UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Beam coupling in holograms stored in LiNbO₃ 1981

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
UBC_1981_A7 W66.pdf
UBC_1981_A7 W66.pdf [ 4.2MB ]
Metadata
JSON: 1.0065627.json
JSON-LD: 1.0065627+ld.json
RDF/XML (Pretty): 1.0065627.xml
RDF/JSON: 1.0065627+rdf.json
Turtle: 1.0065627+rdf-turtle.txt
N-Triples: 1.0065627+rdf-ntriples.txt
Citation
1.0065627.ris

Full Text

BEAM COUPLING IN HOLOGRAMS STORED IN L i N b 0 3 by R a n d a l l J . Woods B . S c , U n i v e r s i t y o f B r i t i s h Columbia, 1974 M . S c , U n i v e r s i t y o f Western O n t a r i o , 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department o f E l e c t r i c a l E n g i n e e r i n g UNIVERSITY OF BRITISH COLUMBIA We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE.UNIVERSITY OF BRITISH COLUMBIA J u l y , 1981 © Randall J . Woods In 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 of the requirements f o r an advanced degree at the 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 study. 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 copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be a llowed without my w r i t t e n p e r m i s s i o n . Department of cjufruu»j\ LS^^V^O^I^U^ The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date }2/ fr/ S / 7<n i i ABSTRACT The p o t e n t i a l use o f p h o t o r e f r a c t i v e c r y s t a l s i n o p t i c a l d a t a h a n d l i n g a p p l i c a t i o n s has r e c e i v e d c o n s i d e r a b l e a t t e n t i o n d u r i n g the p a s t decade. The p h o t o r e f r a c t i v e e f f e c t i n LiNbOgiFe i n v o l v e s the p h o t o l i b e r a t i o n and inhomo- geneous r e d i s t r i b u t i o n o f e l e c t r o n s which a r e s u b s e q u e n t l y t r a p p e d i n the new d i s t r i b u t i o n . The r e s u l t i n g e l e c t r i c f i e l d s s e t up a s i m i l a r d i s t r i b u t i o n o f v a r i a t i o n s i n the r e f r a c t i v e i ndex a l l o w i n g the f o r m a t i o n o f t h i c k phase h o l o - grams. In t h e case o f an elementary hologram ( t h a t w r i t t e n by a s i m p l e i n t e r f e r e n c e p a t t e r n between two e q u a l i n t e n s i t y beams o f monochromatic l i g h t ) , the h o l o g r a p h i c g r a t i n g may be d i s p l a c e d from the f r i n g e p a t t e r n t h a t wrote i t . T h i s phase s h i f t g i v e s r i s e t o a r e d i s t r i b u t i o n o f energy between t h e two beams, c a l l e d beam c o u p l i n g . Measurements o f t h e beam c o u p l i n g can be used t o o b t a i n a v a l u e f o r t h e phase s h i f t , and t h i s i n t u r n i s u s e f u l i n s t u d y i n g t h e e l e c t r o n t r a n s p o r t mechanisms which g i v e r i s e t o the s h i f t . P r e v i o u s attempts t o measure t h e beam c o u p l i n g have not y i e l d e d r e p r o d u c i b l e r e s u l t s and have c o n s e q u e n t l y n o t been v e r y u s e f u l i n d e t e r m i n i n g t h e phase s h i f t o r s t u d y i n g i t c a u s e s . The p r e s e n t s e t o f experi m e n t s have shown t h a t t h e phase s h i f t i s not c o n s t a n t i n time d u r i n g the w r i t i n g p r o c e s s , and r e p e a t e d measurements as t h e hologram was w r i t t e n showed t h a t t h i s v a r i a t i o n i s a l i n e a r f u n c t i o n o f time f o r any p a r t i c u l a r e x periment. The causes o f t h i s v a r i a t i o n can be a t t r i b u t e d t o the n a t u r e o f the w r i t i n g p r o c e s s , t h e r m a l e x p a n s i o n and m e c h a n i c a l c r e e p o f the o p t i c a l components. Repeated experiments showed t h a t t h e v a l u e o f the s h i f t e x t r a p o l a t e d t o t=0 o f the w r i t i n g p r o c e s s was a c o n s t a n t f o r t h e c r y s t a l u s e d . Because the holograms were w r i t t e n under c o n d i t i o n s which made the b u l k p h o t o - v o l t a i c e f f e c t t h e dominant p r o c e s s , i t was p o s s i b l e t o c a l c u l a t e the t r a n s p o r t l e n g t h due t o t h i s e f f e c t . The v a l u e a r r i v e d a t was 13±3 nm. TABLE OF CONTENTS A b s t r a c t L i s t o f V a r i a b l e s L i s t o f F i g u r e s Acknowledgements 1. I n t r o d u c t i o n 2. O p t i c a l Holography 3. D i f f r a c t i o n o f C o u p l e d Waves 4 . P h o t o r e f r a c t i v e E f f e c t 5. P h o t o c u r r e n t and B u l k P h o t o v o l t a i c E f f e c t 6. Hologram W r i t i n g i n LiNbOg C r y s t a l s 7. Appara t u s 8. Measurements and A n a l y s i s 9. Summary R e f e r e n c e s Appendices A: LiNbO^ C r y s t a l Data B: PZT Phase S h i f t e r C: M i c h e l s o n I n t e r f e r o m e t e r D: A p p l i c a t i o n s o f Holography E: Programs i v L IST OF VARIABLES A a m p l i t u d e o f s u b j e c t beam w i t h r e f e r e n c e n o r m a l i z e d BPE " b u l k p h o t o v o l t a i c e f f e c t " d c r y s t a l t h i c k n e s s d. .,' p i e z o e l e c t r i c m o duli (d. ., c o n t r a c t e d form) 13k ^ 13 D,D^ d i e l e c t r i c d i s p l a c e m e n t v e c t o r , t e n s o r element E a p p l i e d e l e c t r i c f i e l d EO " e l e c t r o - o p t i c " E f i e l d due t o space charge i n c r y s t a l s s c E v i r t u a l f i e l d v E D d r i f t f i e l d f ^ , f 2 r e a l c o n s t a n t s (eq. 6.4, 6.6) FP " f r i n g e p a t t e r n " g(x) volume g e n e r a t i o n r a t e o f f r e e e l e c t r o n s g Q c o n s t a n t o f g e n e r a t i o n r a t e e q u a t i o n ( e q . 6.6) HG " h o l o g r a p h i c g r a t i n g " i , p h o t o c u r r e n t Pb I l i g h t beam, i n t e n s i t y I ,1 t r a n s m i t t e d r e f e r e n c e and s u b j e c t beam i n t e n s i t i e s ( e q . 3.8) R S J p h o t o c u r r e n t d e n s i t y (eq. 4.1) J Q average c o n d u c t i o n c u r r e n t d e n s i t y (eq. 6.11) J .. / J , , J see eq. 4.1 e1 e<> r k f r i n g e p a t t e r n g r a t i n g v e c t o r H f r i n g e s p a c i n g i n HG i+SL £ + ' £ * e l e c t r o n mean f r e e p a t h s ( e q . 5.2-5.4) AJL d i s p l a c e m e n t o f i i o n (eq. 5.3) l e n g t h o f c r y s t a l a l o n g c - a x i s ( e q . 6.10) e l e c t r o n t r a n s p o r t l e n g t h due t o BPE d i f f u s i o n t r a n s p o r t l e n g t h m o d u l a t i o n r a t o ( e q . 6.1) e f f e c t i v e m o d u l a t i o n r a t o ( e q . 5.11 a,b,c) i n d e x o f r e f r a c t i o n , n. = /e.. see e q . 3.1 a m p l i t u d e o f i n d e x m o d u l a t i o n i n g r a t i n g o r d i n a r y / e x t r a o r d i n a r y i n d e x o f r e f r a c t i o n i m p u r i t y c o n c e n t r a t i o n s c a t t e r i n g p r o b a b i l i t i e s ( e q . 5.11c) p r o b a b i l i t i e s ( e q . 5.11c) p o l a r i z a t i o n t e n s o r change i n spontaneous p o l a r i z a t i o n u n i t c h a r g e a m p l i t u d e o f r e f e r e n c e wave r e f e r e n c e beam ( e q . 3.3) = ( r / s ) 2 c o n t r a c t e d q u a d r a t i c EO m a t r i x element q u a d r a t i c EO m a t r i x element c o n t r a c t e d l i n e a r EO m a t r i x element a m p l i t u d e o f s i g n a l ( s u b j e c t ) wave s u b j e c t beam ( e q . 3.3) p h o t o - i o n i z a t i o n c r o s s s e c t i o n ( e q . 5.7) f r i n g e v i s i b i l i t y ( v o l t a g e ) u n i t v e c t o r ( f i g u r e 2 . 2 ) u n i t v e c t o r ( f i g u r e 2 . 2 ) v i t i l charge on t h e i i o n (eq. 5.3) l i n e a r EO c o e f f i c i e n t a c r y s t a l l i g h t a b s o r p t i o n c o n s t a n t e p e r m i t t i v i t y o f f i r s t space e_̂ _. t e n s o r r e l a t i v e , p e r m i t t i v i t y ( s t r a i n m a t r i x i n Appendix B) T) g r a t i n g d i f f r a c t i o n e f f i c i e n c y 0 a n g l e o f i n c i d e n t beam r e l a t i v e t o c r y s t a l normal i n s i d e c r y s t a l 0 as above o u t s i d e c r y s t a l o 0^ a n g l e o f i n c i d e n c e o f beams from normal t o m i r r o r s M^Mg K a n i s o t r o p y c o n s t a n t (eq. 4.1) g r a t i n g c o u p l i n g c o n s t a n t ( e q . 3.4) X o p t i c a l w a velength i n s i d e c r y s t a l X Q o p t i c a l wavelength i n f r e e s p a c e U j ^ degree o f coherence between two i n t e r f e r i n g beams £ quantum e f f i c i e n c y p e l e c t r o n d e n s i t y p , p c r y s t a l e l e c t r o n d e n s i t y i n l i g h t , dark p t r a p p e d charge d e n s i t y s c a.. s t r e s s t e n s o r ( a . , c o n t r a c t e d form) Jk D T f r e e e l e c t r o n l i f e t i m e 4> phase o f r e f e r e n c e beam w i t h r e s p e c t t o s u b j e c t beam, o r e q u i v a l e n t l y , phase mismatch between HG and FP <b phase s h i f t due t o BPE P <j>̂  phase s h i f t due t o f i n i t e d i f f u s i o n t r a n s p o r t l e n g t h w o p t i c a l f r e q u e n c y Q a n g l e between d i r e c t i o n o f p o l a r i z a t i o n o f two i n t e r f e r i n g beams. v i i L IST OF FIGURES Page 2.1 Simple hologram w r i t i n g and r e a d i n g g e o m e t r i e s 4 2.2 An e l e m e n t a r y hologram formed i n a medium o f t h i c k n e s s "d" 4 2.3 S p a t i a l f i l t e r 7 2.4 Hologram f o r m a t i o n u s i n g a m p l i t u d e d i v i s i o n 7 3.1 S u p e r p o s i t i o n o f f r i n g e p a t t e r n on hologram g r a t i n g 10 4.1 Change i n b i r e f r i n g e n c e i n d u c e d w i t h a s i n g l e l a s e r beam i n L i N b o ^ 16 4.2 Chen's p o s t u l a t e d space charge f i e l d 16 5.1 P h o t o c u r r e n t i n a LiNbO^ C r y s t a l 21 5.2 Asymmetric p h o t o d e l o c a l i z a t i o n model i n LiNbOg 24 5.3 (a) P h y s i c a l mechanism o f c o l l e c t i v e Franck-Condon 26 r e l a x a t i o n model (b) C o o r d i n a t e c o n f i g u r a t i o n diagram f o r Franck-Condon 26 r e l a x a t i o n 7.1 E x p e r i m e n t a l a p p a r a t u s 36 8.1 Beam Geometry d u r i n g hologram w r i t i n g 40 8.2 Sample o u t p u t from e x p e r i m e n t a l run 41 8.3 (a,b,c) N o r m a l i z e d d a t a from F i g u r e 8.2 42 8.4 V a r i a t i o n o f phase s h i f t w i t h exposure 45 8.5 Examples o f c o u p l i n g d a t a from Young e t a l . ( 1 9 7 9 ) 46 v i i i 8.6 P l o t s o f t h e o r e t i c a l c o u p l i n g from e l G u i b a l y (1979) 47 8.7 Time development o f g r a t i n g c u r v a t u r e 49 8.8 S i m u l a t e d c o u p l i n g development f o r time v a r y i n g phase s h i f t 51 A . l T a b l e o f c r y s t a l d a t a 61 A.2 The i n d i c a t r i x f o r a p o s i t i v e u n i a x i a l c r y s t a l 61 A. 3 A p p l i c a t i o n o f a f i e l d t o change the i n d e x o f a L i N b O 3 c r y s t a l 61 B. l S t r e s s a., on a c r y s t a l 65 B.2 V e r n i t r o n PZT p i e z o e l e c t r i c element 65 B. 3 Time r e s p o n s e o f PZT d i s c t o s t e p i n v o l t a g e 68 C. l M i c h e l s o n i n t e r f e r o m e t e r 71 D. l Comparative p e r f o r mance o f h o l o g r a p h i c memories (a) a c c e s s time v s . c o s t p e r s t o r e d b i t 79 (b) a c c e s s t i m e v s . s t o r a g e c a p a c i t y 80 D.2 Schematic o f a volume h o l o g r a p h i c s t o r a g e d e v i c e 81 ix ACKNOWLEDGEMENTS I would l i k e t o thank Dr. Lawrence Young f o r s u g g e s t i n g t h e r e s e a r c h t o p i c and f o r h i s guidance d u r i n g t h e c o u r s e o f t h e r e s e a r c h . Thanks a l s o t o A l MacKenzie f o r h e l p w i t h the d r a f t i n g and Kathy Brindamour and G a i l Hrehorka f o r p r e p a r i n g t h e m a n u s c r i p t . 1 1. INTRODUCTION I t i s p o s s i b l e (Chen e t a l . , 1968) t o s t o r e pure phase holograms i n c e r t a i n f e r r o e l e c t r i c c r y s t a l s . P h y s i c a l l y , the hologram i s composed o f a r e f r a c t i v e i n d e x d i s t r i b u t i o n w i t h t h e v a r i a t i o n i n t h e in d e x b e i n g t y p i c a l l y one p a r t i n 10 ~" and e f f e c t i n g t h e r e p r o d u c t i o n o f a h o l o g r a p h i c image t h r o u g h Bragg d i f f r a c t i o n o f t h e r e f e r e n c e beam. The s t o r a g e o f t h e o r i g i n a l l i g h t d i f f r a c t i o n p a t t e r n as a r e f r a c t i v e i ndex d i s t r i b u t i o n i s a r e s u l t o f t h e p h o t o r e f r a c t i v e e f f e c t . In b r i e f , t h i s e f f e c t i s brought about by t h e p h o t o - l i b e r a t i o n o f e l e c t r o n s i n t r a p s w i t h i n t h e c r y s t a l , which t h e n m i g r a t e under d r i f t , d i f f u s i o n and a p r o c e s s c a l l e d the b u l k p h o t o v o l t a i c e f f e c t . Through s u c c e s s i v e e x c i t a t i o n s and r e - e n t r a p m e n t s , the e l e c t r o n s approach an inhomo- geneous d i s t r i b u t i o n which i s d e t e r m i n e d by the l i g h t d i f f r a c t i o n p a t t e r n and c e r t a i n c h a r a c t e r i s t i c s o f the t r a n s p o r t mechanisms. The r e s u l t i n g e l e c t r i c f i e l d d i s t r i b u t i o n r e s u l t s i n a r e f r a c t i v e i n d e x d i s t r i b u t i o n due t o t h e e l e c t r o - o p t i c e f f e c t , and t h i s becomes the p h y s i c a l r e c o r d i n g o f the hologram. T h i s r e c o r d i n g may be s u b s e q u e n t l y removed by u n i f o r m i l l u m i n a t i o n o r h e a t i n g of the c r y s t a l , which removes the r e f r a c t i v e i n d e x m o d u l a t i o n t h r o u g h o p t i c a l o r t h e r m a l e x c i t a t i o n and u n i f o r m r e d i s t r i b u t i o n o f the e l e c t r o n s . S t a e b l e r and Amodei (1972) n o t e d t h a t i f the f r i n g e p a t t e r n formed by two i n t e r f e r i n g p l a n e waves i s l a t e r a l l y d i s p l a c e d from a d i f f r a c t i o n g r a t i n g upon which t h e f r i n g e s a r e i n c i d e n t , then t h e two beams w i l l become c o u p l e d . T h i s r e s u l t s i n a t r a d i n g of energy between them (Ch. 3 ) . There a r e s e v e r a l p r o c e s s e s which c o u l d r e s u l t i n the hologram g r a t i n g b e i n g s h i f t e d by some phase a n g l e w i t h r e s p e c t t o t h e f r i n g e p a t t e r n . Young e t a l . (1979) showed t h a t a phase s h i f t o f from 0 t o TT/2 r a d i a n s can a r i s e i f d r i f t i s an i m p o r t a n t means o f e l e c t r o n t r a n s p o r t . The t r a n s p o r t l e n g t h due t o the b u l k p h o t o v o l t a i c e f f e c t can a l s o r e s u l t i n a phase s h i f t , 2 depending on th e v a l u e of the t r a n s p o r t length, w i t h r e s p e c t t o t h e g r a t i n g l i n e s p a c i n g . (More c o r r e c t l y , a t h i c k phase hologram has a "plane s p a c i n g " , b u t t h e more common term i s used here.) The beam c o u p l i n g a r i s i n g from the phase s h i f t due t o t h e b u l k p h o t o - v o l t a i c e f f e c t can be s t u d i e d by e i t h e r a l l o w i n g f o r o t h e r s o u r c e s o f s h i f t i n subsequent c a l c u l a t i o n s o r d e v i s i n g an e x p e r i m e n t a l arrangement i n which t h e y a r e n e g l i g i b l e . There have been s e v e r a l r e p o r t e d attempts t o s t u d y t h e n a t u r e o f t h i s c o u p l i n g and draw c o n c l u s i o n s c o n c e r n i n g t h e p r o c e s s e s which g i v e r i s e t o i t , but t o date t h e r e has been c o n s i d e r a b l e d i f f i c u l t y i n a c h i e v i n g r e p r o - d u c i b l e r e s u l t s . The o b j e c t of t h e p r e s e n t work was t o f u r t h e r i n v e s t i g a t e t h e c o u p l i n g w i t h an aim t o d e t e r m i n i n g t h e s o u r c e s o f d i f f i c u l t y and whether t h e y c o u l d be a v o i d e d by changes i n t h e a p p a r a t u s , o r t h e i r e f f e c t s c i r c u m v e n t e d by a p p r o p r i a t e d a t a a n a l y s i s . From an e n g i n e e r i n g s t a n d p o i n t , the u l t i m a t e g o a l o f i n v e s t i g a t i o n s such as t h i s i n t o t h e p h y s i c a l p r o c e s s e s a t work i n f e r r o e l e c t r i c c r y s t a l s d u r i n g i l l u m i n a t i o n i s t o e n a b l e t h e d e s i g n and c o n s t r u c t i o n of e l e c t r o - o p t i c d e v i c e s u s i n g t h e s e m a t e r i a l s . The main a p p l i c a t i o n of t h e s e d e v i c e s w i l l be i n t h e g e n e r a l a r e a o f d a t a h a n d l i n g and w i l l make use o f t h e h i g h speed and r e l a t i v e immunity t o e l e c t r i c a l i n t e r f e r e n c e o f f e r e d by o p t i c a l r a t h e r t h a n e l e c t r i c a l t r a n s m i s s i o n . S p e c i f i c a p p l i c a t i o n s i n c l u d e two and t h r e e dimen- s i o n a l d i s p l a y s , p a t t e r n and c h a r a c t e r r e c o g n i t i o n image p r o c e s s i n g , h o l o - g r a p h i c i n t e r f e r o m e t r y and h o l o g r a p h i c d a t a s t o r a g e . These are d i s c u s s e d i n Appendix D. 3 2 . OPTICAL HOLOGRAPHY Du r i n g the t h r e e decades s i n c e i t s i n c e p t i o n , w a v efront r e c o n s t r u c - t i o n , or hologr a p h y , has undergone c o n s i d e r a b l e development and seen an i n - c r e a s i n g l y b r o a d range o f a p p l i c a t i o n . I t i s c h a r a c t e r i z e d , i n i t s s i m p l e s t form, by the f o r m a t i o n o f a r e c o r d e d image which i s not t h e o b j e c t ' s image", but r a t h e r t h e i n t e r f e r e n c e p a t t e r n between t h e o b j e c t ' s i n c o m i n g wavefront and t h a t o f a phase r e l a t e d second beam c a l l e d the r e f e r e n c e beam, which i s o f t e n (but not n e c e s s a r i l y ) a p l a n e wave. A f t e r t h e image has been formed on t h e r e c o r d i n g medium, s h i n i n g o n l y the r e f e r e n c e on i t r e s u l t s i n t h e r e c o n s t r u c - t i o n o f t h e o r i g i n a l o b j e c t ' s image, which w i l l e x h i b i t t h e depth and p a r a l l a x p r o p e r t i e s n o r m a l l y a s s o c i a t e d w i t h t h a t o b j e c t (from a l i m i t e d a n g l e o f v i e w ) . T h i s i s p o s s i b l e because, u n l i k e a common photograph which p r e s e r v e s o n l y am- p l i t u d e i n f o r m a t i o n a l o n g the s u r f a c e o c c u p i e d by the f i l m , the hologram i s an i n t e r f e r e n c e p a t t e r n between two phase r e l a t e d beams, from which b o t h t h e o r i g i n a l a m p litude and phase i n f o r m a t i o n can be r e c o r d e d ( F i g . 