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UBC Theses and Dissertations

Classical signal detection theory and reconstruction problems in holographic imaging systems Ghandeharian, Hossein 1980

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CLASSICAL SIGNAL DETECTION THEORY AND RECONSTRUCTION PROBLEMS IN HOLOGRAPHIC IMAGING SYSTEMS by HOSSEIN GHANDEHARIAN B.Sc . ( P h y s i c s ) , The U n i v e r s i t y of Tehran, I r a n , 1970 .Sc.(E.E.)» The U n i v e r s i t y of Manitoba, Winnipeg, Man. 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES ( Department of E l e c t r i c a l Engineering ) We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA February 1980 (£) Hossein Ghandeharian, 1980 In p resent ing t h i s t he s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re ference and study. I f u r t h e r agree that permiss ion f o r ex tens i ve copying of t h i s t he s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t he s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my w r i t t e n permi s s ion . ^ E l e c t r i c a l E n g i n e e r i n g Department of „ _ The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P lace Vancouver, Canada V6T 1W5 Date A p r i l 3, 19 8 0 ABSTRACT A new " c i r c u i t " model i s developed to study non-linear e f f e c t s i n holography. The model l i n k s c l a s s i c a l s i g n a l detection theory to holography, for i t c l e a r l y shows that the reconstructed images i n a t h i n hologram can be computed from mathematical formulas obtained for the out-put of non-linear detectors. In preparation for hologram a n a l y s i s , the r e s u l t s for the (time-) autocorrelation of e l e c t r i c a l vth-law devices i n response to s i g n a l plus noise are extended to memoryless non-linear detectors with a r b i t r a r y char-a c t e r i s t i c s . Mathematical p a r a l l e l s are next established between holo-graphy and the non-linear detection of s i g n a l s , and these are incorporated i n the model. The ready-made formulas for e l e c t r i c a l detectors apply d i -r e c t l y to give formulas for the (space-)autocorrelation of holograms of d i f f u s e objects. The autocorrelation function predicts d i s t o r t e d m u l t i -pie images, t h e i r r e l a t i v e p o s i t i o n s , o r i e n t a t i o n s , widths, and strengths. The m u l t i p l i c i t y of images i s due to the generation of harmonics; the background h a l o - l i k e noise components added to the f a i t h f u l images are mainly due to the m u l t i p l i c i t y of the f i r s t harmonic i t s e l f . The analysis i s further expanded to include a s t i l l more gen-e r a l case i n which the r e f l e c t e d l i g h t from the object i s depolarized. A decrease i n signal-to-noise r a t i o (decrease i n f r i n g e v i s i b i l i t y ) ; a loss of information, and an augmentation of non-linear d i s t o r t i o n could be expected. A simple way of reducing these e f f e c t s i s suggested. F i n a l l y , holography with more than one reference beam i s studied. An exact formulation i s given for double-reference-beam holograms. For multiple-reference-beam holograms, only approximate closed forms are pre-i i sented. I t i s shown that the addition of extra reference beams during the recording step of the hologram may amplify the f a i t h f u l images with-out increasing t h e i r background noise s i g n i f i c a n t l y . Experiments confirm the t h e o r e t i c a l expectations. i i i TABLE OF CONTENTS Page Abs t r a c t .". i i Table of Contents .. . i v L i s t of Figures v i i L i s t of Symbols i x Acknowledgement x v i I INTRODUCTION 1.1. Background and Mo t i v a t i o n s '...I 1.1.1. Holography and Corrmunioation Theory "...2 1.1.2. Transform Method and Its Applications in Holography: A Brief Review ,.~.. 3 1.1.3. Transform Method and the Kozma Model; Previous analysis and Their Drawbacks ."..5 A. Large Reference-to-Object I r r a d i a n c e R a t i o 8 B. A r b i t r a r y Reference-to-Object I r r a d i a n c e R a t i o 9 1.2. This Thesis . 10 1.2.1. A New Model for the Non-linear Hologram; Remedy for the Kozma Model Drawbacks 10 1.2.2. Output of N o n - l i n e a r i t i e s with Arbitrary C h a r a c t e r i s t i c s . . . 1 1 1.2.3. P o l a r i z a t i o n Effects in Holography 11 1.2.4. Multiple-Reference-Beam Holography and Suppression of Ihtermodulation Noise 13 1.2.5. Summary 13 I I AUTOCORRELATION OF OUTPUT OF MEMORYLESS NON-LINEARITIES WITH ARBITRARY CHARACTERISTICS 2.1. I n t r o d u c t i o n 15 2.2. D e r i v a t i o n of Formulas f o r h , 18 Case 1: A Shifted g(X) 20 Case 2: g(X) with Cut-off or Limiting 21 Case 3: Combination of Case 1 and Case 2 22 2.3. Examples 22 (a) Symmetrical Peak Clipping 22 (b) Symmetrical Center Clipping 24 (c) A Quantizer 25 2.4. S i g n a l X S i g n a l , Noise X Noise, and S i g n a l X Noise Output of a N o n - l i n e a r i t y 27 k 2.4.1. Narrowband Inputs and Expansion of R^t^t^) 27 2.4.2. Output Correlation of a Detector for Narrowband Noise 30 2.5. F i n a l Remarks 32 2.6. Summary 32 i v Page I I I A NON-LINEAR SYSTEMS MODEL FOR HOLOGRAPHIC PROCESS 3.1. I n t r o d u c t i o n 33 3.2. S t a t i s t i c s of a Coherently I l l u m i n a t e d D i f f u s e Object 34 3.3. S t a t i s t i c s of F r e s n e l D i f f r a c t i o n F i e l d of a D i f f u s e Object 36 3.4. S t a t i s t i c a l P r o p e r t i e s of a Hologram of a D i f f u s e Object 38 3.4.1. Transmittanee Autocorrelation of a Linear Hologram 38 3.4.2. Analogy Between a Linear Hologram and a Square-Law Envelope detector 41 3.4.3. Hologram Imaging Properties via Autocorrelation Function. ..43 3.4.4. Autocorrelation of Transmittanee of a Non-linear Hologram: 3.4.5. Physical Interpretation of the Terms Involved in R (r ,r ).49 Examples f o r t(E) and the Corresponding A 52 (a) Gaussian t (E) 52 (b) L o r e n t z i a n t (E) 54 a (c) A polynomial approximation 54 (d) Sum of Laguerre polynomials 55 (e) L i n e a r phase hologram 55 3.5. D i s c u s s i o n of the Assumptions 56 3.6. Summary 57 IV DEGRADATION OF HOLOGRAPHIC IMAGES DUE TO DEPOLARIZATION OF REFLECTED LIGHT 4.1. I n t r o d u c t i o n 59 4.2. D e p o l a r i z a t i o n of I l l u m i n a t i o n and Loss of Information 60 4.3. Experimental R e s u l t s 62 4.4. Detector's N o n - l i n e a r i t y and D i s t o r t i n g E f f e c t s of the Cross-P o l a r i z e d Component 65 4.5. Summary 69 V MULTIPLE-REFERENCE-BEAM NON-EINEAR HOLOGRAPHY WITH APPLICATIONS IN SUPPRESSION OF INTERMODULATION NOISE 5.1. I n t r o d u c t i o n 71 5.2. P r e l i m i n a r y Considerations 72 v Page 5.3. Transmittanee A u t o c o r r e l a t i o n of a Double-Reference-Beam Hologram 75 5.4. P h y s i c a l I n t e r p r e t a t i o n of Terms Involved i n R q 78 5.5. Suppression of Intermodulation D i s t o r t i o n of F i r s t - O r d e r Images .82 5.6. Experimental R e s u l t s 86 5.7. Holograms Made w i t h More than Two Reference Beams 89 5.8. White L i g h t Reconstruction of SRB Hologram vs. DRB Hologram 93 5.9. Summary 98 VI CONCLUSIONS 6.1. Summary 99 6.2. Further P o s s i b l e Extensions 102 REFERENCES 103 v i Figure Page LIST OF FIGURES 1.1 Film n o n - l i n e a r i t y and i t s representation as a full-wave transfer c h a r a c t e r i s t i c of a memoryless non-linear element, a) The t - E curve; b) the dependence of t' on E'; c) the function g(E'), 7 1.2 The Kozma " c i r c u i t " model for analysing non-linear holography 7 2.1 Gradual peak c l i p p e r 23 2.2 Abrupt peak c l i p p e r '. 23 2.3 Linear center c l i p p e r 25 2.4 Quantizer 26 3.1 Arrangement for recording an o f f - a x i s hologram of a transparent object 39 3.2 Reconstructed images of non-linear single-reference-beam holograms. a) Oth harmonic, b) 1st and 2nd harmonics, c) 2nd harmonic enlarged and more i n focus, d) 1st and 2nd harmonics of a hologram recorded with severe n o n - l i n e a r i t y 51 3.3 Hologram imaging properties v i a R^C r^ , ^ ) . Harmonic images are centered on axes enclosing angles subtended by m(a^Xd/ 2 T r ) ;m=0,1,2 .. 53 4.1 Laser illuminated m e t a l l i c star, a) Both p o l a r i z a t i o n components present; b) p a r a l l e l p o l a r i z e d component; c) cross-polarized component 63 4.2 E f f e c t s of using a rotated reference beam on holographic image O O O q u a l i t y . The angles of the r o t a t i o n were: a) 0 , b) 30 , c) 60 , and d) 90° 64 5.1 Hologram imaging properties v i a R ( r ^ r ^ ) . Rough sketches of some of the images predicted by R ,for two recording geometries: a) DRBH, b) SRBH ? 80 2 2 2 5.2 Reconstructed images of non-linear DRBH; S^ >> S£ and a . Images are mainly due to: a) R , b) R and R . c) Lower part of (b) nxSp nxs^ nxs^xs2 enlarged and more i n focus, d) Upper part'of (b)' enlarged and more i n focus 87 2 2 2 2 5.3 Reconstructed images of non-linear DRBH; S^~ S£ + o" , and S£ i s s l i g h t l y larger than a 2 . Images are mainly due to: a) R , b) nxs2 R and R , c) R , and the 2nd harmonic of R nxs^ nxs^xs2 nxs^xs2 nxs^ v i i Figure Page 5.4&5 Images of n o n - l i n e a r double- and triple-reference-beam holograms ... 92 5.6 White l i g h t r e c o n s t r u c t e d images of a SRB hologram. The l a r g e angle ••'. between the object and reference has caused a l a r g e c o l o r d i s p e r s i o n of the images 95 5.7 White l i g h t r e constructed images of a SRB hologram. The c o l o r d i s -persionof the images i s g r e a t l y reduced by b r i n g i n g the reference beam c l o s e r to the object 96 5.8 White l i g h t r e c o n s t r u c t e d images of a DRB hologram 97 v i i i LIST OF SYMBOLS c o e f f i c i e n t s i n a power-series expansion of g(X) unit vectors ( N =a N ; N=a.TN; N =a N ; N =a N ) c c c N p p p s s s constants = a a p q a power series i n |u (Ar)| that replaces A i n pq R (r.,r„) for depolarizing d i f f u s e objects o i l constants constants =e a r ( v + i ) / 2 v + 1 r ( i - ^ V ) r ( i + ^ ) m z z distance along o p t i c a l axis from object (image) to f i l m constants double-reference-beam holography a general random input at a n o n - l i n e a r i t y Ee(r^), input v a r i a b l e at point r ^ ; i=l,2 exponential time-averaged i n t e n s i t y of coherently added object and reference waves (the input exposure at f i l m plane) 2 bias exposure (S T) time-averaged i n t e n s i t y of the object wave component cross-polarized to reference wave i x X T (p) object's irradiance d i s t r i b u t i o n obj Im{.J. ' imaginary part of 1 j integer j imaginary number (/-l) J ( T ) Fourier transform of I , .(p) r obj H / k integer k^ constant c o e f f i c i e n t s i n a Lagurre polynomial representation f o r t (E) a. l; 11=1+1 integers L (•) Laguerre polynomial m integer M c ( s ^ , S 2 ) j o i n t c h a r a c t e r i s t i c function of E^(r) M^Cs^jS^) j o i n t c h a r a c t e r i s t i c function of e(r) M^Cs^jS^) j o i n t c h a r a c t e r i s t i c function of exposure E(r) at f i l m plane MgjCs^jS^) j o i n t c h a r a c t e r i s t i c function of exposure v a r i a t i o n E'(r) at f i l m plane M (s, ,s n) j o i n t c h a r a c t e r i s t i c of E (r) p i I p MRBH multiple-reference-beam holography n(.) narrowband normal process (n(t) represents a narrowband noise; n(r) represents d i f f u s e object f i e l d at f i l m plane) n; N; N^; ^ integers N(.) amplitude of n(.) (N(r) represents d i f f u s e object wave amplitude at f i l m plane) N(.) complex presentation of amplitude and phase of n(.) x i N^C.) cosine quadrature component of n(.) ( N(.) cos|<})(.) | ) N g(.) sine quadrature component of n(.) ( N(.) sin|<|>(.) | ) N (r) complex object f i e l d component cross-polarized to reference wave,at f i l m plane N (r) complex object f i e l d component p a r a l l e l -p o l a r i z e d to reference wave,at f i l m plane 0 (p ) deterministic amplitude of d i f f u s e f i e l d immediately i n front of object p; plsp+1 integers p(.,.) j o i n t p r o b a b i l i t y density function P integer; number of reference beams P(t) random amplitude of a sin u s o i d a l s i g n a l P. =P(t ±); 1=1,2 q integer r integer r s p a t i a l v a r i a b l e over hologram plane R (.,.) cosine quadrature component of R^(.,.) c R (.,.) sine quadrature component of R (.,.) cs n R ^ C r ^ , ^ ) autocorrelation of amplitude transmittanee of a linear hologram R m k ( t l' t2 ) ^ t W V h mk ( t2 }] R (.,.) autocorrelation of n(.) R (.,.) part of output autocorrelation that is due to nxn interaction of n(.) with itself x i i R (...) part of output autocorrelation that i s due to nxs i n t e r a c t i o n of „n(.) with s(.) R^(.,.) autocorrelation of N(.) (E [l^N*] ) R^Cr^r^) autocorrelation of hologram transmittanee R , . ( P T , P 0 ) autocorrelation of the f i e l d over a d i f f u s e obj 1 I object ( ^ ^ ( p p ( S C P J - P ^ ) R (r, ,r„) part of R (r.,r„) that i s due to interference s^xs2 1 2 o 1 / of s^(r) and S2(r) R (...) output autocorrelation of a non-linear device y R (t ,t ) ^output autocorrelation of a non-linear detector R^-Cr^,^) autocorrelation of random part of hologram transmittanee ( EJ^ S ^ ] ) Re{.} 1 r e a l part of ' s integer s(.) sine wave ( s(t) represents a s i n u s o i d a l s i g n a l ; s(r) represents reference wave at f i l m plane ) s (r) sine wave presentation of reference beams at P f i l m plane; p=l,2,...,P S constant amplitude of pth reference beam P S (r) comolex representation of amplitude and phase P of pth reference wave SRBH single-reference-beam holography t time v a r i a b l e t _E amplitude transmittanee vs. exposure a t (E) approximate function f i t to t^-E curve a t (r) amplitude transmittanee of hologram a t bias transmittanee b t ' ( r ) v a r i a t i o n of hologram transmittanee X l l l mean amplitude transmittance of hologram U(t) step function U(p) complex f i e l d function over d i f f u s e object v(.) phase of x(.)=s(.)+n(.) V(.) amplitude of x(.)=s(.)+n(.) x(.) sum of a narrowband process and sinusoid(s) ( n(.)+? s (.) ; p=l,2,...,P ) P=l P y(.) output of a non-linear device (y=g(X)) z coordinate along o p t i c a l axis Z(.) output of a non-linear detector 01^ o f f - a x i s radian frequency of pth reference beam B constant Y = a Xd/27r P P gamma function 6(,) Dirac d e l t a function 1 p=q 6 Kronecker delta =!J ' pq 0 p-q Ar = r l _ i : 2 e Neumann factor m 1 m=0 2 m*0 angle of incidence of reference wave at f i l m plane 0(.,.) = t a n -1 \ a < - > R c (...) X wavelength of the i l l u m i n a t i n g source A constant (complex number with p o s i t i v e imaginary part) x i v constant normalized autocorrelation function of N (r) c exponent of vth-law n o n - l i n e a r i t y random part of hologram transmittance s p a t i a l v a r i a b l e i n object plane _P(t.) /2a(t i) standard deviation of d i f f u s e object complex f i e l d at f i l m plane standard deviation of n(.) (a.=o ( t . ) : i=l,2 ) x n 1 E27rAr/Ad exposure time phase of s(t) phase of n(.) ( <j)(r) s i g n i f i e s phase of object f i e l d at f i l m plane ) random phase over d i f f u s e object complex v a r i a b l e i n Fourier or Laplace transform plane an a r b i t r a r y large c a r r i e r frequency radian frequency of i l l u m i n a t i n g l a s e r source ' p or q whichever smaller' ' expected value of ' xv ACKNOWLEDGEMENT The encouragement and assistance from my supervisor, Professor Michael P. Beddoes, i s g r a t e f u l l y acknowledged. I would l i k e to thank him p a r t i c u l a r l y f o r h i s constructive c r i t i c i s m of the thesis i t s e l f , and for suggesting s t y l i s t i c changes i n the o r i g i n a l manuscript. Preliminary work was done at the Un i v e r s i t y of Manitoba under the supervision of Professor Wolfgang M. Boerner, and h i s constant support and encouragement i s g r a t e f u l l y acknowledged. Appreciation i s expressed to the members of my supervisory commit-tee, Professors L. Young, E.V. J u l l , and C F . Schwerdtfeger; t h e i r questions, c r i t i c i s m and advice have been very h e l p f u l . I am indebted to many s t a f f members of the E l e c t r i c a l Engineering Department for t h e i r assistance, advice, and tolerance during the course of th i s work, p a r t i c u l a r l y Mr. A l McKenzie for helping me with some of the draw-ings. The se c r e t a r i e s i n the E.E. O f f i c e have been most h e l p f u l , s p e c i a l l y with regard to typing the thesis and some of the associated p u b l i c a t i o n s . I acknowledge my fellow graduate students whose friendship has been a source of constant moral support, p a r t i c u l a r l y Doug Dean and Peter Driessen who also proofread the manuscript. The author g r a t e f u l l y acknowledges the f i n a n c i a l support provided by the National Research Council of Canada i n the form of a research a s s i s t -antship. x v i I. INTRODUCTION 1.1. Background and Motivations Holographic techniques have led to extensive achievements i n d i f f e r e n t f i e l d s of o p t i c a l computing and data processing '[la]. There have also been some achievements of great i n t e r e s t i n the domain of three-dimensional (3-D) imagery [la.]. These include applications i n 3-D movie and di s p l a y , archeology and art knowledge, and arc h i t e c t u r e . Other important a p p l i c a t i o n s , recently under extensive research, are for simple construction of low cost, l i g h t weight, and e a s i l y manage-able o p t i c a l elements [ l b ] , l i k e mirrors and lenses. There i s also a r a p i d l y growing number of applications i n other regions of the electromagnetic spectrum, namely, microwave [2] and radio frequency. In the f i e l d of acoustics, u l t r a s o n i c holography i s gaining s p e c i a l attention [ 3 j , p a r t i c u l a r l y due to i t s a b i l i t y to form 3-D images of objects which are opaque to l i g h t waves but transmit u l t r a -sonic waves. This feature has introduced important applications i n medicine, and i n the areas of underwater and underground s u r v e i l l a n c e and exploration. The promise implied i n the above l i s t i s large, but many applications are not e f f i c i e n t l y r e a l i z e d outside research centers, mainly due to noise oriented f a c t o r s , some of which are not yet s u f f i c -i e n t l y analyzed. Furthermore, while holography i s a mature d i s c i p l i n e , some basic t h e o r e t i c a l problems are yet to be better understood; and the e f f e c t s due to the non-linear storage medium, to which t h i s thesis i s addressed, i s one such problem. A storage medium i s an indispen-sable component i n holography. A common medium i s the photographic 1 f i l m , and although the theory developed i n t h i s t h e s i s i s addressed to the problem of f i l m o v e r load, t h i s by no means need be a r e s t r i c -t i o n . A hologram i s a r e c o r d i n g of the i n t e r f e r e n c e p a t t e r n c r e a t e d by an " o b j e c t " wave and a "r e f e r e n c e " or "background" wave. The spa-t i a l transmittance of the f i l m i s a f f e c t e d by the exposure E ( f i e l d i n t e n s i t y times exposure ti m e ) . The v a r i a t i o n s of transmi t t a n c e de-termine how the hologram d i f f r a c t s l i g h t . A d i r e c t r e l a t i o n between the v a r i a t i o n s i n exposure and transmi t t a n c e i s u s u a l l y provided by a t -E (amplitude transmittance v s . exposure) curve ( F i g . 1.1a). This 3. r e l a t i o n i s a no n - l i n e a r one. Many i n v e s t i g a t o r s have t r i e d to give experimental and t h e o r e t i c a l accounts f o r the n o n - l i n e a r e f f e c t s i n holography [4-18] to determine the l i m i t a t i o n s these e f f e c t s impose, and some have t r i e d to f i n d u s e f u l a p p l i c a t i o n s f o r them f9,17,18]. Most of these works d e a l w i t h the subj e c t matter i n the framework of communication theory. 1.1.1. Holography and Communication Theory From the e a r l y stages of the advent of holography the s i m i -l a r i t i e s between t h i s process and some well-known concepts i n commun-i c a t i o n theory were c l e a r l y noted [19] , i n a time when the a p p l i c a -t i o n s of c l a s s i c a l one-dimensional s i g n a l d e t e c t i o n theory to a n a l y s i s and processing of o p t i c a l images had already proven to be very f r u i t f u l ['20,21] . In the domain of o p t i c a l holography, Kozma [4] was the f i r s t to make use of a well-known technique i n communication theory t o ana-l y z e some of the e f f e c t s of the f i l m n o n - l i n e a r i t i e s i n the r e c o r d i n g 2 of the i n t e r f e r e n c e p a t t e r n of a weak d e t e r m i n i s t i c o b j e c t wave and a strong plane reference wave (see S e c t i o n 1.1.2). The method used by Kozma and l a t e r by some other research workers to study n o n - l i n e a r e f f e c t s i n holography i s the 'transform' or ' c h a r a c t e r i s t i c f u n c t i o n ' method [22,Ch.l3]. When the object i s assumed to be an ensemble of sources w i t h known amplitudes and phases the d e t e r m i n i s t i c form of the method i s employed [5,6]. However, when d i f f u s e i l l u m i n a t i o n i s used, the a n a l y s i s r e q u i r e s a s t a t i s t i c a l treatment [6,7,9-15,23]. The key concepts and equations f o r a s t a t -i s t i c a l a n a l y s i s of d i f f u s e i l l u m i n a t i o n were i n i t i a t e d by Goodman [23] i n 1965. L a t e r Knight [6] developed a f u l l treatment of the e f f e c t s of f i l m n o n - l i n e a r i t i e s on wavefront r e c o n s t r u c t i o n of d i f f u s e o b j e c t s ( i . e . a subject transparency backed by a d i f f u s i n g screen, or a r e -f l e c t i n g o b j e c t w i t h a rough surface) i n h i s Ph.D. t h e s i s , a c o n c i s e r e p o r t of which was given by Goodman and Knight [7] . Other s t u d i e s were reported i n [5,8-16]. These s t u d i e s have been confined e i t h e r to cases of l a r g e r e f e r e n c e - t o - o b j e c t i r r a d i a n c e r a t i o s or to p a r t -i c u l a r forms of n o n - l i n e a r i t i e s . The most extensive study of the sub-j e c t i s Knight's [ 6 ] . However, even t h i s treatment l a c k s f l e x i b i l i t y , accuracy, and completeness. This w i l l be discussed a f t e r a b r i e f review of the b a s i c s of the usual method of a n a l y s i s . The p r i n c i p a l ideas i n the f o l l o w i n g review come from [4], [22,Chs. 12 and 1 3 ] , 124, Ch. 8] , a n d " l 7 ] , to which the reader i s r e f e r r e d f o r a d d i t i o n a l d e t a i l s . 