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Beam coupling in holograms stored in LiNbO₃ Woods, Randall J. 1981

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BEAM COUPLING IN HOLOGRAMS STORED IN L i N b 0  3  by  Randall  J . Woods  B.Sc,  University of B r i t i s h  Columbia,  M.Sc,  U n i v e r s i t y o f Western  Ontario,  1974 1978  A T H E S I S SUBMITTED IN P A R T I A L F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D  SCIENCE  in  THE FACULTY Department  OF GRADUATE  of Electrical  STUDIES  Engineering  U N I V E R S I T Y OF B R I T I S H COLUMBIA  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g to the required standard  THE.UNIVERSITY  OF B R I T I S H COLUMBIA  July,  ©  1981  R a n d a l l J . Woods  In  presenting  requirements  this thesis f o r an  of  British  it  freely available  agree for  that  for  that  Library  shall  for reference  and  study.  I  for extensive  copying  be  her or  shall  copying of  g r a n t e d by  not  be  cjufruu»j\  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Date  7<n  }2/  fr/  S/  of  LS^^V^O^I^U^ Columbia  make  further this  thesis  head o f  this  It  my  is  thesis  a l l o w e d w i t h o u t my  permission.  Department of  the  representatives. publication  the  University  the  h i s or  f i n a n c i a l gain  the  that  p u r p o s e s may by  f u l f i l m e n t of  degree at  I agree  permission  department or understood  advanced  Columbia,  scholarly  in partial  written  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  applications  has r e c e i v e d  photorefractive  effect  geneous r e d i s t r i b u t i o n distribution. variations  considerable  i n LiNbOgiFe  crystals i n optical  attention during  involves  the past  i n the r e f r a c t i v e  index  fields  allowing  handling  decade.  The  t h e p h o t o l i b e r a t i o n and inhomo-  o f e l e c t r o n s which a r e subsequently  The r e s u l t i n g e l e c t r i c  data  trapped  i n t h e new  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 the formation  o f t h i c k phase  holo-  grams. In  the case  interference the  pattern  holographic  This  phase  called value  o f an e l e m e n t a r y h o l o g r a m between  shift  gives  f o r t h e phase  beam c o u p l i n g  not  been v e r y The  shift,  present  to a redistribution  and t h i s give  rise  to the s h i f t .  the  nature of the writing process, components.  t h e phase  this  p a r t i c u l a r experiment.  Repeated  thermal  experiments  Previous  t h e h o l o g r a m s were w r i t t e n  voltaic  effect  under  t h e dominant p r o c e s s , effect.  The v a l u e  light),  wrote i t .  a  the electron t o measure  consequently i t causes.  t h e phase  and repeated  to obtain  attempts  or studying  shift  i snot  measurements as t h e f u n c t i o n o f time f o r  v a r i a t i o n can be a t t r i b u t e d t o  expansion  and mechanical  showed t h a t  t o t=0 o f t h e w r i t i n g p r o c e s s  due t o t h i s  i n studying  r e s u l t s and have  shift  of this  that  can be u s e d  variation i s a linear  The causes  Because  length  i s useful  the w r i t i n g process,  showed t h a t  simple  o f e n e r g y b e t w e e n t h e two beams,  s e t o f e x p e r i m e n t s h a v e shown t h a t  any  extrapolated  i n turn  i n determining  during  h o l o g r a m was w r i t t e n  optical  the f r i n g e pattern  have not y i e l d e d r e p r o d u c i b l e  useful  i n time  from  M e a s u r e m e n t s o f t h e beam c o u p l i n g  mechanisms which  the  constant  rise  by a  i n t e n s i t y beams o f m o n o c h r o m a t i c  g r a t i n g may b e d i s p l a c e d  beam c o u p l i n g .  transport  two e q u a l  (that written  the value  was a c o n s t a n t  conditions  i t was p o s s i b l e  creep o f the  of the s h i f t  f o r the crystal  w h i c h made t h e b u l k  used.  photo-  to c a l c u l a t e the transport  a r r i v e d a t was 1 3 ± 3 nm.  T A B L E OF CONTENTS  Abstract List  of Variables  List  of  Figures  Acknowledgements  1.  Introduction  2.  Optical  3.  Diffraction  4.  Photorefractive  5.  Photocurrent  6.  Hologram W r i t i n g  7.  Apparatus  8.  Measurements and A n a l y s i s  9.  Holography o f Coupled  Waves  Effect  and Bulk  Photovoltaic  i n LiNbOg  Effect  Crystals  Summary  References  Appendices  A:  LiNbO^ C r y s t a l  B:  PZT  C:  Michelson  D:  Applications  E:  Programs  Data  Phase S h i f t e r Interferometer of Holography  iv  L I S T OF V A R I A B L E S  A  amplitude  BPE  "bulk  d  crystal  d. .,'  of subject  photovoltaic  D,D^  dielectric  E  applied  E  field  real  2  constants  g(x)  volume  g  constant  Q  HG  light ,1  S  J  ( e q . 6.4, 6.6)  generation  rate  of generation  .. / J  transmitted  reference  photocurrent  density  ( e q . 6.6)  , ,J  e<>  conduction  current  (eq.  r  fringe spacing  i n HG  +  density  s e e e q . 4.1  H £ '£  beam i n t e n s i t i e s  (eq.4.1)  grating  AJL  equation  and s u b j e c t  fringe pattern  +  rate  electrons  grating"  k  i SL  of free  beam, i n t e n s i t y  average  Q  e1  i n crystals  photocurrent  I  J  charge  pattern"  "holographic  , Pb  J  element  field  "fringe  R  tensor  field  FP  I  vector,  field  due t o s p a c e  drift  D  f ^ , f  i  form)  13  displacement  electric  virtual  v  E  ( d . ., c o n t r a c t e d  "electro-optic"  s c  E  normalized  effect"  moduli  ^  EO  reference  thickness  piezoelectric  13k  beam w i t h  *  e l e c t r o n mean displacement  vector  free paths of i  ( e q . 5.2-5.4)  i o n (eq.5.3)  6.11)  ( e q . 3.8)  length of c r y s t a l along  c-axis  ( 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 diffusion  transport  modulation rato effective index  length  ( e q . 6.1)  modulation rato  of refraction,  ( e q . 5.11 a , b , c )  n. = / e . .  s e e e q . 3.1 amplitude of index  modulation i n grating  ordinary/extraordinary index impurity  refraction  concentration  scattering  probabilities  probabilities polarization  ( e q . 5.11c)  ( e q . 5.11c) tensor  change i n spontaneous unit  of  polarization  charge  amplitude of reference reference =(r/s)  beam  wave  ( e q . 3.3)  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 q u a d r a t i c EO m a t r i x contracted  linear  s u b j e c t beam  element  EO m a t r i x  amplitude of s i g n a l  element  ( s u b j e c t ) wave  ( e q . 3.3)  photo-ionization cross fringe  element  visibility  s e c t i o n ( e q . 5.7)  (voltage)  unit  vector  (figure 2 . 2 )  unit  vector  (figure 2 . 2 )  vi  til c h a r g e on t h e i linear  EO  i o n ( e q . 5.3)  coefficient  a  crystal  e  permittivity  e_^_.  tensor  T)  grating  0  angle o f incident  0  light  absorption of f i r s t  space  relative, permittivity  diffraction  a s above o u t s i d e  o  constant  0^  angle  K  anisotropy  (strain matrix  beam r e l a t i v e  to crystal  of incidence  constant  o f beams f r o m n o r m a l t o m i r r o r s  optical  wavelength i n s i d e  XQ  optical  wavelength  Uj^  d e g r e e o f c o h e r e n c e b e t w e e n two i n t e r f e r i n g  £  quantum  p  electron  sc  trapped  charge  T  free  4>  phase  ( e q . 3.4)  crystal  i n freespace  beams  density electron  stress  constant  efficiency  crystal  a.. Jk  M^Mg  ( e q . 4.1)  X  p  normal i n s i d e  crystal  coupling  ,p  B)  efficiency  grating  p  i n Appendix  tensor  electron  density  dark  density (a., D  contracted  form)  lifetime  of reference  equivalently,  i n light,  beam w i t h  respect  to subject  <b P  phase s h i f t  due t o BPE  <j>^  phase s h i f t  due t o f i n i t e  w  optical  Q  a n g l e between  diffusion  transport  frequency  interfering  beam, o r  p h a s e m i s m a t c h b e t w e e n HG a n d F P  direction  beams.  of polarization  o f two  length  crystal  vii LIST  OF  FIGURES  Page 2.1  Simple  hologram  2.2  An  2.3  Spatial  2.4  Hologram  3.1  Superposition  4.1  Change i n b i r e f r i n g e n c e  induced with a single  4.2  Chen's p o s t u l a t e d  charge  5.1  P h o t o c u r r e n t i n a LiNbO^  5.2  Asymmetric  5.3  (a)  elementary hologram  and  reading geometries  formed  4  i n a medium o f t h i c k n e s s  "d"  filter  4 7  formation using  amplitude  of fringe pattern  space  division  on h o l o g r a m  7  grating  laser  10  beam i n L i N b o ^  field  16 16  Crystal  21  p h o t o d e l o c a l i z a t i o n model i n LiNbOg  24  Physical  mechanism o f c o l l e c t i v e  relaxation (b)  writing  Franck-Condon  26  f o r Franck-Condon  26  model  Coordinate configuration  diagram  relaxation  7.1  Experimental apparatus  8.1  Beam G e o m e t r y d u r i n g h o l o g r a m  8.2  Sample  8.3  (a,b,c) Normalized data from  8.4  Variation  8.5  Examples o f c o u p l i n g  output from  of phase  36  writing  experimental run  shift  Figure  41 8.2  with exposure  data from  40  Young e t a l . ( 1 9 7 9 )  42 45 46  viii  8.6  Plots of  8.7  Time development  8.8  Simulated  A.l  Table  A.2  The  A. 3  A p p l i c a t i o n of  B. l  S t r e s s a.,  B.2  V e r n i t r o n PZT  piezoelectric  B. 3  Time r e s p o n s e  of  C. l  Michelson  D. l  Comparative performance of holographic  D.2  of  theoretical of  c o u p l i n g from e l Guibaly  indicatrix  on  a  f o r time  v a r y i n g phase s h i f t  field  to  uniaxial  change the  crystal  index  of a LiNbO  crystal  PZT  disc  61  3  crystal  61  65  element  to  step  65  i n voltage  interferometer  access  time  vs.  cost per  (b)  access  time  vs.  storage  of  51  61  (a)  Schematic  49  data  for a positive a  47  grating curvature  c o u p l i n g development  crystal  (1979)  68  71  memories  stored b i t  79  capacity  a volume h o l o g r a p h i c  storage  80 device  81  ix  ACKNOWLEDGEMENTS  I research  would  topic  like  t o thank D r . Lawrence Young f o r s u g g e s t i n g t h e  and f o r h i s guidance  Thanks a l s o  during the course  t o A l MacKenzie  Kathy Brindamour and G a i l  Hrehorka  f o r help with  for preparing  of the research.  