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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 I N 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 o f B r i t i s h Columbia, 1 9 7 1  A THESIS. SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  i n t h e Department of E l e c t r i c a l Engineering  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA ' AUGUST, 1 9 7 3  In p r e s e n t i n g an  this  thesis in partial  advanced degree at the  the  Library  University  s h a l l make i t f r e e l y  f u l f i l m e n t of the  of B r i t i s h Columbia, I agree  a v a i l a b l e f o r r e f e r e n c e and  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 extensive for by  s c h o l a r l y p u r p o s e s may his representatives.  be  g r a n t e d by  thesis for financial  written  permission.  gain  Department  Date  the  Head o f my  Columbia  s h a l l not  be  for  that  study.  copying of t h i s  thesis  Department  I t i s understood that copying or  of t h i s  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  requirements  or  publication  allowed without  my  ABSTRACT A computer-controlled  e l l i p s o m e t e r was  on the b i r e f r i n g e n c e changes i n d u c e d i r r a d i a t i o n w i t h an a r g o n - i o n  u s e d to o b t a i n d a t a  i n l i t h i u m niobate c r y s t a l s  laser,,  The  i n s t r u m e n t was  set to  by take  r e a d i n g s on a r e c t a n g u l a r g r i d o f p o i n t s b e f o r e 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 l i g h t beams.  p e r f o r m e d 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  R e s u l t s a r e compared w i t h the p r e d i c t i o n o f the model  o f Chen i n w h i c h he p o s t u l a t e d the p r e s e n c e o f an i n t e r n a l f i e l d E the d i r e c t i o n o f the c - a x i s o f t h e c r y s t a l .  q  in  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 f r o m t r a p s by the l a s e r l i g h t , d r i f t u n d e r the i n f l u e n c e of E  q  along the c - a x i s before being retrapped.  space-charge f i e l d  The  s e t between p o s i t i v e i o n i s e d c e n t r e s and  resulting trapped  e l e c t r o n s g i v e s r i s e t o the o b s e r v e d b i r e f r i n g e n c e changes v i a t h e e l e c t r o - o p t i c e f f e c t o f the c r y s t a l ,  linear  Approximate m a t h e m a t i c a l models  based on Chen's t h e o r y a r e u s e d t o s o l v e the p r o b l e m 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 a r e compared w i t h  i  experiments.  TABLE OF CONTENTS Page Abstract  . . . o . o  Table o f Contents  i . . » . o .  . . . • • * . .  . . . . . .  L i s t of I l l u s t r a t i o n s . . . . . . . . .  i i i  Acknowledgement I. II.  IV.  1  THE GENERAL EXPERIMENTAL ARRANGEMENT . . . . . . . . .  3  1 2 3 4  Introduction Ellipsometer setu p . . O p t i c a l damage p r o c e d u r e . ..... R e f r a c t i v e Index Measurements f r o m E l l i p s o m e t e r Readings . . . . . . . . . .  3 3 6  Balancing Procedure . . . . . . . . . . . . . . .  13  1 Experiments . . . . 2 Qualitative Explanation . . . 3 E l e c t r o - o p t i c e f f e c t i n LiNbOj 4 Q u a n t i t a t i v e M o d e l l i n g o f O p t i c a l Damage, . . NARROW SLIT EXPERIMENT . . .  3  9  15  CIRCULAR LASER BEAM EXPERIMENT  1.1 1.2 1.3 2.1  V.  T  INTRODUCTION .  '5 III.  i i  . . . . . . . .  15 15 21 25 29  S l i t perpendicular to the c - a x i s . . . . . . . . . Theory. . . . . . . . . . . . . . . . . . . . . . Experimental Results. . . . . . . . . . . . . . . S l i t p a r a l l e l to c-axis . . . . . . . . . . . .  29 30 36 36  0  Theory. . . . . . . . . . . . . . . . o o . . . .  36  F u r t h e r Experiment t o 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  CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . .  47  APPENDIX  49  . . . . . . . . . . . . . . .  57  REFERENCES .  ii  LIST OF ILLUSTRATIONS Figure II-l II-2  Page Rate of cooling of crystal after baking to 500°C for l/2 hour .....<>».. Computer-Ellipsometer interface.  4  Dashed lines  indicate connection to the computer interface . „  5  II-3  Sample holder for LiNbO,, crystal,  .  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 . .  III- l  Map  of  A(n  - n ) .  e  r  .  0  .  .  0  0  .  0  0  .  0  .  0  .  0  .  .  10 16  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 beam The space-charge f i e l d E shown i n this figure would explain the plot of A{n -n ) of Figures 1,2,3 . °. 0  III-5  s c  6  III- 6  IV- 1  20  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 f i e l d and (c) the y component of the internal field .  26  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 »  IV-3  Plot.of g(x), n(x) and E ( x ) , the steady state solution of the problem formulated i n section  33  sc  1.2  IV-4  19  o » o . « « . o o . « . o o . o * . « o o o o  Plot of A(n -n ) along the c-axis for a s l i t width of 0.06+. 0.002 cm „ . . . . „ e  0  iii  o.  35  37  Figure IV-5  Page F o r a s u f f i c i e n t l y narrow s t r i p o f l i g h t p l a c e d a c r o s s t h e c r y s t a l as shown, d i f f u s i o n w o u l d be the dominant mechanism a f f e c t i n g the movement o f e l e c t r o n s i n a d i r e c t i o n perpendicular to the c-axis  38  IV-6  P l o t o f h ( x ) , h (x) and h (x)  40  IV-7  P l o t o f g ( x ) , n(x) and E ( x ) , t h e s t e a d y 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  P l o t o f (n'-n ) a l o n g t h e c - a x i s b e f o r e 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 ( d o t t e d l i n e ) •>....  44  P l o t o f A ( n - n ) 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  V a r i a t i o n s of E along the c-axis to s a t i s f y t h e c r i t e r i o n t h a t j'Edx = 0 a l o n g the c - a x i s  46  IV-8  IV-9  IV-10  s c  e  0  iv  ACKNOWLEDGEMENT I am most grateful to Dr. L. Young f o r his helpful suggestion and guidance during the course of this work. The f i n a n c i a l 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. F i n a l l y I wish to thank Miss Norma Duggan and Miss Betty Cockburn for help i n typing the thesis.  v  I.  INTRODUCTION  Modern d a t a p r o c e s s i n g systems use a h i e r a r c h y o f s t o r a g e d e v i c e s r a n g i n g from c o r e s , d i s k s , tapes to s e m i c o n d u c t o r  memories.  Mass o p t i c a l memories' " 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 1  powerful a d d i t i o n to t h i s h i e r a r c h y .  V a r i o u s t y p e s o f 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 r e a d - w r i t e o p t i c a l memory based on hologram s t o r a g e  on  an e r a s a b l e medium and compound a d d r e s s i n g by pages and w i t h i n pages. An a r r a y o f l i g h t v a l v e s composes the page to be s t o r e d .  The  direction  o f i l l u m i n a t i o n o f the page composer, r e s u l t i n g from d e f l e c t i o n o f a l a s e r beam, determines  the l o c a t i o n on the s t o r a g e medium a t w h i c h a  h o l o g r a m o f the page w i l l be r e c o r d e d (and the p r e v i o u s r e c o r d e r a s e d ) . The  page composer and s e n s o r a r r a y a r e 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 a d d r e s s a b l e memory.  T h i s type o f o p t i c a l memories seems to  be o f p a r t i c u l a r promise as an e x t e n s i o n o f t h i s h i e r a r c h y where l a r g e Q amounts o f i n f o r m a t i o n ( > 10 (~  ^sec)  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  on a page by page b a s i s . T h i s t h e s i s i s concerned  w i t h s t u d y i n g the mechanism by w h i c h  i n f o r m a t i o n i s s t o r e d i n one o f t h e p o s s i b l e c a n d i d a t e s f o r the medium - l i t h i u m n i o b a t e ( L i N b O ^ ) .  