1.1.2. Transform Method and Its applications in Holography:A Brief Review The b a s i c a n a l y t i c procedure i s the transform or c h a r a c t e r -3 i s t i c f u n c t i o n method f o r o b t a i n i n g the c o r r e l a t i o n f u n c t i o n R q of the output of a 'memoryless' n o n - l i n e a r i t y w i t h t r a n s f e r c h a r a c t e r i s t i c g ( . ) , i n response to a random input e. This method i s based on a purely mathematical device and w i l l be explored f u r t h e r i n the f o l l o w i n g chap-t e r s . Mathematically expressed [22, P. 288], CO R 0 ( r ; r 2 ) = ff g ( e 1 ) g ( e 2 ) p ( e , e 2 ) d e 1 d e 2 (1.1a) — 0 0 1 6+j°° , = 2 // f ( s x ) f ( s 2 ) M e ( s 1 , s 2 ) d s 1 d s 2 (1.1b) (27TJ) <5-j°° where g(.) i s the t r a n s f e r c h a r a c t e r i s t i c of the n o n - l i n e a r i t y w i t h f ( . ) i t s F o u r i e r or Laplace transform (note that t h i s transform has no r e a l p h y s i c a l s i g n i f i c a n c e , f o r i t i s not a "spectrum" of a time or space f u n c t i o n i n the u s u a l sense of the word) , e^=e(r_^) denotes the input v a r i a b l e at point r ^ , p(e^, e 2) i s the j o i n t p r o b a b i l i t y d e n s i t y func-t i o n and M (s.., s„) i s the j o i n t c h a r a c t e r i s t i c f u n c t i o n of and e„ e 1 2 J 1 2 expressed as a f u n c t i o n of the complex v a r i a b l e s s^ and s^. Equation (1.1b) i s the fundamental equation of the transform method of a n a l y s i s of n o n - l i n e a r devices i n response to random i n p u t s . In the f o l l o w i n g when we t a l k of an " i n p u t " i n general terms, i t w i l l be denoted by e ( . ) , otherwise i t i s replaced by the n o t a t i o n s p e c i f i c a l l y employed to de-note the q u a n t i t y taken to be the " i n p u t " . Furthermore, the indepen-dent v a r i a b l e r i n general i s a v e c t o r i n an a r b i t r a r y space. For the above r e l a t i o n s h i p , g(.) i s assumed to be a r e a l func-t i o n (e.g. the t -E curve). In p r a c t i c e , however, the t r a n s f e r charac-cL t e r i s t i c might be a complex f u n c t i o n , to take i n t o account the phase s h i f t s imposed by the device on the input. In that case f ( s ^ ) and 4 and f ( s 2 ) , r e s p e c t i v e l y , w i l l be the Laplace transforms of g(e^) and g*(e^) , where shows t h a t . t h e f u n c t i o n i s complex, and the * i m p l i e s the f u n c t i o n i s the complex conjugate. In the process of holography the "output" i s considered t o be the transmittance of the hologram. Once a f u n c t i o n approximation to the r e c o r d i n g medium's n o n - l i n e a r i t y has been adopted, i t s Laplace transform f ( . ) may be computed. The two remaining steps are t o d e t e r -mine M e(s^, s 2 ) , and t o perform the complex i n t e g r a t i o n of Equation (1.1b) to o b t a i n the a u t o c o r r e l a t i o n . o f the transmittance of the h o l o -gram. This c o r r e l a t i o n f u n c t i o n then could be used to p r e d i c t some important p r o p e r t i e s of the hologram images i n c l u d i n g t h e i r i r r a -diance d i s t r i b u t i o n , s i n c e the s c a l a r d i f f r a c t i o n of the r e c o n s t r u c t -ing beam by a t h i n hologram transparency could be assumed as the a c t i o n of a l i n e a r system (e.g. Sherman[25]). l.l.3. Transform Method and the Kozma Model; Previous Analysis and  Their Drawbacks Employing the c h a r a c t e r i s t i c f u n c t i o n method of a n a l y s i s , Knight [6] uses Kozma's [4] model f o r n o n - l i n e a r i t y (see F i g . 1.2), and t h i s i s the source of some co m p l i c a t i o n s and inaccuracy i n h i s work. Kozma used the t -E curve i n e x a c t l y the same way as the c h a r a c t e r i s t i c curves of n o n - l i n e a r e l e c t r o n i c devices are being t r e a t e d (e.g. c h a r a c t e r i s t i c curve of a vacuum tube). A b i a s p o i n t i s u s u a l l y chosen on the center p o r t i o n of the smoothest p a r t of the curve and a s m a l l v a r i a t i o n a l a n a l y s i s i s a p p l i e d when the input v a r -i a t i o n i s not exceeding the l i m i t s of t h i s smooth p a r t . However, when the v a r i a t i o n exceeds the l i n e a r l i m i t s , a f u n c t i o n (e.g. an e r r o r 5 f u n c t i o n l i m i t e r , a polynomial or a vth-law f u n c t i o n ) i s chosen that f i t s the curve best on both sides of the b i a s p o i n t . Considering the chosen f u n c t i o n as a (full-wave) t r a n s f e r c h a r a c t e r i s t i c of a no n - l i n e a r device the transform method i s employed to c h a r a c t e r i z e the output. (A ful l - w a v e n o n - l i n e a r device operates on both p o s i t i v e and negative amplitudes, whereas a half-wave n o n - l i n e a r device operates only on the p o s i t i v e amplitudes.) In Kozma's model ( F i g . 1.2), the b i a s transmittanee i s pro-vided by a uniform reference wave exposure. The " i n p u t " to the non-l i n e a r element, then, i s taken to be the " v a r i a t i o n of exposure", which c o n s i s t s of two terms; the exposure due to the object wave alone, and the exposure due to the i n t e r f e r e n c e of the object wave w i t h the uniform reference wave. The l a t t e r term i s capable of r e c o n s t r u c t i n g the two r e a l and v i r t u a l image r e p l i c a s of the o b j e c t . To be more s p e c i f i c [24], w i t h an exposure time T, an object wave N(r) exp {-j(j)(r)}, and a coherent i n t e r a c t i n g uniform plane reference wave S exp ( - j a r ) , the " t o t a l exposure" E, may be w r i t t e n , E(r) = S 2T + N 2 ( r ) T + 2SN(r) cos{ar - (|>(r)}T = E b + E^(r) (1.2) 2 This exposure i s regarded as c o n s i s t i n g of a uniform exposure = S T c o n t r i b u t e d by the reference plus v a r i a t i o n E about E^, where, E ( r ) = N 2(r)T. + 2SN(r) cos{ar - (j)(r)}T (1.3) The amplitude transmittanee t ( r ) of the developed transparency i s r e -a presented by a constant " b i a s " transmittanee t ^ introduced by the r e -6 t a(E) F igure 1 .1 . F i l m n o n - l i n e a r i t y and i t s r e p r e s e n t a t i o n as a f u l l - w a v e t r a n s f e r c h a r a c t e r i s t i c of a memoryless n o n - l i n e a r element, a) the t -E cu rve ; b) the dependence of t ' on E ' ; c) the f u n c t i o n g ( E ' ) . E(r) + g(E) t a(r) g ( E ) f + F i gu re 1.2. The Kozma " c i r c u i t " model f o r ana l y s i n g n o n - l i n e a r holography. 7 ference, plus v a r i a t i o n s of transmittance t ( r ) . Thus, t a ( r ) = t b + t ' ( r ) . (1.4) I t i s the f u n c t i o n a l dependence of t on E ( i . e . the f u n c t i o n t ( E ) ) which determines the l i n e a r i t y of the process. U s u a l l y t h i s f u n c t i o n a l dependence i s represented by g(E ) , where, g ( E ' ) = - / ( E ' ) (1.5) Figure 1.1 i l l u s t r a t e s - t y p i c a l dependencies 0 f t on E , t (.) on E , and f i n a l l y g(.) on E . As can be seen ( F i g . 1.1c) the t r a n s f e r c h a r a c t e r i s t i c g ( E ) i s defined from -°° to the value of b i a s E^." However, Kozma's model r e q u i r e s a f u l l - w a v e c h a r a c t e r i s t i c (- 0 0 < E < » ) ; something that i s not p h y s i c a l l y r e a l i z a b l e . Furthermore, the f o l l o w i n g c o n s i d e r a t i o n s w i l l show the reasons f o r the co m p l i c a t i o n s i n v o l v e d i n t h i s model of a n a l y s i s when an a r b i t r a r y r e f e r e n c e - t o - o b j e c t i r r a d i a n c e r a t i o i s assumed. A. Large Reference-to-Object I r r a d i a n c e R a t i o Under the assumption of a refe r e n c e beam much stronger than the object beam, the v a r i a t i o n of the exposure due to the object wave alone w i l l be very small and can be neglected to s i m p l i f y the form of the i n p u t , e = E to-a s i n g l e s i n u s o i d a l l y v a r y i n g i n t e r f e r e n c e term. This approximation has the e f f e c t of g r e a t l y s i m p l i f y i n g the mathe-matics i n v o l v e d i n c h a r a c t e r i z i n g the output of the n o n - l i n e a r d e v i c e , which i s considered to be the v a r i a t i o n of tra n s m i t t a n c e of the h o l o -gram due t o the v a r i a t i o n of exposure E . In t h a t case, s i n c e f o r a 8 d i f f u s e object N(r) and (|)(r) are s t a t i s t i c a l l y i d e n t i c a l to the en-velope and phase of a narrowband Gaussian process [ 6 , 7, 26] , the i n -put E ~ 2TSN(r)cos{ar - ())(r)} obeys the s t a t i s t i c s of a narrowband Gaussian random process. The j o i n t c h a r a c t e r i s t i c f u n c t i o n of the process E , K^.^ (s]_> s^),^ r e a d i l y obtained and the e v a l u a t i o n of I n t e g r a l (1.1b) i s reasonably simple [22.1 I t i s shown [6, 7, 22] that the a u t o c o r r e l a t i o n of the transmittanee v a r i a t i o n t (E ), may be expressed as a sum of powers of the a u t o c o r r e l a t i o n of E , which i s e x p r e s s i b l e i n terms of the F o u r i e r transforms of the object i r r a -diance. However, the above approximation, i n gene r a l , i s not v a l i d . A high f r i n g e v i s i b i l i t y (high modulation depth) i s a t t a i n a b l e only when the object and reference i r r a d i a n c e s at the f i l m have roughly eqUal strengths. As a r e s u l t of t h i s increase i n modulation depth the r e -constructed images w i l l be b r i g h t e r , but the no n - l i n e a r e f f e c t s w i l l be more pronounced. This case may be f u r t h e r contrasted w i t h that of a very weak o b j e c t , f o r which f i l m g r a i n noise i s g e n e r a l l y the l i m i t -ing f a c t o r [27]. B. A r b i t r a r y Reference-to-Object I r r a d i a n c e R a t i o Under the c o n d i t i o n of a u n i t y r e f e r e n c e - t o - o b j e c t i r r a d i a n c e r a t i o or an a r b i t r a r y r a t i o , the ob j e c t - o b j e c t i n t e r m o d u l a t i o n term, 2 N ( r ) T , can no longer be neglected and, t h e r e f o r e , the exposure v a r i a -t i o n E', can no longer be tre a t e d as a Gaussian process. In that case, f i n d i n g an expression f o r M^,..(s^, s^) and hence the e v a l u a t i o n of I n t e g r a l (1.1b) becomes a formidable task. Besides, r e s u l t s obtained f o r power-law n o n - l i n e a r i t i e s [ 6 , 7J are u n n e c e s s a r i l y complex. F i n -a l l y , note that the Kozma model l i m i t s the a n a l y s i s to holography w i t h 9 a uniform irradiance reference wave (that accounts for the constant bias transmittanee t, ) b 1.2. This Thesis 1:2.1. A New Model for the Non-linear Hologram;.  Remedy for the Kozma Model Drawbacks An examination of Integrals ( 1 . 1 ) i n conjunction with Figures 1 . 1 a -1.1c shows that the complications i n using the Kozma model originate, from dealing with the undefined region i n g(E') = -t'(E') ( F i g . 1 . 1 c ) : from E^ to 0 0• That i s , with E' as the "input", the contribution of the negative range of E' can be dealt with only by using a two-sided Laplace transform, and one i s forced to f a b r i c a t e some value for g(E') from E, to »• The straightforward b approach would be to use the exposure E as input. The t o t a l contribution then would be evaluated from the one-sided Laplace transform (as E i s always p o s i -tive) . One would expect that the mathematical treatment might be more complex than Knight's treatment. Chapter I I I w i l l prove the opposite. The problem i s shown mathematically to be completely analogous to the c l a s s i c a l problem of the passage of e l e c t r i c a l s i g n a l and noise through n o n - l i n e a r i t i e s . S p e c i f i -c a l l y , i t w i l l be shown that the "output" (the hologram transmittanee) can be taken as equivalent to the r e s u l t of passing the sum of a narrowband process and a sinusoid through a non-linear detector. ( If the reference i s a d i f f u s e 10 source, the input w i l l be the sum of two narrowband normal processes.) Using well-known techniques, more accurate solutions are obtainable and the computation e f f o r t i s gr e a t l y decreased. Such a d i r e c t a p p l i -c ation of s i g n a l detection theory to holography was previously impos-2 s i b l e because of the object-object intermodulation term, N (r)T i n Equation (1.3). 1.2.2. Output of Non-linearities with Arbitrary Char act-eristics The problem of the passage of s i g n a l and noise through non-l i n e a r i t i e s has received considerable a t t e n t i o n i n the past, i n com-munication theory, and solutions have been worked out for some s p e c i f i c non-linear c h a r a c t e r i s t i c s , i n c l u d i n g vth-law devices. But, since i n t h i s t hesis we are interested i n a very general treatment of the problem of non-linear e f f e c t s i n holography, closed form solutions f o r n o n - l i n -e a r i t i e s with a r b i t r a r y c h a r a c t e r i s t i c s w i l l be e s s e n t i a l and most con-venient. This w i l l be the task of Chapter I I of t h i s t h e s i s . It w i l l e s s e n t i a l l y deal with the problem i n e l e c t r i c a l communications of f i n d i n g closed form solutions f o r the c o r r e l a t i o n f u nction of the output of memory-l e s s non-linear devices with a r b i t r a r y c h a r a c t e r i s t i c s . U t i l i z i n g a power-series expansion for the transf e r c h a r a c t e r i s t i c g(.). of the non-l i n e a r i t y l o r j o i n t c h a r a c t e r i s t i c function of the input M e(s^, S 2 ) ] , we generalize upon material reported i n c l a s s i c a l works on vth-law de-vices [ 2 2 , 28J. This allows us to give ready-made formulas for autocor-r e l a t i o n functions of the output of n o n - l i n e a r i t i e s i n response to s i g n a l and noise. 1.2.3. Polarization Effects in Holography The a n a l y s i s of Chapter I I I i s very general i n the sense that 11 the irradiances of reference and object, the type of hologram, and the nature of the film's n o n - l i n e a r i t y are taken to be a r b i t r a r y . (Note, how-ever,that the hologram is..assumed t o be thin,large,and t o have a very f i n e grain emulsion). But, i n previous works as well as i n Chapter I I I , the in t e r a c t i n g wave amplitudes over the hologram plane are considered as scal a r q u a n t i t i e s ; or the p o l a r i z a t i o n of the scattered f i e l d by the ob-j e c t i s assumed to be the same and p a r a l l e l to that of the reference beam. In general, these assumptions are not well j u s t i f i e d and the e f f e c t s due to a change i n the p o l a r i z a t i o n of the r e f l e c t e d f i e l d from the object should also be included. This i s an important topic because depolariza-t i o n i s responsible for some degradations i n image q u a l i t y , whether the hologram i s recorded l i n e a r l y or not. The analysis method of Chapter I I I i s s u f f i c i e n t l y general to allow consideration of p o l a r i z a t i o n changes due to s c a t t e r i n g of the f i e l d by the object. Chapter IV gives the math-ematical techniques for evaluating the autocorrelation function for the transmittanee of a non-linearly recorded hologram of a d i f f u s e object when an object cross-polarized component i s present. It i s shown that a depolarizing object not only causes a reduction i n signal-to-noise r a t i o , and for some objects, a probable loss of information from those segments of the object that have large curvatures, but also f o r t i f i e s the d i s t o r -t i o n of the images due to the n o n - l i n e a r i t y of the recording medium. This new d i s t o r t i n g e f f e c t appears as a m u l t i p l i c a t i v e factor and i s de-pendent on the irradiance d i s t r i b u t i o n of the object's cross-polarized component. A simple way of decreasing these degradation e f f e c t s i s sug-gested. 12 1.2.4. Multiple-Ret'erenae-Beam Holograph/ and Suppression  or Intexmodulation Noise Chapter V w i l l study the nonlinear e f f e c t s i n holography i n a s t i l l more general case, where several strong r e f l e c t o r s accompany the d i f f u s e object and/or a multiple-reference-beam holographic r e -cording i s employed (e.g. holography with two reference beams). An important aspect of the analysis of Chapter V i s i t s deve-lopment of applications of multiple-reference-beam holography i n r e -ducing the nonlinear d i s t o r t i o n s , without much of a loss i n hologram e f f i c i e n c y . More s p e c i f i c a l l y , as the analysis of Chapter III w i l l show, i n a nonlinearly recorded hologram, the d i s t o r t i o n of the f i r s t order images i s due to those intermodulation terms that produce extra d i s -torted images i n the same d i r e c t i o n as the f i r s t order ones. These extra images produce a n o n f i l t e r a b l e background noise that i s reduc-i b l e only at the cost of d i f f r a c t i o n e f f i c i e n c y of the hologram, by having a reference wave much stronger than the object wave. Thus, the important question a r i s e s whether i t i s possible to reduce the background noise without s u f f e r i n g much l o s s i n e f f i c i e n c y . The analysis i n Chapter V of nonlinear e f f e c t s i n double-and m u l t i p l e -reference-beam holography w i l l show the answer i s p o s i t i v e . .1.2-S' Summarij-In summary, the main contributions of t h i s t h e s i s are as follows: 1. Formulation of closed form solutions for the output cor-r e l a t i o n function of memoryless non-linear devices with a r b i t r a r y 13 t r a n s f e r c h a r a c t e r i s t i c s i n response to s i g n a l and noise (Chapter I I ) . 2. Development of a new mathematical model f o r the process of holography. The model allows many of the " c l a s s i c a l " r e s u l t s of s i g -n a l d e t e c t i o n theory to be d i r e c t l y a p p l i e d to holography. (Chapter I I I ) 3. A b e t t e r understanding of the n o n - l i n e a r e f f e c t s i n h o l o -graphic imaging, through a more accurate, more g e n e r a l , and f l e x i b l e f o r m u l a t i o n of the a u t o c o r r e l a t i o n of transmittanee of holograms of d i f f u s e o b j e c t s . (Chapter I I I ) 4. An i n q u i r y i n t o probable holographic image degradations encountered w i t h d e p o l a r i z i n g o b j e c t s , and a simple way to decrease them (Chapter I V ) . 