t h e d r a f t i n g and  the  manuscript.  1  1.  It certain  ferroelectric  refractive one  i s possible  part  index  INTRODUCTION  (Chen e t a l . , 1968) t o s t o r e  crystals.  distribution  i n 10 ~" a n d e f f e c t i n g  Physically,  with  diffraction  pattern  photorefractive  the reproduction  liberation drift,  of electrons  diffusion  successive  field  excitations  This  distribution  which  this  within  called  or  thermal  which  excitation Staebler  interfering  upon w h i c h  and t h i s  This  results  being  Young e t a l . arise length  shifted  (1972) n o t e d  that  i s an i m p o r t a n t  The r e s u l t i n g  a phase  could  with  shift  effect  due t o t h e  illumination  or heating  pattern  from a d i f f r a c t i o n  optical  become  formed by grating  coupled.  (Ch.3 ) . result  respect o f from  means o f e l e c t r o n  due t o t h e b u l k p h o t o v o l t a i c  electric  recording o f the hologram.  i f the fringe  o f e n e r g y b e t w e e n them  b y some p h a s e a n g l e  d i f f r a c t i o n p a t t e r n and  t h e two beams w i l l  which  an inhomo-  of the electrons.  displaced  then  approach  under  Through  index modulation through  redistribution  processes  light  of the  effect.  index d i s t r i b u t i o n  the r e f r a c t i v e  ( 1 9 7 9 ) showed t h a t  i f drift  the electrons  mechanisms.  p l a n e waves i s l a t e r a l l y  are several  through  about by t h e photo-  the bulk photovoltaic  removed by u n i f o r m  are incident,  of the o r i g i n a l  i s brought  becomes t h e p h y s i c a l  and uniform  i n a trading  There grating  removes  typically  image  i s a result  of a  the c r y s t a l , which then migrate  i n a refractive  a n d Amodei  the fringes  distribution  effect  of the transport  r e c o r d i n g may b e s u b s e q u e n t l y  the c r y s t a l ,  of a holographic  i s determined by t h e l i g h t  results  effect,  of  two  i n traps  index  i s composed  i n the index being  The s t o r a g e  and r e - e n t r a p m e n t s ,  characteristics  electro-optic  In b r i e f ,  and a process  geneous d i s t r i b u t i o n certain  beam.  as a r e f r a c t i v e  effect.  the hologram  the variation  Bragg d i f f r a c t i o n o f the reference  pure phase holograms i n  i n the hologram  t o the fringe  0 t o TT/2 r a d i a n s c a n  transport.  can also  pattern.  result  The  transport  i n a phase  shift,  2  d e p e n d i n g on spacing. the  the  more common t e r m  voltaic  can  of  this  to  i t , but  to  date  an  avoided  appropriate  as  during  this  the  into  general  has  to  Appendix  experimental  concerning  the  present the  due  grating  spacing",  the  w o r k was  apparatus,  or  to  line  but  of  photo-  shift  i n which the  which  in  they nature  give  i n achieving  rise  repro-  f u r t h e r i n v e s t i g a t e the  difficulty  their  bulk  study  processes  to  the  sources  attempts  of  to  arrangement  difficulty  sources  standpoint,  p h y s i c a l processes  i s to enable materials. of  data  transmission.  the The  and  effects  the  ultimate  goal  whether  they  circumvented  by  design  and  and  c o n s t r u c t i o n of  will  make u s e  these of  i n t e r f e r e n c e o f f e r e d by  Specific  the  holographic  crystals  electro-optic  devices w i l l high  optical  a p p l i c a t i o n s i n c l u d e two  character  of i n v e s t i g a t i o n s  a t work i n f e r r o e l e c t r i c  main a p p l i c a t i o n of  handling  to e l e c t r i c a l  i n t e r f e r o m e t r y and D.  of  i n the  d i s p l a y s , p a t t e r n and  graphic  to the  "plane  allowing for other  several reported  determining  changes  a  phase s h i f t  been c o n s i d e r a b l e  object  the  area  immunity  electrical  either  d e v i s i n g an  engineering  using these  relative  sional  aim by  from the  draw c o n c l u s i o n s  The  illumination  devices  s t u d i e d by  there  respect  data a n a l y s i s .  From an such  be  length, w i t h  here.)  There have been  results.  be  transport  t h i c k phase hologram has  i s used  c o u p l i n g and  coupling with could  a  c a l c u l a t i o n s or  negligible.  ducible  the  beam c o u p l i n g a r i s i n g  effect  subsequent are  of  (More c o r r e c t l y ,  The  in  value  and  speed  rather three  r e c o g n i t i o n image p r o c e s s i n g , data  storage.  These  are  be and  than dimen-  holo-  discussed  in  3  2.  During tion,  the three  or holography,  creasingly  rather  decades  has undergone  since  of a recorded  the interference pattern  i t s i n c e p t i o n , wavefront  considerable  broad range o f a p p l i c a t i o n .  f o r m , by t h e f o r m a t i o n but  O P T I C A L HOLOGRAPHY  image w h i c h  o f a p h a s e r e l a t e d s e c o n d beam c a l l e d  (but  not n e c e s s a r i l y ) a plane  tion  of the o r i g i n a l  properties This  normally  object's  information  interference original  along  that  pattern  amplitude  the surface  object  i n the reconstruc-  e x h i b i t t h e depth and p a r a l l a x (from  a limited  angle  and phase  information  h o l o g r a m made b y r e c o r d i n g  section  o f two p l a n e ,  same a m p l i t u d e grating  with  plane  ference  (Fig. 2.2) .  c a n be r e c o r d e d  i s accomplished by f i r s t  only  the hologram  view). ami s an  ( F i g . 2.1). c o n s i d e r i n g an elemen-  the interference pattern  As t h e d i a g r a m  the grating vector  o f two a r b i t r a r y patterns  by t h e f i l m ,  formed by t h e i n t e r -  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  o f t h e two beams.  interference  occupied  of  b e t w e e n two p h a s e r e l a t e d beams, f r o m w h i c h b o t h t h e  One c a n s e e how t h i s  the  on i t r e s u l t s  i s often  formed on t h e  i s p o s s i b l e b e c a u s e , u n l i k e a common p h o t o g r a p h w h i c h p r e s e r v e s  plitude  tary  beam, w h i c h  A f t e r t h e image h a s b e e n  image, w h i c h w i l l  image",  incoming wavefront and  the reference  the reference  associated with  i s not the object's  between t h e o b j e c t ' s  wave.  and seen an i n -  I t i s characterized, i n i t s simplest  that  r e c o r d i n g medium, s h i n i n g o n l y  development  reconstruc-  shows, t h i s  perpendicular  The g e n e r a l  wavefronts  in a  plane  t o t h e beam b i s e c t o r a n d i n  a n d more c o m p l e x  c a n be l o o k e d  of a l l the plane  results  beams o f t h e  case  a t as j u s t  of the i n t e r -  t h e sum o f t h e  waves c o n s t i t u t i n g t h e i r  Fourier  com-  ponents. The  formation  characteristics is  o f a hologram  and t h e d e s c r i p t i o n of s e v e r a l o f i t s  depends upon t h e degree o f c o h e r e n c e o f t h e i l l u m i n a t i o n . I t  useful- t o d i v i d e the t o p i c of coherence  i n t o two p a r t s : s p a t i a l  (lateral)  4  FIGURE  2.1  Simple hologram geometries, of  w r i t i n g and r e a d i n g  " a " shows t h e p r e s e n c e  both the reference and subject  beams, w h i c h f o r m a n 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 g e o m e t r y i n w h i c h t h e o b j e c t beam i s a b s e n t b u t is reconstructed v i a diffraction of the  FIGURE  2.2  An e l e m e n t a r y h o l o g r a m  formed  i n a medium o f t h i c k n e s s ' d ' (as  viewed from t h e top i n  these experiments).  +x  r e f e r e n c e beam.  5  coherence, oscillate As  and t e m p o r a l  (longitudinal)  ( o r c a n be so a d j u s t e d )  a consequence, they  a plane on  the plane,  of w r i t i n g  and thus  longitudinal  distance  mode p o s s e s s e s  present.  phase path  n o t be attain  system  wave r a t i o ,  ideal  m  a  x  and I  beams  problem  any o t h e r p o i n t  f o r the purpose  v  i s closely  oscillating  coherence,  coherence  beams d i f f e r  related  to i t s  i n o n l y one  s e v e r a l modes a r e  l e n g t h , AL , d e t e r m i n e s H  b y more t h a n  f r i n g e s when b r o u g h t  high  diffraction to display  coherent.  the  If the  the coherence  length  together, and thus  a  efficiencies,  i t i s necessary  optimal fringe  visibility,  that the  V, o r s t a n d i n g  i s o f t h e form  ^  n  I max I  - I . mm + I .  