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Ellipsometric investigation of the mechanism of hologram storage in lithium niobate Wong, William K. Y. 1973

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ELLIPSOMSTRIC INVESTIGATION OF THE MECHANISM OF HOLOGRAM STORAGE IN LITHIUM NIOBATB by W i l l i a m K.Y. Wong , B.A.So., U n i v e r s i t y of B r i t i s h Columbia, 1971 A THESIS. SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l E n g i n eering We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA ' AUGUST, 1 973 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a 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 r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p urposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date ABSTRACT A computer-controlled e l l i p s o m e t e r was used to o b t a i n data on the b i r e f r i n g e n c e changes induced i n l i t h i u m niobate c r y s t a l s by i r r a d i a t i o n w i t h an argon-ion laser,, The instrument was set to take readings on a r e c t a n g u l a r g r i d of p o i n t s before and a f t e r i r r a d i a t i o n . I r r a d i a t i o n was performed w i t h c i r c u l a r and w i t h narrow r e c t a n g u l a r l i g h t beams. R e s u l t s are compared w i t h the p r e d i c t i o n of the model of Chen i n which he p o s t u l a t e d the presence of an i n t e r n a l f i e l d E q i n the d i r e c t i o n of the c - a x i s of the c r y s t a l . According to h i s theory, e l e c t r o n s , p h o t o e x c i t e d from t r a p s by the l a s e r l i g h t , d r i f t under the i n f l u e n c e of E q along the c - a x i s before being retrapped. The r e s u l t i n g space-charge f i e l d set between p o s i t i v e i o n i s e d centres and trapped e l e c t r o n s g i v e s r i s e to the observed b i r e f r i n g e n c e changes v i a the l i n e a r e l e c t r o - o p t i c e f f e c t of the c r y s t a l , Approximate mathematical models based on Chen's theory are used to s o l v e the problem w i t h narrow r e c t a n -g u l a r l i g h t beams. R e s u l t s are compared w i t h experiments. i TABLE OF CONTENTS Page A b s t r a c t . . . o . o i Table of Contents . . » . o . . . . • • * . . . . . . . . i i L i s t of I l l u s t r a t i o n s . . . . . . . . . i i i Acknowledgement T I. INTRODUCTION . 1 I I . THE GENERAL EXPERIMENTAL ARRANGEMENT . . . . . . . . . 3 1 I n t r o d u c t i o n 3 2 E l l i p s o m e t e r s et u p . . 3 3 O p t i c a l damage procedure. . . . . . 6 4 R e f r a c t i v e Index Measurements from E l l i p s o m e t e r Readings . . . . . . . . . . 9 '5 B a l a n c i n g Procedure . . . . . . . . . . . . . . . 13 I I I . CIRCULAR LASER BEAM EXPERIMENT 15 1 Experiments . . . . . . 15 2 Q u a l i t a t i v e E x p l a n a t i o n . . . . . 15 3 E l e c t r o - o p t i c e f f e c t i n LiNbOj 21 4 Q u a n t i t a t i v e M o d e l l i n g of O p t i c a l Damage, . . . . 25 IV. NARROW SLIT EXPERIMENT . . . . . 29 1.1 S l i t p e r p e n d i c u l a r to the c - a x i s . . . . . . . . . 29 1.2 Theory. . . . . . . . . . . . . . . . . . . . . . 30 1.3 Experimental R e s u l t s . . . . . . . . . . . . . . . 36 2.1 S l i t p a r a l l e l to c - a x i s 0 . . . . . . . . . . . . 36 Theory. . . . . . . . . . . . . . . . o o . . . . 36 3 F u r t h e r Experiment to i n v e s t i g a t e i n t e r n a l f i e l d . 43 V. CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . 47 APPENDIX . . . . . . . . . . . . . . . 49 REFERENCES . 57 i i LIST OF ILLUSTRATIONS Figure Page II- l Rate of cooling of crystal after baking to 500°C for l/2 hour . . . . . < > » . . 4 II-2 Computer-Ellipsometer interface. Dashed lines indicate connection to the computer interface . „ 5 II-3 Sample holder for LiNbO,, crystal, . 0 . 0 0 0 0 . 7 II-4 Temperature controlling box . . . o . . o o . o » 8 II- 5 Resolution of light into components as i t passes through the ellipsometer. . . . 0 0 . . 10 III- l Map of A(n - n ) . 0 . . 0 0 . . . . . 16 r e 0 III-2 Sectional view through the c-axis of Figure 1 . <> 17 III-3 Sectional view through the b-axis of Figure 1 . . 18 III-4 A(n -n ) versus exposure time for a circular beam0 19 III-5 The space-charge field E s c shown in this figure would explain the plot of A{n -n ) of Figures 1,2,3 6. °. 20 III- 6 Three-dimensional plots showing the spatial distribution of (a) the space charge with h i l l s and valleys signifying positive and negative charge, (b) the x- component of the internal field and (c) the y component of the internal field . 26 IV- 1 For a sufficiently narrow strip of light placed across the crystals as shown, drift would be the dominant mechanism affecting the movement of electrons along the c-axis ...<,.<,<>.... 29 IV"" 2 3?l0*fc O f l l ( x ) o o o o o o o * o o o o o « o * o o » 33 IV-3 Plot.of g(x), n(x) and E s c(x), the steady state solution of the problem formulated in section 1.2 o » o . « « . o o . « . o o . o * . « o o o o 35 IV-4 Plot of A(ne-n0) along the c-axis for a s l i t width of 0.06+. 0.002 cm „ . . . . „ o . 37 i i i Figure Page IV-5 For a s u f f i c i e n t l y narrow s t r i p of l i g h t placed across the c r y s t a l as shown, d i f f u s i o n would be the dominant mechanism a f f e c t i n g the movement of e l e c t r o n s i n a d i r e c t i o n p e r p e n d i c u l a r to the c - a x i s 38 IV-6 P l o t of h ( x ) , h (x) and h (x) 40 IV-7 P l o t of g ( x ) , n(x) and E s c ( x ) , the steady s t a t e s o l u t i o n o f the problem formulated i n s e c t i o n 2.2 . . . „ 42 IV-8 P l o t of (n'-n ) along the c- a x i s before d e p o s i t i n g e l e c t r o d e s ( s o l i d l i n e ) and a f t e r d e p o s i t i n g e l e c t r o d e s and s h o r t i n g them (dotted l i n e ) • > . . . . 44 IV-9 P l o t of A ( n e - n 0 ) f o r a c i r c u l a r damaging beam placed a t 3 p o s i t i o n s along the c - a x i s . . . . 45 IV-10 V a r i a t i o n s of E along the c - a x i s to s a t i s f y the c r i t e r i o n that j'Edx = 0 along the c - a x i s 46 i v ACKNOWLEDGEMENT I am most grateful to Dr. L. Young for his helpful suggestion and guidance during the course of this work. The financial support of this work by the Canadian Defence Research Board (DRB Grant 5501-67) i s most gratefully acknowledged. Grateful acknowledgement for financial support i s also given to the National Research Council (scholarship awarded 1972-1973) and the University df B r i t i s h Columbia (fellowship awarded 1971-1972). I wish to thank Mr. W. Cornish and Mr. M. Thewalt for many helpful suggestions and technical assistance. Finally I wish to thank Miss Norma Duggan and Miss Betty Cockburn for help i n typing the thesis. v I . INTRODUCTION Modern data processing systems use a h i e r a r c h y of storage devices ranging from cores, d i s k s , tapes to semiconductor memories. Mass o p t i c a l memories'1" are c u r r e n t l y under a c t i v e c o n s i d e r a t i o n as a powerful a d d i t i o n to t h i s h i e r a r c h y . Various types of o p t i c a l memories have been proposed and i n some cases r e a l i z e d . One p a r t i c u l a r type i s the l a r g e s c a l e read-write o p t i c a l memory based on hologram storage on an erasable medium and compound addressing by pages and w i t h i n pages. An a r r a y of l i g h t v a l v e s composes the page to be s t o r e d . The d i r e c t i o n of i l l u m i n a t i o n of the page composer, r e s u l t i n g from d e f l e c t i o n of a l a s e r beam, determines the l o c a t i o n on the storage medium at which a hologram of the page w i l l be recorded (and the previous r e c o r d erased). The page composer and sensor a r r a y are p h y s i c a l l y combined w i t h an e l e c t r i c a l l y addressable memory. This type of o p t i c a l memories seems to be of p a r t i c u l a r promise as an extension of t h i s h i e r a r c h y where l a r g