These authors have also studied source inversion from noisy data and have concluded that the backpropagated image field is very close to the ideal solution for noisy data; to achieve better performance we must sacrifice spatial

352

[V,§ 4

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

invariance that is characteristic of imaging from closed or infinite open holograms. In this section we shall establish basic properties of radiating and nonradiating source fields, develop solutions to the inverse source problem for scalar and electromagnetic waves, and generalize these solutions to scalar waves in inhomogeneous media.

4.1. RADIATING AND NONRADIATING SOURCES

Consider solutions YSto the reduced scalar wave equation with source ps localized in some volume V given by eq. (2.2). If V is a sphere of radius R, we can expand all fields outside in terms of spherical wave functions (DEVANEY and WOLF [ 19731). In spherical polar coordinates the Green function can be expanded in the form ik Gf(r,r’)=-ho(lr- r’l) 4A = ik

C a0

jn(kr’)h?)(kr) Ynm(8,$1 Y,*,(B’,

m=-n

n=O

$’I,

(4.1)

where the j, are spherical Bessel functions, hi1)are spherical Hankel functions, and

Y,,

=

in( - 1)”( - 1)”

r2

1

+ l)(n - m)! 4n(n + m)!

n

P, (cos e) eim@

(4.2)

are spherical harmonics, P, denoting Legendre polynomials. Since {Y,,} is a complete orthonormal set, we can expand the source distribution in terms of the spherical harmonics, with the expansion coefficients given by

Ps nm(r’)

=

JonJoZn

dB’ d$’ s i n e ps(r’)Y,*,(B’, @‘).

(4.3)

The radiated field is then given by

ys(4= ik

c

n=O

o

n

1 h\”’(kr) Ynm(4$1

m=-n

joR

d r r2ps nm(r)jn(kr). (4.4)

From eq. (4.3) we see that we can represent ps in the form (4.5)

v, 8 41

353

IMAGING SOLUTIONS TO INVERSE SOURCE PROBLEMS

Let us also define the partial source

n=O

where

an =

m= -n

joR

d r r 2j,2(kr)

(4.7a)

and (4.7b) The rest of the source distribution, that is, the difference between p, and is n=O

p ~ ,

m=-n

The partial sources are designated as p ~ the , radiating source, and hR, the nonradiating source. Several properties of radiating and nonradiating sources follow. More and WOLF [1973] and by detailed proofs have been given by DEVANEY BLEISTEINand COHEN[ 19771. The radiating sources p~ satisfy the homogeneous wave equation: Property I:

(V'

+ k ' ) p ~ ( r )= 0 .

It follows that the Fourier transform of propagating wave numbers. Property 11:

h ( K )=

J

(4.9) p~

contains only components at

d VpR(r)e-iK"' = P,(ks) 6,(K - ks) ,

(4.10)

V

where Is1 = 1 and So is the three-dimensional delta function. Similarly, we see from eq. (4.8) that the fields due to nonradiating sources have no radiating components

JoR

d r r 2 hR nm(r)jn(kr),

Property 111:

hR nm

Property IV:

h R ( k s ) = 0 for Is1 = 1 .

=

0

=

(4.11) (4.12)

It follows from properties 11 and IV that j dl.2 PR(ks) PZR(ks) = O (where the integral extends over the unit sphere) so that the radiating and

354

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V.§ 4

nonradiating sources are orthogonal: Property V:

S d VpR(r) p&(r)

=

0.

(4.12a)

The field due to the nonradiating sources vanishes outside the volume V and must be related to the sources by Property VI:

-hR(r)=

(8’ + k 2 ) Q(r) ,

(4.13)

where Q is any function that vanishes identically outside V. The real image space in the imaging step must be source free, from which it follows that Property VII:

(V’ + k 2 ) Yb(r)= 0 ,

(4.14)

where eqs. (2.7) and (2.8) were used. Equation (4.14) can also be established by direct substitution of eqs. (2.7) and (2.8) into the scalar wave equation (2,l). Some of the results have been extended to inhomogeneous media by DEVANEY and PORTER[1985]. A necessary and sufficient condition for a source to be nonradiating is that it satisfies the homogeneous generalized imaging equation Property VIII:

jv

d V’ p&(r’) ImG(r, r’) = 0 ,

(4.15)

where Im denotes the imaginary part. Since nonradiating sources must generate a field that is zero outside the source volume, we can extend property VI: Property IX:

- h R ( r ) = [V’

+ k2n2(r)]Q(r) .

(4.16)

Here Q is a continuous function that possesses continuous first derivatives inside the volume V. The set of nonradiating sources is very broad, fundamentally limiting our ability to image arbitrary source distributions. Consider, as an example, the nonradiating source in free space producing the field

QW = jdW6-

-

,

(4.17)

where 6- ,(R - r)

=

0

r>R,

1

O
(4.18)

IMAGING SOLUTIONS TO INVERSE SOURCE PROBLEMS

355

The nonradiating source producing this field is

(4.19) where 6,(x) = d6,Jdx. DEVANEY and SHERMAN [ 19821 have shown that a spherically symmetrical source will not radiate if

(4.20) This fact can be verified by means of substitution of eq. (4.19) into eq. (4.20) and integrating by parts. An inspection of eq. (4.19) shows that the source consists of a dipole and a monopole layer located on the spherical shell r = R.

4.1.1. Electromagnetic waves For a detailed treatment of nonradiating currents we refer the reader to a and WOLF[ 19731. Here we simply summarize the relevant paper by DEVANEY properties of such sources. Let j ( K ) be the Fourier transform of J(r),

j ( K )=

s

(4.21)

dVJ(r) e-iX",

and let j,(ks) be the transverse part of j(ks), with Is1 = 1, j,(ks)

= -s

(4.22)

x s x @s).

Only the transverse Fourier components of the current distribution with and WOLF k = w/c produce a propagating electromagnetic wave (DEVANEY [ 19731). Therefore the radiating portion of a current distribution is given by Property I:

I,(K) =

-s x s

x I ( ~ s 6,(~ ) - ks).

(4.23)

All radiating current sources satisfy the homogeneous reduced vector wave equation, Property 11:

V x V x JR(r) - k2JR(r)= 0 .

(4.24)

The electromagnetic field produced by a nonradiating current source must vanish outside the source volume V. From this result BLEISTEINand COHEN [ 19771 have inferred that nonradiating sources are in the null space of the propagating fields,

356

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

Property 111:

J N R nm =

0=

[V, 0 4

$)

drdOd$r2 sin$J,(kr)Y;,JO,

A necessary and sufficient condition for the current distribution J to be nonradiating is that the transverse Fourier components vanish at 1s 1 = 1:

Property IV: j , NR(kS) = o for

I S I=

I

.

(4.26)

Finally, the nonradiating current and its electric field Q(r) are related by Property v :

c

JNR(r)= -[V x V x Q(r) - k2Q(r)] . (4.27) i4 xk

Here Q is any twice continuously differentiable function whose value is identically zero outside the volume V.

4.2. INVERSE SOURCE PROBLEM FOR SCALAR WAVES

All the information about the radiating sources and none of the information about the nonradiating sources is contained in the wave fields outside the object. It follows that backpropagation should enable us to find the radiating source distribution. In this section we study the image field given by ,-

(4.28)

- ik Kbr(rrr f ) = -jo(klr 2K

-

r f I).

(4.29)

To solve for the source ps we seek solutions to the integral equation (4.30) We have shown in the preceding section that the radiating source distribution satisfies the homogeneous scalar wave equation (4.9). The image field Ybr also satisfies the homogeneous wave equation, as can be verified by direct sub-

v, I 41

IMAGING SOLUTIONS TO INVERSE SOURCE PROBLEMS

357

stitution (eq. 4.14). Therefore we expect that eq. (4.30) can be solved for the radiating sources. Consider the source volume V to be a sphere of radius R centered at the origin of a spherical coordinate system (r, 0, 9).We can decompose the source into radiating and nonradiating sources,

P(r) = PRO.) + P N R W

(4.31)

3

where the radiating sources are represented by eq. (4.6). Using the expansion n-Om-

-n

for the spherical Bessel function in terms of spherical harmonics, the image field for radiating sources is (4.33) where on = 4n

jom

(4.34)

drr2j,2(kr)

For the nonradiating sources, solutions to eq. (4.30) have Q = 0. Any nonradiating source is also a solution to this inverse problem. We can solve this inverse problem for the sources uniquely if we can restrict the class of solutions to the radiating sources alone. Suppose we minimize the source energy (PORTERand DEVANEY [ 1982a1) (4.35) which becomes on substituting from eq. (4.31) +

E=

J

~ v ( I ~ +, IPNRI’ I ’ +h

V.

& R

+g

h R ) .

(4.36)

The orthogonality of radiating and nonradiating sources (cf. eqs. 4.12a) requires that (4.37) The radiating source distribution is the minimum energy solution p&r) inverse source problem.

to the

358

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V,8 4

The preceding results, restricted to a source confined to a sphere, can be generalized to source distributions confined to any finite volume by using an angular spectral expansion for both the source and the image kernel given by eq. (2.66):

Ybr(r) = 8A2 - ik j4zdG?y$(ks)exp(-iks.r).

(4.38)

This equation can be inverted for any p,(ks) with I s I = 1. From property I1 for the radiating sources given by eq. (4. lo), we see that we have inverted Ybr and that this is the minimum energy solution to the inverse problem. We can generalize some of these results to inhomogeneous media by seeking solutions to the integral equation

7 s,

ompm(r)= -

dV’ prn(r’)ImG(r, r’) .

(4.39)

The reader interested in more details of these results is referred to a paper by DEVANEYand PORTER[1985]. The kernel ImG(r,r’) is a symmetrical function of r, r’ and is everywhere bounded, since it satisfies the scalar wave equation (V2 + k2n2)ImG = 0. Furthermore, the kernel is non-negative definite for any piecewise continuous function f ( r ) localized in the volume V:

I

dVdV‘ ImG(r, r ’ ) f ( r ) f * ( r ’ 2 ) 0.

(4.40)

Using Mercer’s theorem (COURANT and HILBERT [ 1962]), we can expand the kernel in eigenfunctions (4.41) defined over the source volume V. Without loss of generality we assume that the pm are orthonormal, (4.42) where hmm, is the Kronecker delta. The totality of the eigenfunctions is complete for square integrable functions defined in the volume V if the zero eigenvalue is included. We choose the Green function G(r, r‘) to be the one which satisfies the outgoing radiation condition; that is, G(r, r’) = G (r, r’), which satisfies the +

v, B 41

IMAGING SOLUTIONS TO INVERSE SOURCE PROBLEMS

359

Sommerfeld radiation condition. For values of r’ lying within the source volume V, we can expand the Green function into the series (4.43)

where (4.44)

because the pm are orthonormal. We conclude from eq. (4.43) that the @m are the fields generated by the eigenfunctions pm, which then play the role of sources. Moreover, for am = 0 the pm are nonradiating sources that satisfy the homogeneous holographic imaging equation (4.15). Consequently, for these values of m the @m(r)must vanish outside the volume V. Finally, we note that [ 19641). the @m satisfy the Sommerfeld radiation condition (SOMMERFELD Let us now expand the source p(r) using the eigenfunctions p,: (4.45)

(4.46)

If we now substitute the expansion for ps(r) into the integral representation for the field radiated by an arbitrary source ps, (4.47)

(4.48)

where the @m are defined by eq. (4.44). The @m in eq. (4.48), for which am = 0, vanish outside the source region V. Thus we can partition Ysinto the sum of a nonradiating field (4.49)

and a radiating field (4.50)

360

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V,8 4

The nonradiating component YNRis generated by the projection of the source ps onto the nonradiating source eigenfunctionsp,,,, which are solutions to the homogeneous generalized holographic imaging equation, whereas the radiating component YR is generated by the projection of the source onto the radiating source eigenfunctions p,,, having non-zero eigenvalues. Because the nonradiating component YNRmust vanish outside the source volume V, the field in this region will only depend on the projections of the source onto the radiating source eigenfunctions. It then follows that only this component of the source can be reconstructed from field measurements performed outside V. Any nonradiating components of the source cannot be determined from field measurements alone and must be deduced from auxiliary information or additional constraints imposed on the inverse source problem. The projection of the source onto the radiating source eigenfunctionscan be readily determined from the generalized holographic imaging equation (2.34). If we substitute eqs. (4.41) and (4.45) into eq. (2.34), we obtain (4.5 1)

(4.52) Hence the amcan be determined for all values of m such that a,,, > 0. These are, of course, the expansion coefficientsof the projection of the source onto the radiating source eigenfunctions. Denoting this projection by m( and the we have projection of ps onto the nonradiating source eigenfunctions by hR, the result that Ps(~)= &(r) + hR(r)

(4.53)

7

(4.54) is uniquely determined from the generalized holographic imaging equation and hR is a nonradiating source. We now show that p~ is the minimum energy solution to the inverse source problem. We simply note that any solution to the inverse source problem can Morebe decomposed into the sum m( and any nonradiating component hR. over, we must have r

P

P

IMAGING SOLUTIONS TO INVERSE SOURCE PROBLEMS

36 1

which follows immediately from the orthogonality of the pm. We conclude that ~dVl~l’b[dVIPs/’,

(4.56)

where p, is any solution to the inverse source problem. This establishes that pR(r) is the minimum energy solution.

4.3. INVERSE SOURCE PROBLEM FOR ELECTROMAGNETICWAVES

Backpropagation can be used to find the electric source currents if we assume that there are no isolated magnetic current fields. For free space the wave equation for the electric field (2.42) is i4 nk

V x V x E - k2E=-Je,

(2.42’)

c

and the image field is (2.49‘) where, if we combine eqs. (4.29) and (2.52), (4.57) An equation for the eigencurrents can be found by substituting eq. (4.57) into eq. (2.49’), yielding

Eb(r) =

1 2 (I + kz VV) c

JV

dVj,(kIr

-

r’ l).lz(r’).

(4.58)

We require solutions of the form Eb(r)

=

-(- 2k2 I + c

1 k2 vv

-

)

*

aJ,*(r).

(4.59)

This yields the eigenfunction equation

aJ,(r) =

s

d Vj,(k I r - r’ I) Je(r’).

(4.60)

362

GENERALIZED HOLOGRAPHY~INVERSEPROBLEMS

[V, B 4

Referring to eq. (4.32), we see that the eigenfunctions have the form

(4.61) where JRnm are constant vectors. Since the currents in eq. (4.61) are the complete set of currents which satisfy property I1 (eq. 4.24), that equation represents the set of radiating currents. The nonradiating currents also satisfy eq. (4.60), but since they are in the null space of the propagating fields (property 111), they have t~ = 0. Any nonradiating current is a solution to eq. (4.60).This non-uniqueness can be avoided if we restrict the class of solutions to the set of radiating sources. As before, we minimize the source energy

(4.62) which implies that E N = ~ v d Y ( l J R+ ~ 2IJNR12+ J R . J &

+ Jg-JNR).

(4.63)

Properties I and IV guarantee that the radiating and nonradiating currents are orthogonal, yielding P

L

(4.64)

4.4. COMPUTATIONAL SOLUTIONS FOR INVERSE PROBLEMS

In the absence of noise the image field from the generalized backpropagator can be used to extract the entire radiating source distribution by operating on the image field with an object restorer. This inverse filter contains higher-order terms that grow exponentially large and has a point source response that is spatially variant in contrast to the generalized backpropagator, whose point response is spatially invariant. Because imaging is a wavelength-limited process, larger spatial scales are imperfectly imaged but are present in the image field. A similar object restorer has been proposed for band-Limited imaging systems in which the object is recovered by analytical continuation. (See BARNES

v, o 41

363

IMAGING SOLUTIONS TO INVERSE SOURCE PROBLEMS

[ 19661, BERSHAD[ 19691.) When noise is present in the recorded data, it is necessary to truncate the higher-order terms to avoid noise dominance of the object estimate. For non-zero image noise the backpropagator faithfully reproduces object detail greater than $L so that, for a large object of linear dimension L (L/L% l), the object restorer does not significantly improve the image. However, for linear objects comparable to the wavelength, the use of an inverse filter can improve the reconstruction (TRICOLES,ROPE and HAYWOOD [ 19811). Our discussion closely follows PORTERand DELANEY[ 1982bI.