5. A f u r t h e r g e n e r a l i z a t i o n of the a n a l y s i s of n o n - l i n e a r e f f e c t s i n holography to the cases o f a d e p o l a r i z i n g d i f f u s e object (Chapter I V ) , and/or more complex o b j e c t s w i t h more than one reference wave present. (Chapter V) 6. E x p l o r a t i o n and suggested a p p l i c a t i o n s f o r double (or m u l t i p l e ) - reference - beam holography; e.g. i t may be employed to suppress n o n - l i n e a r n o i s e . (Chapter V) 14 I I . AUTOCORRELATION OF OUTPUT OF MEMORYLESS NON-LINEARITIES  WITH ARBITRARY CHARACTERISTICS 2.1. I n t r o d u c t i o n The 'transform' or ' c h a r a c t e r i s t i c f u n c t i o n ' method of Bennett and Rice {29, 30, 311, and Middleton J28, 32] i s a well-known technique that has been widely used f o r g a i n i n g i n s i g h t i n t o the response of a non-l i n e a r i t y to s i g n a l s and noise. A d e t a i l e d and c l e a r explanation of t h i s method and i t s a p p l i c a t i o n s to the determination of the a u t o c o r r e l a t i o n f u n c t i o n s of 'memoryless' n o n - l i n e a r i t i e s i s given i n Chapter 13 of Da-venport and Root [22]. The r e s u l t s given i n these works f o r the output c o r r e l a t i o n f u n c t i o n s of n o n - l i n e a r i t i e s are general i n the sense that the t r a n s f e r c h a r a c t e r i s t i c of the n o n - l i n e a r i t y has been u n s p e c i f i e d , although the s o l u t i o n s are worked out fo r some s p e c i f i c c h a r a c t e r i s t i c s i n c l u d i n g vth-law devices. F o l l o w i n g the n o t a t i o n of [22], l e t the input to the n o n - l i n e a r i t y be, .x(t) = n(.t) + P ( t ) cos (ID t +$•), (2.1) where n ( t ) , the input n o i s e , i s assumed f o r the moment to be a sample f u n c t i o n of a r e a l Gaussian random process, P ( t ) i s a low-frequency r a n -dom process independent of n ( t ) , and where $ , a random v a r i a b l e indepen-dent of P and n, i s uniformly d i s t r i b u t e d over [0, 2TT] . The output c o r -r e l a t i o n f u n c t i o n R ( t ^ , t^) of a no n - l i n e a r device w i t h the t r a n s f e r c h a r a c t e r i s t i c y = g(X), i n response to x ( t ) , can be shown to be, OO OO £ R y ( t l ' t 2 ) = E . r r E l h . (.t. ) h . (t „ ) J R k ( t . , O cos mu, T , (2.2) J m=0 k=0 k l mk 1 mk 2 n 1 2 c 15 where T = - t ^ , ^nC.t^» t^) i s the c o r r e l a t i o n f u n c t i o n of the input n o i s e , ^^'-roVl^ "*"s t* i e Neumann f a c t o r , the averaging E[.] i s w i t h respect to the input s i g n a l modulation, and 2 2 = "537 f„ f ( u ) <» I [ < J exp 1-^—] do), (2.3) mk 1 ZTTJ c m x 2 where f(cj) i s the t r a n s f e r f u n c t i o n of the n o n - l i n e a r i t y , which i s r e -l a t e d to g(X) by a F o u r i e r or Laplace transform (one-sided or two-sided, depending on the nature of the n o n - l i n e a r i t y ) , c i s a contour of i n t e g r a -2 t i o n , P. = P ( t . ) , a. i s the v a r i a n c e of n ( t ) at t . , and I [.] i s a mod-' x v ±'' x x' mL J i f i e d B e s s e l f u n c t i o n of the f i r s t k i n d . Looking back at (2.2) & (2.3), i t seems that f o r each d i f f e r -ent d e v i c e , one has to go through the elaborate process of e v a l u a t i n g i n t e g r a l (.2.3) i n a d d i t i o n to the e v a l u a t i o n of f ( w ) , the Laplace t r a n s -form of g(X), to o b t a i n R ( t ^ , t^). However, the s t u d i e s conducted by s e v e r a l i n v e s t i g a t o r s have l e d to some s i m p l i f i c a t i o n s of the problem. Baum [33], on the b a s i s of r u l e s governing F o u r i e r transforms, e s t a b l i s h e d c e r t a i n r u l e s f o r the c a l c u l a t i o n of the output a u t o c o r r e l a t i o n f u n c t i o n s of n o n - l i n e a r devices whose t r a n s f e r c h a r a c t e r i s t i c s are r e l a t e d to each other by elementary transformations such as a d d i t i o n , d i f f e r e n t i a t i o n , s h i f t i n g e t c . These r u l e s were obtained f o r the cases i n which the La-place transform r e l a t i o n s h i p of g(X) and f(w) can be replaced by a Four-i e r transform, and i n which the input i s noise alone. I t was a l s o shown that i n the case of unmodulated s i n u s o i d plus narrow-band n o i s e w i t h sym-metric spectrum about s i g n a l frequency the elementary r u l e s could apply to the 'envelope' of the a u t o c o r r e l a t i o n f u n c t i o n . L a t e r , Weiner et a l 134] s t a t e d (without proof) that the r e s u l t s of £33J could be extended to account f o r a l l of the s p e c t r a l zones generated at the output of the 16 device when e x c i t e d w i t h s i g n a l plus n o i s e . Although the a p p l i c a t i o n s of the r e s u l t s of {33] and [34] could to a great extent s i m p l i f y the c a l c u l a t i o n s of the c o r r e l a t i o n f u n c t i o n s of c e r t a i n d e v i c e s , they do not provide a compact and ready-made formula f o r an a r b i t r a r y t r a n s f e r c h a r a c t e r i s t i c . Moreover, sometimes the elemen-t a r y operations that have to be performed on the a u t o c o r r e l a t i o n of the output of the b a s i c d e v i c e ( s ) , could become q u i t e t e d i o u s , and almost im-p o s s i b l e to manage. On the other hand, i t can be shown that general and e a s i l y manageable formula can be obtained f o r the c o r r e l a t i o n f u n c t i o n s of a r b i t r a r y n o n - l i n e a r i t i e s i f one chooses to evaluate the fundamental i n t e g r a l s f o r ( 2 . 6 a and 2.6b> below) simply by employing the power s e r i e s expansion f o r e i t h e r the j o i n t c h a r a c t e r i s t i c f u n c t i o n of the input M(u)^, 1^2), or the t r a n s f e r c h a r a c t e r i s t i c of the n o n - l i n e a r i t y g(X). An example of the use of the series-expansion method was given by Shutter-l y [35] , i n which expressions were derived f o r the output time f u n c t i o n and a u t o c o r r e l a t i o n f u n c t i o n i n terms of weighted averages of the non-l i n e a r c h a r a c t e r i s t i c and i t s d e r i v a t i v e s . For the case of s t a t i o n a r y Gaussian noise and unmodulated s i n u s o i d , he obtained the f o l l o w i n g ex-pr e s s i o n f o r the c o e f f i c i e n t s h ^, _ E l g ( k ^ + 2 r ) ( x ) ] pm +2r  m k r=0 (m+r)! p! 2 m + 2 r where g(k+m+2r) denotes the (k+m+2r)th d e r i v a t i v e of g(X) , and where E[.J i n d i c a t e s the s t a t i s t i c a l average with respect to n o i s e ; thus, 2 2 TT r (p) t M A ,°° (p) , , -| -x /2a , . E [g r (x)] = f_m g ^ ' ( x ) 1 e dx (2.5) /2ira As can be seen, although the e v a l u a t i o n of h ^ i s s i m p l i f i e d to some 17 extent, i t i s s t i l l dependent on the e v a l u a t i o n of weighted averages of the n o n - l i n e a r c h a r a c t e r i s t i c and i t s d e r i v a t i v e s . An a l t e r n a t i v e tech-nique i s suggested here which gives expressions f o r h , which are d i r e c t -l y dependent on the d e r i v a t i v e s of g(X) evaluated at o r i g i n . These are the c o e f f i c i e n t s of the power-series expansion of g(X). The a n a l y s i s simply e x p l o i t s the mathematical method used to o b t a i n the c o e f f i c i e n t s h k f° r a vth-law n o n - l i n e a r i t y as reported w i t h great c l a r i t y i n [22 & 28J. As a consequence, the extension of the a n a l y s i s to cover the cases of nearly-Gaussian noise becomes s t r a i g h t f o r w a r d w i t h the a i d of a method reported by Bowen [36J. 2.2. D e r i v a t i o n of Formulas f o r h ^ The fundamental equations f o r o b t a i n i n g the a u t o c o r r e l a t i o n f u n c t i o n of the output of a n o n - l i n e a r device are given by, R (t t 2 ) = / / gC^) g ( x 2 ) p ( x x , x 2 ) dx± d x 2 (2.6a) y - c o t2 iTJ) 2 J c f (u)^)dw^ f(w2)da32 M x ( u 1 , a>2) c (2.6b) where, pCx^, .x2) i s the j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n and M x(oi^, u>^) the j o i n t c h a r a c t e r i s t i c f u n c t i o n of x^ and x 2 expressed as a f u n c t i o n of the complex v a r i a b l e s O J ^ and u^. A power-series expansion of g(x^) i n '(2. '6a) g i v e s , Vv * = l o l 0 v 2 • ( 2 - 7 ) where, 18 0 0 V l Vo I = a a // x„ z p ( x i s x 0 ) dx., dx„, v l v 2 v l v2 lo, ^ 2 1 2 1 2' (2.8) and where a i s the c o e f f i c i e n t of X V l i n the expansion of g(X). I f i the expansion i s a Mac l a u r i n s e r i e s expansion, then, V . V . ! V . 1 1 dX 1 (2.9) x = 0 Upon performing s i m i l a r mathematical manipulations which l e d to (2.2) (see, f o r example, [22J , Chapter 13), i n c l u d i n g the replacement of x^ i n terms of i t s Laplace transform, i . e . , G O = v. x X V .+1 CJ.X (2.10) R (t.. , t 0 ) could be w r i t t e n as, y- 1 2 CO CO W = ^ n v \ kT E I h m k ( tl> \k ^ 2 ) J ^  ( t r t 2 ) cos m V • m=0 k=0 (2.2) w i t h , 2 2 0 0 v^!a _^ o\o) TI A. N _ S. i u : to i I (uP.) expO—=—) du (2.11) x 2TTJ ^ ( f o r a half-wave n o n - l i n e a r i t y ) . N o t i c e that f o r full-wave n o n - l i n e a r i t i e s of e i t h e r odd or even symmetry h ^ w i l l have non-zero value only when (m + k) i s a c c o r d i n g l y odd or even and i s twice the h ^ value of the corresponding half-wave n o n - l i n -e a r i t y . (A half^-wave n o n - l i n e a r device operates only on the p o s i t i v e amplitudes, w h i l e a ful l - w a v e one operates on both p o s i t i v e and negative 19 amplitudes.) The i n t e g r a l i n (2.11) i s evaluated i n [22 & 28] and hence, , ,„ v f v 1F 1I(m+k-v,)/2;m+l;-p 2J Ca 172) 1 \k Cti?  = vl=0 Kmkv [ c f . (2.4)],(2.12a) " r J l - (m+k-v.)/2] where, i 3 1 1 a v. J p . -r, v. i i P. K = _ J : , p. =•— , (2.12b) m k V • • -r- k 1 /2 a 2ml (o-./JIy ± and where ^ F l Ca;c;z) i s the confluent hypergeometric f u n c t i o n . The c o e f f i c i e n t s h ^ of (2.12) are obtained on the premises that g(X) i s continuous over the r e g i o n -°° to 0 0 (or ,0 to 0 0) and x ( t ) i s supposed to be a zero-mean process. However, very o f t e n t h i s i s not the case, i . e . , gQO might be discontinuous at some po i n t s and/or x ( t ) might have a non-zero mean. The f o l l o w i n g w i l l cover those cases. Case 1: A Shifted g(X) Let us assume that g(X) undergoes a s h i f t (and/or x ( t ) has a nonrzero mean v a l u e ) . This s h i f t i s equivalent to m u l t i p l i c a t i o n o f the integrand i n (2.11) by exp(-bai), where b i s the amount of s h i f t . The new i n t e g r a l f o r h ^(-t^) may be evaluated the same way as i n t e g r a l (2.11) i f exp(-ba)) i s replaced by i t s power-series expansion. The new c o e f f i c i e n t s h , a r e , mk v.-l 2 (-by (a. 1/2) 1 -F. [(m+k+£-v.)/2;m+l;-p ] oo co 1 J_ X 1 -L h , (t.) = 1 K . z. _ — m k 1 v.=0 m V 1=0 V. rIl-(m+k+iL-v i)/2] (2.13) 20 Case 2: g(X) with Cut-off ov Limiting The second case we consider i s a half-wave g(X) w i t h cut-roff at p o i n t X = c. The power-series expansion of g(X) i s now f o r the i n -t e r v a l 0 to c. The Laplace''transform of a vth-law n o n - l i n e a r i t y w i t h c u t - o f f is, given by [37J , c V + 1 j F (v+1; v+2; -CUJ)/(V+1), (2.14) which w i l l reduce to (2.10) when c °° , i . e . , a vth-law n o n - l i n e a r i t y without c u t - o f f . Replacing v ! / u ) V + 1 w i t h (2.14) i n (2.11) and employing the power-series r e p r e s e n t a t i o n of ^ F ^ i n the new i n t e g r a l enable one to evaluate h , i n a s i m i l a r manner as i n the previous cases: mk v. (-1) c X - L 1 F 1 [(m+Wl)/2;mHipf] h„, ( t . ) = I Cc \ , /v,!) 1 1 1 m k " 1 v^.Q m k V 1 £=0 (v±+£l)Jl! r l l-(m+k+U)/2J (o±/fI) (2.15) where 11 = (.1+1) I f , i n s t e a d of a c u t - o f f , g(X) i s saturated at p o i n t c, then v!/to V +^ i n (2.11) should be replaced by, c v 1F 1(V;V+1;-COJ)/W (2';.16) I f i n (2.15) a l l £l's are replaced by Z and the numerator i s m u l t i p l i e d by v., an expression f o r the c o e f f i c i e n t s h . ( t . ) f o r the saturated l mk i case w i l l be obtained. 21 Case 3: Combination of Case 1 and Case 2 v+1 I f both s h i f t and c u t - o f f are present, then v!/u) i n (2.11) should be replaced by [37], ( c - b ) ^ 1 e b w 1V± [v+l;v+2;(b-c)o)J/(v+l) (2.17) Using the approaches i n Case 1 and Case 2 one f i n d s , V i oo oo ( - l ) p + ( l ( c - b ) p l ( b ) q V ± p-u.q-u ( V i + p l ) p ! q ! r I 1 _ ( I I 1 + k + q + p l ) / 2 ] $ i [(m+k+q+pl)/2;m+l;p1] — — — (2.18) ( 0 j / / 2 ) q + p l where p l = (p+1). I f , i n s t e a d of c u t - o f f a s a t u r a t i o n i s present, the c o e f f i c i e n t s ^ m ^ ( t ^ ) may be obtained by r e p l a c i n g a l l p i ' s by p and mul-t i p l y i n g the numerator by v^. Almost any n o n - l i n e a r i t y w i t h d i s c o n t i n u i t i e s (and/or w i t h non-zero mean input) could be represented by some combination of the above mentioned cases and the c o e f f i c i e n t s h . of the n o n - l i n e a r i t y w i l l be mk the sum of the c o e f f i c i e n t s h , f o r each i n d i v i d u a l of that combination. mk As a p a r t i a l check on our r e s u l t s the f o l l o w i n g examples are worked out. 2.3. Examples (a) Symmetrical Peak Clipping We s h a l l consider the gradual peak c l i p p i n g and the abrupt peak c l i p p i n g whose c h a r a c t e r i s t i c s are shown i n Figures 2.1 and 2.2. 22 For the gradual ; clipper of Figure2. l",g(X) i s defined as follows: g(X) = 1 - e -dx2 -dx -1 X > 0 X < 0 d > 0 (2.19) The coefficients a of (2.9) in this case w i l l be , 1 v. 9 x = 0 (1+v.12) 1 v 12 K J d for even v.>0 otherwise (2.20) Thus, coefficients fr^ f° r t n e gradual peak clipper of (2.19) w i l l be twice C2.12) for (m+k) odd (because of the odd symmetry of g(X))with 23 a given by (2.20); i . e . , h ^ has non-zero values only when i s even V i and greater than zero, and when m+k i s odd. For the abrupt c l i p p e r of Figure 2.2 g(X) = g + (X) - g + ( - X ) , (2.21) where, g + a ) = aX X < c a X > c 0 X 0 (2.22) The only non-zero c o e f f i c i e n t a i s a,^  = a. A d i r e c t a p p l i c a t i o n of V i + (.2.15) w i t h v = 1, and replacement of i l l ' s w i t h I give h ^ f o r g + ( X ) . Hence, the c o e f f i c i e n t s f o r g(X) of (2.21) w i l l be, ' mk 2h + mk f o r (m+k) odd otherwise. Furthermore, s i n c e T(-n) = °° f o r n = 0, 1,2,.... and t h e r e f o r e , odd values of £ make h , = 0, I may be replaced by '21. mk Cb) Symmetrical Center Clipping The t r a n s f e r c h a r a c t e r i s t i c of a symmetrical center c l i p p e r i s shown i n Figure",2.3. 24 1 / a ( x - b ) -b X y b Va(x+b) Figure 2.3. A l i n e a r center c l i p p e r A d i r e c t a p p l i c a t i o n of (2.13) with v ± = 1 and a± = a w i l l give h j ^ . Hence, 2h mk for (m+k) odd otherwise This, time, even values of I greater than zero w i l l make h m k zero, and, therefore, one can write, ??+1 2 a « l k + _ i ^ + 2 K m k i E O ^ f o r (m+k) odd 1 *~ u C2X,+l)!rll-(m+k+2JD/2](ai//2)^X' 0 otherwise, (2.23) where 5 i s the Kronecker d e l t a = 1 i f p = q 0 i f p f q (o) A Quantizer F i n a l l y , we s h a l l consider an N-level quantizer. The character-i s t i c of such a quantizer i s shown i n Figure2.4. This quantizer may be 25 defined as f o l l o w s i n terms of s h i f t e d step f u n c t i o n s : g(X) N l I r = 1 N 2 S = 1 A U(X-B ) r r C U(-X-D ) s s X > 0 X < 0 N = N x + N 2 (2.24) where U(X) i s a step f u n c t i o n . A d i r e c t a p p l i c a t i o n of (2.13) w i t h v. = 0. b = B and a, = A w i l l g i v e h , f o r A U(X-B ). S i m i l a r l y , I * r l r & mk r r C2.13) w i t h v . = 0, b = D and a. = ( - l ) m + k C w i l l g i v e h f o r C U(-X-D ) 1 s JL S XDIC S" S The c o e f f i c i e n t s h , f o r g ( X ) , then, w i l l be the a l g e b r a i c sum of h ,'s U|K. mic f o r a l l the s t e p - f u n c t i o n s i n v o l v e d . In the case of a symmetrical quan-t i z e r , i . e . , when A. = C , B. = D. and N.. = N„, one a r r i v e s a t : 3 3 3 3 1 2 m N/2 A. p. J 1 I BT 1F 1[(m+k+2A)/2; m+l.-p^] ml Lo.//2) «,=0:2£!rll-(m+k+2Ji)/2J (a.1/2) 21 3 = 1 0 N R r=1 A 1 + A 2 DN 2 D 2 I t g(x) J X B l B 2 B N 1 C 1 + C 2 N 2 s=1 f o r m+k odd otherwise (2.25) Figure 2.4. A q u a n t i z e r . 26 2.4. Signal X Signal, Noise X Noise and Signal X Noise  Output of a Non-linearity Equation C 2.2) that gives an expression for R (t.. , t_) i s y x z not i n i t s most convenient form for our purposes. A better form i s ob-tained by expanding i t into three components [22, P.292, Eqs. (13.62) -(13.64)], as follows: R ( t , , t 0 ) = 'Z e R ( t , , t„) cos mu T , sxs 1' 2 n m mo 1 2 c m = u (2.26) k = 1 R 6 x n ( t r t 2 ) = 2 ^ Z , R n i k ( t r t 2 ) R n <V t 2 ) C O S m V > ( 2' 2 8 ) m = 1 k = 1 where, \ k ( t r V = E { h m k ( t l ^ m k ( t 2 ) } ' ( 2 ' 2 9 ) and where subscripts (s x s ) , (n x n), and (s x n) s i g n i f y , respectively, the interactions of signal with i t s e l f , noise with i t s e l f , and signal with noise. k 2.4.1. Narrowband Inputs and an- Expansion of R (t^t^) When the noise process n(t) i s a r e a l narrowband normal pro-cess, (2.26)-(2.28) would be more revealing i f an expanded form of ( t ^ , ty) were used. More s p e c i f i c a l l y , a sample function n(t) of a narrowband r e a l normal process may be expressed i n the form, 27 . n(t) = N(t) cos[co t + (|)(t)], or equivalently, (2.30) nCt) = Re {N(t)exp(joj t)} = Re{N(t)expI j(])(t)Jexp(ja) t)} = N (t) C O S C J t - N (t) sinio t c c s c (2.31) where Re{.} denotes 'the r e a l part' (and Im{.} w i l l denote 'the imaginary p a r t ' ) , w c i s the mean radian frequency of the sp e c t r a l band, NCt) = N(t) explj(|)tt)j, N (t) = Re{N(t)} = N(t) cos(j)(t), c (2.32) (2.33) N (t) = Im{N(t)} = N(t) sin(|)(t), (2.34) hence,. NCt) = I N * 00 + N^Ct^172, c s (2.35) and, (J)(t) = tan -1 y t ) NjtT c (2.36) and where the envelope N(t) and phase (JKt) are slowly varying functions of time. For a normal n ( t ) , N(t) w i l l be Rayleigh-distributed and (j)(t) w i l l be uniformly d i s t r i b u t e d between 0 and 2IT. For the c o r r e l a t i o n function R (t.. , t.) of a narrowband nor-n 1 2 mal process, equations s i m i l a r i n form to (2.30) - (2.36) can be obtained, i . e . R n C t 1 , t 2 ) = 1/2 R N C t 1 , t 2 ) cos -IUCT + G ( t 1 , t 2)J , (2.37) 28 or equivalently, R n ( t 1 5 t 2) = 1/2 Re{R^(t 1 , t 2) e x p ( j u > c T ) } = 1/2 R e ^ C t ^ , t 2)exp I j G C ^ , t 2 ) J e x p ( j o ) c T ) } = R (t, , t j c o s w T — R ( t r , t )sino) x (2.38) c 1 2 c c s l z c where, R^( t ; L, t 2) = E{NCt1)N*Ct2) = V V t 2 > e x P ^ 0 ( t l ' t 2 ) J ( 2 ' 3 9 ) R ( t l , t 2) = 1/2 Re{R^(t 1, t 2 ) } = 1/2 R N ( t 1 , t ^ c o s e ^ , t 2) (2.40) c R c s ( t 1 , t 2) = 1/2 Im{R^(t 1, t2)} = 1/2 R N ( t 1 , t ^ s i n G ^ , t 2) (2.41) Hence, W V = ^ I ' V + R L ( t l ' l 2 » 1 / 2 ( 2 ' 4 2 ) and, - l r R c a ( t r t 2 ) 0 ( t t )=taxv I R ( t t ) J . . (2.43) 1 x- c 1 2 With an R n ( t 1 > t 2 ) a s i n ( 2-37), an expansion for could be found as [22, P. 298, Eq.(13.87)]. 29 R;< ti» t2> • „2k-l k-2 £ '(fc-rl'r' C O S l ( k _ 2 r ) ( a ) c T + @ ) J + ~^~2 2(|!) r=0 k-1 f o r k even k 2 V t l ' t 2 ) I' ( k _ ^ , r ! cosI (k-2r)( W cx +0 ) j ,2k-l r=0 f o r k odd (2.44) The above expansions w i l l be used i n the f o l l o w i n g s e c t i o n and throughout the t h e s i s . 2.4.2. Output Correlation of a Detector for Narrowband Noise When a non- l i n e a r device i s followed by a low-pass zonal f i l t e r a 'detector' i s formed. Because of the f i l t e r , only those terms i n the expansion of the s p e c t r a l d e n s i t y of the device output which are concen-t r a t e d about zero frequency can c o n t r i b u t e to the detector output. With.the a i d of (2.26) - (2.28) and (2.44), and a f t e r some a l -gebraic manipulations the components of the c o r r e l a t i o n f u n c t i o n of the output o f i t h e detector Z, i n response to s i g n a l plus narrowband n o i s e , may be w r i t t e n as, V a x a ^ W = R 0 0 ( t l ' t 2 ) ' (2.45) V n x n / V V " I V V V ^ ^ ^ ^ ( 2 . 4 6 ) k=2 ( k, ) 22k (k even) 30 co k RZ ( s X n ) ( W ' I ' I . k=l m=l W W ,k+mr. ,k-m(. 2k-1 [(m+k) even] 1 2 ' H 2 ' J RJT ( t 1 , t 2 ) c o s m 0 ( t 1 , t 2 ) C2.47) Up t i l l now the s i g n a l has been assumed to be an amplitude mod-ula t e d s i n u s o i d w i t h § as a random v a r i a b l e uniformly d i s t r i b u t e d between 0 and 2TT. The c h a r a c t e r i s t i c f u n c t i o n of such a s i g n a l i s [22,P.290, Eq. (13.47)]. M (GO ,o)_) = T e E{I ( ^ P j l (oi„P„) Jcosmw x s 1 ? 2 u m m ' 1 1 m 2 2 c m=0 (2.48) where cosmw x i s due t o , c E{cosm(aJ t, +$ ) cosn(o) t 0 + $) } =-c i , c z . when m ^ n — cosmoj x when m = n e c m (2.49) I f we took the s i g n a l to be phase modulated, too, i . e . s ( t ) = P ( t ) COSICJ t + 8§(t)] C (2.50) where 6 i s the modulation index, then — cosw x i s to be replaced by m 1.38,Eq. (14)1 E{cosmIaj t , + B'$(t.,)l cosn[u t„ + c 1 1 c z 6'4(t-)]} = — R(mFM) z e m (2.51) where R(mFM) s i g n i f i e s that the replacement i s equivalent to the auto-c o r r e l a t i o n of an FM wave whose c a r r i e r frequency and modulation index 31 are m u l t i p l i e d by m. I f , f o r example, 8$(t) = at,•then R(mFM) = cosm(a) x + ax). When a«ui^, the output a u t o c o r r e l a t i o n f u n c t i o n of the detector i n response to s ( t ) + n ( t ) ' w i l l s t i l l be given by (2.45) - (2.47), but w i t h ©(t^jt^) i n (2.47) now being replaced by [ax + 0 ( t ^ , t 2 ) ] . 2.5. F i n a l Remarks The r a t h e r complicated expressions (2.12), (2.13), (2.15), and (2.18) f o r h k m a y t>e s i m p l i f i e d i n the s p e c i a l cases of very small or very l a r g e input s i g n a l - t o - n o i s e r a t i o s . To achieve t h i s the appropriate expansion of the confluent hypergeometric f u n c t i o n ^F^ i s used. For a d e t a i l e d example of such an approximation, the reader i s r e f e r r e d to [22,PP. 305-308J. 2.6. Summary An important c l a s s of n o n - l i n e a r i t i e s i s that based on the v t h -law c h a r a c t e r i s t i c . Mathematical methods already employed i n the a n a l y s i s of t h i s c l a s s of n o n - l i n e a r i t i e s can be e x p l o i t e d i n a r a t h e r s t r a i g h t -forward manner to ob t a i n ready-made formulas f o r a u t o c o r r e l a t i o n func-t i o n s of a r b i t r a r y c h a r a c t e r i s t i c s i n response to noise plus s i g n a l . 32 I l l A NON^LINEAR SYSTEMS MODEL  FOR HOLOGRAPHIC PROCESS 3.1. I n t r o d u c t i o n The F o u r i e r i n t e g r a l works w i t h any p h y s i c a l system i n which cause and e f f e c t are ' l i n e a r l y ' r e l a t e d ; and t h i s has been the main cause f o r b e a u t i f u l . j j a r a l l e l s between many problems i n o p t i c s and e l e c t r i c a l engineering. The p a r a l l e l s , however, have not been l i m i t e d to l i n e a r systems; an example being the treatment of re c o r d i n g media f o r l i g h t i n -t e n s i t y (e.g. a photographic p l a t e ) , i n the same way as a combination of l i n e a r and n o n - l i n e a r e l e c t r o n i c elements. D.H. K e l l y [21] suggested and used such analogy i n h i s three-stage model f o r the process of photography. K e l l y used the H u r t e r - D r i f f i e l d curve (a p l o t of photographic d e n s i t y versus the l o g a r i t h m of exposure) as the ' t r a n s f e r c h a r a c t e r i s t i c ' of the non- l i n e a r part of h i s model. For holography, i t i s more convenient to use a p l o t that w i l l provide a d i r e c t f u n c t i o n a l r e l a t i o n s h i p between amplitude transmittance and exposure. Kozma [4] used such a p l o t (t -E 3. curve) i n h i s c i r c u i t model f o r the process of holography. The input-output r e l a t i o n s h i p , o r ' t r a n s f e r c h a r a c t e r i s t i c ' o f the non - l i n e a r d e t e c t i n g medium i s of major importance i n the holographic pro-cess. Photographic emulsion i s commonly taken as the d e t e c t i n g medium, but t h i s i s by no means e s s e n t i a l ; i t i s only necessary that an i n p u t -output r e l a t i o n s h i p s i m i l a r to t (E) i s known f o r the detector i n use. The new model developed here allows n o n - l i n e a r e f f e c t s i n h o l o -graphy to be stud i e d w i t h few l i m i t i n g assumptions. This study t r e a t s the ' t o t a l exposure', E, as the input e to the no n - l i n e a r element g ( . ) . The t r a n s f e r c h a r a c t e r i s t i c of the n o n - l i n e a r i t y , g ( E ) , then, i s a f u n c t i o n 33 that best f i t s the t -E curve i n the p o s i t i v e h a l f - p l a n e and i s defined a f o r the f u l l range of i t s argument E > 0. (Perhaps i t i s h e l p f u l to be reminded that negative exposure i s n o n - e x i s t e n t , and i n (1*1), ^'(e^je^) = p ( E 1 , E 2 ) = 0 f o r E 1 and E 2 < 0, and that the l i m i t s of the i n t e g r a l w i l l be 0 and 0 0 ) . In that case, f ( s ) must be a one-sided Laplace (or F o u r i e r ) transform of the f u n c t i o n g ( E ) , and the problem of a g(e) = g(E') w i t h an undefined s e c t i o n i s circumvented. In other words, a half-wave non-l i n e a r model i s used f o r the t -E curve r a t h e r than a p a r t i a l l y f a b r i c a t e d full-wave c h a r a c t e r i s t i c about a b i a s p o i n t as used by o t h e r s , and i n doing so the accuracy of the model i s improved. The model a l s o w i l l pro-v i d e an extremely u s e f u l d i r e c t p a r a l l e l between holography and the pro-blem of n o n - l i n e a r d e t e c t i o n of signal(s)embedded i n noise of communica-t i o n theory. This d i r e c t analogy was p r e v i o u s l y obscured. 