e Q amounts of i n f o r m a t i o n ( > 10 b i t s ) must be f a i r l y r a p i d l y a c c e s s i b l e (~ ^ s e c ) on a page by page b a s i s . This t h e s i s i s concerned w i t h s t u d y i n g the mechanism by which i n f o r m a t i o n i s s t o r e d i n one of the p o s s i b l e candidates f o r the storage medium - l i t h i u m niobate (LiNbO^). L i t h i u m niobate has many advantages as f a r as holographic storage i s concerned: holographic d i f f r a c t i o n e f f i c i e n c y as h i g h as 80f> has been achieved i n l i t h i u m niobate doped 2 w i t h i r o n ; storage times of many days are p o s s i b l e at o r d i n a r y room temperature and erasure can be achieved e i t h e r o p t i c a l l y or by h e a t i n g o 3 the c r y s t a l to 300 C ; the holograms s t o r e d can a l s o be ' f i x e d ' t h e r m a l l y 4 so that the p a t t e r n s are o p t i c a l l y non-erasable ; no b l e a c h i n g of the 2 hologram i s r e q u i r e d . The mechanism by-which i n f o r m a t i o n i s s t o r e d i n l i t h i u m niobate i s v i a ' o p t i c a l damage 1, that i s , l i t h i u m niobate responds to l a s e r l i g h t of the appr o p r i a t e wavelength w i t h change i n the r e f r a c t i v e index. Although t h i s e f f e c t i s d e t r i m e n t a l to the o p e r a t i o n of t h i s m a t e r i a l i n e l e c t r o - o p t i c modulation and second harmonic genera t i o n , 6 7 Chen and others have used t h i s e f f e c t to s t o r e t h i c k phase holograms i n l i t h i u m niobate. With these f a c t s i n mind, the purpose of the t h e s i s i s to i n v e s t i g a t e the mechanism of holographic storage i n l i t h i u m n iobate by study i n g the o p t i c a l damage u s i n g a computer-controlled e l l i p s o m e t e r . Chapter I I gives a b r i e f d e s c r i p t i o n of the experimental set up, the o p t i c a l damage procedure and shows how the e l l i p s o m e t e r readings are r e l a t e d to r e f r a c t i v e index measurements. In Chapter I I I , the patterns of r e f r a c t i v e index change due to a c i r c u l a r damaging l a s e r beam i s investigated.. The r e s u l t s are e x p l a i n e d q u a l i t a t i v e l y u s i n g Chen's model. A b r i e f review of the e l e c t r o - o p t i c e f f e c t i n l i t h i u m niobate i s then g i v e n . F i n a l l y , numerical methods based on King's a n a l y s i s of o p t i c a l damage i n KTN are used to t r e a t t h i s problem q u a n t i t a t i v e l y and the r e s u l t s compared w i t h experiments. I n Chapter IV, the pa t t e r n s of r e f r a c t i v e index change due to a narrow s t r i p of l a s e r l i g h t i s investigated'. A s i m p l i f i e d theory based on Chen's model i s solved a n a l y t i c a l l y u s i n g Laplace transform method. This method i s used since i t can e a s i l y be modifi e d to solve f o r any other l i g h t d i s t r i b u t i o n s . The r e s u l t s are then compared w i t h experiments. F i n a l l y , the r e s u l t s of an experiment r e l a t i n g to the proposed i n t e r n a l f i e l d are discussed. The c o n c l u s i o n s that can be drawn from the above s t u d i e s are remarked upon i n Chapter V. 3 II. THE GENERAL EXPERIMENTAL ARRANGEMENT 1. Introduction The lithium niobate crystal used i n the experiments was supplied by Harshaw Chemical Company and i t s dimensions are 2.0 x 1.5 x 3 2 / 0.3 cm . The 2.0 x 1.5 cm faces are polished f l a t to 1/2 wavelength and parallel within one minute at sodium 'D' light within 1 mm of the edge or better. The optical damage was performed by exposing the crystal to a Coherent Radiation argon-ion laser (maximum output=500mW) for specified periods of time. The power of the laser light was measured by a Jordan optical power wattmeter,, The resulting optical damage was then probed by placing the crystal i n an automated ellipsometer. Before each experiment, the crystal was heated to 500°C for half an hour and then cooled slowly., This treatment was shown to anneal out any space-charge f i e l d resulting from optical damage of the previous experiment. Figure 1 shows the rate of cooling 2. Ellipsometer set up The Rudolph' Thin Film Ellipsometer (type 43603-200E), used i n ' this project, was interfaced to a DEC PDP-8/E computer. Only minor additions were required to the existing interface. Figure 2 shows the basic set up> Light from a Spectra Physics helium-neon laser (model 133) passed through a green f i l t e r which cut down the intensity of the laser light so that this probing beam would not optically damage the crystal. (The crystal could be l e f t exposed to this reduced laser light for several hours without showing any sign of being optically damaged.) Figure 1: Rate of c o o l i n g of c r y s t a l a f t e r baking to 500°C f o r 1/2 hour. o - C o - —0 GREEN FILTER SHAFT ENCODER MOTOR c — n CRYSTAL QWP POLARIZER ANALYZER PHOTO MULTIPLIER VARIABLE GAIN i AMPLIFIER o Figure 2: Computer-Ellipsometer i n t e r f a c e . Dashed l i n e s i n d i c a t e connection to the computer i n t e r f a c e . 6 A f t e r p a s s i n g through the p o l a r i s e r , the l i g h t was l i n e a r l y p o l a r i s e d . The quarter-wave p l a t e (QWP) converted the l i n e a r l y p o l a r i s e d l i g h t i n t o e l l i p t i c a l l y p o l a r i s e d l i g h t which was then i n c i d e n t normal to the c r y s t a l . Any l i g h t emerging from the a n a l y s e r went i n t o a photo-m u l t i p l i e r and then i n t o a v a r i a b l e g a i n a m p l i f i e r i n t e r f a c e d to the computer. Since the r e f r a c t i v e index i n l i t h i u m niobate i s temperature 9 dependent , the temperature must be kept constant d u r i n g ~ measurement. The c r y s t a l , h e l d i n a sample holder ( f i g u r e 3), was pla c e d i n a p l e x i g l a s s box w i t h openings at opposite ends of the box to a l l o w the probing l a s e r beam to enter and leave ( f i g u r e 4 ) . R e s i s t o r s were used as h e a t i n g elements and together w i t h a f a n , a Yellow Springs Instruments p r o p o r t i o n a l temperature c o n t r o l l e r and a t h e r m i s t o r as sensing element, the temperature was maintained at 32 +_ 0.05°C. Throughout the whole experiment, the temperature was f u r t h e r monitored by i n s e r t i n g i n t o the box an iron-constantan thermocouple and connecting the output to a s t r i p c h a r t r e c o r d e r . Stepping motors were mounted to the s h a f t of the sample holder i n such a way that the c r y s t a l c o u l d be moved i n two mutually p e r p e n d i c u l a r d i r e c t i o n s normal to the l i g h t beam. 3. O p t i c a l damage procedure Before o p t i c a l damage, the c r y s t a l was placed i n the set up shown i n f i g u r e 2. A f t e r alignment of the c r y s t a l so the the probing l a s e r beam passed normally through i t , the e l l i p s o m e t e r was balanced. The b a l a n c i n g procedure w i l l be d e s c r i b e d i n a l a t e r s e c t i o n . Stepping motors then moved the c r y s t a l to a d i f f e r e n t p o s i t i o n and the Opening Figure 3- Sample holder f o r LiNbO c r y s t a l . TO STEPPING MOTORS Figure .4: Temperature c o n t r o l l i n g box. co 9 same procedure was repeated. Continuing i n t h i s way, a map " of the r e f r a c t i v e index v a r i a t i o n s throughout the c r y s t a l was obtained. The data was simu l t a n e o u s l y recorded on paper tapes as the e l l i p s o m e t e r was t a k i n g r e a d i n g . The c r y s t a l was then removed from the set up and exposed to the a r g o n - i o n l a s e r f o r a s p e c i f i e d p e r i o d o f time. I t was then r e p l a c e d i n the e l l i p s o m e t e r and the b a l a n c i n g r o u t i n e repeated. A re f e r e n c e mask attached to the sample h o l d e r enabled the c r y s t a l to be scanned a t the same g r i d - p o i n t s before and a f t e r the o p t i c a l damage. I n t h i s way a three-dimensional map of the r e f r a c t i v e index v a r i a t i o n s a f t e r o p t i c a l damage was again obtained and the data a g a i n recorded on paper tapes. The d i f f e r e n c e between the two s e t s of readings gave us a data map of the induced change i n the r e f r a c t i v e index o f the c r y s t a l . A program was w r i t t e n f o r the PDP-8/E to s u b t r a c t the corresponding numbers on the two paper tapes. The r e s u l t i n g data (on paper tapes) was then copied onto 9-track magnetic tapes i n the IBM 360/67 computer u s i n g programs provided by the UBC Computing Centre. A p e r s p e c t i v e view of the induced r e f r a c t i v e index change was then p l o t t e d u s i n g the Computing Centre's program UBC PERSP. 