4.4.1. One-dimensional objects Line sources of finite length contain only radiating sources, which can be completely reconstructed in the absence of noise. The image field, produced by the generalized backpropagator for a closed surface, of a real line source PW

=

6 ( x ) KY)

(4.65)

contained in the interval IzI < $ L is (eqs. 2.7, 2.13)

sink[x2 + y2 + (z - z’)2]’/2 dz’ [x’ + y2 + (z - z’)2]’/2

.

(4.66)

Along the z-axis eq. (4.66) simplifies to Y,,,(o, 0, z) = - $i Y

sin [k(z - z ’ ) ] dz’ n(z - z ‘ )

,

(4.67)

We see from eq. (4.67) that, as L + 0 (k + a),the backpropagator reproduces q(z) exactly along the z-axis; that is, limA+,, Y(z) = q(z). The same object, but imaged from an open surface, n - /3> 8” > fl (see 3 2.3), has the image field

J

-+L

n(z - z’

(4.68)

on the z-axis, where 0 = k cosp. (See BERSHAD[ 19691 for a related treatment of a finite-aperture, band-limited optical system.) When /3 = 0, eq. (4.68) reduces to the image field produced by the closed hologram. When /3 = f n + 6 where 6 4 1, eq. (4.68) reduces to the image field produced by small-angle, coherent imaging systems.

364

GENERALIZED HOLOGRAPHY/lNVERSE PROBLEMS

[V,D 4

Equations (4.67)and (4.68)are Fredholm integral equations of the first kind, relating the unknown source q(z') to the known image field !P(z). As is customary when dealing with such equations, we shall look for solutions in terms of the eigenfunctions fo the associated homogeneous integral equation

(4.69) in the interval 1 z ( < $ L. Note that the q(z) which satisfy eq. (4.69), with the exception of the scale factor a, are perfectly imaged by the backpropagator. The eigenfunctionsof this equation are simply related to the prolate spheroidal wave functions discussed by SLEPIANand POLLAK[ 19611,LANDAUand POLLAK [ 1961, 19611,and SLEPIAN[ 1964, 19651.These functions have been used by RUSHFORTH and HARRIS[ 19681 in the theory of image restoration with small-angle apertures. We summarize some properties of the prolate spheroidal functions using the normalization of Slepian, Landau and Pollak, which differs slightly from that of some papers in the optical literature. Given any L > 0 and 62 > 0, as well as a countably infinite set of solutions (u,,, u,, u 2 , . . .), the eigenvalues of eq. (4.69)can be found such that a,, > ul > a, * .). The eigenfunctions are scaled versions of solutions of the equation

(1 - 242)-

d2u du - 2u- + ( x - 22z2)u = 0 , dz2 dz

(4.70)

where 22 = 62L and x is a parameter that can take on only certain discrete positive values 0 < xo(z) < xl(z) < ~ ~ (<2* ) - . Corresponding to each xv, v = 0, 1,2,. .,,there is a unique solution that is the angular, prolate spheroidal wave function S,,(T, z) defined by FLAMMER [ 19571. The solutions to the integral equation (4.69),in terms of these functions, are

-

(4.71)

The eigenfunctions are orthogonal in the interval I zI

u,(z) u,,(z) dz = a,S,,. ,

< $ L, (4.72)

and complete. Here Svv, is the Kronecker delta function. The eigenfunctionsare

IMAGING SOLUTIONS TO INVERSE SOURCE PROBLEMS

365

continued onto the entire real line by eq. (4.69) and are given by eq. (4.71). The u(z) are orthonormal on the infinite interval; that is,

s,c

u,(z) u,.(z) dz

=

aV,, .

(4.73)

The eigenvalues have the following limiting behavior, derived by SLEPIAN [ 19651 for large values of z: limm 0, = (1

+ errb)- ,

(4.74)

v = {”T+ K

bln(2J.l)

(4.75)

7-

where

The braces denote the “largest integer in”, and In is the natural logarithm. The behavior of the eigenfunctions is illustrated in fig. 2 of SLEPIAN[ 19651 for small values of z. The behavior for large values of z is shown in fig. 13 here. Representing the source as an infinite sum of eigenfunctions yields (4.76) 0

Fig. 13. Eigenvalues for reconstructingone-dimensionalobjects. The eigenvalues are plotted as a function of v normalized by 2L/A for various values of T = nL/A.

366

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V.B 4

which, after substitution into eq. (4.67), yields the following expression for the on-axis image field:

c m

Y(z) =

0

a,(z) a,u,(z) .

(4.77)

It follows from eq. (4.77) and the orthogonality condition (4.72) that the object set (a,) can be extracted from a noise-free image by the operation Y(z) u,(z) dz .

(4.78)

We conclude that the object (ao * aM) located in the interval IzI < :L is perfectly restored by the object restorer with output R(z) and kernel h,,(z, z’ ): R(z) =

s,, tL

dz‘ h,(z,

z’) Y(z’) ,

(4.79) (4.80)

A block diagram of the image system with the ideal, noise-free processor is shown in fig. 14. The stop limits the image field spatially before processing, as required by the restoration operation defined by eq. (4.78). The processor can be realized only if the object set can be truncated because the kernel h,(z, z’) diverges as M --+ co. RUSHFORTH and HARRIS[ 19681have shown that additive noise is amplified i f M > 2L cos u/A; thus a heuristic maximization of detail in the restored image is achieved by truncating the processor at M = 2L cos a l l .

IMAGE SYSTEM

OBJECT RESTORE

I Fig. 14. Block diagram of the image system with the ideal, noise-free processor.

When the image is noisy and the object is long compared with the wavelength, the truncated object restorer does not significantly improve the image quality over that of the focused image. As an example, consider the special case for

v, o 41

361

IMAGING SOLUTIONS TO INVERSE SOURCE PROBLEMS

which the noise is stationary with band-limited autocorrelation function A(z

- z ’ )=

No sindyz - z ’ ) n(z - z ’ )

(4.8 1)

It follows from the Karhunen-Loeve theorem (DAVENPORT and ROOT [ 19581) that the noise process n ( z ) can then be expanded into the eigenfunctions u(z), (4.82)

where the expansion coefficients are uncorrelated. In particular we find that ( 4 ,n,*> = No

L, 9

(4.83)

where the angle brackets denote an ensemble average. The output coefficients of the truncated, ideal, noise-free processor, when the preceding noise process is present, are

(4.84)

where the processor output R ( z ) = Z bvuvcontains the line source. Referring to fig. 13, we see that the eigenvalues ov+ 0 as v+ 00, so that the noise contribution nv/ovis greatly amplified for sufficiently large v. It is difficult to choose the appropriate truncation without specifying the object. Let us consider the case for which the a, are chosen from an ensemble for which (lav12> = Eo

(4.85)

such that Eo/NoB 1. We want to truncate the processor at some value vo where object and noise contributions to the output are about the same. This implies that (4.86)

If we truncate the processor at M = vo for which the preceding equality is satisfied, we find from eq. (4.75) that the highest-order eigenvalue v, included

368

GENERALIZED HOLOGRAPHYllNVERSE PROBLEMS

in the processor is (4.87)

For all v > v, we have a,’ < N,/Eo. For some values of v only slightly larger than vo the noise will dominate the object contribution to the processor output. For c; = 0.01 N,/E, we find the highest-order eigenvalue v1 (the value of M for truncating the processor) to be (4.88)

Table 1 gives v, and v1 for values of Eo/No and 2L/I. We see that when v1 is slightly greater than v,, this leads to a 100-fold increase in the contribution of noise energy to the processor output; this implies that a, decays rapidly for v > 2L/1, which allows us to set a practical truncation limit for large LI1:

M

=

2L/3,.

(4.89) TABLE1 Truncation limits for one-dimensional inverse filter

VI

12 102 103 1003 1004 1007

10 101 102 1001 1002 1004

2L -

E -

1

NO

10 100 100 1000 1000 1000

10 10 100 10 100

104

The processor output and input signal-to-noise levels are then about equal. For all v < M we see from fig. 13 that the eigenvalues are close to unity, yielding the very good approximation h(z, z’)E hM(z, z’) =

M

C u,(z) u,(z’),

(4.90)

0

which, when substituted into eq. (4.79), yields Y(z)= R(z) =

M 0

u,u,(z).

(4.91)

IMAGING SOLUTIONS TO INVERSE SOURCE PROBLEMS

369

To a good approximation the output of the backpropagator and the object processor h, are then identical. Any possible improvement obtained by using the inversefilter carries with it the further disadvantagethat the object processor has a spatially varying point object response. If we use the backpropagator for an open surface, then M is truncated at cos fi. The backpropagator for closed recording surfaces smaller values (2~71) provides the most complete inversion of the object field for large A and non-zero noise if the averages ( I a , I 2 , are essentially the same. However, if the average object term ( l a , l z ) decreases rapidly as v increases, it may be possible to truncate the processor at some value M < 2L/1. Inversion of a recorded field on an open hologram then can be nearly as accurate as inversion of data recorded on a closed surface. 4.4.2. Two-dimensional objects Multidimensional objects cannot, in general, be uniquely determined from their image fields. Since nonradiating sources produce fields that vanish identically outside their source volume, these sources cannot be reconstructed by any imaging system. Moreover, since an image is formed in a source-free region, the image field must satisfy the homogeneous wave equation. An exact image of an object contained in a region A (see fig. 15) can be found only if the source distribution satisfies the homogeneous wave equation; that is,

(V2 + k2)p(r)= 0

(4.92)

so that p(r) must be a perfect radiator (property I of § 4.1). We shall find that, when eq. (4.92) holds, a noise-free inversion can be achieved by forming an image with the generalized backpropagator and then using an object restorer closely related to the processor for one-dimensional objects. Also see PORTER and DEVANEY [ 1982bl. For treating two-dimensional objects we follow the analysis of Q 2.2, except that the kernel for the backpropagator is now taken as (4.93) with J, being the Bessel function of zero order. The real image field is given by

Ybr(r)= - $i

jA

dS’ p*(r’)J,(k Ir - r’ I),

(4.94)

where A is the cross-sectional area bounding the object shown in fig. 15. and the source p have different units, it is convenient to Because the field YbYbr

370

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V.8 4

Fig. 15. Two-dimensional source geometry where A is the object, C is the contour enclosing the object, and S, is the hologram.

work with the image field Y(r) = i2D Ybr(r) =

I*

dS' p * ( r ' ) D Jo(k Ir - r' I).

(4.95)

where D is a real normalization constant, to be chosen. We search for source distributionsthat are perfectly reproduced in the area A in the absence of noise. Such sources must then generate the image field Y(r) = a*p*(r) and, thus, must satisfy the integral equation (4.96)

where c is a dimensionless eigenvalue. We shall restrict our attention to the case for which the area A of the source is a circle of radius R, centered at the origin. To contain the eigenfunctions u, of eq. (4.96), we note that (4.97)

where E , = 1if v = 0 and E , = 2 otherwise. Substituting eq. (4.97) into eq. (4.96) and interchanging the order of summation and integration, we conclude that the set of eigenfunctions of eq. (4.96) has the form (4.98)

v, B 41

371

IMAGING SOLUTIONS TO INVERSE SOURCE PROBLEMS

where the superscript c is used for the cosine and s for the sine. We select the normalization constant F, so that

jAdS

' ) ( r )~ $ 7 '. ) ( r )= 6,h,,,

.

(4.99)

We fix D so that a, -, 1 as R -,GO. Substituting u, from eq. (4.98) into eq. (4.96), we eventually find that a,

=

KD

s,"

RO d r r Ji(kr) + BD nk

as R , 4 00. Setting D for arbitrary R,,

=

k / R , yields a,

=

(4.100) 1 for large values of R,. Then we h d ,

d r r Jt(kr)

(4.101)

We still need to determine the normalization constant for the eigenfunctions; substituting eq. (4.98) into eq. (4.99) yields

(4.102) giving the eigenfunction

(

ul"*s)(r)= ( z ) 1 ' 2 J , ( k r ) cos v+ sin v+

}

,

(4.103)

The behavior of the eigenvalues for large kR, is of interest here. Using Debye's asymptotic approximation to Bessel's function (see SOMMERFELD [ 1964]), we can approximate the eigenvalues by when

v < kR, - 2 , (4.104)

6,x

kR,

exp [ 2( v 2 - k2R;)'/' - 2 v cosh - ' v/kR,] 'V - k 2 R i

when

v > kR,

+2.

(4.105) These expressions are valid at v = kR, k 2 if (kRo)'l2 >> 1. At v = kR,

+ n,

312

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

where n is some integer greater than 2, we find that 1

c,+-exp[ 2n

-(2n)3/2(k~o)-'/2]

(4.106)

for large kR,, whereas at v = kRo - n we find that 6,+

(2)'".

(4.107)

A plot of the eigenvalues is shown in fig. 16 for kRo = 10, kRo = 100 and kRo = 1000. For large values of kRo we further approximatethe eigenvalues by

lo

v > kRo.

(4..108)

vlkR,

Fig. 16. Eigenvalues for two-dimensionalobjects. The eigenvalues are plotted as functions of v normalized by & for various values of &.

The radiating sources are the set of objects that can be constructed from the eigenfunctions, with weights { B}, (4.109)

v, B 41

IMAGING SOLUTIONS TO INVERSE SOURCE PROBLEMS

313

The image field Y(r)produced by such a source is given by eq. (4.99, which simplifies to

Y(r)=

c o,BY* co

'1 u',".')(r)

0

.

(4.1 10)

Equation (4.110) is similar in form to the image field for one-dimensional objects given in eq. (4.77). The approximatebehavior of the image field for large kRo is given by (4.111)

where we have used eq. (4.108). The upper limit of the s u m is understood to be the largest integer vo < kR,. For integers v < vo the eigenvalues approach unity at a slower rate than those of the one-dimensional image system. If B, = 0 for v > M,the optimum processor, in the absence of noise, is (4.112)

Using the block diagram shown in fig. 14 with the stop to limit the image space to r < R, we find the output of the object restorer in the presence of noise to be R(r) =

c (Bf. M

0

')

+ n,/o,) uv*')(r),

(4.113)

where the noise is of the form n(r) =

co

C n,~f.~)(r),

(4.114)

0

with the expansion coefficients having the covariance matrix (4.115)

This result is consistent with white background noise introduced during the imaging process. Since the eigenvalues do not decay as rapidly as those of the one-dimensional problem, we expect to find source distributions for which it is desirable to use the inverse filter. The generalized backpropagator may not yield an acceptable approximation to the inverse problem if the signal-to-noise ratio is high. How-

314

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V. § 5

ever, the reader should keep in mind that the object restorer is space dependent, whereas the backpropagator has a spatially invariant point response function.