3.2. S t a t i s t i c s of a Coherently I l l u m i n a t e d D i f f u s e Object Most ob j e c t s have rough surfaces i n the v i s i b l e p o r t i o n of the electromagnetic spectrum. Many of these might be considered as c o n f i g -u r a t i o n s of very many i n d i v i d u a l s c a t t e r e r s w i t h . randqin p o s i t i o n s and o r i e n t a t i o n s . . The' surface of the object can be regarded as a c o l l e c t i o n of secondary point sources. Each source appears to emit a s p h e r i c a l wave of random phase.An object w i t h a very rough surface or a subject transparency backed by a d i f f u s i n g screen i s r e f e r r e d to as a " d i f f u s e o b j e c t 1 . The complex f i e l d f u n c t i o n of the l i g h t on the sur-face of a d i f f u s e object may be represented by, U(p) = 0(p) exp [ J i K p ) J ( 3 - D w i t h a d e t e r m i n i s t i c amplitude 0 ( p ) , but a random phase ty(p). p i s a 34 p o s i t i o n yector. The random phase of each, secondary source i s considered i n most cases to be uniformly d i s t r i b u t e d between 0 and 2ir or i n general, between some constant C and C + 2ir, i . e . {39, PP. 146-151]. pGjO = CC < ty < C + 2TT) (3.2) 'Furthermore, i f there i s a very large number of s c a t t e r e r s , we might also assume that the space phase f l u c t u a t i o n s for any two points of the object are s t a t i s t i c a l l y independent. On the basis of the above assumptions i t follows that, pGl^ ,^ ') = v(ty±) pOJ>2) = 7^  x 17 = O-j) bit The autocorrelation function of the d i f f u s e object f i e l d could now be obtained as, R o b j ( p 1 , P 2 ) = E { 0 ( p 1 ) e x p I j ^ ( p 1 ) J 0 ( p 2)expij . K p 2 ) ] } 2TT = 0 ( P l ) 0 ( p 2 ) ; ff exp j [ ^ ( P l ) - ij)(p 2)jdi|) 1di() 2 4ir 0 = 0 ( p 1)0 ( p 2)6 ( p 1 -,p 2) = T-ohi(P1)^(P1 ~ P 2) 0.4) 2 where 6(_.) i s the Dirac d e l t a function and I Q (p) = 0 (p). The assumption of the many scatterers and t h e i r independence enable the Central Limit Theorem to be applied : "Roughly speaking, the Central Limit Theorem states that whenever a random process can be represented by a l i n e a r superposition of a large number of e s s e n t i a l l y independent random e f f e c t s , i t s s t a t i s t i c s w i l l 35 a s y m p t o t i c a l l y approach normal Cor Gaussian).' 1 [40, PP. 362-366]' Therefore, i t f o l l o w s that the d i f f u s e object f i e l d f u n c t i o n has normal r e a l and imaginary components. The assumption of a uniformly d i s t r i b u t e d phase between 0 and 2TT makes these components independent [39, P. 123]. The mean -value of the f u n c t i o n i s assumed to be zero, s i n c e , E { 0 ( p i e x p I j K p ) J = 0(.p) x ^ f2^ exp(j^) d^ = 0 (3.5) 3.3. S t a t i s t i c s of F r e s n e l D i f f r a c t i o n F i e l d of a D i f f u s e Object From a 'systems' p o i n t of view the d i f f r a c t i o n of s c a l a r mono-,'chromatic waves by an object i s equivalent to the operation of a l i n e a r system, w i t h the f i e l d d i s t r i b u t i o n as the input and the d i f f r a c t i o n f i e l d as the output. For example, f o r F r e s n e l d i f f r a c t i o n the impulse response of the f i l t e r i s that of a s h i f t - i n v a r i a n t 'quadratic phase' f i l t e r [25], [41, PP. 199-200 and PP. 411-412]. That i s , o m i t t i n g a p r o p o r t i o n a l i t y f a c t o r , h(p;d) = exp(jup 2/Ad) (3.6) where d i s the d i s t a n c e of the output plane from the d i f f r a c t i n g aper-t u r e , X i s the wavelength of the i l l u m i n a t i o n . Let s c a l a r monochromatic f i e l d d i f f r a c t e d by a d i f f u s e object i n the F r e s n e l region be denoted by, Re{N(r)exp(jfit)} = Re{N(r ) e x p I j(j)(r)]exp(jftt) } = N(r)cosT.ftt + (|)(r)J (3.7) 36 where Re{.} denotes 'the r e a l p a r t ' (and Im{ . } w i l l s i g n i f y 'the imag-i n a r y p a r t ' ) , N(.r) i s the magnitude and (j)(r) I s the phase of the complex f i e l d N(r) , 0, i s the temporal r a d i a n frequency of the coherent wave, t i;s the time v a r i a b l e , and r i s the space v a r i a b l e over the observation plane z = d. Using the input-output r e l a t i o n s h i p s of the a u t o c o r r e l a t i o n f u n c t i o n s i n l i n e a r s h i f t - i n v a r i a n t systems {41, Ch.,8J one can r e a d i l y o b t a i n the f o l l o w i n g expression f o r the a u t o c o r r e l a t i o n of N ( r ) , ^ C r r r 2 ) = expl (x*-**)} h^) (3.8) where J ( . ) d e n o t e s the ' F o u r i e r transform of I , . ( p ) , and T = 2iTAr/Ad, obj r wi t h Ar = r ^ - . I f the s t a t i s t i c s of a random input to a l i n e a r system are nor-mal, the s t a t i s t i c s of the output are a l s o normal. Hence, the f o l l o w i n g conclusions can be made: N(r) has Gaussian r e a l and imaginary components [N(r)cos(()(r) and N(r)sin(()(r)] w i t h zero means, which are independent at any s i n g l e p o i n t . N(r) i s Rayleigh d i s t r i b u t e d and (|)(.r) i s uniforml y 2 d i s t r i b u t e d between 0 and 2ir. The va r i a n c e of N(r) , o" , i s equal to the mean i r r a d i a n c e of the f i e l d , a 2 = |J(0)| = const. (3.9) A somewhat d i f f e r e n t treatment of the above t o p i c , but i n a more elaborate and elegant f a s h i o n , i s given by Goodman [26] , to which the reader i s r e f e r r e d f o r a d d i t i o n a l d e t a i l s . 37 3 . 4 . S t a t i s t i c a l Properties of a Hologram of a Dif f u s e Object 3.4.1. Transmittanee Autocorrelation of a Linear Hologram To make a hologram of the object at plane z = d, one has to bring another coherent wave into interference with the object wave and record a time-averaged i n t e n s i t y of the r e s u l t i n g pattern on a recording medium (e.g. a photographic emulsion), placed at plane z = d (see F i g . 3 . 1 ) . Let the coherent i n t e r f e r i n g wave (reference wave) be a u n i -form plane wave incident at the recording plane at an anglev . n . We may denote t h i s wave by, Re{S(.r)exp(jfit) } = Refeexp(jar)exp(jfit)} =S cos(fit + ar) ( 3 . 1 0 ) where S i s a constant amplitude and, a = 2ir s x n n ( 2 . 1 1 ) The t o t a l f i e l d over the hologram plane, the time-averaged i n t e n s i t y of which i s to be recorded, thus becomes, N(.r)cos.[nt + (|)(r)] +S cos(Qt + ar) = V(r)cos[fit + v ( r ) ] where, V(r) = {S 2 + N 2 ( r ) + 2SN(r)cos.[ar - (|)(r)J } 1 / 2 and, ( 3 . 1 2 ) ( 3 . 1 3 ) v(.r) = tan { N ( r ) s i r i ( K r ) -S s i n a r N ( r ) c o s ( K r ) + S . c o s a r } ( 3 . 1 4 ) 38 .1. Arrangement for recording an o f f - a x i s hologram of a transparent object. 39 With an exposure time T, the exposure E w i l l be, E = TV 2 = T { S 2 + N 2 ( r ) + 2SN(r)cos{ar - (j)(r)J } (3.15) The assumption of a hologram being l i n e a r l y recorded i m p l i e s t h a t , g(E) = c'E = c'TV 2 = cV 2 (3.16) c where c' = TjT'is a constant f a c t o r . The a u t o c o r r e l a t i o n of the transmittance of such hologram w i l l be, V ( r l ' r 2 } = ° 2 { + P ' 2 ) 2 + '^' 2 + S 2 [ e x p ( j a A r ) R | ( r - ^ r ^ + e x p ( - j a A r ) R ^ ( r 1 , r 2 ) ] } (3.17a) or e q u i v a l e n t l y , R L H ( r l ' r 2 ) = c 2 { ( s 2 + " 2 ) 2 + ^ <Ar> + 2S 2 R N ( A r ) c o s I a A r - e ^ , ^ ) ] } (3.17b) where, R^(Ar) = i R ^ C r ^ r ^ l 2 = [Re{R^Cr 1,r 2)}J 2 + [Im{R^(r ; L,r 2) } ] 2 (3.18) and, e C r l S r 2 ) = t a n " 1 I m { V r l ' V 2 ) } . (3.19) 40 3.4.2. Analogy Between a Linear Hologram and a Square-Law Envelope  Detector Before going f u r t h e r , l e t us consider a problem i n s i g n a l d e t e c t i o n theory t h a t , as w i l l be shown, i s , from a mathematical p o i n t -of-view, e x a c t l y the same as the one j u s t discussed. Let the input to a half^-wave square-law device followed by an i d e a l low-pass f i l t e r be, x ( t ) = s ( t ) + n ( t ) (3.20) where, s ( t ) = S cos(o) t + at) (3.21) c w i t h a<<w , and c n ( t ) = N(t)cos(io t + ([>(t).] (3.22) c i s a narrowband normal process. As before, one may express x ( t ) i n terms of an envelope and phase, i . e . x ( t ) = V(t)cos{o) ct + v(.t)J (3.23) where 'V(t) and -v(.t) are, r e s p e c t i v e l y , given by (3.13) and (3.14), a f t e r r e p l a c i n g r w i t h t . At the output of the d e v i c e , we w i l l have, y ( t ) = gjx(.t)j =• 2 ax x > 0 (3.24) 0 • x < 0 where a i s a s c a l i n g constant. A f t e r some a l g e b r a i c manipulation, y ( t ) = J {S + N Ct) + 2SN(.t)cosIat - (j)(t)] + 41 + S 2cos2(w t + at) + N 2(t)cos2[w t + A ( t ) j c . c + SN(t)cos[2a) ct + at + §tt)]} (3.25) The i d e a l low-pass f i l t e r i s assumed to pass without d i s t o r -t i o n the low-frequency part of i t s input but to f i l t e r out completely the high-frequency components. Thus, at the output of the f i l t e r , we w i l l have, Z ( t ) = -| {S 2 + N2(„t) + 2SNCt)cos[at - (j)(t)]} = f v 2 (3.26) That i s , the cascade of a square-law device and an i d e a l low-pass f i l t e r i s a c t i n g as a square-law envelope d e t e c t o r . Furthermore, the s i m i l a r i t y of t h i s system to th a t of a l i n e a r hologram i s obvious; that i s , w i t h i n a constant f a c t o r , and aside from an a r t i f i c i a l d i f f e r e n c e of the symbols used to denote the independent v a r i a b l e s i n the two cases (r f o r space and t f o r t i m e ) , the. output of the square-law envelope detector i n response to x ( t ) i s equal to the transmittance of a l i n e a r hologram [cf.(3.26) w i t h (3.15) and (3.16)]. The constant f a c t o r i s 4c/a. The a u t o c o r r e l a t i o n f u n c t i o n of Z ( t ) could be obtained w i t h the a i d of (2.45) - (2.47), (2.29), and (2.12) w i t h v± = 2 and a 2 = a 2/2: oo H 2 9 n Ok k R Z ^ 1 ' V = H00 + I 7 i ^ 2 k V W k=2 L2'} (k even) k TT2 co H + l I — Rjj C t 1,t 2)cosm(aAt --0) , , .' ,k+m. ,/k-m. , „2k^l k=l ; -m==l (-y-) (-~2-) ! 2 (m+k even) (3.27) 42 where, H i (S/cx) 1 F 1 [ v — 2 — . ; m+1; . - .- | J m k Tn!(.or/2) k 2 rll^Cm+k-2)/2;] (3.28) Since | i'(-n) | - 0 0 f o r n = 0,1,2, , the c o e f f i c i e n t vanishes f o r m+k > 2. Hence, H z ( . t 1 , t 2 ) - r 6-2 2 X F 2 t - l ; l ; - \ ) o-4 + R ^ ( A t ) 1 F 1 ( 0 ; l ; - ^ >' a" a 2 S 2 + 2S R NCAt)cos(aAt - 0) F 1(0;2; - ) a 2 a IT Cl+^2 ><c t 2) 2 + ^ CAfc) + 2 S 2 R N ( A t ) c o s ( a A t - 0) a • (3.29) which, i s , w i t h i n a constant f a c t o r and a f t e r r e p l a c i n g t w i t h r , equal to the a u t o c o r r e l a t i o n of the transmittance of a l i n e a r hologram. The 2 2 2 constant f a c t o r i s 4 c /a , as expected. A more general analogy between holography and n o n - l i n e a r d e t e c t i o n of s i g n a l s and noise w i l l be given s h o r t l y . But, before t h a t , the r o l e of the transmittance a u t o c o r r e l a t i o n i n the p r e d i c t i o n of h o l o -graphic image i n f o r m a t i o n i s to be understood. 3.4.3. Hologram Imaging Properties via Autocorrelation Function Comparing the equations f o r the transmittance of a l i n e a r hologram and i t s a u t o c o r r e l a t i o n f u n c t i o n {cf. (3.15) and (3.16) w i t h (3-17)J shows a c l o s e resemblance between the two; each term i n one has an analogous term i n the other, from which almost the same in f o r m a t i o n can be d e r i v e d . For example, the o s c i l l a t i n g term i n both equations pre-43 d i e t s two r e p l i c a s of the object on the planes z = d and z = -d, when r e c o n s t r u c t i n g the hologram w i t h a plane wave (e.g. at normal angl e ) . More e x p l i c i t l y , by using the input-output r e l a t i o n s h i p s of the auto-c o r r e l a t i o n f u n c t i o n s i n l i n e a r s h i f t i n v a r i a n t systems (see the d i s c u s s i o n i n S e c t i o n 3.3 ), one could show that the c o n t r i b u t i o n of such a term as exp(jaAr) to the a u t o c o r r e l a t i o n of the f i e l d d i f f r a c t e d by the hologram at plane z = d, i s given by, R r i ( p l ' p 2 ) = V j ^ l " Y ) 6 ( P 1 " P 2 } ( 3 ' 3 0 ) where s u b s c r i p t r i stands f o r " r e a l image". Equation (3.30) i m p l i e s t h a t , at the plane z = d, the c o n t r i b u t i o n of the term exp(jaAr) i s a r e p l i c a of the object i n v e r t e d and s h i f t e d sidewise by y = aXd/2ir [ c f . ( 3 . 4 ) ] . A s i m i l a r operation over R^ j e x p ( - j a A r ) , however, w i t h negative d i n the impulse response of (3.6), y i e l d s , R v i ( P l » P 2 ) = W P 1 " Y > 6 ( P 1 - P 2 } ' ( 3 ' 3 1 ) that i s a r e p l i c a of the object s h i f t e d side-wise by y. Note the d i f f r a c t e d f i e l d i s t r a v e l l i n g i n the p o s i t i v e z d i r e c t i o n . Therefore, to have an image at the plane z = -d i m p l i e s that the image i s v i r t u a l (hence, the s u b s c r i p t v i f o r ' v i r t u a l image'). I f we evaluated the c o n t r i b u t i o n of R^ exp(-jaAr) at the plane z = d, we would get, 2exp ( p 2 - p22)] exp(jaAp) J (4|) , (3.32) 2 which has a mean i r r a d i a n c e equal to 2a . The e f f e c t of (3.32) i s the r e d u c t i o n of the r e a l image c o n t r a s t . I t can a l s o be shown that the second term i n (3.17) i s the ana-44 logue of the ambiguity term [41, P. 41'OJ , Isecond term i n (3.15)J. This term gives r i s e to an image, the c o r r e l a t i o n f u n c t i o n of which i n the Fraunhofer r e g i o n i s , 1 ^ I o b j ( p 1 + x ) 6 ( P 2 - p 2 ) ' (3.33) that i s , an image centered on the o p t i c a l a x i s w i t h an i r r a d i a n c e d i s -t r i b u t i o n equal to an a u t o c o r r e l a t i o n (or a u t o - c o n v o l u t i o n , s i n c e I i s a p o s i t i v e r e a l f unction) of the object i r r a d i a n c e . obj 3.4.4. . Autocorrelation of Transmittanee of a Non-linear Hologram: R^ir^r^) When the ' t o t a l exposure', E, i s taken as the ' i n p u t ' , e, to the r e c o r d i n g medium's n o n - l i n e a r i t y , the i n t e g r a l (1.1a) that gives R ( r ^ j r ^ ) w i l l be , CO R Q ( r 1 , r 2 ) = // g ( E 1 ) g ( E 2 ) p ( E 1 , E 2 ) d E 1 d E 2 (3.34) A power^series expansion of g(E^) i n (3.34) g i v e s , OO CO where R (r.. , r 0 ) = E n E n a a I o 1 2 p=0 q=0 p q pq E 1 P E 2 q P ( E 1 ' V d E l d E 2 p q i j (3.35) (3.36) and where a i s the c o e f f i c i e n t of X i n the expansion of g(X). I f the n expansion i s a M a c l a u r i n s e r i e s expansion, then, i = h- ^ — i g ( x ) j n n! d x n (3.37) X = 0 Using the t r a n s f o r m a t i o n E = TV , the i n t e g r a l (3.36) may be transformed t o , 45 T = T P+q 0 v 2 p Y2^ p(y, ,y 7 ) . dy. d y . (.3.38) pq n ± z J-Z J. z where, P(V 1,V 2) = 4f 2 ^ P(E 1,E 2) . (142], PP. 307-8) (3.39) F i r s t l e t 2p = 2q = v . 1 w i l l be * H pq I = T V / / V V P( Vi »Vo) d V i d V 9 ( 3- 4 0> vv o 1 2 1 2 l z But I could be thought of as the autocorrelation of the output of a half-wave vth-law device with V as i t s input, i . e . 0 V < 0 (v. even) This can be stated d i f f e r e n t l y i f a vth-law detector (a vth-law device followed by a low-pass zonal f i l t e r ) i s used as the model, rather than a vth-law device. That i s , I gives (within a constant factor) the auto-vv c o r r e l a t i o n function of a vth-law detector with the following as i t s input, x(r) = V(r)cos[o) r + v ( r ) ] = N(r)cosru) r + (j)(r)] +s*"cos(w r + ar) c c ' c (3.42) where <D i s an a r b i t r a r y c a r r i e r frequency so chosen that the v a r i a t i o n s of N(r) , (|)(r) and cosar are slow compared to those of cosco^r". C w c i s a purely f i c t i t i o u s quantity t h a t . i s introduced s o l e l y for mathematical purposes; i t does not have any phys i c a l s i g n i f i c a n c e . ) With N(r) being Rayleigh d i s t r i b u t e d , and (|)-(-r) being uniformlyedistributed^between 0 and 2TT, x(r) w i l l be the sum of a narrowband normal process and a sinusoid. 46 The output of a half-wave vth-law device to t h i s input i s , CO y ( r ) = £ C(v,m) V V ( r ) cos[mu>cr + m v ( r ) ] , (3.43) m=0 (.[22], P. 283, Eq. (13.22)) where, e aT(v+l) , Q / / N m , (3.44) C(v,m) = ([2 2 ] , P. 286, Eq.(13.36)) e i s the Neumann factor,and a i s - a s c a l i n g constant.The•low-pass f i l t e r , m then, gives at i t s output, Z ( r ) = C(v,0) V V ( r ) (3.45) ( [ 2 2 ] , P. 287, Eq.(13.37a)) w i t h , C,(v,0) = r ( v + 1 ) = ^ — (3.46) 2 v + 1 r 2 ( i + f ) " 2 V + 1 ( | ! ) 2 This i s i n accordance w i t h our e a r l i e r statement that the e v a l u a t i o n of I [see (3.40)] i s ( w i t h i n a constant f a c t o r ) equivalent to the e v a l -u a t i o n of the a u t o c o r r e l a t i o n f u n c t i o n of the output of a vth-law detec-t o r . This analogy i n con j u n c t i o n w i t h the r e s u l t s of Chapter I I ( i . e . the a p p l i c a t i o n s of the mathematical procedures f o r vth-law n o n - l i n e a r i t i e s to a r b i t r a r y c h a r a c t e r i s t i c s ) provides the f o l l o w i n g model f o r a d i r e c t e v a l u a t i o n of R ( r , ,r„): o 1 2 1. A power-series approximation f o r t (E) i s employed, e.g. a a polynomial i n exposure N t (E) = T a E n (3.47) a L n n=0 47 2. E i s replaced by x and the r e s u l t i s taken as the t r a n s f e r c h a r a c t e r i s t i c g(x) of a non- l i n e a r d e t e c t o r , e.g. g(x) = I a * n = I a '-x n=0 v=0 (v even) ( 3 . 4 8 ) 3. Equations (2.45) - (2.47) i n conj u n c t i o n w i t h (2.29) and (2.12) provide the c o r r e l a t i o n f u n c t i o n of the output of the detector i n response to x ( r ) o f (3.42), which i s then d i v i d e d by the f a c t o r s , [C(2p,0)C(2q,0)/T P + q] to give the f o l l o w i n g general expression f o r R ( r ^ , ^ ) 2q OO CO p=0 q=0 2 P 2q k=2 (k even) k=l m=l [ (k-hn)even] J 2H . R* (Ar)cos[m(aAr - . 6 ) ] *• pqkm 1ST ( 3 . 4 9 ) where, H pqkm A T P ^ ( P I ) 3 ( Q ! ) 3 ( S 2 / . 0 2 ) » F . m + 1 ; - ^ ) F ( H ^ ; m f l ; - ^ ) pq X x z /g^ x x z 0 ( m ! )2 (k+m ) ! (k=m ) ! r ( 1_ E ± ^ 2 £ ) r ( 1 _ 2 A = a a -pq p q' 2p 2q> (3.50) means '2p or 2q whichever s m a l l e r ' , and (;;) denotes the confluent hypergeometric function defined by the series, °° ( y ) .u1 .- , i v 2 1 ? 1 (y;g;u) = ±Z o ci)~lT - 1 + g l ! + g(g+D 2! - t < 3 ' 51) 4 8 Note that when u i s a negative i n t e g e r = -n, F^ w i l l terminate a f t e r n + 1st term. For t (E) = c'E, that i s when a hologram i s assumed to be Si l i n e a r l y recorded, c' p = q = 1 a = a p q (3.52) otherwise, and hence, H , w i l l have a non-zero value only when p = q = 1. In that pqmk J case, (3.49) reduces to (3.17), the r e s u l t e a r l i e r obtained f o r the auto-c o r r e l a t i o n of a l i n e a r hologram. 3.4,5. Physical Interpretation of the Terms Involved in R (r^r^) I f we express the transmittance of the hologram i n terms of i t s mean and i t s random part as, t (r) = T + 5(r) (3.53) C l CL the a u t o c o r r e l a t i o n of t (r) may be w r i t t e n as, a. R o ( r l ' r 2 ) = T a + ( 3 * 5 4 ) where, R 5 ( r 1 5 r 2 ) = E { 5 1 5 2 } . (3.55) A comparison of (3.54) w i t h (3.49) shows that the f i r s t term i n (3.49) i s the square of the transmittance mean va l u e , i . e . T2 = I I A P ! q ! a 2 ( p + ^ - F r - p ; ! ; - ^ F ( - q ; l ; - \ ; a p=0 q=0 M 1 1 ° 2 1 1 ° 49 which i s r e s p o n s i b l e f o r the u n d i f f r a c t e d p o r t i o n of the tran s m i t t e d wave i n the r e c o n s t r u c t i o n process (e.g. the b r i g h t spot on the o p t i c a l a x i s ) . In a d d i t i o n , the a u t o c o r r e l a t i o n f u n c t i o n of the random part of t r a n s -mittance may be expressed as, V V V " W ' l ' V + Rnxs< rl>V <3'57> where R n x n i s given by the second part i n (3.49) and i t represents that part of transmittance that i s due to the i n t e r a c t i o n of the object wave wi t h i t s e l f . R n x s i s given by the t h i r d part of (3.49) and i t represents that p a r t of transmittance that i s due to the i n t e r a c t i o n of the object wave w i t h the reference wave. A comparison of (3.17) w i t h (3.49) shows that each term i n the input (exposure) has i t s analogue i n the output ( t r a n s m i t t a n c e ) . In other words, the e f f e c t of the n o n - l i n e a r i t y i s the generation of some e x t r a terms f o r each i n d i v i d u a l term i n the input exposure. For k = 1, R nxs gives the usual r e a l and v i r t u a l images, w h i l e f o r k > 1, i t generates d i s t o r t e d images. Some of these d i s t o r t e d images are l o c a t e d i n the same d i r e c t i o n as the f i r s t (those f o r which k > 1, but m = 1) and, t h e r e f o r e , by overlapping could cause d i s t o r t i o n of the f a i t h f u l images. The order of k gives the width of the corresponding image r e l a t i v e to that of the f a i t h f u l image. For example, a term such as R^(Ar)cos(aAr - 0 ) , where „•. -k = 3 and m = 1, w i l l be r e s p o n s i b l e f o r a d i s t o r t e d image i n the same d i r e c t i o n as the f a i t h f u l image due to R^(Ar)cos(aAr - 0 ) , but 3 times l a r g e r . These terms f o r which m = 1, but k = 3,5, ... t 2 q ] ~~ ^ ' c a u s e a background h a l o - l i k e n o i s e that could be troublesome when the v a r i a t i o n of exposure i s l a r g e due to a very strong object wave [ c f . F i g . 