4. R e f r a c t i v e Index Measurements from E l l i p s o m e t e r Readings The p o l a r i s e r and a n a l y s e r are Glan-Thompson prisms mounted i n graduated c i r c l e s which were r o t a t e d by ste p p i n g motors. D e c i t r a k s h a f t encoders (TR-5HC-CW / D ) which were mounted on the analy s e r and p o l a r i s e r converted the angles to BCD f o r input to the computer. The ste p p i n g motors and s h a f t encoders were both m u l t i p l e x e d to one device code. The e l l i p s o m e t e r measured P and A, the azimuths of the e l e c t r i c f i e l d v e c t o r t r a n s m i t t e d by the p o l a r i s e r and an a l y s e r r e s p e c t i v e l y . A f t e r passing through the p o l a r i s e r , the l i g h t i s plane p o l a r i s e d i n the d i r e c t i o n P (Figure 5 ). R e s o l v i n g t h i s i n t o components along the slow and f a s t a x i s of the QWP, we get E F = cos(Q-P) E g = sin(Q-P) ( l ) where Q i s the azimuth of the f a s t a x i s of the QWP. A f t e r passing through the QWP. the slow a x i s component of l i g h t i s delayed by 6 r e l a t i v e to the f a s t a x i s (s i s the r e t a r d a t i o n of the QWP) and assuming no a t t e n u a t i o n due to the QWP we have Ep 1 = E p E s« = E s exp -j6 (2) R e s o l v i n g t h i s i n t o components along the X and Y a x i s ( n g and n^ a x i s of the LiNbO^) we then get E x = Ej,' cos Q + Eg' s i n Q ^ ) = E ' s i n Q - Eg' cos Q 11 LiNbO^ i s a negative u n i a x i a l c r y s t a l , that i s , n > n and 3 ' o e hence a f t e r passing through the c r y s t a l , E i s delayed by cf> r e l a t i v e A to E y E x« = E x exp E y' = E y ( 4 ) To be extinguished by the analyser, the l i g h t must be plane p o l a r i s e d , or s t a t i n g i t i n another way (phase of E ) - <f> = (phase of E ) (5) A 1 S u b s t i t u t i n g ( l ) and ( 2 ) i n t o ( 3 ) , we have E = cos (Q-P) cos Q + s i n (Q-P) s i n Q exp - j 6 A . E y = cos (Q-P) s i n Q - s i n (Q-P) cos Q exp - j & or E x = C + C 2 exp - j 6 E y = C 5 + C 4 exp - j 6 (6) where = cos (Q-P) cos Q = s i n (Q-P) s i n Q = cos (Q-P) s i n Q ° 4 = ~ s i n ^ ~ P ^ c o s ^ Now exp - j <$ = cos 5 - j s i n <$ Hence equation ( 6 ) can be w r i t t e n E x = (C + C 2 cos 6 ) _ j c 2 s i n 6 E y = (Q + C 4 cos 6 ) - j C 4 s i n g From equations ( 7 ) we get - c 2 tan <f>x = tan (phase of E ) = C l + °2 C 0 S < S - C, ( 7 ) tan <j>v = tan (phase of Ev) = ——, _ • y Y' c_ + c. coss 3 4 From equation (5) we have tan (phase of E„ - phase of E ) = tan A> (8) A I 12 tan 4> v - tan <f> or J 1 + tan <j) x tan * y = t a n * (9) F i n a l l y s u b s t i t u t i n g (8) i n t o (9) and s i m p l i f y i n g we get s i n 6 s i n 2(Q-P) t a n * = = s i n 6 tan 2(Q-P) cos 2 (Q-P) For a p e r f e c t QWP 6' = 90° and our QWP was set w i t h Q = -45° and hence t a n * = tan (-|- - 2P) Hence * =- — - 2P Z nir n=0,l,2, I f <J>.,P^  are the phase change and p o l a r i s e r reading before o p t i c a l damage as the l a s e r l i g h t t r a v e r s e d the c r y s t a l and are the phase change and p o l a r i s e r reading a f t e r o p t i c a l damage, then <f> . = - • £ - . 2P . ± nir 1 2 1 • f - - J - 2Pf h i n=0,1,2,3. or (*f - «J.±) = 2(P ± - P f) The phase change * i s r e l a t e d to the r e f r a c t i v e index of the c r y s t a l by 2n * = — d (n -n ) X 6 where d = t h i c k n e s s of c r y s t a l as t r a v e r s e d by the l a s e r beam X = vacuum wavelength of the l a s e r l i g h t n g= e x t r a o r d i n a r y r e f r a c t i v e index n Q= o r d i n a r y r e f r a c t i v e index (n - i i ) i s c a l l e d the b i r e f r i n g e n c e of the c r y s t a l . Hence ' 2TC <j>. = — d (n -n ) X 6 ° 2TI chp = — d [ (n -n ) + A(n -n ) 1 ^ , e 0 e 0 13 where A(n -n ) i s the change i n b i r e f r i n g e n c e due to o p t i c a l damage, e o S u b t r a c t i n g we get 271 <4>f " * ±) = — d A ( n e _ n o ) K Combining equations (lO) and ( l l ) we f i n a l l y get A(n e - n o ) = . (12) ixd Equation ( l 2 ) i s the b a s i c equation r e l a t i n g the r e f r a c t i v e index change due to o p t i c a l damage and the p o l a r i s e r r e a d ings. 5. Balancing Procedure A f t e r the balan c i n g program i s loaded i n t o the computer and s t a r t e d , the v a r i o u s p o i n t e r s are i n i t i a l i s e d . The computer then w a i t s u n t i l an i n s t r u c t i o n i s typed on the keyboard of the t e l e t y p e . There are s e v e r a l i n s t r u c t i o n s which can be i s s u e d ; undecoded i n s t r u c t i o n s are ignored. Under normal b a l a n c i n g , the i n s t r u c t i o n BE i s i s s u e d (BE = balance e l l i p s o m e t e r ) and the p o l a r i s e r i s balanced f i r s t . The program determines which way i t must step the stepping motor of the p o l a r i s e r i n order to minimise the e r r o r s i g n a l of the p h o t o m u l t i p l i e r . I t steps the motor i n that d i r e c t i o n u n t i l a set of 64 p h o t o m u l t i p l i e r readings are taken (one a f t e r each step) and summed. The motor continues stepping u n t i l the e r r o r s i g n a l goes through a minimum and s t a r t s to inc r e a s e again. A second sum of readings are then taken as the motor steps and i s c o n t i n u o u s l y updated to c o n t a i n o n l y the 64 most recent readings. When t h i s second sum equals the f i r s t sum found on the other s i d e of the minimum, the balance p o i n t , which i s midway between the two equal sums, has been found. The an a l y s e r i s then balanced i n the same 14 manner. The p o l a r i s e r i s then again balanced. The p o l a r i s e r and ana l y s e r readings are' then p r i n t e d out on the t e l e t y p e . I I I . CIRCULAR LASER BEAM EXPERIMENT 1. Experiments F i g u r e 1 shows a map of the change i n b i r e f r i n g e n c e A(n -n ) e o when the c r y s t a l i s damaged by the argon-ion l a s e r (output=200 mW, r a d i u s of beam=0.5 mm) f o r 10 seconds. A s e c t i o n a l view of the p l o t through the c - a x i s o f the c r y s t a l i s shown i n f i g u r e 2. One n o t i c e s t h a t A(n -n ) r e v e r s e s s i g n near the beam edge w h i l e i t remains negative 6 0 i n s i d e the beam diameter. A s e c t i o n a l view at r i g h t angles to the c - a x i s i s shown i n f i g u r e 3 and A ( n e ~ n Q ) stays negative a l l the way. F i g u r e 4 shows a p l o t of A ( n g - n o ) versus exposure time f o r a c i r c u l a r damaging beam (radius=0.5 mm). Ain^-n^) i s l i n e a r i n time up to an exposure time of 20 seconds. T h i s corresponds to an exposure energy d e n s i t y of 500j/cm' which agrees w e l l w i t h . p r e v i o u s l y r e p o r t e d r e s u l t s . 