L i t h i u m n i o b a t e has many a d v a n t a g e s  as f a r as h o l o g r a p h i c s t o r a g e i s concerned: e f f i c i e n c y as h i g h as  80f>  storage  holographic  diffraction  has been a c h i e v e d i n l i t h i u m n i o b a t e doped  2 w i t h i r o n ; s t o r a g e t i m e s o f many days a r e p o s s i b l e a t o r d i n a r y room temperature  and e r a s u r e can be a c h i e v e d e i t h e r o p t i c a l l y o r by h e a t i n g  the c r y s t a l to 300  o 3 C ; the holograms s t o r e d can a l s o be  'fixed' thermally  4 so t h a t the p a t t e r n s a r e o p t i c a l l y n o n - e r a s a b l e  ; no b l e a c h i n g o f t h e  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 n i o b a t e i s v i a ' o p t i c a l damage , t h a t i s , l i t h i u m 1  niobate  responds t o l a s e r l i g h t o f t h e a p p r o p r i a t e w a v e l e n g t h w i t h change i n t h e 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  o f 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 m o d u l a t i o n and second harmonic 6 Chen  generation,  7 and o t h e r s  l i t h i u m niobate. investigate  have used t h i s e f f e c t t o s t o r e t h i c k phase holograms i n W i t h these f a c t s i n mind, t h e purpose o f t h e t h e s i s i s t o  t h e mechanism o f h o l o g r a p h i c  s t o r a g e i n l i t h i u m n i o b a t e by  s t u d y i n g t h e o p t i c a l damage u s i n g a c o m p u t e r - c o n t r o l l e d  ellipsometer.  Chapter I I gives a b r i e f d e s c r i p t i o n o f the experimental  s e t up,  the o p t i c a l damage procedure and shows how t h e e l l i p s o m e t e r r e a d i n g s a r e r e l a t e d t o r e f r a c t i v e i n d e x measurements. I n C h a p t e r I I I , the p a t t e r n s o f r e f r a c t i v e i n d e x change due t o a c i r c u l a r damaging l a s e r beam i s i n v e s t i g a t e d . . q u a l i t a t i v e l y u s i n g Chen's model.  The r e s u l t s a r e e x p l a i n e d  A b r i e f review o f 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 n i o b a t e i s then g i v e n .  F i n a l l y , n u m e r i c a l methods based  on K i n g ' s a n a l y s i s o f o p t i c a l damage i n KTN a r e u s e d t o t r e a t t h i s p r o b l e m q u a n t i t a t i v e l y and t h e r e s u l t s compared w i t h  experiments.  I n C h a p t e r I V , the p a t t e r n s o f r e f r a c t i v e i n d e x change due t o a narrow s t r i p o f l a s e r l i g h t i s i n v e s t i g a t e d ' .  A s i m p l i f i e d t h e o r y based  on Chen's model i s s o l v e d a n a l y t i c a l l y u s i n g L a p l a c e  t r a n s f o r m method.  T h i s method i s u s e d s i n c e i t can e a s i l y be m o d i f i e d t o s o l v e f o r any o t h e r light distributions.  The r e s u l t s a r e t h e n compared w i t h  experiments.  F i n a l l y , t h e r e s u l t s o f an experiment r e l a t i n g t o t h e p r o p o s e d 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 t h a t c a n be drawn f r o m t h e above s t u d i e s a r e  remarked upon i n C h a p t e r V.  3  II. 1.  THE GENERAL EXPERIMENTAL ARRANGEMENT  Introduction The l i t h i u m niobate c r y s t a l used i n the experiments was  supplied by Harshaw Chemical Company and i t s dimensions are 2.0 x 1.5 x 3 0.3 cm .  The 2.0 x 1.5 cm  2  / faces are polished f l a t to 1/2 wavelength  and p a r a l l e l within one minute at sodium 'D' l i g h t within 1 mm of the edge or better.  The o p t i c a l damage was performed by exposing the c r y s t a l  to a Coherent Radiation argon-ion l a s e r (maximum output=500mW) f o r s p e c i f i e d periods of time.  The power of the l a s e r l i g h t was measured by  a Jordan o p t i c a l power wattmeter,,  The r e s u l t i n g o p t i c a l damage was then  probed by placing the c r y s t a l i n an automated ellipsometer.  Before each  experiment, the c r y s t a l was heated to 500°C f o r h a l f an hour and then cooled slowly.,  This treatment was shown to anneal out any space-charge  f i e l d r e s u l t i n g from o p t i c a l 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 '  t h i s project, was interfaced to a DEC PDP-8/E computer. additions were required to the e x i s t i n g i n t e r f a c e .  Only minor  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 i n t e n s i t y of the l a s e r l i g h t so that t h i s probing beam would not o p t i c a l l y damage the c r y s t a l . (The c r y s t a l could be l e f t exposed to t h i s reduced laser l i g h t f o r several hours without showing any sign of being o p t i c a l l y damaged.)  F i g u r e 1:  Rate o f c o o l i n g o f c r y s t a l a f t e r b a k i n g t o 500°C f o r 1/2 h o u r .  SHAFT  ENCODER  MOTOR  o  -C  o-—  0  c—n  PHOTO MULTIPLIER GREEN  CRYSTAL  FILTER  QWP  VARIABLE GAIN  POLARIZER  F i g u r e 2:  ANALYZER  i  o  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 c o n n e c t i o n t o t h e computer i n t e r f a c e .  AMPLIFIER  6  A f t e r p a s s i n g t h r o u g h the p o l a r i s e r , the l i g h t was The q u a r t e r - w a v e p l a t e (QWP)  c o n v e r t e d the l i n e a r l y p o l a r i s e d  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 w h i c h was the  crystal.  linearly polarised. light  then i n c i d e n t normal t o  Any l i g h t emerging from the a n a l y s e r went i n t o a p h o t o -  m u l t i p l i e r and t h e n 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. S i n c e t h e r e f r a c t i v e i n d e x i n l i t h i u m n i o b a t e i s temperature 9 dependent , the t e m p e r a t u r e must be k e p t c o n s t a n t d u r i n g ~ measurement. The c r y s t a l , h e l d i n a sample h o l d e r ( f i g u r e 3 ) , was p l a c e d i n a p l e x i g l a s s box w i t h openings a t o p p o s i t e ends o f the box to  a l l o w t h e p r o b i n g l a s e r beam to e n t e r and l e a v e ( f i g u r e 4 ) . R e s i s t o r s  were u s e d as h e a t i n g elements and t o g e t h e r w i t h a f a n , a Y e l l o w S p r i n g s I n s t r u m e n t s p r o p o r t i o n a l t e m p e r a t u r e c o n t r o l l e r and a t h e r m i s t o r as s e n s i n g element, the t e m p e r a t u r e was m a i n t a i n e d a t 32 +_ 0.05°C. Throughout  the whole e x p e r i m e n t , the temperature was f u r t h e r m o n i t o r e d  by i n s e r t i n g i n t o the box an i r o n - c o n s t a n t a n thermocouple and c o n n e c t i n g the  o u t p u t to a s t r i p c h a r t r e c o r d e r .  S t e p p i n g motors were mounted to  the  s h a f t o f the sample h o l d e r i n such a way  t h a t t h e c r y s t a l c o u l d be  moved i n two m u t u a l l y 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 p r o c e d u r e B e f o r e o p t i c a l damage, the c r y s t a l was p l a c e d i n the s e t up  shown i n f i g u r e 2.  A f t e r a l i g n m e n t o f the c r y s t a l so t h e the p r o b i n g  l a s e r beam passed n o r m a l l y t h r o u g h i t ,  the e l l i p s o m e t e r was b a l a n c e d .  The b a l a n c i n g p r o c e d u r e w i l l be d e s c r i b e d i n a l a t e r  section.  S t e p p i n g motors t h e n 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  F i g u r e 3-  Sample h o l d e r f o r LiNbO  crystal.  TO  STEPPING  MOTORS  co F i g u r e .4:  Temperature c o n t r o l l i n g  box.  9  same p r o c e d u r e was r e p e a t e d . C o n t i n u i n g i n t h i s way, a map " o f t h e r e f r a c t i v e i n d e x v a r i a t i o n s throughout  t h e c r y s t a l was o b t a i n e d .  The d a t a was s i m u l t a n e o u s l y r e c o r d e d on paper t a p e s a s t h e 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 t h e n removed from t h e s e t up and exposed t o t h e 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 t i m e . I t was t h e n r e p l a c e d i n t h e e l l i p s o m e t e r and t h e b a l a n c i n g r o u t i n e repeated.  A r e f e r e n c e mask a t t a c h e d t o t h e sample h o l d e r e n a b l e d t h e  c r y s t a l t o be scanned a t t h e same g r i d - p o i n t s b e f o r e and a f t e r t h e o p t i c a l damage.  I n t h i s way a t h r e e - d i m e n s i o n a l map o f t h e r e f r a c t i v e  i n d e x v a r i a t i o n s a f t e r o p t i c a l damage was a g a i n o b t a i n e d and t h e d a t a a g a i n r e c o r d e d on p a p e r t a p e s .  The d i f f e r e n c e between t h e two s e t s o f  r e a d i n g s gave u s a d a t a map o f t h e i n d u c e d change i n t h e 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 t h e PDP-8/E t o s u b t r a c t  t h e c o r r e s p o n d i n g numbers on t h e two paper t a p e s .  