max  mm  a r e t h e maximum a n d minimum i n t e n s i t i e s V i s , i n general, a function  b e t w e e n t h e two i n t e r f e r i n g  (P  radiation  formed.  fringes.  of p o l a r i z a t i o n two  m  temporal  interference  be a r r a n g e d  which  interference |Pl2(T)|  ( T E M Q Q ) .  a t any p o i n t on  phase with  between p o i n t s which a r e s t i l l  _  T  the l i g h t  is.in  Though a l a s e r  The ( l o n g i t u d i n a l )  not exhibit  To  where  i s not a  o f t h e beam i s u s e f u l  of a laser's  l e n g t h s o f two s p l i t  hologram w i l l  optical  width  coherence.  a l o n g t h e beam a x i s  they w i l l  of propagation  gas l a s e r s  t r a n s v e r s e mode  and so t h i s  T h i s means t h a t  the entire  spectral purity  o f i t s temporal  generally  coherent,  commercial  a holgram. The  degree  them.  normal t o t h e d i r e c t i o n  Most  only i n the lowest  are s p a t i a l l y  when w r i t i n g h o l o g r a m s w i t h  coherence.  - as f o l l o w s  o f the degree  of  coherence  beams, t h e a n g l e ft b e t w e e n t h e d i r e c t i o n s  o f t h e two beams, a n d t h e r a t i o  < 1)  of the  R^. o f t h e i n t e n s i t i e s  of the  6  2|u  1 2  V  (T)|^cosfi R  The  geometry o f t h e p r e s e n t  and  thus  I  +  1  s e t of experiments  i s such  that  °>=0  a n d | u( T) | =  o  1  f o r our purposes  where  R  =(r/s)  V = R^+l To  optimize  fraction  r,s  by  a lens  only  pattern,  and passed lowest  ing  from t h e hole  lenses A  and  cause  i t requires  less  from  spatial  such  r i n g s from dust  frequencies  a pinhole  sources  beam e x p a n s i o n  of the former  only  from  varying  characteris-  as m u l t i p l e  reflections  spots.  The f o r m e r  beam i n t o a  to split are that  division  a n d wave-  more u n i f o r m i l l u m i n a t i o n .  t h e beam b e f o r e i t (1) r e q u i r e s i s filtered.  or after a smaller  stationary with  f i l split-  The s e c o n d o p elements.  of a sharp holographic  fairly  subject  i s g e n e r a l l y p r e f e r r e d be-  and p r o v i d e s  the s p l i t t e r  remain  emerg-  diameter of the  o n e s e t o f beam e x p a n d i n g o p t i c a l  interference fringe pattern  focus  through  of the slowly  i s required t o divide the laser  It. i s e s s e n t i a l t o t h e f o r m a t i o n the  thus a l l o w i n g  characteristic  I t c a n b e shown t h a t  t o be made i s w h e t h e r  requires  This i s  i s u s e d t o c o l l i m a t e t h e s p h e r i c a l wave  out noise  a n d (2) t h e n o i s e  however,  must  cross-section.  T h e two m e t h o d s a v a i l a b l e a r e a m p l i t u d e  The advantages  aperture  u s e d , means t h a t we  a t that point,  and b l o c k i n g t h e h i g h e r  a s shown i n F i g . 2.4.  further choice  tion,  frequencies  ( F i g . 2.3).  wave.  division,  ter  spatial  beam s p l i t t e r  reference  tering.  a pinhole  and d i f f r a c t i o n  front  A  through  filter  i n the d i f -  f i l t e r i n g . * The beam i s b r o u g h t t o a p o i n t  A following lens  o f 10 ym w i l l  setup  a n d s i g n a l waves  at a l l points  beam, p r e f e r a b l y o f G a u s s i a n  frequency  distribution,  of noise.  from  i n the experimental  symmetric  tic  order  which,  by s p a t i a l  the very  Gaussian  of reference  V i t i s s e e n t h a t we m u s t make R^ u n i t y  have an a x i a l l y achieved  = amplitudes  image  respect  that  t o the  7  laser  FIGURE  2.3  Effect  of a spatial  FIGURE  2.4  Hologram  geometry  division  shown  filter  using  on t h e l i g h t  intensity  amplitude d i v i s i o n  i n F i g . 2.1a).  profile.  (as opposed  to wavefront  8  r e c o r d i n g medium t h r o u g h o u t minimize  the  total  est  spatial  the  wavelength of  of  the  setup  was  included  this  the  from as  the  being  relative  supports,  being  used.  care  and  beam s p l i t t e r  source i t . part  of  the  In the of  the  In  In the  which  present  the  filter  or  problem.  to provide  highto  be  degree  hologram  their  steel  out  the  the  and  resting  building  especially  i n the  minimized  o f t e n more p r a c t i c a l l y ,  experiments, a  i n order  to  random v a r i a t i o n s i n  r e c o r d i n g medium m u s t be  set of  must  i s u s u a l l y taken  fringe pattern  disturbances,  or,  one  work, r e l a t i n g  granite, concrete  acoustic  general,  bench v i b r a t i o n s exceeds the  o f t e n used to  disturbance  system  process.  recorded,  to prevent  and  current  o c c a s i o n a l l y troublesome  the  p o s i t i o n s of  e t c . are  thermal  of  magnitude of  pattern  greater  Airborne  p a t h s between the eliminating  the  duration  t a b l e s , o f t e n made o f  sand, pneumatic  vibrations.  light  even  Massive  that  i n the  the to  required  position. on  frequency  beam c o u p l i n g  grating  time  the  Michelson  continuous  beam  by protecting  interferometer monitoring  of  9  3.  This incident  chapter  be  seen  written  later  Though that  i n LiNbOg, Consider  An  coherent  surface We  then  Applying  n  they  to  grating  of thickness  d  here  grating  yields useful applicable  to certain  and  results, i t will t o holograms  conditions.  (an e l e m e n t a r y  thick  phase  transmission  .  (3.1)  R a n d S, a r e s y m m e t r i c a l l y  Bragg  (Staebler  (Figure 3.1):  c o s [(2ir/Ji)x]  make a n a n g l e  Si i s g i v e n  0 relative  conditions  i n c i d e n t upon t h e  t o the Z-axis  p r e v a i l so that  X i n the c r y s t a l  inside  the c r y s t a l .  the r e l a t i o n s h i p  and t h e phase  grating  by  9  X  =  2 H sin  R  =  R ( z ) exp  {-i[(2Tr c o s 0 / X ) z +  (TT/£)X]>  S  =  S ( z ) exp {-i[(2ir c o s G / X ) z -  (TT/A)X]}  (3.2)  write  coupled  wave t h e o r y  =  (Kogelnik,  t h e volume o f t h e g a t i n g  Tfnj/X  coupled  . =  L  constant  equations i s  1969) we  i s given  , ,  ~ i < S(z) g  dS(z) K  cos 0 i s the grating  the anisotrppy  these  x  perfect  — dz  with  o f t h e i n t e r a c t i o n o f one o r two  given  i s restricted  a sinusoidal  dR(z)  K  the development  the o p t i c a l wavelength  beams w i t h i n  where  the theory  WAVES  a s i n u s o i d a l t h i c k phase  but rather  =  so t h a t  wavelength  We  with  beams o f l i g h t ,  assume t h a t  between  with  OF COUPLED  i t s a p p l i c a t i o n i s not generally  hologram) i n a c r y s t a l  Two  deals  beams o f l i g h t  A m o d e i , .1972).  DIFFRACTION  T  -  L  find  dz coupling  K t o be i n t r o d u c e d  that  the v a r i a t i o n of the  by t h e e x p r e s s i o n s .  =  (3.3)  , ,  ,  - i K R(z)  (3.4)  g constant later).  ( n o t t o be  confused  The g e n e r a l  solution  10  fringe  pattern  hologram grating  * X  D o  3.1  Fringe pattern crystal been the to  +  3  c- axis  FIGURE  o  I  c a u s e d by  thickness  written. hologram  The  i n t e r f e r e n c e o f R(0)=1 and  'd' u p o n w h i c h fringe pattern  grating.,--which  the c r y s t a l  a hologram  n o r m a l made by  by  ^  compared  g r a t i n g of spacing  i s d i s p l a c e d by  i s described  S(0)=e  § along  equation  '1*  to  has  the x-axis  from  8 i s the  angle  3.1.  t h e i n c i d e n t beams i n s i d e t h e  crystal.  11  IK  R(z)  =  ae  Z  -IK Z  + be  g  g  (3.5) IK  S(z)  The  coefficients  i.e.  t h e phase  =  -ae  Z  -IK Z  + be  g  g  a and b a r e determined  and a m p l i t u d e  from  the boundary c o n d i t i o n s a t z =  o f t h e beam a t t h e f i r s t  I n t h e g e n e r a l c a s e we n o r m a l i z e  R(0) = 1 a n d t h e n  0,  surface of the crystal.  s e t S(0) = Ae  This  gives  R(z)  =  S(z)  R  1  =  S q u a r i n g t h e s e we  I  c o s (K z) - i A e ^ s i n ( K z) g g - i s i n ( K z ) + Ae g  X  (3.6)  ^ cos (K Z) g  get the i n t e n s i t i e s ,  =  cos  2  ( K z) + A g  2  sin  2  ( K z ) - A s i n (2K Z ) s i n g g  <h  =  sin  2  ( K z) + A g  2  cos  2  ( K z ) + A s i n (2K z ) s i n & g g  (3.7) I  Although  S  these equations  beam's p h a s e  differs  relationship  of fixed An  The  coupling tions  beams  [R(0) = 1,  during writing,  with  (3,7)  I  reduce  grating  given by  applies  T h e more c o m p l i c a t e d c a s e  i n this  work, u s e s  I f the i n c i d e n t  >  well  o f beam  the boundary c o n d i -  beams a r e b a l a n c e d  (A = 1),  to  =  1  - s i n (2K Z ) s i n  =  1  + s i n (2K  g  <f>  (3.8)  s i s immediately  describe the equivalent  S(0) = A] a n d a m o v a b l e  experiment.  which 1  R  and one  r e p r e s e n t e d b y S(0) = 0 a n d R(0) = 1, h a s b e e n  R(0) = 1 a n d S(0) = Ae ^ .  equations  can a l s o  i s fixed  n ^ c o s [ (2 it/ I) x+ <}>]  =  and t h e o r y agrees  I  It  t o t h e c a s e where t h e g r a t i n g  t h e o t h e r ' s b y (j>, we  from  c a s e o f beam r e a d o u t ,  studied  the  refer  Z)  s i n <b  g obvious  that  f o r t h e c a s e o f cj) = 0 ( i . e . when t h e g r a t i n g  12  lines  up  with  the  c a s e where t h e (<J> of the  0)  grating  results  grating  fringe pattern),  displacement.  