2.1). One can see how t h i s i s a c c o m p l i s h e d by f i r s t c o n s i d e r i n g an elemen- t a r y hologram made by r e c o r d i n g the i n t e r f e r e n c e p a t t e r n formed by t h e i n t e r - s e c t i o n o f two p l a n e , n o n - p a r a l l e l , monochromtic and p h a s e - r e l a t e d beams o f t h e same amplitude ( F i g . 2 . 2 ) . As t h e diagram shows, t h i s r e s u l t s i n a p l a n e g r a t i n g w i t h the g r a t i n g v e c t o r p e r p e n d i c u l a r t o t h e beam b i s e c t o r and i n th e p l a n e of the two beams. The g e n e r a l and more complex case o f the i n t e r - f e r e n c e o f two a r b i t r a r y w a v e f r o n t s can be l o o k e d a t as j u s t t h e sum o f t h e i n t e r f e r e n c e p a t t e r n s of a l l the p l a n e waves c o n s t i t u t i n g t h e i r F o u r i e r com- p o n e n t s . The f o r m a t i o n o f a hologram and the d e s c r i p t i o n of s e v e r a l o f i t s c h a r a c t e r i s t i c s depends upon t h e degree o f coherence o f t h e i l l u m i n a t i o n . I t i s u s e f u l - t o d i v i d e t h e t o p i c o f coherence i n t o two p a r t s : s p a t i a l ( l a t e r a l ) 4 FIGURE 2.2 An e l e m e n t a r y hologram formed i n a medium o f t h i c k n e s s 'd' (as v i e w e d from t h e top i n t h e s e e x p e r i m e n t s ) . + x FIGURE 2.1 S i m p l e hologram w r i t i n g and r e a d i n g g e o m e t r i e s , " a " shows t h e p r e s e n c e o f b o t h t h e r e f e r e n c e and s u b j e c t beams, w h i c h f o r m an i n t e r f e r e n c e p a t t e r n on t h e r e c o r d i n g medium, "b" shows t h e r e a d i n g geometry i n wh i c h t h e o b j e c t beam i s a b s e n t b u t i s r e c o n s t r u c t e d v i a d i f f r a c t i o n o f th e r e f e r e n c e beam. 5 coh e r e n c e , and tempor a l ( l o n g i t u d i n a l ) c o h e r e n c e . Most commercial gas l a s e r s o s c i l l a t e ( or can be so a d j u s t e d ) o n l y i n t h e lo w e s t t r a n s v e r s e mode ( T E M Q Q ) . As a consequence, t h e y a r e s p a t i a l l y c o h e r e n t , and so t h i s i s not a pr o b l e m when w r i t i n g holograms w i t h them. T h i s means t h a t t h e l i g h t a t any p o i n t on a p l a n e normal t o t h e d i r e c t i o n o f p r o p a g a t i o n i s . i n phase w i t h any o t h e r p o i n t on t h e p l a n e , and thu s t h e e n t i r e w i d t h o f t h e beam i s u s e f u l f o r t h e purpose o f w r i t i n g a holgram. The s p e c t r a l p u r i t y o f a l a s e r ' s r a d i a t i o n i s c l o s e l y r e l a t e d t o i t s degree o f i t s t e m p o r a l c o h e r e n c e . Though a l a s e r o s c i l l a t i n g i n o n l y one l o n g i t u d i n a l mode p o s s e s s e s i d e a l t e m p o r a l coherence, s e v e r a l modes a r e g e n e r a l l y p r e s e n t . The ( l o n g i t u d i n a l ) coherence l e n g t h , AL , de t e r m i n e s t h e H d i s t a n c e a l o n g t h e beam a x i s between p o i n t s which a r e s t i l l c o h e r e n t . I f t h e phase p a t h l e n g t h s o f two s p l i t beams d i f f e r by more t h a n t h e coherence l e n g t h t h e y w i l l n ot e x h i b i t i n t e r f e r e n c e f r i n g e s when brought t o g e t h e r , and thu s a hologram w i l l not be formed. To a t t a i n h i g h d i f f r a c t i o n e f f i c i e n c i e s , i t i s n e c e s s a r y t h a t t h e o p t i c a l system be a r r a n g e d t o d i s p l a y o p t i m a l f r i n g e v i s i b i l i t y , V, o r s t a n d i n g wave r a t i o , which i s o f the form I - I . _ max mm I + I . max mm where T m a x and I m ^ n a r e the maximum and minimum i n t e n s i t i e s of t h e i n t e r f e r e n c e f r i n g e s . V i s , i n g e n e r a l , a f u n c t i o n o f t h e degree o f coherence | P l 2 ( T ) | between t h e two i n t e r f e r i n g beams, the a n g l e ft between t h e d i r e c t i o n s o f p o l a r i z a t i o n o f t h e two beams, and t h e r a t i o R̂ . o f t h e i n t e n s i t i e s o f t h e two beams ( P v < 1) - as f o l l o w s 6 V 2 | u 1 2 ( T ) | ^ c o s f i R I + 1 The geometry o f t h e p r e s e n t s e t o f experiments i s such t h a t °>=0o and | u( T) | = 1 and t h u s f o r our purposes where R = ( r / s ) V = R^+l r , s = a m p l i t u d e s o f r e f e r e n c e and s i g n a l waves To o p t i m i z e V i t i s seen t h a t we must make R^ u n i t y a t a l l p o i n t s i n t h e d i f - f r a c t i o n p a t t e r n , which, i n the e x p e r i m e n t a l s e t u p used, means t h a t we must have an a x i a l l y symmetric beam, p r e f e r a b l y o f G a u s s i a n c r o s s - s e c t i o n . T h i s i s a c h i e v e d by s p a t i a l f r e q u e n c y f i l t e r i n g . * The beam i s brought t o a p o i n t f o c u s by a l e n s and p a s s e d t h r o u g h a p i n h o l e a t t h a t p o i n t , thus a l l o w i n g t h r o u g h o n l y t h e v e r y l o w e s t s p a t i a l f r e q u e n c i e s c h a r a c t e r i s t i c o f the s l o w l y v a r y i n g G a u s s i a n d i s t r i b u t i o n , and b l o c k i n g t h e h i g h e r s p a t i a l f r e q u e n c i e s c h a r a c t e r i s - t i c o f n o i s e . A f o l l o w i n g l e n s i s used t o c o l l i m a t e the s p h e r i c a l wave emerg- i n g from t h e h o l e ( F i g . 2.3). I t can be shown t h a t a p i n h o l e d iameter o f t h e o r d e r o f 10 ym w i l l f i l t e r o ut n o i s e from such s o u r c e s as m u l t i p l e r e f l e c t i o n s f rom l e n s e s and d i f f r a c t i o n r i n g s from d u s t s p o t s . A beam s p l i t t e r i s r e q u i r e d t o d i v i d e the l a s e r beam i n t o a s u b j e c t and r e f e r e n c e wave. The two methods a v a i l a b l e a r e am p l i t u d e d i v i s i o n and wave- f r o n t d i v i s i o n , as shown i n F i g . 2.4. The former i s g e n e r a l l y p r e f e r r e d be- cause i t r e q u i r e s l e s s beam ex p a n s i o n and p r o v i d e s more u n i f o r m i l l u m i n a t i o n . A f u r t h e r c h o i c e t o be made i s whether t o s p l i t the beam b e f o r e o r a f t e r f i l - t e r i n g . The advantages of t h e former a r e t h a t i t (1) r e q u i r e s a s m a l l e r s p l i t - t e r a p e r t u r e and (2) the n o i s e from the s p l i t t e r i s f i l t e r e d . The second op- t i o n , however, r e q u i r e s o n l y one s e t o f beam expanding o p t i c a l e l e m e n t s . It. i s e s s e n t i a l t o the f o r m a t i o n o f a sharp h o l o g r a p h i c image t h a t t h e i n t e r f e r e n c e f r i n g e p a t t e r n remain f a i r l y s t a t i o n a r y w i t h r e s p e c t t o the 7 l a s e r FIGURE 2.3 E f f e c t o f a s p a t i a l f i l t e r on the l i g h t i n t e n s i t y p r o f i l e . FIGURE 2.4 Hologram geometry u s i n g a m p l i t u d e d i v i s i o n (as opposed t o w a v e f r o n t d i v i s i o n shown i n F i g . 2 . 1 a ) . 8 r e c o r d i n g medium thr o u g h o u t the d u r a t i o n of t h e p r o c e s s . I n g e n e r a l , one must m i n i m i z e the t o t a l time t h a t the magnitude of bench v i b r a t i o n s exceeds t h e h i g h - e s t s p a t i a l f r e q u e n c y i n the p a t t e r n b e i n g r e c o r d e d , which i s u s u a l l y t a k e n t o be t h e wavelength o f t h e l i g h t b e i n g u s e d . In t h e p r e s e n t work, r e l a t i n g t h e degree of beam c o u p l i n g t o the r e l a t i v e p o s i t i o n s of the f r i n g e p a t t e r n t o t h e hologram g r a t i n g r e q u i r e d even g r e a t e r c a r e t o p r e v e n t random v a r i a t i o n s i n t h e i r p o s i t i o n . M a s s i v e t a b l e s , o f t e n made of g r a n i t e , c o n c r e t e o r s t e e l and r e s t i n g on sand, pneumatic s u p p o r t s , e t c . a r e o f t e n u s ed t o f i l t e r o ut b u i l d i n g v i b r a t i o n s . A i r b o r n e t h e r m a l and a c o u s t i c d i s t u r b a n c e s , e s p e c i a l l y i n t h e beam p a t h s between the beam s p l i t t e r and r e c o r d i n g medium must be m i n i m i z e d by e l i m i n a t i n g t h e s o u r c e o f t h e d i s t u r b a n c e o r , o f t e n more p r a c t i c a l l y , p r o t e c t i n g t h e s e t u p from i t . In t h e c u r r e n t s e t o f e x p e r i m e n t s , a M i c h e l s o n i n t e r f e r o m e t e r was i n c l u d e d as p a r t of the system i n o r d e r t o p r o v i d e c o n t i n u o u s m o n i t o r i n g o f t h i s o c c a s i o n a l l y troublesome p r o b l e m . 9 3. DIFFRACTION OF COUPLED WAVES T h i s c h a p t e r d e a l s w i t h t h e t h e o r y o f t h e i n t e r a c t i o n o f one o r two i n c i d e n t beams of l i g h t w i t h a s i n u s o i d a l t h i c k phase g r a t i n g ( S t a e b l e r and Amodei, .1972). Though the development g i v e n h e r e y i e l d s u s e f u l r e s u l t s , i t w i l l be seen l a t e r t h a t i t s a p p l i c a t i o n i s not g e n e r a l l y a p p l i c a b l e t o holograms w r i t t e n i n LiNbOg, but r a t h e r i s r e s t r i c t e d t o c e r t a i n c o n d i t i o n s . C o n s i d e r a s i n u s o i d a l g r a t i n g (an el e m e n t a r y t h i c k phase t r a n s m i s s i o n hologram) i n a c r y s t a l o f t h i c k n e s s d ( F i g u r e 3 . 1): An = n x c o s [(2ir/Ji)x] . (3.1) Two c o h e r e n t beams o f l i g h t , R and S, a r e s y m m e t r i c a l l y i n c i d e n t upon t h e s u r f a c e so t h a t t h e y make an a n g l e 0 r e l a t i v e t o the Z - a x i s i n s i d e t h e c r y s t a l . We assume t h a t p e r f e c t Bragg c o n d i t i o n s p r e v a i l so t h a t t h e r e l a t i o n s h i p between the o p t i c a l wavelength X i n the c r y s t a l and the phase g r a t i n g wavelength Si i s g i v e n by X = 2 H s i n 9 (3.2) We the n w r i t e R = R(z) exp {-i[(2Tr cos 0 / X ) z + ( T T / £ ) X ] > S = S(z) exp { - i [ ( 2 i r cos G / X ) z - ( T T / A ) X]} (3.3) A p p l y i n g c o u p l e d wave t h e o r y ( K o g e l n i k , 1969) we f i n d t h a t t h e v a r i a t i o n o f t h e beams w i t h i n t h e volume o f t h e g a t i n g i s g i v e n by t h e e x p r e s s i o n s dR(z) . , , dS(z) . , , , — L = ~ i < S(z) KT-L = - i K R(z) (3.4) dz g dz g where K = T f n j / X cos 0 i s the g r a t i n g c o u p l i n g c o n s t a n t (not t o be c o n f u s e d w i t h the a n i s o t r p p y c o n s t a n t K t o be i n t r o d u c e d l a t e r ) . The g e n e r a l s o l u t i o n t o t h e s e c o u p l e d e q u a t i o n s i s 10 f r i n g e p a t t e r n * - X c - a x i s h o l o g r a m g r a t i n g D o I 3 o + FIGURE 3.1 F r i n g e p a t t e r n c a u s e d by i n t e r f e r e n c e o f R(0)=1 and S(0)=e ^ compared t o c r y s t a l t h i c k n e s s 'd' upon which a hologram g r a t i n g o f s p a c i n g '1* has been w r i t t e n . The f r i n g e p a t t e r n i s d i s p l a c e d by § a l o n g t h e x - a x i s from t h e hologram grating.,--which i s d e s c r i b e d by e q u a t i o n 3.1. 8 i s t h e a n g l e t o the c r y s t a l normal made by the i n c i d e n t beams i n s i d e t h e c r y s t a l . 11 I K Z - I K Z (3.5) R(z) = ae g + be g I K Z - I K Z S ( z ) = -ae g + be g The c o e f f i c i e n t s a and b a r e dete r m i n e d from t h e boundary c o n d i t i o n s a t z = 0, i . e . t h e phase and a m p l i t u d e o f the beam a t t h e f i r s t s u r f a c e o f t h e c r y s t a l . In t h e g e n e r a l case we n o r m a l i z e R(0) = 1 and then s e t S(0) = Ae T h i s g i v e s R(z) = cos (K z) - i A e 1 ^ s i n ( K z) g g S(z) = - i s i n ( K z) + Ae X ^ cos ( K Z) g g (3.6) S q u a r i n g t h e s e we get t h e i n t e n s i t i e s , I = c o s 2 ( K z) + A 2 s i n 2 ( K z) - A s i n (2K Z) s i n <h R g g g (3.7) I = s i n 2 ( K z) + A 2 c o s 2 ( K z) + A s i n (2K z) s i n & S g g g A l t h o u g h t h e s e e q u a t i o n s r e f e r t o t h e case where t h e g r a t i n g i s f i x e d and one beam's phase d i f f e r s from the o t h e r ' s by (j>, we can a l s o d e s c r i b e t h e e q u i v a l e n t r e l a t i o n s h i p o f f i x e d beams [R(0) = 1, S(0) = A] and a movable g r a t i n g g i v e n by An = n ̂  cos [ (2 it/ I) x+ <}>] The case o f beam r e a d o u t , r e p r e s e n t e d by S(0) = 0 and R(0) = 1, has been w e l l s t u d i e d and t h e o r y a g r e e s w i t h e x p e r i m e n t . The more c o m p l i c a t e d c a s e o f beam c o u p l i n g d u r i n g w r i t i n g , which a p p l i e s i n t h i s work, uses t h e boundary c o n d i - t i o n s R(0) = 1 and S(0) = Ae 1^ >. I f the i n c i d e n t beams a r e b a l a n c e d (A = 1), t h e e q u a t i o n s (3,7) reduce t o I = 1 - s i n (2K Z) s i n <f> R g I = 1 + s i n (2K Z ) s i n <b s g (3.8) I t i s i m m e d i a t e l y o b v i o u s t h a t f o r t h e case o f cj) = 0 ( i . e . when t h e g r a t i n g 12 l i n e s up w i t h the f r i n g e p a t t e r n ) , t h e r e s h o u l d be no energy t r a n s f e r . The c ase where the g r a t i n g i s d i s p l a c e d a l o n g the x - a x i s from the f r i n g e p a t t e r n (<J> 0) r e s u l t s i n energy t r a n s f e r , the amount o f which depends upon the amount o f g r a t i n g d i s p l a c e m e n t . Q u a l i t a t i v e l y , a s m a l l s h i f t o f t h e g r a t i n g towards the S-beam s i d e enhances the S-beam, and v i c e - v e r s a . A l s o , the s t r e n g t h of the g r a t i n g a t a p a r t i c u l a r time i n f l u e n c e s t h e amount o f c o u p l i n g , as i s shown i n t h e s i n (2<gZ) f a c t o r i n the l a s t term. T h i s i s r e l a t e d t o t h e d i f f r a c t i o n e f f i c i e n c y o f t h e g r a t i n g by ( S t a e b l e r and Amodei, 1972) n = s i n 2 ( K Z) ( a t z=d) (3.9) g We d e f i n e n f o r a l o s s l e s s c r y s t a l as I (d) I (d) _ s _ s 1 1 " I R ( ° ) " I s ( d ) + I R ( d ) where t h e r i g h t hand s i d e i s u s e d i n t h e s e e x p e r i m e n t s . The f o r e g o i n g t h e o r y a p p l i e s t o any l o s s l e s s p u r e l y s i n u s o i d a l t h i c k phase t r a n s m i s s i o n g r a t i n g . However, t h i s i s o n l y an a pproximate d e s c r i p t i o n o f t h e g r a t i n g a c t u a l l y formed by i l l u m i n a t i n g a LiNbOg:Fe c r y s t a l w i t h two i n t e r f e r i n g p l a n e waves, and the d e s c r i p t i o n becomes l e s s a c c u r a t e as t h e i l - l u m i n a t i o n c o n t i n u e s . The r e a s o n f o r t h i s l i e s i n the d e t a i l e d p h y s i c a l n a t u r e o f the w r i t i n g mechanism w i t h i n t h e c r y s t a l . T h e r e f o r e , b e f o r e c o n s i d e r i n g t h e s p e c i f i c case of hologram w r i t i n g i n a L i N b O 3 c r y s t a l , we w i l l f i r s t d e a l w i t h t h e b a s i c p r o c e s s e s i n v o l v e d , namely the p h o t o r e f r a c t i v e e f f e c t and the b u l k p h o t o v o l t a i c e f f e c t . 13 4. PHOTOREFRACTIVE EFFECT A number o f f e r r o e l e c t r i c c r y s t a l s have been f o u n d t o undergo minute changes i n t h e i r i n d i c e s o f r e f r a c t i o n upon exposure t o s u f f i c i e n t l y i n t e n s e l i g h t o f an a p p r o p r i a t e w a velength. T h i s phenomenon, c a l l e d t h e p h o t o r e f r a c - t i v e e f f e c t , was f i r s t c o n s i d e r e d t o be something o f a n u i s a n c e because i t l e d t o e v e n t u a l d e g r a d a t i o n o f performance i n e a r l y e l e c t r o - o p t i c (EO) d e v i c e s u s i n g t h e s e c r y s t a l s . However, because t h e y had such d e s i r a b l e ( i . e . h i g h ) EO c o e f f i c i e n t s , work was done t o f i n d o u t how t o e l i m i n a t e t h i s p r o b l e m i n t h e c r y s t a l s , which l e d t o a b e t t e r u n d e r s t a n d i n g o f th e e f f e c t . The p h o t o r e f r a c t i v e e f f e c t t a k e s p l a c e i n t h e f o l l o w i n g manner. L i g h t i n t h e a p p r o p r i a t e wavelength range (450-550nm i n t h e case o f LiNbOg) i n c i d e n t upon the c r y s t a l causes p h o t o e x c i t a t i o n of e l e c t r o n s i n deep t r a p s , which a r e a s s o c i a t e d w i t h d e f e c t s i t e s and i m p u r i t i e s . The e l e c t r o n s then m i g r a t e under the i n f l u e n c e o f a number o f d i f f e r e n t mechanisms f o r some p e r i o d o f time u n t i l t h e y a r e r e c a p t u r e d by t r a p s a t o t h e r l o c a t i o n s w i t h i n the c r y s - t a l . Depending upon t h e i n t e n s i t y p a t t e r n o f t h e i n c i d e n t l i g h t , t h e end r e - s u l t w i l l be an uneven r e d i s t r i b u t i o n o f t h e e l e c t r o n d e n s i t y , r e s u l t i n g i n a p a t t e r n o f e l e c t r i c f i e l d s w i t h i n the c r y s t a l . F i n a l l y , t h e l i n e a r EO e f f e c t ( i n which an a p p l i e d e l e c t r i c f i e l d may i n d u c e a s m a l l change i n the index of r e f r a c t i o n - see Appendix A) r e s u l t s i n t h i s f i e l d p a t t e r n b e i n g t r a n s l a t e d i n t o a r e f r a c t i v e i n d e x d i s t r i b u t i o n which i s r e l a t e d i n some unique f a s h i o n t o t h e i n t e n s i t y p a t t e r n o f i n c i d e n t l i g h t . The t r a p p e d c h a r g e / r e f r a c t i v e i n d e x p a t t e r n can be removed i n a t l e a s t t h r e e ways: i ) by a l l o w i n g the dark c o n d u c t i v i t y o f the c r y s t a l t o s l o w l y r e t u r n the d i s t r i b u t i o n t o i t s o r i g i n a l s t a t e ; 14 i i ) by s t r o n g l y i l l u m i n a t i n g the e n t i r e c r y s t a l w i t h a wavelength t o which i t i s p h o t o s e n s i t i v e ( t h i s can be f a s t , b u t measures must be t a k e n t o ensure t h a t p a r a s i t i c holograms do n o t f o r m ) ; i i i ) by h e a t i n g t h e c r y s t a l i n an oven (200°C f o r one hour was f o u n d t o be e f f e c t i v e ) . The l a s t method, f o l l o w e d by two hours of g r a d u a l c o o l i n g i n t h e oven ( w i t h t h e c - a x i s f a c e s s h o r t e d d u r i n g h e a t i n g and c o o l i n g t o c o n t r o l p y r o - e l e c t r i c e f f e c t s ) was adopted i n t h i s work. As mentioned above, p h o t o e x c i t a t i o n of e l e c t r o n s i n deep t r a p s a s s o - c i a t e d w i t