2. Q u a l i t a t i v e E x p l a n a t i o n I n order to e x p l a i n o p t i c a l damage, Chen"^ had to p o s t u l a t e the presence of an i n t e r n a l f i e l d E q of unknown o r i g i n i n the d i r e c t i o n of the c - a x i s of the c r y s t a l . Conduction e l e c t r o n s , photoexcited from t r a p s by l a s e r l i g h t of the a p p r o p r i a t e wavelength, d r i f t under the i n f l u e n c e of t h i s f i e l d along the c - a x i s of the c r y s t a l f o r some d i s t a n c e before being retrapped. Assuming t h a t thermal detrapping i s n e g l i g i b l e , the e l e c t r o n s s t a y trapped. The space-charge f i e l d E set up between SC the p o s i t i v e i o n i s e d centres i n the i l l u m i n a t e d area and the trapped e l e c t r o n s then g i v e s r i s e to the observed r e f r a c t i v e index change v i a the l i n e a r e l e c t r o - o p t i c e f f e c t of the c r y s t a l . 16 &(ne-rh) X 1 0 5 * c - a x i s ( c m ) Figure 2: S e c t i o n a l view through the c - a x i s of F i g u r e 1. Figure 4: A ( n e - n o ) versus exposure time f o r a c i r c u l a r beam. 20 c-axis F i g u r e 5*. The space-charge f i e l d E s c shown i n t h i s f i g u r e would e x p l a i n the p l o t of A ( n e - n 0 ) of F i g u r e s 1,2,3. Figure 5 i n d i c a t e s s i t u a t i o n f o r a c i r c u l a r damaging beam. I t i s obvious that the space-charge f i e l d (shown here as l i n e s of f o r c e ) ' E reverses s i g n along the c-a x i s but not at r i g h t angles to i t . 11 Johnston proposed an a l t e r n a t e theory i n which the photo-e x c i t a t i o n of e l e c t r o n s by the damaging l a s e r l i g h t c r e a t e s an excess of empty donor s i t e s w i t h i n the i l l u m i n a t e d area and leads to a l o c a l i n c r e a s e i n the macroscopic p o l a r i s a t i o n w i t h an accompanying p o l a r i s a t i o n charge p., = -V, P / 0. The e l e c t r i c f i e l d , due to |Op i s t h e r e f o r e created a n t i p a r a l l e l to the spontaneous p o l a r i s a t i o n which causes the f r e e c a r r i e r s to d r i f t along the c - a x i s u n t i l they become retrapped i n shallow traps outside the i l l u m i n a t e d area. The f i e l d , due to the p o l a r i s a t i o n charge, r e p l a c e s the i n t e r n a l f i e l d E o p o s t u l a t e d by Chen. Steady s t a t e i s reached when the e l e c t r i c f i e l d r e s u l t i n g from the g r a d i e n t of the p o l a r i s a t i o n i s balanced by the 21 space-charge f i e l d caused by the e l e c t r o n s which have moved out from the i l l u m i n a t e d area. The s p a t i a l l y v a r y i n g p o l a r i s a t i o n r e s u l t i n g from the change i n p o l a r i s a t i o n d e s c r i b e d above and the e x c i t a t i o n , m i g r a t i o n and t r a p p i n g o f e l e c t r o n s induces a s p a t i a l v a r i a t i o n i n the 12 r e f r a c t i v e index v i a the e l e c t r o - o p t i c e f f e c t . According to Amodei , the amount of charge t r a n s f e r i n v o l v e d according to Johnston's theory i s too l a r g e to be p r a c t i c a l l y r e a l i z a b l e . 3. E l e c t r o - O p t i c e f f e c t i n LiFbCu The e l e c t r o - o p t i c e f f e c t i s de f i n e d as the change i n the r e f r a c t i v e index of a m a t e r i a l when a f i e l d i s a p p l i e d to i t . For an i s o t r o p i c medium, the d i e l e c t r i c p r o p e r t i e s at o p t i c a l f r e q u e n c i e s are g i v e n by D - - (1) E q = p e r m i t t i v i t y o f f r e e space e = d i e l e c t r i c constant of the medium D = displacement E = e l e c t r i c f i e l d The r e f r a c t i v e index n i s d e f i n e d as n = /e~. For an a n i s o t r o p i c medium equation ( l ) has to be re p l a c e d by D. = e e . .E i o i j I t can be shown th a t f o r t h i s case two waves, of d i f f e r e n t v e l o c i t i e s , may i n gene r a l propagate through the c r y s t a l f o r a given wave normal. Each wave has i t s own r e f r a c t i v e index. The p r i n c i p a l r e f r a c t i v e i n d i c e s n.^ n^, are then a c c o r d i n g l y defined as where e ^ , E j , e ^ are the p r i n c i p a l d i e l e c t r i c constants. The o p t i c a l p r o p e r t i e s of a c r y s t a l are o f t e n described i n terms of the index e l l i p s o i d ( i n d i c a t r i x ) . The equation of t h i s s u r f a c e i s 2 2 2 X l X2 X3 = 1 2 2 2 n l n 2 n3 where the co-ordinates x^ are p a r a l l e l to the axes of the e l l i p s o i d and n^ are the p r i n c i p a l r e f r a c t i v e i n d i c e s . For d e s c r i b i n g the e l e c t r o - o p t i c e f f e c t where an e l e c t r i c f i e l d i s a p p l i e d , the general equation of the i n d i c a t r i x then becomes J [ - ^ - 6 . . + Z. E. + R. .. -B.-E, •+ ] x.x = 1 (2) L n 2 i j i j k k l j k l k l J i j i , j , k , l i j where the i n d i c e s i , j , k , 1 run from 1 to 3. The Z. ., and R. .... are I J K i j k i l i n e a r and q u a d r a t i c e l e c t r o - o p t i c tensor components r e s p e c t i v e l y . The i n d i c e s i , j can be interchanged as can k and 1, and the u s u a l c o n t r a c t i o n s can be made r . «—> Z/. .\. and R < — • R/..w, ,s mk U j ; k mn . ( i j ) ( k l ; where m and n run from 1 to 6 and m i s r e l a t e d to ( i j ) and n to ( k l ) as f o l l o w s : W l l , 2+-*22, 3<-*33, 4*->23, 5+-+13, 6WL2. For the case of the l i n e a r e l e c t r o - o p t i c e f f e c t equation ( 2 ) becomes I [ —5- 6. . + Z. .. E. ] x . x . = 1 23 The form of the r ^ matrix of electro-optic coefficients for LiNbO^ (class 3m) i s 0 0 0 0 12 •22 0 13 "23 '33 L42 '51 61 0 0 0 0 0 Symmetry requires r 23 "13' x"51 ~ r42' x'22 - i12 ~A'61* Writing out i n f u l l , the equation of the. indicatrix for "13 * r 2 2 = 3.4 x 10 T _ , » 30.8x 10' 33 '42 r, = - r , „ = - r , -10 cm/Volt -10 cm/Volt -10 cm/Volt f 10 cm/Volt LiFbO„ then becomes 3 + r n oE„ + r n„E„ ] x n n., 12 2 13 3 + r„„E 0 + r 0 _ E , ] x 0 n,. 22 2 '23 3 '33 3 2 £3 + r„E„ ] x" + 2r^,E.x nx^ + 2r„0E„x^x., + 2r^ nE nx.x, = 1 "42 2 2 3 • 5 1 1 1 3 where E^, E 2, E^ are the electric f i e l d strength components i n the x l * X2* z 3 d l r e c f l ° n s respectively. For a f i e l d (either applied or internal) paralle 1 to x^ ( the c-axis of the crystal) the indicatrix becomes + r13 E3 ] x l + [ n. + r 2 3 E 3 ] x 2 + [ 2 n3 + r 3 3 E 3 ] x! Thus there are modifications only to the axis length but no rotation of the principal axes of the index e l l i p s o i d . The indicatrix has the following important properties. I f a ->• wavefront has i t s normal i n a certain direction P, then the two wave-fronts normal to P which may be propagated through the crystal have 24 r e f r a c t i v e i n d i c e s equal to the semi-axes of the e l l i p s e obtained i n the f o l l o w i n g way. Draw through the o r i g i n of the i n d i c a t r i x a s t r a i g h t -> l i n e i n the d i r e c t i o n of P . Draw the c e n t r a l s e c t i o n of the i n d i c a t r i x p e r p e n d i c u l a r to i t . This w i l l be the e l l i p s e whose major and minor axes are the r e s p e c t i v e i n d i c e s . For the experimental set up, a l i g h t wave (the probing helium-neon l a s e r ) i s propagating i n the d i r e c t i o n and w i t h space-charge f i e l d E_ i n the x„ d i r e c t i o n , the equation f o r the x 0 = 0 s e c t i o n of 3 3 i 2 the i n d i c a t r i x i s + r _ E ] + [ " ~ - * r E ] x? = 1 13 3 J 1 u 2 ^33 3 J 3 1 3 LiNbO„ i s an u n i a x i a l c r y s t a l w i t h x„ as the p o l a r a x i s and 3 3 n l = n o = V n 3 = n e Hence i 2 1 2 [ + T ^ E J ] x" + [ — — + r 5 3 E 3 ] x 5 = 1 n o n e The e f f e c t of the f i e l d E^ i s thus to change the index of r e f r a c t i o n f o r a wave p o l a r i s e d along x, so t h a t the new index (n + An ) i s g i v e n 1 o o b y 1 2 , 1 , „ . 