at  the  (2<gZ) f a c t o r of  the  n  define  =  sin  I  "  11  I  the  grating  interfering lumination of  the  by  (  I  _ "  )  side  I  s  (  basic  theory  grating.  The  of  effect.  shift  and  reason  involved,  of  the  grating  the  coupling,  i s r e l a t e d to  Amodei,  pattern  depends upon the  Also,  amount o f  fringe  the  amount  towards  strength as  of  the  i s shown  in  diffraction  1972)  z=d)  (3.9)  s  (d)  +  I  R  ( d )  i n these to  experiments. any  lossless purely an  sinusoidal  approximate  i l l u m i n a t i n g a LiNbOg:Fe  for  the  this  lies  crystal.  in a LiNbO  namely  the  i n the  c r y s t a l with  crystal,  two  the  detailed physical  Therefore,  3  thick  description  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  hologram w r i t i n g  processes  which  However, t h i s i s o n l y  the  from the  The  as  applies  a c t u a l l y f o r m e d by  continues.  photovoltaic  )  i s used  p l a n e waves, and  case  d  energy t r a n s f e r .  vice-versa.  This  (at  no  x-axis  small  the  (Staebler  w r i t i n g mechanism w i t h i n  specific the  (d)  R °  the  amount o f  term.  ( K Z) g  2  foregoing  phase t r a n s m i s s i o n of  s  r i g h t hand The  last  lossless crystal  _  the  influences  i n the  grating  n for a  where t h e  along  S-beam, a n d  a p a r t i c u l a r time  efficiency  s h o u l d be  Qualitatively, a  enhances the  grating  We  i s displaced  i n energy t r a n s f e r ,  S-beam s i d e  sin  there  we  photorefractive  before will  i l nature  considering first  e f f e c t and  deal the  the  with  bulk  13  4.  A changes light tive to  number o f  in their  of  an  eventual  using  was  i n the  which  led to  are  migrate under  the  of  they  time u n t i l  tal.  will  pattern  be  of  which  an  the  a refractive  three i)  by the  range  sites  trapped  within  electric  index  and  number o f by  traps  A)  eliminate of  the  i n the  at other of  the  incident  the  photorefrac-  ( i . e . high)  case of  electrons  the  The  traps,  electrons  then  mechanisms f o r  some  locations within  the  small  the  change  pattern  is related in  the  period crys-  end  index pattern  linear i n the  being  EO  the  distribution  dark to  conductivity of its original  the  state;  a  effect  index  of  translated  some u n i q u e  fashion  can  be  removed i n  crystal  to  slowly  at  ways: allowing  re-  resulting in  light.  charge/refractive  EO  LiNbOg)  i n deep  incident light,  field  devices  f o l l o w i n g manner.  Finally, a  i t led  problem i n  electron density,  induce  which  of  intense  effect.  different  crystal.  may  this  minute  because  desirable  impurities.  results in this  distribution  of  of  the  field  the  nuisance  (450-550nm i n t h e  intensity pattern  fields  intensity pattern  least  a  recaptured  Appendix  to  takes place  uneven r e d i s t r i b u t i o n  see  The  defect  a  such  causes p h o t o e x c i t a t i o n  with  applied  -  effect  i n f l u e n c e of  electric  refraction into  an  out  understanding  wavelength  Depending upon t h e  sult  (in  how  find  sufficiently  i n e a r l y e l e c t r o - o p t i c (EO)  done t o a better  are  something of  t h e y had  crystal  associated  t o be  However, b e c a u s e  appropriate  found t o undergo  T h i s phenomenon, c a l l e d  performance  photorefractive  upon the  c r y s t a l s have been  wavelength.  of  EFFECT  r e f r a c t i o n upon e x p o s u r e t o  considered  work was  which The  incident  first  crystals.  coefficients,  Light  indices of  degradation  these  crystals,  ferroelectric  appropriate  effect,  PHOTOREFRACTIVE  return  to  14  ii)  by  iii)  strongly  which  i t i s photosensitive  taken  to  by  electric  trons et  last  the  with  defect  panied  by  upon the  adopted  Peterson  The  optical  and  doping with refractive  percent.  et  under  iron,  iron, iron  defects  of  the  not  m e a s u r e s must  be  form); h o u r was  found  A  the  to  i s present  as  a l . , 1973; and  Fe^  +  expect  the  and  the  the  defects The  pyro-  and  i n the iron  Fe^  +  a  with  impurity's  et  +  iron  an  al.,  of  a  though  Fe^  impurities  to  molebeing being  neither  state  occupied  +  that  photo-  possibly the  1972,  showed  best  to  specific  found  trivalent  (Fe^ )  studied  the  and  gamma  impurities  the  +  have  accom-  increase  Fe^  sites,  be  t o make  hundredths  1975)  niobium  the  of  first  (Phillips  the  (Peterson  seen t o  influence  (and  elec-  LiNbOj to  e t a l . , 1974)  few  Rauber,  s t a t e of  pure  developed  commonly u s e d  asso-  irregularities)  was  impurities  of  i n deep t r a p s  c r y s t a l s was  Glass  divalent state  oxidation/reduction  control  in effect  lattice  studies  other  studies As  the  f o r making f r e e  processes  devices  usually i n concentrations  trap  electrons  lattice  number o f  most  cooling in  cooling to  in  by  empty  and  exposing nominally  e f f e c t s u p o n EO  with  is certain.  of  photorefractive  Mickami et  effect  gradual  by  located  s t i t u t e s , an  heating  sensitivity.  manganese, c o p p e r  Mossbauer  of  (geometric  ( D i s c h l e r and  findings  hours  various  i n oxygen v a c a n c i e s  that  do  but  to  work.  the  a l . , 1972)  properties  a l . , 1971;  two  located  these  fast,  a wavelength  (200°C f o r one  during  photoconductive  decreasing  The  by  r e s u l t i n g increase  sensitivity,  date) being  with  impurities i s responsible  photoelectric effect.  et  be  holograms  oven  in this  Stoichiometric  increased  the  can  above, p h o t o e x c i t a t i o n  sites  (Phillips  ways o f of  i n an  shorted  available for migration  studied  find  parasitic  crystal  faces  mentioned  irradiation.  use  c-axis  a l . , 1972).  been  (this  method, f o l l o w e d  e f f e c t s ) was As  ciated  the  that  entire crystal  effective).  The (with  ensure  heating  be  oven  i l l u m i n a t i n g the  of  (Fe^ ) +  con-  trap,  one  would  would  affect  15  the  photorefractive sensitivity  LiNbOg one  i n a i r or  effective  oxygen a t  method of  number o f  traps),  (Clark  a l . , 1973).  et  crystal  to  with  state  and  destroyed  while  aiding optical  annealing Smith et with  a  stoichiometry  by  a process  oxide to  general  d e s c r i p t i o n put  charge lem  fields  determining  attributed "bulk  to  the  an  photo-voltaic effect".  to the  trivalent  (1975) t h a t h e a t i n g iron  thermal  i n v o l v i n g the  heating  centres decay of  of a  of  the  to the  Fe  a  d e c a d e . Though t h e y  at the  beginning  of  this  hologram  under  the the  Following  migration.  or  non-  the p h o t o r e f r a c t i v e a l l agree  section  are  there  has  I t has  i n f l u e n c e of  with  (i.e.  been  the  photospace  some  prob-  been v a r i o u s l y  some f i e l d ,  summaries o f  -  contact  a n i s o t r o p i c re-entrapment, r e s u l t i n g effect),  several  and  a  new  published  models: i)  The tive  Internal Field effect  crystal,  to  the  Model.  drift  origin  i n the of which  b a s e d upon a d j u s t a b l e refringence section the  of  change  induced a  Chen  by  crystal.  (1969) a t t r i b u t e d  presence was  not  of  an  the  internal  clear.  His  a laser As  beam i n c i d e n t o n  shown i n F i g u r e  i n b i r e f r i n g e n c e changed  4.1,  sign along  photorefrac-  field  in  conclusions  compensator measurements o f  changes  a small i t was the  Z  in  crystals.  past  +  crystal  two  of  state  field  t r a n s p o r t between t h e  a p h y s i c a l model of  be  i n c r e a s i n g the  Other methods i n c l u d e  c h a n g e v i a EO  drift  (i.e.,  +  shown t o  composition  p r e c i s e nature  diffusion,  been  the  the  index  decreased  Annealing  which a l t e r s  traps, migration,  causing  has  going  Cornish  a,b).  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  construct  forward  iron  powder r e d u c e d  much l a r g e r c r y s t a l ,  attempts  the  erasure.  defects to  1974  state to Fe^  shown b y  2  w r i t i n g and  Amodei,  e t a l . , 1971)  2 +  t r a p s , which  h a v e b e e n made d u r i n g  from  of  in Li C0g  process  excitation  96%  packed  lithium  Several  Fe  been  and  but  the  I t has  lattice  a l . , 1968)  (Peterson  much a s  shallow  (removing  similar  600°C  oxidizing as  540°C w h i l e  ( S t a e b l e r and  the were  in bi-  central  found  that  c - a x i s , but  not  I  FIGURE  4.1  Change i n b i r e f r i n g e n c e o b s e r v e d by Chen c-axis along  beam I diam eter  (1969).  due t o a r a d i a l l y  induced with The s o l i d symmetric  a single line  laser  represents  beam i n L i N b O g t h e change  beam, a n d t h e d a s h e d l i n e  as  along the  t h e change  the b-axis.  f +c-axis  FIGURE Space in  4.2 charge f i e l d  postulated  the b i r e f r i n g e n c e .  by Chen t o a c c o u n t f o r t h e o b s e r v e d  change  17  the  b-axis.  