h d e f e c t s i t e s and i m p u r i t i e s i s r e s p o n s i b l e f o r making f r e e e l e c - t r o n s a v a i l a b l e f o r m i g r a t i o n under the v a r i o u s p r o c e s s e s i n e f f e c t ( P e t e r s o n e t a l . , 1972). S t o i c h i o m e t r i c d e f e c t s ( g e o m e t r i c l a t t i c e i r r e g u l a r i t i e s ) have been s t u d i e d ( P h i l l i p s e t a l . , 1972) by e x p o s i n g n o m i n a l l y pure L i N b O j t o gamma i r r a d i a t i o n . The r e s u l t i n g i n c r e a s e i n l a t t i c e d e f e c t s was seen t o be accom- p a n i e d by i n c r e a s e d p h o t o c o n d u c t i v e s e n s i t i v i t y . The i n f l u e n c e o f i m p u r i t i e s upon the o p t i c a l p r o p e r t i e s o f p h o t o r e f r a c t i v e c r y s t a l s was f i r s t s t u d i e d t o f i n d ways of d e c r e a s i n g t h e e f f e c t s upon EO d e v i c e s d e v e l o p e d t o make s p e c i f i c use o f t h e p h o t o e l e c t r i c e f f e c t . A number of s t u d i e s ( P h i l l i p s e t a l . , 1972, P e t e r s o n e t a l . , 1971; Mickami e t a l . , 1973; G l a s s e t a l . , 1974) showed t h a t d o p i n g w i t h i r o n , manganese, copper and o t h e r i m p u r i t i e s i n c r e a s e t h e p h o t o - r e f r a c t i v e s e n s i t i v i t y , w i t h the most commonly used (and the b e s t f o u nd t o date) b e i n g i r o n , u s u a l l y i n c o n c e n t r a t i o n s o f a few hundredths o f a mole- p e r c e n t . The i r o n i s p r e s e n t as F e ^ + and F e ^ + w i t h the F e ^ + p o s s i b l y b e i n g l o c a t e d i n oxygen v a c a n c i e s ( D i s c h l e r and Rauber, 1975) and t h e F e ^ + b e i n g l o c a t e d by Mossbauer e f f e c t s t u d i e s i n t h e niobium s i t e s , though n e i t h e r of t h e s e f i n d i n g s i s c e r t a i n . As the i r o n i m p u r i t y ' s t r i v a l e n t s t a t e ( F e ^ + ) con- s t i t u t e s , an empty t r a p and the d i v a l e n t s t a t e ( F e ^ + ) an o c c u p i e d t r a p , one would ex p e c t t h a t the o x i d a t i o n / r e d u c t i o n s t a t e o f t h e i r o n i m p u r i t i e s would a f f e c t 15 t h e p h o t o r e f r a c t i v e s e n s i t i v i t y ( S t a e b l e r and Amodei, 1974 a , b ) . A n n e a l i n g LiNbOg i n a i r o r oxygen a t 600°C ( P e t e r s o n e t a l . , 1971) has been shown t o be one e f f e c t i v e method of o x i d i z i n g the F e 2 + s t a t e t o F e ^ + ( i . e . , i n c r e a s i n g t h e number of t r a p s ) , w i t h as much as 96% o f t h e i r o n g o i n g t o t h e t r i v a l e n t s t a t e ( C l a r k e t a l . , 1973). I t has been shown by C o r n i s h (1975) t h a t h e a t i n g o f t h e c r y s t a l t o 540°C w h i l e packed i n L i 2 C 0 g powder r e d u c e d i r o n c e n t r e s t o t h e F e + Z s t a t e and d e s t r o y e d s h a l l o w t r a p s , which d e c r e a s e d t h e r m a l decay o f a hologram w h i l e a i d i n g o p t i c a l w r i t i n g and e r a s u r e . Other methods i n c l u d e f i e l d a n n e a l i n g (removing l a t t i c e d e f e c t s t o reduce p h o t o r e f r a c t i v e s e n s i t i v i t y - Smith e t a l . , 1968) and a p r o c e s s i n v o l v i n g t h e h e a t i n g o f a c r y s t a l i n c o n t a c t w i t h a s i m i l a r but much l a r g e r c r y s t a l , which a l t e r s t h e c o m p o s i t i o n o r non- s t o i c h i o m e t r y by l i t h i u m o x i d e t r a n s p o r t between t h e two c r y s t a l s . S e v e r a l attempts t o c o n s t r u c t a p h y s i c a l model of the p h o t o r e f r a c t i v e p r o c e s s have been made d u r i n g t h e p a s t decade. Though t h e y a l l a g r e e w i t h t h e g e n e r a l d e s c r i p t i o n p u t f o r w a r d a t the b e g i n n i n g o f t h i s s e c t i o n ( i . e . p h o t o - e x c i t a t i o n from t r a p s , m i g r a t i o n , a n i s o t r o p i c r e - e n t r a p m e n t , r e s u l t i n g s p a c e charge f i e l d s c a u s i n g an index change v i a EO e f f e c t ) , t h e r e has been some p r o b - lem d e t e r m i n i n g the p r e c i s e n a t u r e of t h e m i g r a t i o n . I t has been v a r i o u s l y a t t r i b u t e d t o d i f f u s i o n , d r i f t under the i n f l u e n c e of some f i e l d , and a new " b u l k p h o t o - v o l t a i c e f f e c t " . F o l l o w i n g a r e summaries o f s e v e r a l p u b l i s h e d models: i ) The I n t e r n a l F i e l d Model. Chen (1969) a t t r i b u t e d the p h o t o r e f r a c - t i v e e f f e c t t o d r i f t i n t h 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 i n t h e c r y s t a l , the o r i g i n o f which was not c l e a r . H i s c o n c l u s i o n s were based upon a d j u s t a b l e compensator measurements o f changes i n b i - r e f r i n g e n c e i n d u c e d by a l a s e r beam i n c i d e n t on a s m a l l c e n t r a l s e c t i o n o f a c r y s t a l . As shown i n F i g u r e 4.1, i t was f o u n d t h a t the change i n b i r e f r i n g e n c e changed s i g n a l o n g t h e c - a x i s , but n o t I beam I diam eter FIGURE 4.1 Change i n b i r e f r i n g e n c e i n d u c e d w i t h a s i n g l e l a s e r beam i n LiNbOg as o b s e r v e d by Chen (19 6 9 ) . The s o l i d l i n e r e p r e s e n t s t h e change a l o n g t h e c - a x i s due t o a r a d i a l l y symmetric beam, and t h e dashed l i n e t h e change a l o n g t h e b - a x i s . f +c -ax is FIGURE 4.2 Space charge f i e l d p o s t u l a t e d by Chen t o acc o u n t f o r the o b s e r v e d change i n the b i r e f r i n g e n c e . 17 t h e b - a x i s . He assumed an e l e c t r o - o p t i c e f f e c t and p o s t u l a t e d (1) t h e p r e s e n c e of empty t r a p s a v a i l a b l e t o c a p t u r e e l e c t r o n s and f u l l t r a p s a v a i l a b l e t o donate them upon p h o t o - i o n i z a t i o n , and (2) t h e p r e s e n c e of an i n t e r n a l e l e c t r i c f i e l d , E i n » d i r e c t e d o p p o s i t e t o t h e d i r e c t i o n o f spontaneous p o l a r i z a t i o n . P h o t o e x c i t e d e l e c - t r o n s w i l l t h e r e f o r e d r i f t i n the d i r e c t i o n o f t h e c - a x i s b e i n g r e p e a t e d l y t r a p p e d and r e - e x c i t e d u n t i l t h e y a r e o u t o f t h e i l l u m i - n a t e d volume, where t h e y are f i n a l l y t r a p p e d . Thus a space charge d e v e l o p s ( F i g u r e 4.2) which r e s u l t s i n t h e c r e a t i o n o f an e l e c t r i c f i e l d and v a r i a t i o n i n the i n d e x of r e f r a c t i o n by t h e e l e c t r o - o p t i c e f f e c t . O b s e r v a t i o n o f s h o r t - c i r c u i t p h o t o c u r r e n t s was c o n s i d e r e d t o be f u r t h e r e v i d e n c e o f the e x i s t e n c e of E ^ n . A p y r o e l e c t r i c o r i g i n o f t h e f i e l d , a r i s i n g from non-uniform h e a t i n g o f t h e c r y s t a l d u r i n g i l l u m i n a t i o n , was d i s m i s s e d because the d i r e c t i o n o f the p h o t o c u r r e n t d i s a g r e e d w i t h the s i g n of dP/dT ( n e g a t i v e ) . i i ) The P o l a r i z a t i o n Model. J o h n s t o n (1970) put f o r w a r d a model i n which t h e p h o t o r e f r a c t i v e e f f e c t ( o r more s p e c i f i c a l l y , t h e as y e t u n e x p l a i n e d E^ n) was a t t r i b u t e d t o p h o t o i n d u c e d v a r i a t i o n s i n t h e m a c r o s c o p i c p o l a r i z a t i o n . In t h i s mechanism, i n c i d e n t l i g h t e x c i t e s e l e c t r o n s i n t o the c o n d u c t i o n band (where t h e y a r e f r e e t o move) c a u s i n g a d e c r e a s e i n the d e n s i t y o f f i l l e d t r a p s i n t h e i l l u m i n a t e d volume and thus a l o c a l change i n p o l a r i z a t i o n . With the l a s e r o f f , t h i s change i s f r o z e n i n and t h e r e s u l t a n t d i v e r - gence of the p o l a r i z a t i o n i s accompanied by a permanent f i e l d which g i v e s r i s e t o a change i n i n d e x v i a the EO e f f e c t . One o f s e v e r a l d i f f i c u l t i e s w i t h t h i s model i s t h a t i t does not e x p l a i n t h e s t e a d y 18 s t a t e , s h o r t c i r c u i t p h o t o c u r r e n t o b s e r v e d i n a c r y s t a l under f u l l i l l u m i n a t i o n . i i i ) P y r o e l e c t r i c F i e l d Model. A second attempt (Amodei and S t a e b l e r 1972 b) t o a t t r i b u t e t h e f i e l d t o a p o l a r i z a t i o n e f f e c t , t h i s time a r i s i n g d u r i n g the c o o l i n g o f t h e c r y s t a l from a h i g h temperature ( e . g . d u r i n g c r y s t a l f o r m a t i o n ) was not s u p p o r t e d by e x p e r i m e n t a l r e s u l t s ( C o r n i s h e t a l . , 1976) i n which no r e l a x a t i o n i n t h e ob- s e s r v e d p h o t o c u r r e n t was e f f e c t e d a f t e r h e a t i n g t h e c r y s t a l t o remove the remanent p o l a r i z a t i o n by t e m p o r a r i l y i n c r e a s i n g t h e c o n d u c t i v i t y and c o o l i n g w i t h and w i t h o u t a s h o r t c i r c u i t , nor by r e s u l t s ( G l a s s e t a l . , 1974) i n which l o n g p e r i o d s o f i l l u m i n a t i o n r e s u l t e d i n no d e t e c t a b l e d e c r e a s e i n t h e p h o t o c u r r e n t as would be e x p e c t e d by th e e f f e c t s o f p h o t o c o n d u c t i v i t y . i v ) B u l k P h o t o v o l t a i c E f f e c t . R e a soning t h a t any b u i l t - i n i n t e r n a l f i e l d would be e v e n t u a l l y r e l a x e d by p h o t o c o n d u c t i v i t y o v e r l o n g p e r i o d s of i l l u m i n a t i o n (which i s not o b s e r v e d ) , G l a s s e t a l . (1974, 1975 a) were l e d t o p o s t u l a t e t h e e x i s t e n c e o f a new t r a n s - p o r t mechanism, c a l l e d the " b u l k p h o t o v o l t a i c e f f e c t " . The e l e c - t r o n s which c o n t r i b u t e t o t h e p h o t o c u r r e n t (and hence t o t h e p h o t o - r e f r a c t i v e e f f e c t ) a r e t r a p p e d i n asymmetric p o t e n t i a l w e l l s (the F e 2 + i o n s ) . Note i n t h i s r e g a r d t h a t the N b - F e 2 + d i s t a n c e s i n t h e ±c-axis d i r e c t i o n s a r e d i f f e r e n t . Upon e x c i t a t i o n , t h e s e e l e c t r o n s have a g r e a t e r p r o b a b i l i t y o f moving a l o n g t h e + c - a x i s and t h e r e - f o r e g i v e r i s e t o a net e l e c t r o n c u r r e n t J e j « The d i s p l a c e m e n t of t h e i o n i z e d i m p u r i t y a l o n g t h e - c - a x i s due t o Franck-Condon r e l a x a t i o n g i v e s r i s e t o a f u r t h e r c u r r e n t , J&2" A t h i r d c u r - r e n t , J r , r e s u l t s from asymmetric r e t r a p p i n g o f the e x c i t e d e l e c - 19 t r o n by a n o t h e r i o n i z e d i m p u r i t y ( F e 3 + ) . The s t e a d y s t a t e p h o t o - c u r r e n t d e n s i t y , Jp^» can t h e r e f o r e be e x p r e s s e d as t h e sum J , = J , + J 0 ~ J = KCCI (4. ph e l e^ r where I i s t h e i n c i d e n t l i g h t i n t e n s i t y , a i s t h e a b s o r p t i o n c o n - s t a n t a t t h a t w a v e l e n g t h , and K a c o n s t a n t ( t h e " a n i s o t r o p y con- s t a n t " ) r e l a t e d t o t h e p h y s i c a l n a t u r e o f t h e t r a p . T h i s model w i l l be d i s c u s s e d i n g r e a t e r d e t a i l l a t e r . 20 5. PHOTOCURRENT AND BULK PHOTOVOLTAIC EFFECT It has been noted (Chen 1969) that l i g h t incident upon poled f e r r o - e l e c t r i c s i n g l e c r y s t a l s causes a small current to flow between electrodes attached to opposite c r y s t a l faces without the a p p l i c a t i o n of an external f i e l d . Though i n i t i a l l y thought to be caused by b u i l t - i n f i e l d s due to, for instance, e l e c t r i c moments a r i s i n g from cooling the c r y s t a l from a high tem- perature, subsequent experimentation has led to the conclusion that a new e f f e c t , the "bulk photovoltaic e f f e c t " i s responsible. i ) Photocurrents Figure 5-1 shows an example of a photocurrent caused by uniformly i l l u m i n a t i n g with an Argon-ion laser (514.5nm at 17.5 mW/cm2) a face of an Fe doped LiNbOg c r y s t a l (perpendicular to the c-axis) and measuring the current passing between aluminum electrodes placed upon the c-faces. The transient component i s the p y r o e l e c t r i c current caused by a change i n the spontaneous dipole moment as the incident l i g h t r a i s e s the temperature. The opposite e f f e c t can be seen as the c r y s t a l cools a f t e r the l i g h t i s turned o f f . The current component remaining a f t e r the p y r o e l e c t r i c current has become n e g l i g i b l e i s the photocurrent, which has the following properties: a) the photocurrent strength i s l i n e a r l y r e l a t e d to the incident beam i n t e n s i t y and also depends upon wavelength (Cornish, 1975); b) i t does not change over long periods of time under constant i l l u m i n a - t i o n (10 2 hours or more); c) i t does not depend upon the thermal h i s t o r y of the c r y s t a l ( i . e . the e f f e c t i s not changed by heating and cooling the c r y s t a l under ei t h e r open or short c i r c u i t conditions. FIGURE 5.1 Graph o f the p y r o e l e c t r i c c u r r e n t (PEC) and the p h o t o e l e c t r i c c u r r e n t (PC) which r e s u l t s when an expanded l a s e r beam i s d i r e c t e d a t a LiNbO c r y s t a l . The c u r r e n t i s measured a c r o s s t h e c - a x i s (see t e x t ) . 22 A si m p l e m a t h e m a t i c a l e x p r e s s i o n f o r the p h o t o c u r r e n t d e n s i t y ( G l a s s e t a l . , 1974 b) i s g i v e n by J . = KOtf (5.1) ph where I i s t h e i n c i d e n t beam i n t e n s i t y , a i s t h e a b s o r p t i o n , and K i s t h e p h o t o c u r r e n t c o n s t a n t , which i s dependent upon the n a t u r e o f t h e c r y s t a l ( i m p u r i t i e s , d e f e c t s , e t c . ) . i i ) The B u l k P h o t o v o l t a i c E f f e c t The o b s e r v a t i o n o f s t e a d y - s t a t e , s h o r t - c i r c u i t p h o t o v o l t a i c c u r r e n t s i n r e g u l a r c r y s t a l s c o n t a i n i n g no m a c r o s c o p i c e l e c t r i c f i e l d s , c o n c e n t r a t i o n g r a d i e n t s o r p n - l i k e j u n c t i o n s l e d t o a s e a r c h f o r some b u l k p r o p e r t y o f a c r y s t a l which c o u l d a c c o u n t f o r such an unexpected a n i s o t r o p i c e f f e c t . The s e v e r a l mechanisms p r o p o s e d , which a r e n o t m u t u a l l y e x c l u s i v e , a r e b r i e f l y o u t l i n e d h e r e . The asymmetric p h o t o - d e l o c a l i z a t i o n model ( G l a s s e t a l . , 1974) c e n t r e s on t h e f a c t t h a t a L i N b 0 3 : F e c r y s t a l has a unique p o l a r a x i s a l o n g which t h e Fe-Nb d i s t a n c e s a r e d i f f e r e n t i n t h e +c and - c d i r e c t i o n s (see Appen- d i x A ) . C o n s e q u e n t l y , t h e r e i s a l a r g e r o v e r l a p between t h e o r b i t a l s o f t h e F e 2 + i o n w i t h the Nb i o n i n t h e +c d i r e c t i o n t h a n those i n the - c d i r e c t i o n , which r e s u l t s i n a p r e f e r r e d d i r e c t i o n o f t r a n s f e r o f e l e c t r o n s f o l l o w i n g o p t i - c a l e x c i t a t i o n . T h i s g i v e s r i s e t o a c u r r e n t , J e l / g i v e n by J e l = ^ £ ( P + V P - 0 • . ( 5 ' 2 ) where £ .is t h e quantum e f f i c i e n c y , h u i s the quantum o f l i g h t energy, P+»P_ a r e p r o b a b i l i t i e s o f charge t r a n s f e r i n t h e +c and - c d i r e c t i o n s , 23 t h e e l e c t r o n mean f r e e p a t h s a l o n g t h e s e d i r e c t i o n s . T h i s i s f o l l o w e d by Franck-Condon r e l a x a t i o n o f t h e i o n s , w i t h t h e i r n e t d i s - p lacement a l o n g the p o l a r a x i s g i v i n g a f u r t h e r c u r r e n t , J e 2 / J „ = E,(-—)z.Mi. p r o d u c t summed o v e r a l l i ( 5 . 3 ) e^ Mia)-' 1 1 +*Vi where A£^ i s the d i s p l a c e m e n t o f t h e i i o n , z ^ i s the charge o f t h e i i o n . Upon s u c c e s s i v e s c a t t e r i n g , t h e motion o f t h e p h o t o - l i b e r a t e d e l e c t r o n s becomes random and th e y no l o n g e r c o n t r i b u t e t o a n e t c u r r e n t i n t h e absence o f an a p p l i e d f i e l d . However, i f t h e p r o b a b i l i t y o f a subsequent r e c o m b i n a t i o n a t an i m p u r i t y i s dependent upon the d i r e c t i o n from which the e l e c t r o n a r r i v e s , then we have a r e c o m b i n a t i o n c u r r e n t , as f o l l o w s : Jri - i C*; P; - * : P 1 ) ( 5 . 4 > and a n o t h e r Franck-Condon r e l a x a t i o n c u r r e n t J 9 = z\ A*. ( 5 . 5 ) r2 Miu)'' i i where t h e pri m e d q u a n t i t i e s V+l p^, £', p'and i n t h e r e c o m b i n a t i o n terms a r e analogous t o the e x c i t a t i o n q u a n t i t i e s . I f we sum a l l o f t h e s e c u r r e n t s we g e t J , = J + J = KOCI (5.6) P h £1,2 r i , 2 where < = r K P x - ^ p + r P! - r p 1 ] + [ z . - z ; ] hu L + + - - + + - - J hoj L l I J and v a r i e s w i t h the photon energy and environment o f t h e a b s o r b i n g c e n t r e . See F i g u r e 5.2. 24 O Nb 5* O Li elec t ron o o #0 OO a) exci tat ion c - a x i s P' OO P: O o o b) thermal recombination FIGURE 5.2 Asymmetric p h o t o - d e l o c a l i z a t i o n model i n a LiNbOg c r y s t a l . In (a) an e l e c t r o n i s p h o t o e x c i t e d from an F e 3 + i m p u r i t y and has p r o b a b i l i t i e s o f p + and p_ of.moving a l o n g t h e +c and - c a x i s d i r e c t i o n s r e s p e c t i v e l y . In (b) i s shown t h e Franck-Condon s h i f t and e l e c t r o n r e c o m b i n a t i o n p r o b a b i l i t i e s . 