2 (n • M f X l " ( ~ + ) ^ - , n 0 0 0 Since An Q << n Q we can make the approximation 1 An Q -2 i 2An [ 1 + 1 « -i- t 1 - ° ] / ^ . s2 2 J ^ 2 l.n + An J n n n n o o' 0 o o o Therefore 2An ° = r _ E , 3 13 3 3 n r n„E„ o 13 3 or An = -o 2 S i m i l a r l y f o r the wave p o l a r i s e d along x. 3 n r ^ - E , e 33 3 A i e 2 E, 3 / 3 3 o 13 e At the wavelength of the helium-neon l a s e r (6328 2.) Hence A(n -n ) = — — (n r n _ - n r,_) e o' 2   33 n = 2.2918 n = 2.2012 o e r 1 3 = 8.6 x 10~ 1 0 cm/v r ^ = 30.8 x 10~ 1 0 cm/v S u b s t i t u t i n g we get A(n -n ) = -1.13 x 10~ 8 E^ -,. . e o 3 4. Q u a n t i t a t i v e M o d e l l i n g of O p t i c a l Damage Q King c a r r i e d out a q u a n t i t a t i v e a n a l y s i s of the l i g h t induced r e f r a c t i v e index change i n KTN where a s t a t i c e x t e r n a l f i e l d has to be a p p l i e d i n order f o r o p t i c a l damage to occur. He based h i s a n a l y s i s on Chen's theory. N e g l e c t i n g d i f f u s i o n e f f e c t s , e l e c t r o n s which are photo-e x c i t e d d r i f t and become retrapped and t h e i r c o n c e n t r a t i o n s are assumed to remain small compared to the donor and tr a p d e n s i t i e s . Furthermore, K i n g assumed t h a t the d r i f t l e n g t h of e l e c t r o n s i s s m a l l compared to the sc a l e of r e f r a c t i v e index change so that the e l e c t r o n c o n c e n t r a t i o n always remains p r o p o r t i o n a l to the l i g h t i n t e n s i t y and i s given by: 2 P t t i o f 2 ^ 2 x o - 2{x +y ) n f t(x,y) = [ g—^ e x P t urjvr r 0 o where = power of l a s e r beam hv = photon energy 26 (c) Ey(x,y) F i g u r e 6: Three-dimensional p l o t s showing the s p a t i a l d i s t r i b u t i o n of (a) the space charge w i t h h i l l s and v a l l e y s s i g n i f y i n g p o s i t i v e and negative charge, (b) the x component of the i n t e r n a l f i e l d and (c) the y component of the i n t e r n a l f i e l d 27 e l e c t r o n l i f e time r = l a s e r beam r a d i u s o a = o p t i c a l a b s o r p t i o n at the damaged wavelength The equation of c o n t i n u i t y r e q u i r e s and j = qn u (E + E ) (3) ^ 0 0 sc where p = charge d e n s i t y q = e l e c t r o n charge u = m o b i l i t y of e l e c t r o n E = i n t e r n a l f i e l d 0 -»-E = space-charge f i e l d s c Equations ( l ) , (2) and (3) together w i t h Poisson's equation V.E = V,(E + E ) = V.E o sc sc e e o r enables us to s o l v e f o r E : . sc Now, the formal s o l u t i o n to (4) i s K c ^ ' V = 2 ^ T v x , y " P ^ ' . y ' . ' ) £ n l ? l d*'<*' where r i s the d i s t a n c e from (x', y') to (x, y ) . The above equations have been solved u s i n g f i n i t e d i f f e r e n c e methods . Appendix A o u t l i n e s the method used and a l s o c o n t a i n s a copy of the program w r i t t e n to solve the above equations. F i g u r e 6 show a three-dimensional p l o t of p(x,y), E ^ ( x , y ) r E (x,y) a f t e r 12 i t e r a t i o n s u s i n g the f o l l o w i n g constants O U T = 10 "L m^/V P = 10~ 5 Watts 0 = 2.08 x 1 0 1 5 c y c l e s / s e c h = 1.2 x 10 m r = 10~4m o E = 4 x 1 0 5 v/m o At =1.0 sec e e = 2.83 x 10""1CF/m o r ' where E^(x,y) and E (x,y) are the x and y components of the space-charge f i e l d E sc One n o t i c e s that E (x,y) indeed shows the same f e a t u r e as Figure. 1. Along the x d i r e c t i o n , . E (x,y) r e v e r s e s s i g n at the beam edge whereas along the y d i r e c t i o n E (x,y) stays unchanged i n s i g n . K i n g was able to o b t a i n numerical values of aux s i n c e E i s o a known constant i n h i s case. For the case of LiNbO^, both ayr and E are unknowns and hence unless aut and E can be determined o o independently, i t i s not p o s s i b l e to s o l v e f o r these two q u a n t i t i e s from the numerical a n a l y s i s . 29 IV. NARROW SLIT EXPERIMENT 1.1 S l i t p e r p e n d i c u l a r to the c - a x i s The holographic storage mechanism i n l i t h i u m niobate can f u r t h e r be i n v e s t i g a t e d by u s i n g d i f f e r e n t geometry f o r the damaging l a s e r beam. R e f e r r i n g to Fi g u r e 1, i f a narrow s t r i p of l a s e r l i g h t i s place d across the c r y s t a l i n a d i r e c t i o n p erpendicular to that of the proposed i n t e r n a l f i e l d E q , then, f o r a s u f f i c i e n t l y narrow s l i t , we expect a one-dimensional problem along the c - a x i s where d r i f t due to E q would be the dominant mechanism a f f e c t i n g the movement of photo-e x c i t e d e l e c t r o n s . This i s a si m p l e r problem to analyse than the previous case where the f a c t t h a t i t i s a two-dimensional problem makes i t more d i f f i c u l t to s o l v e . Furthers, i n the a n a l y s i s by King, he has to assume that the d i f f u s i o n and d r i f t lengths are both small so that n (x,y) always remains p r o p o r t i o n a l to the l i g h t i n t e n s i t y . No such assumption, i s r e q u i r e d i n the one-dimensional case. c-axis SLIT EXPOSURE' A. i- -- + -|- "I- + T LiNb03 F i g u r e 1: For a s u f f i c i e n t l y narrow s t r i p of l i g h t placed'across the c r y s t a l as shown,'drift would be the dominant mechanism a f f e c t i n g the movement of e l e c t r o n s along the c - a x i s . 30 1.2 Theory A c o n s i d e r a b l e s i m p l i c a t i o n i s obtained when the problem i s made s p a t i a l l y one-dimensional. F u r t h e r s i m p l i f i c a t i o n i s obtained i f the exposure (time x i n t e n s i t y ) , i s kept small enough f o r the e f f e c t s c a u s i n g s a t u r a t i o n to be ne g l e c t e d . This means: (1) the space-charge f i e l d E g c set up between the p o s i t i v e i o n i s e d centres i n the i l l u m i n a t e d area and the trapped e l e c t r o n s i s neglected compared to the i n t e r n a l f i e l d E . 0 (2) the t r a p occupancy i s considered to be only s l i g h t l y perturbed, so th a t (a) the r a t e of r e l e a s e of e l e c t r o n s from t r a p s a t a given p o i n t remains p r o p o r t i o n a l to the l i g h t i n t e n s i t y a t that p o i n t , and (b) the r a t e o f capture of conduction band e l e c t r o n s by t r a p s i s p r o p o r t i o n a l ' to t h e i r c o n c e n t r a t i o n . Using these assumptions, the treatment i s a p p l i c a b l e to the i n i t i a l l i n e a r p a r t of the hologram-writing process i l l u s t r a t e d i n Figure 4 of Chapter I I I 0 We have two simple cases. (1) W i t h a narrow s t r i p of l a s e r l i g h t p e r p e n d i c u l a r to the c - a x i s . I n t h i s case, the motion of e l e c t r o n s along the c - a x i s i s due c h i e f l y to d r i f t r e s u l t i n g from E . 0 (2) W i t h a narrow s t r i p of l a s e r l i g h t p a r a l l e l to the c - a x i s . I n t h i s case, d i f f u s i o n i s the dominant mechanism a f f e c t i n g the movement of e l e c t r o n s i n a d i r e c t i o n p e r p e n d i c u l a r to the c - a x i s Under the above assumptions, the f o l l o w i n g equations governing o p t i c a l damage can be w r i t t e n down: J = neu (E + E ) + eD V n <\, ney I ( l ) n o sc n n o v ' an1 n • 1 -». at = ~ T + 7 v .J + g ( i i g h t ) (2) £ = - v - (3) • V '\c = P / e o e (4) Equation ( l ) i s the c u r r e n t d e n s i t y equation where the d i f f u s i o n term i s n e g l e c t e d . Equations (2) and (3) are the c o n t i n u i t y equations f o r the concentra-t i o n of e l e c t r o n s and the c u r r e n t d e n s i t y r e s p e c t i v e l y . Equation (4) i s Poisson's equation. For the one-dimensional problem, these equations s i m p l i f y to j ( x , t ) = n e M E (5) • ' n o =- J ^ L + I " + e ( l i g n t ) ( 6 ) dt T e 3 x 8p(x,t) = _ 3j(x,t> /rj\ 3t 9x 3E sc P ( x ^ ) (8) ax E E O where j ( x , t ) = e l e c t r o n c u r r e n t d e n s i t y E = i n t e r n a l f i e l d 0 n ( x , t ) = c o n c e n t r a t i o n of e l e c t r o n s i n the conduction band e = e l e c t r o n charge p(x,t) = charge d e n s i t y E Q = p e r m i t t i v i t y of f r e e space e = d i e l e c t r i c constant of medium g( l i g h t . ) = r a t e of