3.2b. w i t h 3 .2d] The d i f f r a c t i o n order of the images i s i n d i c a t e d by m. For example, 50 (b) (c) F i g u r e 3.2. R e c o n s t r u c t e d images o f n o n - l i n e a r s i n g l e - r e f e r e n c e - b e a m h o l o g r a m s . a) Oth h a r m o n i c , b) 1 s t and 2nd h a r m o n i c s , c) 2nd harmonic e n l a r g e d and more i n f o c u s , d) 1 s t and 2nd h a r m o n i c s o f a h o l o g r a m r e c o r d e d w i t h s e v e r e n o n - l i n e a r i t y . 51 the f a i t h f u l images and t h e i r background halos c o n s t i t u t e the f i r s t order images ( f i r s t harmonics, m = 1). In the same manner, second-order images could be considered as the t o t a l e f f e c t of a l l terms f o r which m = 2 (but k = 2,4, ... [~P] - 2) [ F i g . 3.2c], and so on. I t i s worth mention-zq i n g that a p a r t i a l overlap of these harmonic images could occur i f the separation angles between the reference and object waves are not l a r g e enough. (See a l s o Chapter V). I f the r e c o n s t r u c t i n g plane wave i s i l l u m -i n a t i n g the hologram at a normal angle, the images w i l l be centered on axes e n c l o s i n g the angles subtended by my(m = 1,2,3 ...) ( F i g . 3.3). Therefore, i f the separation angles are l a r g e enough, the background noise of the usual r e a l and v i r t u a l images w i l l be due only to the m u l t i p l i c i t y of the f i r s t harmonic. For k = 2, R gives an image, the i r r a d i a n c e d i s t r i b u t i o n nxn of which i s an autoconvolution of the object i r r a d i a n c e . S i m i l a r l y , f o r any value of k, R gives an image w i t h an i r r a d i a n c e d i s t r i b u t i o n equal nxn to a (k - 1) - f o l d c o n v o l u t i o n of the object i r r a d i a n c e ( i n the Fraunhofer r e g i o n ) . Thus, the image due to R , which i s centered on the o p t i c a l ° ° nxn a x i s (m = 0 ) , i s the r e s u l t a n t e f f e c t of a l l the images due to k = 2,4,6 .. [ 2 p ] . This image i s the zeroth harmonic ( F i g . 3.2a). zq F i n a l l y , l e t us consider some commonly employed f u n c t i o n s f o r t ( E ) , and evaluate the c o e f f i c i e n t s A . Further d i s c u s s i o n s on R (r,..,xn) • ' pq o 1' 2 are deferred to Chapter V, where comparisons are made between s i n g l e - vs m u l t i p l e - reference beam holography. Examples f o r t(E) and the Corresponding A^ n, (a) Gaussian t (E) 3. Let the t -E curve be approximated by [13] 2 OPTICAL AXIS, IMAGE DUE TO RjJ(fcr) AMD MEAN TRANSMITTAMCB VIRTUAL IMAGE DUE TO (Ar) co» (C^r VIRTUAL IMAGE . * DUE TO R^(Ar)co«(2c*Ar-2©l 1 Figure 3 . 3 . Hologram imaging properties v i a R (r,,r»). o J- 2 Harmonic images are centered on axes enclosing angles subtended by m(a Xd/2Tr), m=0,l,2,... 53 E t (E) = exp( r) a b 2 E > 0 (3.58) A i n t h i s case w i l l be, pq pq ( - I / O (p+q)/2 •(p/2).:(q/2>! f o r p and q both even otherwise (3.59) (b) L o r e n t z i a n t (E) • a Another example of p r a c t i c a l use could be the f u n c t i o n [12] t (E) = g 2/(E 2 + e2) , E > 0 a (3.60) where g i s a parameter used to f i t t - E to the experimentally derived cl data. For A i n t h i s case, one f i n d s , pq A =H pq ( _ 1 / g y p + q ) / 2 For both p and q even otherwise (3.51) (c) A polynomial approximation I f we used a polynomial curve f i t t i n g i n the form [6,7] N t (E) = y a E (3.52) a L n n=0 the c o e f f i c i e n t s A would be pq 54 A pq a a f o r p and q < N P q (3.63) otherwise (d) Sum of Laguerre polynomials To increase the accuracy of the commonly used polynomial approx-imation f o r t - E curve, V e l z e l [15] proposed the f o l l o w i n g f u n c t i o n , which 3. i s a sum of Laguerre polynomials i n the form, t (E) = exp(-yE) £ k L (yE). (3.64) a. IT Ii n As a s p e c i f i c example of t h i s general model, he gave a crude numerical approximation as t (E) = (1 + 2.1 E)exp(-2E). (3.65) c l For t h i s f u n c t i o n A w i l l be, pq A p q = ( - 2 ) P + q ( l - 1 . 0 5 p ) ( l - 1 . 0 5 q ) / p ! q ! (3.65) (e) L i n e a r phase hologram The above approximations were only concerned w i t h amplitude transmittance vs exposure c h a r a c t e r i s t i c s . In p r a c t i c e , however, the t r a n s f e r c h a r a c t e r i s t i c might be a complex f u n c t i o n , to take i n t o account the phase s h i f t s impressed by the device on the input (e.g. as a consequence of v a r i a t i o n s of f i l m t h i c k n e s s [24]). An example of t h i s i s a l i n e a r phase hologram, f o r which the t r a n s f e r c h a r a c t e r i s t i c may be shown by an exponential [l1,14,15] t(E) = exp(-AE), E > 0 (3.67) where A i s a complex number w i t h a p o s i t i v e r e a l p a r t , i . e . 55 A = g + j f w i t h g > 0 (3.68) A i n t h i s case w i l l be pq A p q = ( " ^ ) P ("^* ) q ' P ! q ! ( 3 ' 6 9 ) 3.5. D i s c u s s i o n of the Assumptions In the foregoing a n a l y s i s , i t was assumed that the hologram emulsion recorded a l l the s p a t i a l frequencies of the input exposure. This might be reasonable f o r a l a r g e area emulsion of a very f i n e g r a i n s t r u c t u r e . Otherwise, the b a n d - l i m i t i n g e f f e c t s of the medium due to i t s f i n i t e aperture and l o s s of modulation caused by a frequency dependent o p t i c a l d i f f u s i o n , must a l s o be accounted f o r . This n e c e s s i t a t e s a mod-i f i c a t i o n of the model of a n a l y s i s . A reasonable m o d i f i c a t i o n i n t h i s case seems to be the a d d i t i o n of a band-pass f i l t e r centered at w^, preceding the n o n - l i n e a r d e t e c t o r . The t r a n s f e r f u n c t i o n of the f i l t e r i s given by the product of the p u p i l f u n c t i o n of the r e c o r d i n g medium wi t h i t s modulation t r a n s f e r f u n c t i o n [21; and 24, Sec. 8-5], The input to the n o n - l i n e a r d e t e c t o r of the model of a n a l y s i s , then, w i l l be the output of the band-pass f i l t e r i n response to x ( r ) ; t h i s output remains as a sum of a narrowband normal process plus s i n u s o i d ( s ) . We d i d not consider the 'adjacency e f f e c t s ' . During the develop-ment, the formation of the changes i n the o p t i c a l p r o p e r t i e s of the medium at a c e r t a i n point i s determined not only by the exposure at that p o i n t , but a l s o by the exposure at adjacent regions. K e l l y [21], i n h i s three-stage model of photographic process, suggested that the ad-jacency e f f e c t s are, f o r the most p a r t , l i n e a r , and thus could be r e p r e s -ented by the operation of a l i n e a r f i l t e r . Therefore, more accurate ex-pressions f o r the a u t o c o r r e l a t i o n f u n c t i o n of the hologram transmittanee 56 may be obtained i f the i n f l u e n c e of t h i s l a t t e r f i l t e r were a l s o taken i n t o account. For a very t h i n hologram developed i n a concentrated dev-eloper w i t h constant a g i t a t i o n , however, the adjacency e f f e c t s become n e g l i g i b l e {21]. In t h i s chapter the detector's n o n - l i n e a r i t y i s assumed con-tinuous and the complex f i e l d d i f f r a c t e d by the object over the h o l o -gram plane i s assumed a zero-mean c i r c u l a r complex normal process [26]. O c c a s i o n a l l y , n e i t h e r assumption may be v a l i d but such cases can be cover-ed by a d i r e c t extension of the r e s u l t s from Chapter I I i n conj u n c t i o n w i t h [36]. L a s t l y , the i n t e r a c t i n g wave amplitudes over the hologram plane have been tr e a t e d as s c a l a r q u a n t i t i e s ; and any change of p o l a r i -z a t i o n i n the r e f l e c t e d l i g h t from the object has been ignored. In the next chapter, the model w i l l be extended to cover n o n - l i n e a r e f f e c t s i n holography when a c r o s s - p o l a r i z e d component i s present i n the r e f l e c t e d l i g h t from the obj e c t . 3.6. Summary By a d i r e c t a p p l i c a t i o n of t h e ' c h a r a c t e r i s t i c f u n c t i o n ' method of communication theory, a ready-made general expression was ob-tained f o r the transmittance a u t o c o r r e l a t i o n of holograms of d i f f u s e o b j e c t s , recorded on t h i n and f i n e - g r a i n emulsions. The expression (3.49) i s general i n the sense that the i r r a d i a n c e s of reference and o b j e c t , the type of the hologram, and the nature of the recording medium's n o n - l i n -e a r i t y are taken to be a r b i t r a r y . This expression can be d i r e c t l y used to acquire some informa t i o n about the i r r a d i a n c e d i s t r i b u t i o n over the image plane. The e f f e c t of the n o n - l i n e a r i t y i s the generation of d i s -t o r t e d m u l t i p l e images. Some of these d i s t o r t e d images are l o c a t e d i n 57 the same d i r e c t i o n as the f a i t h f u l ones, and thus produce n o n f i l t e r a b l e background noise. While the m u l t i p l i c i t y of the images i s due to the gen-eration of higher harmonics of the reference-object intermodulation term, the background h a l o - l i k e noise of the f a i t h f u l images i s due to the mul-t i p l i c i t y of t h e - f i r s t harmonics themselves. 58 IV DEGRADATION OF HOLOGRAPHIC IMAGES DUE TO  DEPOLARIZATION OF REFLECTED LIGHT 4.1. I n t r o d u c t i o n In holography, the i n t e r a c t i n g wave amplitudes over a hologram plane are u s u a l l y considered as s c a l a r q u a n t i t i e s . In other words, the i l l u m i n a t i n g beam i s considered to be l i n e a r l y p o l a r i z e d and the question of whether the p o l a r i z a t i o n of the s c a t t e r e d f i e l d i s d i f f e r e n t from that of the i l l u m i n a t i n g beam i s ignored; i . e . i t i s supposed that the s c a t t e r e d f i e l d i s a l s o l i n e a r l y p o l a r i z e d and p a r a l l e l to the reference wave. Under these assumptions, the v e c t o r form of the time-averaged i r r a d i a n c e of the i n t e r f e r e n c e of two coherent waves, i . e . I ( r ) = S ( r ) - S * ( r ) + N ( r ) - N * ( r ) + S ( r ) - N * ( r ) + S * ( r ) - N ( r ) , (4.1) has been w r i t t e n i n the s c a l a r form I ( r ) = S 2 + N 2 ( r ) + S ( r ) N * ( r ) + S*(r)N(r) = S 2 + N 2 ( r ) + 2SN(r)cos [ a r - J J K r ) ] (4.2) where N(r) = a^, N(r) = a^N(r)exp[j(])(r) ] i s the complex vector of the ob-j e c t ' s s c a t t e r e d f i e l d at the hologram plane, S(r) = a g S ( r ) = a g S e x p ( j a r ) i s the complex f i e l d v e c t o r of a plane reference wave i n c i d e n t on the hole gram plane at an angle n = s i n ^(aA/2iT), a^ and a g are the u n i t v e c t o r s , (j)(r) i s the object's phase f u n c t i o n , the arrow shows that the f u n c t i o n i s a vector f u n c t i o n , the t i l d e shows that i t i s complex valued; absence of these signs shows that the f u n c t i o n i s a r e a l s c a l a r f u n c t i o n ; e.g. N(r) w i l l mean the magnitude of the complex f u n c t i o n N ( r ) . The assumptions which y i e l d (4.2) are seldom j u s t i f i e d and the exact s o l u t i o n of (4.1) should be used, s i n c e the s c a t t e r e d f i e l d , i n gener a l , i s not l i n e a r l y p o l a r i z e d and/or p a r a l l e l to the i n c i d e n t f i e l d . The purpose of the f o l l o w i n g i s to study some of the probable e f f e c t s of these changes i n p o l a r i z a t i o n on the reconstructed images. 4.2. D e p o l a r i z a t i o n of . i l l l u m i n a t i o r i ,and Loss of Information We l e t the source beam be l i n e a r l y p o l a r i z e d , whereas the sc a t t e r e d complex f i e l d vector N(r) i s decomposed i n t o two orthogonal components a N (r ) and a N (r) p o l a r i z e d p a r a l l e l and normal to the f i e l d p p c c ->- -> ~ S(r) = a ^ S ( r ) . Then,(4.1) may be w r i t t e n i n the f o l l o w i n g form: I ( r ) = S 2 + N 2 ( r ) + N 2 ( r ) + 2SN ( r ) c o s [ a r + (t) (r) ]. (4.3) p c p P For an exposure time T, the t o t a l exposure i s given by E(r) = T I ( r ) = E p ( r ) + E c ( r ) ( 4 > 4 ) 2 where E (r ) = TN (r ) and E (r ) denotes the product of T with the remain-c c p 2 der of (4.3). As can be seen, the background noise power TN (r) = 2 2 T[N (r) + N ( r ) ] remains the same w h i l e the s i g n a l 2TSN (r)cos[ctr - (t) ( r ) ] p c P P i s reduced i n power w i t h respect to the case i n which the s c a t t e r e d f i e l d was not de p o l a r i z e d (decrease i n f r i n g e v i s i b i l i t y ) . Furthermore, i t has been s t a t e d by some authors [43,44], that i n the b a c k s c a t t e r i n g of l i n e a r -l y p o l a r i z e d waves at an extended o b j e c t , the s c a t t e r e d f i e l d of p a r a l l e l p o l a r i z a t i o n i s determined mainly by the segments of the surface of the object having small c u r v a t u r e s , whereas the cross pol-arjiz at i o n component of the s c a t t e r e d f i e l d i s caused by the s c a t t e r i n g o f the waves at seg-ments of the surface w i t h l a r g e curvatures (edges, d i s c o n t i n u i t i e s ) [44]. 60 This i s by no means the only mechanism r e s p o n s i b l e f o r the d e p o l a r i z a t i o n of the s c a t t e r e d f i e l d . But, w i t h such a d e p o l a r i z a t i o n process, the r e -constructed images w i l l c o n t a i n only segments w i t h small curvatures; the i n f o r m a t i o n about the segments w i t h l a r g e curvatures, that are c a r r i e d by the c r o s s - p o l a r i z e d component, w i l l be l o s t . This i s due to the f a c t that two waves which are p o l a r i z e d i n mutually perpendicular d i r e c t i o n s cannot i n t e r f e r e [45, 46], Although l o s s of i n f o r m a t i o n from c r o s s -p o l a r i z a t i o n may be r a r e i n o p t i c a l holography, t h i s d i s c u s s i o n i s meant to point out a p o s s i b l e mechanism f o r image degradation i n holography. The above-mentioned degradation e f f e c t s can be almost e l i m i n a t e d simultaneously, simply by a proper r o t a t i o n of the l i n e a r l y p o l a r i z e d reference wave, which i n e f f e c t assigns a part of the reference power to record the i n f o r m a t i o n c a r r i e d by the cross p o l a r i z e d component. The proper angle of the r o t a t i o n i s determined by the power r a t i o of the two components of the object f i e l d . The best f r i n g e v i s i b i l i t y i s obtained when the corresponding components of the object and reference wave have the same power. [The f r i n g e v i s i b i l i t y i s given by FV = (E - E . ) / b ° max mm (E + E . ). FV w i l l have i t s maximum value (=1) i f E . = 0 , i . e . max mxn min when the strengths of the two i n t e r f e r i n g beams are equal.] Note that t h i s procedure w i l l work best w i t h evenly s t r u c t u r e d o b j e c t s , where the d e p o l a r i z a t i o n c h a r a c t e r i s t i c s of the regions under i l l u m i n a t i o n are a l -most the same. For objects w i t h s e c t i o n s of unequal d e p o l a r i z a t i o n char-a c t e r i s t i c s , only those s e c t i o n s f o r which the high f r i n g e v i s i b i l i t y con-d i t i o n i s f u l f i l l e d w i l l have improved image, w h i l e the image of some other s e c t i o n s might be degraded. For the l a t t e r o b j e c t s the above-men-tioned procedure might s t i l l provide a reasonable compromise. 61 4.3. Experimental Results A p a r t i a l i n s i g h t i n t o the above-mentioned t h e o r e t i c a l expec-t a t i o n s may be gained by c o n s i d e r i n g the f o l l o w i n g observations. Figure 4.1 shows a l a s e r i l l u m i n a t e d m e t a l l i c s t a r , which was the object of our experiments. Each t i p of the s t a r has two s e c t i o n s , one w i t h grooves and the other smooth. The smooth-parts of the lower t i p s are dark, s i n c e the specular r e f l e c t i o n s of these p a r t s were beyond the reach of the ca-mera. (For the same reason the grooves appear as dark and b r i g h t bands). The parts which appear through the c i r c u l a r opening i n the middle of the s t a r and between i t s lower t i p s are parts of a support which t o t a l l y d i f -fused the i l l u m i n a t i o n . Figure 4.1b shows the p a r a l l e l - p o l a r i z e d and Figure 4.1c the c r o s s - p o l a r i z e d component. I t i s c l e a r that w h i l e the smooth s e c t i o n s send only p a r a l l e l - p o l a r i z e d l i g h t , the grooved p a r t s and the support are sending l i g h t of both p o l a r i z a t i o n s . A comparison of Figures 4.1a and 4.1b suggests that f o r t h i s object the p a r a l l e l - p o l a r i z e d component shares nearly' a l l the i n f o r m a t i o n c a r r i e d by the c r o s s - p o l a r i z e d component, since f i l t e r i n g t h i s component i n Figure 4.1b does not a f f e c t the appear-ance of the object to a l e v e l e a s i l y d e t e c t a b l e by the eye. However, the problem of the decreased s i g n a l - t o - n o i s e r a t i o due to the change of p o l -a r i z a t i o n of i l l u m i n a t i o n remains to be resolved. This could be done by r o t a t i o n of the reference wave to a proper angle, as suggested at the end of Section 4.2. Figures 4.2a - 4.2d show the e f f e c t s of using a r o -tated reference wave on holographic image q u a l i t y . The compromised pro-per angle i n t h i s case seemed to be around 30°(Fig. 4.2b), as i n c r e a s i n g the angle d i s t r i b u t e d the power of the reference wave unevenly between the two components of the object wave, r e s u l t i n g i n a decrease i n the (a) Both p o l a r i z a t i o n components present. (c) (d) Figure 4.2. E f f e c t s of using a r o t a t e d reference beam on holographic image q u a l i t y . The angles of the r o t a t i o n were, f o r ( a ) 0°, (b) 30° , (c) 60°, and (d) 90°. 64 f r i n g e v i s i b i l i t y . The angles of r o t a t i o n f o r the holograms that pro-duced images of Figures 4.2c and 4.2d were 60° and 90°, r e s p e c t i v e l y . A comparison of Figures 4.1c and 4.2d shows that a r o t a t i o n of 90° pro-duces a hologram of the c r o s s - p o l a r i z e d component, where the i n f o r m a t i o n c a r r i e d only by the p a r a l l e l - p o l a r i z e d component i s l o s t , that i s , a sim-p l e c o n f i r m a t i o n of the f a c t that two waves which are p o l a r i z e d i n mut-u a l l y perpendicular d i r e c t i o n s cannot i n t e r f e r e . Figures 4.2a - 4.2d could a l s o simulate a dramatized case i n which the s t a r i s imagined to have the r o l e s of i t s two components interchanged. With such an object a normal hologram w i l l have the poor image of Figure 4.2d and a r o t a t i o n of the reference wave through 60° improves the image d r a s t i c a l l y . Although the above experiments were done i n the o p t i c a l r e -gion, s i m i l a r e f f e c t s have a l s o been observed i n microwave holography [47]. The two degradation e f f e c t s mentioned above are not the only p o s s i b l e e f f e c t s , but as w i l l be shown i n the f o l l o w i n g s e c t i o n , when the hologram i s n o n - l i n e a r l y recorded, the d i s t o r t i o n s of the images due to the d e t e c tor's n o n - l i n e a r i t y are f o r t i f i e d i f a c r o s s - p o l a r i z e d com-ponent i s present. 4.4. Detector's N o n - l i n e a r i t y and D i s t o r t i n g E f f e c t s of the Cross- P o l a r i z e d Component The a p p l i c a t i o n of the a u t o c o r r e l a t i o n f u n c t i o n of the h o l o -gram transmittance i n the study of n o n - l i n e a r e f f e c t s i n holography was explained i n Chapter I I I . In t h i s s e c t i o n , the a u t o c o r r e l a t i o n f u n c t i o n of transmittance of a n o n - l i n e a r l y recorded hologram of a d i f f u s e object w i l l be derived f o r the a c t u a l and more general s i t u a t i o n when the s c a l a r form of (4.2) i s no longer considered and (4.1) has to be used i n s t e a d . 65 On the b a s i s of the assumptions about the s t a t i s t i c s of a d i f -fuse object (see Chapter I I I ) , N (r) and N^(r) are c i r c u l a r complex Gaus-2 2 s i a n processes [26] w i t h zero means and variances a and a I t i s a l s o assumed that the two components are unc o r r e l a t e d [48], and s i n c e they are c i r c u l a r complex Gaussian, they are s t a t i s t i c a l l y independent. The auto-c o r r e l a t i o n of the output of the n o n - l i n e a r i t y i s , then, given by [22, PP. 288-289]. e+j°° ~ e+j 0 0 „ R o ( r l ' r 2 ) = 2 1 f ( s ! ) d s i / f ( s 2 ) d s 2 M E ( s 1 , s 2 ) (4.5) (2-rrj) e-j°° e - j 0 0 where M g ( s ^ > s 2 ^ t n e i o i n t c h a r a c t e r i s t i c f u n c t i o n of E( r ^ ) and E ( r 2 ) . Mg(s^,s 2) may be expressed as, M E ( s 1 , s 2 ) = M ( s 1 , s 2 ) M c ( s 1 , s 2 ) , (4.6) where M (s.,,s„) and M (s,,s„) are the j o i n t c h a r a c t e r i s t i c f u n c t i o n s of p 1 2 c 1 2 E^(r^) and E ^ ( r 2 ) , and E c ( r ^ ) and E c ( r 2 ) , r e s p e c t i v e l y , being expressed as f u n c t i o n s of the complex v a r i a b l e s s^ and s2« Equation (4.6) holds due to the f a c t that the j o i n t c h a r a c t e r i s t i c f u n c t i o n f o r a sum of s t a -t i s t i c a l l y independent random processes i s equal to the product of t h e i r r e s p e c t i v e j o i n t c h a r a c t e r i s t i c f u n c t i o n s . The f i r s t task f o r e v a l u a t i n g (4.5) i s to o b t a i n some expression f o r Mg(s^,s 2) or e q u i v a l e n t l y some expressions f o r M^(s^,s 2) and M c ( s ^ , s 2 ) . We s t a r t w i t h M c ( s ^ , s 2 ) . One way of e v a l u a t i n g M c ( s ^ , s 2 ) i s to get the Laplace transform of p ( E c ^ , E c 2 ) , the j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n of E ( r , ) and E (r„),"as f o l l o w s . p(E ,,E „) i s shown to be (see f o r c 1 c 2 . c l c2 example, Ref. 26, Eq. (2.94), P.43). 