The r e s u l t i n g d a t a  (on paper t a p e s ) was then c o p i e d onto 9-track magnetic tapes i n t h e IBM 360/67 computer u s i n g programs p r o v i d e d by t h e UBC Computing C e n t r e . A p e r s p e c t i v e view o f t h e i n d u c e d r e f r a c t i v e i n d e x change was then p l o t t e d u s i n g t h e Computing C e n t r e ' 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 a r e Glan-Thompson p r i s m s mounted i n g r a d u a t e d c i r c l e s w h i c h were r o t a t e d b y s t e p p i n g motors. D e c i t r a k s h a f t encoders  (TR-5HC-CW/D)  w h i c h were mounted on t h e a n a l y s e r and  p o l a r i s e r c o n v e r t e d t h e a n g l e s t o BCD f o r i n p u t t o t h e computer. The s t e p p i n g motors and s h a f t encoders were b o t h m u l t i p l e x e d t o one d e v i c e code. The e l l i p s o m e t e r measured P and A, t h e azimuths o f t h e 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 b y the p o l a r i s e r and a n a l y s e r r e s p e c t i v e l y .  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 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 ( F i g u r e 5 ) .  Resolving this into  components a l o n g the slow and f a s t a x i s o f t h e QWP, E  = cos(Q-P)  F  E  g  = sin(Q-P)  where Q i s the a z i m u t h o f the f a s t a x i s o f t h e QWP. t h r o u g h the QWP.  we g e t (l) After passing  the slow a x i s component o f l i g h t i s d e l a y e d by 6  r e l a t i v e t o the f a s t a x i s (s i s the r e t a r d a t i o n o f 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 a l o n g t h e X and Y a x i s ( n and n^ a x i s g  o f t h e LiNbO^) we t h e n g e t E  x  = Ej,' cos Q + Eg' s i n Q = E ' s i n Q - Eg' cos Q  ^)  11  LiNbO^ i s a n e g a t i v e u n i a x i a l c r y s t a l , t h a t i s , n > n and 3 ' o e hence a f t e r p a s s i n g through the c r y s t a l , E  i s d e l a y e d by cf> r e l a t i v e A  to  E  y  E« x  = E  x  exp  E ' = E y  (4)  y  To be e x t i n g u i s h e d by the a n a l y s e r , t h e l i g h t must be p l a n e or s t a t i n g i t i n a n o t h e r  polarised,  way  (phase o f E ) - <f> = (phase o f E ) Substituting E  (5)  1  A  ( l ) and ( 2 ) i n t o ( 3 ) , we have = cos (Q-P) cos Q + s i n (Q-P) s i n Q exp - j 6  A.  or  E  y  = cos (Q-P) s i n Q - s i n (Q-P) cos Q exp - j &  E  x  = C  E  y  = C  where  5  + C  2  exp - j 6  + C  4  exp - j 6  (6)  = cos (Q-P) cos Q  = s i n (Q-P) s i n Q  = cos (Q-P) s i n Q Now  exp - j <$ = cos 5  °4  ~  =  s  i  ^ ~  n  P  ^  c  o  s  ^  - j s i n <$  Hence e q u a t i o n ( 6 ) can be w r i t t e n E  x  = (C  + C  2  cos 6  ) _ j c  2  sin 6  E  y  = (Q  + C  4  cos 6  ) - j C  4  sing  (7)  From e q u a t i o n s ( 7 ) we g e t t a n <f>x = t a n (phase o f E )  - c  =  °2 - C, = ——, _ • c_ + c. coss C  t a n <j> = t a n (phase o f E ) y Y' v  v  l  2  3  +  C 0 S < S  4  From e q u a t i o n (5) we have tan  (phase o f E„ - phase o f E ) = t a n A> A  I  (8)  12  t a n 4>  v  or  - t a n <f> J  1 + t a n <j)  Finally substituting  (8)  tan*  (9)  y  i n t o (9)  sin6 tan*  =  tan *  x  and s i m p l i f y i n g we g e t  s i n 2(Q-P)  =  = sin6  t a n 2(Q-P)  cos 2 (Q-P)  F o r a p e r f e c t QWP tan*  6' = 90° and our QWP was s e t w i t h Q = -45° and hence =  Hence  t a n (-|- - 2P)  * =- — - 2P Z nir  n=0,l,2,  I f <J>.,P^ a r e the phase change and p o l a r i s e r r e a d i n g b e f o r e  optical  damage as the l a s e r l i g h t t r a v e r s e d t h e c r y s t a l and  a r e t h e phase  change and p o l a r i s e r r e a d i n g a f t e r o p t i c a l damage, t h e n <f> . = - • £ - . 2P . ± nir 1 2 1 n=0,1,2,3. • or  - - J - 2P  f  (*  f  h i  f  - «J.) = 2(P ±  ±  - P ) f  The phase change * i s r e l a t e d to t h e r e f r a c t i v e i n d e x o f t h e 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 o f 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 w a v e l e n g t h o f t h e l a s e r n=  extraordinary refractive  n=  ordinary refractive  g  Q  light  index  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 o f t h e c r y s t a l . Hence  ' <j>. =  2TC — d ( n -n ) X ° 6  2TI  chp = ^  —  ,  d [ ( n -n ) + A(n -n ) e 0 e 0  1  13  where A ( n -n ) i s t h e change i n b i r e f r i n g e n c e due t o o p t i c a l damage, e o S u b t r a c t i n g we g e t 271  <4> " * ) = f  — K  ±  d  A  ( e n  _ n  o  )  Combining e q u a t i o n s ( l O ) and ( l l ) we f i n a l l y g e t (12)  A(n -n ) = . e  o  ixd Equation ( l 2 ) i s the basic equation r e l a t i n g the r e f r a c t i v e index change due t o o p t i c a l damage and t h e p o l a r i s e r r e a d i n g s . 5.  Balancing  Procedure  A f t e r t h e b a l a n c i n g program i s l o a d e d i n t o t h e computer and s t a r t e d , t h e v a r i o u s p o i n t e r s a r e i n i t i a l i s e d . The computer t h e n 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 t h e keyboard  o f the teletype.  There  a r e s e v e r a l i n s t r u c t i o n s which c a n 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 , t h e i n s t r u c t i o n BE i s i s s u e d  (BE = b a l a n c e e l l i p s o m e t e r ) and t h e p o l a r i s e r i s b a l a n c e d f i r s t . The program d e t e r m i n e s w h i c h way i t must s t e p t h e s t e p p i n g motor o f t h e 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 o f the p h o t o m u l t i p l i e r . I t s t e p s t h e motor i n t h a t d i r e c t i o n u n t i l  a s e t o f 64 p h o t o m u l t i p l i e r  r e a d i n g s a r e t a k e n (one a f t e r each s t e p ) and summed. The motor c o n t i n u e s stepping u n t i l increase again.  t h e e r r o r s i g n a l goes t h r o u g h a minimum and s t a r t s t o A second  sum o f r e a d i n g s a r e t h e n t a k e n as t h e motor  s t e p s and i s c o n t i n u o u s l y updated t o c o n t a i n o n l y t h e 64 most r e c e n t r e a d i n g s . When t h i s second  sum e q u a l s t h e f i r s t sum found on t h e o t h e r  s i d e o f t h e minimum, t h e b a l a n c e p o i n t , w h i c h i s midway between t h e two e q u a l sums, has been found.  The a n a l y s e r i s then b a l a n c e d i n t h e same  14  manner.  The p o l a r i s e r i s then a g a i n b a l a n c e d .  a n a l y s e r r e a d i n g s are' then p r i n t e d out on the  The  p o l a r i s e r and  teletype.  III. 1.  CIRCULAR LASER BEAM EXPERIMENT  Experiments F i g u r e 1 shows a map  o f t h e change i n b i r e f r i n g e n c e A(n -n ) e o  when t h e c r y s t a l i s damaged by t h e a r g o n - i o n l a s e r (output=200 r a d i u s o f beam=0.5 mm)  f o r 10 s e c o n d s .  mW,  A s e c t i o n a l view o f t h e p l o t  t h r o u g h t h e c - a x i s o f t h e c r y s t a l i s shown i n f i g u r e 2.  One  notices  t h a t A ( n -n ) r e v e r s e s s i g n n e a r t h e beam edge w h i l e i t r e m a i n s n e g a t i v e 6  0  i n s i d e t h e beam d i a m e t e r .  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 ~ n ) s t a y s n e g a t i v e a l l the way. e  Q  Figure 4  shows a p l o t o f A ( n - n ) v e r s u s e x p o s u r e t i m e f o r a c i r c u l a r damaging g  beam ( r a d i u s = 0 . 5 mm). o f 20 s e c o n d s .  o  Ain^-n^)  i s l i n e a r i n time up t o an exposure time  T h i s c o r r e s p o n d s t o an exposure energy d e n s i t y o f  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 In the of  o r d e r t o e x p l a i n o p t i c a l damage, Chen"^ had t o p o s t u l a t e  p r e s e n c e o f an i n t e r n a l f i e l d E the c - a x i s of the c r y s t a l .  q  o f unknown o r i g i n i n t h e d i r e c t i o n  Conduction 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 l a s e r l i g h t o f t h e a p p r o p r i a t e w a v e l e n g t h , d r i f t under t h e i n f l u e n c e o f t h i s f i e l d a l o n g t h e c - a x i s o f the c r y s t a l f o r some d i s t a n c e b e f o r e b e i n g r e t r a p p e d . Assuming t h a t t h e r m a l d e t r a p p i n g i s n e g l i g i b l e , the  electrons stay trapped.  