generation of e l e c t r o n s due to l a s e r l i g h t 32 For the problem of holography g ( l i g h t ) may be a complicated e x p r e s s i o n of the s p a t i a l v a r i a b l e which may be known only gr a p h i c a l l y , , As i n d i c a t e d below, the Laplace transform method i s p a r t i c u l a r l y s u i t a b l e f o r s o l v i n g the problem of a r b i t r a r y g ( l i g h t ) . Thus, although the o r i g i n a l problem can e a s i l y be solved u s i n g elementary methods, the Laplace transform method i s used. The b i l a t e r a l Laplace transform / of a f u n c t i o n f ( x ) i s defi n e d as: [ f ( x ) ] = / f ( x ) e " B X dx = F ( s ) —oo I n the steady s t a t e 9n(x,t) = 0 at S u b s t i t u t i n g equation ( 5 ) i n t o ( 6 ) and t a k i n g Laplace transform ( w i t h r e s p e c t to the s p a t i a l v a r i a b l e x) of the r e s u l t i n g equation we get N(s) 0 = - -1- s u E N(s) + G-(s) T n o where n N(s) =1 [ n ( x , t ) ] G(s) =£[g(light)] Hence ^ N(s) = G ( s ) = H(s) G ( S ) E [ -s + ] y E x n o and n ( x , t ) = h(x) * g ( l i g h t ) where h(x) * g ( l i g h t ) = / g(y) h(x-y) dy = c o n v o l u t i o n of g ( l i g h t ) -co and h(x) and h(x) i s the i n v e r s e Laplace transform of H(s) and i s given by u(-x) x h(x) = exp u E u E T n o n o 33 where u(x) i s a u n i t step f u n c t i o n . h(x) i s shown i n Figure 2. f/> fx) F i g u r e 2: P l o t of h ( x ) . Thus we see that f o r a r b i t r a r y g ( l i g h t ) , n ( x , t ) i s obtained by the c o n v o l u t i o n of h(x) and g ( l i g h t ) , e i t h e r g r a p h i c a l l y or a n a l y t i c a l l y . For the i r r a d i a t i n g geometry of F i g u r e 1, g ( l i g h t ) i s g i v e n by g ( l i g h t ) = A -d/2 < x < d/2 where A i s a constant, the c o n v o l u t i o n can be c a r r i e d out a n a l y t i c a l l y . E (x) i s then obtained from n ( x , t ) by combining equations (5),(7) and (8) sc to y i e l d n ( x , t ) e E t y m ( \ o K n E s c ( x ) = - : C a r r y i n g out the computations we f i n a l l y get: 34 x d n(x, t ) = 2 AT exp — s i n h : ' x < -d/2 y E T 2y E x n o n o - d x .. = AT [ 1 - exp : exp ] - d/2 < x < d/2 2 y E T y E T Hn o n o 0 x > d/2 2 ey x d E (x) = - ~ E tAx exp s i n h z < -d/2 s c e e 0 y E x 2 y E x 0 n o n o , 1 - d x = - A e y E t [ l - exp exp ] -d/2< x< e e n o 2 y E x y E x o n o n o = 0 x > g ( x ) , n(x) and E (z) are p l o t t e d i n F i g u r e 3 f o r the case of a s l i t SC p e r p e n d i c u l a r to the c - a x i s . d/2 36 1.3 Experimental R e s u l t s Figure 4 shows a p l o t of -A(n -n Q) versus d i s t a n c e along the c - a x i s f o r a s l i t width of 0.06 _+ 0.002cm (Power = 200mW). Comparing t h i s p l o t w i t h the p l o t of E (x) of Fi g u r e 3- , one n o t i c e s two anomalies: s c A(n f e-n )' remains s l i g h t l y p o s i t i v e on one end of the s l i t and the maximum of A ( n g - n o ) does not e x a c t l y occur at the s l i t edge as p r e d i c t e d by theory. The anomalies are probably due to experimental e r r o r s . From F i g u r e 4 and the theory, a value of p T E = 0.09 + 0.005 cm is. obtained. ' n o -" 2.1 S l i t p a r a l l e l to c - a x i s On the ether hand, i f the s l i t of l i g h t i s placed across the c r y s t a l i n the d i r e c t i o n of the c - a x i s (Figure 5 ); then, again f o r a s u f f i c i e n t l y narrow s l i t , we expect a one-dimensional problem i n a d i r e c t i o n p e r p e n d i c u l a r to the c - a x i s and si n c e there i s no f i e l d i n t h i s d i r e c t i o n , d i f f u s i o n i s now the dominant mechanism governing the movement of e l e c t r o n s . However, e l e c t r o n s are not expected to move very f a r by d i f f u s i o n and hence i t was not s u r p r i s i n g that e x p e r i m e n t a l l y we have so f a r observed no e f f e c t w i t h t h i s geometry. The d i f f u s i o n l e n g t h i s probably too s m a l l to be r e s o l v e d under our present experimental i n s t r u m e n t a t i o n . However, the one-dimensional probelm has been solved a n a l y t i c a l l y i n the next s e c t i o n . 2.2 Theory The assumptions made to s o l v e t h i s problem are the same as the previous case except that d i f f u s i o n , and not d r i f t , dominates the movement of e l e c t r o n s . Except f o r equation ( l ) , the remaining three equations remain unchanged. Figure 4: P l o t of A(n e-n 0) along the c - a x i s f o r a w i d t h of 0.06 + 0.002 cm. 28 A c - a x i s 4 4 + 4-+ + -f +-+• O •Z.//V60 '5 •» < — S L I T E X P O S U R E F i g u r e 5". 'For a s u f f i c i e n t l y narrow s t r i p of l i g h t placed across the c r y s t a l as shown, d i f f u s i o n would be the dominant mechanism a f f e c t i n g the movement of e l e c t r o n s i n a d i r e c t i o n p e r p e n d i c u l a r to the c - a x i s . 39 J = eD 3E 3n St 3t sc 3n n 3x n 1 3J e 11 3x 3x (1) (2) (3) (4) E £ O Again, the steady s t a t e s o l u t i o n i s solved u s i n g Laplace transform method. S u b s t i t u t i n g equation ( l ) i n t o (2) and t a k i n g Laplace transform ( w i t h respect to the s p a t i a l v a r i a b l e x) of the r e s u l t i n g equation we get N(s) 0 = - + D s N(s) + G ( s ) where N(s) = ^ [ n ( x ) ] G(s) = X, [ g ( l i g h t ) ] Hence W(s) = - T (D T s — 1) n G ( s ) = H(s) G ( S ) and n(x) = h(x) * g ( l i g h t ) where h(x) i s the i n v e r s e Laplace transform of H(s) and i s given by h ( x ) = 2 V D T - x x [ exp J u(x) + exp j u(-x) ] n V/ITT v n V n = h x ( x ) + h 2 ( x ) h ( x ) , h-^x) and h,-,(x) are p l o t t e d as shown i n F i g u r e 6« 41 Thus we see that f o r a r b i t r a r y g ( l i g h t ) , n(x) i s obtained by the c o n v o l u t i o n of h(x) and g ( l i g h t ) , e i t h e r g r a p h i c a l l y or a n a l y t i c a l l y . N o t i c e that h(x) i s made up of two simple e x p o n e n t i a l s , h^(x) and h ^ x ) , so that the c o n v o l u t i o n i s e s s e n t i a l l y the same as the previous case. For the i r r a d i a t i n g geometry of Fi g u r e 5 , g ( l i g h t ) i s gi v e n by g ( l i g h t ) = A -d/2 < x < d/2 Performing the c o n v o l u t i o n i n t e g r a l s s e p a r a t e l y f o r h-^(x) and h (x) we get x d n(x) = A t exp s i n h ( • ) x < -d/2 VB x 2VD T n n - d x = A T [ 1 - exp • — cosh ] -d/2 < x < d/2 2 N/T~r s/B~T n v n - x d = A T exp s i n h x > d/2 V B T 2\/B T v n n Having solved f o r n ( x ) , E (x) can again be solved. Combining equations sc ( l ) , ( j ) and (4) we have E ( x ) - - ^ sc 9x e e 0 S u b s t i t u t i n g f o r n(x) from above we have -eAt x d E g c ( x ) = y ^ T exp — - s i n h ( — ) x < -d/2 eAt - d x V D n T exp s i n h ( ^ ) -d/2 < x < d/2 e 0£ 2\/ B T V / D T eAt - x d e x P ~ ~ Z T s i n h ( " ) x > d/2 0 V / F T 2fB g ( x ) , n(x) and E (x) are p l o t t e d i n F i g u r e 7 . Figure 7: P l o t of g ( x ) , n(x) and E s c ( x ) , the steady s t a t e s o l u t i o n of the problem formulated i n s e c t i o n 2.2 . 43 3. F u r t h e r Experiment to i n v e s t i g a t e i n t e r n a l f i e l d The b u i l t - i n f i e l d p o s t u l a t e d by Chen may f u r t h e r be i n v e s t i -gated as f o l l o w s . I f one scans along the c - a x i s of the c r y s t a l so as to get a map of the b i r e f r i n g e n c e (n - n Q ) , then any v a r i a t i o n s i n (n -n Q) along the c - a x i s may g i v e an i n d i c a t i o n of the v a r i a t i o n s i n E . The v a r i a t i o n s i n (n -n Q) may al s o be the r e s u l t of non-stoichiometry caused 14 by changes i n the composition of the c r y s t a l d u r i ng the growth process. A p l o t of (n -n ) along the c - a x i s i s shown i n f i g u r e 8 ( s o l i d l i n e ) . The po i n t of zero f i e l d i s undetermined. By p u t t i n g e l e c t r o d e s across opposite end of the c r y s t a l and s h o r t i n g the e l e c t r o d e s , we expect to make the average f i e l d J*Edx along the c-ax i s zero. I f the v a r i a t i o n s i n (n -n ) Q 0 are not due to non-stoichiometry, then from the curve shown one would expect t h a t the p o i n t of zero f i e l d would be roughly i n the middle of the c r y s t a l ( s i n c e (n^-n^) i n c r e a s e s q u i t e smoothly along the c - a x i s ) . ( n e ~ n ) a f t e r s h o r t i n g the e l e c t r o d e s i s a l s o shown i n f i g u r e 8 as a dotted l i n e . I t can be seen that (n -n ) remains almost unchanged. (The maximum s h i f t i n (n -n ) e o . e o i s 8 . 