25 whereas the above model i n v o l v e s e l e c t r o n t r a n s p o r t between i m p u r i t y i o n s , t h e p h o t o r e f r a c t i v e e f f e c t has been o b s e r v e d (von der L i n d e e t a l . , 1974, 1975 a, 1975 b, 1976 and Ohmori e t a l . , 1977) i n h i g h p u r i t y c r y s t a l s as w e l l . The C o l l e c t i v e Franck-Condon R e l a x a t i o n Model a t t r i b u t e s t h i s t o Franck-Condon r e l a x a t i o n of e x c i t e d s t a t e s f o l l o w i n g o p t i c a l e x c i t a t i o n . The p h y s i c a l mechanism of the model i s shown i n F i g u r e 5.3a, where an a n i o n and c a - t i o n c o n s t i t u t e a u n i t c e l l , and arrows r e p r e s e n t the " d i r e c t i o n o f motion" t o the n e x t s t a t e . The ground s t a t e ( i ) i s t r a n s f e r r e d t o t h e Franck-Condon s t a t e ( i i ) by o p t i c a l e x c i t a t i o n . The c a t i o n then s h i f t s a d i s t a n c e Ax t o the r e l a x e d e x c i t e d s t a t e ( i i i ) a c c o r d i n g t o the c o o r d i n a t e c o n f i g u r a t i o n diagram ( F i g u r e 5.3b). Up t o t h i s p o i n t t h e r e has been both a d i r e c t i o n a l e l e c t r o n i c charge t r a n s f e r c u r r e n t , J e e , and an i o n i c r e l a x a t i o n c u r r e n t , J £ e « Next the c e l l goes t o t h e ground Franck-Condon s t a t e ( i v ) as t h e e l e c t r o n r e t r e a t s t o t h e a n i o n and t h e n t o the ground s t a t e (v) as the c a t i o n assumes i t s o r i g i n a l l o c a - t i o n i n t h e l a t t i c e . These motions c o n t r i b u t e e l e c t r o n i c and i o n i c r e l a x a t i o n c u r r e n t s , J e r and J ^ r / which c a n c e l out t h e p r e v i o u s two c u r r e n t s i f t h e e l e c t r o n and c a t i o n r e t u r n t o t h e i r o r i g i n a l l o c a t i o n s . However, i f t h e e l e c t r o n ends up combining w i t h an a n i o n i n a n e i g h b o u r i n g c e l l as shown i n ( v ) , a non-zero c o n t r i b u t i o n t o the n e t c u r r e n t r e s u l t s . F urthermore, i f c o h e r e n t r e l a x a t i o n of t h e l a t t i c e i n v o l v i n g a n o n i s o t r o p i c r e c o m b i n a t i o n p r o b a b i l i t y ( r e c o m b i n a t i o n t o the r i g h t i n ( v ) , f o r i n s t a n c e ) i s p r e s e n t , the summed e f f e c t w i l l be a b u l k s t e a d y - s t a t e p h o t o c u r r e n t . A t h i r d model i s the P h o t o f l u c t u a t i o n Model ( F r i d k i n , 1977), which a t t r i b u t e s the BPE_in-s.ome oxygen o c t a h e d r a f e r r o e l e c t r i c s ( i . e . , BaTiOg, LiNbOg) t o p h o t o i n d u c e d p o l a r i z a t i o n f l u c t u a t i o n s . L i g h t a b s o r p t i o n i n n-type f e r r o e l e c t r i c c r y s t a l s r e s u l t s i n the p r o d u c t i o n of f r e e and t r a p p e d e l e c t r o n s . A t r a p p e d e l e c t r o n can i n t e r a c t w i t h an o p t i c a l phonon t o c r e a t e a p h o t o i n d u c e d f l u c t u a t i o n o f p o l a r i z a t i o n l o c a l i z e d near the t r a p (Chanussot, 1974). W i t h i n 26 FIGURE 5.3a Physical mechanism of the c o l - l e c t i v e Franck-Condon r e l a x a t i o n 0 ; o model showing i n successive time anion Cation electron frames: ( i ) photo-excitation from the ground state to an excited Franck-Condon state; ( i i ) c a t i o n H " " ® * s h i f t to the "relaxed excited a state"; ( i i i ) e l e c t r o n t r a n s f e r leading to the ground Franck- Condon state; (iv) ca t i o n s h i f t to the ground s t a t e . o> O o O o (io O o - O o (.ii) O o O o M O -o O *-o c - a x i s (v> o o O o FIGURE 5.3b Coordinate configuration diagram fo r the Franck-Condon model show- ing the f i r s t two events described above. >- O CC LU z LU R E S . G.5. Q 27 t h e volume of the f l u c t u a t i o n t h e r e o c c u r s a change i n spontaneous p o l a r i z a - t i o n , AP, which g i v e s r i s e t o an e l e c t r i c f i e l d AE » AP/E. Symmetry r e q u i r e s t h a t a l l p h o t o i n d u c e d f l u c t u a t i o n s be i n the same d i r e c t i o n . A b u l k p h o t o - v o l t a i c e f f e c t r e s u l t s from the f r e e p h o t o - e x c i t e d e l e c t r o n s c o u p l i n g t o t h e s e f l u c t u a t i o n s and moving i n the d i r e c t i o n o f AE. T h i s model s u g g e s t s t h a t a d d i - t i o n a l P.ayleigh - s c a t t e r i n g of l i g h t s h o u l d be p r e s e n t , and t h i s has been ob- s e r v e d i n BaTiOg, though not i n LiNbOg (Chanussot e t a l . , 1977). The P o l a r i z e d I m p u r i t i e s Model (von B a l t z , 1977) assumes t h a t t h e randomly d i s t r i b u t e d ground s t a t e i m p u r i t i e s which p r o v i d e p h o t o c o n d u c t i o n e l e c t r o n s a r e p o l a r i z e d due t o an asymmetric s h o r t - r a n g e p o t e n t i a l . The p h o t o - c u r r e n t t h e n a r i s e s from an asymmetry i n the photo c r o s s - s e c t i o n f o r i o n i z a t i o n from t h e ground s t a t e t o a d e l o c a l i z e d f i n a l s t a t e , s(fi) £ s(-ft), where s(°») i s a p h o t o - i o n i z a t i o n c r o s s - s e c t i o n r e s u l t i n g i n t h e e j e c t i o n o f an e l e c t r o n i n a d i r e c t i o n ft. The p h o t o c u r r e n t d e n s i t y i s g i v e n by j = KOI = SSSSE ( 5 . 7 ) ph hoi where N i s th e i m p u r i t y c o n c e n t r a t i o n , Lp i s t h e e l e c t r o n t r a n s p o r t l e n g t h due t o the p h o t o v o l t a i c e f f e c t , and a = N / s(Q) d2Q, (5.8) T h i s model p r o v i d e s an e x p l a n a t i o n f o r t h e d i f f e r e n t s p e c t r a l p r o p e r t i e s o f t h e p h o t o v o l t a i c c u r r e n t f o r l i g h t p o l a r i z e d p a r a l l e l and p e r p e n d i c u l a r t o the d i r e c t i o n o f spontaneous p o l a r i z a t i o n , and f o r o b s e r v e d changes o f s i g n o f t h e c u r r e n t , as i n BaTiOg (Koch e t a l . , 1976). A l s o , i t does not p r e d i c t t h e g e n e r a t i o n of a m a c r o s c o p i c dark c u r r e n t due t o t h e r m a l e x c i t a t i o n which, though not o b s e s r v e d , i s p r e d i c t e d by t h e asymmetric p h o t o d e l o c a l i z a t i o n model. E a r l y s t u d i e s o f the b u l k p h o t o v o l t a i c e f f e c t l e d t o t h e f o l l o w i n g 28 e x p r e s s i o n r e l a t i n g p h o t o c u r r e n t d e n s i t y , J p ^ , t o i n c i d e n t l i g h t i n t e n - s i t y , I : J . (x) = Kctl(x) (5.9) ph However, more r e c e n t work (Young e t a l . , 1979) has p o i n t e d o u t t h a t t h i s s i m p l e e x p r e s s i o n p r e d i c t s n e i t h e r beam c o u p l i n g d u r i n g h ologram w r i t i n g nor a depen- dence o f the r e c o r d i n g m a t e r i a l r e s p o n s e upon the s p a t i a l f r e q u e n c y o f t h e i n c i d e n t l i g h t . Both of t h e s e a r e o b s e r v e d c h a r a c t e r i s t i c s of holograms w r i t - t e n i n f e r r o e l e c t r i c c r y s t a l s . As w i l l be d i s c u s s e d l a t e r , t h e beam c o u p l i n g d u r i n g w r i t i n g i s a r e s u l t o f a phase s h i f t between the h o l o g r a p h i c g r a t i n g b e i n g w r i t t e n and t h e d i f f r a c t i o n p a t t e r n w r i t i n g i t . A l s o , t h e "ac" p a r t o f t h e c u r r e n t i s s m a l l e r f o r f r i n g e s w i t h h i g h e r s p a t i a l f r e q u e n c i e s . The s e a r c h f o r a more r e a l i s t i c e x p r e s s i o n f o r t h e p h o t o v o l t a i c c u r - r e n t s t a r t s w i t h t h e c h o i c e of a s e t of assumptions c o n c e r n i n g t h e m i c r o s c o p i c mechanisms i n v o l v e d . The f i r s t a ssumption i s t h a t p h o t o e l e c t r o n s a r e p r e f e r e n - t i a l l y e j e c t e d a l o n g t h e d i r e c t i o n of t h e + c - a x i s . I t t h e n f o l l o w s t h a t we can d e f i n e a q u a n t i t y , L^, w h i c h ' i s t h e mean d i s t a n c e t r a v e l l e d by t h e e l e c t r o n b e f o r e b e i n g r e t r a p p e d o r h a v i n g i t s motion randomized. The second assumption d e a l s w i t h t h e manner of t r a p p i n g . Young e t a l . d e a l t w i t h t h r e e p o s s i b i l i - t i e s : a) c o n t i n u o u s s c a t t e r i n g - c l o s e l y spaced s c a t t e r i n g c e n t r e s a l o n g t h e c - a x i s g r a d u a l l y randomize t h e d i r e c t i o n o f m o t i o n o f t h e p h o t o - l i b e r a t e d e l e c t r o n s , b) f i x e d t r a n s p o r t l e n g t h - t h e p h o t o - l i b e r a t e d e l e c t r o n s t r a v e l a f i x e d d i s t a n c e , L , a l o n g t h e + c - a x i s and ar e t h e n t r a p p e d or have t h e i r m o t i o n r a n d o m i z e d a t t h a t p o i n t , 29 c) d i s c r e t e s c a t t e r i n g - i d e n t i c a l s c a t t e r i n g c e n t r e s spaced L p a p a r t a l o n g the c - a x i s and each h a v i n g a p r o b a b i l i t y p ' < l of s c a t t e r i n g t h e e l e c t r o n and p r o b a b i l i t y p = 1-p' o f not a f f e c t i n g i t s m o t i o n . Once the second assumption i s d e c i d e d upon, t h e c o n s t r u c t i o n o f a m a t h e m a t i c a l model f o r the p h o t o c u r r e n t i n v o l v e s f i n d i n g an e x p r e s s i o n f o r t h e p h o t o c u r r e n t d e n s i t y l i g h t i m pulse response and t h e n f o r m i n g the c o n v o l u t i o n o f t h i s w i t h t h e a p p r o p r i a t e i n c i d e n t l i g h t p a t t e r n , i n t h i s case a s i n u s o i d a l l y v a r y i n g g r a t i n g i n the + c - a x i s d i r e c t i o n : I ( x ) = I (1 + cos Kx) (5.10) o The r e s u l t s o f t h e s e c a l c u l a t i o n s g i v e (Young e t a l . 1979), f o r t h e t h r e e c a s e s : a) c o n t i n u o u s s c a t t e r i n g J (x) = -KCCI [ l + m" cos (Kx-<b ) 1 (5.11a) p o L P where £>L_ K = hoi -1/2 »' = rat1 + ( *%)* 4> = t a n - 1 (KL ) P • P b) f i x e d t r a n s p o r t l e n g t h J (x) = -KOI [ l + m" cos (Kx-<j>) ] * (5.11b) p o L x. where £aL K = • E hw KL - . r P i . . . s l n ^ m' = m s i n e — — i s i n c ( x ) = — — 1 2 i r J TIX 30 KL c) d i s c r e t e s c a t t e r i n g c e n t r e s J (x) = - K a l [ l = m' cos (Kx-d> ) ] (5.11c) p o P J where q ? L K - p' hco 1/2 2TTJ «•* ' e " e p' KL m* = mp s i n c [ — ~ \ ( l + p 2 - 2p cos KL ) KL p s i n KL <, = 2 + t a n - l [- P _ ] y p 2 L l - p cos KL 1 P Note t h a t i n a l l t h r e e c a s e s , l ( x ) i n eq. 5 . 9 has been r e p l a c e d by I Q p l u s a s p a t i a l l y v a r y i n g (ac) component which i s phase s h i f t e d by an amount In a l l t h r e e c ases the a n i s o t r o p y c o n s t a n t , K, which g i v e s t h e s t r e n g t h o f t h e b u l k p h o t o v o l t a i c e f f e c t , v a r i e s l i n e a r l y w i t h Lp. A l s o , t h e phase s h i f t <(>p i s such t h a t i t goes t o z e r o as Lp goes t o z e r o . The f u n c t i o n a l depen- dence of m' upon the s p a t i a l f r e q u e n c y of the f r i n g e p a t t e r n , K, shows t h a t h i g h s p a t i a l f r e q u e n c i e s r e s u l t i n a s m a l l e r "ac" c u r r e n t and t h u s l e s s e f f i - c i e n t hologram w r i t i n g . I t s h o u l d be p o i n t e d out t h a t i t i s q u i t e p o s s i b l e t h a t some we i g h t e d c o m b i n a t i o n o f a l l t h r e e of t h e s e c a s e s (and p o s s i b l y o t h e r s ) may be a p p l i c a b l e t o hologram f o r m a t i o n i n a r e a l c r y s t a l , and t h u s d e t a i l e d n u m e r i c a l work may n o t r e v e a l one o f them t o be c l e a r l y and s o l e l y r e s p o n s i b l e f o r t h e g r a t i n g f o r m a t i o n . However-, t h e y a l l agree on s e v e r a l i m p o r t a n t p o i n t s , and t h u s p r o - v i d e a u s e f u l - s t a r t i n g p o i n t f o r e x p e r i m e n t a l a n a l y s i s . 31 6. HOLOGRAM WRITING IN LiNbOg CRYSTALS Having c o n s i d e r e d beam c o u p l i n g i n an " i d e a l " t h i c k phase g r a t i n g i n Chapter 3, and t h e b a s i c p r o c e s s e s i n v o l v e d i n w r i t i n g a hologram i n Cha p t e r s 4 and 5, we now c o n s i d e r the t h e o r y o f hologram w r i t i n g i n LiNbOg. In t h e f i r s t t r e a t m e n t s o f t h i s problem ( e . g . S t a e b l e r and Amodei, 1972) n e i t h e r t h e feedback e f f e c t o f t h e r i s i n g space c h a r g e nor t h e e f f e c t o f s e l f d i f f r a c t i o n of t h e w r i t i n g beams on the hologram g r a t i n g were t a k e n i n t o a c c o u n t . A l s o , t h e c o n c l u s i o n s c o n c e r n i n g t h e r e l a t i v e phase between t h e FP and HG r e s t e d on an i m p l i c i t , though u n r e a l i z e d , assumption o f s h o r t t r a n s p o r t l e n g t h o f t h e m i g r a t i n g e l e c t r o n . I t was shown t h a t i f d i f f u s i o n was t h e main means o f e l e c t r o n t r a n s p o r t , the hologram g r a t i n g i s s h i f t e d by TT/2 w i t h r e - s p e c t t o t h e f r i n g e p a t t e r n . I f d r i f t i s t h e o n l y means o f e l e c t r o n t r a n s p o r t t h e r e i s no phase s h i f t (and c o n s e q u e n t l y no energy t r a n s f e r ) . L a t e r work (Young e t a l . , 1979) showed t h a t i f t h e d r i f t t r a n s p o r t l e n g t h i s n o t much l e s s t h a n t h e f r i n g e s p a c i n g t h e n some phase s h i f t i s o b s e r v e d , and as t h e d r i f t l e n g t h approaches o r exceeds the f r i n g e s p a c i n g , one o b t a i n s a l s o t h e ir/2 s h i f t e d HG a t t h e i n i t i a l s t a g e o f r e c o r d i n g . T h i s a n a l y s i s was a p p l i c a b l e o n l y t o t h i n l a y e r s o f o p t i c a l c r y s t a l s o r t o t h e i n i t i a l s t a g e s o f r e c o r d i n g when energy and phase r e d i s t r i b u t i o n between the two beams may be n e g l e c t e d . S h o r t l y a f t e r w a r d s , a dynamic t h e o r y o f hologram r e c o r d i n g was p u t f o r w a r d (Ninomiya, 1973) based upon t h e s o l u t i o n of n o n l i n e a r wave e q u a t i o n s w i t h the d i e l e c t r i c c o n s t a n t dependent upon t h e l i g h t i n t e n s i t y . Whereas t h i s and s e v e r a l s u c c e e d i n g p apers used a p o s t u l a t e d dependence o f r e f r a c t i v e i ndex upon i n t e n s i t y , Kukhtarev e t a l . (1979) d e r i v e d i t from t h e m a t e r i a l e q u a t i o n s . Though t h e i r t h e o r y i n c l u d e s t h e p h o t o v o l t a i c c u r r e n t , t h e i r subsequent e x p e r i - ments were c o n d u c t e d on c r y s t a l s and under c o n d i t i o n s where t h e e f f e c t s o f 32 d i f f u s i o n dominate, and thus do not c o n f i r m the a c c u r a c y o f t h e i r p r e d i c t i o n s v i s - a - v i s t h e BPE. The t h e o r e t i c a l development g i v e n h e r e f o l l o w s Moharam e t a l . (1979) and w i l l be a p p l i e d t o t h e i n i t i a l s t a g e s o f hologram f o r m a t i o n b e f o r e t h e e f f e c t s o f feedback and s e l f - d i f f r a c t i o n become s i g n i f i c a n t . The f o l l o w i n g c o n d i t i o n s and assumptions h o l d : i ) a r b i t r a r y e l e c t r o n t r a n s p o r t l e n g t h , i i ) e x t e r n a l s h o r t on c r y s t a l , i i i ) u n i f o r m i l l u m i n a t i o n o f c r y s t a l w i t h geometry as i n F i g u r e 6.1. The e x p r e s s i o n f o r the f r i n g e p a t t e r n formed by t h e two i n t e r f e r i n g beams i s : I ( x ) = I (1 + m cos kx) (6.1) o where m = 1 ( m o d u l a t i o n r a t i o f o r I R = I s ) / k = 4TT s i n 6/A ( f r i n g e p a t t e r n wave v e c t o r ) . The c o n d u c t i o n c u r r e n t d e n s i t y d i s t r i b u t i o n a l o n g t h e x - a x i s i s J ( x ) = qD 8 p ( . X , t ) + q u p ( x , t ) E ( x , t ) + J ( x , t ) (6.2) dx s c p w i t h t h e t h r e e terms r e p r e s e n t i n g d i f f u s i o n , d r i f t (due t o the p h o t o - i n d u c e d space charge f i e l d E s c ( x , t ) ) , and t h e BPE. The f r e e e l e c t r o n c o n c e n t r a - t i o n i s t h e sum o f t h e dark c o n c e n t r a t i o n and p h o t o e x c i t e d c o n c e n t r a t i o n : p ( x , t ) = p + p ( x , t ) (6.3) Reasoning t h a t f o r u n i f o r m i l l u m i n a t i o n , t h e p h o t o v o l t a i c c u r r e n t d e n s i t y equa- t i o n s h o u l d reduce t o t h e e x p r e s s i o n o b t a i n e d by G l a s s e t a l . (1974 b) (see eq. 5.1), Young e t a l . (1979) p u t f o r w a r d t h e f o l l o w i n g e x p r e s s i o n (see eq. 5.11): 33 J (x) = -KCII [ l + mf, (k,L ) cos (kx-A ) 1 (6.4) p o L 1 P P where 0 < f ^ (k,Lp) < 1 i s a r e a l f u n c t i o n which can be s e t e q u a l t o u n i t y f o r our p u r p o s e s . They a l s o p o i n t e d out t h a t i f t h e p h o t o v o l t a i c t r a n s p o r t l e n g t h i s n o t s m a l l w i t h r e s p e c t t o t h e f r i n g e s p a c i n g , t h e n t h e iced term i s n o t s t r i c t l y c o r r e c t . The p h o t o e x c i t e d e l e c t r o n c o n c e n t r a t i o n i n e q . 6.3 f o l - lows from 8 p L ( X ' t } , , p L ( X ' t } , 1 3 J ( x , t ) 0 - — - = g(x) _ + _ _ _ (6.5) where g(x) i s the volume g e n e r a t i o n r a t e , g(x) = g Q [ l + m f 2 ( k » L p ) cos ( k x - ^ ) ] (6.6) and S p j ^ f x j t j / S t » 0 c o n s t i t u t e s an assumption t h a t t h e p h o t o e x c i t e d e l e c t r o n c o n c e n t r a t i o n i s time i n v a r i a n t f o r s t e a d y i l l u m i n a t i o n . We combine e q . 6.6 w i t h t h e c o n t i n u i t y e q u a t i o n f o r t r a p p e d charge d e n s i t y , P s c / due t o c a r - r i e r m i g r a t i o n 3p_ s c 3 J ( x , t ) (6.7) a t 8x and w i t h P o i s s o n ' s e q u a t i o n 8 E ( x , t ) p * £ = _ 2 £ (6.8) 3x e t o g e t ( a f t e r i n t e g r a t i n g o v e r space and time) , t E ( x , t ) = - - f J ( s , t ) d t + A ( t ) • (6.9) sc e J o The c o n s t a n t of i n t e g r a t i o n comes from the assumption ( i i above) t h a t t h e r e i s a s h o r t a c r o s s the c r y s t a l , L f E ( x , t ) dx = 0 (6.10) J sc o 34 from which we get T t E (x,t) = / f j ( x , t ) - J (x,t) ] dt . (6.11) sc e •* L o J o Here, JQ i s the average conduction current density. Solving the above we can derive [Moharam et a l . ( 1 9 7 9 ) , E l Guibaly (1979)] an expression f o r the space charge f i e l d : 1/2 qg tm [(kL ) 2 + (kL ) 4 J E (x,t) = E cos (kx-(j) -<j>,) (6.12) 8 k 1 + ( k L , ) 2 P d d (for the case of large scale d r i f t ) , where -1 ( k L * > 2 * -1 E ° *d - t a n " k l — = T A N V p v kT' T' = absolute temperature. E = k D B q k B = B ° l t z r n a n , s const. L JE "V yx E„ = — i s the v i r t u a l f i e l d 1/2 T, = (TD) d i f f u s i o n transport length d <j) i s the phase s h i f t due to the f i n i t e photovoltaic transport length. Knowing the d i s t r i b u t i o n f o r the space charge f i e l d we can f i n d the r e f r a c t i v e index d i s t r i b u t i o n which constitutes the gr a t i n g (see Appendix A), and then use the r e s u l t s of Chapter 4 to r e l a t e t h i s to the beam coupling. 