He  a s s u m e d an  the  presence  of  empty t r a p s  traps  available  presence to  the  trons  of  will  field  and  be  trapped  4.2)  Observation  the  of  during  illumination,  The  which the  was  illuminated laser  of  gence  of  the  gives  rise  to  difficulties  electrons  E  i »  direction until  finally results  index  they are  trapped. i n the  of  the  out  of  the  existence  of  of  Thus a  refraction  by  E^ .  elec-  being  the  illumi-  space  of  the  A  n  an  charge  electric  electro-optic  was  considered  pyroelectric  was  of  dismissed  with  the  sign  Johnston effect  attributed  the  change i n  a  dP/dT  (1970) p u t  to  direction  the  f o r w a r d a model the  variations  density  local  of  change i n and  i s a c c o m p a n i e d by index v i a i s that  the  EO  (where t h e y filled  light free  in  resultant  a permanent effect.  i t does not  One  explain  to  the  in polarization. the  yet  in  are  traps  in as  t h i s mechanism, i n c i d e n t band  crystal  (negative).  photoinduced  i s frozen  t h i s model  the  ( o r more s p e c i f i c a l l y ,  In  the  thus  polarization  of  conduction  in  change  because  the  opposite  c-axis  creation  full  (2)  Photoexcited  of  (1)  and  and  directed  n  s h o r t - c i r c u i t photocurrents  decrease  with  capture  postulated  the  into  this  a  and  a r i s i n g from non-uniform h e a t i n g of  volume and  off,  the  polarization.  electrons  move) c a u s i n g a  the  i n the  Model.  n  macroscopic  excites  which  disagreed  E^ )  in  are  photorefractive  unexplained the  field,  Polarization  field,  re-excited  evidence  origin  photocurrent  ii)  drift  and  variation  further  to  spontaneous p o l a r i z a t i o n .  therefore  (Figure  effect. to  of  electric  v o l u m e , where t h e y  develops  available  effect  d o n a t e them u p o n p h o t o - i o n i z a t i o n ,  internal  direction  repeatedly nated  an  to  electro-optic  With  diver-  field of the  which  several steady  18  state,  short  circuit  photocurrent  observed  in a  crystal  under  full  illumination.  iii)  Pyroelectric Field 1972  b)  to  arising (e.g.  a t t r i b u t e the  during  during  the  crystal  (Cornish  sesrved  photocurrent  et  and  (Glass  resulted  i n no  e x p e c t e d by  field  was  of  (1974,  1975  detectable  which  refractive Fe  2 +  have a fore of  of  to a  net  J , r  gives results  rise  to  temperature experimental  crystal  increasing  short  obto  the  circuit,  nor  by  of i l l u m i n a t i o n  photocurrent  any  as  observed), the  would  be  photovoltaic  that  the  long  et a l . a  new  trans-  The  elec-  hence t o the  potential wells  Nb-Fe  the  distances  2 +  +c-axis  J j«  The  e  - c - a x i s due  further current,  from asymmetric  of  effect".  (and  over  Upon e x c i t a t i o n , t h e s e  electron current the  internal  Glass  existence  of moving along  along  built-in  photoconductivity  i n asymmetric  regard  a  the  long periods  photocurrent  different.  ionized impurity  relaxation  i s not  "bulk  trapped  probability  time  photoconductivity.  r e l a x e d by  Note i n t h i s  rise  a  Reasoning that  the  by  this  r e l a x a t i o n i n the  heating  without  Staebler  effect,  supported no  and  a high  temporarily  i n which  contribute to the  greater  by  led to postulate  d i r e c t i o n s are  give  the  rent,  were  e f f e c t ) are  ions).  ±c-axis  and  i l l u m i n a t i o n (which a)  not  from  decrease i n the  eventually  mechanism, c a l l e d  trons  was  effected after  a l . , 1974)  effects  crystal  i n which  c o o l i n g with  et  the  w o u l d be  periods  port  the  a l . , 1976)  (Amodei  to a polarization  formation)  Bulk P h o t o v o l t a i c E f f e c t . field  second attempt  remanent p o l a r i z a t i o n  conductivity results  A  c o o l i n g of  results  remove t h e  iv)  Model.  to  J 2"  (the in  the  electrons  and  there-  displacement Franck-Condon A  &  retrapping of  photo-  the  third  cur-  excited  elec-  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 current J  ,  ph  density, =  J  , +  el  Jp^» J  e^ 0  (Fe  can t h e r e f o r e ~  J  r  =  where I i s t h e i n c i d e n t l i g h t  3 +  ).  The  steady state  be e x p r e s s e d a s t h e sum  KCCI  (4.  intensity,  a i s the absorption  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 stant") will  r e l a t e d t o the physical nature of the trap.  be d i s c u s s e d  i n greater  photo-  detail  later.  This  concon-  model  20  5.  I t has  PHOTOCURRENT AND  BULK PHOTOVOLTAIC EFFECT  been noted (Chen 1969)  t h a t l i g h t i n c i d e n t upon p o l e d  ferro-  electric  s i n g l e c r y s t a l s causes a s m a l l c u r r e n t t o flow between e l e c t r o d e s  attached  to opposite  field.  c r y s t a l f a c e s without the a p p l i c a t i o n of an  Though i n i t i a l l y  thought to be  external  caused by b u i l t - i n f i e l d s due  to, for  i n s t a n c e , e l e c t r i c moments a r i s i n g from c o o l i n g the c r y s t a l from a h i g h temperature,  subsequent e x p e r i m e n t a t i o n  e f f e c t , the i)  has  l e d t o the c o n c l u s i o n t h a t a  new  "bulk p h o t o v o l t a i c e f f e c t " i s r e s p o n s i b l e .  Photocurrents F i g u r e 5-1  i l l u m i n a t i n g with  shows an example of a p h o t o c u r r e n t  an Argon-ion l a s e r (514.5nm a t 17.5  doped LiNbOg c r y s t a l  caused by  mW/cm ) a f a c e of an 2  ( p e r p e n d i c u l a r t o the c - a x i s ) and measuring the  p a s s i n g between aluminum e l e c t r o d e s p l a c e d upon the c - f a c e s .  The  component i s the p y r o e l e c t r i c c u r r e n t caused by a change i n the d i p o l e moment as the e f f e c t can be  i n c i d e n t l i g h t r a i s e s the temperature.  seen as the c r y s t a l c o o l s a f t e r the  n e g l i g i b l e i s the p h o t o c u r r e n t , the p h o t o c u r r e n t i n t e n s i t y and b)  i t does not tion  c)  (10  2  strength  which has  Fe  current  transient  spontaneous  The  opposite  l i g h t i s turned  c u r r e n t component r e m a i n i n g a f t e r the p y r o e l e c t r i c c u r r e n t has  a)  uniformly  off.  The  become  the f o l l o w i n g p r o p e r t i e s :  i s l i n e a r l y r e l a t e d t o the  a l s o depends upon wavelength ( C o r n i s h ,  change over l o n g p e r i o d s  i n c i d e n t beam 1975);  of time under constant  illumina-  hours or more);  i t does not depend upon the thermal h i s t o r y of the c r y s t a l e f f e c t i s not  changed by h e a t i n g  open or s h o r t c i r c u i t  conditions.  and  ( i . e . the  c o o l i n g the c r y s t a l under e i t h e r  F I G U R E 5.1 Graph o f the p y r o e l e c t r i c (PC) w h i c h r e s u l t s crystal.  current  (PEC) a n d t h e p h o t o e l e c t r i c  when a n e x p a n d e d  The c u r r e n t  i s measured  laser  across  beam i s d i r e c t e d the c-axis  current  a t a LiNbO  (see t e x t ) .  22  A et  simple  mathematical  a l . , 1 9 7 4 b) i s g i v e n J  . ph  =  expression  f o r the photocurrent  density  (Glass  by  KOtf  (5.1)  w h e r e 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 the absorption,  and  K i s the photocurrent upon t h e n a t u r e  ii)  The B u l k P h o t o v o l t a i c The  in  regular  gradients crystal  observation  which  could  which  of steady-state,  no m a c r o s c o p i c  junctions account  short-circuit electric  l e d t o a search  f o r such  which dix Fe  which cal  on t h e f a c t  that  Consequently,  i o n with  2 +  fields,  currents  concentration  property  of a  an u n e x p e c t e d a n i s o t r o p i c e f f e c t .  asymmetric p h o t o - d e l o c a l i z a t i o n model a LiNb0 :Fe  t h e Fe-Nb d i s t a n c e s  A).  etc.).  The  exclusive, are b r i e f l y  here. The  centres  photovoltaic  f o r some b u l k  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 outlined  i s dependent  (impurities, defects,  Effect  crystals containing or pn-like  constant,  of the crystal  t h e Nb  3  are different  there  J  This  el  where  gives  = ^ £  +  V  £ . i s t h e quantum hu  to a current,  P  a  r  e  ( s e e Appen-  l  of the  i n the -c direction,  / given  following  opti-  by  • .  (  efficiency,  probabilities  -c d i r e c t i o n s ,  e  - 0  i s t h e quantum o f l i g h t  P+»P_  J  along  between t h e o r b i t a l s  direction of transfer of electrons  P  axis  i n t h e +c a n d - c d i r e c t i o n s  i s a larger overlap  rise  (  e t a l . , 1974)  has a unique p o l a r  i o n i n t h e +c d i r e c t i o n t h a n t h o s e  results i n a preferred  excitation.  crystal  (Glass  energy,  o f c h a r g e t r a n s f e r i n t h e +c a n d  5  '  2  )  23  the  electron  mean f r e e p a t h s  along  these  directions. This  i s followed  placement  by F r a n c k - C o n d o n  along the polar  J  „  giving a further  E,(-—)z.Mi.  =  e^  axis  product  1 1  Mia)-'  relaxation of the ions, current, J  summed o v e r  with their  e  net dis-  /  2  a l l i  (5.3)  +*Vi  where A £ ^ i s t h e d i s p l a c e m e n t z^  i s the charge  of the i  of the i  ion,  ion.  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  random a n d t h e y  no l o n g e r c o n t r i b u t e  applied  However, i f t h e p r o b a b i l i t y o f a s u b s e q u e n t  field.  