2 x 10 ^ which corresponds to a s h i f t i n the i n t e r n a l f i e l d of 800V/cm.) This i n d i c a t e s that the average f i e l d j E d x along the c - a x i s i s almost zero even before s h o r t i n g the e l e c t r o d e s . By r e p e a t i n g the c i r c u l a r damaging beam experiments but w i t h the beam placed i n 3 places along the c - a x i s where E q i s negative, zero and p o s i t i v e , one would expect then t h a t A^^-n^) would a l s o reverse s i g n a t the two spots where E q r e v e r s e s i g n and remains zero where E Q ' i s zero. The experimental r e s u l t s are shown i n f i g u r e 9 where i t i s obvious t h a t A ( n g - n o ) d i d not re v e r s e s i g n . This could be e x p l a i n e d i f the i n t e r n a l f i e l d v a r i e s along the c - a x i s as shown i n f i g u r e 10 where to meet the requirements of = 0 w i t h the .electrodes shorted, the shaded area must be equal before putting s electrodes after putting * electrodes and short/ng them i 0 0.5 c - axis (cm) 10 Figure 8: Plot of (n e-n D) along the c-azis before depositing electrodes (solid line) and after depositing electrodes and shorting them (dotted l i n e ) . Figure 10: V a r i a t i o n s of E along the c-axis to s a t i s f y the c r i t e r i o n t h a t J"Edx = 0 along the c - a x i s . V. CONCLUSIONS The o p t i c a l damage process i n l i t h i u m n iobate has been i n v e s t i g a t e d . The automated e l l i p s o m e t e r i s used f o r stud y i n g the process because the b i r e f r i n g e n c e of the c r y s t a l can r e a d i l y be determined and the balancing procedure would be extremely t e d i o u s i f done by hand. F u r t h e r , the angles of the p o l a r i s e r and analy s e r can be read to a s e n s i t i v i t y of +_ 0.01° which corresponds to 10 ^ i n (n -n ) f o r the specimen thi c k n e s s used. I t was found that c o n s i s t e n c y of experimental r e s u l t s was obtained by baking the c r y s t a l and c o o l i n g i t s l o w l y before each experiment so as to anneal out the space-charge f i e l d c r e a t e d during o p t i c a l damage of previous experiments. The c o o l i n g of the c r y s t a l a f t e r baking had to be c a r r i e d out sl o w l y , otherwise random p y r o e l e c t r i c f i e l d might be f r o z e n i n the c r y s t a l . To v e r i f y Chen's model of the ex i s t e n c e of an i n t e r n a l f i e l d - E o i n the d i r e c t i o n of the c - a x i s of the c r y s t a l , the c i r c u l a r damaging beam experiment was c a r r i e d out. The s p a t i a l d i s t r i b u t i o n of Ain^-n^) can be exp l a i n e d q u a l i t a t i v e l y u s i n g Chen's model. For t h i s i r r a d i a t i o n geometry, A(n^-n^) i n c r e a s e s l i n e a r l y w i t h time up an an exposure time of 20 seconds which corresponds to an exposure energy d e n s i t y of 500J"/cm^. Based on King's a n a l y s i s of o p t i c a l damage i n KTN, the problem f o r the c i r c u l a r damaging beam was solved u s i n g f i n i t e d i f f e r e n c e methodo The r e s u l t i n g p l o t of E (x,y) shows the name f e a t u r e s as the experimental p l o t of A(n -n ). x e o No numerical comparison of experimental and t h e o r e t i c a l r e s u l t s are p o s s i b l e s i n c e there are two unknowns, au.T and E , to be determined from one ' o' experiment. To i n v e s t i g a t e the id e a of an i n t e r n a l f i e l d , an experiment u s i n g 48 a narrow s t r i p of l i g h t has been c a r r i e d out. When the s t r i p i s perpen-d i c u l a r to the c - a x i s , d r i f t would be the dominant mechanism a f f e c t i n g the movement of e l e c t r o n s . An a n a l y s i s of t h i s problem has been c a r r i e d out based on Chen's model. Compared to King's a n a l y s i s , no approximation as to the form of n Q ( x , y ) i s necessary. Numerical comparison between theory and experiment gave a value of 0 . 0 9 ± 0 . 0 0 5 cm f o r the product U_T E » The case f o r the s l i t p a r a l l e l to the c - a x i s where d i f f u s i o n i s the dominant mechanism has a l s o been s o l v e d a n a l y t i c a l l y . However, no e x p e r i -mental r e s u l t s have been obtained. This i s not unexpected s i n c e the d i f f u s i o n l e n g t h of e l e c t r o n s i s probably too s m a l l to be r e s o l v e d w i t h our present experimgntal set up c From the above s t u d i e s , i t can be concluded t h a t Chen's model i s b a s i c a l l y c o r r e c t but the v a r i o u s parameters of the process need f u r t h e r i n v e s t i g a t i o n . F u r t h e r work i n t h i s area i n v o l v e s the a c t u a l storage of holograms both on pure l i t h i u m n iobate c r y s t a l s and c r y s t a l s doped w i t h i r o n ; the l a t t e r are u s e f u l as f a r as holographic storage i s concerned s i n c e the w r i t i n g time and the w r i t i n g energy are l e s s than the undoped c r y s t a l s . As f a r as the engineering aspects of the process are concerned, f u t u r e work should concentrate on improving the s e n s i t i v i t y to o p t i c a l damage and the f i x i n g of the holograms s t o r e d . 4y APPENDIX • S o l u t i o n of ' O p t i c a l Damage' Problem f o r a C i r c u l a r Laser Beam F i n i t e d i f f e r e n c e methods were used to solv e the f o l l o w i n g equa-t i o n s which were expl a i n e d i n Chapter I I I . If: = - V. [q n Qy ( E q + E) ] (1) E(x,y,t) = - * c V [ / / p U ' . y ' . t ) to|r| dx'dy'] (2) o J where 2P ax 2(.x + y ; n Q ( x , y ) = Y~ exp - £ ( 3) Trhvr r o o The p a r t i a l d i f f e r e n t i a l equations are approximated by f i n i t e d i f f e r e n c e equations through the tra n s f o r m a t i o n scheme below. The value of a v a r i a -b l e z ( x , y , t ) are then determined at a d i s c r e t e mesh of p o i n t s i n (x,y,t) space. The p a r t i a l d e r i v a t i v e i s approximated by 5 p p ( x i , y . > t k + A t ) - P ( x l t y 1 t t k ) dt At Let A = q n y(E + E) = A x + A y M o o x y J 2P ax c 1 = (qu) ~ i r h v r o 2 [ ( ( 1 0 - i ) A x ) 2 + ( ( 1 0 - j ) A y ) 2 ] A exp = exp — r o For an i n t e r n a l f i e l d E i n the x d i r e c t i o n o n l y , that i s o J 50 E = E x o o we have A = q n u(E + E ) x ^ o x o A = q n uE y o y Hence V^^j'V = c i A e x p (vV tVv^'V + V A y ( x . , y j 5 t k ) = C± A exp (x.,y..) [ E y ( x i , - y j , t k ) ] The i n d i c e s i , j both run from 1 to 20 and hence there are 400 mesh poi n t s ( f i g u r e 1). (*o> y0) • fxo>y2o) I 1 J (X20>Yo^ (x20>y20' Figure 1: Square g r i d of mesh p o i n t s used i n the i t e r a t i o n scheme. For any v e c t o r q u a n t i t y A = x + y 9A 9A V . A = — 2 + 9x 9y Hence Equation (1) becomes transformed i n t o the f o l l o w i n g d i f f e r e n c e equation P ( x ±,y.,t k+At) - p(x ,y ,t^) A t A (x. , - ,y ., t, ) - A (x. ,y ., t. ) A (x-. ,y .,., , t. ) - A (x. ,y. ,t, ) x r+1 ,i k x l j k y i > J 3 + l ' k y v 1 3 k Ax Ay In the i t e r a t i o n scheme a square g r i d i s used, that i s , Ax = Ay = h Hence we have p C x ^ y . ^ + A t ) = P ( x i 5 y . , t k ) -^JL t A x ( x . + 1 , y . , t k ) - A x ( x . , y . , t k ) + A y ( x i ' Y 3 + l ' V " V\^y^ )] (4) Equation (2) can be separated i n t o two component equations as; E x ( x , y , t ) = - g [//P(x',y',t) £n|r| dx'dy'] (5) o E y ( x , y , t ) = " l y " (*'' >fc) dx'dy'] (6) Now -3- £n r = -5- i n ( x - x ' ) 2 + ( y - y ' ) 2 = ( x x )  (x-x') +(y-y') S i m i l a r l y , A_ to 1+1 = (y-y') 3 y ( x i x ' ) 2 + ( y - y ' ) 2 Equations (5) and (6) therefore becomes 52 i ( v x ' } E (x, ,y.,t ) = - — — — [ / / p ( x \ y ? , t ) 2 x 1 J k 2 7 T V ( x . - x ' ) 2 + ( y . - y ' ) Z (y, - y') dx'dy'] (7) _1 ' J o (x.