35 7. A P P A R A T U S F i g u r e 7-1 i s a sche m a t i c o f t h e e n t i r e e x p e r i m e n t a l s e t u p used i n t h i s s t u d y . A l l o p t i c a l components i n c l u d i n g the l a s e r were p l a c e d on a mas- s i v e cement bench f o r m e c h a n i c a l i s o l a t i o n from b u i l d i n g v i b r a t i o n s , and m e t a l s t r i p s were cemented t o the t o p s u r f a c e t o p r o v i d e a s u r f a c e f o r magnetic com- ponent s t a n d s . Slamming doors and r o l l i n g c a r t s down t h e h a l l s t i l l had an e f f e c t on the r e a d i n g s so most e x p e r i m e n t a t i o n was done i n t h e e v e n i n g . The l i g h t s o u r c e was a S p e c t r a P h y s i c s Model 166 a r g o n - i o n l a s e r o p e r a t e d CW a t 514.5nm, a t which i t has a maximum s p e c i f i e d o u t p u t power o f 800mW ( f o r T E M Q Q mode), though o n l y 515mW was o b t a i n e d . I t was g e n e r a l l y o p e r a t e d f o r 30 minutes b e f o r e runs t o a l l o w i t t o r e a c h a s t a b l e o u t p u t power. The beam was d i r e c t e d by m i r r o r Ml i n t o a p l e x i g l a s and aluminum "greenhouse" which s e r v e d t o i s o l a t e i t from t h e r m a l a i r c u r r e n t s which cause random f l u c t u a t i o n s i n t h e phase p a t h l e n g t h . Those p a r t s o f t h e beam p a t h where t h e beam was s e p a r a t e d and then brought t o g e t h e r t o form an i n t e r f e r e n c e p a t t e r n were f u r t h e r i s o l a t e d from a i r c u r r e n t s by b e i n g e n c l o s e d i n paper t u b e s f o r most o f t h e i r l e n g t h . A s m a l l f r a c t i o n o f the beam was s p l i t o f f by BSj b e f o r e t h e hologram s e t u p and used i n a M i c h e l s o n i n t e r f e r o m e t e r (BS£, M 2, M ^ , M^, sc r e e n ) f o r t h e pu r p o s e o f f o r m i n g an i n t e r f e r e n c e p a t t e r n o f v i s i b l e d i m e n s i o n s . T h i s a l l o w e d s i m u l t a n e o u s v i s u a l m o n i t o r i n g o f bench v i b r a t i o n s , which was u s e f u l i n d i a g - n o s i n g o c c a s i o n a l problems e x p e r i e n c e d i n f o r m i n g holograms. The main p o r t i o n o f t h e beam c o n t i n u e d on t h r o u g h a s p a t i a l f i l t e r t o a c o l l i m a t i n g l e n s p o s i t i o n e d a t s l i g h t l y more than i t s f o c a l l e n g t h from t h e f i l t e r p i n h o l e so t h a t t h e beam converged t o a p o i n t a t t h e beam s p l i t t e r BSg h a l f way between the l e n s and c r y s t a l . T h i s was done f o r two r e a s o n s : FIGURE 7.1 ( e x p e r i m e n t a l a p p a r a t u s ) The l a s e r beam i s d i r e c t e d i n t o t h e p l e x i g l a s s "greenhouse" by m i r r o r . Beam s p l i t t e r B S 1 d i v e r t s a p o r t i o n o f t h e beam f o r use i n a M i c h e l s o n i n t e r f e r o m e t e r ( B S 2 , M 3, M^, M 2, l e n s s c r e e n ) t o p r o v i d e a m a c r o s c o p i c d i f f r a c t i o n p a t t e r n on t h e s c r e e n t o m o n i t o r bench v i b r a - t i o n . The r e s t o f t h e beam p a s s e s t h r o u g h a s p a t i a l f i l t e r and c o l l i - m a t i n g l e n s , p a s t a s h u t t e r S 1 t o t h e beam s p l i t t e r BSg. Then two beams o f e q u a l i n t e n s i t y a r e r e f l e c t e d from M 5 and M & towards t h e c r y s t a l where t h e y come t o g e t h e r t o form a f r i n g e p a t t e r n . The i n t e n s i t i e s o f the c o u p l e d beams t r a n s m i t t e d by the c r y s t a l a r e m o n i t o r e d by t h e p h o t o - meters Pg and P R. The f r i n g e p a t t e r n a t t h e c r y s t a l c a n be s h i f t e d p a r a l l e l t o t h e c r y s t a l c - a x i s by a d j u s t i n g t h e v o l t a g e on t h e p i e z o - e l e c t r i c d i s c upon which m i r r o r Mg i s mounted. Opening s h u t t e r s and S 2 causes both beams t o pas s ( w r i t i n g ) , whereas o p e n i n g o n l y s h u t t e r l e t s o n l y the r e f e r e n c e beam t h r o u g h ( r e a d i n g / e r a s i n g ) . The beams be- tween t h e m i r r o r s and M g and t h e c r y s t a l a r e e n c l o s e d i n paper t u b e s t o f u r t h e r reduce a i r c u r r e n t p r o b l e m s . J 37 i ) i t p e r m i t t e d the i n s e r t i o n o f s m a l l a p e r t u r e s i n both beam pa t h s i m m e d i a t e l y a f t e r s p l i t t i n g by BS^ i n o r d e r t o b l o c k o f f s e c o n d a r y beams a r i s i n g from i n t e r n a l r e f l e c t i o n i n t h e s p l i t t e r , which i n t e r - f e r e d w i t h the main beams; i i ) t h e c o n t i n u o u s l y v a r i a b l e beam s p l i t t e r g i v e s a more n e a r l y u n i f o r m a m p l i t u d e i n t h e two r e s u l t i n g beams i f the i n c i d e n t beam has a s m a l l c r o s s s e c t i o n . T h i n geometry was adopted i n p r e f e r e n c e t o one i n which two s p a t i a l f i l t e r s were p l a c e d a f t e r BS^ t o a v o i d h a v i n g t o spend too much time t u n i n g t h e f i l t e r s i n o r d e r t o b a l a n c e beam i n t e n s i t i e s a t t h e c r y s t a l . The r e s u l t i n g beam d i v e r - gence o f the new geometry (< 20 mR) d i d not s e r i o u s l y a f f e c t t h e assumption o f i n t e r f e r i n g p l a n e waves. A f t e r b e i n g s p l i t , the beams a r e r e f l e c t e d from m i r r o r s M 5 and M g towards a p o i n t where th e y i n t e r s e c t , f o r m i n g a d i f f r a c t i o n p a t t e r n o f v e r t i c a l p l a n e s i n t h e c r y s t a l h e l d i n t h e c r y s t a l h o l d e r . One o f the two m i r r o r s i s mounted on a PZT ceramic d i s c , thus p r o v i d i n g the c a p a b i l i t y t o e l e c t r i c a l l y v a r y one of the phase p a t h l e n g t h s w i t h r e s p e c t t o t h e o t h e r . The beam i n t e n s i t i e s emerging from the c r y s t a l a r e the n m o n i t o r e d by two s i l i c o n p-n j u n c t i o n p h o t o d e t e c t o r s , one an A l p h a m e t r i c s Model P1110S d r i v - i n g a Model DC1010 d i s p l a y , and the o t h e r a model PS1101S d r i v i n g a Model 1030 d i s p l a y . A K e i t h l e y 602 e l e c t r o m e t e r was co n n e c t e d a c r o s s the c r y s t a l f a c e s p e r p e n d i c u l a r t o the c - a x i s t o measure the p h o t o c u r r e n t . The PZT d i s c was d r i v e n by a F l u k e Model 405B h i g h v o l t a g e power s u p p l y , which i s d i s c r e t e l y v a r i a b l e i n 0.1V s t e p s w i t h i n a ±3000V r a n g e . A Moseley A u t o g r a f Model 7100 BM two c h a n n e l s t r i p c h a r t r e c o r d e r was used t o m o n i t o r any two of the f o u r r e l e v a n t p a r a m e t e r s : PZT v o l t a g e ( t h r o u g h a 10:1 v o l t a g e d i v i d e r ) , e l e c t r o m e t e r o u t p u t and t h e two p h o t o d e t e c t o r s . 38 A DC v o l t a g e s u p p l y was u s e d t o d r i v e t h e two s h u t t e r s and S^t w h i c h a l l o w e d e i t h e r b o t h beams ( w r i t i n g ) , t h e r e f e r e n c e beam ( r e a d i n g ) o r n e i t h e r beam t o r e a c h t h e c r y s t a l . L a s t l y , t o e r a s e t h e h o l o g r a m s , t h e c r y s t a l was s h o r t c i r c u i t e d a c r o s s t h e c - f a c e s w i t h a p l a t i n u m w i r e mesh t o p r e v e n t damage fr o m s t r e s s e s o f p y r o e l e c t r i c o r i g i n , a n d . p l a c e d i n a h o l d e r w h i c h was p l a c e d i n an aluminum box t o e n s u r e even and g r a d u a l h e a t i n g and c o o l i n g i n t h e o v e n . 39 8. MEASUREMENTS AND ANALYSIS 8.1 Measurements In t h e f o l l o w i n g we make the assumption o f a l o s s l e s s d i e l e c t r i c by s e t t i n g I (z=0) + I (z=0) = I (z=d) + I (z=d). The d i f f r a c t i o n e f f i c i e n c y i s R s R s t h e n c a l c u l a t e d from measurements when I (z=0) as s I g ( z = d ) n = I (z=d) + I (z=d) ( 8 , 1 ) s R The amount o f c o u p l i n g , C and C , between t h e two beams when I (z=0) = I (z=0) R s R s r e s u l t s i n n o r m a l i z e d v a l u e s o f I (z=d) I R ( z = d ) C s = [I (z=d) + I (z=d)J/2 a n d °R = [I (z=d) + I_(z=d) J/2 ( 8 ' 2 )  L s R J L s R J The c r y s t a l u s e d (#6 - see Appendix A) was 0.1 mole p e r c e n t i r o n doped and a n t i r e f l e c t i o n c o a t e d t o reduce c o m p l i c a t i o n s i n t h e f r i n g e p a t t e r n due t o i n t e r n a l r e f l e c t i o n . . F i g . 8.2 i s a photocopy o f s t r i p c h a r t r e c o r d e r o u t p u t (1.25 cm/min) o f a p a i r o f p h o t o d e t e c t o r s m o n i t o r i n g t h e i n t e n s i t i e s o f t h e t r a n s m i t t e d sub- j e c t and r e f e r e n c e beams, I g ( z = d ) and I R ( z = d ) as a hologram i s r e c o r d e d The f o l l o w i n g f e a t u r e s a r e p r e s e n t : i ) L i n e s ' C and 1D' a r e t h e t r a n s m i t t e d s u b j e c t and r e f e r e n c e beam i n t e n s i t i e s r e s p e c t i v e l y . These s t a r t o u t e q u a l (5mW/cm2) b u t t h e i r i n t e n s i t i e s d i v e r g e as time p r o g r e s s e s due t o beam c o u p l i n g . i i ) At r e g u l a r i n t e r v a l s the i n c i d e n t s u b j e c t beam i s i n t e r r u p t e d by a s o l e n o i d o p e r a t e d s h u t t e r c a u s i n g l" s(z=d) t o drop ('a' i n f i g u r e ) and I R ( z = d ) t o remain unchanged. As t h e hologram d e v e l o p s , how- e v e r , some o f t h e r e f e r e n c e beam i s d i f f r a c t e d i n t o t h e s u b j e c t beam p a t h and the i n t e n s i t y o f I s( z=a) g r a d u a l l y i n c r e a s e s ( l i n e FIGURE 8.1 Beam geometry d u r i n g hologram w r i t i n g (eq. 8 . 1 , 8.2). 41 FIGURE 8.2 Development of hologram, showing: i ) beam coupling (C,D) ' i i ) diffraction efficiency for momentary reading (a,A,b»B) i i i ) "nulling voltage" (E) 42 F I G U R E 8.3c ) * i i i i i i i 0 1 2 3 4 5 6 7 E x p o s u r e (J/cm) 43 'A') and t h a t o f I R ( z = d ) d e c r e a s e s ( l i n e ' B*). P o i n t s on l i n e s 'A* and *B' a r e i n s e r t e d i n t o e q u a t i o n 8.1 t o o b t a i n v a l u e s f o r t h e d i f f r a c t i o n e f f i c i e n c y as a f u n c t i o n o f ti m e . i i i ) A l s o , t h e i n t e r v a l s o f the c o u p l e d beams were b a l a n c e d a t r e g u l a r i n t e r v a l s [ l R ( z = d ) s e t e q u a l t o I s ( z = d ) ] by v a r y i n g the s p a t i a l phase r e l a t i o n s h i p <J> between t h e f r i n g e p a t t e r n and hologram g r a t - i n g . T h i s was a c h i e v e d by a p p l y i n g t h e a p p r o p r i a t e v o l t a g e t o the PZT d i s c (see F i g . 7.1), and i s r e p r e s e n t e d by t h e s p i k e s 'E* on F i g . 8.2 w i t h the c o r r e s p o n d i n g v o l t a g e s w r i t t e n above. F i g u r e 8.3a i s a graph o f t h e development o f t h e n o r m a l i z e d d i f f r a c - t i o n e f f i c i e n c y ( eq. 8.1) from F i g . 8.2. The x - a x i s u n i t s a r e exposure ( j o u l e s / c m 2 ) which can be c a l c u l a t e d from t h e i n c i d e n t i n t e n s i t i e s [ i (z=0) = I r ( Z = 0 ) = 5.0 mW/cm2], the a n g l e s o f i n c i d e n c e from t h e normal (6^ = ± 1 9 ° ) and th e c h a r t r e c o r d e r speed (1.25 cm/min.). F i g u r e 8.3b i s a graph o f t h e development o f the amount of beam c o u p l i n g ( l i n e s * C and 'D' i n F i g . 8.2 n o r - m a l i z e d w i t h e q u a t i o n s 8.2). F i g u r e 8.3c i s a graph o f t h e v o l t a g e r e q u i r e d t o n u l l the c o u p l i n g . I t was seen i n Chapter 3 t h a t a s i m p l e s i n u s o i d a l f r i n g e p a t t e r n i n c i d e n t upon a s i m p l e s i n u s o i d a l h o l o g r a p h i c g r a t i n g w i l l r e s u l t i n beam coup- l i n g and a t r a n s f e r o f energy from one beam t o the o t h e r (eq. 3.8) i f t h e r e i s a non-zero s p a t i a l phase <{> between t h e f r i n g e s and g r a t i n g ( F i g . 3.1). (Here, " s i m p l e " means t h a t t h e r e i s no v a r i a t i o n i n e i t h e r the f r i n g e p a t t e r n or t h e g r a t i n g i n t h e Z d i r e c t i o n . ) T h e r e f o r e , whether o r n o t we a r e a c t u a l l y d e a l i n g w i t h s i m p l e g r a t i n g s and f r i n g e p a t t e r n s , an e f f e c t i v e phase s h i f t (J>E can be d e f i n e d which i s t h e s h i f t which would a c c o u n t f o r t h e c o u p l i n g i f i n f a c t s i m p l e f r i n g e s and g r a t i n g s were p r e s e n t . fye can be o b t a i n e d from t h e data i n two ways: 44 i ) d i r e c t c a l c u l a t i o n from the " n u l l i n g v o l t a g e " V: V 6 = — — x 360 (degrees) e V_ 2TT where V 2 u = 1180 v o l t s i s the v o l t a g e which ca u s e s a f u l l 2K phase s h i f t , and i s d e t e r m i n e d by t h e p i e z o e l e c t r i c c o e f f i c i e n t o f the phase s h i f t e r (Appendix B) and by t h e a n g l e 8p ( F i g . 7.1), and i i ) c a l c u l a t i o n s from the d a t a i n f i g u r e s 8.3 a and b i n s e r t e d i n t o e q u a t i o n s 3.8, 3.9 and 8.1. These are p l o t t e d i n F i g . 8.4 a l o n g w i t h t h e r e s u l t s from two o t h e r e x p e r i m e n t a l r u n s . The t h r e e f e a t u r e s o f note a r e t h a t t h e v a r i a t i o n i n phase w i t h exposure i s a p p r o x i m a t e l y l i n e a r o v e r t h e range o f o b s e r v a t i o n , t h a t a l l l i n e s have a p p r o x i m a t e l y t h e same i n t e r c e p t a t Exp. = 0, and t h a t t h e s l o p e s of t h e l i n e s v a r y not o n l y w i t h d i f f e r e n t s e t s o f d a t a b u t a l s o w i t h the means o f c a l c u l a t i o n ( i and i i a b o v e ) . The c o u p l i n g shown i n F i g . 8.3b can be compared t o t h a t o f F i g . 8.5, which shows t h a t th e development o f c o u p l i n g can be d i f f e r e n t f o r n o m i n a l l y i d e n t i c a l e x p e r i m e n t s . However, i t i s p o s s i b l e t o e x t r a c t u s e f u l i n f o r m a t i o n from t h e c o u p l i n g d e s p i t e i t s i r r e g u l a r b e h a v i o u r . T h i s e n t a i l s a c l o s e r e x a m i n a t i o n o f th e r e a s o n s f o r the time v a r i a t i o n o f t h e c a l c u l a t e d phase s h i f t s . 8.2 A n a l y s i s o f Phase S h i f t Measurements F i g u r e 8.4, which shows t h a t th e phase s h i f t v a r i e s w i t h t i m e , demon- s t r a t e s why graphs l i k e F i g . 8.6 g i v i n g t h e p r e d i c t e d c o u p l i n g f o r v a r i o u s c o n s t a n t v a l u e s of the s h i f t a r e not o b s e r v e d e x p e r i m e n t a l l y . There are s e v e r a l a p p a r e n t r e a s o n s why t h e phase d i f f e r e n c e between t h e g r a t i n g and f r i n g e p a t t e r n v a r i e s as the hologram i s w r i t t e n . 45 FIGURE 8.4 V a r i a t i o n o f phase s h i f t w i t h exposure f o r t h r e e r u n s : i ) Dashed l i n e s c a l c u l a t e d u s i n g e q u a t i o n s 3.8, 3.9, and t h e measured v a l u e s f o r the d i f f r a c t i o n e f f i c i e n c y and c o u p l i n g , i i ) S o l i d l i n e s c a l c u l a t e d d i r e c t l y from n u l l i n g v o l t a g e o f PZT s h i f t e r . R - B E A M 0 5 10 EXPOSURE ( J / c m 2 ) 15 FIGURE 8.5 Time development o f beam c o u p l i n g f o r f i v e n o m i n a l l y i d e n t i c a l e x p e r i m e n t s . The beam i n t e n s i t i e s a r e n o r m a l i z e d by d i v i d i n g each by o n e - h a l f t h e i r sum. From Young e t a l . (1979). 47 EXPOSURE ( J / c m 2 ) FIGURE 8.6 Beam i n t e n s i t i e s c a l c u l a t e d from a computer model f o r d i f f e r e n t assumed v a l u e s o f the phase s h i f t <j> ( c o n s t a n t ) a s s o c i a t e d w i t h t h e b a l k p h o t o v o l t a i c e f f e c t . From e l G u i b a l y ( t h e s i s ) . 48 8.2.1 G r a t i n g Bending I t has been shown ( S t a e b l e r e t a l . , 1972) t h a t t h e phase o f t h e h o l o - gram g r a t i n g v a r i e s a l o n g the z - a x i s as time (exposure) i n c r e a s e s . T h i s i s i l l u s t r a t e d i n F i g . 8.7, which uses d a t a o b t a i n e d from a program d e v e l o p e d by Moharam (1978) and l a t e r m o d i f i e d by e l G u i b a l y (1979) f o r the case of a c r y s - t a l 1mm t h i c k w i t h an a n g l e o f i n c i d e n c e o f 19° and v i r t u a l f i e l d o f 55 kV/cm. The s o l i d l i n e s g i v e t h e v a r i a t i o n o f phase w i t h depth i n t o the c r y s t a l f o r t e n s u c c e s s i v e and e q u a l l y spaced time samples. The dashed l i n e s a r e t h e c o r r e - s p o n d i n g " e f f e c t i v e phase s h i f t s " , where $eff i s a measure o f the d i s - p lacement o f a s t r a i g h t g r a t i n g which would produce t h e same c o u p l i n g , . a s d e t e r m i n e d by th e program i n Appendix E. T h i s i s the q u a n t i t y which i s mea- s u r e d by n u l l i n g t h e c o u p l i n g u s i n g the PZT s h i f t e r and assuming no o t h e r v a r i a t i o n s i n t h e s h i f t t a k e p l a c e . 8.2.2 Thermal E x p a n s i o n From the p h o t o c u r r e n t measurements showing the p y r o e l e c t r i c c u r r e n t ( F i g . 5.1) and t h e v a l u e o f t h e p y r o e l e c t r i c c o n s t a n t (Appendix A) one can show t h a t d u r i n g t h e s e experiments the c r y s t a l h e a t e d up by s e v e r a l t e n t h s o f a degree c e n t i g r a d e . H e a t i n g the c r y s t a l by 0.5°C w i l l expand i t a l o n g t h e x - d i r e c t i o n by about 1/20 of a g r a t i n g w a velength. T h i s e xpansion would take p l a c e m o s t l y d u r i n g t h e f i r s t minute o f i l l u m i n a t i o n , and t h e r e f o r e a f f e c t o n l y t h e f i r s t r e a d i n g ( i f any) i n t h e p r e s e n t e x p e r i m e n t s . Thermal e x p a n s i o n and c o n t r a c t i o n may e x p l a i n t h e l a c k o f s u c c e s s met w i t h i n i t i a l l y i n t h i s work i n attempts t o s t u d y - c o u p l i n g a t r e g u l a r i n t e r v a l s d u r i n g w r i t i n g by d e c r e a s i n g t h e beam i n t e n s i t y b e f o r e a d j u s t i n g t h e PZT s h i f t e r so as not t o w r i t e w h i l e s h i f t i n g . -160.0 -358.0 -356.0 -354.0 -352.0 -350.0 -348.0 -3 46.0 -3 44.0 -342 0 -3 40 0 PHRSE S H I F T (DEC) FIGURE 8.7 Time development of hologram curvature: s o l i d l i n e s - curves generated by Moharam's model (see Appendix E) dashed l i n e s - effe c t i v e displacement of equivalent simple grating 50 8 . 