0 5. Determination of structure of weakly scattering objects Weak scatterers are characterizqd by a linear relationship between the object and its scattered field in the Born approximation (WOLF[ 19691, BORN and WOLF [ 19751, WOLF and PORTER[ 19861) or between the object and the complex phase of the scattered field in the Rytov approximation (DEVANEY [ 19811). Since the scattered field contains the necessary information about a filtered version of the object, reconstruction is generally possible for all detail larger than about a wavelength. Many applications in medical imaging, geophysical tomography, and optical tomography exploit the weakly inhomogeneous nature of these objects. Our purpose here is to show that image formation by backpropagation of the scattered field provides a general framework encompassing all of these applications. A general approach to inversion by backpropagation is an elaboration of the two-step holographic reconstruction process illustrated schematically in fig. 17. The scattering object is illuminated by a set of known incident fields, very often plane or spherical waves, from different directions (see also fig. 1). The scattered field caused by each illumination is measured or recorded on a noncontacting surface outside the object as illustrated in fig. 17. An image is generated by establishing conjugate sources, defined as the complex conjugate of the scattered field and its normal gradient, on the recording surface as shown in fig. 17. Usually this image is a blurred or diffused replica of the object with the important feature that the image is concentrated in space. The images due to each incident wave illumination are summed, forming a filtered version of the scatterer. This version is low-pass, since only details larger than a wavelength are reproduced and are distorted because all spatial frequency components are not imaged with precisely the same weighting. In the final step this distortion is removed by deconvolution, by spectral weighting, or by any iterative technique as proposed by DEVANEY [ 19841. We shall see that there is a fundamental duality between reconstruction from monochromatic, wide-angle incident waves and reconstruction from wideband, narrow-angle incident waves. This duality is most easily revealed by the angular spectrum representation of the image field and the object. The output of the summation step must contain all the angular spectral components llyl < k,, where ko is the highest frequency component contained in the incident waves.

v, s 51

STRUCTURE DETERMINATION OF WEAKLY SCA'lTERING OBJECTS

315

Fig. 17. Inversion process showing the three steps of recording, imaging, and filtered reconstruction.

We proceed by recalling some of the basic results for weak scattering of scalar waves. We then consider the angular spectra for the scattered field for an object illuminated by a set of plane waves and develop the general reconstruction method. In $ 5.3 we develop the algebraic reconstruction method for arbitrary recording surfaces, and in $ 5.4 we show that filtered backpropagation is a special case of the general method. In $ 5.5 we present a reconstruction method for wide-band illumination that shows the duality between angular coverage and bandwidth. Two inversion methods suitable for wide-band illumi5.6 and 5.7. nation signals are reviewed in

316

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

5.1. WEAK SCATTERING IN THE BORN APPROXIMATION

We shall restrict our attention to the first Born approximation, since the Rytov approximation leads to similar results for the complex phase of the scattered field. Consider a monochromatic incident plane wave

Yi(r, t ) = Yi(r, o)e-’”‘

(5.1)

incident on a scattering object occupying a finite domain in free space. We assume that the macroscopic properties of the object are nondispersive and characterized by the real or complex index of refraction n(r). The total field consists of the sum of the scattered and incident fields,

Yt(r, 4 = Ydr, 4 + Y&,

4,

(5.2)

and must satisfy the time-independent wave equation

[V’ + k2n2(r)] Yt(r, o)= 0 , where k

= w/c. Defining the

(5.3)

scattering potential as

F(r, o)= k2[n’(r) - 11,

(5.4)

the time-independent wave equation is

(V’ + k2) Ys(r, o)=

- F(r,

o)Yt(r, o),

(5.5)

where we have assumed that the incident field is source free in the volume of interest. In the first Born approximation we assume that the scattered field is small compared with the incident field such that

I ys(r, o)l 4 I yi(r, 011

(5.6)

9

which implies that the magnitude of the scattering potential, IF I, must also be small. In the weak scattering limit we have in the first Born approximation

(V’ + k2) Ys(r, o)= - F(r, o)Yi(r, o).

(5.7)

We can formally connect the inverse scattering and inverse source problems by noting that the equivalent source for the scattered field is

p k , 0)= W, w) yi(r, a) 9

(5.8)

so that previous solutions discussed in 8 4 are also valid for the scattering problem. However,we now have an additionaldegree of freedom, since we may choose the incident field from a large class { Yi}that satisfies the homogeneous

v, I 51

STRUCTURE DETERMINATION OF WEAKLY SCATTERING OBJECTS

311

wave equation. Additional information can be obtained about the scattering potential F by making many individual scattering experiments. It is important to note that the first Born-approximate solutions are linear because if

S a --+

Y'sa

7

yib

ysb

7

then

5.2. DETERMINATION OF THE SCATTERING POTENTIAL FOR MONOCHROMATIC WAVES

WOLF [ 19691 has shown that the weak scattering problem can be solved if the object is illuminated by the set of plane waves containing all possible illumination angles where the scattered fields for each illumination experiment are recorded on a planar hologram. All spectral components contained in a sphere of radius 2k (Ewald limiting sphere) can, in principle, be determined. DEVANEY [1982a] has introduced an integral operator that performs the inversion, yielding a band-limited version of the scattering potential. This result can be obtained from the backpropagated image field by substituting the plane wave incident field Yi(r) = exp(iks, r)

(5.10)

into eq. (5.8) for the induced scattering sources, which is then Fourier transformed, yielding n

P,(K) = =

J

d3r F(r) exp [ - i(K - ks,) r] V

F(K - ks,) .

(5.11)

This equation for the induced sources can be substituted into eq. (2.66), yielding the image field (5.12) because only the source components on the unit sphere Is/= 1 contribute to the image field (see fig. 18). By varying the angle of the incident field, we sweep

378

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

Fig. 18. Fourier transform of the scattering function for monochromatic waves. The solid circle, centered at k, is a slice through the spherical shell of Fourier components obtainable for one illumination direction. The dotted circle is a slice through the sphere of possible illumination directions. The center of the Fourier component shell is swept around the ks, shell to reconstruct all Fourier components inside a sphere of radius 2k.

out the spectral region defined by F ( K ' ) where (5.13)

IK'I G 2 k .

To extract the scattering function, consider that the Fourier transform (eq. 5.12) holds over all space, as shown in $2.6, given by

Ybr(K)= - ink

s,.

d a F * ( k s - ks,) 6(ks + K ) ,

(5.14)

where 6(ks t K) is understood to be the three-dimensional delta function. Since 1st = 1, we have Ybt6,(~) = - ink-

'F(K - ks,) S(K - k ) ,

(5.15)

permitting us to extract the scattering function inside the Ewald limiting sphere by scanning over all possible real unit vectors so (also see LANGENBERG [ 19871). 5.3. DETERMINATION O F THE SCATTERING POTENTIAL FROM BROAD-BAND SIGNALS - TIME AND ANGLE SCAN DUALITY

The spatial Fourier transform of the image field at a single frequency for a given incident plane wave is given by eq. (5.15). Figure 19 shows contours on

v, 4 51

STRUCTURE DETERMINATION OF WEAKLY SCATTERING OBJECTS

379

Fig. 19. Fourier transform of the scattering function for wideband pulses. Each circle represents the countour on which the scattering potential can be determined for a given frequency.

which the scattering potential can be determined for a set of wavenumbers up to k,,,. For a continuous range of frequencies 0 < w w,,, we can find all values of F ( K )inside the sphere of radius k,,,. Suppose F(K)is spatially band limited by B. It follows that if k,, > B, there is a discrete set of plane wave directions {so} sufficientto envelop all spectral components ofF. LANGENBERG [ 19871 has derived a frequency diversity backpropagation algorithm. It is assumed that the scattering potential F i s independent of frequency. The duality discussed here holds only when the frequency dependence of Fcan be ignored. Consider the limiting case when k,, % B and the object is illuminated by a broad-band, planar pulse S(a) with direction so. Each frequency component has the image field

-=

Ybr(r,w) =

ikS(w)

-- z.

S,.

~ * ( k-s ks,) exp ( - iks * r)

(5.16a)

A frequency-balanced, time-dependent image field can be found by weighting each frequency component by w/S(w), yielding tjbr(r, t )

=

-1

16d

dw$

j4zd62F*(ks - ks,)exp[ -i(ks.r

+ at)]. (5.16b)

Evaluating the image field at t

= -so *

r/c, defining K' = k(s - so), and

380

[V, 5 5

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

recalling that k $ b h

=

w/c, and d3K' = k2 dk do, we find that

- ic2 - so rlc) = - d 3 K P*(K') exp( - iK' r) 16n3

(5.17a)

or $zr(r, - so * r/c) = +ic2F(r) .

(5.17b)

By comparing this result with eq. (5.15), we see that we can extract the scattering function either by illuminating the scatterer with all possible plane waves at a single frequency or by the dual process of illuminatingthe object with all frequencies but incident from only one direction.

5.4. INVERSE SCATTERING FOR ARBITRARY RECEIVER GEOMETRIES HOLOGRAPHIC TOMOGRAPHY

-

Holographic tomography (PORTER[ 19861) is a procedure for solving the inverse scattering problem at a single frequency when the object can be illuminated over a range of angles. In its most elegant form we illuminate the object by a plane wave and record the field and its normal gradient over a closed surface surrounding the object. Figure 17 (above in Q 5 ) illustrates the steps involved in the reconstruction. First, we illuminate the object F(r) with a plane wave. We measure both the scattered field Ys and its normal gradient, R V Ys,on the surface S,. Backpropagation is achieved by establishing the conjugate sources as shown in fig. 17. The image contains an incomplete description of the object because we have used only one direction of illumination. Second, we repeat this process for another illuminating plane wave so and add the image fields. If this process is followed for all possible illumination directions, the sum over all images is a distorted low-pass filtered version of the scattering potential F. This distortion is a weighting in wavenumber space that can be easily removed. We start from eqs. (2.8) and (5.8) with the plane wave incident field (5.18)

Yi(r) = exp(iks, * r) so that, after backpropagation, the image field is

Yb,(r) =

1"

-

d V' F*(r') exp( - iks, r ' ) Kbr(r,r') .

(5.19)

v, 8 51

STRUCTURE DETERMINATION OF WEAKLY SCAlTERING OBJECTS

381

Tomographic reconstruction or inversion consists of removing the plane wave phase either by weighting the image field or by suitable illumination of the image. The weighted images are summed over the range of directions so. Considering the set of solid angles Go : (0 < Go < 4 4 , we can write the tomographic reconstruction as T(r) =

J04n

-

dG0 Ybr(r)exp(iks, r) .

(5.20)

Substituting eq. (2.13) for K,, into eq. (5.19) and that result into eq. (5.20), we obtain T ( r ) = -2ik

[ dV’F*(r’)j,2(klr-r’I),

(5.21)

Jv

where we have used the relation (STRATTON jo(kr) = 4 jo’dO(sinO)exp(ikrcos8). An alternative to eq. (5.21) is the expression

T ( r ) = -2ikJvdV’F*(r’)[ sinklr k l - r’I r-r‘l

]’

,

(5.23)

This inversion method does not give a precise band-limited reconstruction because we have not filtered prior to backpropagation. To 6nd the relationship between T and F, we Fourier-transform eq. (5.23), obtaining

Qks)

=

1 s<2, E*(ks) 4~k’s 0 s>2, -1

~

(5.24)

which is the band-limited, weighted version of F that we have been seeking. Some details on the derivation of eq. (5.24), as well as the behavior of the appropriate post filter for correcting the reconstruction, are given by PORTER [ 19861.

5.5. HOLOGRAPHIC TOMOGRAPHY WITH PLANAR SCREENS

We discuss the case of tomographic inversion with planar screens and three-dimensional objects for comparison with the treatment by DEVANEY [ 1982b] and KAK [ 19851. The holographic approach results in a different

382

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V, 8 5

method for introducing the prefilter common in tomography. For the planar screen, tomographic inversion is a method for inverting the 3D Fourier transform in a coordinate system where the curvilinear surfaces are circles. The prefilter arises from mapping these circles into Cartesian coordinates. An expression for the prefilter is derived explicitly in the following analysis. We closely follow the development of PORTER[ 19861.

Fig. 20. Coordinate system for planar screen holographic tomography.The screen is located at z = Z,. Other illumination directions are generated by the rotation 0, as shown.

We will assume the recording plane is oriented normal to the z-axis with an incident plane wave in the + z-direction, as shown in fig. 20. The procedure is to solve for the image field and then generalize the solution for any plane wave by suitable rotations. We only need a screen located at z = + 2,;we assume the backscattered field is negligible for weak scatterers. The scattered field on the screen at z = z h is d3r' F(r') exp(ikz')

exp(ik1r - r t I ) . Ir-r'l

We define the two-dimensional Fourier transform pair

and

(5.25)

v, I 51

STRUCTURE DETERMINATION OF WEAKLY SCA’ITERING OBJECTS

383

We now determine the Fourier transform of the scattered field; in addition, we ignore contributions from evanescent waves. Considering only the radiating Fourier components, we can write the free-space Green function as (5.28)

where P is the region p 2 + q2 C 1, s = pf + qg + mi, m = + (1 - p 2 - q2)”’ and where we have ignored the contribution of the nonradiating wavenumbers; 2, 9 and i are unit vectors along the x-, y-, and z-directions, respectively. The Fourier transform of the field and its gradient in the z-direction are Y s ( P ,4, Z,)

ix mk

= -exp(ikmZh) p(kp, kq,

d Ys (p, q, Z,,) dz

~

=

k(m - 1)) ,

ikm Ys.

(5.29)

(5.30)

5.5.1. Holographic image formation In generalized holography we record the field Ys and its normal gradient n^ * V Y s .Reconstruction proceeds by establishing the conjugate sources on the surface. To perform an undistorted tomographic inversion of the scatterer, we wish to establish a filtering operation on the recorded fields. We modify the sources normally specified in generalized holography as follows:

(5.31) (5.32) where H(p, q) is a filter function to be determined later. The backpropagated image field in the object space is given by Yb,(r) = -

s,.

-

d2r“ [a,$ V” Gf(r, r ” ) - a,,G,(r, r”)] ,

(5.33)

where the Green function is

(5.34)

384

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

[V,8 5

and (5.35)

s = (p,q, -m).

Substituting the Green function given by eq. (5.35) and the surface sources given by eqs. (5.31) and (5.32), we find the image field YbYbr(r) = - ink

s jp

@ ! ! exp [ik(px + qy)] exp( - ikzm) m

P*( - kp, - kq, - k(m - 1)).

(5.36)

5.5.2. Tomographic reconstruction

To reconstruct a band-limited image of the object, we remove the illuminating plane wave and sum over all illumination directions, as shown in 5 5.4. Removing the illuminating plane wave yields eik YbYbr(r) = - ink

dpdqexp[ik(px

+ qy)] exp[ -ik(m

- 1)z]

1s jp

It is convenient to note the identity eib Ybr = - ink

ds3 exp(iks r ) P*(- ks)

where 6, is the Dirac delta function ds3 = dp dq dw; P denotes the domain < 2, K Z = pz + q2, and H ( K ) = H(p, 4). To reconstruct the scattering function, we first rotate the recording screen and the illuminating plane wave in the K,w plane. This is accomplished by a change in coordinates for the term

pz + q 2 + W’

A(K, W ) = H ( K )

6,(W

- 1 + JCT)

JcT

(5.39)

9

making the rotation W =

w, case, - K, sine,,

K

=

wo sine,

+ K, case,.

(5.40)

STRUCTURE DETERMINATION OF WEAKLY SCATI'ERING OBJECTS

385

It is convenient to express K, w in polar coordinates by defining wo = s c o s 8 ,

KO = s s i n 8 .

(5.41)

We then have A(s, 0) = H(s) sin ($

+ 0))

After some manipulation the root of So is found to be = 2 cos(eo

+ e) G Jz .

(5.43)

Reconstruction of the scattering function is limited to the spatial frequencies available from a single screen recording forward scattered components [eq. ( 5 W 1 (5.44)

Recording forward- and backscattered components on two screens would permit reconstruction of Fourier components s < 2, as noted in eq. (5.24). If we choose the prefilter consistent with conventional tomography then K H ( K ) = ~. 4 n2

(5.45)

The tomographic reconstruction requires a coherent summation of image fields (eq. 5.20), which far this case implies an integration over the set of rotations. The integral of A over 0, has the value (5.46) because eq. (5.43) has two solutions. To complete the reconstruction, we rotate the screen and illumination about the +o axis, yielding -1 d +o d 0, sin +o A(s, 0) = -. nz

(5.47)

386

[V. § 5

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

The tomographic reconstruction is

(5.48) which is the low-pass version of the scattering function F*(r), eq. (5.44).