The s p a c e - c h a r g e f i e l d E  s e t up between SC  the  p o s i t i v e i o n i s e d c e n t r e s i n the i l l u m i n a t e d a r e a and the t r a p p e d  e l e c t r o n s t h e n g i v e s r i s e t o the o b s e r v e d r e f r a c t i v e i n d e x change v i a t h e 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  &(n -rh) e  X 1 0  5  *  c - a x i s  F i g u r e 2:  ( c m )  S e c t i o n a l view t h r o u g h the c - a x i s o f F i g u r e 1.  F i g u r e 4: A ( n - n ) v e r s u s exposure time f o r a c i r c u l a r beam. e  o  20 c-axis  F i g u r e 5*.  The space-charge f i e l d E would e x p l a i n  Figure 5 indicates  shown i n t h i s  figure  the p l o t o f A ( n - n ) o f F i g u r e s e  0  s i t u a t i o n f o r a c i r c u l a r damaging beam.  o b v i o u s t h a t the space-charge f i e l d E  s c  1,2,3. It is  (shown h e r e as l i n e s o f f o r c e ) '  r e v e r s e s s i g n a l o n g the c - a x i s but n o t a t r i g h t a n g l e s t o i t .  11 Johnston  proposed an a l t e r n a t e  t h e o r y i n w h i c h the p h o t o -  e x c i t a t i o n o f 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 e x c e s s o f 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  a r e a and l e a d s to a l o c a l  i n c r e a s e i n the m a c r o s c o p i c p o l a r i s a t i o n w i t h an p o l a r i s a t i o n charge p.,  = -V, P / 0.  accompanying  The e l e c t r i c f i e l d ,  due t o |Op  i s t h e r e f o r e c r e a t e d a n t i p a r a l l e l t o the spontaneous p o l a r i s a t i o n  which  causes t h e f r e e c a r r i e r s t o d r i f t a l o n g the c - a x i s u n t i l t h e y become r e t r a p p e d i n s h a l l o w t r a p s o u t s i d e the i l l u m i n a t e d a r e a . The f i e l d , due t o the p o l a r i s a t i o n c h a r g e , 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.  S t e a d y s t a t e i s r e a c h e d when t h e 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 b a l a n c e d by the  21  s p a c e - c h a r g e f i e l d c a u s e d by t h e e l e c t r o n s w h i c h have moved o u t f r o m the  illuminated  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  resulting  f r o m t h e 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 t h e 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 i n d u c e s  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 the amount o f charge t r a n s f e r too  large  3.  Electro-Optic  effect.  A c c o r d i n g t o Amodei  ,  i n v o l v e d according to Johnston's theory i s  t o be p r a c t i c a l l y r e a l i z a b l e . 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 d e f i n e d as t h e change i n t h e  r e f r a c t i v e index of a material For  when a f i e l d i s a p p l i e d  to i t .  an i s o t r o p i c medium, t h e d i e l e c t r i c p r o p e r t i e s a t o p t i c a l  f r e q u e n c i e s a r e g i v e n by  -  D  E  -  (1)  = p e r m i t t i v i t y of free  q  e  space  = d i e l e c t r i c c o n s t a n t o f t h e medium  D = displacement E = electric field The r e f r a c t i v e i n d e x n i s d e f i n e d as n For  =  /e~.  an a n i s o t r o p i c  medium e q u a t i o n ( l ) has t o be r e p l a c e d by  D. = e e . .E i o i j I t c a n be shown t h a t f o r t h i s case two waves, o f d i f f e r e n t v e l o c i t i e s , may i n g e n e r a l p r o p a g a t e  t h r o u g h the c r y s t a l f o r a g i v e n wave n o r m a l .  Each wave has i t s own r e f r a c t i v e i n d e x . i n d i c e s n.^ n^,  The p r i n c i p a l r e f r a c t i v e  a r e t h e n a c c o r d i n g l y d e f i n e d as  where e ^ , E j ,  e ^ arethe 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  o f a c r y s t a l are often described i n  terms o f t h e i n d e x e l l i p s o i d ( i n d i c a t r i x ) .  The e q u a t i o n o f t h i s  surface i s 2 l  X  2 X  2 l  n  n  2 X  2 2 2  n  3  =  2  3  1  where t h e c o - o r d i n a t e s x ^ a r e p a r a l l e l t o t h e axes o f t h e e l l i p s o i d and n^  are thep r i n c i p a l r e f r a c t i v e indices. For  describing  f i e l d i s applied,  J  t h e g e n e r a l e q u a t i o n o f t h e i n d i c a t r i x t h e n becomes  [ - ^ - 6 . . + Z.  L  n  ij  2  i,j,k,l  t h e 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  E. + R. .. -B.-E, •+  ijk k  ljkl  ] x.x = 1  k l  J  (2)  i j  ij  where t h e i n d i c e s i , j , k , 1 r u n f r o m 1 t o 3 . The Z. ., and R. .... a r e I J K  ijki  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 t e n s o r components r e s p e c t i v e l y . i n d i c e s i , j c a n be i n t e r c h a n g e d as c a n k and 1, and t h e u s u a l can  The  contractions  be made r . « — > Z/. .\. mk Uj;k  and  R <—• mn  R/..w, , s . (ij)(kl;  where m and n r u n f r o m 1 t o 6 and m i s r e l a t e d t o ( i j ) and n t o ( k l ) as follows:  W l l , 2+-*22, For  3<-*33,  4*->23,  5+-+13, 6 W L 2 .  t h e case o f t h e 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 c o e f f i c i e n t s f o r LiNbO^ (class 3m) i s 0  12  0  13  •22  0  0  0  L  '51 61 Symmetry requires r  -10  "23  "13  '33  r  0  0  0  0  cm/Volt  -10 cm/Volt = 3.4 x 10  -10 T _ , » 30.8x 10' cm/Volt 33 f 10 cm/Volt  0  42  2 2  *  '42  r, =1 2- r , „ ~='61* -r, "13' "51 ~ 42' '22  23  x  r  x  -i  A  Writing out i n f u l l , the equation of the. i n d i c a t r i x f o r LiFbO„ then becomes 3 + r E „ + r „E„ ] x 12 2 13 3 n o  n.,  n  2 + r „ E „ ] x" 3 '33 3  + r„„E + r _ E , ] x 22 2 '23 3 0  n  n,.  n  0  n  where E^, E , E^ are the e l e c t r i c f i e l d strength X  2*  z  3  d  l  r e c  fl°  n s  respectively.  n  components i n the  2  l*  0  + 2r^,E.x x^ + 2r„ E„x^x., + 2 r ^ E x . x , = 1 "42 2 2 3 • 5 1 1 1 3  £  x  0  For a f i e l d (either applied or  i n t e r n a l ) paralle 1 to x^ ( the c-axis of the c r y s t a l ) the i n d i c a t r i x becomes  +  r  13 3 E  ]  x  l  +  Thus there are modifications  + r  [  2 3  E  3  ] x  + [  2  n.  2 n  +  r  3 3  E  3  ] x!  3  only to the axis length but no r o t a t i o n of  the p r i n c i p a l axes of the index e l l i p s o i d . The i n d i c a t r i x has the following important properties.  If a  ->•  wavefront has i t s normal i n a c e r t a i n d i r e c t i o n fronts normal to  P  P, then the two wave-  which may be propagated through the c r y s t a l have  24  r e f r a c t i v e i n d i c e s e q u a l t o t h e semi-axes o f t h e e l l i p s e o b t a i n e d i n the f o l l o w i n g way. Draw t h r o u g h t h e o r i g i n o f t h e 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 o f P . perpendicular to i t .  Draw t h e c e n t r a l s e c t i o n o f t h e i n d i c a t r i x  T h i s w i l l be t h e e l l i p s e whose major and minor  axes a r e t h e r e s p e c t i v e i n d i c e s . F o r t h e e x p e r i m e n t a l s e t up, a l i g h t wave ( t h e p r o b i n g h e l i u m neon l a s e r ) i s p r o p a g a t i n g i n the d i r e c t i o n and w i t h s p a c e - c h a r g e f i e l d E_ i n t h e x„ d i r e c t i o n , t h e e q u a t i o n f o r t h e x = 0 s e c t i o n o f 3 3 i 0  2  the i n d i c a t r i x i s  + r_E ] + [ 13 3 1 J  "~- * r E ^33 3 3  u  2  1  ] x? 3  J  =  1  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 t h e p o l a r a x i s and 3 3 n  l  =  n  o  V  =  n  Hence  3  =  n  e  i  2  [  2  1  + T ^ E J ] x" + [ — — n  o  + r  5 3  E  ] x  3  5  =  1  e  n  The  e f f e c t o f t h e f i e l d E^ i s t h u s t o change t h e i n d e x o f r e f r a c t i o n  for  a wave p o l a r i s e d a l o n g x, so t h a t t h e new i n d e x ( n 1  b  2  1  y  • M f,  (n 0  Since  X l  1  ,  "  ~  (  ,  Q  Q  /  Therefore  ^  o  .  2  ^  0  we c a n make the a p p r o x i m a t i o n 1  l.n  . )  n  0  A n << n  „  +  + An ) i s g i v e n o o  An  2 n  s2  + An J o'  [ 1+  0  2An 3  °  = r_E, 13 3  n  -2  Q  J1  o  ^«  i -in o 2  2An  t 1-  n  °] o  3 n r „E„ o 13 3 n  or  An = o  2  S i m i l a r l y f o r t h e wave p o l a r i s e d a l o n g x. 3 n r^-E, e 33 3 A i  e  2  E, A(n -n ) = — 3 — / (3n r _ - n3r , _ ) e o' 2 e 33 oo 13 e  Hence  n  A t t h e w a v e l e n g t h o f the helium-neon l a s e r (6328 2.) n  o  r  1 3  = 2.2918  n  e  = 8.6 x 1 0 ~  10  = 2.2012  r ^ = 30.8 x 1 0 ~  cm/v  10  cm/v  S u b s t i t u t i n g we g e t A(n -n ) = - 1 . 