2 . 3 B u l k Motion of C r y s t a l and O p t i c a l Components As shown by t h e s l o p e s i n F i g . 8.4, t h e v a r i a t i o n o f phase w i t h t i m e can be caused by a sideways ( x - a x i s ) motion o f th e c r y s t a l o f about lOnm/min. Motion o f t h e m i r r o r and l e n s e s i n t h e o p t i c a l system may r e s u l t i n an e q u i v a - l e n t e f f e c t by s h i f t i n g the f r i n g e p a t t e r n . D i r e c t measurement o f t h e motion o f t h e f r i n g e p a t t e r n from t h e M i c h e l s o n I n t e r f e r o m e t e r ( F i g . 7.1) shows t h a t such motion p e r s i s t s f o r days a f t e r the adjustment o f t h e p o s i t i o n o f any com- ponent i n th e beam p a t h due t o t h e r e s i d u a l m e c h a n i c a l c r e e p o f t h e com- p o n e n t s . The e f f e c t o f the above problems i s t o g i v e t h e e) l i n e s i n F i g . 8.4 a non-zero s l o p e . Because t h i s s l o p e was t h e r e s u l t o f u n c o n t r o l l a b l e and q u a n t i t a t i v e l y u n p r e d i c t a b l e p r o c e s s e s , t h e c o u p l i n g t h a t f o l l o w s from i t i s a l s o u n p r e d i c t a b l e . T h i s a c c o u n t s f o r t h e wide v a r i a t i o n i n r e s u l t s i n F i g . 8.5. However, as seen i n F i g . 8.4, t h i s i s due t o t h e phase s h i f t u n d e r g o i n g a l i n e a r change w i t h i n c r e a s i n g time ( o r exposure) i n s t e a d o f r e m a i n i n g c o n s t a n t . E x t r a p o l a t i o n back t o t=0 g i v e s a f i g u r e f o r t h e phase s h i f t o f t h e i n i t i a l l y s t r a i g h t g r a t i n g b e f o r e t h e above mentioned p r o c e s s e s have had a chance t o cause a change i n t h e r e l a t i v e p o s i t i o n s o f the g r a t i n g and f r i n g e p a t t e r n . A l i n e a r r e g r e s s i o n o f t h e phase n u l l i n g v o l t a g e d a t a (from which t h e l i n e s i n F i g . 8.4 were o b t a i n e d ) g i v e s the v a l u e fy0 = 5.0° ± 0 . 7 ° . Because i t i s a d i r e c t measurement o f t h e phase s h i f t , t h e l i n e g i v e n by t h e PZT phase s h i f t e r i s t h e more r e l i a b l e o f the two. The l i n e o b t a i n e d from e q u a t i o n s 3.8 and 3.9 g e t s p r o g r e s s i v e l y l e s s a c c u r a t e as t h e g r a t i n g d e p a r t s from the " s i m p l e " g r a t i n g geometry assumed i n the d e r i v a t i o n o f t h e e q u a t i o n s . In any c a s e , t h e two methods agree a t t=0 t o w i t h i n ± 1 . 4 ° . F i g . 8.8 i s a graph o f t h e c o u p l i n g which would be e x p e c t e d f o r t h e case o f sjiCt^O) = 25° and dtp/dt = -10, -5, 0, 5 and 10 degrees p e r u n i t t i m e . FIGURE 8.8 S i m u l a t e d c o u p l i n g f o r c a s e s where <))( ( d e g / ( J / c m 2 ) ) . = 25° and f o r v a r i o u s v a l u e s o f <j> 52 The c r o s s o v e r shown i n the f i r s t two c a s e s was o c c a s i o n a l l y o b s e r v e d d u r i n g an e x p e r i m e n t a l r u n , i n d i c a t i n g t h a t t h e c r e e p was i n t h e d i r e c t i o n o p p o s i t e t o t h e phase s h i f t due t o t h e b u l k p h o t o - v o l t a i c e f f e c t . T h i s was sometimes s t r o n g enough t o g i v e t h e i m p r e s s i o n o f r e v e r s e d c o u p l i n g . 8.3 C a l c u l a t i o n o f P h o t o v o l t a i c T r a n s p o r t Length Young e t a l . _ (1979) used the o b s e r v e d d i f f r a c t i o n e f f i c i e n c y and t h e measured r a t i o o f t h e e x c i t i n g beam i n t e n s i t i e s a t an exposure o f 1.2 j o u l e s / c m 2 t o c a l c u l a t e t h e phase s h i f t u s i n g e q u a t i o n s 3.8 and 3.9, and a r r i v e d a t a f i g u r e o f 1 0 . 4 ° . From t h i s t h e y o b t a i n e d a v a l u e f o r t h e p h o t o - v o l t a i c t r a n s p o r t l e n g t h , L p , of 24nm. These c a l c u l a t i o n s can now be r e p e a t e d f o r t h e p r e s e n t s e t o f d a t a u s i n g <j>Q = 5.0°. The t o t a l phase s h i f t i s g i v e n by *o " *p + *D = t a n " 1 (icL ) + t a n " 1 (E /E ) p D v where kxT' Ep = i s t h e d i f f u s i o n e q u i v a l e n t f i e l d , E = 45 ± 5k V/cm 2 i s t h e v i r t u a l f i e l d , v From t h i s we can show t h a t L p = 13nm ± 3nm. The d i f f e r e n c e between t h i s v a l u e and t h a t o b t a i n e d by Young e t a l . (1979) can be e x p l a i n e d by a p o s s i b l e d r i f t i n t h e phase s h i f t d u r i n g t h e time taken f o r the exposure t o r e a c h 1.2 j o u l e s / c m 2 i n t h e i r e x p e r i m e n t . To r e s t a t e the r e s u l t s , o f t h i s c h a p t e r , the e l e c t r o n t r a n s p o r t l e n g t h due t o t h e b u l k p h o t o v o l t a i c e f f e c t , L p , has been measured as 13nm. T h i s m a n i f e s t s i t s e l f i n t h e hologram as a d i s p l a c e m e n t o f t h e w r i t t e n g r a t i n g a l o n g 53 t h e + c - a x i s w i t h r e s p e c t t o t h e l i g h t f r i n g e p a t t e r n , w h i c h i n t u r n g i v e s r i s e t o beam c o u p l i n g between t h e t r a n s m i t t e d r e f e r e n c e and s u b j e c t beams. 54 9. SUMMARY S t u d i e s of t h e hologram w r i t i n g p r o c e s s i n f e r r o e l e c t r i c c r y s t a l s and models t o p r e d i c t t h e i r e v a l u a t i o n i n time and space c e n t r e around o b s e r v a t i o n s o f t h e time development o f the d i f f r a c t i o n e f f i c i e n c y and beam c o u p l i n g . Though c o n s i d e r a b l e s u c c e s s has been a c h i e v e d i n p r e d i c t i n g t h e f ormer q u a n t i t y i n computer models, no such s u c c e s s has met attempts t o match ex p e r i m e n t w i t h model p r e d i c t i o n s of the development of c o u p l i n g . C o n s e q u e n t l y , some d i f f i - c u l t y has been e n c o u n t e r e d i n th e c a l c u l a t i o n o f such parameters as t h e p h o t o - v o l t a i c t r a n s p o r t l e n g t h , as t h i s r e q u i r e s an e x p e r i m e n t a l d e t e r m i n a t i o n o f t h e phase s h i f t which g i v e s r i s e t o t h e c o u p l i n g . The p r e s e n t work has been aimed a t measuring t h e phase s h i f t d i r e c t l y t h r o u g h the use o f a PZT phase s h i f t e r and comparing t h i s t o c a l c u l a t i o n s o f t h e s h i f t from the measured d i f f r a c t i o n e f f i c i e n c y and beam c o u p l i n g . I t has been shown t h a t the s h i f t between the g r a t i n g and f r i n g e p a t t e r n does n o t remain a c o n s t a n t q u a n t i t y d u r i n g the w r i t i n g p r o c e s s . T h i s v a r i a t i o n i s due t o p r o c e s s e s which can be n e i t h e r c o m p l e t e l y c o n t r o l l e d nor q u a n t i t a t i v e l y p r e d i c t e d . However, the v a r i a t i o n ( a t l e a s t w i t h t h e a p p a r a t u s u s e d i n t h e s e e x p e r i m e n t s ) i s n e a r l y l i n e a r , and i t was f o u nd t h a t b o t h methods o f m e a s u r i n g t h e phase s h i f t y i e l d t h e same r e s u l t when the b e s t - f i t l i n e s a r e e x t r a p o l a t e d t o t=0. T h i s i n t e r c e p t g i v e s t h e v a l u e o f the phase s h i f t b e f o r e i t has been a f f e c t e d by the s e v e r a l p r o c e s s e s which l a t e r d i s g u i s e t h e o r i g i n a l v a l u e . F o r t h e LiNbOg c r y s t a l used, a phase s h i f t due t o the b u l k p h o t o v o l t a i c e f f e c t o f 2.9° was o b t a i n e d , which c o r r e s p o n d s t o a t r a n s p o r t l e n g t h o f about 13nm. To e s t a b l i s h - t h e g e n e r a l i t y o f t h i s a n a l y s i s i t would be u s e f u l t o make measurements on s e v e r a l c r y s t a l s of d i f f e r e n t d o p i n g and under a v a r i e t y of c o n d i t i o n s . In any c a s e , i t i s c l e a r t h a t w i t h o u t a p p a r a t u s d e s i g n e d t o 55 h o l d t h e o p t i c a l components v e r y s t i l l f o r p e r i o d s of s e v e r a l m i n u t e s o r h o u r i t w i l l n o t be p o s s i b l e t o match up e x p e r i m e n t a l r e s u l t s w i t h computer model p r e d i c t i o n s o f t h e c o u p l i n g . 56 BIBLIOGRAPHY Cathey, W. Thomas: Optical Information Processing and Holography. John Wiley and Sons (1974). Caulfield, H.J. (editor): Handbook of Optical Holography (ch. 10). Academic Press (1979). Chanussot, G: Ferroelectrics 8, p. 671 (1974). Channussot, Fridkin, Godefroy and Jannot: App. Phys. Lett. 31/ p. 3 (1977). Chen, F.S.: J. App. Phys. 40, p. 3389 (1969). Chen, LaMacchia, Fraser: App. Phys. Lett. 13, p. 223 (1968). Clark, Disalvo, Glass, Peterson: J. Chem. Phys. 59, p. 6209 (1973). Col l i e r , Burckhardt and Lin: Optical Holography. Academic Press, Inc. (1971). Cornish, William D.: The Photorefractive Effect in Lithium Niobate. Ph.D. thesis, Dept. of Elec. Eng., U.B.C. (1975). Cornish, Moharam and Young: J. Appl. Phys. 47, p. 1479 (1976). Dischler, Rauber: Solid State Comm. 17, p. 953 (1975). el Guibaly, Fayez H.F.: The Photorefractive Effect i n Lithium Niobate and i t s Applications. Ph.D. thesis, Dept. of Elec. Eng., U.B.C. (1979). Fridkin, V.M.: Appl. Phys. 13, p. 357 (1977). Glass, von der Linde, Negran: App. Phys. Lett. 25, p. 233 (1974). Glass, von der Linde, Auston and Negran: J. Electr. Mat. 4, p. 915 (1975). Jacobs, Sargent, Sculley: Adaptive Optics and Short Wavelength Sources (vol. 6 of "Physics of Quantum Electronics"). Addison-Wesley Publishing Co., Inc. (1978). Johnston, W.D.: J. App. Phys. 41, p. 3279. Koch, Munser, Ruppel, Wurfel: Ferroelectrics 13, p. 305 (1976). Kogelnik, H.: Bell Tel. Tech. J. 48, 2909 (1969). Kukhtarev, Markov, Odulov: Opt. Commun. 23^, p. 338 (1977). Marcatelli, E.A.J.: Appl. Opt. 19, p. 1468 (1980). Marom, Friesem, Wiener-Avnear (editors): Applications of Holography and Optical Data Processing. Pergamon Press (1976). 57 Mikami, Ishida: Opt. Comm. 9, p. 354 (1973). Moharam, M. Gamal: Hologram Storage by the Photorefractive Effect. Ph.D. thesis, Dept. of Elec. Eng., U.B.C. (1978). Moharam, Gaylord, Magnusson, Young: J. Appl. Phys. 5J), p. 5642 (1979). Ninomiya, Y.: J. Opt. Soc. Am. 63, p. 1124 (1973). Nye, J.F.: Physical Properties of Crystals (7th printing). Oxford University Press (1976). Ohmori, Yamaguchi, Yoshino, Inuishi: Japan J. App. Phys. 1 L 6 , p. 181 (1977). Peterson, Glass, Negran: App. Phys. Lett. 19, p. 130 (1971). Peterson, Glass, Carnevale, Bridenbaugh: J. Am. Ceram. Soc. 56, 278 (1973). Ph i l l i p s , Amodei, Staebler: RCA Rev. 33, p. 94 (1972). Smith, H.M. (editor): Holographic Recording Materials (vol. 20 of "Topics i n Applied Physics"). Springer-Verlag (1977). Smith, Fraser, Denton, Rich: J. App. Phys. 39, p. 1600 (1968). Staebler, Amodei: Ferroelectrics 3y 107 (1972a). Staebler, P h i l l i p s : App. Opt. 13, p. 788 (1972a). Staebler, Amodei: J. App. Phys. 43, 1042 (1972b). Staebler, P h i l l i p s : App. Phys. Lett. 24, p. 268 (1974b). Tamir, T. (editor): Integrated Optics (2nd ed.) (vol. 7 of "Topics i n Applied Physics"). Springer-Verlag (1979). Turner, E.H.: Appl. Phys. Lett. 8, p. 303 (1966). von Baltz, R.: Phys. Stat. Sol. (b) 89, p. 419 (1978). von der Linde, Glass and Rogers: Appl. Phys. Lett. 25, p. 155 (1974). von der Linde, Glass and Rogers: Appl. Phys. Lett. 26, p. 22 (1975a). von der Linde, Glass: Appl. Phys. 8y p. 85 (1975b). von der Linde, Glass: Ferroelectrics JL0, p. 5 (1976). Young, Wong, Thewalt, Cornish: Appl. Phys. Lett. 2!4, 264 (1974). Young, Moharam, El Guibaly, Lun: J. Appl. Phys. 50(6), 4201 (1979). 58 APPENDIX A LiNbOg C r y s t a l Data A . l S o u r c e s o f C r y s t a l s The c r y s t a l s used i n t h i s work were c o m m e r c i a l l y p r e p a r e d by C r y s t a l T e chnology I n c . , Mountain View, C a l i f o r n i a , and by Harshaw Chemical Co., S o l o n , O h i o . They were grown by the C z o c h r a l s k i t e c h n i q u e , i n which a p o l i n g e l e c t r i c f i e l d i s a p p l i e d and the b o u l e i s r o t a t e d as i t i s withdrawn from t h e m e l t . The c o m p o s i t i o n o f t h e m e l t i s e i t h e r s t o i c h i o m e t r i c o r congruent, where t h e former c o n t a i n s more L i than t h e l a t t e r (49.0 mole% L i 0 2 v s . 48.6 mole% LiO£ r e s p e c t i v e l y ) . T a b l e A - l l i s t s the r e l e v a n t data f o r each c r y s t a l u s e d . I t s h o u l d be r e a l i z e d t h a t o v e r t h e p a s t f i v e y e a r s o r so, t h e s e c r y s t a l s have been s u b j e c t t o a number o f p r o c e s s e s , not always o f a w e l l r e c o r d e d n a t u r e , and as a r e s u l t can e x h i b i t c h a r a c t e r i s t i c s which may not be r e p r o d u c i b l e i n a n o t h e r nomi- n a l l y i d e n t i c a l c r y s t a l . A.2 M i s c e l l a n e o u s P r o p e r t i e s L i t h i u m n i o b a t e i s a f e r r o e l e c t r i c c r y s t a l w i t h a d i s t o r t e d p e r o v s k i t e s t r u c t u r e (ABOg). As a s t a b l e f e r r o e l e c t r i c a t room t e m p e r a t u r e , i t e x h i b i t s an e l e c t r i c d i p o l e moment even i n t h e absence o f an e l e c t r i c f i e l d . I f h e a t e d above i t s t r a n s i t i o n t e m p e r a t u r e , or C u r i e p o i n t (T = 1470° k f o r LiNbOg) i t l o s e s i t s moment. Below T , the degree o f p o l a r i z a t i o n can be made t o v a r y a l o n g a h y s t e r e s i s l o o p as t h e e x t e r n a l l y a p p l i e d f i e l d i s v a r i e d ( d i f f i c u l t a t lew tem- p e r a t u r e ) . The c r y s t a l has a rhombohedral s t r u c t u r e ( p o i n t group 3m w i t h a = .5499, a = 55°52') (Nassau e t a l . , 1966), and c o v a l e n t bonds p r e d o m i n a t e . I t can be thought of as b e i n g composed of s h e e t s o f oxygen i n a p p r o x i m a t e l y hexa- g o n a l c l o s e p a c k i n g w i t h the r e s u l t i n g o c t a h e d r a l i n t e r s t i c e s b e i n g o n e - t h i r d o c c u p i e d by L i , o n e - t h i r d by Nb, and t h e r e m a i n i n g o n e - t h i r d empty. There a r e 59 two s e p a r a t e v a l u e s each f o r the L i - 0 and Nb-0 d i s t a n c e s . The p r e s e n t s e t o f experim e n t s was done w i t h l i g h t o f X = 514.5nm, which f a l l s w i t h i n t h e r e g i o n o f r e l a t i v e t r a n s p a r e n c y between 350nm and 5000nm. The i n d i c e s o f r e f r a c t i o n a t 500nm and 25°C a r e n =2.34 and n _ = 2 . 2 4 . The d i e l e c t r i c c o n s t a n t s a r e e „ o r = 78 ( p e r p e n d i c u l a r t o c - a x i s ) and e r = 32 ( p a r a l l e l t o c - a x i s ) . The p y r o e l e c t r i c c o e f f i c i e n t i s 1 0 ~ 2 yC/(m 2deg) a t 100°C. I t m e l t s a t 1260°C and has c o e f f i c i e n t s o f . e x p a n s i o n o f 16.7 x 10~^/deg C. a l o n g t h e a - a x i s and 2 x 10~^/deg C. a l o n g the c - a x i s . A.3 E l e c t r o - o p t i c B e h a v i o u r I t i s u s u a l l y s u f f i c i e n t t o c o n s i d e r t h e p e r m i t t i v i t y c o e f f i c i e n t s o f a c r y s t a l t o be c o n s t a n t s , u n a f f e c t e d by t h e s t r e n g t h o f an a p p l i e d e l e c t r i c f i e l d . However, i t i s the case f o r some c r y s t a l s ( i n c l u d i n g LiNbOg) t h a t t h e r e i s a s m a l l b u t d e t e c t a b l e h i g h e r - o r d e r e f f e c t whereby r e a d i l y a t t a i n a b l e f i e l d s cause a change i n t h e p e r m i t t i v i t y . At o p t i c a l f r e q u e n c i e s t h i s i s e q u i v a l e n t t o a change i n t h e i n d e x o f r e f r a c t i o n , and i s t h e b a s i s o f t h e e l e c t r o - o p t i c e f f e c t . The g e n e r a l t e n s o r e x p r e s s i o n f o r the d i e l e c t r i c p r o p e r t i e s o f an a n i s o t r o p i c c r y s t a l a t o p t i c a l f r e q u e n c i e s i s D. = e e. .E. i o ID 3 where e i s t h e t e n s o r r e l a t i v e p e r m i t t i v i t y r e l a t i n g t h e a p p l i e d e l e c t r i c f i e l d E t o t h e d i e l e c t r i c d i s p l a c e m e n t D. Both the p o l a r i z a t i o n and d i r e c t i o n o f p r o - p a g a t i o n w i t h r e s p e c t t o t h e c r y s t a l axes a f f e c t t h e p r o p a g a t i o n o f an e l e c t r o - magnetic wave i n a c r y s t a l , and i n g e n e r a l two waves of d i f f e r e n t v e l o c i t i e s may pr o p a g a t e f o r a g i v e n wave normal. T h e i r r e f r a c t i v e i n d i c e s may be found by t h e e l l i p s o i d , , c a l l e d t h e o p t i c a l i n d i c a t r i x , d e f i n e d by the f o l l o w i n g e q u a t i o n : 60 f — ) = 1 , n. = / e.. (e.. o f d i a g o n a l i z e d m a t r i x ) v n . ' 1 i i i i l where we have used t h e E i n s t e i n summation c o n v e n t i o n t h a t any i n d e x which o c c u r s t w i c e i n t h e same term i s t o be summed. I f t h e e l l i p s o i d i s c l e a v e d t h r o u g h i t s c e n t r e p e r p e n d i c u l a r t o t h e wave normal, t h e l e n g t h s o f t h e axes o f t h e r e s u l t i n g e l l i p s e y i e l d t h e d i r e c t i o n s o f p o l a r i z a t i o n o f t h e two waves (see F i g u r e A - 2 ) . The p r e s e n c e o f an e l e c t r i c f i e l d r e s u l t s i n d e f o r m a t i o n o f t h e i n d i c a t r i x , which i s now d e f i n e d by t h e e q u a t i o n ( eq. A . l ) f n T 2 + Z . . E + R . ., „ E, E + . ..1 x.x. = 1 i,j,k,£ = 1,2,3 *• i j 13k k ink£ k £ ' i j where z.. and R.. a r e t h e 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 c o e f f i c i e n t s , i j k i]k£ Because t h i r d and f o u r t h rank t e n s o r s a r e i n c o n v e n i e n t t o w r i t e o u t , c o n t r a c t i o n s t o second rank form a r e pe r f o r m e d a c c o r d i n g t o t h e f o l l o w i n g c o n v e n t i o n s : rmk "* Z ( i j ) k mn (i;j)(k£) where m,n = 1,2,...,6 and m i s r e l a t e d t o i j and n t o k£ by the r u l e s 1+11, 2->-22, 3-»-33, 4-»-23, 5*13, 6*12. We a r e i n t e r e s t e d i n t h e s p e c i f i c case o f LiNbOg, which i s a u n i a x i a l c r y s t a l e x h i b i t i n g t h e e l e c t r o - o p t i c e f f e c t . I t s h i g h degree o f symmetry r e s u l t s i n most o f the e l e c t r o - o p t i c c o e f f i c i e n t s r m k b e i n g z e r o , and t h e r e s t b e i n g i n t e r d e p e n d a n t . The m a t r i x form and n u m e r i c a l v a l u e s o f i t s elements a r e ( T u r n e r 1966) 61 C r y s t a l Number Dimens: a .on (n b nm) c Polished Faces Iron Doping (mole %) Composition of the Melt 1 6 1 5 ( 2 ) -,(3) 3 10 20 10 b a undoped 0.1% congruent s t o i c h i o m e t r i c (1) according to numbering of E l Guibaly (1979). (2) broken - h a l f used was 7 x 3 x 20 mm. (3) nominal - actual measurement i s 0.69 mm. FIGURE A-1: Table of C r y s t a l Data o p t i c a x i s FIGURE A-2 The i n d i c a t r i x for a p o s i t i v e u n i a x i a l c r y s t a l . FIGURE A-3 App l i c a t i o n of an e l e c t r i c f i e l d Eg to a l t e r the indices of refrac- t i o n seen by a wave propagating i n the x^ d i r e c t i o n . L i N b 0 3 i n d i c e s n o c r y s t a l a x e s d i a g r a m a x e s 62 - r 22 r 1 3 22 r 1 3 0 r 3 3 hZ 0 0 0 0 0 13 22 «t2 33 8.6 x 1 0 " 1 0 cm/Volt 3.4 x 28 x " 30.8 x " 0 0 0 0 :12 :22 The f a c t t h a t LiNbOg i s u n i a x i a l can be u s e d t o o b t a i n t h e i n d e x change t o t h e a p p l i e d f i e l d f o r the s p e c i a l case o f a wave p r o p a g a t i n g i n t h e x 2 d i r e c t i o n t h r o u g h a c r y s t a l w i t h t h e f i e l d a p p l i e d i n t h e x^ d i r e c t i o n ( s e e F i g u r e A - 3 ) . Because t h e i n d i c a t r i x here i s an e l l i p s o i d of r e v o l u t i o n , two o f th e p r i n c i p a l axes a r e e q u a l : n 0 = n l = n 2 n = n o The i n d i c a t r i x ( eq. 1) t h u s r e d u c e s t o ( V 2 - r 2 2 E 2 + r 1 3 E 3 ) X l 2 + ( V 2 + r 2 2 E 2 + r 1 3 E 3 ) X 2 2 + ( V 2 + r 3 3 E 3 3 ) X 3 2 + 2 (" r22 E 1 > 1 X 2 + 2 ( r 1 + 2 E 2 ) x 2 x 3 + 2 ( r ( + 2 E 1 ) x 3 x 1 = 1 ( A . 