5.6. INVERSE SCATTERING BY IMAGE FILTERING

Generalized backpropagation has been applied in the seismic context by ESMERSOY and LEVY[ 19861 to estimate the velocity potential of an acoustic medium. They have developed a two-step image filtering method, which is, essentially, an implementation of the time duality inversion discussed in 3 5.3, consisting of illuminating the scatterer by a single wide-band plane wave and recording the scattered field along a curved contour, which may or may not completely surround the scatterer. In the first step the wave traces recorded at individual receiver locations along the contour are time filtered and then [ 19851, LANGENBERG [ 19871) backpropagated. Wave migration (CLAERBOUT is similar to this first step. In the second step a space invariant filtering operation, depending only on the direction of the incident plane wave, is necessary to obtain the velocity potential. The filtered, backpropagated field is imaged at z(r) = so r/c corresponding to a plane wave source and a homogeneous background, yielding the image function

(5.49) where S(o) represents the source frequency dependence. We define the velocity potential as

F(r) k2

(5.50)

=-

and use the kernel for a wedge-shaped open hologram

L ( r , r ' , a)=

411

do'' exp[ -ikdcos(O" - O)],

(5.51)

P,

we denote

given by eq. (2.26). (To avoid confusion with wedge angles

v, B 51

387

STRUCTURE DETERMINATION OF WEAKLY SCA’ITERING OBJECTS

Esmersoy and Levy’s image function by isgnw K J r , r ’ , w ) = -~ 471

s

fi -B

PI.) Noting that we can rewrite

d0“ exp[ - iks” (r - r ’ ) ] ,

(5.52)

where s “ is the unit vector in the direction making an angle 0“ from the wedge axis (fig. 4), we have the image function d r ’ y ( r ’ ) h(r - r ’ ),

P,(r) =

(5.53)

V

s

where h(r) is the point response of the space invariant filter,

h(r) =

1s l g d O ” 8 n2

a,

dk Ikl exp[ik(s” - s o ) . r ] .

(5.54)

--oo

ESMERSOY and LEVY[ 19861 have shown that

h(P) =

(5.55)

where I A 1 is the range - fl< I A I c P expressed in the coordinates of

p

=

k(s - so).

(5.56)

If the hologram is closed, then 0 < I A I c 271 and there are no missing spatial frequency components. Emerson and Levy have studied the performance of the image filter for various receiver array coverages. The reader is referred to that paper for a detailed discussion, where it is shown that inversion performance, when the coverage is incomplete, is dependent on the structure of the scatterer. For their examples best results are obtained when the receiver array covers the greatest range of scattering angles. and LEVY Consider the example shown in fig. 21a, taken from ESMERSOY [ 19861. The scatterer is a cylinder, approximately one wavelength in diameter at the highest frequency in the illuminating normally incident wide-band pulse. The scattered field, shown in fig. 21b, has an “h” part and a “v” part, denoting the horizontal and vertical arrays. Figure 22 presents the projections of the cylindrical scatterer as obtained with infinite arrays and with finite arrays. Figure 22a shows the projections obtained with an infinite horizontal array (315 > $ > 225) and an infinite vertical array (360 > $> 315). These projections are obtained with a slant stack inversion method, also discussed by

388

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

Fig. 21. Geometry and waveforms for scattering example: (a) a cylindrical object illuminated by a plane wave; (b) the scattered fields received by the horizontal “h” and the vertical “v” arrays. (Courtesy C. Esmersoy.)

ESMERSOY and LEVY[ 19861. The projections obtained by inverting the waveforms obtained with the finite horizontal array are shown in fig. 22b, and the projections for the combined finite horizontal and vertical arrays are shown in fig. 22c. Figure 23 shows the image-filtered reconstruction of the cylindrical scatterer, with the result for the horizontal array shown in fig. 23a and the combined horizontal and vertical array in fig. 23b.

v, 4 51

STRUCTURE DETERMINATION OF WEAKLY SCATTERING OBJECTS

Fig. 22. Projections of the velocity potential determined by slant stack method. The origin for the projections is chosen at the center of the object. The horizontal axis is the projection angle @correspondingto the unit vector fi, where the angle is measured clockwise from the x-axis. The vertical axis s is the offset of the slice from the center of the cylinder: (a) the true projections; (b) the estimates of the projections obtained with only the horizontal array; (c) the projections obtained with the combined horizontal and vertical arrays. (Courtesy C. Esmersoy.)

Fig. 23. Inversion for the cylinder using the image filtering method: (a) the inversion using the horizontal array data only; (b) the use of data from both the vertical and horizontal arrays. (Courtesy C. Esmersoy.)

390

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

5.7. INVERSE SCATTERING FOR WIDE-BAND, POINT-SOURCE

ILLUMINATION

It is useful to consider inversion of a weakly scattered field when the scatterer is illuminated by a point source (ESMERSOY, ORISTAGLIO and LEVY[ 19851). This example compares closely with examples from seismics, sonar, and radar imaging. A single wide-band point source is sufficient for obtaining all projections of the velocity function when the scattered field is collected on a closed surface surrounding the scatterer. Our starting point is the image field of a scatterer illuminated by an arbitrary incident wave (eqs. 2.7, 5.8)

Ybr(r,o)=

1.

d V’ F*(r’, a)!P,*(r, o)Kbr(r, r’ o).

ESMERSOY,ORISTAGLIO and

(5.57)

LEVY[ 19851 have treated this problem in two

dimensions, where

Kbr(r, r’o)

=

-$ sgn(k) J,(k

Ir - r’ I) ,

F(r’, o)= k2y(r),

(5.59) (5.60)

yielding the real part of the image field as Re[Ybr(r,o)] = - i k 2

1”

dV’ y ( r r ) J o ( k ~ r - r s ~ ) J o ( k ~ r - r ’(5.61) ~),

where r, denotes the point-source location. Consider the filtered version of Y,, evaluated at zero time, p(r) = -41r - rsl

(5.62)

which can be written in terms of the real part of Yb,, p(r)

=

-81r - r,]

(5.63)

v, I 51

STRUCTURE DETERMINATION OF WEAKLY SCATTERING OBJECTS

39 1

On substituting from eq. (5.61) one finds that ?(r)= l r - r s l JvdVr y(r’)jomdkkJo(klr’- r s l ) J o ( k [ r - r ’ l ) b(lr’ - rsl - Ir- r’l)

(5.64)

after applying the suitable identity (ERDELYI[ 19541). To show that 9 gives the projections of y(r), we reconsider eq. (5.64) using the (p, 5 ) coordinate system with origin at r, as shown in fig. 24. Noting that i represents the orientation of the line I r - r, I and that its length is r, we can rewrite eq. (5.64) as $J(i, r) = r

2

+

t2)1/2

- [ ( p - 4 2 + 52]1/2} , (5.65)

Fig. 24. Coordinate system for wideband point source illumination.

where (p, t, 4 denotes the coordinates of the object for a given direction i. Integrating over p to remove the 6, function yields, after some manipulation, $J(i, 2r) =

Jm

dty(r,t;O r Z O ,

-m

=o

r=O.

(5.66)

Since the point source must be located outside the object, we have y(0,O; +) = 0, so that the result 9 (r, 0) = 0 does not affect the reconstructed velocities. The velocity function y can be evaluated by any tomographic reconstruction algorithm including the backprojection or projection-slice methods.

392

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

16. Conclusion We have shown that a theory of holography, generalized to arbitrary recording surfaces,can serve as a framework for a physical understanding of solutions to inverse source and inverse scattering problems. We have also shown that conventional holography is an approximate implementation of the theory of generalized holography.Important developments discussed here include extensions of conventional holography to nonhomogeneous media and to electromagnetic waves. The concepts of radiating and nonradiating sources have been shown to be crucial to the understanding of the inverse source problem. We have analyzed classes of sources whose radiated field can be inverted for the source structure. The inverse scattering problem is considerably broader because of additional freedom permitted by the use of several distinct illuminations. We have seen that one can perform the inversion either by broad-band illumination or by a tomographic approach, taking advantage of many illumination directions. Some practical implementationschemes have been presented for the inverse scattering problem. Active research in these areas is currently being pursued. Extensions of generalized holography to attenuating media appear to be promising. A substantial effort is also being made to develop algorithms for inversion based on backpropagation that will be more accurate and more efficient than the conventional methods.

Acknowledgements The author is deeply indebted to A. Devaney who has participated in and encouraged the developmentof these ideas over the last several years. K. Liang has graciously assisted in the numerical analysis of the image field for open holograms. Special thanks to the Editor for his suggestions which have materially improved the manuscript. 0. Gara has patiently and accurately guided this paper through the preparation of its many drafts. Finally, thanks to Charlotte Lin for her constant support and gentle pressure during the writing of this paper.

VI

APPENDIX A

393

Appendix A. Field representations for surface sources A.l. SCALAR WAVES

The generalized backpropagator is launched from a surface coincident with the recording surface s h by setting up a monopole and a dipole layer on the surface as shown in fig. 25. The volume inside Sh is the red image space with field Yr, whereas the outside volume is the virtual image space with field Y,,. The field and its normal gradient are discontinuous across the source layers, with the discontinuities given by

ii.(VYr - VY,,) = a,,

(A. 1)

Yr - Y,, = 6,.

(A.2)

Fig. 25. Monopole and dipole layers on the surface S,.

In the following paragraphs we find an expression for the exterior and interior fields radiated by these sources. Consider the extinction theorem (WATERMAN[ 19691) for the exterior field when the space is illuminated by the incident field !Pi, dS“ =

{:(r)

[ YvV”Gf(r,r ” ) - Gf(r, r”)V’Yv]

outside s h , inside s h

9

394

GENERALIZED HOLOGRAPHY/INVERSE PROBLEMS

where G, is the free space Green function Gf(r, r ” ) =

exp(ik1r - r”1) 4n1r-r”l ’

64-41

and where the normal n^ points outward. The extinction theorem is derived from Green’s theorem, subject to the Sommerfield radiation condition that the scattered field Yv - Yibe outgoing at infinity. For the interior or real image space the theorem gives -

s,,

d S ” 1 . [ YrV”Gf(r,r ” ) - Gf(r, rN)Vf’Yr]

=I

0 outside s h , Yr inside s h .

For the second step or reconstruction problem the incident field is zero. The field everywhere, except on s h , is found by adding eqs. (A.3) and (A.5), yielding the reconstruction field: Y(r) =

Js

dS”[o,G,(r, r ” ) - a,n^-V”G,(r,r”)] .

(A4

h

Now consider a variation on the extinction theorem for scattered fields given by eq. (2.2). We prove that d S “ n^. [G,(r, r ” ) V ” Ys(r”) - Ys(r”) V”G,(r, r”)]] Ys(r) outside s h , = I0 inside s h . Using Green’s theorem to convert the left side of eq. (A.7) to a volume integral and then interchanging orders of integration yields

-I I, dV’ ps(r’)

dV“ [G,(r’, r ” ) b,(r - r ” ) - Gf(r, r “ ) 6,(r‘ - r”)]

for the left side. Noting that r’ is inside find

sh

but r can be inside or outside, we

- Gf(r, r ‘ ) outside Sh , inside Sh , giving eq. (A.7).

VI

395

REFERENCES

A.2. ELECTROMAGNETIC WAVES

To develop expressions for electromagnetic image fields, we need to represent the propagating field in terms of discontinuities in electric and magnetic fields across the surface S , shown in fig. 25. The background medium (see 8 2.6) is identical in the radiating step and the imaging step. No sources are present in either the real or virtual image space. The electric field propagates according to eq. 2.42, and the dyadic Green function r(r,r ’ ) is given by eq. (2.43). As before, we start from the extinction theorem given by PAITANYAKand WOLF[ 19721, as

+ [V” x

r(r, r”)] * [ i x E,(r”)}.

(A. 9)

We may write a similar equation for the virtual image space: EJr)

inside s h outside S ,

d S ” [ r ( r , r ” ) .[ i x

V ” x E,(r”)]

+ [V”x r(r,r ” ) ]* [ i x E,(r”)} .

(A. 10)

Combining equations for both the real and the virtual image spaces yields E(r) =

jS + [V”

- H,)] [ i x (E, - E,)]} .

dS“{ikr(r, r ” ) . [ti x ( H ,

h

x

r(r,r ” ]

(A. 11)

We can express the discontinuities in electric and magnetic fields in terms of “virtual” surface currents K,, K,

-411 ii x (E” - E,) = -K* C

I

(A. 12)

References BARNES,C. W., 1966, J. Opt. SOC.Am. 56, 575-578. BERSHAD,N. J., 1969, J. Opt. SOC. Am. 59, 157-162. BLEISTEIN,N., and J. K. COHEN,1977, J. Math. Phys. 18, 194-201. N. N., 1973, Inverse Scattering, NTlS No. AD-A910661 (Department of the Navy, BOJARSKI, Naval Air Systems Command).

396

GENERALIZED

HOLOGRAPHY/INVERSE PROBLEMS

BOJARSKI, N.N., 1981, J. Math. Phys. 22, 1647-1650. BOJARSKI, N. N., 1982, IEEE Trans. Antennas & Propag. AP-30, 975-979. BORN,M., and E. WOLF,1975, Principles of Optics, 5th Ed. (Pergamon Press, New York) pp. 480-484. BREMMER, H., 1951, Commun. Pure & Appl. Math. 4, 61-74. CAULFIELD, H. J., and S. Lu, 1970, The Applications of Holography (Wiley, New York) pp. 31-49. CLAERBOUT, J. F., 1985, Imaging the Earth’s Interior (Blackwell Scientific Publications, Oxford). COLLIER, R. J., C. B. BURCKHARDT and L. H. LIN, 1971, Optical Holography (Academic Press, New York) pp. 273-276. COURANT,R., and D. HILBERT,1962, Methods of Mathematical Physics, Vol. 2 (WileyInterscience, New York). DAVENPORT, W. B., and W. L. ROOT, 1958, Random Signals and Noise (McGraw-Hill, New York) ch. 6. DEVANEY, A. J., 1981, Opt. Lett. 6, 374. DEVANEY, A. J., 1982a, Opt. Lett. 7, 111-112. DEVANEY, A. J., 1982b, Ultrason. Imaging 4, 336-350. DEVANEY, A. J., 1984, IEEE Trans. Geosci. & Remote Sensing GE22, 3-13. DEVANEY, A. J., and R. P. PORTER,1985, J. Opt. SOC.Am. A 2, 2006-2012. DEVANEY,A.J., and G.C. SHERMAN,1982, IEEE Trans. Antennas & Propag. AP-30, 1034-1037. DEVANEY, A. J., and E. WOLF,1973, Phys. Rev. D 8, 1044-1047. ERDELYI, A., ed., 1954, in: Bateman Manuscript Project. Tables of Integral Transforms, Vol. 2 (McGraw-Hill, New York). ESMERSOY, C., and B. C. LEVY,1986, Proc. IEEE 74,466-475. ESMERSOY, C., M. L. ORISTAGLIO and B. C. LEVY,1985, J. Acoust. SOC.Am. 78, 1052-1057. FISCHER, M., and K. LANGENBERG, 1984, IEEE Trans. Antennas & Propag. AP-11,1080-1088. FITZPATRICK, G., 1979, Proc. IEEE 67,536-553. FLAMMER, C., 1957, Spheroidal Wave Functions (Stanford Univ. Press, Stanford, CA). GOODMAN, J. W., 1968, Introduction to Fourier Optics (McGraw-Hill, New York). IWATA,K., and R. NAGATA,1975, Jpn. J. Appl. Phys. 14,379-383. KAK, A. C., 1985, Tomographic imaging with diffracting and nondiffracting sources, in: Array Signal Processing, ed. S. Haykin (Prentice-Hall, Englewood Cliffs, NJ) ch. 6. 1982, IEEE Trans. Sonics & Ultrason. SU-29,213-217. KENUE,S.K.,and J. F. GREENLEAF, and G. WADE, 1985, IEEE Trans. Sonics & Ultrason. SU-32, LAN,C. Q., G.T. FLESHER 562-565. LANDAU, H. J., and H. 0. POLLAK,1961, Bell Syst. Tech. J. 40,65-84. LANDAU, H. J., and H. 0. POLLAK,1962, Bell Syst. Tech. J. 41, 1295-1336. LANGENBERG, K. J., 1987, Applied inverse problems for acoustic, electromagnetic, and elastic wave scattering, in: Basic Methods of Tomography and Inverse Problems, ed. P.C. Sabatier (IOP Publishing, Bristol, England). MITTRA,R., and P. L. RANSOM,1967, Imaging with coherent fields, in: Polytechnic Institute of Brooklyn Microwave Research Institute Symp. Series, Vol. XVII (Polytechnic Press of the Polytechnic Institute of Brooklyn) pp. 619-647. MORSE,P. M., and H. FESHBACK, 1953, Methods of Theoretical Physics (McGraw-Hill, New York). MUELLER, R. K., M. KAVEKand G. WADE,1979, Proc. IEEE 67, 567-587. PATTANAYAK, D. N., and E. WOLF, 1972, Opt. Commun. 6,217-220. PORTER,R. P., 1969a, The Theory of Holographic Image Formation with Scalar Electromagnetic Waves, PbD. Thesis (Northeastern Univ.).