1 3 x 1 0 ~ E^ e o 3 8  -,. . 4.  Q u a n t i t a t i v e M o d e l l i n g o f 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 o f t h e l i g h t i n d u c e d  r e f r a c t i v e i n d e x 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 t o be a p p l i e d i n o r d e r f o r o p t i c a l damage t o o c c u r . Chen's t h e o r y .  He based h i s a n a l y s i s on  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 r e t r a p p e d  and t h e i r c o n c e n t r a t i o n s  t o r e m a i n s m a l l compared t o t h e donor and t r a p d e n s i t i e s .  a r e assumed Furthermore,  K i n g assumed t h a t t h e d r i f t l e n g t h o f e l e c t r o n s i s s m a l l compared t o t h e s c a l e o f r e f r a c t i v e i n d e x change so t h a t t h e e l e c t r o n c o n c e n t r a t i o n r e m a i n s p r o p o r t i o n a l t o t h e l i g h t i n t e n s i t y and i s g i v e n by: 2 P t t i  o  o  n (x,y) = [ g—^ P t urjvr  f  e x  ft  0  where  = power o f l a s e r beam hv  = photon energy  +y )  2 ^  - 2{x  r o  2 x  always  26  (c) E (x,y) y  F i g u r e 6:  T h r e e - d i m e n s i o n a l p l o t s showing t h e s p a t i a l d i s t r i b u t i o n o f (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 n e g a t i v e charge, (b) the x component o f the i n t e r n a l f i e l d and ( c ) the y component o f the i n t e r n a l f i e l d  27  electron l i f e r  = l a s e r beam r a d i u s  o  a The  time  = o p t i c a l a b s o r p t i o n a t the damaged w a v e l e n g t h  equation of c o n t i n u i t y r e q u i r e s  and where  j = qn u (E + E ) ^ 0 0 sc 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 -»-  E Equations  (3)  = internal  0 sc  = space-charge f i e l d  ( l ) , (2) and V.E  field  (3) t o g e t h e r w i t h P o i s s o n ' s  = V,(E o + E sc ) = V.E sc  e n a b l e s us to s o l v e f o r E : . sc Now, t h e f o r m a l s o l u t i o n to (4) K c ^ ' V  = 2 ^ T  methods .  above e q u a t i o n s  e e o r  is v x  ,y  where r i s the d i s t a n c e f r o m ( x ' , y') The  equation  "P^'.y'.')  £  n  l l ?  d  *'<*'  to ( x , y ) .  have been s o l v e d u s i n g f i n i t e d i f f e r e n c e  A p p e n d i x A o u t l i n e s the method u s e d and a l s o c o n t a i n s  copy of the program w r i t t e n to s o l v e the above e q u a t i o n s . show a t h r e e - d i m e n s i o n a l  p l o t of  p(x,y), E ^ ( x , y )  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 c o n s t a n t s OUT  = 10 "^m/V  P  = 10~  0  L  5  = 2.08  Watts x 10  1 5  cycles/sec  r  a  Figure 6  E (x,y) a f t e r  12  h  = 1.2 x 10  r  =  o  E  10~ m 4  = 4 x 10  o  m  5  v/m  At  =1.0  sec  e e o r  = 2.83 x 10"" F/m ' 1C  where E ^ ( x , y ) and E ( x , y ) a r e t h e x and y components o f t h e s p a c e charge f i e l d E sc One n o t i c e s t h a t E ( x , y ) i n d e e d shows t h e same f e a t u r e as F i g u r e . 1.  Along the x direction,. E (x,y) r e v e r s e s s i g n a t the  beam edge whereas a l o n g t h e y d i r e c t i o n E ( x , y ) s t a y s unchanged i n sign. K i n g was a b l e to o b t a i n n u m e r i c a l v a l u e s o f aux s i n c e E  is o  a known c o n s t a n t i n h i s c a s e . E  o  F o r t h e case o f LiNbO^, b o t h  a r e unknowns and hence u n l e s s a u t  and E  o  can be  ayr  and  determined  i n d e p e n d e n t l y , i t i s n o t p o s s i b l e t o s o l v e f o r t h e s e two q u a n t i t i e s from the numerical  analysis.  29  IV. 1.1  NARROW SLIT EXPERIMENT  S l i t perpendicular to the c-axis The h o l o g r a p h i c s t o r a g e mechanism i n l i t h i u m n i o b a t e c a n  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 l a s e r beam.  geometry f o r t h e damaging  R e f e r r i n g t o F i g u r e 1, i f a narrow s t r i p o f l a s e r l i g h t i s  placed across the c r y s t a l i n a d i r e c t i o n perpendicular to that of the proposed i n t e r n a l f i e l d E  q  , t h e n , f o r a s u f f i c i e n t l y narrow  s l i t , we e x p e c t a o n e - d i m e n s i o n a l due t o E  q  problem a l o n g t h e c - a x i s where d r i f t  w o u l d be t h e dominant mechanism a f f e c t i n g the movement o f p h o t o -  excited electrons.  T h i s i s a s i m p l e r problem t o a n a l y s e than t h e p r e v i o u s  c a s e where t h e f a c t t h a t i t i s a t w o - d i m e n s i o n a l d i f f i c u l t to solve.  problem makes i t more  Furthers, i n the a n a l y s i s by K i n g , he has t o assume  t h a t t h e d i f f u s i o n and d r i f t l e n g t h s a r e both s m a l l so t h a t n ( x , y ) a l w a y s remains p r o p o r t i o n a l to t h e l i g h t i n t e n s i t y . i s r e q u i r e d i n the one-dimensional  No such assumption,  case.  c-axis  SLIT  EXPOSURE' i- -- +  -|- "I-  +  A.  T LiNb0  F i g u r e 1:  3  F o r a s u f f i c i e n t l y narrow s t r i p o f l i g h t p l a c e d ' a c r o s s t h e c r y s t a l as s h o w n , ' d r i f t would be t h e dominant mechanism a f f e c t i n g the movement o f e l e c t r o n s a l o n g t h e 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 o b t a i n e d when t h e problem i s  made s p a t i a l l y o n e - d i m e n s i o n a l .  Further s i m p l i f i c a t i o n i s obtained i f  t h e e x p o s u r e ( t i m e x i n t e n s i t y ) , i s k e p t s m a l l enough f o r t h e e f f e c t s c a u s i n g s a t u r a t i o n t o be n e g l e c t e d . T h i s means: (1)  the space-charge f i e l d E  g c  s e t up between t h e p o s i t i v e i o n i s e d c e n t r e s  i n t h e i l l u m i n a t e d a r e a and t h e t r a p p e d e l e c t r o n s i s n e g l e c t e d compared to  the i n t e r n a l f i e l d E . 0  (2)  t h e t r a p occupancy i s c o n s i d e r e d t o be o n l y s l i g h t l y p e r t u r b e d , so t h a t (a) t h e r a t e o f r e l e a s e o f e l e c t r o n s f r o m t r a p s a t a g i v e n p o i n t r e m a i n s p r o p o r t i o n a l t o t h e l i g h t i n t e n s i t y a t t h a t p o i n t , and (b) t h e r a t e o f c a p t u r e o f c o n d u c t i o n band e l e c t r o n s by t r a p s i s proportional' to t h e i r concentration. Using these assumptions, the treatment  i s applicable 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 s i m p l e  cases.  (1) W i t h a narrow s t r i p o f l a s e r l i g h t p e r p e n d i c u l a r t o t h e c - a x i s . I n t h i s c a s e , t h e m o t i o n o f e l e c t r o n s a l o n g t h e c - a x i s i s due c h i e f l y t o 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 o f l a s e r l i g h t p a r a l l e l t o t h e c - a x i s .  In this  c a s e , d i f f u s i o n i s t h e dominant mechanism a f f e c t i n g t h e movement o f e l e c t r o n s i n a d i r e c t i o n perpendicular to the c-axis Under t h e above a s s u m p t i o n s , t h e f o l l o w i n g e q u a t i o n s o p t i c a l damage c a n be w r i t t e n down:  governing  J  = neu  an  n  1  at  =  £  •  V  (E + E ) + eD V n <\, ney I o sc n no  n  ~ T  •1 7  +  =  P  /  e  v  -». v .J + g(iight)  (2)  = - v -  '\c  (l) '  (3)  o  (4)  e  E q u a t i o n ( l ) i s the c u r r e n t d e n s i t y e q u a t i o n where t h e d i f f u s i o n  term  i s neglected. Equations  (2) and (3) a r e the c o n t i n u i t y e q u a t i o n s f o r t h e c o n c e n t r a -  t i o n o f e l e c t r o n s and t h e 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 . E q u a t i o n (4) i s P o i s s o n ' s e q u a t i o n . For the one-dimensional j(x,t) • '  =  ne  =-  these equations s i m p l i f y to  E no  M  J^L  dt  8p(x,t) 3t  problem,  (5)  +  T  I " e3x  +  e  (lignt)  /rj\  _ 3j(x,t> 9x  =  3E sc ax  ( 6 )  (8)  P( ^) x  E E O  where  j(x,t)  =  electron current density  E  =  internal  n(x,t)  =  c o n c e n t r a t i o n o f e l e c t r o n s i n t h e c o n d u c t i o n band  e  =  electron  =  charge d e n s i t y  =  p e r m i t t i v i t y o f f r e e space  =  d i e l e c t r i c c o n s t a n t o f medium  0  p(x,t) E e  Q  g( l i g h t . ) =  field  charge  r a t e o f g e n e r a t i o n o f e l e c t r o n s due t o l a s e r  light  32  F o r t h e problem o f h o l o g r a p h y  g ( l i g h t ) may be a c o m p l i c a t e d e x p r e s s i o n  o f t h e s p a t i a l v a r i a b l e w h i c h may be known o n l y g r a p h i c a l l y , ,  As i n d i c a t e d  below, t h e L a p l a c e t r a n s f o r m 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 o f a r b i t r a r y g ( l i g h t ) .  