66 P(E 1 5 E ,) - e x p [ - ( E c l + E c 2 ) /<E> ( 1 - j u J 2 ] c l CZ ~ : : r <E> ( i - | y c | ) 2 ( E c r E c 2 ) 1 / 2 l ^ c l • I ( 7 —) (4.7) ° ^ V ^ - I ^ l > 2 where I (.) i s a modified Bessel f u n c t i o n of the f i r s t k i n d , <E > = 2Ta , o c c and where u I = |u (r,,r„)I = lu (r,-r„)| i s the modulus of the norma-c 1 ' c l z ' ' c l z ' l i z e d a u t o c o r r e l a t i o n f u n c t i o n (complex coherence f a c t o r ) of the c r o s s -p o l a r i z e d component, i . e . *N < r r r 2 ) y c ( r l ' r 2 ) = ~^~2 • (4-8) a c P c ( o r ) i s [see Eq.(3.8)], r e l a t e d to the radiance d i s t r i b u t i o n of c the o b j e c t ' s c r o s s - p o l a r i z e d component I^(p), by a F o u r i e r transform. Using the i d e n t i t y [49] 1/2 °° ( l - b ) " 1 e x p ( - ( g + h ) I ^ b - ) I o ( ) = I b n L n ( g ) L n ( h ) , (4.9) n=0 a more convenient form f o r p(E ., ,E „) i s obtained as [50] c l c2 E +E » E E P<Ed»Ec2> - V T « P ( - i - r 2 - ) I >V ) <E > c n=0 c c c (4.10) where l ^ C - ) I s the Laguerre •polynomial. M c ( s ^ , s 2 ) , the Laplace transform of P(^c^»^c2^» now, may be obtained w i t h the a i d of Equations (5,9.3.2), P. 55 of Reference 51 as, 67 °° s. s u (Ar) 1 L 1 c c c l c 2 < E >* n=0 ( s 1 + l / < E > ) n + 1 ( s 9 + l / < E > ) n + 1 C ± c z c (c f . R e f . 5 2 ) , where Ar = ^ - r 2 To o b t a i n an e x p r e s s i terms of i t s j o i n t moments, i . e . [22,PP.52-54] pression for M ^ s - ^ s ^ , one may express i t i n P q co co s!, s„ : ( S l ,s 0) = 1 T < E p 1 E q„> - f - ^ - (4.12) P' 1' 2' p £ 0 q £ Q P l P2 P!q! where, < E p l Ep2»'= E p l Ep2 V V V V <4-1 3 ' o o The above i n t e g r a l i s the same as I n t e g r a l (3.36) that was evaluated i n Section 3.4.4. I t was shown that <EP E q >can be expressed i n terms of p l p2 harmonics of the a u t o c o r r e l a t i o n f u n c t i o n of the p a r a l l e l - p o l a r i z e d com-ponent, the f i r s t harmonic of which i s r e s p o n s i b l e f o r the usual v i r t u a l and r e a l images and t h e i r background h a l o - l i k e noise. The higher harmon-i c s were shown to be r e s p o n s i b l e f o r the higher order d i s t o r t e d images. Each harmonic i t s e l f c o n s i s t s of a number of members. I t i s important to mention that without a c r o s s - p o l a r i z e d component present, the d i s t o r t -ing background noise of the f a i t h f u l images i s due to the m u l t i p l i c i t y of the f i r s t harmonic i t s e l f . To gain a p a r t i a l i n s i g h t i n t o the nature of the e x t r a d i s t o r t i n g e f f e c t s due to the presence of E ^ ( r ) , a s u b s t i t u t i o n of (4.11) and (4.12) i n t o (4.5) i s made. Thus, R ( r 1 5 r j = V I A (Ar) .<EP1 E q > (4.14) o v 1' T Lr. L n pqn v p l p2 p=U q=U 68 where, - lu (Ar) 2 n s f P f ( S l ) . .. s f q f ( s 2 ) A (Ar) = T J — L _ /„ ds. — ± ~r / ds„ — ± 2 c (4.15) I f i t were not f o r the presence of the c r o s s - p o l a r i z e d component, the co-e f f i c i e n t A (Ar) would have been a constant given by, pqn d [gOO] — Ig(x) ] pq P!q! d x P d x<l = a a (4.16) p q |x=0 [cf.(2.9) or (3.37)]. Comparing (4.15) w i t h (4.16) shows that the presence of c r o s s -p o l a r i z e d component introduces a new d i s t o r t i n g f a c t o r which i s dependent on the i r r a d i a n c e d i s t r i b u t i o n of the object's c r o s s - p o l a r i z e d component. ( R e c a l l that I'v^ H ± s given, w i t h i n a constant f a c t o r , by the F o u r i e r transform of the object's c r o s s - p o l a r i z e d radiance d i s t r i b u t i o n . ) Since | u c ( A r ) | < 1, a good approximation to (Ar) i s obtained i f only the f i r s t two terms i n (4.15) are considered, i . e . the terms f o r which n = 0 and 1. 4.5. Summary This chapter b r i e f l y studied some probable e f f e c t s on hologra-phic image q u a l i t y due to p o l a r i z a t i o n changes of the i l l u m i n a t i n g beam a f t e r s c a t t e r i n g by an ob j e c t . These e f f e c t s i n c l u d e a decrease i n s i g -n a l - t o - n o i s e r a t i o (decrease i n f r i n g e v i s i b i l i t y ) of the hologram and, fo r some o b j e c t s , a l o s s of inf o r m a t i o n of those segments of the object that have l a r g e curvatures (edges, d i s c o n t i n u i t i e s , e t c . ) . The auto-69 c o r r e l a t i o n function of the transmittanee of a non-linear ly recorded holo-gram of a depolarizing d i f f u s e object was. obtained. It i s shown that the change of p o l a r i z a t i o n of the i l l u m i n a t i n g beam by the object not only reduces the signal-to-noise r a t i o , but also augments the d i s t o r t i o n of the images due to the n o n - l i n e a r i t y of the recording medium. This new d i s t o r t i n g e f f e c t , which appears as a m u l t i p l i c a t i v e f a c t o r , i s depen-dent on the radiance d i s t r i b u t i o n of the object's cross-polarized component. These degradation e f f e c t s can be reduced e a s i l y i f a ro-tated reference beam i s used, on the condition that the angle of r o t a -t i o n would d i s t r i b u t e the reference power between the two orthogonal components of the object wave proportional to the strengths of these components. Despite the importance of the subject matter, not enough atten-t i o n has been paid to i t to date and we hope t h i s study w i l l i n i t i a t e further i n v e s t i g a t i o n i n t h i s f i e l d . 70 V. MULTIPLE-REFERENCE-BEAM NON-LINEAR HOLOGRAPHY WITH  APPLICATIONS IN SUPPRESSION OF INTERMODULATION  BACKGROUND NOISE 5.1. I n t r o d u c t i o n This chapter i s concerned w i t h n o n - l i n e a r e f f e c t s i n m u l t i p l e -reference-beam holography (MRBH) versus single-reference-beam holography (SRBH), and i t explores a p o s s i b i l i t y of improving ^reconstructed f a i t h f u l images by employing a d d i t i o n a l reference beams during the re c o r d i n g step of the hologram. T h e o r e t i c a l improvements are shown through e v a l u a t i o n of the transmittanee a u t o c o r r e l a t i o n f u n c t i o n s of the holograms. The object to be imaged i s assumed to be a d i f f u s e object (or a d i f f u s e ob-j e c t accompanied w i t h strong n o n - d i f f u s e r e f l e c t o r s ) . Previous s t u d i e s of the n o n - l i n e a r e f f e c t s i n holography w i t h d i f -fuse objects (see Ch. I l l , or [7] and [53]) showed that the h a l o - l i k e noise around the f a i t h f u l images i s due to those i n t e r m o d u l a t i o n terms that produce e x t r a d i s t o r t e d images i n the d i r e c t i o n s of the f a i t h f u l images. In SRBH t h i s background noise can be reduced, at the cost of the f r i n g e v i s i b i l i t y and d i f f r a c t i o n e f f i c i e n c y of the hologram, by keeping the obje c t i r r a d i a n c e s u f f i c i e n t l y low so that the v a r i a t i o n of exposure i s confined to an almost l i n e a r r e g i o n of the t - E curve. Thus, the important question a r i s e s whether i t i s p o s s i b l e to reduce the intermodulation d i s t o r t i o n without much l o s s i n e f f i c i e n c y . An a f f i r m a -t i v e answer to t h i s question i s contained i n the f o l l o w i n g study of non-l i n e a r a f f e c t s i n double-reference beam holography (DRBH) and MRBH. The d e r i v a t i o n of the formulas f o r the problem under study i s done i n two p a r t s . F i r s t l y , an exact f o r m u l a t i o n i s given f o r DRBH. Secondly, 71 the a n a l y s i s i s extended to the general case of MRBH, but only • approxi-mate closed forms are presented. In both cases i t i s shown that a favour-able r e d i s t r i b u t i o n of the intermodulation d i s t o r t i o n i s p o s s i b l e . As before, the a n a l y t i c procedure w i l l be that of the transform method of communication theory i n connection w i t h the study of vth-law devices. 5.2. P r e l i m i n a r y Considerations I t has already been e s t a b l i s h e d t h a t , from a mathematical point of view, r e c o r d i n g a hologram i s analogous to the passage of e l e c t r i c a l s i g n a l s through a non-linear e l e c t r o n i c d e t e c t o r . I t was shown that the transmittance of a hologram of a d i f f u s e object can be taken to be e q u i -v a l e n t to the r e s u l t of passing the sum of a r e a l narrowband normal pro-cess and a s i n u s o i d ( i f the reference i s a plane wave) or two r e a l nar-rowband normal processes ( i f the reference i s a l s o a d i f f u s e source), through a half-wave n o n - l i n e a r d e t e c t o r . (Ch. I I I ) . More s p e c i f i c a l l y , the problem i n i t s o r i g i n a l form i s concerned w i t h the exposure E as the input to a n o n - l i n e a r i t y whose t r a n s f e r char-a c t e r i s t i c i s a f i t to the t - E curve, e.g. a polynomial i n exposure, t (E) = a + a,E + a„E 2 + a„E 3 + ... E > 0 (5.1) a o 1 l. 5 The output i s the amplitude transmittance t (E) of the hologram. The 3. e q u i v a l e n t problem i n communication theory i s concerned w i t h the passage of an input x, which c o n s i s t s of a r e a l narrowband noise plus s i n u s o i -d a l s i g n a l ( s ) , through a memoryless no n - l i n e a r device followed by an i d e a l low-pass f i l t e r . The t r a n s f e r c h a r a c t e r i s t i c of the device, g ( . ) , 2 i s obtained by r e p l a c i n g E, i n t ( E ) , w i t h x , e.g. 3. 2 4 6 g(x) = a + a l X + a 0 x + a„x + .... ° o 1 2 J 72 N = I a v / 2 x V ' ( 5 ' 2 ) v=0 V / (v even) The quantity of main concern i s the autocorrelation function of the out-put. This problem i n communication theory had been evaluated and closed form solutions for the output c o r r e l a t i o n functions have been given when the input x consists of one s i g n a l plus noise and for vth-law devices, e.g. a half-wave vth-law device, where, g(x) = V (5.3) ax x > 0 x < 0 It has been shown (Ch. II,[54]), that the mathematical procedure for an a r b i t r a r y type of n o n - l i n e a r i t y i s b a s i c a l l y the same as that of the vth-law devices. In t h i s chapter, too, without any loss of gen e r a l i t y , the non-linear c h a r a c t e r i s t i c of the recording medium i s taken to be of (half-wave) vth-law type, with v being an even integer. Furthermore, the reference beams (and the g l i n t s on the object, i f any) are presented as i f they produced uniform plane waves on the hologram plane; the exten-sion of the analysis to cases where some, or a l l , of the reference waves are s p h e r i c a l i s straightforward. With the above as background, our study i s the equivalent of the following c l a s s i c a l problem of s i g n a l detection theory: f i n d the auto-c o r r e l a t i o n function of the output of a half-wave vth-law detector i n response to the input, P x(r) = Y s (r) + n(r) (5.4) - i P P=l . 73 where, s (r) = S C O S ( O J r + a r ) , (p = 1, 2, P), (5.5) p p c p ' ' r ' ' and n ( r ) i s a sample f u n c t i o n of a r e a l narrowband zero-mean normal pro-cess , ri(r) = N ( r ) c o s [ u r + ())(r)] (5.6) where i s an a r b i t r a r y c a r r i e r frequency so chosen that the v a r i a t i o n s of N ( r ) , (|)(r), and cosa^r are slow compared to those of cosu^r. s^,S2, '. .. . ,s axe f i c t i t i o u s r e p r e s e n t a t i o n s of the P reference beams, w i t h constant amplitudes S^jS^,••••, S p, i n c i d e n t on the hologram plane at angles, rip = s i n ^(Actp/ ^ T r ) (p = 1, 2, . . . . , P) (5.7) S i m i l a r l y , n ( r ) i s a f i c t i t i o u s r e p r e s e n t a t i o n of the o b j e c t ' s f i e l d at the hologram plane w i t h i t s amplitude, N ( r ) , R a y l e i g h d i s t r i b u t e d and i t s phase, (()(r), u niformly d i s t r i b u t e d between 0 and 2TT. With these no-t a t i o n s the r e a l p r e s e n t a t i o n of the s c a l a r monochromatic f i e l d d i f -f r a c t e d by a d i f f u s e object i n the F r e s n e l r e g i o n can be w r i t t e n as, Re{N(r)exp(jfit)} = Re{N(r)exp [j<|)(r) ]exp(jftt) } (5.8) = N ( r ) c o s [ ( f i t + (j)(r)] The r e a l p r e s e n t a t i o n of the reference waves may be given as, R e { S p ( r ) e x p ( j f i t ) } = S pcos(ftt + oy:) , (p = 1,2, . . . ,P) (5.9) 74 5.3. Transmittanee Autocorrelation of a Double-Reference-Beam Hologram With x(r) of (5.4) the input to the detector and P = 2, the auto-c o r r e l a t i o n of the hologram transmittanee, R q , may be divided into s i x terms; R = T 2 + R + R + R + R + R (5.10) o a s^xs2 nxn nxs^ nxs^ nxs^xs2 where T i s the mean transmittanee: and where each index of the remain-a ing terms i s i n d i c a t i v e of those signals that are e s s e n t i a l l y (but not always s o l e l y ) responsible for the corresponding term. For example, R represents that part of the transmittanee autocorrelation that s l x s 2 i s e s s e n t i a l l y due to the i n t e r a c t i o n of the two reference waves. Without going into d e t a i l s of the algebraic manipulation, but r e f e r r i n g the interested reader to the r i c h l i t e r a t u r e on c l a s s i c a l s i g n a l detection theory [22, 55-58], the f i n a l r e s u l t s are evaluated as follows, ^ = H000 ( 5 - U ) S 1 X S 2 1=1 v/2 „ R , = y C 0 . cosU(cx - a )Ar] (5.12) s:xs. . - 0££ 1 z R nxn k= 2 (k even) R v v „ 2 .k n X S l k=l 1=1 I I H j J 0 R^ (Ar)cos[)l(a 1Ar - 0)] (5.14) 1 1 1 [(k+£)even] 75 v-m k „ , R = 1 1 Hlf, Rj (Ar)cos[m(a 9Ar - 9)] (5.15) nxs„ , ^  ij, kOm TM z 2 k=l m=l [(k+m)even] v-£-m v-k-m v-k-£ „ , £a.. -ma„ R I I I B k t a < t a i ; ( 4 r ) = o . [ ( « ^ ) ( - T = - » r - 9 ) ] =1 £=1 m=J I ( | £ - m | + k ) e v e n ] n x s l x s 2 k=l £=1 m=l v-£-m k-m k-£ „ , £a +ma + I I I Hk£m \ ( A r ) c o s ( £ + m ) ( — > r + 9)] k=l £=1 m=l I(k+£+m)even] for v >_ 4 (5.16) where, ^ ( A r ) = | R ^ ( r i ; r 2 ) | , (5.17) -1 ^ R ^ C r ,r )} 0 = 0(r ; r 2 ) = tan ^ — - — - — , (5.18) Re{R^(r 1,r 2)} [k+£+m)/2] ! [(k-£-m)/2] ! ( 5 i g ) W [k+£-m)/2] ! [(k-£+m)/2] ! and where, H 2„ = hf„ / C 2(v,0)[(k+£+m)/2] ! [(k-£-m)/2] ! 2 2 k 1 (5.20) k£m k£m with C(v,0) = T(v+1) / 2 V + 1 r 2 ( l+v/2) ([22], p.. 287) (5.21) and h = a r ( v f l ) . Jc-v-1 i ( u S l ) I (a )S 2)exp(a 2a) 2/2)dc J, (5.22) k5.m 2m . c + a l m z n £ irj + 76 f o r which a closed form s o l u t i o n may be obtained as, -s 2 T,, ,-,wo2, 2.m/2.n2. 2.1/2 , „ ,J2. 2 . i _ /k+£+m+2i-v .... 1. ar( v + l ) ( S 2 / a ) (S±/a ) v-k-£-m (S 2/a ) 2  h k £ m " V. ( a2. )(k-v)/2 2(v+l-k) J Q i r ( m + 1 ) !r[l-(k+il+m+2i-v)/2] (5.23) where I (.) denotes a modified Bessel function of the f i r s t kind and or-q der q,T(.) i s the gamma function and i s replaceable by the f a c t o r i a l of i t s argument minus one when the argument i s a p o s i t i v e integer, e.g. T(v+1) = v!; when the argument i s a nggative integer the value of the gamma function i s °°.a i s the s c a l i n g constant of the vth-law n o n - l i n e a r i t y , 2 2 and a E 2O^ . ^1^''^ ^s t* l e confluent hypergeometric function defined by the s e r i e s , 0 0 (M) z r / . i \ 2 / \ Y r 1 , H Z , Vl(y+1) Z . / r o / \ ,F. (u;v;z) = ) 7—c r = 1 + — -77- + ') • 1 ( TT + (5.24) I P ' ' ' •^_ (v) r ! v 1! v(v+l) 21 r=0 r Note that when u i s a negative integer = -q, ^ F^ w i l l terminate a f t e r q + 1st term . c,+ i n i n t e g r a l (5.22) i s a contour of in t e g r a t i o n that l i e s to the r i g h t of the imaginary axis i n the GO plane (cf. [22], Ch.13). In solving (5.22), the power-series expansion of I m(S 2oj) has been employed [59]: S o m - (u)S 9/2) 2 1 i (us2) = (/) I — 2 ; <5-25) m 1 1 i=0 i!(m+i)! replacing I (u)S„) i n (5.22) with the s e r i e s gives: m I a r ( v + l ) ( S 0 / 2 ) m - ( S „ / 2 ) 2 1 k+m+2i-v-l 2 2 h = I V —4 r r r - /- I„(u)S 1)ex P(a Zo) Z/2)du> \lm 2TTJ ± £ 0 i!(m+i)! c+ ly V * n (5.26) 77 A d e t a i l e d and p r e c i s e procedure f o r s o l v i n g such i n t e g r a l s as i n (5~. 26) i s given i n Chapter 13 of Davenport and Root [22]. F o l l o w i n g that procedure gives the r e s u l t of (5.23). Equations (5.11) - (5.16) now may be used to p r e d i c t some important p r o p e r t i e s of the hologram images i n c l u d i n g t h e i r i r r a d i a n c e d i s t r i b u -t i o n s . 5.4. P h y s i c a l I n t e r p r e t a t i o n of Terms Involved i n R Q Examples are given i n Figure 5.1 to i l l u s t r a t e p h y s i c a l l y the s i g -n i f i c a n c e of the terms i n Equation (5.10). In Figure 5.1a, one of the reference waves emanates from a small opening beside the o b j e c t . This pinhole opening and object,which i s a d i f f u s e r cut to the shape of the number 2,are i n a plane p a r a l l e l to the plane of the hologram ( l e n s l e s s F o u r i e r transform r e c o r d i n g geometry [24, PP. 227-230]). The s p a t i a l r a d i a n frequency of the i n t e r f e r e n c e of t h i s reference wave and some poi n t of the object on the same, h o r i z o n t a l - l i n e i s ' r e f e r r e d to as a^. The i n t e r f e r e n c e of the other reference wave w i t h the same point on the object i s chosen to produce orthogonal f r i n g e s w i t h the spa-t i a l frequency a^. The images of t h i s point due to the i n d i v i d u a l r e -ference waves w i l l be l o c a t e d at coordinates (y 2,0) a n d (0,Y-^)> where Y = a Ad/2Tr; w i t h d the observation d i s t a n c e , and p = 1, 2. Several P P images are sketched approximately to s c a l e . The images would be produced by such terms as, R N ( A r ) c o s ( a 2 A r - 0) of ; R N ( A r ) e x p [ - j (c^Ar - 0 ) ] of R ; R N ( A r ) e x p { - j [ ( a x - 2a 2)Ar + 0) ] } . ^ ( A r ) e x p - j [ ( 2 ^ - a^Ar - 0 ] } , and 78 R N ( A r ) e x p { - j [ 2 a ± - 3a 2>Ar - 9]} of the f i r s t part of R n x s x s ; R^(Ar)exp{-j [ a x + c t 2 ) A r - 29]} of the second part of x g ^ ; exp[-j£(a1 - a„)Ar] of R f o r 1 = 1 and 2 (not i n d i c a t e d on the f i g -1 I s l x s 2 2 ur e ) ; and the mean transmittanee T . For rough sketches of such terms as R^(Ar) of R n x n and R^(Ar)exp[-j(2a^Ar - 20)] of R n x g see Figure 5.1(b). 2 The f i r s t term i n (5.10), T , the mean squared transmittanee of cL the hologram i s r e s p o n s i b l e f o r the u n d i f f r a c t e d p o r t i o n of the t r a n s -mitted l i g h t . I t i s the b r i g h t spot at (0,0) i n Figure 5.1a. The second term, R , i s the a u t o c o r r e l a t i o n f u n c t i o n f o r the S 1 X S 2 sum of v/2 s i n u s o i d a l g r a t i n g s w i t h frequencies £(a^ - a ^ ) ; 1=1, 2, ... v/2. This term i s r e s p o n s i b l e f o r the array of the p o i n t - l i k e images ( i n the Fraunhofer region) that are formed i n d i r e c t i o n s d i c t a t e d by £ ( Y ^ - Y 2 ) f F i S - 5.1a, at ( - Y 2 » Y 1 ) a n d ( - 2 Y 2 > 2 Y 1 ) ] . (Note t h a t , i n gen-e r a l , a 's and y 's must be t r e a t e d as vect o r s . ) P P The t h i r d term, R produces an image on the o p t i c a l a x i s ( F i g . nxn 5.1b). The i r r a d i a n c e d i s t r i b u t i o n of t h i s image, which sometimes i s c a l l e d the ambiguity f u n c t i o n , i s p r o p o r t i o n a l to £ [ ( k - l ) - f o l d con-k=2 v o l u t i o n of the object i r r a d i a n c e ] , (k even) The f o u r t h and f i f t h terms, R and R , represent r e s p e c t i v e l y nxs^ n x s 2 the c o n t r i b u t i o n s from the a c t i v e involvement of the object wave with the f i r s t reference wave (nxs^), and the object wave w i t h the second reference wave (nxs^). For Z = 1, k = 1, and m = 1, k = 1 each i s r e s -p o n s i b l e f o r a p a i r of f a i t h f u l images (the usual r e a l and v i r t u a l images), w h i l e f o r k > 1 they generate d i s t o r t e d images. Some of these d i s t o r t e d 79 (a) 1 OPTICAL AXIS, IMAGE DUE TO RJJ(fcr) AMD MEAN TRANSMITTANCB VIRTUAL IMAGE DUE TO RJJ (Ar ) c o i (ca^&r-O) V I R T U A L IMAGE . D U E T O (b) Figure 5.1. Hologram imaging p r o p e r t i e s v i a R ^ r ^ r ^ ) .Each term i n R^ s i g n i f i e s an image,and i t i n d i c a t e s the r e l a t i v e s t r e n g t h , p o s i t i o n , o r i e n t a t i o n , and width of that image.Rough sketches of these images along w i t h the terms s i g n i f y i n g them are given f o r two r e c o r d i n g geometries: a)DRBH, the geometry i s b a s i c a l l y the normal o f f - a x i s holography(Fig. 3 .1),and the a d d i t i o n a l background wave o r i g i n a t e s from a hol e i n the plane of the o b j e c t ( l e n s l e s s F o u r i e r transform geometry) ; b) SRBH. 8 0 images are l o c a t e d i n the same d i r e c t i o n as the f a i t h f u l ones "(those f o r which k > 1, but I = 1 and m = 1). This mechanism was shown to be r e s -p o n s i b l e f o r the h a l o - l i k e noise around the f a i t h f u l images reconstructed from n o n - l i n e a r l y recorded SRB holograms (see F i g . 