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PORTER,R. P., 1969b, Phys. Lett. A 29, 193-194. PORTER,R. P., 1970, J. Opt. SOC.Am. 60,1051-1059. PORTER,R. P., 1971, J. Opt. SOC.Am. 61, 789-796. PORTER,R. P., 1986, J. Acoust. SOC.Am. 80, 1220-1227. PORTER,R. P., and A. J. DEVANEY, 1982a, J. Opt. SOC.Am. 72, 327-330. PORTER,R. P., and A. J. DEVANEY, 1982b, J. Opt. SOC.Am. 72, 1707-1713. PORTER,R. P., and W. C. SCHWAB, 1971, J. Opt. SOC.Am. 61, 664A. RUSHFORTH, C. K., and R. W. HARRIS,1968, J. Opt. SOC. Am. 58, 539-545. G. C., 1967, J. Opt. SOC.Am. 57, 1490-1498. SHERMAN, SHEWELL, J. R., and E. WOLF, 1968, J. Opt. SOC.Am. 58, 1593-1603. SLEPIAN,D., 1964, Bell Syst. Tech. J. 43, 3009-3057. SLEPIAN, D., 1965, J. Math. Phys. 44,99-140. SLEPIAN, D., and H. 0. POLLAK,1961, Bell Syst. Tech. J. 40,65-84. SOMMERFELD, A., 1964, Partial Differential Equations in Physics (Academic Press, New York) p. 189. STONE,R., 1981. Ultrason. Imaging 11, pp. 385-398. STRAITON,J. A., 1941, Electromagnetic Theory (McGraw-Hill, New York) p. 410. STROKE,G. W., 1966, An Introduction to Coherent Optics and Holography (Academic Press, New York). TAI,C. T., 1971, Dyadic Green’s Function in Electromagnetic Theory (Educational Publishers, Scranton). G., E. L. ROPEand R. A. HAYWARD, 1981, IEEE Trans. Antennas & Propag. AP-29, TRICOLES, 320-326. WATERMAN, P. C., 1969, J. Acoust. SOC.Am. 45, 1417. WOLF,E., 1969, Opt. Commun. 1, 153-156. WOLF, E., and R. P. PORTER,1986, Radio Sci. 21, 627-634.

AUTHOR INDEX

A ABELES,F., 289,309 AGRANOVICH, V. M., 230,243,244, 289, 309 N. N., 230,240,245,248-251, AKHMEDIEV, 261, 263, 267, 271-273, 280,309, 310 AKOPYAN,V. S . , 125, 158 ALANAKYAN, Yu. R.,243,310 ALFERNES, R.,83, 102 Ai.IPP1, A,, 297, 304,310 ALTAMIRANO, H., 91, 95, 102 AMES,W. F., 242,283,310 ANDERSON, J. A., 91,94, 101 ANDRES,P., 44, 99, 101, I05 ANKELE,G., 41, 56, 107 AOKI, S., 61, 102 ARISTOV,V. V., 45, 61, 76, 101, 102 J., 230, 235, 236, 239, 245, 251, ARIYASU, 280, 283,310,312,313 ARLOT,P., 297, 307,310, 313 ARMITAGE, J. D., 98, 105 ASAKURA, T., 16, 104 ASHKIN,A,, 305,310, 312 ASSA,A., 65, 102 ASSANTO,G., 297,300,310 AS'TAKHOV, A. YA., 128, 153, I58 AUSTIN,S., 30, 103 AZZAM,R.M. A., 120, 158

B BABICHENKO, V. S., 230, 243, 244,309 BALLANTYNE, J. M., 297, 300,310 BALLIK,E. A,, 74, 102 BAR-ZIV,E., 25, 27, 62, 102 BARANOV, V. K., 163, 180,225 BARNES,C. W., 359,395 BARNETT,M. E., 213,214,225 BARTELT,H. O., 91, 92, 95, 96, 102 BASSETT,I. M., 191, 193, 197,201-203, 216, 218-220,224,225,226 BAZADZE,M. A,, 103 BEAUCHAMP, K. G., 56, 102 399

BELANGER,P.-A.,113, 117, 118, 158, 160 BENCZE,GY. L., 126-128, 131, 132, 136, 138, 139, 144-147, 153, 158 BENNION,I., 231,297, 300-302, 307,310,311 BENTON,S. A., 66, 102 BERGMANN, L., 4, 5 , 102 BERKHOER, A. L., 247,310 BERNARDO, L. M., 66, 102 BERRY,D. H., 66, 104 BERSHAD,N. J., 363,395 BERTOLOTTI,M., 230,297, 304, 305,310, 312 BETSER,A. A., 65, 102 BIALOBRZESKI, P., 17, 20, 102 BICKEL,W. S., 119, 158 BIEDERMANN, K., 41,102 BLEISTEIN, N., 318, 326, 351, 353, 355,395 BLISS, E. S., 285, 310 BLUME,S., 4,46, 102 BOARDMAN, A. D., 230,231,245,246,248, 251, 252,266-268,270-274,279,283, 284, 289,292,310,312,313 BODEM,F., 170,225 BOEGEHOLD,H., 163, 169,225 BOIKO,B. B., 304,310 BOJARSKI,N. N., 318,326,396 BOLONGINI,N., 57,91,92, 102, 107 BOLTAR,K.O., 271-273,310 BORN,M., 11,44, 100, 102, 205,225, 326, 331, 374,396 BOSECK,S., 45, 104 BOURGEON,M. H., 80,91, 97, 102 G., 30, 102 BOUWHUIS, BRANNEN,E., 113, 159 BREAZEALE, M. A,, 29, 103 BREMMER,H., 317,396 BRENEMAN, E. J., 204,225 BRENNER,K. H., 76, 102 BRIGUET,V., 297, 307,311 O., 25,26,46, 53-59, 70, 102, BRYNGDAHL, 105, 118, I58

400

AUTHOR INDEX

BURCH, J. M., 24, 59, 102 BURCKHARDT, C. B., 338,396 BURTON, R. A,, 91, 102 BYRD,P. F., 271,272,310 C

CADA,M., 229,310 CAGLIOTTI, E., 301,310 CALLIGARIS, F., 30, 102 CAMPBELL, D. K., 123, 158 CARTER, G. M., 229,297,298, 300,310 CARTWRIGHT, S., 91, 102 S. L., 74, 103 CARTWRIGHT, CAULFIELD, H. J., 337,396 CESS, R. c., 189,209,226 F. H., 121, 158 CHAFEE, CHAN,J. H. C., 74, 102 CHANDRASEKHAR, S., 205,225 CHANG, B. J., 18, 83, 84, 102 CHANG, T. Y., 41,107 CHAVCHANIDZE, V. V., 37,56, 103, 107 CHAVEL, P., 69-71,102 CHELLI,H., 297, 307,312 CHEMLA, D. S., 236,310 CHEN,Y. J., 229, 297, 298, 300,310 CHENG,Y. S., 84, 102 CHERNYAK, V. YA.,230,243,244,309 CHILWELL, J. T., 230, 255,258, 266, 269, 270,312,313 R. A., 119, 158 CHODZKO, CHOUDHURY, A., 149, 160 J., 229,310 CHROSTOWSKI, CHUGUI, Yu. V., 20, 102 CHURIN, E. G., 116, 127, 136, 159 CIUTI,P., 30, 102 J. F., 320,386,396 CLAERBOUT, COHEN, J. K., 318,326, 351, 353, 355,395 Y., 15, 19, 20, 102, 104 COHEN-SABBAN, COLBERT, H. M., 29,72,102 COLLARES-PEREIRA, M., 183,225 COLLIER, R. J., 338, 396 COOK,B. D., 29, 73, 102 CORNEIO, A., 65,66,91.94,95, 102, 105 CORNEIO-RODRIGUEZ, A., 92-95, 102, 106 COURANT, R., 325, 335, 358,396 COWLEY, J., 3,4, 7,9, 14, 15, 25-27, 29, 35, 37,39, 102, 103 J. M., 65, 104 COWLEY, Cox, I. J., 149, 150, 160 CROVA, M. A., 88, 103

CULLEN, T. J., 297,302,310,313 CUSH, R.,297, 307,310,311

D DAINO,B., 301,310 DALLAS, W. J., 66, 103 DAMMANN, H., 39,49,56, 103 DANILENKO, Yu. K., 125, 158 DANILOV, V. A., 123-125, 158-160 DAVENPORT, W. B., 367,396 A., 44, 103 DAVILA, DAVIS,D., 18, 24, 107 DE VELIS, J. B., 103 DECK,R. T., 240,312 DECKERS, CH., 35, 37,41, I03 V. P., 159 DEGTJARJOVA, DELISLE, C., 300, 312 DENISYUK, YU. N.,37, 103 DENTE,G. C., 120, 159 L. N., 30, 103 DERIUGIN, DERRICK, G. H., 189, 191, 193, 197, 201-203,225,226 A. J., 318-320,330, 335, DEVANEY, 351-355, 357, 358, 363,369, 374, 377, 381,396,397 DICKE, R. H., 224,226 DOBOSZ,M.,25.27, 106 DURELLI, A. J., 141, 159 DYSON,J., 114, 115, 159 DZHILAVDARI, I. Z., 304,310 DZIEDZIC, J. M., 305,310

E EBBENI, J., 5 , 32-34,15, 83, 103 EBERSOLE, J. F., 51, 103 EBRALIDZE, T. D., 25,26, 103 EDGAR,R. F., 16.22, 70, 103 EDMONDS, W. P., 113, 119, 159 EGAN,P., 230, 231, 266-268, 270-274, 279, 310,313 EGOROV, G. S., 121, 122, 159 ELEONSKII, V. M., 271-273,310 A., 391,396 ERDELYI, ERKO,A. I., 61, 76, 102 ESCHBACH, R., 26,53,56-58, I05 ESMERSOY. C., 318-320,386-388,390,396 EU, J. K. T., 49-51, I03

AUTHOR INDEX

F FEDYANIN, V. K., 230, 271, 272,310, 312 FEIT,M.D., 285,310 FERGUSON, J. R.,120, 159 A., 297, 304,310 FERRARI, FESHBACK, H., 323,396 FINK,D., 120, 159 FISCHER, M., 325,396 G., 319,396 FITZPATRICK, C., 364,396 FLAMMER, FLANDERS, D. C., 30,61, 103 FLECK,J. A., 285,310 G. T., 319,396 FLESHER, FORTENBERRY, R.M.,236,297, 300,310, 313 G., 80,91,97,102 FORTUNATO, FOURE,J. C., 66, 103 M.,92, 103 FRANCON, FRIEDMAN, M.D., 271,272,310 FUJI,H., 76, 104 FUJIWARA, H., 35, 36, 103 FURLAN, W. D., 66, 106

G GABEL, A., 236,313 GABOR, D., 222,225 GABRIELLI, I., 30, 102 GALBRAITH, I., 236,312 A. D., 9,40,107 GALPERN, GANCI,S., 20, 103 GARAVAGLIA, M.,57, 107 M. P., 88, 103 GARBE, GARDNER, C., 241,310 GARIBASHVILI, K. A., 56, 107 GARMIRE, E., 297, 307,311 GAUTHIER, R.C., 229,310 GAVRIKOV, V. F., 119, 159 A., 205,225 GERSHUN, W. M., 301,310 GIBBONS, GIBES, H. M., 229,310,311 U. J., 297, 300,310 GIBSON, GLATT,I., 65, 66, 68, 103, 104 GLAUBER, R.J., 220,225 GLENN,R.,300, 301,311 GOLUB,M. A,, 123, 125, 159, 160 GOMEZ-REINO, C., 166,226 GONCHARSKII, A. V., 124, 125, 159 GOODMAN, J. W., 224,225, 338,396 GOODMAN, N. B., 178,226 GOODWIN, M. J., 231,297, 300-302,310, 311

40 1

GORDON, J. P.,231,240, 304, 306, 313

GoRi, F., 5 , 76, 81, 86, 91, 103 GOSSARD, A. C., 229,310 GREENE, J., 241, 3IO

GREENLEAF, J. F., 319,396 GREENSPAN, D., 242,283,311 GROH,G., 39,56, 103 GROUSSON, R.,35, 37, 103 GROVER, C. P., 76, 103 M., 231,285,288, 289,311 GUBBELS, GUIGAY, J. P., 25, 27, 103

H

HAMILTON, D. K.,149, 150, 160 HANE,K., 30, 76, 103 HARDING, K. G., 74, 103 HARGROVE, L. E., 30, I03 HARPER, D. A., 214,215,225 HARRIS,R.W., 364, 366, 397 S., 30, 76, 103 HATTORI, HAUS,H. A., 297,302,31I HAWRYLUK, A. M.,61, 103 HAYWARD, R.A., 318, 363,397 HECKENBERG, N. R.,236,312 HERMANN, J. P., 304,312 HERSCHEY, R.,76, 104 HETHERINGTON 111, W. M., 231, 279,313 HIEDEMANN, E., 29, 103 HIEDEMANN, E. A., 30, 103 HIGBIE,J., 121, 159 HILBERT, D., 325,335,358,396 HILDEBRAND, R.H., 214,215,225 HINTERBERGER, H., 163, 178, 180, 187,225, 226 HIRSCH,P. M.,116, 159 HOLLAND, W. R., 271,311 HOPKINS,H. H., 10, 103 HOTTEL,H., 175, 189,225 HOWELL, J. R.,189,226 I J., 99,101 IBARRA, ICHIOKA, Y.,73, 106 IKEDA,K., 95, 103 IMAI,Y., 35, 103 G., 4,43,44, 57, 100, 103, INDEBETOUW, 104

R.E., 9,25, 26, 53, 104 IOSELIANI, IPPEN, E. P., 297, 302,311 IRONSIDE, C. N., 297, 302,310, 313 ISHIGURO, K., 66, 105

402

AUTHOR INDEX

IVANOVA, G. A., 45, 101 IWATA,K., 319,396

J JAHNS,J., 5 , 66, 75, 76, 81, 82, 86, 91, 92, 95,96, 102, 104 JAIN,R.K., 236,311 JANNSON, I., 170,225 JAROSZEWICZ, Z., 59,61, 104 JENSEN,S. M., 229, 301,311 JEROMINEK, H., 300,312 JEWELL, J. L., 229,31 I JINGHONG, T., 84, 104 JOHNSTON, A., 229,312 JONES,C. K. R. T., 230,280-282,31 I JORDAN, J. A., 116, 159 JOYEUX,D., 15, 19, 20, 102, 104 JOZWICKI, R.,14, 15, 33, 104 S., 66, 76, 104 JUTAMULIA,