Thus, a l t h o u g h t h e o r i g i n a l p r o b l e m  can e a s i l y be s o l v e d u s i n g e l e m e n t a r y  methods, t h e L a p l a c e  transform  method i s used. The b i l a t e r a l L a p l a c e t r a n s f o r m /  of a function f ( x ) i s  defined as: [ f(x) ] =  / f(x) e"  B X  dx = F ( s )  —oo  In the steady s t a t e  9n(x,t) = 0  at  S u b s t i t u t i n g e q u a t i o n ( 5 ) i n t o ( 6 ) and t a k i n g L a p l a c e t r a n s f o r m  (with  r e s p e c t t o t h e s p a t i a l v a r i a b l e x ) o f t h e r e s u l t i n g e q u a t i o n we g e t N(s)  0 = -  -1-  T  where  s u E N ( s ) + G-(s) n o  n N(s) =1 G(s)  [n(x,t)]  =£[g(light)]  Hence  ^ N(s) =  G ( s ) = H(s) G ( S )  E and  [ -s +  y E x n o  ]  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 o f g ( l i g h t )  -co  and h ( x )  and h ( x ) i s t h e i n v e r s e L a p l a c e t r a n s f o r m o f H ( s ) and i s g i v e n by u(-x) h(x) =  x exp  u  E no  u E T no  33  where u ( x ) i s a u n i t s t e p f u n c t i o n .  h ( x ) i s shown i n F i g u r e 2.  f/>  F i g u r e 2:  fx)  P l o t of h ( x ) .  Thus we see t h a t f o r a r b i t r a r y g ( l i g h t ) , n ( x , t ) i s o b t a i n e d by t h e c o n v o l u t i o n o f 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 o r a n a l y t i c a l l y . F o r t h e i r r a d i a t i n g geometry o f F i g u r e 1, g ( l i g h t ) i s g i v e n by g(light) = A  -d/2 < x < d/2  where A i s a c o n s t a n t , the c o n v o l u t i o n c a n be c a r r i e d o u t a n a l y t i c a l l y . E  sc  (x) i s t h e n o b t a i n e d f r o m n ( x , t ) by c o m b i n i n g e q u a t i o n s  to y i e l d m E  sc  n(x,t) e E  ( \ ( x )  =  -  :  o  ty n K  C a r r y i n g o u t the c o m p u t a t i o n s we f i n a l l y g e t :  (5),(7) and (8)  34  x n(x, t ) =  =  2 AT exp — y  E T no  : 2y  '  x  ..  exp  ] y  - d/2 < x <  x >  2 ey (x) = -  x ~  E t A x exp  ee  s c  0  1 A ee o =  e y n  E o  n  t [ l - exp  o  , x  exp 2yEx n o  ]  -d/2<  y E x n o x > d/2  0  g ( x ) , n ( x ) and E  ( z ) a r e p l o t t e d i n F i g u r e 3 f o r t h e case o f a s l i t SC  perpendicular  z < -d/2 2yEx  - d  = -  d/2  d sinh  y E x no  0  d/2  E T n o  0  E  < -d/2  Ex n o  x  - d : 2y E T Hn o  1 - exp  [  AT  d sinh  to the c - a x i s .  x<  36  1.3  Experimental Results Figure 4  shows a p l o t o f - A ( n - n ) v e r s u s d i s t a n c e a l o n g Q  the c - a x i s f o r a s l i t w i d t h o f 0.06 _+ 0.002cm (Power = 200mW).  Comparing  ( x ) o f F i g u r e 3- , one n o t i c e s two a n o m a l i e s :  t h i s p l o t w i t h the p l o t o f E sc  A(n -n )' remains s l i g h t l y p o s i t i v e on one end o f t h e s l i t and t h e maximum fe  o f A ( n - n ) does n o t e x a c t l y o c c u r a t t h e s l i t edge as p r e d i c t e d by t h e o r y . g  o  The anomalies a r e p r o b a b l y due t o e x p e r i m e n t a l e r r o r s . From F i g u r e 4 and t h e t h e o r y , a v a l u e o f p T E = 0.09 + 0.005 cm i s . o b t a i n e d . ' n o -" 2.1 S l i t p a r a l l e l t o c - a x i s On t h e e t h e r hand, i f t h e s l i t o f l i g h t i s p l a c e d a c r o s s t h e c r y s t a l i n the d i r e c t i o n o f t h e c - a x i s ( F i g u r e 5 ) ; t h e n , a g a i n f o r a s u f f i c i e n t l y narrow s l i t , we e x p e c t a o n e - d i m e n s i o n a l  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 t o the c - a x i s and s i n c e t h e r e i s no f i e l d i n this direction, diffusion movement o f e l e c t r o n s .  i s now  t h e dominant mechanism g o v e r n i n g t h e  However, e l e c t r o n s a r e n o t e x p e c t e d  t o move  v e r y f a r by d i f f u s i o n and hence i t was n o t s u r p r i s i n g t h a t e x p e r i m e n t a l l y we have so f a r o b s e r v e d 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 p r o b a b l y t o o s m a l l t o be r e s o l v e d under o u r p r e s e n t  experimental  instrumentation.  However, t h e o n e - d i m e n s i o n a l  a n a l y t i c a l l y i n t h e next 2.2  probelm has been s o l v e d  section.  Theory The assumptions made t o s o l v e t h i s p r o b l e m a r e t h e same as  the p r e v i o u s case except t h a t d i f f u s i o n , a n d n o t d r i f t ,  dominates t h e  movement o f e l e c t r o n s . Except f o r e q u a t i o n ( l ) , t h e r e m a i n i n g e q u a t i o n s r e m a i n unchanged.  three  F i g u r e 4:  P l o t o f A ( n - n ) a l o n g the c - a x i s f o r a w i d t h o f 0.06 + 0.002 cm. e  0  28  A c - a x i s  4 O  4 + 4-  •Z.//V60'5  +  + -f ++•  •»  <  —  E  S  X  L  P  I  O  T  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 o f l i g h t p l a c e d a c r o s s t h e c r y s t a l as shown, d i f f u s i o n would be t h e dominant mechanism a f f e c t i n g t h e movement o f e l e c t r o n s i n a d i r e c t i o n perpendicular to the c - a x i s .  39  3n 3x  J = eD n n  3n St  (1)  1 3J (2)  e  11  (3)  3x  3t 3E  sc 3x  (4) E  O  £  A g a i n , the s t e a d y s t a t e s o l u t i o n i s s o l v e d u s i n g L a p l a c e t r a n s f o r m method.  S u b s t i t u t i n g e q u a t i o n ( l ) i n t o (2)  and t a k i n g L a p l a c e t r a n s f o r m  ( w i t h r e s p e c t t o t h e s p a t i a l v a r i a b l e x) o f t h e r e s u l t i n g e q u a t i o n we g e t  0  where  Hence  =  N(s)  -  + D s  N(s) =  ^[n(x)]  G(s) =  X,  N(s) + G ( s )  [g(light)] -  T  W(s) =  G ( s ) = H(s)  (D  T  s  G(S)  — 1)  n and  n(x) = h ( x ) * g ( l i g h t )  where h ( x ) i s t h e i n v e r s e L a p l a c e t r a n s f o r m o f H ( s ) and i s g i v e n by  - x  T  h(x) = 2  V  [ exp D n  J  V/ITT v  x u(x)  +  exp  n  = h (x) + h (x) x  2  h ( x ) , h-^x) and h,-,(x) a r e p l o t t e d as shown i n F i g u r e 6«  u(-x) ]  j  V  n  41  Thus we see t h a t f o r a r b i t r a r y g ( l i g h t ) , n ( x ) i s o b t a i n e d by t h e c o n v o l u t i o n o f 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 o r a n a l y t i c a l l y . N o t i c e t h a t h ( x ) i s made up o f two s i m p l e e x p o n e n t i a l s , h ^ ( x )  and h ^ x ) ,  so t h a t t h e c o n v o l u t i o n i s e s s e n t i a l l y t h e same as t h e p r e v i o u s  case.  F o r t h e i r r a d i a t i n g geometry o f F i g u r e 5 , g ( l i g h t ) i s g i v e n by g(light) = A Performing  -d/2 < x < d/2  t h e 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 g e t x  n(x) =  A  exp  t  VB  d • 2VD  sinh (  x  n  )  n - d x • — cosh 2 /T~r s/B~T n n  = A T [ 1 - exp  N  ]  -d/2  n  v  Having solved f o r n ( x ) , E  sc  < x  < d/2  d sinh  V B  -d/2  v  - x = A T exp  (l),  x <  T  x  2\/B  T  n  >  d/2  T  ( x ) c a n a g a i n be s o l v e d .  Combining  equations  ( j ) and (4) we have  E  (x) -  -  ^  9x  e e  sc  0  S u b s t i t u t i n g f o r n ( x ) from above we have  E g c  ( ) x  -eAt =  y  ^ T  x -  exp —  eAt  d —  sinh (  - d V  D  T n  eAt  e x  0  g ( x ) , n ( x ) and E  sinh ( 2\/ B  0  T  V / F T  ^ V  <  -d/2  )  / D  -d/2 < x < d/2  T  d  x  P ~~ZT  x  x  exp  e £  )  s  i  n  h  (  " 2fB  (x) are p l o t t e d i n Figure 7.  )  x >  d/2  F i g u r e 7:  P l o t o f g ( x ) , n ( x ) and E ( x ) , the s t e a d y s t a t e s o l u t i o n of the problem f o r m u l a t e d i n s e c t i o n 2.2 . s c  43  3.  F u r t h e r Experiment t o i n v e s t i g a t e i n t e r n a l  field  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 g a t e d as f o l l o w s .  