impurity  t o a net current  i s d e p e n d e n t upon t h e d i r e c t i o n from  we h a v e a r e c o m b i n a t i o n J  and  of the photo-liberated  another  ri - i  as  C*; P ; - * :  where t h e p r i m e d  Miu)''  i  quantities  P 1  , P  the e l e c t r o n  =  J  h  £  a t an  arrives,  )  then  ( .4> 5  current  (5.5)  p ^ , £', p'and  +l  i n the recombination  terms a r e  quantities.  we sum a l l o f t h e s e J  recombination  i  V  t o the e x c i t a t i o n If  i n t h e absence o f an  z\ A*.  =  9  r2  becomes  follows:  Franck-Condon r e l a x a t i o n  J  analogous  current,  which  electrons  +  1,2  currents  J r  =  we g e t  KOCI  (5.6)  i,2  where  <  =  r hu  and  varies  Figure  with  5.2.  K L  P +  the photon  x  +  ^  p  + r  - -  energy  +  P! -  r  p  +  -  -  and environment  ] +  1  J  [z.- ; ] z  hoj  L  l  I  J  of the absorbing centre.  See  24  O Nb  O Li  5*  electron  oo a)  #0  excitation  c -  axis  P'  P:  OO b) thermal  FIGURE  electron +  In  recombination  O  oo  5.2  Asymmetric  p  OO  photo-delocalization  i s photoexcited  a n d p_ o f . m o v i n g a l o n g ( b ) i s shown  probabilities.  model i n a LiNbOg  from an F e  3  +  impurity  t h e +c a n d - c a x i s  the Franck-Condon  shift  crystal.  In (a) an  and has p r o b a b i l i t i e s o f  directions  and e l e c t r o n  respectively.  recombination  25  whereas the ions, 1975  the a,  photorefractive  1975  b,  The  tion the  1976  mechanism of  next by  Ohmori e t  relaxation  constitute  (ii)  and  e f f e c t has  The  of  the  a unit  state.  cell,  ground  and  The  ( i i i )according to  5.3b).  Up  this point  goes t o  anion tion  current,  and  i n the  e  ,  the  and  r  and  e n d s up  non-zero  e  lattice.  electron  electron a  J  e  of  t h e r e has  and  J^ /  which  r  cation  state  the  to  lattice  steady-state  third  (v)  the  the  to  as  out  the  the  an net  anion  model i s the  fluctuation  of  crystals results  Photofluctuation  i n the  i n t e r a c t with  polarization  Ax  and  and  to  Next  the  to  state relaxed  (Figure charge  the  retreats  to  the  i t s original locaionic  currents  relaxation  i f  However, i f cell  as  Furthermore, recombination  i s present,  ca-  motion"  diagram  electron  two  results.  Model  photoinduced p o l a r i z a t i o n f l u c t u a t i o n s .  can  e  assumes  previous  The  the  shown i n i f  (v),  coherent  probability  the  summed  effect  photocurrent.  BPE_in-s.ome o x y g e n o c t a h e d r a  electron  J£ «  electronic  LiNbOg) t o  trapped  the  cation  the  A  well.  Franck-Condon  distance  current,  instance)  1974,  to  anion  directional electronic  attributes  ferroelectric  the  configuration  a nonisotropic  (v), for  al.,  this  " d i r e c t i o n of to  a  w h e r e an  i n a neighbouring  current  et  c r y s t a l s as  their original locations.  involving  right in  the  ( i v ) as  contribute  cancel  combining with  will  A  ionic relaxation  return  to  a bulk  coordinate  state  Linde  impurity  optical excitation.  5.3a,  shifts  been both a  These motions  (recombination be  an  ground  contribution  relaxation  the  ground Franck-Condon  then to  currents, the  the  J  following  then  between  Model a t t r i b u t e s  (i) i s transferred cation  der  i n high purity  arrows r e p r e s e n t  state  state to  states  transport (von  m o d e l i s shown i n F i g u r e  excited  cell  been o b s e r v e d  a l . , 1977)  excited  optical excitation.  transfer  electron  C o l l e c t i v e Franck-Condon Relaxation  Franck-Condon physical  above model i n v o l v e s  ferroelectrics (i.e.,  production an  ( F r i d k i n , 1977),  of  Light free  o p t i c a l phonon t o  l o c a l i z e d near  the  trap  BaTiOg,  absorption and  in  trapped  create  (Chanussot,  which  a  n-type  electrons.  photoinduced  1974).  Within  26  FIGURE 5 . 3 a P h y s i c a l mechanism o f the c o l l e c t i v e Franck-Condon 0;  o  anion  model showing i n s u c c e s s i v e t i m e  Cation  electron  frames: the  H "  "  ®  O o  o>  *  O o  a  ( i ) photo-excitation  Franck-Condon s t a t e ;  ( i i ) cation  s h i f t t o the " r e l a x e d  excited  state";  ( i i i ) electron  transfer  l e a d i n g t o the ground F r a n c k (iv) cation  t o the ground s t a t e .  (io  O o -  O o  (.ii)  O  O  M  O -o  O *-o  (v>  o o  O o  o c-axis  >O  CC  FIGURE 5.3b  LU  Coordinate configuration  diagram  for  the Franck-Condon model show-  ing  the f i r s t  two events  from  ground s t a t e t o an e x c i t e d  Condon s t a t e ;  o  relaxation  z  LU  RES.  described  above.  G.5.  Q  shift  27  the  volume of  tion, that  AP,  the  which  f l u c t u a t i o n there  gives  rise  a l l photoinduced  voltaic  to  and  moving  i n the  tional  P.ayleigh - s c a t t e r i n g o f  served  i n BaTiOg, though not The  randomly  current  Polarized  are  then  from the  p o l a r i z e d due arises  ground  state  a photo-ionization direction  ft.  The  j  f r o m an  where N  =  model p r o v i d e s  =  i n LiNbOg  generation  of  though not  obsesrved,  (Koch  of  this  has  density  Baltz,  photo  1977)  that been  assumes t h a t  photo-  s(fi) £ s(-ft), where  e j e c t i o n of  an  s(°»)  electron  is  in  a  by  ( 5  .7)  concentration, length  due  to  the  (5.8)  2  f o r the  different spectral properties  polarized parallel  by  the  and  perpendicular  f o r observed  a l . , 1976).  dark c u r r e n t  the  the  for ionization  d Q,  et  ob-  The  hoi  impurity  addi-  photoconduction  cross-section  state,  i s given  these  et a l . , 1977).  r e s u l t i n g i n the  is predicted  studies  and  to  SSSSE  light  a macroscopic  Early  present,  short-range p o t e n t i a l .  spontaneous p o l a r i z a t i o n , and i n BaTiOg  model s u g g e s t s  asymmetric i n the  photo-  coupling  to  an  requires  bulk  which provide  direction  as  This  A  impurities  current  current,  be  (von  photovoltaic of  Symmetry  state  explanation  for  » AP/E.  electrons  (Chanussot  Model  i n spontaneous p o l a r i z a -  same d i r e c t i o n .  i s the e l e c t r o n t r a n s p o r t photovoltaic effect, and  an  AE  AE.  delocalized final  / s(Q)  N  i n the  should  asymmetry  i s the  Lp  a  light  photocurrent  KOI  field  d i r e c t i o n of  cross-section  =  ph  to a  a change  free photo-excited  Impurities  d i s t r i b u t e d ground  electrons  electric  f l u c t u a t i o n s be  e f f e c t r e s u l t s from the  fluctuations  This  an  occurs  Also,  due  to  i t does  thermal  asymmetric  bulk p h o t o v o l t a i c  changes of not  of  to  the  sign  of  predict  the  the  the  e x c i t a t i o n which,  photodelocalization  e f f e c t l e d to  the  model.  following  28  expression sity,  relating  photocurrent  . (x) ph  =  H o w e v e r , more r e c e n t expression  the  incident  light.  work  recording Both  in ferroelectric  during being  writing is a written  current  and  of  As  the  tially  ejected  define  a  before  being  along  quantity,  will  be  pattern  inten-  The the  L^,  of  a  first  of  d i r e c t i o n of  or  the  having  spatial  expression  assumptions  the  that  +c-axis.  mean d i s t a n c e  a  the  beam  holographic Also,  simple depen-  the  holograms  the  writ-  coupling grating  "ac"  part  of  frequencies.  f o r the  photovoltaic  concerning  the  i s that photoelectrons I t then travelled  randomized.  Young e t  this  frequency of  later,  between the  i t s motion  trapping.  spatial  writing i t .  assumption  which'is  manner o f  set  out  hologram w r i t i n g nor  discussed  higher  f o r a more r e a l i s t i c  pointed  c h a r a c t e r i s t i c s of  shift  f o r f r i n g e s with  choice  during  observed  a phase  diffraction  retrapped  the  of  has  r e s p o n s e upon the  crystals.  the  a l . , 1979)  beam c o u p l i n g  are  result  search  s t a r t s with  with  (Young e t  these  mechanisms i n v o l v e d .  deals  incident light  (5.9)  material  i s smaller  The rent  to  Kctl(x)  predicts neither  dence of  the  Jp^,  I: J  ten  density,  The  a l . dealt with  microscopic are  follows by  the  preferen-  t h a t we  assumption  possibili-  ties: a)  continuous c-axis  gradually  liberated b)  fixed  s c a t t e r i n g - c l o s e l y spaced  d i r e c t i o n of  motion  of  the  along  the  photo-  electrons,  transport  distance,  randomize the  scattering centres  L  length  , along  the  - the  photo-liberated  +c-axis  motion randomized a t that  point,  and  are  then  electrons trapped  or  can  electron  second three  cur-  travel have  a  fixed  their  29  c)  discrete along  the  electron Once t h e model  density the  second  for  the  light  in  c-axis and  results  and  assumption  impulse  the  each h a v i n g a  incident  =  =  1-p'  i s decided involves and  light  probability of  not  finding  pattern,  an  spaced  p'<l  of  affecting its  upon, the  then  centres  construction expression  forming  the  in this  case  apart the  motion. a  mathematical  the  photocurrent  convolution a  p  scattering  of  for  L  of  sinusoidally  this  with  varying  direction: I  these  p  response  +c-axis  of  identical scattering  probability  I(x)  The  -  photocurrent  appropriate  grating  scattering  (1  o  +  cos  calculations  Kx)  (5.10)  give  (Young e t  a l . 