K

KAFRI,O., 37, 62, 66, 68, 102-104 KAIJUN,H., 66,104 KAK,A. C., 381,396 KAKICHASHVILI, SH. D.,45, 104 K A L E S ~ S K41,42, I, 54 KALESTY~~SKI, A., 37, 39-43, 53, 54, 56, 104, 107 KAMIYA,T., 4, 30,46,47, 104, 107 KAMIYAMA, M., 30, 104 KAPLAN,A. E., 230, 231, 233, 234, 240, 304-306,311,313 K A R , A. J., 236,312 KARAGULEFF, C., 236,313 KARNEEV, S. V., 123, 125, 159, 160 KASANSKII,N. L., 160 KATYL,R. H., 95,104 KAVEK,M., 319,396 KENUE,S. K., 319,396 KEREN,E., 37,62,65, 102,104 KHAN,M. E. R.,120, 158 V. A., 116, 160 KHANOV, KHOO,I.C., 231, 259,297, 302-304,313 KIKUTA,S., 61, 102 KIM,B. G., 297,307,311 KING,M. C., 66, 104 KIRIJANOV, V. P., 116, 136, 159, 160 KIRKBY, C. J. G., 297, 307,310,311 KISHVARADI, A., 128, 131, 132, 153, 158 KITAYAMA, K., 301,311 KLAGES,H., 69, 104

KLIMCHUK, V. V., 56, 107 KLIMOV,A. N., 159 KLIMOV, Yu. G., 120, 159 KOBAYASHI, S., 66, 107 KOCK,M., 39,56, 103 KODATE,K., 30, 104 KOGELNIK, 10 KOGELNIK, H., 104, 223,225, 229, 253,31 I KOKOULIN, F.I., 116, 136,159, 160 KOLODZIEJCZYK, 42, 54 KOLODZIEJCZYK, A., 59,61, 104 KOLODZIEJCZYK, A., 104 KOLOKOLOV, A. A., 242,31 I KOMISSARUK, V. A., 65, 104 W. A., 30, 103 KOMOCKIJ, KOMOV,G. M., 128, 158 KONITZ,H., 9, 25, 29,45, 104 KORIAKOVSKII, A. s., 66, 104 V. I., 230,240,310 KORNEEV, KORONKEVITCH, V. P., 20,102, 116, 127, 136, 159, 160 KOSTER,A., 297,306, 307,311-313 KOTHIYAL, M. P., 66, 104 KOZAK,S., 20, 106 KRASINSKI, J., 66, 104 KREUZER, J. H., 160 KRITCHMAN, E. M., 188,225 KRIVENKOV, B.E., 20, 102 KRUSKAL, M., 241,310 KUCHIBHATLA, D., 297, 300,310 KUMAR,A., 230,31 I KUPRENYUK, V. I., 119,160 KUSAKA, Y., 108 KUSHWAHA, M. S., 251, 311 KUZMENKO, Yu. V., 230,240,310

L LAN,C. Q., 319,396 H. J., 364,396 LANDAU, LANDAU, L. D., 120, 159 LANDGRAVE, J. E. A., 44,103 U., 230, 235-237, 240,249, 251, LANGBEIN, 275,280,284,287,295,311,313 P., 59,104 LANGENBECK, LANGENBERG, K., 325,396 LANGENBERG, K. J., 318, 378, 379, 386,396 LASH,R.,45, 104 LAVES, A., 297,302, 31I LAU,E., 5 , 75, 104 LAVAL,S., 297, 306, 307, 311, 312 LAVOIE,L., 118, 159

AUTHOR INDEX

F., 230.235-237,240, 249, 251, LEDERER, 275, 280,284, 287,291,292,294-296, 311-313

LEDGER,A. M., 118, 160 LEE, 10 LEGER,J. R.,69, 70, 72, 104 LEIFER,I., 98, 104 LEINE,L., 230, 280, 284, 287, 311 LEITH.E. N., 76, 83, 84, 102, 104, 107 LENOUVEL, M. L., 88,104 LEONBERGER, F. J., 297,302,311 LESEM,L. B., 116, 159 LEUKOVA, G. A,, 127, 136, 159 LEUNG,K. M., 230,311 LEVY,B. C., 318-320, 386-388, 390, 396 LI, KAMWA, 297,302,313 LI, T., 104,223,225 LI, Y., 92, 95, 102 LIAO,C., 229, 297, 299, 311, 313 LICHTENBERG, A. J., 170,225 LIEBES,S., 120, 159 LIFSHITZ,E. M., 120, 159 LIGHTMAN, A. J., 91, 102 LIM, C. S., 66, 104 LIN,J. A., 65, 104 LIN, L. H., 338,396 LIN,T. W., 66, 104 LIND,R. C., 236,311 LINFOOT,E. H., 39, 105 LIT, J. W.Y., 113, 120, 159 LITVAK,A. G., 230,233,234, 290, 311 LIU, C. Y. C., 49, 50, 103 LIVNAT,A,, 65, 104 LOHMANN,A. W., 4, 5, 7, 25,43,49-51, 56, 62, 65, 66, 68, 69, 75, 76, 81, 82, 86, 98-100, 102-105, 123, 160 LOKHMATOV, A. I., 116, 160 LOMTEV,A. I., 230,244,31 I LOPEZ-RIOS,T., 289,309 LOUDON,R.,221,225 LOVE,J. D., 223,226 Lu, S., 337,396 LUGIATO,L. A,, 229,311 LUKOSZ,W., 297, 307, 311 LUNEBURG, R.K., 185,225 LUQUE,A., 182, 184,226

M MACGOVERN, A. J., 83,98, 105 MAI, Xu, 229, 230, 258,297,299-301.310, 312, 313

403

MAIER,A. A., 301,311 MAKER,P. D., 233,311 MALACARA, D., 25,65, 66, 103, 105 MALLICK,S., 10, 35, 37, 103, 105 MALONEY,P. J., 304, 305, 312 MANDEL,L., 219,225 MANDEL,P., 229, 311 MARADUDIN, A. A., 230,234-236,244,245, 251, 277, 278, 290-292, 294, 296, 310, 311,313 MARCUSE,D., 105,240,252,253,256, 304, 311 MARCZENKO, W. M., 66,104 MARGALIT,E., 66, 104 MARTINOT,P., 297, 306, 307,311 V. V., 61, 76, 102 MARTYNOV, MASON,N. J., 297,302,313 MASON,S. B., 119, 158 MATHEW,J. G. H., 236,312 MAXWELL, J. C., 185,226 MAZILU,D., 230, 236, 251, 255, 280, 283-286,291, 296,311,312 MCCALL,S. L., 229, 310 A. W., 160 MCCULLOUGH, MCCURRY,R. E., 75, 105 MCINTIRE,W., 202,226 MCLEOD,J. H., 112, 160 MEGRELISHVILI, R. SH., 103 MENZEL,CH., 37,105 MENZEL,E., 37, I05 MERRILL,D. P., 66, 102 MERTENS,R., 30, 103 MIHALACHE, D., 230,236,244,246,248, 251,252,255-257,271,272,280, 283-286,291,292,294,296,310-312 MIKHALTSOVA, I. A., 116, 127, 136, 159, 160 MIKHLYAEV, S. V., 20, 102 MILLER,D. A. B., 229,236, 310,312 MILLS,D. L., 243,289,312 MILLS,D. R., 201,225,226 MINANO,J. C., 184487,226 MIRONOV, V. A., 230, 233, 234, 290,311 MITTRA,R.,317,396 MIURA,R., 241,310 MIYAGI,M., 230, 233, 234, 271,312 MIZUSHIMA, Y., 30, 31, 107 MOLLEDO,A. G., 182,226 MOLONEY,J. V., 230,231, 236, 239, 277, 278,280-285,288,289,292,302,310-313 MONTEMAYOR. . V. J... 240.312 . MONTGOMERY, C. G., 224,226

404

AUTHOR INDEX

MONTGOMERY, W. D., 3, 5, 7, 9, 41, 100, 105

MOODIE,A., 3, 4, 7, 9, 14, 15, 25-27, 29, 35,37, 39, 102, 103

MOON,P., 226 MORRIS,J. R., 285,310 MORSE,P. M., 323,396 MOSHREFZADEH, R., 297, 300, 301,310,312 MUCHANOV, B. F., 121, 160 MUELLER,R. K., 319,396 MUMLADZE, V. V., 56, 107 MUMOLA,P. B., 160 MURATA,K., 66, 105 MURTY,M. V. R. K., 91,94,95, 102, 105 N NAGATA,R., 319,396 NAKANO, Y.,66, 105 NALIVAIKO, V. I., 116, 160 NATH,4 NATH,N. S. N., 4,29, 106 NAUMIDI,L. P., 125, 158 NAZMITDINOV, R. G., 230,312 NEHME,M. A., 240,313 NESRULLAJEV, A. N., 297, 304,310 NEVIERE,M., 297,306,313 NIEDERMAYER, T. P., 153, 158 NIEMEIER,TH.,48, 105 NING,X.,184, 188, 214,226 NISHIDA,S., 230,233, 234, 271,312 NISHIMURA, T., 66,105 NITRAI,G., 128, 131, 132, 153, 158 NOLTING,H. P., 252,312 NOVIKOV, I. D., 120, 160 0

OGALLAGHER, J. J., 184, 188,202,226 OGLAND,J. W.,119, 137,160 OHNISHII,K., 14,66, 108 OHTSUKA, Y.,35,103 OJEDA-CASTANEDA, J., 4,43,44, 56, 76, 82, 86,91,92,98-100,

101, 102, 104,105

ORISTAGLIO, M. L., 319, 320,390,396 OSTER,G., 62, I05

P

PACKROSS,B., 26,53, 56-58, 105 PALTCHIKOVA, I. G., 116, 127, 136, 138, 147, 158-160

PAPPU,S. V., 45, 52, 107 PARAIRE,N., 297, 306,307,312,313

PARDO,F., 297, 307,312 PARFJANOWICZ, G., 21, 106 PARIS,D. P., 49, 105, 123, 160 PARKS,V. J., 141, 159 PARRY,G., 66,107 PASSNER, A., 229,310 PATELA,S., 300,312 PATON,B. E., 229,310 'PATORSKI,K., 5, 10, 11, 13, 14, 17, 20, 21, 24-27, 29-34, 41, 51, 52, 56, 59, 61, 62, 65, 66, 73, 14, 77, 78, 80, 84-87, 91-96, 102, 105-108 PATTANAYAK, D. N., 395,396 PENNINGTON, K. S., 51,90, 106 PESCHEL,T., 236,311 PETERS,W. N., 118, 160 PETROV,N. S., 304,310 PEITIGREW,R. M., 5,75,76, 83, 106 PEYGHAMBARIAN, N., 229,311 PIRANI,P., 297, 307,31 I PISKUNOV, A. K., 119, 159 PLOKE,M., 163, 180,226 PLUMMER,W. W., 119, 158 POLESHCHUK, A.G., 116, 127, 136, 138, 147,158-160 POLITCH,J., 65, 102 POLLAK,H. O., 364,396,397 PONATH,H. E., 230.235-237,249, 275, 295, 311 POOLE,S. B., 48, 105 POPOV, V. V., 123-125, 158, 159 PORTER,351 PORTER,R. P., 318-320, 322, 327, 330, 331, 335, 337, 341, 345, 354, 357, 358, 363, 369, 374, 380-382,396,397 PORTER,R. W., 91,94, 101 POST, D., 59, 106 PRISE,M. E., 229, 312 PROKHOROV, A.M., 123-125, 159 PURCELL,E. M., 224,226

R

RABAL,H. J., 66, 106 RABL,A., 175, 178, 183, 187,202,225,226 RAMAN,4 RAMAN,C. V., 4, 29, 106 RAMASWAMY, R. V., 240,313 RAMISHVILI, N. M., 37, 56, 103, 107 RANSOM,P. L., 317,396 RAYCES, J. L., 113, 160 RAYLEIOH, LORD,3,7, 57, 105

AUTHOR INDEX

REICHELT, A,, 27, 49, 106 REINISCH, R., 297, 306,313 REYNOLDS, G. O., 103 RIES,H., 221,226 RILEY, W. A,, 29, 73, 106 Rioux, M., 113, 117, 118, 158, 160 ROBERTS, J. S., 297, 302, 313 ROBERTSON, H. J., 160 ROBLIN,M . L., 5, 10, 15, 37, 75, 80, 83, 88-90, 105, 106 ROBSON,P. N., 297, 302,313 ROGERS,G. L., 3, 4, 7, 25, 26, 37, 45, 53, 98, 104, 106 RONCHI, v., 65.91, 93, 106 ROOT,W. L., 367,396 ROPE,E. L., 318, 363,397 ROSIN,S., 119, 160 N. N., 231,240, 304, 312 ROZANOV, Ruiz, J. M., 184,226 RUSHFORD, M . C., 229,311 RUSHFORTH, C. K., 364,366,397 S SAGATELYAN, D. M., 124, 159 SAIGA, N., 73, 106 SAKUMA, H., 66, 107 SALBUT, L., 41, 56, 66, 106 SALET,G., 95, 106 SARGENT111, M . , 301,312 SARID, D., 301,310,312 SARMA,L. V. A. R., 45, 52, 107 SCHARDIN, H., 91, 107 SCHATZEL, K., 66, 107 SCHUMACHER, B. W., 170,226 SCHWAB, W. C., 318,397 SCIAMMARELLA, C. A., 18,24,41, 107 SCRYL,I. I., 128, 158 SEATON,C. T., 229-231, 235, 236, 239, 244-246,248, 251,252,255, 258, 259, 266, 269, 270, 277-280, 283-285,288, 289, 291, 292, 294, 296, 297, 299-304, 307,310-313 SEDUKHIN, A. G., 116, 127, 136, 159 SERCEEV, V. V., 119, 160 SEITE, D., 230,312 SGULIM, s., 62, 102 A. B., 241,313 SHABAT, SHANNON,C. R., 42, 5 I , 107 SHCHEGLOV, V. A,, 119, 159 A.M., 116, 136, 159, SHCHERBACHENKO, 160

405

SHELTON, G. L., 51,90, 106 SHEN,T. P., 230,236, 239, 277, 278, 313 SHENOY,R. S., 45, 52, 107 SHEPPARD, C. J. R., 149, 150, 160 SHERMAN, G. C., 337, 355,396,397 SHERSTOBITOV, V. E., 119, 160 J. R., 317,350,397 SHEWELL, SHIBATA, N.,297, 307, 31I SHIVAROVA, A., 230,310 SHOEMAKER, R. L., 229, 258, 266, 269, 270, 297, 299,311-313 SIBILIA, c., 230. 297, 304, 305,310, 312 SICRE,E. E., 56, 57, 66, 82, 91, 92, 98, 99, 102, 105-1 07 SIDOROVA, V. I., 128, 158 SIECEL, H., 189,226 SIGELMANN, R. A,, 66, 103 S i L v A , D. E., 37, 62, 65, 66, 68, 105, 107 SIPE, J. E., 229,253,291, 292, 294, 296,312, 313 SIROHI, R. S., 66, 104 SISAKYAN, E. V., 159 SISAKYAN, I. N., 123-125, 158-160 SITCH,J. E., 297, 302,313 SLEPIAN, D., 364, 365, 397 SMIRNOV, A. P., 9, 25, 26, 33, 35, 37, 39-41, 66, 76, 107 SMITH,H. I., 30, 61, 103 SMITH,P. W., 231,236, 240, 304-306,310, 312,313 SMITH,S. D., 229, 230, 255, 258, 266, 269, 270,311-313 SMOLINSKA, B., 37, 39-43, 53, 54, 56, 104, 107