I f one scans a l o n g t h e c - a x i s o f t h e c r y s t a l so a s t o  g e t a map o f the b i r e f r i n g e n c e ( n - n ) , then any v a r i a t i o n s i n ( n - n ) Q  Q  a l o n g t h e c - a x i s may g i v e an i n d i c a t i o n o f t h e 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 ) may a l s o be t h e r e s u l t o f n o n - s t o i c h i o m e t r y  caused  Q  14 by changes i n t h e c o m p o s i t i o n o f t h e c r y s t a l d u r i n g t h e growth p r o c e s s . A p l o t o f (n -n ) a l o n g t h e 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 ) . p o i n t o f zero f i e l d i s undetermined.  The  By p u t t i n g e l e c t r o d e s a c r o s s o p p o s i t e  end o f t h e c r y s t a l and s h o r t i n g t h e e l e c t r o d e s , we e x p e c t t o make t h e average f i e l d  J*Edx a l o n g t h e c - a x i s z e r o .  I f t h e v a r i a t i o n s i n (n -n ) Q  a r e n o t due t o n o n - s t o i c h i o m e t r y , t h e n f r o m t h e c u r v e shown one w o u l d  0  expect  t h a t t h e p o i n t o f z e r o f i e l d would be r o u g h l y i n t h e m i d d l e o f t h e 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 s m o o t h l y a l o n g t h e c - a x i s ) .  (  n e  ~  n  ) after  s h o r t i n g t h e 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 d o t t e d l i n e . I t can be seen t h a t (n -n ) r e m a i n s almost unchanged. (The maximum s h i f t i n ( n -n ) e o . e o i s 8 . 2 x 10 ^ w h i c h c o r r e s p o n d s  t o a s h i f t i n t h e i n t e r n a l f i e l d o f 800V/cm.)  T h i s i n d i c a t e s t h a t t h e average f i e l d even b e f o r e s h o r t i n g t h e e l e c t r o d e s . beam e x p e r i m e n t s where E  q  j E d x along the c - a x i s i s almost  By r e p e a t i n g t h e c i r c u l a r damaging  b u t w i t h t h e beam p l a c e d i n 3 p l a c e s a l o n g t h e c - a x i s  i s n e g a t i v e , z e r o and p o s i t i v e , one would e x p e c t t h e n t h a t A ^ ^ - n ^ )  w o u l d a l s o r e v e r s e s i g n a t the two s p o t s where E z e r o where E ' i s z e r o .  q  r e v e r s e s i g n and r e m a i n s  The e x p e r i m e n t a l r e s u l t s a r e shown i n f i g u r e 9 where  Q  i t i s obvious t h a t A ( n - n ) d i d not r e v e r s e s i g n . g  if  zero  o  T h i s c o u l d be e x p l a i n e d  t h e i n t e r n a l f i e l d v a r i e s a l o n g t h e c - a x i s a s shown i n f i g u r e 10 where  to meet t h e r e q u i r e m e n t s shaded a r e a must be e q u a l  of  = 0 w i t h the .electrodes shorted, the  s  before putting electrodes  after *  electrodes  putting and short/ng  them  i  0  0.5 c-  Figure 8:  axis  10 (cm)  Plot of ( n - n ) along the c-azis before depositing electrodes ( s o l i d and a f t e r depositing electrodes and shorting them (dotted l i n e ) . e  D  line)  F i g u r e 10:  V a r i a t i o n s o f E a l o n g the c - a x i s to s a t i s f y t h e c r i t e r i o n J"Edx = 0 a l o n g the c - a x i s .  that  V. The  CONCLUSIONS  o p t i c a l damage p r o c e s s  investigated.  i n l i t h i u m n i o b a t e has been  The automated e l l i p s o m e t e r i s u s e d f o r s t u d y i n g t h e  p r o c e s s because t h e b i r e f r i n g e n c e o f t h e c r y s t a l c a n r e a d i l y be d e t e r m i n e d and t h e b a l a n c i n g p r o c e d u r e would be e x t r e m e l y  t e d i o u s i f done by hand.  F u r t h e r , t h e a n g l e s o f the p o l a r i s e r and a n a l y s e r c a n be r e a d t o a s e n s i t i v i t y o f +_ 0.01° w h i c h c o r r e s p o n d s t o 10 ^ i n ( n -n ) f o r t h e specimen t h i c k n e s s used. I t was found t h a t c o n s i s t e n c y o f e x p e r i m e n t a l  r e s u l t s was  o b t a i n e d by b a k i n g t h e c r y s t a l and c o o l i n g i t s l o w l y b e f o r e each e x p e r i m e n t so a s t o a n n e a l  o u t t h e space-charge f i e l d c r e a t e d d u r i n g o p t i c a l damage  of previous experiments.  The c o o l i n g o f t h e c r y s t a l a f t e r b a k i n g had t o be  c a r r i e d o u t s l o w l y , o t h e r w i s e 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  crystal. To v e r i f y Chen's model o f t h e e x i s t e n c e o f an i n t e r n a l  field-E o  i n t h e d i r e c t i o n o f t h e c - a x i s o f t h e c r y s t a l , t h e c i r c u l a r damaging beam e x p e r i m e n t was c a r r i e d o u t .  The s p a t i a l d i s t r i b u t i o n o f Ain^-n^)  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(n^-n^)  c a n be  F o r t h i s i r r a d i a t i o n geometry,  i n c r e a s e s l i n e a r l y w i t h t i m e up an an exposure t i m e o f 20 seconds  w h i c h c o r r e s p o n d s t o an exposure energy d e n s i t y o f 500J"/cm^.  Based on  K i n g ' s a n a l y s i s o f o p t i c a l damage i n KTN, t h e problem f o r t h e c i r c u l a r damaging beam was s o l v e d 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 o f E (x,y) shows t h e name f e a t u r e s as t h e e x p e r i m e n t a l p l o t o f A ( n -n ) . x e o No n u m e r i c a l  comparison o f e x p e r i m e n t a l  s i n c e t h e r e a r e two unknowns, au.T ' experiment.  and t h e o r e t i c a l r e s u l t s a r e p o s s i b l e  and E , t o be d e t e r m i n e d f r o m one o'  To i n v e s t i g a t e t h e i d e a o f an i n t e r n a l f i e l d , an e x p e r i m e n t u s i n g  48  a narrow s t r i p o f l i g h t has been c a r r i e d o u t .  When t h e s t r i p i s p e r p e n -  d i c u l a r t o t h e c - a x i s , d r i f t would be t h e dominant mechanism a f f e c t i n g t h e movement o f e l e c t r o n s .  An a n a l y s i s o f t h i s problem has been c a r r i e d o u t  based on Chen's model.  Compared t o K i n g ' s a n a l y s i s , no a p p r o x i m a t i o n as  to the form of n ( x , y ) i s necessary. Q  and experiment The  N u m e r i c a l comparison  between t h e o r y  gave a v a l u e o f 0 . 0 9 ± 0 . 0 0 5 cm f o r t h e p r o d u c t U_T E »  case f o r t h e s l i t  p a r a l l e l t o t h e c - a x i s where d i f f u s i o n i s t h e  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 . m e n t a l r e s u l t s have been o b t a i n e d .  However, no e x p e r i -  This i s not unexpected  since the  d i f f u s i o n l e n g t h o f e l e c t r o n s i s p r o b a b l y t o o s m a l l t o be r e s o l v e d w i t h o u r present experimgntal s e t up  c  From t h e above s t u d i e s , i t can be c o n c l u d e d 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 b u t t h e v a r i o u s parameters  o f t h e p r o c e s s 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 a r e a i n v o l v e s t h e a c t u a l s t o r a g e o f holograms b o t h on pure l i t h i u m n i o b a t e 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 ; t h e l a t t e r a r e u s e f u l as f a r as h o l o g r a p h i c s t o r a g e i s c o n c e r n e d s i n c e t h e w r i t i n g time and t h e w r i t i n g energy crystals.  a r e l e s s t h a n t h e undoped  As f a r as t h e e n g i n e e r i n g a s p e c t s o f t h e p r o c e s s a r e c o n c e r n e d ,  f u t u r e work s h o u l d c o n c e n t r a t e on i m p r o v i n g t h e s e n s i t i v i t y t o o p t i c a l damage  and t h e f i x i n g o f t h e holograms s t o r e d .  4y  APPENDIX  •  S o l u t i o n o f ' O p t i c a l Damage' Problem f o r a C i r c u l a r L a s e r Beam F i n i t e d i f f e r e n c e methods were used t o s o l v e t h e f o l l o w i n g  equa-  t i o n s w h i c h were e x p l a i n e d i n Chapter I I I .  If: E(x,y,t) = -  = - V. [q n y ( E + E) ] Q  * o  [ / / p U ' . y ' . t ) to|r| dx'dy']  V  c  (1)  q  (2)  J  where n (x,y) = Q  2P ax Y~ exp -  Trhvr  2(.x + y ; £  ( ) 3  r  o  o  The p a r t i a l d i f f e r e n t i a l e q u a t i o n s a r e a p p r o x i m a t e d by f i n i t e e q u a t i o n s through the t r a n s f o r m a t i o n scheme below. ble  The v a l u e o f a v a r i a -  z ( x , y , t ) a r e then determined a t a d i s c r e t e mesh o f 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 a p p r o x i m a t e d by 5 p  p(x ,y. t i  >  k +  A t ) - P(xlty1ttk) At  dt  Let A = q n y ( E + E) = A x + A y o o x y M  c  1  = (qu)  J  2P ax ~ irhvr  o 2[((10-i)Ax) A exp = exp  For  difference  r  2  + ((10-j)Ay) ] — 2  o  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 , t h a t i s o J  50  E  = E x o  o  we have A  A  = q n  x  ^  = q n  y  o  u(E + E ) x o  o  uE y  Hence  V^^j'V i =c  A (x.,y y  j 5  Ae x p(  v V tVv^'V V  t ) = C A exp (x.,y..) ±  k  +  [E (x ,-y ,t )] y  i  j  k  The i n d i c e s i , j b o t h r u n from 1 t o 20 and hence t h e r e a r e 400 mesh points (figure 1).  