-x') +(y.-y ) dx'dy'] (8) i ' "3 The numerical i n t e g r a t i o n s are c a r r i e d out using T r a p e z o i d a l Rule of I n t e g r a t i o n (Figure 2) / n y(x) dx = | [ y Q + 2y]_ + 2 y 2 +•. .. + 2 y n _ 1 + y j (y +y„) = h [- o n + (y, + y 0 + • • •+ yn_j_) 1 1 J2 F i g u r e 2: T r a p e z o i d a l r u l e of i n t e g r a t i o n g i v e n y = y U ) . 53 Let (x - x') B( x \ y ' , x , , y . ) = p ( x \ y ' , t . ) i J ( x ^ ^ ^ + C y j - y ' ) ^ Equation (7) becomes (x,.-x') // p(x',y',t,) ^ dx'dy' ( x i - x ' ) 2 + ( y j - y ' ) ? = / / B ( x ' , y , , x i , y j ) dx'dy' » //dy' / B(x',y',x. i y ) dx' = /dy' [ I Ax [B(x' , y ' x ,y ) ] + a ^ f [ B ( x 1 ' , y n ' , x i , y j ) t B C x ^ . y ^ . x ^ y . ) ] ] The i n t e g r a t i o n w i t h respect to y' i s then s i m i l a r l y c a r r i e d out and equation (8) i s s i m i l a r l y s o l v e d . The i n i t i a l c o n d i t i o n s f o r the i t e r a t i o n scheme are E x ( V ^ ' V 0 * = 0 E y ( x . , y . , V 0 ) = 0 p ( x i ' y j ' t k = 0 ) = 0 E = constant o A copy of the F o r t r a n program based on the above i t e r a t i o n scheme i s appended. The program evaluates p ( x , y , t ) , E x ( x , y , t ) and Ey(x,y,t) a t a d i s c r e t e mesh of po i n t s (x,y) f o r a s p e c i f i e d number of i t e r a t i o n s , p, E and E are "then p l o t t e d out as a p e r s p e c t i v e drawing x y using the program UBC PERSP i n the UBC Computing Centre F i l e . 54 F O R T R A N I V G C T M P I L E K HA I N 12-01-72 0) :24:55 PAGE C001 0001 000 2 01 MENS ION EX I 21 ,21 ) , EY( 21 , 21 ) , AX I ?1 , ?1 ) , AY( 21 t 21 ) ,RHO( 21 , 21 ) , CAEXP(20,20) , 13(2 0 ,20) ,C(20 , 2 0 ) , X ( 2 0 , 2 0 ) , Y ( 2 C , 2 C ) , I ( 2 0 , 2 C I REAL PO 000 3 ' 000''-, 0005 INTEGER P,4,R WRITE 10,2) 2 _ FORMAT (1H1, S3HC.ALC.ULATI ON OF THE SPACE CHARGE FIELD C 10 3 A T E 'WHEN EXPOSED TO "LASER" BEAM) C READ IN INITIAL VALUES FOR VARIOUS PARAMFTERS C ANT=PRODUCT OF IN LITHIUM N C P0= POWER Or THE LASER BEAM C FRE 0 = FRE QUE NC Y OF THE LASER BEAM _C_F0=KADIUS OF THE LASER BEAM C H=S I 7. E O F THE GRID SQUARE IN METRES C E0= VALUE OF THE INTERNAL ELECTRIC FIELD C.0ELT=INCREMENT OF THE TIME OF ITERATION I N VOLTS/M 0006 000 7 COO 8 0009 0010 C EPS I L = PERM[TT IV ITY OF THE L II HI U NIOBATE READ (5,1) A NT,PO,FR cQ,H,RO,E 0,DE L T , E PS IL,Q 1 FORMAT (8E3.3, I2) WRITE ( 6, 902 ) ANT , P0~, FR E Q, H, RO, EO,DELT , EPS 11.» Q 902 FORMAT (1H0, 3(E9.3,3X) ,I 2) Cl=(2 .0 *P0* {1 .6 E-19)*ANT 1/( 3. 14159*(6.625E-34) I 0011 0012 001.3 0014 C2 = ( -1. • 0 ) / ( 2 • 0* 3 . 1415 9* EP S 1 L ) W R I T E (6,609) C1.C2 _609 FORMAT (1110, El 0 . 3 , E1 0 . 3 ) _ C IN! T! Ai 1 SE ' T H E " VARTUUS"" ARRAY S P=l R=0 0016 0017 001 0_ 0019 0020 0021 00 10 1=1,21 DO 10 J=l ,21 _EX( Ij J )_=0__0 _ EYI I ,J )=0.0 AX(I,J1=0.0 AY(I ,J)=0.0 0022 002 3 00 24" 00 2 5 00 2 6 10 RHO(I,J)=0.0 C START CALCULATION OF RHO 0 0 2 0 1=1,20 20 DO  0 J= l ,20 20 A E X P (I,JI= EXP (-40 DO 5 0 1=1,20 2.O'M ( ( (1C- I ) * *2)+( ( 10-J1**2) » / ( R 0 * R 0 ) ) * H * H ) 0027 0028 0029 6030' 0031 0032 50 DO 5 0 J = l ,20 AX( I ,J )=AEXP( I ,J ) * (EX( I ,J )+EO) AY! I ,J)= A E X P ( I , J ) * E Y ( I , J ) 6 0 DO  J=l ,20 AX(21,J ) =• AX (19 ,J ) 60 AY(J,21)=AY(J,19) 0033 0034 0035 0036 0037 DO 6 5 1=1,20 DO 65 J = l ,20 J>_5_ RHO ( I , J) =__H0 ( U J ) -C l * ( ( 1 AX( 1*1, J ) -A X( I , JJJ + (_AY ( I , J + 1 )-C*DEl. T WRITE (6,31 P 3 FORMAT (1H0, 20 UNO. OF IT ERATIONS= ,12) A Y ( I , J ) ) ) / H I 00 33 _003 9 004 0" !~EVA L U A T T G N OF EX AND EY U S I N G lTTtRXTlVc Fu'RWCS" 1=0 _ 9 0 0 _ I = I + 1_ _____ KR I T E ("6, 962) I 55 FORTRAN) IV G "COMPILER MAIN 12-01-72 01:24:55 PAGE 00C2 962 FORMAT (1H0, 4;-l I = ,12) IF (I .FQ. 211 GG TO 600 J=0 500 J = J+"l 00 7 0 L=l,20 DO 7 0 M = l , 20 _ _ ' _ IF ( I . EQ. ' L " T A \ ' D . ' J ".EO. '•'•) GCi TO 7 ' i ~ C4=P. HO ( L , M) / ( ( ( I-L ) **2 + ( J-M ) **2 ) *H) BJ L , M_=f_-*J I _L) C ( I , M) =C4*fJ-H) GO TO 7 0 __7_1_ _ B ( L , M) = 0. 0 _ ~ C"(L,.M)=0.0 " " " " " " ~ " 7 0 CONTINUE C INITIALISE THE TEMPORARY PARAMETERS 00 55 BTEMP 1 = 0.0 0056 BTEMP2=0.0 j005 7__ BTEMP3=0.0 _ _ j _ _ 0058 "BTEMP4'="0'".0~ " " "~ . " ~ 0059 BTEHP5=0.0 0060 CTE _P 1 _£?__2 0061 CTEMP2=0.0 0062 CTEMP3=0.0 _0063__J CTEMP4=0.0 006 4 ' " " CTcMP5"=0.0 0065 DO 80 L=2,19 0066 DO 8 0 M=2,19  0067 8 T E M P1 = B T E M P1+ 0(L» H ) 0068 80 C. TEM PI = C TEMP1 +C ( L , M) 0069_ DO 90_N=2,19 . _ ""0070 " B T " E " ' M P 2 = 'B T E M ' P _ 4-B"('M";"n " " " C071 BTE:MP3 = BTEMP 3+ B ( M , 20 ) 007.2 3T EMP4 = BT EMP4 + B (1 , N )  0073 B T E M p 5 = B T E MP 5 + tt(2 0,N) 0074 CTEMP2=CTEMP2+C(N,1) CO7 5 CT E M P3 = CT E.M °3 +C_( H , 20 ) • "00 7 6"' " CTh M!>4 -"CTEMPV+'C I ! ,M 0077 90 CTEMP5=CTEMP5+C(2Q,N) 0078 E X (I , J) = C.2*H*H» ( B T E MP 1 + 0.. 5* ( B T EMP2 + BT EMP3+BT E MP 4 + BT EMP 5 ) + 0. 25* ( B ( 1 C l ) + B ( l ,201+3(2 0, l) + B ( 2 0TT6TrJ : 0079 100 EY ( I , J ) = C2*H*H* ( CTEMP1 + 0. 5 * < C T E.MP2 + CT EMP 3 + CTE MP4+CTEMP5 ) +0. 25*( C( 1 C,l H C( 1 ,20)+C120,1 I +C(20,2C ) I ) CO 8 0 I f f J . E;). 2 1)' GO' TC '9C0 ~ ' C0S1 GO TO 500 0082 008 3 IF <R . EO. 9) GO TO 601: : 0084 GO TO 449 _0085 601 _ DO'534 1 = 1,20 . 0086" ' WRITE ( 6, 533) (RHOfl,J), J=),20) 0087 . 533 FORMAT (1H0, 10 ( E1 0 . 3 ,2 X ) / 1 0 ( E10 . 3, 2X ) ) 0088 534 CONTINUE ' 0089 00 543 1 = 1, 20 ' ~ ~ ~ 0090 WRITE (6, 544) ( E X ( I , J ) , J = 1,20 ) _009i 54 4 FOR " A T ( 1H0, 10 ( E 1 0. 3 , 2 X ) / 10 ( E 10. 3, 2X ) ) C092 543 CONT INJE" 0041 0042 004 3 004 4 '004 5 004 6 004 7" 004 8 0049 0050 00 51 005 2 00 5 3 00 54 r FORTRAN IV G fXM°I LER MAIN 12-01-72 01: 24: 55 PAGE 0 003 0093 0094 v 0095 5 55 DO 153 1=1,20 WRITE (6,555) ( F Y ( I , J ) , J = 1 , 2 0 ). . FORMAT (IH), 101E10.3,2X1 /1CIE10.3, 2X) ) r 0096 C09-V 0093 553 CONT I NJS DO P83 I=1.20 DO 8 33 J= 1,2.1 0099 0100 0101 0 03' Z(1,J)=RHC(I,J) DO 931 1=1,20 DO 931 J-1,20 01C2 0103 0104 931 Y d , J)=EX(l,J) DO 2 17 1=1,20 DO 217 J=l,20 0105 0106 0107 21 / X(I,J)=EY(I,J) CALL PE-^ S (Z, 20, 20, 20,1 .0 , 0. 6, 10.0, CALL PLOT (20.0,0.0,-3) 45. 0 ,10.0, 10.0) 0108 0109 0) 10 CALL PERS (Y,20,20,20,1.0,0.6,10.0, CALL PLOT(20.0,0.0,-3) CALL PF.RS ( X, 20 , 20,20,1 .0 ,0. 6, 1 0.0, 45. C 45.0 ,10.0, ,10.0, 10.0) 10.0) 0111 0112 0113 CALL PL OT NQ P=P+1 IF (P .EQ. 10) GO TO 907 0114 Oi l 5 0116 937 GO TO 40 STOP END TOTAL MEMORY REQUIREMENTS C057BE BYTES C0MP1LF TIME = 5.7 SECONDS 5 7 REFERENCES 1. W.C. Stewart, R.S. Mezrich, L.S. Cosentino, E0M. Nagle, F.S. Wendt and R.D. Lohman, RCA Review, ___  3 (1973). 2. W. Phillips, J.J. Amodei and D.L. Staebler, RCA Review, __5_5, 94 (1972). 3. J.J. Amodei and D.L. Staebler, RCA Review, ^gt 71 (l97l). 4. D.L. Staebler and J.J. Amodei, Ferroelectrics, J_5, 107 (1972). 5. A. Ashkin, O.D. Boyd,. J.M. Dziedzic, R«G. Smith, A.A. Eallman, J.J. Levinstein and K. Nassau, Appl. Phys. Letters, % 72 (l966). 6. F.S. Chen, J.T. LaMacchia and D.B. Fraser, Appl. Phys. Letters, 13» 223 (1968). 7. T.K. Gaylord, T.A. Rabson and F.K. Tittel, Appl. Phys. Letters, 20, 47 (1972). 8. S.R. King, T.S. Hartwick and A.B. Chase, Appl. Phys. Letters, 21, 312 (1972). 9. G.D. Boyd, W.L. Bond and H.L. Carter, J. Appl. Phys., ___8, 1941 (1967). 10. F.S. Chen, J. Appl. Phys.., 40, 3389 (l969). 11. W.D. Johnston, Jr., J. Appl. Phys., 4_1, 3279 (l970). 12. J.J. Amodei, RCA Review, _52, 185 (l97l). 13. J.F. Nye, Physical Properties of Crystals (Oxford University Press, Oxford, I960). 14. J.E. Midwinter, Appl. Phys. Letters, 11, 128 (l967). 

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