A. W., 223,226 SNYDER, SNYDER, M. A., 69, 70, 72, 104 SOARES,0. D. D., 66,102 SOCHACKI, J., 166,226 SODHA,M . S., 230,311 SOIFER, V. A., 123-125, 159, 160 SOLDATENKOV, I. S., 116, 160 SOMMERFELD, A,, 359, 371,397 SONEK, G. J., 297,300,310 SOROKO, L. M., 126-128, 131-133, 136-139, 141, 142, 144-148, 150-153, 155,158, 160 SPARROW, E. M., 189,209,226 SRINIVASAN, v., 66, 104 SRIVASTAVA, R., 240,313 STEGEMAN, G. I., 229-231,233,235,236, 239. 244-246,248,251-255,258, 259, 266,269,270,277-280, 283-285, 288,

406

AUTHOR INDEX

289, 291, 292, 294, 296, 297, 299-304, 307,310-313 G. N., 160 STEINBERG, STEPANOV, N. S., 121, 122,159 STEPANOV, V. V., 124, 125, 159 STEPHENS, N. W. F., 98,104 STEWARD, G. C., 115, 160 STEWART, W. J., 231,297, 302,310 STIENING, R.,214, 215,225 STONE,R.,325, 397 STONER, J. A., 119, 160 STORCK,E., 27,49, 106 STRAND, T. C., 69-71, 102 J. A., 381,397 STRATTON, STRAUBEL, R., 169,226 STREIBL, N., 4, 43, 100, 105 STRICKER, J., 66, 107 STROKE,G. W., 342, 345,397 SUBBASWAMY, K. R., 243, 289,312 SUDOL,R., 5, 61, 76, 81, 82, 86, 91, 107 SUZUKI, T., 51, 52, 59, 62, 66, 67, 106, 108 SVENSON, B., 244,297, 300,301, 307,310, 312,313 G. J., 37,76, 83, 84, 107 SWANSON, SWEENEY, D. W., 123,158 SZAPIEL, S., 5, 10, 11, 13, 107 SZWAYKOWSKI, P., 8, 25,26, 32-34,41, 54, 56, 59-61.66, 73, 87, 106, 107

T

TAGHIZADEH, M. R.,229,312 TAI. C. T., 332, 397 TAKEDA, M., 66,107 TAKENADA, H., 30, 104 TALBOT,H. F., 3,6, 107 G. G., 116,160 TARASOV, TEPICHIN, E., 44, I05 TERHUNE, R.,233,311 THEOCARIS, P. S., 41, 75, 97, 107 THOMPSON, B. J., 5,42, 76, 81, 82, 86,91, 107 L., 301,313 THYLEN, TIEN,P. K., 252,256,313 G., 226 TIMOSHENKO, TOMLINSON, W. J., 230, 231, 234, 235, 240, 251, 290, 304-306,312,313 TOOLEY,F. A. P., 229,312 TORI], Y.,30, 31, 107 TOTIA,H., 255,256,312 TREMBLAY,R., 113, 117, 118, 120, 159, 160, 300,312

TRICOLES, G., 318, 363, 397 TRILLO,S., 301, 310 S. K., 298,310 TRIPATHY, TRUNDLE, C., 297, 307, 310 TURNER, E. B., 119, I58 TWARDOWSKI, T., 230,245,246,248, 251, 252,310, 312

U UCHIDA,Y., 30, 103 ULRICH,R.,4,46-48, 56, 105, 107, 252, 312 USHIODA, S., 289,313 G. V., 160 UVAROV,

v

VACH,H., 231, 259, 297, 302-304,313 J. D., 229,244, 258, 266, 269, 270, VALERA, 291,292,294,296,297,299, 301,311-313 A. R.,56, 107 VANYAN, VARSHNEY, R. K., 240,313 VASIN,A. G., 160 VEDERNIKOV, V. M., 116, 160 VEG, Y A N , 128, 158 VENKATESAN, T. N. K., 229,310 VIJUKHIN, V. N., 116, 160 P., 297, 306,313 VINCENT, VITRANT, G., 297,307,310,313 VOGES, E., 46, 107

W WABNITZ, S., 301, 310 WACHTER, CH., 230, 240, 280, 284, 287, 311, 313 WADA,H., 66, 107 WADE,G., 319,396 WALKER, A. C., 302,313 WALLIS,R. F., 235,245, 251, 277, 278, 310, 313 WANG,S., 301,311 M., 62, 105 WASSERMAN, WATERMAN, P. C., 393,397 WEINBERG, F.J., 84, 107 WEISEI.,H., 3. 107 WELFORD, W.T., 163, 170, 171, 173, 175, 176, 178-182, 184, 185, 187, 204, 212, 213, 218,221,226 WERLICH, H. W.. 123, 160 WIEGMANN, W., 229,310 WILL,P. M., 51, 90, 106 WILSON, T., 149, 160 WINFUL, H. G., 230,258, 297, 311,312

AUTHOR INDEX

WINKELMANN, A., 3, 107 WINSTON,R., 163, 170, 171, 173, 175, 176, 178-185, 187, 188, 202, 204, 212-215, 218, 221,225, 226 WINTHROP,J. T., 3, 7, 9, 14, 24-26, 53, 80, 81, 108 WITTEKOEK,S., 30, 102 WOLF,E., 11,44, 100, 102, 166, 205,225, 226, 317-319, 322, 326, 331,335, 337, 350-353, 355, 374, 377, 395,396,397 WOLFF,U., 27,49, 106 WOLFKE,M., 3, 108 WOOD, N. B., 84, 107 WORTHINGTON, C. R., 3, 7,9, 14, 24-26, 53, 80, 81, 108 WRIGHT,E. M., 230, 231, 245, 246, 248, 251, 252,277,27a,283-285,288,289,292, 297, 300, 301, 310-313 WRONKOWSKI, L., 25,27, 106, 108 WYANT,J. C., 91, 92, 108

Y YAO, S. S., 236,313 YOKOZEKI,S., 14, 51, 52, 59, 62, 66, 67, 105, 106, 108 Yu, F. T. S., 66, 104 Yu, M. Y.,230, 244, 313 YURLOV,Yu. I., 116, 127, 136, 159. 160

Z ZAKHAROV, V. E., 241,247, 310,313 ZANKEL,K. L., 29, 72, 102 ZANONI,R., 230, 246,248, 251, 252,312 ZAVGORODNEVA, S. I., 119, 160 ZELDOVITZ, YA. B., 120, 160 ZEMBUTSU, s., 297,307,311 ZIJL,H., 205,226 ZINKE,M., 49, 103 ZWERLING, C., 62, 105

407

SUBJECT INDEX A aberration theory, 165 acousto-optics, 4, 101 Airy disk, 117 axicon, 111, 112, 117, 118 - kinoform, 116 - W-, 119, 121

F Fabry-Perot cavity, 57 Fabry-Perot etalon, 57 Fabry-Perot interferometer, 43,44, 100 Fermat principle, 19 Fourier transformation, 8, 136 Fresnel diffraction, 3-5, 23, 25, 29, 30, 32, 45, 49, 77 Fresnel images, 24,26, 78, 79, 81, 85 Fresnel-Kirchhoff diffraction integral, 1 1, 14, 19

B Bessel function, 73, 115

- spherical, 325, 335, 352 bistable switch, 229 blackbody radiation, 221 Born approximation, 376

G Gaussian beam, 10, 12-14, 16, 117, 240 geometrical optics, 167, 220, 222 grating, - binary amplitude, 18, 54, 86 - circular diffraction, 114 - phase, 73, 86 - shearing interferometry, 91 gravitational lens, 120, 121 Green function, 323, 327, 329, 330, 332, 345, 348,349,351,352,358, 359, 383, 384, 395 Green’s theorem, 324, 326, 331, 333, 394

C coherence time, 219 - theory of partial, 5 , 35 computer-generated hologram, 123 concentrator, - compound elliptical, 174, 187, 188 - compound parabolic, 171, 172, 180, 182 - compound triangular, 186 coupled-mode theory, 229 Crank-Nicolson scheme, 242

D diffraction, diffraction theory, 3 17 - scalar, 7 Dirac delta function, 324, 325

E edge ray principle, 173 eikonal equation, 124 electron microscopy, 4, 101 evanescent wave, 336 Ewald sphere, 378 extinction theorem, 393, 395

H Hale telescope, 215 Heaviside function, 136 Helmholtz equation, 323 Hilbert transform, 136, 137 hologram, 45,50,319, 327, 337-340, 347, 349,377 - closed reflection, 348 - lensless Fourier Transform, 342 - open reflection, 350 - planar Fourier Transform, 345 - point reference, 341 - single-layer reflection, 346 holographic imaging, 317, 318, 334, 337 holographic tomography, 317, 380, 381 holography, 6, 89, 317-319, 392 409

410

SUBJECT INDEX

Hottel string, 189, 200 Huygens-Kirchhoff integral, 338

I image processing, 49 inverse, inverse scattering method, 241, 392 inverse source problem, 319, 351, 356, 357, 361 J Jacobian, 125 Jacobian elliptic function, 271-273

K

Karhunen-Loeve theorem, 367 Kerr effect, 231 Kerr law, Kerr-law cladding, 239, 240, 248,254, 276, 283, 285, 293 Kerr-law medium, 234-238, 243, 244, 246, 252,254,259,271,282, 283, 289,295, 296, 305 Kerr-type nonlinearity, 230,232, 246, 271 Kinoform, I1 I , 116, 136 Kramers-Kronig relation, 285

L

Lagrange invariant, 165, 167, 168 Lambertian radiator, 167, 168, 172, 208 Lamberth source, 173, 190, 191, 195, 197, 204,210 laser, - argon-ion, 300 Nd : YAG, 300 Lau effect, 5, 6, 75-77, 79, 87, 88, 91, 92, 97-99 Legendre polynomial, 352 Linfoot criteria, 39 Liouville’s theorem, 168, 170, 171 liquid crystal MBBA, 239, 256, 277, 295, 302, 303, 307 logic gates, 229 Luneburg lens, 166

-

M Mach-Zehnder interferometer, 52 Maxwell equations, 234, 245, 247, 255, 259, 261, 264-266,268, 271,274 Mercer’s theorem, 358 meso-optical condenser, 151

meso-optical confocal microscope, 149, 150 meso-optical Fourier transform microscope, 128, 152 meso-optical image, 127, 132, 134, 136, 138, 139, 144 meso-optical mirror, .129, 130 meso-optical scanning microscope, 147, 148 meso-optics, 111, 112, 126, 128, 139, 142, 155 moirt deflectometry, 62, 66, 68 moirt effect, 140, 141 moirt fringes, 33, 37, 64, 70, 75, 82, 91 Montgomery ring, 8, 41, 43 N Newton-Picard method, 242 nonimaging optics, 163, 214 nonimaging system, 167, 181 nonlinear coherent coupler, 229 nonlinear directional coupler, 301 nonlinear distributed coupler, 297 nonlinear optical limiter, 297, 302 0 optical bistability, 229, 297, 304, 307 optical communication, 229 optical computing, 229 optical fibre, 185 optical metrology, 48, 68 optical switching, 304, 307 optical testing, 48

P

parabolic approximation, 17, 18 parabolic equation, 240 paraxial approximation, 17, 20, 320 paraxial optics, 165 plasmon, 289, 292, 296 polariton, 289,290, 292, 296 Portor equation, 318 Poynting vector, 248, 253, 256

Q

quantum well, quantum optics, 220 multiple, 279, 303

-

R Rademacher function, 56 radiance, 167, 219 radiance theorem, 168

SUBJECT INDEX

Raman-Nath parameter, 29, 73 Raman-Nath thin phase grating approximation, 32 Ronchi test, 65, 92, 93, 95 Rytov approximation, 374, 376

S

sampling theorem, 5 1 self-focusing, 230, 237,242, 257, 259, 260, 265,274, 308 self-imaging, 3.4, 6, 8-10, 14, 18, 20, 25, 29, 33, 34, 39,41,43,45,47,48, 50-52, 55-57, 61, 65, 69, 75-77, 82, 85, 91, 100, 101 - nonparaxial, 19 self-imaging of linear periodic objects, 21 self-pulsing, 307 signal processing, 229

41 1

hell's law, 182 Sommerfeld radiation condition, 331, 359, 394 spatial coherence, 34 spatial filtering, 6, 41, 49 spherical wave, 342

T

Talbot effect, 3, 7, 9, 75, 99 Talbot interferometry, 62, 65, 66, 68, 91, 96 tomography, 319, 320, 374 - acoustic, 319

W

Wigner distribution function, 92

CUMULATIVE INDEX - VOLUMES I-XXVII 11, 249 ABELBS,F., Methods for Determining Optical Parameters of Thin Films VII, 139 ABELLA,I. D., Echoes at Optical Frequencies XVI, 71 ABITBOL,C. I., see J. J. Clair ABRAHAM, N. B., P. MANDEL,L. M. NARDUCCI,Dynamical Instabilities and Pulsations in Lasers xxv, 1 AGARWAL, G. S., Master Equation Methods in Quantum Optics XI, 1 AGRAWAL, G. P., Single-longitudinal-mode Semiconductor Lasers XXVI, 163 AGRANOVICH, IX, 235 V. M., V. L. GINZBURG, Crystal Optics with Spatial Dispersion IX, 179 ALLEN, L., D. G. C. JONES,Mode Locking in Gas Lasers IX, 123 AMMANN, E. O., Synthesis of Optical Birefringent Networks ARMSTRONG, J. A., A. W. SMITH,Experimental Studies of Intensity Fluctuations VI, 211 in Lasers XI, 247 ARNAUD,J. A., Hamiltonian Theory of Beam Mode Propagation BALTES,H. P., On the Validity of Kirchhofl’s Law of Heat Radiation for a Body XII, 1 in a Nonequilibrium Environment BARAKAT, R., The Intensity Distribution and Total Illumination of AberrationFree Diffraction Images I, 67 XXI, 217 BARRETT, H. H., The Radon Transform and its Applications BASHKIN,S., Beam-Foil Spectroscopy XII, 287 BASSE’IT,I. M., W. T. WELFORD,R. WINSTON,Nonimaging Optics for Flux XXVII, 161 Concentration VI, 53 BECKMANN, P., Scattering of Light by Rough Surfaces BERRY,M. V., C. UPSTILL,Catastrophe Optics: Morphologies of Caustics and XVIII, 259 their Diffraction Patterns XXVII, 227 BERTOLOTII, M., see D. Mihalache XVI, 357 BEVERLY 111, R. E., Light Emission from High-Current Surface-Spark Discharges IX, 1 BLOOM,A. L., Gas Lasers and their Application to Precise Length Measurements BOUMAN, M. A., W. A. VAN DE GRIND,P. ZUIDEMA,Quantum Fluctuations in Vision XXII, 77 IV, 145 BOUSQUET, P., see P. Rouard XXIII, 1 BROWN,G. S., see J. A. DeSanto 413