fxo>y o)  (*o>y )  2  0  •  I  1 J ( 20>Yo^ X  ( 20>y20' x  F i g u r e 1: Square g r i d o f mesh p o i n t s u s e d i n t h e i t e r a t i o n scheme.  F o r any v e c t o r q u a n t i t y A =  9A .  V  +  9x Hence E q u a t i o n (1)  y  9A  = — 2  A  x +  9y  becomes t r a n s f o r m e d i n t o t h e f o l l o w i n g  difference  equation P ( x , y . , t + A t ) - p ( x ,y , t ^ ) ±  k  At  A ( x . , - ,y ., t, ) - A (x. ,y ., t. ) x  r+1  ,i k  x  l  j  A (x-. ,y .,., , t. ) - A (x. ,y. ,t, )  k  y  i  >  J  3+l' k  Ax  y  v  1  3  k  Ay  I n the i t e r a t i o n scheme a square g r i d i s used, t h a t i s , Ax = Ay = h Hence we have pCx^y.^+At) = P(x y.,t ) -^JLtA (x. ,y.,t ) - A (x.,y.,t ) + i 5  k  x  y  A  ( x  + 1  i' 3+l'V Y  k  x  " V\^y^ )]  k  (4)  E q u a t i o n (2) can be s e p a r a t e d i n t o two component e q u a t i o n s a s ; E (x,y,t) = x  [//P(x',y',t)  g  £n|r| dx'dy']  (5)  o  ly"  E (x,y,t) = " y  (*'' > )  dx'dy']  fc  Now -3- £n  = -5-  r  in  (x-x')  2  + (y-y')  2  =  ( x  x  )  (x-x') +(y-y') Similarly,  A_ 3 y  Equations  to  1+1 =  (y-y') (xix') +(y-y') 2  (5) and (6) t h e r e f o r e becomes  2  (6)  52  i E (x, ,y.,t ) = - — — — x 1 J k 27T V  (  [//p(x\y ,t  v ' x  }  )  ?  2 (x.-x') +(y.-y') 2  ( y , - y') _1 '  o  J  dx'dy']  dx'dy']  (x.-x') +(y.-y ) i  "3  '  The n u m e r i c a l i n t e g r a t i o n s a r e c a r r i e d out u s i n g T r a p e z o i d a l R u l e o f I n t e g r a t i o n ( F i g u r e 2) /  n  y ( x ) dx = |  = h  F i g u r e 2:  [y + 2 _ + 2y Q  y]  (y +y„) n  [- o  +  2  +•. .. + 2 y _ n  (y, + y J  1  2 +  0  • • •+  + y j  1  y _j_) 1 n  Trapezoidal r u l e of i n t e g r a t i o n given y  =  yU).  (7)  Z  (8)  53  Let (x B(x\y',x,,y.)  = p(x\y',t.)  - x') i  (x^^^+Cyj-y')^  J  E q u a t i o n (7) becomes (x,.-x') ^ (x -x') +(y -y')  // p ( x ' , y ' , t , )  2  i  = //B(x',y ,x ,y ) ,  i  =  /dy' [  I  j  dx'dy' ?  j  dx'dy' » //dy' / B ( x ' , y ' , x .  Ax [B(x' , y '  i y  ) dx'  x ,y ) ] +  a ^f[B(x ',y ',x ,y ) 1  n  i  j  tBCx^.y^.x^y.)]]  The i n t e g r a t i o n w i t h r e s p e c t t o y' i s then  similarly carried  out and e q u a t i o n (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 a r e E  x (V^'V * 0  E  E  0  (x.,y., 0) = 0  y  p  =  V  (  o  x  i' j' k y  t  =  0  )  =  0  = constant  A copy o f t h e F o r t r a n program based on t h e above i t e r a t i o n scheme i s appended.  The program e v a l u a t e s p ( x , y , t ) , E ( x , y , t ) and x  E y ( x , y , t ) a t a d i s c r e t e mesh o f p o i n t s ( x , y ) f o r a s p e c i f i e d number o f iterations,  p, E  x  and E  y  a r e "then p l o t t e d o u t as a p e r s p e c t i v e drawing  u s i n g the program UBC PERSP i n the UBC Computing Centre  File.  54  FORTRAN  0001 000 2 000 3 ' 000''-, 0005  0006 000 7 COO 8 0009 0010 0011 0012 001.3  IV  G  CTMPI L E K  0016 0017 001 0_ 0019 0020 0021 0022 002 3 00 24" 00 2 5 00 2 6 0027 0028 0029 6030' 0031 0032 0033 0034 0035 0036 0037 00 33 _003 9 004 0"  0):24:55  PAGE C001  01 MENS ION EX I 21 ,21 ) , EY( 21 , 21 ) , AX I ?1 , ?1 ) , AY( 21 t 21 ) ,RHO( 21 , 21 ) , C A E X P ( 2 0 , 2 0 ) , 13(2 0 , 2 0 ) , C ( 2 0 , 2 0 ) , X ( 2 0 , 2 0 ) , Y ( 2 C , 2 C ) , I ( 2 0 , 2 C I REAL PO INTEGER P , 4 , R WRITE 10,2) 2 _ FORMAT (1H1, S3HC.ALC.ULATI ON OF THE SPACE CHARGE FIELD IN LITHIUM N C 10 3 A T E 'WHEN EXPOSED TO "LASER" BEAM) C READ IN INITIAL VALUES FOR VARIOUS PARAMFTERS C ANT=PRODUCT OF 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 I N VOLTS/M C.0ELT=INCREMENT OF THE TIME OF ITERATION C EPS I L = PERM[TT IV ITY OF THE L II HI U NIOBATE READ ( 5 , 1 ) A NT,PO,FR c Q , H , R O , E 0,DE L T , E PS I L , Q 1 FORMAT ( 8 E 3 . 3 , I 2 ) WRITE ( 6, 902 ) ANT , P0~, FR E Q, H, RO, EO,DELT , EPS 11.» Q 902 FORMAT (1H0, 3 ( E 9 . 3 , 3 X ) ,I 2) C l = ( 2 . 0 * P 0 * { 1 . 6 E-19)*ANT 1/( 3. 1 4 1 5 9 * ( 6 . 6 2 5 E - 3 4 ) I C2 = ( -1. • 0 ) / ( 2 • 0* 3 . 1415 9* EP S 1 L ) WRITE (6,609) C1.C2 _609 FORMAT (1110, El 0 . 3 , E1 0 . 3 ) _ C  0014  12-01-72  HA I N  IN!  T ! Ai 1 S E ' T H E " V A R T U U S " " A R R A Y S  P=l  R=0 00 10 1=1,21 DO 10 J = l , 2 1 _EX( Ij J )_=0__0 _ EYII,J)=0.0 AX(I,J1=0.0 AY(I,J)=0.0 10 RHO(I,J)=0.0 C START CALCULATION OF RHO 0 0 2 0 1=1,20 200 J = l , 2 0 DO 2 20 A E X P ( I , J I = EXP (- 2.O'M ( ( ( 1 C - I ) * * 2 ) + ( ( 1 0 - J 1 * * 2 ) » / ( R 0 * R 0 ) ) * H * H ) 40 D O 5 0 1=1,20 DO 5 0 J = l ,20 AX(I,J)=AEXP(I,J)*(EX(I,J)+EO) 50 AY! I , J ) = A E X P ( I , J ) * E Y ( I , J ) DO 6 0 0 J=l,20 A X ( 2 1 , J ) =• AX ( 1 9 , J ) 60 A Y ( J , 2 1 ) = A Y ( J , 1 9 ) 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 + 1A Y)-( I , J ) C*DEl. T WRITE (6,31 P 3 FORMAT (1H0, 20 UNO. OF IT ERATIONS= ,12) !~EVA L U A T T G N OF EX AND EY U S I N G lTTtRXTlVc Fu'RWCS" 1=0 _900_  I = I + 1_ KR I  T E ("6, 962)  _____ I  ) ) /HI  55  FORTRAN) IV G "COMPILER 962  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 00 55 0056 j005 7__ 0058 0059 0060 0061 0062 _0063__J 006 4 0065 0066 0067 0068 0069_ ""0070 C071 007.2 0073 0074 CO7 5 "00 7 6"'  500  MAIN  12-01-72  FORMAT (1H0, 4;-l I = ,12) IF (I .FQ. 211 GG TO 600 J=0 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  C4=P. HO ( L , M) / ( ( ( I-L ) **2 + ( J-M  '  "  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 BTEMP 1 = 0.0 BTEMP2=0.0 BTEMP3=0.0 _ _ j "BTEMP4'="0'".0~ " " "~ BTEHP5=0.0 CTE _P 1 _£?__2 CTEMP2=0.0 CTEMP3=0.0 CTEMP4=0.0 " " CTcMP5"=0.0 DO 80 L=2,19 DO 8 0 M=2,19 8 T E M P1 = B T E M P1+ 0(L» H ) 80 C. TEM PI = C TEMP1 +C ( L , M) DO 90_N=2,19 . BT"E"'MP2='BTEM'P_  4-B"('M";"n  BTE:MP3 = BTEMP 3 + B ( M , 20 ) 3T EMP4 = BT EMP4 + B (1 , N ) B T E M p 5 = B T E MP 5 +tt(20,N) CTEMP2=CTEMP2+C(N,1) CT E M P3 = CT E.M °3 +C_( H , 20 )  01:24:55  "  PAGE 00C2  '_ TO  7  '  i  ~  ) **2 ) *H)  ~  "  __ "  .  "  ~  _  "  •  C T h M!>4 -"CTEMPV+'C I ! ,M  "  0077 0078  90  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 ) I f f J . E;). 2 1)' GO' TC '9C0 ~ ' GO TO 500  CO 8 0 C0S1 0082 008 3 0084 _0085 0086" ' 0087 . 0088 0089 0090 _009i C092  CTEMP5=CTEMP5+C(2Q,N) 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  IF <R . EO. 9) GO TO 601: : GO TO 449 601 _ DO'534 1 = 1,20 WRITE ( 6, 533) ( R H O f l , J ) , J=),20) 533 FORMAT (1H0, 10 ( E1 0 . 3 ,2 X ) / 1 0 ( E10 . 3, 2X ) ) 534 CONTINUE 00 543 1 = 1, 20 WRITE (6, 544) ( E X ( I , J ) , J = 1,20 ) 54 4 FOR " A T ( 1H0, 10 ( E 1 0. 3 , 2 X ) / 10 ( E 10. 3, 2X ) ) 543 CONT INJE"  :  . ' ' ~~~  r FORTRAN  v r  0093 0094 0095 0096 C09-V 0093 0099 0100 0101 01C2 0103 0104 0105 0106 0107 0108 0109 0) 10 0111 0112 0113 0114 Oil 5 0116 TOTAL  IV G fXM°I LER  MAIN  12-01-72  01: 24: 55  DO 153 1=1,20 WRITE (6,555) ( F Y ( I , J ) , J = 1 , 2 0 ). . 5 55 FORMAT ( I H ) , 101E10.3,2X1 /1CIE10.3, 2X) ) CONT I NJS 553 DO P83 I=1.20 DO 8 33 J= 1,2.1 0 03' Z(1,J)=RHC(I,J) DO 931 1=1,20 DO 931 J-1,20 Y d , J)=EX(l,J) 931 DO 2 17 1=1,20 DO 217 J=l,20 21 / X ( I , J ) = E Y ( I , J ) CALL PE-^S (Z, 20, 20, 20,1 .0 , 0. 6, 10.0,45. 0 ,10.0, 10.0) CALL PLOT (20.0,0.0,-3) CALL PERS (Y,20,20,20,1.0,0.6,10.0, 45. C ,10.0, 10.0) CALL PLOT(20.0,0.0,-3) 45.0 ,10.0, 10.0) CALL PF.RS ( X, 20 , 20,20,1 .0 ,0. 6, 1 0.0, CALL PL OT NQ P=P+1 IF (P .EQ. 10) GO TO 907 GO TO 40 937 STOP END MEMORY REQUIREMENTS C057BE  C0MP1LF TIME =  5.7  SECONDS  BYTES  PAGE 0 003  57  REFERENCES 1.  W.C. Stewart, R.S. Mezrich, L.S. Cosentino, E M. Nagle, F.S. Wendt 0  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, ^g  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). F.S. Chen, J.T. LaMacchia and D.B. Fraser, Appl. Phys. Letters, 13» 223 (1968).  6.  t  71 (l97l).  7.  T.K. Gaylord, T.A. Rabson and F.K. T i t t e l , 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. 13.  J.J. Amodei, RCA Review, _52, 185 (l97l). 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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