1979),  for  the  three  cases: a)  continuous J  p  scattering  (x)  =  -KCCI o  [l +  L  cos  (Kx-<b ) 1 P  m"  cos  (Kx-<j>) ]  i  sinc(x)  m"  (5.11a)  where £>L_ K  =  hoi  -1/2  b)  fixed  »'  =  rat  4> P  =  tan •  (*%)*  +  1  - 1  (KL  transport  length  J  -KOI  p  (x)  =  o  P  L  )  [l +  *  x.  where  K  =  •  £aL  E  hw  m'  =  m  sine  r 1  KL  P i  —— 2 ir  J  . . .  -  s  =  l  .  ^  n  — —  TIX  (5.11b)  30  KL  c)  discrete scattering J  p  (x)  centres  -Kal  =  [l =  o  m'  cos  (5.11c)  (Kx-d> ) ] P J  where q L ?  K  -  p ' hco  1/2  KL m*  =  <, p  =  mp  sinc[—~\ 2TT  (l + p «•* '  J  KL 2  y  p 2  +  t  a  [-  - l  n  L  Note t h a t  i n a l l three  cases,  spatially  varying  component  all  cases  three  bulk  the  photovoltaic  <(>p i s s u c h dence high  of  that  m'  cient  the  should  be  of  to  formation  not  reveal  formation. vide  a  one  zero  cos  KL  result  in a  p'  )  1  P has  i s phase K,  goes  frequency  KL  P_]  5.9  linearly Lp  cos  e  KL  constant,  as  2p "  of  been  replaced  shifted  which  gives  by the  with  Lp.  Also,  to  zero.  The  the  smaller  an  current  I  the  phase  K,  In of  the  depen-  shows  thus  a  shift  functional  and  plus  Q  amount  strength  fringe pattern, "ac"  by  less  that effi-  writing.  combination hologram  l - p  which  varies  spatial  frequencies  hologram It  effect,  sin  l ( x ) i n eq.  anisotropy  i t goes t o  upon  spatial  (ac)  -  2  e  pointed  a l l three  of  of  in a  them t o  However-, t h e y  out  these real  be  that cases  (and  crystal,  clearly  a l l agree  useful-starting point  i t is quite  and on  and  that  p o s s i b l y others) thus  solely  several  for experimental  possible  may  some be  weighted  applicable  d e t a i l e d n u m e r i c a l work  responsible important  analysis.  f o r the  points,  and  may  grating thus  pro-  31  6.  Having Chapter and  self  the f i r s t  neither  account.  Also,  HG r e s t e d  length  of this  effect  the conclusions  on an i m p l i c i t ,  there  i s no p h a s e  shift  ( Y o u n g e t a l . , 1979)  concerning  If drift  then  HG a t t h e i n i t i a l  layers of optical  and upon  afterwards,  (Ninomiya,  the d i e l e c t r i c several  1973)  papers  i n t e n s i t y , Kukhtarev  Though t h e i r ments were  theory  one o b t a i n s  This  et a l .  theory  of hologram  includes  a postulated  (1979)  was t h e m a i n  derived  the photovoltaic  transport  Later  work  i s n o t much  also  stages  applicable of  recording  neglected.  recording wave  intensity.  was p u t  equations Whereas  dependence o f r e f r a c t i v e i t from  current,  conducted on c r y s t a l s and under c o n d i t i o n s  less  t h e ir/2  t h e two beams may b e  dependent upon t h e l i g h t used  transport  b y TT/2 w i t h r e -  a n a l y s i s was  crystals or to the i n i t i a l between  into  and as t h e d r i f t  b a s e d upon t h e s o l u t i o n o f n o n l i n e a r  constant  succeeding  of short  length  i s observed,  of recording.  a dynamic  taken  p h a s e between t h e FP  i fdiffusion  transport  shift  4  a n d Amodei,  i s t h e o n l y means o f e l e c t r o n  the f r i n g e spacing,  stage  i n Chapters  g r a t i n g were  grating i s shifted  i f the d r i f t  some p h a s e  grating i n  charge nor the e f f e c t of  the relative  I t was shown t h a t  the hologram  showed t h a t  approaches o r exceeds  Shortly  with  space  Staebler  though u n r e a l i z e d , assumption  when e n e r g y a n d p h a s e r e d i s t r i b u t i o n  forward  (e.g.  (and c o n s e q u e n t l y no e n e r g y t r a n s f e r ) .  the fringe spacing  to thin  of the r i s i n g  electron.  to the fringe pattern.  only  problem  o f t h e w r i t i n g beams o n t h e h o l o g r a m  spect  shifted  i n w r i t i n g a hologram  treatments  of the migrating  length  involved  t h i c k phase  o f hologram w r i t i n g i n LiNbOg.  means o f e l e c t r o n t r a n s p o r t ,  than  i n an " i d e a l "  the theory  t h e feedback  diffraction  IN LiNbOg CRYSTALS  beam c o u p l i n g  3, a n d t h e b a s i c p r o c e s s e s  In  and  considered  5, we now c o n s i d e r  1972)  HOLOGRAM WRITING  the material their  this index  equations.  subsequent  experi-  where t h e e f f e c t s o f  32  diffusion  dominate,  vis-a-vis  t h e BPE. The  and  will  effects  and thus  theoretical  do n o t c o n f i r m t h e a c c u r a c y  development  be a p p l i e d t o t h e i n i t i a l o f feedback  conditions i) ii) iii)  The  electron transport  of hologram  formation  become s i g n i f i c a n t .  before the  The f o l l o w i n g  length,  crystal,  illumination  expression  f o l l o w s Moharam e t a l . ( 1 9 7 9 )  hold:  e x t e r n a l s h o r t on uniform  stages  and s e l f - d i f f r a c t i o n  and assumptions  arbitrary  given here  of their predictions  of crystal  f o r the fringe  with  geometry a s i n F i g u r e  p a t t e r n f o r m e d b y t h e two  6.1.  interfering  beams i s : I(x)  =  I  (1 + m c o s k x )  o  (6.1)  where  The  m = 1  (modulation  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 ) .  conduction  current density distribution  J(x)  with  the three  space tion  charge  qD  terms  field  p(x,t)  tion  =  E  8  p  (  . dx X  s c  (x,t)),  5.1), 5.11):  =  p  that f o r uniform  should  reduce  ,  t  )  + qup(x,t)E  and t h e BPE.  for I  R  =I  s  ) /  along the x-axis i s  (x,t) + J (x,t) sc p  representing diffusion,  i s t h e sum o f t h e d a r k  Reasoning  ratio  drift  (due t o t h e  The f r e e  electron  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 +  (6.2)  photo-induced concentra-  concentration:  p (x,t)  (6.3)  illumination,  to the expression  Young e t a l . (1979) p u t f o r w a r d  the photovoltaic current density  obtained  by G l a s s  e t a l . ( 1 9 7 4 b)  the following expression  (see eq.  equa-  (see eq.  33  J  where for  0 < f^  our  length not  (x)  =  (k,Lp)  purposes. i s not  strictly  lows  p  o  [l +  correct.  g(x)  i s the  Spj^fxjtj/St  concentration  p  -  respect  The  L  =  out  to the  cos  that  (kx-A  P  ) 1  be  i f the  fringe  photoexcited  the  rier  migration  (  '  X  —  t  =  (6.4)  set equal  to  unity  photovoltaic transport  spacing,  then  the  iced t e r m  e l e c t r o n c o n c e n t r a t i o n i n eq.  sc  6.3  is fol-  short  (x,t)  L f J  o  , 1 + _  }  3J(x,t) _ _  (6.5)  L p  )  cos  (kx-^) ]  assumption t h a t the  (6.6)  photoexcited  illumination.  charge d e n s i t y ,  We P / s c  electron  combine eq. due  to  6.6  car-  (6.7)  p =  (6.8)  _2£  e  i n t e g r a t i n g over  across  t  rate,  f o r trapped  3x  of  ' _  equation  *£  constant  X  8x  8 E  sc  (  3J(x,t)  Poisson's  E  L  i n v a r i a n t f o r steady  c o n t i n u i t y equation  (after  p  (k»  2  » 0 c o n s t i t u t e s an  at  get  f  Q  3p_  with  , , g(x)  }  -  g [l + m  i s time  with  a  )  P  f u n c t i o n which can  volume g e n e r a t i o n  g(x)  The  (k,L  1  from  where  to  mf,  They a l s o p o i n t e d  8  and  L  < 1 is a real  small with  0  and  -KCII  (x,t)  =  , - e  space and t J  f  J(s,t)  crystal,  E  ( x , t ) dx  sc  dt +  A(t)  •  (6.9)  o  i n t e g r a t i o n comes f r o m the  time)  =  0  the  assumption  ( i i above) t h a t  there  is  (6.10)  34  from which we  get t / f j ( x , t ) - J ( x , t ) ] dt e •* o o  T  E  Here, J  sc  (x,t)  =  L  J  i s the 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 .  Q  derive  [Moharam e t a l . ( 1 9 7 9 ) , E l G u i b a l y  charge  field:  E  (x,t)  8 k  (for  -  E  =  D  t  a  n  -1  * " k l — (  k  L  >  2  =  *N TA  kT' B  f o r t h e space  cos (kx-(j) -<j>,) P  (6.12)  d  where  -1  p  k  can  1/2  2  the case o f l a r g e s c a l e d r i f t ) ,  *d  S o l v i n g the above we  4  2  =  (6.11)  ( 1 9 7 9 ) ] an e x p r e s s i o n  [(kL ) + (kL ) J E 1 + (kL,) d  qg tm  .  T'  q  k  B  °  Vv E  = =  absolute B  °l  t z r n a n , s  temperature. const.  L  E„  =  — JE  T, d  =  (TD)  "V  i s the v i r t u a l  yx  field  1/2 d i f f u s i o n transport  <j) i s the phase s h i f t due t o the f i n i t e  length  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 t h e  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 c o n s t i t u t e s the g r a t i n g (see Appendix and then use the r e s u l t s of Chapter 4 t o r e l a t e t h i s t o t h e beam c o u p l i n g .  A),  35  A P P A R A T U S  7.  Figure  7-1 i s a s c h e m a t i c  this  study.  A l l optical  sive  cement b e n c h  were cemented  ponent  stands.  effect  on t h e r e a d i n g s  light  source  P h y s i c s Model  t o allow  "greenhouse" which  served  fluctuations  pattern tubes  were f u r t h e r  setup  small  and used  purpose  nosing  collimating  filter half  brought  operated  (for f o r 30  a n d aluminum  a i r c u r r e n t s which  t o form  a i r c u r r e n t s by being  T E M Q Q  power.  