3.2). I t has a l s o been e s t a b l i s h e d that an a m p l i f i c a t i o n of the f a i t h f u l images i n SRBH i s u s u a l l y a s s o c i a t e d w i t h stronger background noise. (Note the strong halo introduced by severe n o n - l i n e a r i t y i n Figure 3.2d). However, i t w i l l be shown that i n DRBH (and MRBH) i t i s p o s s i b l e to amplify the f a i t h f u l images without s i g n i f i c a n t l y changing the l e v e l of the background noise. Another i n t e r e s t i n g f e a t u r e of the non - l i n e a r double-reference-beam re c o r d i n g i s .due to the simultaneous i n t e r a c t i o n of the object wave wi t h both of the reference waves. This i n t e r a c t i o n i s represented by the l a s t term i n (5.10), that i s , R . ,and is d i v i d e d i n t o two pa r t s [see Eq. (5.16)]. The i n t e r a c t i o n i s present only when n o n - l i n e a r i t y i s pre-sent: f o r a l i n e a r hologram where v = 2, R = 0, as expected. An ' ° nxs^xs^ important r o l e played by R i s due to the f i r s t part of i t ,which nxs-^xs^ shows the production of f a i t h f u l images. Depending on the value of v, there are s e v e r a l terms which produce these f a i t h f u l images, that i s , those terms f o r which |jt-m| = 1 and k = 1. For example, f o r v = 4 there are two f i r s t - o r d e r terms, that i s , % = 1 and m = 2, and 1=2 and m = 1 w i t h k = 1. In t h i s example, (v = 4 ) , there are not any terms w i t h noise c o n t r i b u t i o n to these f a i t h f u l images due to R ( i . e . no terms w i t h nxs^xs^ k > 1 when |£—m] = 1). Therefore, i f the no n - l i n e a r c h a r a c t e r i s t i c could be approximated w i t h a second order polynomial i n exposure (which would mean v = 2 and 4 ) , n o i s e l e s s images of the object may be recon s t r u c t e d i n regions determined by f a c t o r s (2y^ - Y 2) a n d (Y^ ~~ Z Y 2 ) -The second part of R does not produce l i n e a r images but only nxs .^xs 2 d i s t o r t e d higher order images, i n the regions determined by the factors (SLy^ + mj2). It i s perhaps of i n t e r e s t to r e f e r to the fact that the autocorre-l a t i o n function for the transmittanee of a SRB hologram may be obtained from ( 5 . 1 0 ) by considering i t a s p e c i a l case of DRBH i n which the magni-tude of one of the reference beams, say s„, approaches zero. h i s <u. K. A/in zero for m > 1 , since I (0) ={? m — ^. Therefore, the only non-zero terms — m 1 m = 0 2 i n ( 5 . 1 0 ) are T , R and R , that i s , a nxn nxs R = T 2 + R + R ( 5 . 2 7 ) 0 a nxn nxs^ with S 2 , , _ 2 , 2.1/2 _, , k+fc-y 0 J _ 1 ° 1 av!(S 1 la ) 1 F 1 ( — ^ — ; JH-1; 2 ) hk«, = Try, w o ±i i ~ ( 5 . 2 8 ) J l , ( a 2 ) ( k - v ) / 2 2 v + l - k n i _ ( k + A _ v ) / 2 ] 5 . 5 . Suppression of Intermodulation D i s t o r t i o n of First-Order Images This section w i l l compare the f i r s t harmonic images i n the two cases of SRBH versus DRBH. This comparison w i l l show that the addition of a second reference beam i n the recording of a hologram i s bound to improve the contrast of the f a i t h f u l images compared to those that would have been obtained using one reference beam only. To s i m p l i f y the matter and to present the c e n t r a l ideas and r e s u l t s i n a more t r a c t a b l e fashion, the n o n - l i n e a r i t y i s assumed to have a half-wave vth-law c h a r a c t e r i s t i c with v = 4 . Note that non-linear e f f e c t s are present only for v >_ 4 , and that i n p r a c t i c e when the t - E curve i s approximated by a polynomial i n exposure, the most s i g n i f i c a n t e f f e c t s are due to the terms up to the qua-d r a t i c term ( i . e . v i s to be taken 2 and 4 ) . 82 I t has been shown that the f i r s t harmonics can be s p l i t up i n t o two main p a r t s , one r e s p o n s i b l e f o r the f a i t h f u l images (where i n Eqs. (5.14) and (5.15) k = 1 and £ and/or m = 1 ) , and the other the background halos (where k > 1 but £ and/or m = 1). In the f o l l o w i n g the former i s o f t e n r e f e r r e d to as " s i g n a l " and the l a t t e r as "noise". As u s u a l , i t i s de-s i r e d to increase the s i g n a l - t o - n o i s e r a t i o (to improve the co n t r a s t ) w h i l e keeping the depth of modulation ( f r i n g e v i s i b i l i t y ) high. In SRBH high f r i n g e v i s i b i l i t y i s a t t a i n a b l e when the object and reference i r r a d i a n c e s at the f i l m have roughly equal strengths. I n -cr e a s i n g the i r r a d i a n c e of the object i s u s u a l l y the cause of more pro-nounced n o n - l i n e a r e f f e c t s i n c l u d i n g an increase i n the background n o i s e which could become troublesome [see F i g s . 3.2b - 3.2d]. In DRBH there are three beams i n t e r f e r i n g and a rough d e f i n i t i o n f o r f r i n g e v i s i b i l i t y may be given by assuming the combination of the object wave and one of the reference waves on the r e c o r d i n g medium as one wavefront and the other reference wave as the i n t e r f e r i n g second wavefront. Therefore, a high f r i n g e v i s i b i l i t y w i l l be obtained i f the t o t a l s t r e n g t h ofuthe ob-j e c t wave and one of the reference waves on the hologram plane i s roughly equal to the strength of the other reference wave. In any case, to keep the background noise low the object wave ought to be sm a l l . In SRBH t h i s means a low f r i n g e v i s i b i l i t y , w h i l e i t does not have to be so i n DRBH. I f one of the reference waves i s K times stronger than the object wave the second reference wave should be roughly K + 1 times stronger than the object wave to s a t i s f y the h i g h - f r i n g e v i s i b i l i t y c o n d i t i o n . To be more s p e c i f i c , f o r SRBH, the part r e p r e s e n t i n g the s i g n a l i s given by [see Eqs. (5.14), (5.20), and (5.28), w i t h v = 4 ] , 83 2 5 a 2 S 2 ( a 2 ) 2 [ ^ ( - l ^ - s j / a 2 ) . ] R^(Ar)cos(a^Ar - 0 ) (5.29) and the part r e p r e s e n t i n g the background noise i s given by, 2 4 a 2 S 2 R ^ ( A r ) c o s ( a ] A r - 0) (5.30) (which a l s o shows that i f the t - E curve i s approximated by a second degree polynomial i n exposure, the background noise has a width three times that of the f a i t h f u l images). In DRBH w i t h v = 4 there w i l l be four sets of f a i t h f u l images (two c o n t r i b u t e d by R and R , and the other two by R ). 3 nxs^ n x s 2 ' y nxs^s,^ These images a r e , r e s p e c t i v e l y , presented by [see Eqs. (5.14) - (5.24)], 2 5 a 2 S 2 ( a 2 ) 2 [ ^ ( - l ^ - s j / a 2 ) + S 2 / a 2 ] 2 R N ( A r ) c o s ( a ; L A r - 0 ) , (5.31) (c o n t r i b u t e d by R n x s ) 2 5 a 2 S 2 ( a 2 ) 2 [ ^ ( - l j l j - s j / o 2 ) + S 2 / ( 2 a 2 ) ] 2 R ^ A r J c o s ^ A r - 0 ) , (c o n t r i b u t e d by R ) (5.32) v 3 n x s 2 and, 3 2 2 2 2 2V(Sp(sp R N ( A r ) c o s [ ( 2 a 1 - a 2 ) A r - 0 ] , (5.33) and ,3 2 , A 2 , J 2 a (S^) (S^) R N ( A r ) c o s t a 1 - 2a 2 ) A r + 0] (5.34) ( c o n t r i b u t e d by R ). v 3 n x s ^ x s 2 The background halos f o r the f i r s t two sets of f a i t h f u l images a r e, 84 r e s p e c t i v e l y , given by 2 4 a 2 S 2 ( A r ) c o s ( a 1 A r - 0) (5.35) and 2 4 a 2 S 2 ( A r ) c o s ( a 2 A r - 0) (5.36) Since v i s assumed to be 4, there i s no background noise a s s o c i a t e d w i t h the l a s t two sets of images. As can be e a s i l y seen from a comparison of Equations (5.31) and 2 2 (5.35) w i t h (5.29) and (5.30), i f the same r a t i o S^/o i s used i n both SRBH and DRBH, the background halos of the corresponding f a i t h f u l images w i l l be of the same stre n g t h i n both cases, but the corresponding f a i t h -f u l image i n DRBH i s strengthened by an a d d i t i v e f a c t o r o f , 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ( S 2 / a ^ ) 1 F 1 ( - l ; 2 ; - S ^ / a Z ) + (S^/a r = (S^/a )[2+S^/a -HS^/a ].(5.37) This f a c t o r i s obtained by s u b t r a c t i n g [. ] terms i n (5.29) and (5.31). 2 2 2 2 The s i g n a l strength i s p r o p o r t i o n a l to both (S^/a ) and (.S^/a ), wh i l e 2 2 the corresponding h a l o - l i k e noise i s only p r o p o r t i o n a l to (S ;o )(under the assumption of v = 4). That i s the power of the s i g n a l can be i n -2 2 creased, by i n c r e a s i n g (S^/o ) , independently of the l e v e l of the halo noise. (When v > 4 the background halo c o n t r i b u t e d by R w i l l be de-e J nxs 2 2 pendent on (S^/o ), too, but the e f f e c t s i n most p r a c t i c a l cases are not very s i g n i f i c a n t due t o , among other t h i n g s , the f u r t h e r a m p l i f i c a t i o n of the f a i t h f u l images caused by higher order n o n - l i n e a r i t i e s ) . There-f o r e , the a d d i t i o n of a second reference beam i n the process of re c o r d i n g the hologram not only a m p l i f i e s the f a i t h f u l images without a f f e c t i n g the halo l e v e l (image co n t r a s t improvement), but a l s o provides us with a con-85 t r o l f a c t o r over t h i s a m p l i f i c a t i o n . For example, one may keep the back-2 2 ground noise below some acceptable l e v e l , by a d j u s t i n g the r a t i o (S^/a ) , w h i l e one may c o n t r o l the amount of the inherent a m p l i f i c a t i o n of the s i g -2 2 n a l due to the presence of the second reference ,-'' by a d j u s t i n g (S^/a ). The same arguments apply to the other set of images, w i t h the r o l e s of 2 2 2 2 S., and S„ interchanged. When S., = S~ both sets of images due to R 1 2 ° 1 2 ° nxs^ and R are of the same i r r a d i a n c e . In t h i s l a t t e r case, the strengths n x s 2 of the sets of f a i t h f u l images due to R are a l s o the same. 5.6. Experimental Results I n s i g h t i n t o the above-mentioned t h e o r e t i c a l expectations may be gained by c o n s i d e r i n g the f o l l o w i n g experimental observations. Figure 2 2 5.2 shows the images reco n s t r u c t e d from a hologram f o r which S^ >> 2 and 0 . The object was a d i f f u s e r i n the shape of number 2, and the ex-t r a reference wave was provided through an opening beside i t . Since 2 2 S.. >> S_, the images due to R are much stronger than those due to n x s ^ x s 2 R [ c f . F i g s . 5.2b - 5.2d w i t h 5.2a]. For the same reason, one would n x s 2 expect, from a comparison of (5.33) w i t h (5.34), that out of the f a i t h -f u l images due to R the one centered at coordinates (-Y0,2Y-,) to b n x s 1 x s 2 2 1 be the strongest. This was undoubtedly the case, as i s manifested i n Figures 5.2b and 5.2c. 2 ^ 2 2 2 Next, we consider a case i n which S^ ^ S 2 + a and S 2 i s s l i g h t l y 2 l a r g e r than a ( F i g . 5.3). Images are sharp and e a s i l y observable (an i n d i c a t i o n of high f r i n g e v i s i b i l i t y ) . Non-linear e f f e c t s are a l s o more 2 pronounced, although a was not yet strong enough to produce strong back-2 2 ground noi s e . Since S 0 i s now comparable to S-. , the images due to R and R are c l o s e r i n strengths. The image formed i n the center of n x s 2 86 (d) (b) (c) 2 2 2 Figure 5.2. Reconstructed images of non-linear DRBH ; >> and a . Images are mainly due to : a) R (centered around the o p t i c a l a x i s ) , J nxs„ b) R and R nxs^ nxs^xs2 c) Lower part of (b) enlarged and more i n focus; i t consists of a f a i t h f u l image due to R and the 2nd harmonic of R nxs^xs^ nxs^ d) Upper part of (b) enlarged and more i n focus. 87 (a) F i g u r e 8 8 Figure 5.3b i s due to the f i r s t part of R w i t h I = m = 1, and nxs^xs ^  k = 2, 4, .. . , v - 2. That i s , V ; 2 hkll / c 2( V'°> k I itl , 2 k - l \ C ^ c o s ^ - a 2 ) A r (5.38) k=2 (-1) 2 (k even) which i s a s h i f t e d v e r s i o n of the ambiguity term, R of (5.13). This & £ nxn s h i f t e d ambiguity f u n c t i o n i s producing a stronger image [at coordinates (~Y 2,Y^)] than the o r i g i n a l one produced by & n x n (centered at the o p t i c a l 2 2 a x i s ) , s i n c e f o r t h i s hologram h^-^ > h^QQ [see Eqs. (5.13), (5.16) and (5.23)]. Accomparison of the c e n t r a l regions of Figures 5.3a and 5.3b. c l e a r l y confirms t h i s e xpectation. The i n v e r t e d f a i t h f u l image i n the l e f t s i d e of Figure 5.3c s i g n i f i e s the e f f e c t s of the no n - l i n e a r terms f o r which v > 6 (e.g., a polynomial approximation of t - E curve must at l e a s t be a t h i r d order polynomial i n exposure). The r i g h t s i d e of F i g -ure 5.3c shows a f a i t h f u l image due to R and the second harmonic nxs^xs^ of R n x s • Further experimental r e s u l t s are given i n the f o l l o w i n g sec-t i o n . 5.7. Holograms Made w i t h More than Two Reference Beams When there are more than two reference waves present i n the re c o r d -i n g of the hologram, equations s i m i l a r to those of (5.10) - (5.16) could be w r i t t e n , w i t h new c o e f f i c i e n t s h ^ ^ ^ given as, • 2 2 , ar(v+l) . k-v-1 T / c s T / o \ T / - o \ h k ^ 2 ...*p = - 2 ^ ' c + u \ ( u Sl ) I£2 ( u S2 ) - - - I £ p ( w V e X p (-2—> d w (5.39) 89 2 The intensity in each term w i l l be proportional to h . The terms that are of concern here are those for which one of the subscripts £ 's P is 1 but the others are 0, while k i s 1 for a faithful image and k >_ 3 for i t s background halo-like noise; e.g., one of the faithful images w i l l 2 be proportional to h ^ ^ while the terms contributing to i t s back-2 ground noise each is proportional to h^Q ^ ; k = 3,5, , v - 1. The approximate solutions for these coefficients may be obtained employ-ing a method introduced by Gyi [60] and later used by Shaft [58] to ana-lyse the passage of signal plus noise through a vth-law non-linearity. Without loss of generality, the images due to S- n x s w i l l be taken as the ones of interest. The approximate solutions for h, i n n „ klOO...0 (k = 1,3,...., v - 1) are obtained using the approximate equality, I0(Saj) £ exp[ f - * f - ] (5.40) in (5.39). That i s , h i inn n £ ' • o ^ " 1 ^ (a)S. )exp( -f~ )d<*> (5.41) kl00...0 ' v 2TTJ C - + 1 1 I where a 2 = a 2 + f | s 2 (5.42) t n p=2 2 P that i s , a l l the reference waves except the f i r s t are lumped together with the object wave. The integral (5.41) is the same as the integral in (5.26) with I = 1. Following Davenport and Root [22, P.303] . a r ( V + 1 ) ( S l 2 / 2 ° t 2 ) 1 / 2 v ( 1+k-v , ,2 / 9 2 . r s . ^  h l i n n n y o T, T - 7 T I F I ( —o—;2;-S 1/2o. ) (5.43) klOO...O -u r [ l _ ( 1 + k _ v ) / 2 j ( a 2 / 2 ) ( k - v ) / 2 1 1 2 1 t 90 Hence, taking v = 4 as before, the f a i t h f u l images w i l l be represented by 2 5 a 2 S 2 ( 2 a t 2 ) 1 F 1 2 ( - l ; 2 ; - S 2 / 2 a t 2 ) R N ( A r ) c o s ( a 1 A r - 0) (5.44) and the part representing the background noise i s given by, 2 4a 2S 2R^(Ar)cos(a 1Ar - 0) j (5.45) that i s , equal to that of SRBH or DRBH [cf. (5.30) and (5.35)], and i n -dependent of a l l the other reference waves, i n agreement with our e a r l i e r r e s u l t obtained for DRBH. It could also be shown that when P = 2, that i s , when only two reference beams are present, (5.44) which has been ob-tained through an approximate method, i s equal to (5.31), that was ob-tained e a r l i e r by an exact evaluation of .the i n t e g r a l s involved. Looking back at (5.42), (5.44) and (5.45) one sees that by i n -creasing the number of reference waves the amount of the consequent am-p l i f i c a t i o n of the f a i t h f u l images against t h e i r almost constant back-ground noise and the degree of freedom over t h i s a m p l i f i c a t i o n w i l l be increased. Of course, the number of reference waves that could be accomo-dated without introducing some complications i s not unlimited. Among these complications i s one due to the increase of the intermodulation products with the increase i n the number of the reference waves. These newly produced harmonics could be d i s t u r b i n g i f some of them formed images i n the same regions as the desirable f a i t h f u l images. Examples of t h i s type of problem are given i n Figures 5.4 and 5.5 where the r e -l a t i v e p o s i t i o n s of reference'beams were, not optimal. Note p a r t i c u l a r l y the improvement i n the image q u a l i t y gained by adding thei.third r e f e r -ence wave, i n Figure 5.5. 91 Figure 5.4 Figure 5.5 Figures 5.4 and 5.5. Images of no n - l i n e a r double- and trip l e - r e f e r e n c e - b e a m holograms. The t h i r d reference wave was provided through a hole beside the obj e c t . Careless p o s i t i o n i n g of the reference beams has caused the overlap of some of the newly produced intermodulation products w i t h the main f a i t h f u l ones.Figure 5.5c shows that n o n - l i n e a r i n t e r -modulation of two or more reference beams and the object wave can produce n o i s e l e s s images(cf. F i g . 5 . 2 c ) . Note the improvement i n image q u a l i t y gained by adding the t h i r d reference beam i n Figure 5.5. 92 5.8. White Light Reconstruction of SRB Holograms vs. DRB Holograms It i s desirable to be able to view holographic images using white l i g h t sources. The main requirements for reconstruction of a holo-gram are imposed by the 'transverse' and 'chromatic' coherence of the reconstructing beam. When using an incoherent source, loss of transverse s p a t i a l coherence could be avoided i f a very small source i s used. An extended white l i g h t source can be treated as an incoherent superposi-t i o n of many point sources. Each point source w i l l reconstruct an image that i s displaced r e l a t i v e to the images produced by the others. There w i l l be many p a r t i a l l y over-lapping images that could d i s t o r t the images to an unrecognizable degree, depending on the s i z e of the i l l u m i n a t i n g source. Therefore, to lessen t h i s d i s t o r t i o n the f i r s t requirement i s to have a very small source. The e f f e c t due to the l i m i t e d 'chromatic' coherence of the white l i g h t reconstructing source manifests i t s e l f i n a color dispersion. Each wave-length reconstructs i t s own image, displaced i n p o s i t i o n according to i t s color. A hologram may be considered as a composite g r a t i n g - l i k e s t r u c -ture. I f the angles between the reference and d i f f e r e n t points of the ob-je c t were large during the recording step of the hologram, the s p a t i a l frequencies of the gratings composing the hologram would~be High(small fringe spacings). Consequently, white l i g h t reconstruction of such a hologram would be associated with a large color dispersion; and the r e -constructed images would be d i s t o r t e d beyond recognition. To lessen the color d i s p e r s i o n , therefore, the reference and object should be brought c l o s e r . But making the angles too small w i l l cause overlaps 93 of the f i r s t harmonic images and the ambiguity term; i n case of non-l i n e a r recording, the higher order images w i l l a lso contribute to the overlap. For on-axis holography the color dispersion w i l l be zero at the center, but a l l the harmonic images w i l l be formed i n one l i n e , i n -cluding the usual r e a l and v i r t u a l images (twin image problem of on-axis holography). An example of an i n t o l e r a b l e color d i s p e r s i o n i s given i n Figure 5.6 where the dispersion i s so large that i t makes the recogni-t i o n of the f a i t h f u l images impossible. The almost white-light image i n the center of the photograph i s due to the ambiguity function and the constant b i a s . The color dispersion for t h i s l a t t e r image i s almost zero. For a narrow-width object a compromise might be found where both the afore-mentioned l i m i t a t i o n s are kept from becoming i n t o l e r a b l e . Some improve-ment i s shown i n the SRB hologram of Figure 5.7. The coherent background was provided through a hole beside the object. S t i l l more improvement i s obtained with a DRB hologram, a photograph of the white l i g h t reconstruc-t i o n of which i s given i n Figure 5.8a. (Photographs of laser-reconstructed images of the same hologram are given i n Figure 5.3). A larger view of these images are given i n Figure 5.8b. A comparison of Figures 5.7 and 5.8 shows that the images of the DRB hologram are better than those of the SRB hologram; the f a i t h f u l images i n the DRB hologram are b r i g h t e r , while the troublesome image due to the constant bias and the ambiguity function i s much less intense. In order to eliminate spurious images, Burckhardt [61] used a Venetian b l i n d structure which he placed behind the"'hologram. A Venetian b l i n d e f f e c t might also be responsible for the p a r t i a l blocking of the ambiguity term i n our DRB hologram. That i s , the i n t e r a c t i o n of the two reference beams i n the photographic plate of 94 F i g u r e 5.6. W h i t e l i g h t r e c o n s t r u c t e d images o f a SRB hologram. The l a r g e s e p e r a t i o n a n g l e between t h e o b j e c t and r e f e r e n c e has caused a l a r g e c o l o r d i s p e r s i o n of t h e f a i t h f u l images. 95 F i g u r e 5.7. W h i t e l i g h t r e c o n s t r u c t e d images of a SRB h o l o g r a m ; t h e c o l o r d i s p e r s i o n o f t h e images i s g r e a t l y r e d u c e d by b r i n g i n g t h e r e f e r e n c e beam c l o s e r t o t h e o b j e c t . 9 6 F i g u r e 5.8. W h i t e l i g h t r e c o n s t r u c t e d images o f a DRB h o l o gram. Images a r e b r i g h t e r t h a n t h o s e i n t h e SRB h o l o g r a m o f F i g u r e 5.7, and t h e t r o u b l e s o m e image due t o t h e c o n s t a n t b i a s and t h e a m b i g u i t y f u n c t i o n i s much l e s s i n t e n s e . The m a g n i f i c a t i o n o f t h e images i n F i g u r e 5.8b i s due t o i t s l a r g e r o b s e r v a t i o n d i s t a n c e t h a n t h e o b s e r v a t i o n d i s t a n c e f o r F i g u r e 5.8a. 9 7 f i n i t e thickness w i l l produce a Venetian b l i n d structure. 