2 ) F o r o u r c a s e , E^ = E 2 = 0 and t h u s t h e above r e d u c e s t o ( V 2 + r 1 3 E 3 ) x l 2 + ( n E ~ 2 + r 3 3 E 3 ) x 3 2 = 1 ( A . 3 ) I f we c o n s i d e r f i e l d E 3 as e f f e c t i n g changes An 0 and A n £ i n the r e f r a c t i v e i n d i c e s , we c o u l d w r i t e ( A . 4 ) (n + An ] - 2 E 0 x 2 + fn + An ) ~ 2 E_ x _ 2 = 1 o o' 3 1 >-e e i i E q u a t i n g the c o e f f i c i e n t s of x ^ 2 i n (3) and (4) and s o l v i n g f o r A n 0 / w e g e t 63 V r 1 3 E 3 An = O 2. and s i m i l a r l y f o r t h e x - 2 c o e f f i c i e n t s we get An = - T h i s s i m p l e f o r m u l a f o r t h e index change a l s o p o i n t s o u t t h e l i n e a r r e l a t i o n between An and t h e a p p l i e d f i e l d E f o r t h i s s p e c i a l c a s e . 64 APPENDIX B P i e z o e l e c t r i c Phase S h i f t e r As shown i n F i g u r e 7.1, the phase r e l a t i o n s h i p between t h e o b j e c t and r e f e r e n c e beams was a l t e r e d by a p p l y i n g a v o l t a g e t o the p i e z o e l e c t r i c d i s c upon which t h e r e f e r e n c e beam m i r r o r was mounted. An e x p l a n a t i o n o f t h e e f f e c t and d e t a i l s o f t h e d e v i c e f o l l o w . i ) B a s i c P i e z o e l e c t r i c Theory C e r t a i n c r y s t a l s d e v e l o p an e l e c t r i c moment upon a p p l i c a t i o n o f m e c h a n i c a l s t r e s s , w i t h the moment b e i n g l i n e a r l y p r o p o r t i o n a l t o t h e s t r e s s . T h i s i s c a l l e d t h e " d i r e c t p i e z o e l e c t r i c e f f e c t " , and t h e complementary phe- nomenon, i . e . t h e a p p l i c a t i o n o f an e l e c t r i c f i e l d t o change t h e shape of the c r y s t a l , i s r e f e r r e d t o as t h e " c o n v e r s e p i e z o e l e c t r i c e f f e c t " . Mathe- m a t i c a l l y , t h e p i e z o e l e c t r i c e f f e c t can be r e p r e s e n t e d by a t h i r d rank t e n s o r d . r e l a t i n g t h e s t r e s s and p o l a r i z a t i o n . The s t a t e o f s t r e s s , a, o f a i j k c r y s t a l i s c o m p l e t e l y s p e c i f i e d by a second rank t e n s o r o f n i n e components, o ( j , k = 1,2,3), where a s p e c i f i c component a., r e f e r s , f o r i n s t a n c e , t o a Dk }k f o r c e a p p l i e d i n t h e j d i r e c t i o n a c r o s s t h e . f a c e s p e r p e n d i c u l a r t o t h e k d i r e c - t i o n , as i n F i g u r e B . l . The p o l a r i z a t i o n o f a c r y s t a l i s a v e c t o r q u a n t i t y o f t h r e e com- p o n e n t s , P^. In t h e g e n e r a l c a s e , each o f t h e s e t h r e e components i s a f u n c t i o n o f a l l n i n e s t r e s s components. F o r i n s t a n c e , i n t h e 1 d i r e c t i o n , the p o l a r i z a - t i o n i s g i v e n by P i = d 0 j k ( j , k = 1,2,3) ( B . l ) and t h u s - t h e g e n e r a l e x p r e s s i o n f o r a l l t h r e e d i r e c t i o n s o f p o l a r i z a t i o n i s P. = d. « (B.2) j. i l k -j) 65 FIGURE B.I The s t r e s s t e n s o r element o , r e f e r s t o a f o r c e i n t h e j d i r e c t i o n a c r o s s f a c e s p e r p e n - d i c u l a r t o t h e k d i r e c t i o n . e- A FIGURE B.2 Shape and dimensions o f t h e V e r n i t r o n p i e z o e l e c t r i c element. 66 where d. r e p r e s e n t s the 27 p i e z o e l e c t r i c m o d u l i . In p r a c t i c e , c r y s t a l sym-i j k metry o f t e n r e n d e r s many o f the i n t e r d e p e n d e n t , so t h a t fewer t h a n 27 n u m e r i c a l v a l u e s need be c o n s i d e r e d . E q u a t i o n B.2 i m p l i e s t h a t i f a t e n s i l e s t r e s s , i s changed t o an e q u a l and o p p o s i t e s t r e s s , ^he e f f e c t i s t o r e v e r s e t h e s i g n o f p o l a r i z a t i o n . Furthermore, u n s t a t e d i n e q. B.2 i s t h e f a c t t h a t , i n t h e case o f the c r y s t a l h a v i n g a spontaneous p o l a r i z a t i o n P^O p r e s e n t i n t h e absence o f any s t r e s s , the P^ i n eq. B.2 must be t h o u g h t o f as a change e f f e c t e d i n t h e t o t a l p o l a r i z a t i o n . I f we i g n o r e body t o r q u e s , i t i s i m p o s s i b l e t o a p p l y a s t r e s s o^j w i t h o u t a l s o a p p l y i n g t h e o p p o s i t e s t r e s s °ji' a n d s o w e c a n o n l v p h y s i c a l l y measure t h e i r sum. I t i s c o n v e n t i o n a l (and can be shown t o be n e c e s s a r y ) t o s e t d . = d., . as a r e s u l t o f t h i s , and 13k l k j th u s t h e number o f independent c o e f f i c i e n t s i s r e d u c e d from 27 t o 18. A f u r t h e r s i m p l i f i c a t i o n o f t h e m a t h e m a t i c a l r e p r e s e n t a t i o n i s a c h i e v e d by r e d u c - i n g t h e number o f s u b s c r i p t s from 3 t o 2 a c c o r d i n g t o t h e f o l l o w i n g c o n v e n t i o n : t h e f i r s t s u b s c r i p t remains t h e same, b ut t h e second and t h i r d a r e r e p l a c e d by a s i n g l e d i g i t a c c o r d i n g t o t h e r u l e : 1 1 + 1 2 3 + 4 2 2 + 2 31, 13 + 5 33 + 3 21, 1 2 + 6 F a c t o r s o f 1/2 a r e a l s o i n t r o d u c e d i n t o t h e d ^ j where j = 4,5,6. F o r c o n s i s t e n c y , we make e q u i v a l e n t changes t o t h e s t r e s s t e n s o r t o g e t c l °6 °5 Or 0 o a,, i . e . a. . + a. , j = 1,...,6 b 2 4 3 , k 3 °5 °4 °3 In t h i s new 2 - s u b s c r i p t n o t a t i o n , t h e p i e z o e l e c t r i c m o d u l i t a k e t h e form o f t h e m a t r i x : 67 A21 '12 U d 3 1 l13 e t c . l16 *26 l36—I The r e l a t i o n B.2 i s now w r i t t e n d. .a. ! D 3 i = 1,2,3 j = 1,2,...,6 In t h e experi m e n t s d e a l t w i t h h e r e , t h e .converse p i e z o e l e c t r i c e f f e c t i s u s e d , and we note t h a t i t can be shown (Nye, p. 115) t h a t t h e c o e f f i c i e n t s r e l a t i n g s t r e s s and p o l a r i z a t i o n i n t h e d i r e c t e f f e c t a r e t h e same as t h o s e r e l a t i n g f i e l d and s t r a i n i n t h e con v e r s e e f f e c t . Thus the analogue o f eq. B.2 i s d. .. E. i j k i T h i s t o o can be e x p r e s s e d i n m a t r i x n o t a t i o n by e f f e c t i n g t h e f o l l o w i n g s u b s t i - t u t i o n f o r t h e s t r a i n components: e l l e12 e13 e l l'z< e21 e 2 2 e23 - v e 2 _ e 3 1 e 3 2 e 3 3 _ e 3 - and t h e n w r i t i n g d. . E. I D 1 i = 1,2,3 j = 1,2,...,6 i i ) The " V e r n i t r o n " Lead Z i r c o n i u m T i t a n a t e P i e z o e l e c t r i c Element The p i e z o e l e c t r i c phase s h i f t i n g element used i n t h e s e e x p e r i m e n t s was a p o l e d , t h i c k n e s s - e x p a n d i n g s i n t e r e d c e r a m i c d i s c o f l e a d z i r c o n i u m t i t a n a t e (PZT) w i t h dimensions and c o n v e n t i o n a l axes as shown i n F i g . B.2. The p i e z o e l e c t r i c c o e f f i c i e n t g i v e n i n the V e r n i t r o n spec s h e e t i s d ^ = 285 pm/V ± 20%, which r e l a t e s the s t r a i n i n the 3 d i r e c t i o n t o the a p p l i e d v o l t a g e i n t h e 68 ( i _ J O o - LO c E E IT) o c (U- i <1> C C- LU I v FIGURE B - 3 Response of PZT to an applied voltage: The PZT element was inserted; into • one arm' of;., a • •! Michelson interferometer, and the voltage required •>._ to displace the fringes by one period aas measured. : 69 3 d i r e c t i o n . The v a l u e o b t a i n e d i n t h i s l a b u s i n g a M i c h e l s o n I n t e r f e r o m e t e r i s d 3 3 = 219 pm/V ± 2% ( F i g . B . 2 ) . A lower l i m i t on t h e time r e q u i r e d t o s t e p t h e v o l t a g e on the PZT and t h e r e b y s t u d y t h e c o u p l i n g o f the w r i t i n g beams i s s e t by t h e r e l a x a t i o n t i m e o f t h e PZT ceramic ( F i g . B . 3 ) . That i s , a change i n v o l t a g e a c r o s s the ce r a m i c does not r e s u l t i n t h e ceramic i m m e d i a t e l y assuming i t s new d i m e n s i o n s , b u t r a t h e r i n i t i a t e s an a p p r o x i m a t e l y e x p o n e n t i a l approach t o t h e new d i m e n s i o n . T h i s i s somewhat i n c o n v e n i e n t a s , d u r i n g t h e r e l a t i v e l y l o n g time r e q u i r e d t o s t e p t h e v o l t a g e t h r o u g h t h e r e q u i r e d v o l t a g e range, t h e hologram b e i n g s t u d i e d changes due t o t h e c o n t i n u e d w r i t i n g o f t h e g r a d u a l l y s h i f t i n g f r i n g e p a t t e r n . The o b v i o u s s o l u t i o n o f r e d u c i n g the beam i n t e n s i t y t o i n h i b i t w r i t i n g d u r i n g m e a s u r i n g r a n i n t o problems w i t h t h e r m o e l e c t r i c e f f e c t s , as mentioned e l s e - where. 70 APPENDIX C M i c h e l s o n I n t e r f e r o m e t e r (PZT c a l i b r a t i o n , bench v i b r a t i o n ) In t h i s work a M i c h e l s o n I n t e r f e r o m e t e r was u s e d b o t h t o c a l i b r a t e t h e PZT phase s h i f t e r and t o check on the m e c h a n i c a l s t a b i l i t y o f t h e o p t i c a l bench. F i g u r e C . l i s a s c h e m a t i c diagram o f the d e v i c e geometry. I f |D^-D2I i s l e s s t h a n the t e m p o r a l coherence l e n g t h of the l a s e r , t h e two beams formed by t h e beam s p l i t t e r (BS) and brought back t o g e t h e r by t h e m i r r o r s (M^, M 2) w i l l form a d i f f r a c t i o n p a t t e r n . V a r i a t i o n i n t h e d i s t a n c e | D j - D ^ w i l l cause t h e p a t t e r n f r i n g e s t o move as t h e i n t e r f e r e n c e becomes a l t e r n a t e l y c o n s t r u c t i v e and d e s t r u c t i v e a t any f i x e d p o i n t on the s c r e e n . I f one of the m i r r o r s i s mounted on a PZT s h i f t e r , t h e n v a r y i n g t h e v o l t a g e on t h e s h i f t e r and c o u n t i n g t h e p a s - sage o f f r i n g e s on t h e s c r e e n w i l l p r o v i d e a means o f c a l i b r a t i n g t h e p i e z o e l e c - t r i c c o n s t a n t o f the s h i f t e r (see Appendix B ) . S e t t i n g the i n t e r f e r o m e t e r up on t h e o p t i c a l bench and o b s e r v i n g t h e v a r i a t i o n o f f r i n g e p o s i t i o n p r o v i d e s a means t o measure t h e degree o f m e c h a n i c a l i s o l a t i o n o f f e r e d by t h e bench. Three t y p e s o f v a r i a t i o n were noticed'. The f i r s t was c l e a r l y due t o b u i l d i n g v i b r a t i o n ( o c c u p a n t s , machinery) b e i n g t r a n s - m i t t e d t o t h e o p t i c a l componts by the bench, and was o f t e n accompanied by s lam- ming d o o r s , e t c . T h i s was m i n i m i z e d by c o n d u c t i n g e x p e r i m e n t s a t n i g h t . The second s o u r c e of n o i s e was random v a r i a t i o n s i n t h e d i f f e r e n t i a l p a t h l e n g t h c a u s e d by a i r c u r r e n t s moving i n the s p l i t beam p a t h s and c a u s i n g v a r i a t i o n s i n the phase p a t h l e n g t h s . T h i s was m i n i m i z e d by c o v e r i n g t h e bench w i t h a p l e x i - g l a s s greenhouse and f u r t h e r i s o l a t i n g t h e beams i n c a r d b o a r d t u b i n g . The t h i r d type o f v a r i a t i o n i n p a t h l e n g t h c o n s i s t e d of a slow monotonic i n c r e a s e or d e c r e a s e i n t h e d i f f e r e n t i a l p a t h l e n g t h w i t h a speed o f about one f r i n g e p e r minute o r so and g r a d u a l l y d e c a y i n g o v e r a p e r i o d of days t o a f r i n g e p e r few FIGURE C.1 S c h e m a t i c o f M i c h e l s o n i n t e r f e r o m e t e r as u s e d t o d e t e c t bench v i b r a t i o n and t o c a l i b r a t e p i e z o e l e c t r i c e l e m e n t . 72 h o u r s . T h i s was i mmediately e v i d e n t e v e r y time the equipment was s e t up, and was a t t r i b u t e d t o a g r a d u a l r e l a x a t i o n o f the m i r r o r h o l d e r s a f t e r t h e o p t i c s were p r o p e r l y a l i g n e d . T h i s was t r e a t e d by r e d e s i g n i n g the PZT h o l d e r and a l s o s i m p l y by w a i t i n g a few days a f t e r a l i g n m e n t b e f o r e t a k i n g any s e r i o u s measurements. 73 APPENDIX D A p p l i c a t i o n s o f Holography The s t o r a g e o f holograms i n f e r r o e l e c t r i c c r y s t a l s i s p a r t o f t h e l a r g e r f i e l d o f modern o p t i c a l t e c h n o l o g y c u r r e n t l y u n d e r g o i n g r a p i d d e v e l o p - ment i n a p p l i c a t i o n s c o n n e c t e d t o g e n e r a l i n f o r m a t i o n h a n d l i n g . One example o f the i n t e r f a c i n g o f h o l o g r a p h y t o o t h e r new d e v i c e s i n v o l v e s the r e a l - t i m e r e c o n s t r u c t i o n o f t h r e e - d i m e n s i o n a l images u s i n g h o l o g r a p h i c t e c h n i q u e s t o mix wave components t h a t have p a s s e d t h r o u g h l o n g f i b r e s (Tamir, p . 309). T h i s would make use o f b o t h t h e h i g h d a t a r a t e a t t a i n a b l e a t o p t i c a l f r e q u e n c i e s and t h e r e l a t i v e immunity o f o p t i c a l s i g n a l s t o e l e c t r o m a g n e t i c i n t e r f e r e n c e . F u r t h e r a p p l i c a t i o n s o f h o l o g r a p h i c t e c h n i q u e s w i l l be d i s c u s s e d and an example o f a h o l o g r a p h i c computer memory g i v e n . D e s i r a b l e q u a l i t i e s i n tw o - d i m e n s i o n a l d i s p l a y s ( e . g . 35mm s l i d e s ) a r e h i g h r e s o l u t i o n , h i g h image b r i g h t n e s s , s t o r e d image d u r a b i l i t y , s i m p l i f i e d p r o c e s s i n g and re d u c e d c o s t o f c o p i e s . I t has been shown ( C l a y ; i n C a u l f i e l d ) t h a t 35mm s l i d e s can be c o n s i d e r a b l y improved upon u s i n g focussed-image h o l o - grams, where an image o f the o b j e c t i s formed by a l e n s on a r e c o r d i n g medium and mixed w i t h a r e f e r e n c e beam. Though the i m a g i n a t i o n can come up w i t h many p o s s i b l e uses f o r t h r e e - d i m e n s i o n a l d i s p l a y s , many o f them appear not t o be p r a c t i c a l when c a r e f u l l y s t u d i e d . I t seems t h a t the major p o t e n t i a l use l i e s i n t h e f i e l d o f a r t i s t i c endeavour, but a r t i s t s have been slow t o adopt t h i s new t e c h n i q u e , presumably because o f i t s t e c h n i c a l n a t u r e . A l l o f the c l a s s i c a l i n t e r f e r o m e t r y setups have t h e i r a n a l o g s i n h o l o g r a p h i c i n t e r f e r o m e t r y , w i t h the l a t t e r a l l o w i n g g r e a t l y expanded c a p a b i l i - t i e s ( B r a n d t ; i n C a u l f i e l d ) . These i n c l u d e the use of much l a r g e r a p e r t u r e . 