414

CUMULATIVE INDEX

BRUNNER, W., H. PAUL,Theory of Optical Parametric Amplification and Oscillation xv, 1 BRYNGDAHL, O., Applications of Shearing Interferometry IV, 37 BRYNGDAHL, O., Evanescent Waves in Optical Imaging XI, 167 BURCH,J. M., The Meteorological Applications of Diffraction Gratings 11, 73 BUTTERWECK, H. J., Principles of Optical Data-Processing XIX, 21 1 CAGNAC, B., see E. Giacobino XVII, 85 CASASENT, D., D. PSALTIS,Deformation Invariant, Space-Variant Optical Pattern Recognition XVI, 289 CEGLIO,N. M., D. W. SWEENEY, Zone Plate Coded Imaging: Theory and Applications XXI, 287 CHRISTENSEN, J. L., see W. M. Rosenblum XIII, 69 CLAIR,J. J., C. I. ABITBOL, Recent Advances in Phase Profiles Generation XVI, 71 CLARRICOATS, P. J. B., Optical Fibre Waveguides A Review XIV, 327 COHEN-TANNOUDJI, C., A. KASTLER,Optical Pumping v, 1 COLE,T. W., Quasi-Optical Techniques of Radio Astronomy XV, 187 COLOMBEAU, B., see C. Froehly XX, 63 COURTBS,G., P. CRUVELLIER, M. DETAILLE,M. SAME, Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects xx, 1 CREATH,K., Phase-Measurement Interferometry Techniques XXVI, 349 CREWE,A. V., Production of Electron Probes Using a Field Emission Source XI, 223 CRUVELLIER, P., see C. G. Court& xx, 1 CUMMINS, H. Z., H. L., SWINNEY, Light Beating Spectroscopy VIII, 133 DAINTY,J. C., The Statistics of Speckle Patterns XIV, 1 DANDLIKER, R., Heterodyne Holographic Interferometry XVII, 1 DECKERJr., J. A., see M. Harwit XII, 101 DELANO,E., R. J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters VII, 67 DEMARIA, A. J., Picosecond Laser Pulses IX, 31 DESANTO,J. A., G. S. BROWN,Analytical Techniques for Multiple Scattering from Rough Surfaces XXIII, I DETAILLE, M., see G. Court& xx, 1 DEXTER,D. L., see D. Y. Smith X, 165 DREYHAGE, K. H., Interaction of Light with Monomolecular Dye Layers XII, 163 DUGUAY, M. A., The Ultrafast Optical Kerr Shutter XIV, 161 EBERLY, J. H., Interaction of Very Intense Light with Free Electrons VII, 359 ENGLUND, J. C., R. R. SNAPP,W. C. SCHIEVE,Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity XXI, 355 ENNOS,A. E., Speckle Interferometry XVI, 233 FANTE,R. L., Wave Propagation in Random Media: A Systems Approach XXII, 341 FIORENTINI, A., Dynamic Characteristics of Visual Processes I, 253 FOCKE,J., Higher Order Aberration Theory IV, 1 FRANCON, M., S. MALLICK, Measurement of the Second Order Degree of Coherence VI, 71

-

CUMULATIVE INDEX

415

FRIEDEN, B. R., Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions IX, 311 FROEHLY, C., B. COLOMBEAU, M. VAMPOUILLE, Shaping and Analysis of Picosecond Light Pulses XX, 63 FRY,G. A., The Optical Performance of the Human Eye VIII, 51 GABOR,D., Light and Information I, 109 GAMO,H., Matrix Treatment of Partial Coherence 111, 187 GHATAK,A. K., see M. S. Sodha XIII, 169 GHATAK,A., K. THYAGARAJAN, Graded Index Optical Waveguides: A Review XVIII, 1 GIACOBINO, E., B. CAGNAC,Doppler-Free Multiphoton Spectroscopy XVII, 85 GINZBURG, V. L., see V. M. Agranovich IX, 235 GIOVANELLI, R. G., Diffusion Through Non-Uniform Media 11, 109 GLASER,I., Information Processing with Spatially Incoherent Light XXIV, 389 GNIADEK, K., J. PETYKIEWICZ, Applications of Optical Methods in the Diffraction IX, 281 Theory of Elastic Waves J. W., Synthetic-Aperture Optics VIII, 1 GOODMAN, GRAHAM, R., The Phase Transition Concept and Coherence in Atomic Emission XII, 233 HARIHARAN, P., Colour Holography XX, 263 HARIHARAN, P., Interferometry with Lasers XXIV, 103 HARWIT,M., J. A. DECKERJr., Modulation Techniques in Spectrometry XII, 101 X, 289 HELSTROM, C. W., Quantum Detection Theory VI, 171 HERRIOTT,D. R., Some Applications of Lasers to Interferometry HUANG,T. S., Bandwidth Compression of Optical Images x, 1 JACOBSSON, R., Light Reflection from Films of Continuously Varying Refractive V, 247 Index 111, 29 Apodisation JACQUINOT, P., B. ROIZEN-DOSSIER, JAMROZ,W.,B. P. STOICHEFF,Generation of Tunable Coherent Vacuum-UltraXX, 325 violet Radiation IX, 179 JONES,D. G. C., see L. Allen KASTLER,A,, see C. Cohen-Tannoudji v, 1 XXVI, 105 KHOO,I. C., Nonlinear Optics of Liquid Crystals XX, 155 KIELICH,S., Multi-Photon Scattering Molecular Spectroscopy IV, 85 KINOSITA, K., Surface Deterioration of Optical Glasses KOPPELMANN, G., Multiple-Beam Interference and Natural Modes in Open VII, 1 Resonators 111, 1 KOTTLER,F., The Elements of Radiative Transfer IV, 281 KOTTLER,F., Diffraction at a Black Screen, Part I: Kirchhoffs Theory VI, 331 KOTTLER,F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory XXVI, 227 KRAVTSOV, Yu.A., Rays and Caustics as Physical Objects I. 211 KUBOTA,H., Interference Color XIV, 47 LABEYRIE, A., High-Resolution Techniques in Optical Astronomy XI, 123 LEAN,E. G., Interaction of Light and Acoustic Surface Waves

416

CUMULATIVE INDEX

XVI, 119 LEE,W.-H., Computer-Generated Holograms: Techniques and Applications LEITH,E. N., J. UPATNIEKS, Recent Advances in Holography VI, 1 LETOKHOV, V. S., Laser Selective Photophysics and Photochemistry XVI, 1 LEVI,L., Vision in Communication VIII. 343 LIPSON,H., C. A. TAYLOR, X-Ray Crystal-Structure Determination as a Branch of Physical Optics V, 287 LUGIATO,L. A., Theory of Optical Bistability XXI, 69 MALACARA, D., Optical and Electronic Processing of Medical Images XXII, 1 MALLICK,L., see M. Francon VI, 71 MANDEL,L., Fluctuations of Light Beams 11, 181 MANDEL,L., The Case for and against Semiclassical Radiation Theory XIII, 27 MANDEL,P., see N. B. Abraham xxv, 1 MARCHAND, E. W., Gradient Index Lenses XI, 305 MARTIN,P. J., R. P. NETTERFIELD,Optical Films Produced by Ion-Based Techniques XXIII. 113 MASALOV,A. V., Spectral and Temporal Fluctuations of Broad-Band Laser Radiation XXII, 145 MAYSTRE,D., Rigorous Vector Theories of Diffraction Gratings XXI, 1 MEESSEN,A., see P. Rouard XV, 7 1 MEHTA,C. L., Theory of Photoelectron Counting VIII, 373 MIHALACHE, D., M. Bertolotti, C. Sibilia, Nonlinear wave propagation in planar structures XXVII, 227 MIKAELIAN, A. L., M. I. TER-MIKAELIAN, Quasi-Classical Theory of Laser Radiation VII, 231 MIKAELIAN, A. L., Self-Focusing Media with Variable Index of Refraction XVII, 279 MILLS,D. L., K. R. SUBBASWAMY, Surface and Size Effects on the Light Scattering Spectra of Solids XIX, 43 MIYAMOTO, K.,Wave Optics and Geometrical Optics in Optical Design I, 31 MOLLOW,B. R., Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence XIX, 1 MURATA,K., Instruments for the Measuring of Optical Transfer Functions V, 199 MUSSET,A., A. THELEN,Multilayer Antireflection Coatings VIII, 201 NARDUCCI, L. M., see N. B. Abraham xxv, 1 NETTERFIELD, R. P., see P. J. Martin XXIII, 113 NISHIHARA, H., T. SUHARA,Micro Fresnel Lenses XXIV, 1 OHTSU,M., T. TAKO,Coherence in Semiconductor Lasers XXV, 191 OKOSHI,T., Projection-Type Holography XV, 139 OOUE,S.,The Photographic Image VII, 299 OSTROVSKAYA, G. V., Yu. I. OSTROVSKY, Holographic Methods in Plasma Diagnostics XXII, 197 OSTROVSKY, Yu. I., see G. V. Ostrovskaya XXII, 197 OUGHSTUN, K. E., Unstable Resonator Modes XXIV, 165

CUMULATIVE INDEX

411

XXVII, 1 PATORSKI, K.P., The Self-Imaging Phenomenon and its Applications xv, 1 PAUL,H., see W. Brunner PEGIS,R. J., The Modern Development of Hamiltonian Optics 1, 1 VII, 67 PEGIS,R. J., see E. Delano PERINA, J., Photocount Statistics of Radiation Propagating through Random and XVIII, 129 Nonlinear Media V, 83 PERSHAN, P. S., Non-Linear Optics IX, 281 PETYKIEWICZ, J., see K. Gniadek V, 351 PICHT,J., The Wave of a Moving Classical Electron PORTER,R.P., Generalized Holography with Application to Inverse Scattering and XXVII, 315 Inverse Source Problems XVI, 289 PSALTIS,D., see D. Casasent RISEBERG,L.A., M. J. WEBER,Relaxation Phenomena in Rare-Earth LumiXIV, 89 n escence VIII, 239 RISKEN,H., Statistical Properties of Laser Light XIX, 281 RODDIER,F., The Effects of Atmospheric Turbulence in Optical Astronomy 111, 29 ROIZEN-DOSSIER, B., see P. Jacquinot XXV, 219 RONCHI,L., see Wang Shaomin ROSENBLUM, W. M., J. L. CHRISTENSEN, Objective and Subjective Spherical XIII, 69 Aberration Measurements of the Human Eye XXIV, 39 ROTHBERG, L., Dephasing-Induced Coherent Phenomena IV, 145 ROUARD, P., P. BOUSQUET,Optical Constants of Thin Films xv, 77 ROUARD,P., A. MEESSEN,Optical Properties of Thin Metal Films IV, 199 RUBINOWICZ, A., The Miyamoto-Wolf Diffraction Wave XIV, 195 RUDOLPH, D., see G. Schmahl xx, 1 SAYSSE,M., see G. Court& VI, 259 SAKAI,H., see G. A. Vanasse XXVI, 1 SALEH,B. E. A,, see M. C. Teich XXI, 355 SCHIEVE, W. C., see J. C. Englund XIV, 195 SCHMAHL, G., D. RUDOLPH,Holographic Diffraction Gratings SCHUBERT, M., B. WILHELMI, The Mutual Dependence between Coherence ProXVII, 163 perties of Light and Nonlinear Optical Processes XIII, 93 SCHULZ,G., J. SCHWIDER, Interferometric Testing of Smooth Surfaces xxv, 349 SCHULZ,G., Aspheric Surfaces XIII, 93 SCHWIDER, J., see G. Schulz X, 89 SCULLY, M. 0.. K. G. WHITNEY,Tools of Theoretical Quantum Optics SENITZKY, I. R., Semiclassical Radiation Theory within a Quantum-Mechanical XVI, 413 Framework XXVII, 227 SIBILIA,C., see D. Mihalache XV, 245 SIPE,J. E., see J. Van Kranendonk X, 229 SITTIG,E. K., Elastooptic Light Modulation and Deflection SLUSHER, R. E., Self-Induced Transparency XII, 53

418

CUMULATIVE INDEX

VI, 211 SMITH,A. W., see J. A. Armstrong X, 165 SMITH,D. Y.,D. L. DEXTER,Optical Absorption Strength of Defects in Insulators x, 45 SMITH,R. W., The Use of Image Tubes as Shutters XXI, 355 SNAPP,R. R., see J. C. Englund SODHA,M. S.,A. K. GHATAK,V. K. TRIPATHI, Self Focusing of Laser Beams in XIII, 169 Plasmas and Semiconductors XXVII, 109 SOROKO,L. M., Axicons and Meso-Optical Imaging Devices V, 145 STEEL,W. H., Two-Beam Interferometry XX, 325 STOICHEFF, B. P., see W. Jamroz IX, 73 STROHBEHN, J. W., Optical Propagation Through the Turbulent Atmosphere STROKE, G. W., Ruling, Testing and Use of Optical Gratings for High-Resolution 11, 1 Spectroscopy XIX, 43 SUBBASWAMY, K. R., see D. L. Mills XXIV, 1 SUHARA, T., see H. Nishihara SVELTO,O., Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser XII, 1 Beams XXI, 287 SWEENEY, D. W., see N. M. Ceglio VIII, 133 SWINNEY, H. H., see H. Z. Cummins XXV, 191 TAKO,T., see M. Ohtsu TANAKA, K., Paraxial Theory in Optical Design in Terms of Gaussian Brackets XXIII, 63 XVII, 239 TANGO,W. J., R. Q. TWISS,Michelson Stellar Interferometry TATARSKII, V. I., V. U. ZAVOROTNYI, Strong Fluctuation in Light Propagation XVIII, 207 in a Randomly Inhomogeneous Medium V, 287 TAYLOR, C. A., see H. Lipson XXVI, 1 TEICH,M. C., B. E. A. SALEH,Photon Bunching and Antibunching VII, 231 TER-MIKAELIAN, M. L., see A. L. Mikaelian VIII, 201 THELEN,A., see A. Musset VII, 169 THOMPSON, B. J., Image Formation with Partially Coherent Light XVIII, 1 THYAGARAJAN, K., see A. Ghatak XXIII, 183 TONOMURA, A., Electron Holography XIII, 169 TRIPATHI, V. K., see M. S. Sodha TSUIIUCHI, J., Correction of Optical Images by Compensation of Aberrations and 11, 131 by Spatial Frequency Filtering XVII, 239 TWISS,R. Q., see W. J. Tango VI, 1 UPATNIEKS, J., see E. N. Leith XVIII, 259 UPSTILL,C., see M. V. Berry USHIODA, S.,Light Scattering Spectroscopy of Surface ElectromagneticWaves in XIX, 139 Solids XX, 63 VAMPOUILLE,M., see C. Froehly VI, 259 VANASSE, G. A., H. SAKAI,Fourier Spectroscopy XXII, 77 VAN DE GRIND,W. A., see M. A. Bouman I, 289 VAN HEEL,A. C. S.,Modern Alignment Devices VAN KRANENDONK, J., J. E. SIPE,Foundations of the Macroscopic ElectromagneXV, 245 tic Theory of Dielectric Media

419 VERNIER, P., Photoemission XIV, 245 WANG,SHAOMIN, L. RONCHI,Principles and Design of Optical Arrays XXV, 279 WEBER,M. J., see L. A. Riseberg XIV, 89 IV, 241 WELFORD,W. T., Aberration Theory of Gratings and Grating Mountings WELFORD,W. T., Aplanatism and Isoplanatism XIII, 267 WELFORD, W. T., see I. M. Bassett XXVII, 161 WILHELMI, B., see M. Schubert XVII, 163 WINSTON, R., see I. M. Bassett XXVII, 161 WITNEY,K. G., see M. 0. Scully X, 89 WOLTER,H., On Basic Analogies and Principal Differences between Optical and Electronic Information I, 155 WYNNE,C. G., Field Correctors for Astronomical Telescopes x, 137 YAMAGUCHI, I., Fringe Formations in Deformation and Vibration Measurements Using Laser Light XXII, 271 YAMAJI, K., Design of Zoom Lenses VI, 105 YAMAMOTO,T., Coherence Theory of Source-Size Compensation in Interference Microscopy VIII, 295 YOSHINAGA,H., Recent Developments in Far Infrared Spectroscopic Techniques XI, 77 Yu, F. T. S., Principles of Optical Processing with Partially Coherent Light XXIII, 227 ZAVOROTNYI, V. U., see V. I. Tatarskii XVIII, 207 ZUIDEMA, P., see M. A. Bouman XXII, 77