5.9. Summary By d i r e c t a p p l i c a t i o n of ' c h a r a c t e r i s t i c function' method of communication theory, expressions were evaluated f o r the transmittance autocorrelation functions of multiple-reference-beam holograms of d i f -fuse objects. These expressions can be d i r e c t l y used to acquire useful information about the irradiance d i s t r i b u t i o n over the image plane. Proper use of a d d i t i o n a l reference beam(s), during the recording step of the hologram, i n conjunction with the non-linear nature of the recording media,can improve the contrast of the reconstructed f a i t h f u l images. 98 VI. CONCLUSIONS 6.1. Summary C l a s s i c a l one-dimensional d e t e c t i o n theory has already been ap p l i e d to solve problems i n o p t i c a l imaging, but the extensions to h o l o -graphy have had c e r t a i n b a s i c shortcomings reviewed i n Chapter I. The work i n Chapter I I I removes these shortcomings by i n t r o d u c i n g a new math-ematical model f o r the holographic process. In t h i s model, a monochromatic f i e l d d i f f r a c t e d by a d i f f u s e object over the hologram plane may f i c t i -t i o u s l y be represented by a narrowband normal process, analogous to a narrowband noise i n e l e c t r i c a l communications. S i m i l a r l y , uniform co-herent backgrounds may be represented by s i n e waves. These representa-t i o n s are obtained by r e p l a c i n g fit i n the o r i g i n a l expressions f o r the o p t i c a l waves w i t h co^rjfi i s the frequency of the monochromatic r a d i a t i o n , a) i s an a r b i t r a r y l a r g e c a r r i e r frequency, t i s the time v a r i a b l e , and r i s the space v a r i a b l e . The input to the model, x ( r ) of Equation (3.42), i s represented by the sum of the newly introduced f u n c t i o n s f o r the ob-j e c t and reference waves. The photographic emulsion i s modelled as a memoryless n o n - l i n e a r device followed by an i d e a l low-pass f i l t e r ( i . e . a n o n - l i n e a r d e t e c t o r ) . The t r a n s f e r c h a r a c t e r i s t i c of the no n - l i n e a r element g(x) i s obtained by r e p l a c i n g E i n the approximate equation f o r t (E) w i t h x . A power-series expansion f o r g(x) reduces the mathematics inv o l v e d to that of 'transform' method f o r vth-law n o n - l i n e a r i t i e s . The procedure that makes t h i s a p p l i c a t i o n p o s s i b l e i s developed i n Chapter I I , where general formulas f o r e v a l u a t i o n of c o r r e l a t i o n f u n c t i o n R\'(t t„) b y 1 2 [Eq. (2.2)] of a r b i t r a r y n o n - l i n e a r i t i e s are provided. The c r u c i a l quan-t i t i e s to evaluate are the c o e f f i c i e n t s h ^ v given by the i n t e g r a l (2.3), The s o l u t i o n to t h i s i n t e g r a l f o r a continuous g(x) i s given i n (2.12). For c h a r a c t e r i s t i c s w i t h d i s c o n t i n u i t i e s ; a combination of Equations (2.12), (2.13), (2.15) and (2.18) must be used. Examples are given to i l l u s t r a t e t h i s p o i n t . The q u a n t i t y of main concern i s the a u t o c o r r e l a t i o n f u n c t i o n f o r the output of the low-pass f i l t e r , that i s , the c o r r e l a t i o n of the out-put of the d e t e c t o r s w h i c h i s denoted by ^ ( t ^ j t ^ ) i n the e l e c t r i c a l case and by R ^ r ^ , ^ ) i n the hologram* case. This i s obtained by p i c k i n g out only the low frequency terms i n R '(reviewed i n Sec. 2.4). The f i n a l r e s u l t f o r R ( r ^ i i ^ ) i s given by (3.49). The terms composing R ^ r ^ , ^ ) have i n t e r e s t i n g p h y s i c a l s i g n i f i c a n c e which lead to p r e d i c t i o n s of the nature of the d i s t o r t i o n s produced by the hologram n o n - l i n e a r i t y . The f a i t h f u l image d i s t o r t i o n manifests i t s e l f i n the form of a background h a l o - l i k e noise produced by e x t r a d i s t o r t e d images formed i n the same d i r e c t i o n . The f a i t h f u l image and i t s background noise c o n s t i t u t e the f i r s t harmonic. The equation f o r R n ( r ^ , r 2 ) p r e d i c t s a l s o other d i s -t o r t e d harmonic images that are combinations of higher-order c o r r e l a -t i o n s and convolutions of the object. Such e f f e c t s are observed and r e -ported i n Chapter I I I . The p o l a r i z a t i o n of l i g h t i s u s u a l l y changed a f t e r being s c a t -tered by an o b j e c t . In the r e f l e c t e d l i g h t from most objects i l l u m i n a t e d by l i n e a r l y p o l a r i z e d l i g h t , there i s a c r o s s - p o l a r i z e d component pre-sent that i s not accounted f o r i n the usual p r a c t i c e of holography. In Chapter IV we have t r i e d to give a few p e r s p e c t i v e s on probable e f f e c t s due to the d e p o l a r i z a t i o n of the r e f l e c t e d l i g h t from the o b j e c t . A c e r t a i n degradation e f f e c t w i l l be a decrease i n s i g n a l - t o - n o i s e r a t i o , s ince that part of the object wave that i s orthogonal to the reference 100 wave produces no c o n t r i b u t i o n to the hologram. Such a part of the object wave cannot be reconstructed. When the hologram i s n o n - l i n e a r l y recorded d e p o l a r i z a t i o n w i l l produce m u l t i p l i c a t i v e n o i s e . This noise i s s i g n a l -dependent i n the sense that i t depends on the i r r a d i a n c e d i s t r i b u t i o n of the object's c r o s s - p o l a r i z e d component [ c . f . A p^ n of (4.15) w i t h A of (4.18)]. I t i s suggested that the d e p o l a r i z a t i o n e f f e c t s may be eli m i n a t e d i f the p o l a r i z a t i o n of the reference wave i s changed appro-p r i a t e l y . Some experimental v e r i f i c a t i o n of t h i s matter i s provided i n Sec t i o n 4.3. The theory developed i n Chapters I I and I I I can be a p p l i e d to the i n t e r e s t i n g case of multiple-reference-beam holography (MRBH). The importance of MRBH r e s t s on the improvement i n image q u a l i t y i t o f f e r s . This improvement i s i n terms of a m p l i f i c a t i o n of the de s i r e d f a i t h f u l images against t h e i r background noi s e . Exact equations f o r double-re-ference-beam holography (DRBH) are given i n (5 .10) - (5.16), which can be used to evaluate the o p t i c a l p r o p e r t i e s of the hologram images. Co-e f f i c i e n t s that correspond to h of single-reference-beam holography can be evaluated f o r DRBH and MRBH. The c o e f f i c i e n t s f o r DRBH are denoted by and are given by Equation (5.23). These c o e f f i c i e n t s determine the i n t e n s i t y of reconstructed images i n c l u d i n g the f a i t h f u l ones and t h e i r background noi s e . Approximate closed-form s o l u t i o n s are a l s o given f o r the i r r a d i a n c e of the f a i t h f u l images and t h e i r background noise i n MRBH [Eq. (5.43)]. The p o s s i b i l i t y f o r image improvements i s suggested by a comparison of r e l e v a n t c o e f f i c i e n t s h i n MRBH versus SRBH (Sees. 5.5 and 5.7). Experimental v e r i f i c a t i o n s of the formulas are provided i n S e c t i o n 5.6. L a s t l y some experimental observations are made comparing white 101 l i g h t reconstructed images of DRB vs. SRB holograms. DRBH could pro-duce b r i g h t e r images than SRBH and troublesome images due to the con-stant b i a s and the ambiguity f u n c t i o n are reduced i n the former ( F i g s . 5.7 and 5.8). 6.2. Further P o s s i b l e Extensions In the foregoing treatment the r e c o r d i n g media were assumed to be very t h i n , so that one could t r e a t the input exposure as a two-dimen-s i o n a l " s u r f a c e " process. This assumption i s accurate, f o r example, f o r photoconductor-thermoplastic f i l m s , photopolymers, p h o t o r e s i s t m a t e r i a l s , and photographic emulsions bleached to have only a surface . r e l i e f . A l -though, f o r many experiments, assuming the rec o r d i n g of a hologram as a surface phenomenon may be considered reasonable to a f i r s t approximation, a more c a r e f u l a n a l y s i s of any experiment should i n c l u d e c o n s i d e r a t i o n of the non-zero medium's t h i c k n e s s . To the best of the author's know-ledge there has been no thorough i n v e s t i g a t i o n f o r a t h i c k medium, to date. However, i t i s f e l t t h a t , w i t h some appropriate m o d i f i c a t i o n s , the f o r e -going a n a l y s i s may provide a n a l y t i c a l expressions that could account f o r the combined e f f e c t s due to the n o n - l i n e a r i t y and the thickness of the recording medium. Such a study i s d e s i r a b l e , s i n c e i t i s b e l i e v e d that a b e t t e r understanding of the holographic imaging process i n a t h i c k hologram of a general type object c o u l d . i n s p i r e some new a p p l i c a t i o n s i n holographic image q u a l i t y improvements. The s p a t i a l f i l t e r i n g proper-t i e s of volume g r a t i n g s have already been considered [62]. Thus, double-and multiple-reference-beam holography i n a t h i c k medium might become promising p o s s i b i l i t i e s f o r producing hologr ams w i t h b u i l t — i n f i l t e r s that could d i s c r i m i n a t e , f o r example, against the background noise of the d e s i r a b l e f a i t h f u l images. 102 REFERENCES [ l a ] For example; the Proceedings of OPIEM 79 (Optics Photonics and I c o n i c s Engineering Meeting), Strasbourg , France, November 26-30, 1979; METROP v o l . 210, OPTIMED v o l . 211, IMAGE-3-D v o l . 212, OPTI-COM v o l . 213. Or, Proceedings of the I n t e r n a t i o n a l Conference  on A p p l i c a t i o n s of Holography and O p t i c a l Data Pr o c e s s i n g , Jerusalem, I s r a e l , August 23-26, 1976 (Pergamon, Toronto,1977) [ l b ] M.R. L a t t a , R.V. P o l e : "Design techniques f o r forming 488-nm holo-graphic lenses w i t h r e c o n s t r u c t i o n at 633 nm", Appl.Opt. 18:14, 2418-2421, 1979; B.J. Chang and CD. Leonard: "Dichromated g e l a t i n f o r the f a b r i c a t i o n of holographic o p t i c a l elements", Appl. Opt. 18:14, 2407-2417, 1979; R.V. P o l e , H.W. W e r l i c h , R.J. Krusche, "Holographic l i g h t d e f l e c t i o n " , Appl. Opt., 17:20, 3294-3297, 1978; W.C. Sweatt: "Designing and c o n s t r u c t i n g t h i c k holographic o t p i c a l elements", Appl. Opt., 17:8, 1220-1227, 1978; W.H. Lee:"Holographic g r a t i n g scanners w i t h a b e r r a t i o n c o r r e c t i o n s " , Appl. Opt., 16:5, 1392-1399, 1977; W.C. Sweatt,; "Achromatic t r i p l e t using holographic o p t i c a l elements", Appl. Opt., 16:5, 1390-1391, 1977; J.R. Magarinos, "Large-aperture holographic s p h e r i c a l b e a m - s p l i t t e r m i r r o r s " , J . Opt. Soc. Am., 67:10, 1374A, 1977; J.A. LaRussa and J.R. Magarinos; " O p t i c a l simulator w i t h holographic component", J . Opt. Soc. Am, 67:10, 1374A, 1977, [also a news rep o r t e n t i t l e d " C o n t r o l of Lab Environment Stressed i n HOE Production at Farrand O p t i c a l F a c i l i t y " , Holosphere, 7:4, 1 & 4-6, 1978]; H.R. Manjunath, S.V. Pappu: "Holo-lens o p t i c a l s p a t i a l frequency processor", Appl. Opt., 15:4, 849-850, 1976; 0. Bryngdahl, W.H. Lee: "Laser beam scanning using computer-generated holograms", Appl. Opt., 15:1, 183-194, 1976; R.V. P o l e , H.P. Wollenmann: "Holographic l a s e r beam d e f l e c t o r " , Appl. Opt., 14:4, 976-980, 1975; G.D. M i n t z , D.K. Morland, W.M. Boerner: "Holo-graphic s i m u l a t i o n of p a r a b o l i c m i r r o r s " , Appl. Opt., 14:3, 564-565, 1975. [2] For example; G. T r i c o l e s , N.H. Farhat: "Microwave holography: a p p l i c a t i o n s and techniques", Proc. of the IEEE, 65:1, 108-120, 1977. '[3] For example: the S p e c i a l Issue on A c o u s t i c Imaging, Proc. IEEE, 67:4, 1979. [4] A. Kozma: "Photographic r e c o r d i n g of s p a t i a l l y modulated coherent l i g h t " , J . Opt. Soc. Am., 56:4, 428-432, 1966. [5] A.A. Friesem, J.S. Zelenka: " E f f e c t s of f i l m n o n - l i n e a r i t i e s i n holography", Appl. Opt., 6:10, 1755-1759, 1967. 103 [6] G. R. Knight: E f f e c t s of f i l m n o n - l i n e a r i t i e s i n wavefront- r e c o n s t r u c t i o n imaging, d o c t o r a l d i s s e r t a t i o n , Stanford U n i v e r s i t y , 1967. [7] J.W. Goodman, G.R. Knight: " E f f e c t s of f i l m n o n - l i n e a r i t i e s on wave f r o n t - r e c o n s t r u c t i o n images of d i f f u s e o b j e c t s " , J . Opt. Soc. Am., 58:9, 1276-1283, 1968. [8] 0. Bryngdahl, A. Lohmann: "Non-linear e f f e c t s i n holography",J. Opt. Soc."Am.,.58:10, 1325-1334, 1968. [9] A. Vander Lugt, F.B. Rotz: "The use of f i l m n o n - l i n e a r i t i e s i n o p t i c a l s p a t i a l f i l t e r i n g " , Appl. Opt., 9:1, 215-222, 1970. [10] A. Kozma, G.W. J u l l , K.O. H i l l : "An a n a l y t i c a l and experimental study of n o n - l i n e a r i t i e s i n hologram r e c o r d i n g " , Appl. Opt., 9:3, 721-731, 1970. [11] H. Dammann: "Phase holograms of d i f f u s e o b j e c t s " , J . Opt. Soc. Am., 60:12, 1635-1639, 1970. [12] W.H. Lee, M.O. Greer: "Noise c h a r a c t e r i s t i c s of photographic emulsions used f o r holography", J . Opt. Soc. Am., 61:3, 402-409, 1971. [13] K.O. H i l l , G.W. J u l l : "Holographic noise l e v e l s i n two s i l v e r h a l i d e r e c o r d i n g media", Optica A c t a , 18:10, 729-742, 1971. [14] B. H i l l : " S p a t i a l noise i n o p t i c a l data-storage systems using amplitude f o u r i e r - t r a n s f o r m holograms", J . Opt. Soc. Am., 61:3, 386-398, 1971. [15] C.H.F. V e l z e l : "Image contrast.and e f f i c i e n c y of n o n - l i n e a r l y r e -corded holograms of d i f f u s e l y r e f l e c t i n g o b j e c t s " , Optica A c t a , 20:8, 585-606, 1973. [16] H.J. G e r r i t s e n : "Non-linear e f f e c t s i n image formation", Appl. Phys. L e t t . , 10:9, 239-244, 1967. [17] 0. Bryngdahl, A.W. Lohmann: "Interferograms are image holograms", J . Opt. Soc. Am., 58:1, 141-142, 1968. 104 [18] P. Hariharan: "Intermodulation noise i n amplitude holograms: The e f f e c t of hologram t h i c k n e s s " , Optica A c t a , 26:2, 211-215, 1979. [19] E.N. L e i t h , J . Upatnieks: "Reconstructed wavefronts and communica-t i o n theory", J . Opt. Soc. Am., 52:10, 1123-1130, 1962. [20] P. E l i a s , D.S. Grey, D.Z. Robinson: " F o u r i e r treatment of o p t i c a l processes", J . Opt. Soc. Am., 42:2, 127-134, 1952; P. E l i a s : "Optics and communication theory", J . Opt. Soc. Am., 43:4, 229-232, 1953; E.L. O'Neil: " S p a t i a l f i l t e r i n g i n o p t i c s " , IRE Trans. Inform. Theory, IT-2:2, 56-65, 1956; E.L. O'Neil (ed.): Communication and  Information Theory Aspects of Modern 'Optics (General E l e c t r i c Co., E l e c t r o n i c Lab., Syracuse, N.Y., 1962). [21] D.H. K e l l y : "Systems a n a l y s i s of the photographic process, I. A three stage model", J . Opt. Soc. Am., 50:3, 269-276, 1960. [22] W.B. Davenport, J r . , W.L. Root: Random" S i g n a l and Noise. (McGraw-H i l l , New York, 1958). [23] J.W. Goodman: "Some e f f e c t s of target-induced s c i n t i l l a t i o n on o p t i c a l radar performance", Proc. IEEE, 53:11, 1688-1700, 1965. [24] J.W. Goodman: I n t r o d u c t i o n to F o u r i e r Optics (McGraw-Hill, New York, 1968). [25] G.C.Sherman: " I n t e g r a l - t r a n s f o r m f o r m u l a t i o n of d i f f r a c t i o n theory", J . Opt. Soc. Am., 57:12, 1490-1498, 1967. [26J J.W. Goodman: S t a t i s t i c a l p r o p e r t i e s of l a s e r speckle patterns i n Topics i n A p p l i e d P h y s i c s , volume 9, Laser Speckle and Related  Phenomena, e d i t e d by J.C. Dainty ( S p r i n g e r - V e r l a g , New York, 1975). [27] J.W. Goodman: " F i l m g r a i n noise i n wav e f r o n t - r e c o n s t r u c t i o n imaging", J . Opt. Soc. Am., 57:4, 493-502, 1967. [28] D. Middleton: "Some general r e s u l t s i n the theory of noise through n o n - l i n e a r d e v i c e s " , Quart. Appl. Math., 5:4, 445-498, 1948; a l s o i n [38, PP 248-301]. [29] W.R. Bennett, S.O. Ric e : "Note on methods of computing modulation, products", P h i l . Mag., Ser. 7, 18:422-424, 1934. 105 [30] W.R. Bennet: "Response of a l i n e a r r e c t i f i e r to s i g n a l and n o i s e " , J . Acoust. Soc. Amer., 15:3, 165-172, 1944; a l s o i n [38, PP.190-198]. [31] S.O. Ric e : "Mathematical a n a l y s i s of random n o i s e " , B e l l Syst. Tech. J . , 23:3, 282-332, 1944; 24:1, 46-156, 1945. [32] D. Middleton: "The d i s t r i b u t i o n of energy i n randomly modulated waves", P h i l . Mag., Ser.7, 42:689-707, 1951. [33] R.F. Baum: "The c o r r e l a t i o n f u n c t i o n of gaussian noise passed through n o n - l i n e a r d e v i c e s " , IEEE Trans. Inform. Theory, IT-15:4, 448-456, 1969; a l s o i n [38, PP.79-87]. [34] D.D. Weiner, J.F. Spina, A.W. F i t c h : "On the c o r r e l a t i o n f u n c t i o n of s i g n a l plus gaussian noise passed through n o n - l i n e a r d e v i c e s " , IEEE Trans. Inform. Theory, IT-17:5, 613-614, 1971. [35] H.B. S h u t t e r l y : "General r e s u l t s i n the mathematical theory of random s i g n a l s and noise i n non - l i n e a r d e v i c e s " , IEEE Trans. Inform. Theory, IT-9:2, 74-84, 1963; a l s o i n [38, PP.68-78]. [36] B.A. Bowen: "The transform method f o r n o n - l i n e a r devices w i t h non-gaussian n o i s e " , IEEE Trans. Inform. Theory, IT-13:2, 326-328, 1967. [37] H. Kaufman, G.E. Roberts: " C o r r e l a t i o n f u n c t i o n expansion at the output of a no n - l i n e a r d e v i c e " , E l e c t r o n . Eng. (London), 35:428, 655-658, 1963. [38] A.H. Haddad(ed-): Non-linear Systems (Dowden, Hutchinson, and Ross, Strudsbury, P.A., 1975). [39] P. Beckmann, A. S p i z z i c h i n o : The S c a t t e r i n g of Electromagnetic  Waves from Rough Surfaces. (Pergamon P r e s s , New York, 1963). [40] D. Middleton: An I n t r o d u c t i o n to S t a t i s t i c a l Communication Theory. (McGraw-Hill, New York, 1960). [41] A. P a p o u l i s : Systems and Transforms w i t h A p p l i c a t i o n s i n Optic s (McGraw-Hill, New York, 1968). 106 [42] A. P a p o u l i s : Random V a r i a b l e s and S t o c h a s t i c processes. (McGraw-H i l l , New York, 1965). [43] P.Y. Ufimtsef: Method of Fringe Waves i n P h y s i c a l Theory of D i f f r a c t i o n (Sovetskoye Radio P r e s s , Moscow, 1962). [44] M.L. Varshavchik, V.O. Kobak: "Cross c o r r e l a t i o n of o r t h o g o n a l l y p o l a r i z e d components of electromagnetic f i e l d s c a t t e r e d by an extended o b j e c t " , Radio Eng. E l e c t r o n . Phys., 16:2, 201-205, 1971. [45] G.L. Rogers: " P o l a r i z a t i o n e f f e c t s i n holography", J . Opt. Soc. Am., 56:6, 831, 1966. [46] R.J. C o l l i e r , C.B. Burckhardth, L.H. L i n : O p t i c a l Holography, (Academic P r e s s , New York, 1971). [47] G. T r i c o l e s , E.L. Rope, R.A. Hayward: "Some examples of how wave p o l a r i z a t i o n and surface d i s c o n t i n u i t i e s i n f l u e n c e microwave holo-graphic imaging", presented at the 1979 I n t e r n a t i o n a l IEEE/APS Symposium and N a t i o n a l Radio Science Meeting, S e a t t l e , Washington, June 18-22, 1979. [48] N. George, A. J a i n , R.D.S. M e l v i l l e , J r . : "Speckle d i f f u s e r s and d e p o l a r i z a t i o n " , Appl. Phys. 5:1, 65-70, 1975. [49] G.N. Watson:•"Notes on generating f u n c t i o n s of polynomials: (1) Lagurre Polynomials", J . Lond. Math. S o c , 8:31, 189-192, 1933. [50] Jf.F. B a r r e t t , D.G. Lampard: "An expansion f o r some second-order p r o b a b i l i t y d i s t r i b u t i o n s and i t s a p p l i c a t i o n to noise problem", IRE Trans. Inform. Theory, IT-1:1, 10-15, 1955. [51] G.E. Roberts, H. Kaufman: Tables of Laplace Transforms (Saunders, P h i l a d e l p h i a , 1966). [52] R.C. Waag, K.T. Knox: "Power-spectrum a n a l y s i s of exponential d i f -f u s e r s " , J . Opt. Soc. Am., 62:7, 877-881, 1972. [53] H. Ghandeharian, W.M. Boerner: " A u t o c o r r e l a t i o n of transmittanee of holograms made of d i f f u s e o b j e c t s " . O p t i c a Acta,24:11, 1087-1097, 1977; "Non-linear e f f e c t s i n holography", J . Opt. Soc. Am., 67:10, 1433-1434A, 1977. 107 [54] H. Ghandeharian, M.P. Beddoes: " A u t o c o r r e l a t i o n of output of memoryless no n - l i n e a r devices w i t h a r b i t r a r y c h a r a c t e r i s t i c s " , IEEE Trans. Inform. Theory, IT-24:6, 779-782, 1978. [55] J . J . Jones: " H a r d - l i m i t i n g of two s i g n a l s i n random n o i s e " , IEEE Trans. Inform. Theory, IT-9:1, 34-42, 1963. [56] P.D. Shaft: " L i m i t i n g of s e v e r a l s i g n a l s and i t s e f f e c t s on com-munication system performance", IEEE Trans. Commun. Technol., C0M-13:4, 504-512, 1965. [57] J . J . Sevy: "The e f f e c t of m u l t i p l e CW and FM s i g n a l s passed through a hard l i m i t e r or TWT", IEEE Trans. Commun. Technol., COM-14:5, 568-578, 1966. [58] P.D. Shaft: " S i g n a l s through n o n - l i n e a r i t i e s and the suppression of undesired i n t e r m o d u l a t i o n terms", IEEE Trans. Inform. Theory, IT-18:5, 657-659, 1972. [59] I.S. Gradshteyn, I.M. Ryzhik: Table of I n t e g r a l s , S e r i e s , and P r o - ducts (Academic Press, New York, 1968). [60] M. G y i : "Some t o p i c s on l i m i t e r s and FM demodulaters", Stanford E l e c t r o n Lab., Stanford Univ., S t a n f o r d , C a l i f . , Rep. SEL-65-056, 1965. [61] C.B. Burckhardt: " D i s p l a y of holograms i n white l i g h t " , B e l l Syst. Tech. J . , 45:10, 1841-1844, 1966. [62] U. Langbein, F. Lederer: " S p a t i a l f i l t e r i n g p r o p e r t i e s of volume holograms", Opt. Quant. E l e c t . , 11:1, 29-42, 1979. 108 

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