74 a p p l i c a t i o n t o s t u d i e s i n v o l v i n g random or d i f f u s e w a v e f r o n t s , and p o t e n t i a l f o r m u l t i p l e exposure t e c h n i q u e s ( i n v i b r a t i o n s t u d i e s , f o r i n s t a n c e ) . P a t t e r n and c h a r a c t e r r e c o g n i t i o n a r e c u r r e n t l y main a r e a s of a p p l i e d o p t i c a l d a t a p r o c e s s i n g . The aim h e r e i s t o determine the p r e s e n c e and/or l o c a t i o n o f a r e f e r e n c e p a t t e r n i n an i n p u t image by examining the degree of c o r r e l a t i o n between an i n p u t and a r e f e r e n c e f u n c t i o n . The most common system p e r f o r m s the c o r r e l a t i o n by m u l t i p l y i n g the F o u r i e r t r a n s f o r m s o f the i n p u t and r e f e r e n c e f u n c t i o n s , where t h e l a t t e r i s s t o r e d as t h e c o n j u g a t e F o u r i e r t r a n s - form o f t h e complex r e f e r e n c e f u n c t i o n (which i s , i n e f f e c t , a h o l o g r a m ) . Image p r o c e s s i n g ( g e n e r a l l y , t h e m a n i p u l a t i o n o f m u l t i d i m e n s i o n a l s i g n a l s ) i n v o l v e s image enhancement, i n f o r m a t i o n e x t r a c t i o n , e f f i c i e n t c o d i n g , p a t t e r n r e c o g n i t i o n and computer g r a p h i c s as t h e y r e l a t e t o i n t e r p r e t i n g a e r i a l p h otographs and maps, m e d i c a l , x - r a y s , t e l e v i s i o n images, e t c . O p t i c a l t e c h - n i q u e s a l t e r h i g h e r i n f o r m a t i o n c o n t e n t h a n d l i n g a b i l i t y , and may i n c l u d e t h e use of computer g e n e r a t e d holograms, a s t u d y i n i t s e l f (Catheg; Ch. 9 ) . Holography was o r i g i n a l l y i n v e n t e d t o enhance e l e c t r o n m i c r o s c o p e images. A l t h o u g h the b u l k of h o l o g r a p h i c r e s e a r c h has moved t o o t h e r a r e a s , l i m i t e d s u c c e s s has been a c h i e v e d w i t h h o l o g r a p h i c m i c r o s c o p y (Cox; i n C a u l f i e l d ) . Because a c o n v e n t i o n a l m i c r o s c o p e i s d e s i g n e d t o have a h i g h t r a n s v e r s e m a g n i f i c a t i o n a t the expense o f the depth o f f i e l d , i t i s o f t e n d i f f i c u l t t o see f i n e d e t a i l t h r o u g h o u t a l a r g e volume. T h i s can be overcome t o some e x t e n t by making a hologram of an image a t low m a g n i f i c a t i o n (and good depth of f i e l d ) and l a t e r v i e w i n g i t a t h i g h m a g n i f i c a t i o n w i t h a c o n v e n t i o n a l m i c r o s c o p e . Time v a r y i n g p r o c e s s e s ( e . g . c r y s t a l growth) can be d e t a i l e d by t a k i n g two s u c c e s s i v e holograms and s u p e r p o s i n g them d u r i n g c o n s t r u c t i o n . A h o l o g r a p h i c o p t i c a l element (HOE) can be d e s i g n e d t o t r a n s f o r m any s i n g l e , e n t e r i n g wavefront i n t o any o t h e r s i n g l e , e x i t i n g w a v e f r o n t . They o f f e r t h e a b i l i t y t o p r o v i d e u n u s ual g e o m e t r i c a l c o n f i g u r a t i o n s o r s p e c i a l 75 s p e c t r a l c h a r a c t e r i s t i c s , but t h i s i s u s u a l l y accompanied by t h e a d d i t i o n o f a l a r g e amount o f a b e r r a t i o n t o t h e system (e.g. a s t i g m a t i s m and coma) t o such an e x t e n t t h a t "one s h o u l d r e s o r t t o the use of a HOE o n l y when i t i s i m p o s s i b l e t o use c o n v e n t i o n a l l e n s e s and m i r r o r s " ( C l o s e ; i n C a u l f i e l d ) . Examples a r e elements which must conform t o an u n u s u a l shape such as a helmet v i s o r , d e s i g n o f v e r y l a r g e elements ( i n t h e s p i r i t o f a F r e s n e l l e n s ) , and where l a r g e s u r f a c e s of narrow s p e c t r a l r e f l e c t i v i t y must be used (as i n a i r c r a f t "heads up" d i s p l a y s ) . The c o n s i d e r a b l e number o f o t h e r a p p l i c a t i o n s o f h o l o g r a p h i c t e c h - n i q u e s i n c l u d e : s p e c t r o s c o p y , e s p e c i a l l y a t h i g h speeds; c o n t o u r i n g methods (2-D maps o f 3-D o b j e c t s ) ; m u l t i p l e image g e n e r a t i o n ( e . g . f o r image r e c o r d i n g o r p a r a l l e l o p t i c a l p r o c e s s i n g ) and p a r t i c l e s i z e measurements. F i n a l l y , we d e a l w i t h t h e a p p l i c a t i o n which has r e c e i v e d , most a t t e n - t i o n i n a p p l i e d 'research; d i g i t a l d a t a s t o r a g e . The need f o r l a r g e r , f a s t e r , and l e s s e x p e n s i v e memories f o r data s t o r a g e has l e d t o t h e development o f h o l o g r a p h i c memories i n an attempt t o c a p i t a l i z e on t h e p o t e n t i a l o f o p t i c a l d e v i c e s . We note i n t h i s r e g a r d the r e c e n t d e s c r i p t i o n o f an o p t i c a l s u b p i c o - second gate ( M a r c a t e l l i , 1980). F i g u r e s D l and D2 g i v e p e r f o r m a n c e d a t a on v a r i o u s d e v i c e s i n use as compared t o h o l o g r a p h i c d e v i c e s . I t i s seen t h a t t h e t r a d e - o f f between a c c e s s time and s t o r a g e c a p a c i t y , and between a c c e s s t i m e and memory c o s t i s l e s s s e v e r e f o r h o l o g r a p h i c d e v i c e s than t h e o t h e r s l i s t e d . One can l i s t f o u r b r o a d c a t e g o r i e s o f memory; a r c h i v a l s t o r a g e ( l a r g e , amounts o f da t a o c c a s i o n a l l y a c c e s s e d ) , r e a d - m o s t l y s t o r a g e ( s i m i l a r t o a r c h i v a l but cap- a b l e o f b e i n g o c c a s i o n a l l y a l t e r e d ) , f a s t r e c o r d i n g s t o r a g e ( f o r r a p i d s t o r a g e o f r a p i d l y accumulated d a t a t o be r e a d l a t e r ) , and f a s t r e a d - w r i t e - e r a s e memory (as i n most computer memories). A p o t e n t i a l l y h i g h c a p c i t y , f a s t random a c c e s s h o l o g r a p h i c memory o f the t y p e t o be d e s c r i b e d may r e p r e s e n t a s o l u t i o n t o t h e s e v a r i e d needs, o r 76 occupy a unique position in a hierarchical set of solutions. I n i t i a l design considerations ( H i l l , 1976) indicated that holographic memories would have these five basic features: i) they would be Fourier transform holograms (as opposed to direct images) in order to protect against localized loss of data due to medium imperfections or surface dust; i i ) they would be in a 2-D page format, as the 3-D imaging capability of holograms offers no advantages; i i i ) the information would be in binary code form as opposed to p i c t o r i a l to allow for speed and ease of page composition and bit detection; iv) analysis shows that thick phase holograms would be much more e f f i - cient than absorption or thin holograms; v) moving mechanical parts would be eliminated to improve r e l i a b i l i t y . A schematic of a 3-D storage system using a ferroelectric crystal i s presented in Fig. D.3. The Bragg angle selectivity of such a medium i s used to enable the superposition of multiple holograms (up to 500), so the address of a bit i s XY$. The main optical components are the light source, beam deflectors, page composer, recording medium, and detector array. The light source should provide intense (~1 Watt), collimated, coherent light pulsed or gated at about 1 MHz and in the blue or green part of the spectrum to take advantage of the A - 3 relationship between storage density and wavelength. The argon-ion gas laser meets these requirements, although i t displays a low conversion efficiency of electrical power to optical power. Fast and accurate beam deflectors are needed to position the laser beam for reading, writing and erasing operations. Acousto-optic and electro- optic deflectors are the main candidates, with the latter being faster but suffering, light loss through surfaces and having a high cost. Because 77 e l e c t r o - o p t i c c o e f f i c i e n t s are too s m a l l t o produce l a r g e d e v i a t i o n s t h e y a r e o f t e n c a s c a d e d t o i n c r e a s e the e f f e c t . A d i g i t a l e l e c t r o - o p t i c d e f l e c t o r may be c o n s t r u c t e d by c a s c a d i n g m d e f l e c t o r s t o g i v e 2 m d e f l e c t i o n a n g l e s , and random a c c e s s t i m e s of 0.8 ys. have been a c h i e v e d . The d a t a i n p u t d e v i c e i s l o c a t e d i n the o b j e c t beam p a t h and c o n s i s t s o f an a r r a y o f "open o r s h u t " b i n a r y a p e r t u r e s which i n t e r f e r e w i t h t h e r e f e r e n c e beam t o form a d a t a m a t r i x a t t h e d e t e c t o r a r r a y . The r e q u i r e m e n t s o f t h i s page composer a r e t h a t i t e x h i b i t a h i g h frame speed, h i g h r e s o l u t i o n , a h i g h c o n t r a s t r a t i o , s t a b i l i t y , u n i f o r m i t y , and have f u l l page a d d r e s s i n g and a s u f f i c i e n t l y l a r g e a p e r t u r e t o accomodate a l a r g e number o f b i t s p e r page. A u s e f u l b u t s l o w l i q u i d c r y s t a l page composer has been c o n s t r u c t e d by RCA ( L a b r u n i e e t a l . , 1974). Among a number o f a l t e r n a t e p o s s i b i l i t i e s i s t h e l e a d lanthanum z i r c o n a t e t i t a n a t e (PLZT) b l o c k d a t a composer, which i s f a s t e r than the l i q u i d c r y s t a l v e r s i o n . The r e c o r d i n g m a t e r i a l i s a c e n t r a l component and d e t e r m i n e s t o a l a r g e e x t e n t how f l e x i b l e and e f f i c i e n t the e n t i r e system w i l l be. The c h a r a c t e r i s t i c s o f t h e i d e a l medium are a h i g h s e n s i t i v i t y and l a r g e d i f f r a c t i o n e f f i c i e n c y t o make maximum use o f a v a i l a b l e l i g h t , l o n g l i f e t i m e and n o n v o l a t i l e s t o r a g e , h i g h r e s o l u t i o n t o p r o v i d e h i g h d e n s i t y s t o r a g e , n o n d e s t r u c t i v e r e a d o u t , and an e r a s e - r e w r i t e c a p a b i l i t y . To date no m a t e r i a l f u l f i l l s a l l o f t h e s e r e q u i r e m e n t s . The a r r a y of p h o t o d e t e c t o r s used t o c o n v e r t t h e h o l o g r a p h i c a l l y r e c o n - s t r u c t e d d a t a p a t t e r n i n t o an e l e c t r i c a l s i g n a l would c o n s i s t o f one s e n s i n g p h o t o d i o d e or p h o t o t r a n s i s t o r and one or two a d d r e s s i n g s w i t c h e s f o r each b i t . These p h o t o d i o d e s must e x h i b i t h i g h d e t e c t i v i t y and be a b l e t o s t o r e i n c i d e n t o p t i c a l energy f o r b r i e f p e r i o d s t o a l l o w readout by words. The a r r a y must a l l o w complete random a c c e s s t o a l l words. The t e c h n o l o g y t o c o n s t r u c t l a r g e d e f e c t - 78 f r e e a r r a y s o f t h i s type a l r e a d y e x i s t s i n t h e form o f s i l i c o n - d i o d e - a r r a y camera t u b e s . A l t h o u g h t h e t h e o r e t i c a l s t o r a g e d e n s i t y f o r 2-D systems i s o f t h e o r d e r o f X - 2 , and f o r 3-D o f t h e o r d e r ( n / X ) ^ , p r a c t i c a l l i m i t a t i o n s imposed upon t h e system by o t h e r f a c t o r s d e c r e a s e t h i s l i m i t . F o r i n s t a n c e , o p t i c a l a p e r t u r e e f f e c t s p u t a lower l i m i t on s p o t s i z e , d e t e c t o r n o i s e l i m i t s t h e number o f b i t s p e r page, beam d e f l e c t o r r e s o l u t i o n l i m i t s t h e number o f XY a d d r e s s e s p e r page, and t h e medium r e c o r d i n g range, image c r o s s t a l k and g r a n u - l a r i t y a l l l i m i t t h e number of holograms t h a t can be s u p e r p o s e d . S e v e r a l o p t i c a l memories a r e c o m m e r i c a l l y a v a i l a b l e . The f i r s t com- m e r i c a l nonmechanical o p t i c a l memory was the Megafetch d a t a p r o c e s s o r i n t r o - duced by 3M i n 1974. I t i s a 50 Mbit r e a d - o n l y system c a p a b l e o f 15 Mbaud d a t a r a t e s , and uses a s e t of s e p a r a t e 2-D h o l o g r a p h i c d a t a p l a t e s . 79 10' — i r MAGNETIC TAPE I I I OPTICAL BIT BY BIT » I3M 1360 « HOLOSCAN GRUMMAN MASSTAPE » 3AMPEX TEM ' O UNICOW 10- 10" 10" io- CCD CORES MOS BIPOLAR J L MOVABLE HEAD DISKS FIXED HEAD DISKS DRUMS © MEGAFETCH HOLOGRAPHIC J L J L 10' 10" 10 8 10' 10' STORAGE CAPACITY (bits) 10 FIGURE D-l(a) A comparison of various memory types i n terms of access time and cost per b i t of stored data. (from C a u l f i e l d ) 80 10* OPTICAL BIT BY BIT MAGNETIC TAPE T r 1CT 10- 10~ 10" HOLOGRAPHIC J L MOVABLE HEAD DISKS FIXED HEAD; DISKS DRUMS CCD CORES MOS BIPOLAR l i l t io- 10" io- 10" to -1 MEMORY COST (cents/bit] 10 10' FIGURE D-l(b) A comparison of various memory types i n terms of access time and storage capacity, (from C a u l f i e l d ) POLARIZATION COUIMATING SENSITIVE l t " S HIGK-EFflCIENCY GRATING FIGURE D - 2 A holographic o p t i c a l memory system using three-dimensional (volume) storage (from Caulfield) 82 APPENDIX E A number o f d a t a r e d u c t i o n and s i m u l a t i o n programs were u s e d i n t h i s work. Among t h e s e was the hologram w r i t i n g and r e a d / e r a s e program w r i t t e n by Moharam and m o d i f i e d by e l G u i b a l y f o r t h e case o f a non-zero b u t c o n s t a n t phase s h i f t . T h i s was s l i g h t l y a l t e r e d and used i n t h i s work t o o b t a i n t h e d a t a f o r F i g u r e 8.7 ( s o l i d l i n e s ) , showing the time development o f t h e g r a t i n g c u r v a t u r e . Two o f the r e m a i n i n g programs a r e g i v e n h e r e : (1) "ZERO" i s u s e d t o o b t a i n t h e dashed l i n e s i n F i g u r e 8.7, which shows t h e d i s p l a c e m e n t of t h e s t r a i g h t l i n e g r a t i n g t h a t would produce t h e same c o u p l i n g as the c o r r e s p o n d i n g c u r v e d g r a t i n g . (2) "EXPCUP" i s used t o generate t h e c u r v e s o f F i g u r e 8.8, which shows t h e s o r t o f c o u p l i n g c u r v e s which would a r i s e f o r t h e case o f a non- z e r o i n i t i a l phase (|> f o l l o w e d by a time v a r i a t i o n i n t h e phase (d<j>/dt d d s £ 0 ) . ("SIMDAT" i s a f i l e h o l d i n g t h e c o u p l i n g and d i f f r a c t i o n d a t a measured from t h e e x p e r i m e n t a l r u n shown i n F i g u r e 8.2.) 1 c 2 C 3 C F I L E : "ZERO" 4 C 5 C 6 C ' Z E R O ' INPUTS SELECTED DATA FROM ' H 0 L 0 3 ' (10 TIME 7 C SAMPLES, 12 Z-SEGMENTS) DESCRIBING THE PHASE (PHIHG) 8 C AND AMPLITUDE (AMP) OF THE BENT HOLOGRAM GRATING, 9 C AND MOVES THE GRATING OVER TO A POSITION STRADDLING 10 C THE FRINGE PATTERN SO THAT THE COUPLING IS NULLED 11 C ( I . E . , THE EXITING R AND S BEAMS ARE SET EQUAL. ) 12 C AMPCON IS A CONSTANT CONVERTING THE FIELD AMPLITUDE 13 C OUTPUT FROM ' H 0 L 0 3 ' TO INDEX VARIATION AMPLITUDE 1-1 C FOR USE IN THE SUBROUTINE ' T H R U ' . 15 C 'DO 10' >> TIME STEPS 1G 17 REAL T IME(10 ) . AMP(12, 10) . PH IHG(12 ,10 ) , AMPV(12) . PHIHGV(12) 18 AMPC0N=5.2E-6 19 CALL MODEL(AMPCON,TIME,PHIHG,AMP) 20 WRITE(5,1) 21 1 FORMAT(' TIME ' . ' EFFECTIVE S H I F T ' ) 22 DO 10 1=1,10 23 DO 20 J=1.12 24 A M P V ( J ) ~ A M P ( J , I ) 25 PHIHGV(.J) = PHIHG( J , I ) 26 20 CONTINUE 27 CALL EOUAL(PHIHGV,AMPV.PHIEO) 23 PHIEQ0=PHIEQ*45. /ATAN( 1 .0) 29 WRITE(5,2) TI ME( I ) ,PHIEOD 30 2 FORMAT(' ' , F 4 . 2 , F 1 2 . 2 ) 31 10 CONTINUE 32 STOP 33 END 34 35 36 37 c+ + * + * + * + **-r + *t* + *** + + + ********** + * + *t + ** + + ** + ** + + 38 SUBROUTINE MODEL(AMPCON,TIME,PHIHG,AMP) 39 40 C MODEL READS THE STORED OUTPUT FROM H0L03 IN TWO 12X11 41 C MATRICES ( A P . F ) WHICH GIVE INTERNAL FIELD AMPLITUDE 42 C AND PHASE SHIFT , AND MULTIPLIES THEM BY CONSTANTS 43 C "CONV" TO CONVERT THE PHASE TO RADIANS AND "AMPCON" 44 C TO CONVERT THE FIELD AMPLITUDE TO AN INDEX AMPLITUDE. 45 C "AMPCON" MUST BE GIVEN IN THE CALL STATEMENT. 46 47 REAL T I M E ( 1 0 ) , P H I H G ( 1 2 , 1 0 ) , A M P ( 1 2 . 1 0 ) 48 DO 10 I - 1 , 10 49 READ(5 ,51) T I M E ( I ) , ( P H I H G ( J , I ) . d = 1 . 1 2 ) 50 51 F 0 R M A T ( F 7 . 3 , 1 2 ( F 8 . 3 ) ) 51 10 CONTINUE 52 DO 20 1=1,10 53 READ(5 ,51) T I M E ( I ) , ( A M P ( d , I ) , J * 1 . 1 2 ) 54 20 CONTINUE 00 55 CONV=ATAN(1 . ) / 45 . w 56 DO 30 I 3 1 , 10 57 DO 40 J=1 ,12 • 58 AMP(J , I )=AMP( j , I ) *AMPCON 59 PHIHG(U, I )=PHIHG(J, I ) *CONV 60 40 CONTINUE 61 30 CONTINUE 62 RETURN 63 END 64 65 67 SUBROUTINE EQUAL(PHIHG,AMP,PHIEO) G8 69 C EQUAL DOES A SEARCH FOR THE PHASE ANGLE "PHIEQ" WHICH 70 C WOULD RESULT IN NO COUPLING IF THE HOLOGRAM GRATING 71 C WAS A SIMPLE LINEAR ONE. I . E . , IT IS THE " E F F E C T I V E " 72 C PHASE SHIFT OF THE GRATING MEASURED EXPERIMENTALLY 73 C WHEN THE PZT VOLTAGE IS ADJUSTED FOR ZERO COUPLING. 74 C "DO 10" >>SEARCH STEPS; 'DO 20" >> SHIFTS GRATING BY PHIEQ 75 76 REAL PHIHG(12) ,AMP(12) ,PHIHGT(12) 7 7 PHIGAP = (PHIHG(12)-PHIHG( 1 ) ) / 2 . 78 PHIEQ=PHIHG(1)+PHIGAP 79. DO 10 1 = 1,13 80 DO 20 0=1.12 81 PHIHGT (<J) =PHIHG( J ) - PHI EQ 82 20 CONTINUE 83 CALL THRU(PHIHGT,AMP,RINT , SI NT) 84" PHIGAP=PHIGAP/2. 85 I F ( R I N T . L T . 1 . 0 ) PHIEQ=PHIEQ-PHIGAP 86 I F ( R I N T . G T . 1 . 0 ) PHIEQ=PHIEQ+PHIGAP 87 I F ( A B S ( R I N T - 1 . 0 ) . L E . 0 . 0 0 0 1 ) GO TO 11 88 10 CONTINUE 89 11 RETURN 90 END 91 92 g3 £t*************+++*********+*+***********+***** 94 SUBROUTINE THRU(PHIHG,AMP,RINT,SINT) 95 96 C "THRU" SENDS TWO BEAMS OF EQUAL AND UNIT STRENGTH AT + / - 19 97 C DEGREES TO THE NORMAL THRU THE 12 LAYERS AND GIVES THEIR 98 C INTENSITIES AT THE OUTPUT SIDE ( R I N T , S I N T ) . THE WAVELENGTH 99 C IS 5 1 5 . 5 NM AND LAYER THICKNESS IS DZ. (SEE MOHARAM P74 FOR MATH) 100 101 102 REAL AMP(12) ,PHIHG(12) 103 COMPLEX R . S . R S T O R E . C Z T . S Z T 104 WL=514.5E-7 105 D Z = . 0 6 8 / 1 2 . 106 PI=4 . * A T A N ( 1 . 0 ) 107 T H E T A = 1 9 . * P I / 1 8 0 . 108 109 R = C M P L X ( 1 . , 0 . ) 110 S = C M P L X ( 1 . , 0 . ) 111 DO 10 1=1,12 112 RSTORE =R 113 C=PI *AMP( I ) /WL/COS(THETA) 114 CZT=COS(C*DZ) 115 SZT=SIN(C+DZ) 116 R = R * C Z T - ( 0 . , 1 . ) + S * S Z T * 117 & C M P L X ( C O S ( P H I H G ( I ) ) , - S I N ( P H I H G ( I ) ) ) 118 S = S * C Z T - ( 0 . , 1 . ) *RSTORE*SZT* 119 & CMPLX(COS(PHIHG( I ) ) ,S IN (PH IHG( I ) ) ) 120 10 CONTINUE 121 122 123 124 125 of F i l e 1 2 c* 3 C 4 c* 5 o 7 8 9 10 11 12 13 14 15 1G c 17 c 18 19 20 21 22 5 1 23 24 25 26 10 27 28 1 29 30 31 32 33 34 35 36 C 37 C ' 38 39 40 4 1 42 10 43 44 1 45 46 47 48 49 50 51 52 c 53 c 54 c RINT=REAL(R*CONdG(R)) SINT=REAL(S*CONUG(S)) RETURN END FILE:"EXPCUP" >> TO REDUCE 'SIMDAT' PARAMETERS. REAL RCUP(8),SCUP(8),RDIF(8),SDIF(8) CALL INPUT(RCUP,SCUP,RDIF,SDIF) CALL ANDAT(SDIF.RCUP) CALL SIMDAT(SDIF) STOP END SUBROUTINE INPUT(RCUP,SCUP,RDIF,SDIF) INPUT READS DATA FROM FILE 'SIMDAT'. NORMALIZES THE FIGURES, AND PUTS THEM INTO THE PROPER VECTORS. REAL RCUP(8),SCUP(8),RDIF(8),SDIF(8) READ(5.51 ) (RCUP(I ) ,1 = 1 .8),(SCUP(I),1=1,8), 1 ( R D I F ( I ) , 1 = 1 , 8 ) , ( S D I F ( I ) , 1 = 1,8) FORMAT (8F5.1) DO 10 1=1,8 RCUP(I)=RCUP(I)*2./(RCUP(I)+SCUP(I)) S D I F ( I ) = S D I F ( I ) / ( S D I F ( I ) + R D I F ( I ) ) CONTINUE WRITE (6, 1) ( S D I F ( I ) , I = 1,8),(RCUP(I),I = 1,-8) FORMAT( ' ','SDIF= '.8F7.3./.' RCUP= '.8F7.3) RETURN END SUBROUTINE ANDAT(SDIF,RCUP) "ANDAT" CALCULATES THE PHASE SHIFT FROM THE NORMALIZED COUPLING AND DIFFRACTION EFFICIENCY FOR EACH DATA POINT. REAL SDIF(8),RCUP(8),PHI(8) DO 10 1=2,8 PHI(I)=45./ATAN(1.0)*ARSIN((RCUP(I)- 1 . ) / S I N ( 2 . * A R S I N ( S O R T ( S D I F ( I ) ) ) ) ) CONTINUE WRITE(6, 1) ( P H I ( I ) , I = 2 , 8 ) FORMAT (' ',/,'PHI= ****.** '.8F7.2) RETURN END 03 SUBROUTINE SIMDAT(SDIF) "SIMDAT" STARTS WITH THE OBSERVED VALUES FOR THE DIFFRACTION EFFICIENCY AND CALCULATES THE COUPLING FOR VARIOUS VALUES OF PHI(T=0) AND D/DT (PHI)

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
Canada 6 0
United States 6 1
China 3 24
Japan 2 0
City Views Downloads
Burnaby 6 0
Unknown 3 0
Beijing 3 0
Sunnyvale 2 1
Tokyo 2 0
Ashburn 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}

Share

Share to:

Comment

Related Items