a plexiglas  together  The  CW a t  T h o s e p a r t s o f t h e beam  cause path  an i n t e r f e r e n c e  enclosed i n paper  o f f by B S j b e f o r e  monitoring  t h e hologram  i n t e r f e r o m e t e r ( B S £ , M , M ^ , M^, s c r e e n )  for the  2  o f bench v i b r a t i o n s ,  experienced  i n forming  m a i n p o r t i o n o f t h e beam c o n t i n u e d lens positioned at s l i g h t l y  p i n h o l e s o t h a t t h e beam c o n v e r g e d  way b e t w e e n  output  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  o c c a s i o n a l problems The  laser  com-  had an  i n the evening.  p o w e r o f 800mW  thermal  o f t h e beam was s p l i t  i n a Michelson  visual  from  length.  and metal  length.  fraction  o f forming  simultaneous  a  isolated  f o r most o f t h e i r A  and then  still  I t was g e n e r a l l y o p e r a t e d  i t from  i n t h ephase path  w h e r e t h e beam was s e p a r a t e d  output  by m i r r o r Ml i n t o  to isolate  vibrations,  down t h e h a l l  was done  a stable  used i n  a s u r f a c e f o r magnetic  166 a r g o n - i o n  i t t o reach  beam was d i r e c t e d  setup  w e r e p l a c e d o n a mas-  building  carts  experimentation  515mW was o b t a i n e d .  runs  the laser  from  i t h a s a maximum s p e c i f i e d  mode), t h o u g h o n l y  The  isolation  and r o l l i n g  s o most  was a S p e c t r a  minutes before  including  experimental  t o the top surface t o provide  Slamming doors  514.5nm, a t w h i c h  random  components  f o r mechanical  strips  of theentire  the lens and c r y s t a l .  dimensions.  This  w h i c h was u s e f u l  allowed  i n diag-  holograms. on through  more t h a n  a spatial  i t s focal  filter to  l e n g t h from t h e  t o a p o i n t a t t h e beam s p l i t t e r B S g  T h i s was done  f o r two r e a s o n s :  F I G U R E 7.1 The  laser  .  (experimental  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 " g r e e n h o u s e " b y  Beam s p l i t t e r  Michelson  The r e s t  mating  lens, past  meters  (BS , M , 2  3  pattern  a shutter  S  R  d i s c upon w h i c h m i r r o r  lets  only  to  c-axis  beams t o p a s s  the reference  tween t h e m i r r o r s f u r t h e r reduce  from M  5  and M  &  g  BSg.  towards  Mg i s m o u n t e d .  and  the c r y s t a l  The i n t e n s i t i e s o f by t h e photo-  c a n be  Opening  (reading/erasing).  a i r current  are enclosed  shifted  on t h e p i e z o shutters  ( w r i t i n g ) , whereas o p e n i n g o n l y  and t h e c r y s t a l  colli-  T h e n t w o beams  are monitored  at the crystal  a  bench v i b r a -  filter  by a d j u s t i n g t h e v o l t a g e  beam t h r o u g h  and M  t o provide  t o monitor  through a s p a t i a l  The f r i n g e p a t t e r n  electric  causes both  lens screen)  by t h e c r y s t a l  to the crystal  2  2  t o form a f r i n g e p a t t e r n .  parallel  S  M ,  t o t h e beam s p l i t t e r  1  beams t r a n s m i t t e d  Pg a n d P .  M^,  on t h e s c r e e n  are reflected  come t o g e t h e r  coupled  mirror  d i v e r t s a p o r t i o n o f t h e beam f o r u s e i n a  o f t h e beam p a s s e s  intensity  where t h e y the  1  diffraction  tion.  equal  BS  interferometer  macroscopic  of  apparatus)  and  shutter  T h e beams b e i n paper  tubes  problems.  J  37  i)  i t permitted immediately  the i n s e r t i o n after  beams a r i s i n g fered ii)  with  small Thin  in  variable  i n t e r f e r i n g plane  waves.  being  (< 20 mR)  split,  towards a p o i n t where t h e y i n the c r y s t a l  i n the s p l i t t e r ,  which  a more n e a r l y  t o one i n w h i c h  t o s p e n d t o o much at the crystal.  inter-  uniform  beam h a s a  t h e beams a r e r e f l e c t e d  intersect,  forming  disc,  one o f t h e p h a s e p a t h  lengths  beam i n t e n s i t i e s  two  silicon  p-n j u n c t i o n  ing  a Model  DC1010 d i s p l a y ,  thus  two s p a t i a l time  tuning the f i l t e r s  affect  beam  diver-  the assumption o f  from m i r r o r s M  5  and  M  g  a diffraction pattern of v e r t i c a l One o f t h e two m i r r o r s i s  providing the c a p a b i l i t y with  filters  The r e s u l t i n g  d i d not seriously  held i n the crystal holder.  m o u n t e d o n a PZT c e r a m i c  The  paths  o f f secondary  beams i f t h e i n c i d e n t  i n preference  beam i n t e n s i t i e s  geometry  vary  beam  section.  g e n c e o f t h e new  planes  to block  beam s p l i t t e r g i v e s  BS^ t o a v o i d h a v i n g  t o balance  After  reflection  i n t h e two r e s u l t i n g  g e o m e t r y was a d o p t e d  order  internal  i n both  t h e m a i n beams;  cross  were p l a c e d a f t e r  s p l i t t i n g b y BS^ i n o r d e r  from  the continuously amplitude  of small apertures  to electrically  respect to the other.  emerging from the c r y s t a l  photodetectors, and the o t h e r  are then  one an A l p h a m e t r i c s  monitored  M o d e l P1110S  a model PS1101S d r i v i n g  a Model  by driv-  1030  display. A  K e i t h l e y 602 e l e c t r o m e t e r  perpendicular The supply,  which A  used a  t o t h e c - a x i s t o measure PZT d i s c  was  t o monitor  connected  Autograf  variable Model  across  M o d e l 405B h i g h  7100 BM two c h a n n e l  electrometer  faces  v o l t a g e power  i n 0.1V s t e p s w i t h i n a ± 3 0 0 0 V  output  range.  s t r i p chart recorder  a n y two o f t h e f o u r r e l e v a n t p a r a m e t e r s : divider),  the c r y s t a l  the photocurrent.  d r i v e n by a F l u k e  i s discretely  Moseley  10:1 v o l t a g e  was  a n d t h e two  PZT v o l t a g e  was  (through  photodetectors.  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 t w o s h u t t e r s  a n d 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 erase  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  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 pyroelectric  origin,  and.placed  circuited  damage f r o m s t r e s s e s o f  i n a h o l d e r w h i c h was p l a c e d i n a n a l u m i n u m b o x  t o ensure 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 oven.  39  8.  8.1  MEASUREMENTS AND  ANALYSIS  Measurements In  setting then  t h e f o l l o w i n g we make t h e a s s u m p t i o n  I ( z = 0 ) + I (z=0) = I (z=d) + I ( z = d ) . R s R s  c a l c u l a t e d f r o m m e a s u r e m e n t s when  of a lossless dielectric  The d i f f r a c t i o n  by  efficiency i s  I (z=0) a s  s I (z=d) g  n  The  i n normalized I  C  The  s  =  C  crystal  (z=d)  I  t o reduce  8.2  and r e f e r e n c e  (  z  =  d  )  )  n  °R  d  [ I (z=d) + I _ ( z = d ) J/2 s R  =  L  0.1 m o l e p e r c e n t  complications  (  J  iron  8  '  2  )  doped and  i n the fringe pattern  Lines  1  due t o  ever,  the intensities  are the transmitted  diverge  subject  These s t a r t  output  i n t e r v a l s the incident subject  operated  ( 1 . 2 5 cm/min)  o f the transmitted  sub-  i s recorded  shutter  causing  unchanged.  and reference  out equal  as time p r o g r e s s e s  some o f t h e r e f e r e n c e  beam p a t h  recorder  present:  I (z=d) t o remain R  chart  R  respectively.  intensities  At regular  and  are  a n d D'  intensities  solenoid  monitoring  g  'C  their  of s t r i p  beams, I ( z = d ) a n d I ( z = d ) a s a h o l o g r a m  following features  ii)  a  i s a photocopy  a p a i r of photodetectors  i)  1  reflection.. Fig.  ject  ,  R  (#6 - s e e A p p e n d i x A) was  coated  8  of  J  used  (  a n d C , b e t w e e n t h e two beams when I ( z = 0 ) = I ( z = 0 ) s R s  values  L  internal  R  [ I (z=d) + I ( z = d ) J / 2 s R  antireflection  The  (z=d) + I (z=d) s R  amount o f c o u p l i n g ,  results  of  I  =  2  coupling.  beam i s i n t e r r u p t e d b y a  l" (z=d) t o d r o p  ('a' i n f i g u r e )  As t h e h o l o g r a m  develops,  s  I ( =a) s  (5mW/cm ) b u t  due t o beam  beam i s d i f f r a c t e d  and t h e i n t e n s i t y o f  beam  z  how-  into the subject  gradually  increases  (line  FIGURE  8.1  Beam g e o m e t r y  during  hologram  writing  (eq. 8 . 1 ,  8.2).  41  FIGURE 8.2 Development of hologram, showing: i ) beam coupling (C,D) ' i i ) d i f f r a c t i o n e f f i c i e n c y f o r momentary reading (a,A,b»B) i i i ) " n u l l i n g voltage" (E)  42  FIGURE )  *  0  i  1  i  2  i  3  Exposure  i  4  (J/cm)  i  5  i  6  i  7  8.3c  43  'A') a n d t h a t 'A*  decreases  R  (line  a n d *B' a r e i n s e r t e d i n t o e q u a t i o n  diffraction  iii)  of I (z=d)  Also,  efficiency  the intervals  intervals  s e t equal  R  8.1 t o o b t a i n  PZT  This disc  Fig.  Figure tion  efficiency  (joules/cm ) 2  I (Z=0)  t o I ( z = d ) ] by v a r y i n g s  malized  incident  a  which  from  voltages  at regular  the s p a t i a l  voltage  by t h e s p i k e s  written  gratt o the  'E* o n  above.  F i g . 8.2.  The x - a x i s  units  of incidence  speed  (1.25 cm/min.).  equations  8.2).  Figure  (lines  [ i (z=0) = =  (6^  from t h e normal  Figure  diffrac-  are exposure  c a n be c a l c u l a t e d f r o m t h e i n c i d e n t i n t e n s i t i e s  was  seen  i n Chapter  ± 1 9 ° ) and  8.3b i s a g r a p h o f t h e *C  a n d 'D' i n F i g . 8.2  8.3c i s a g r a p h o f t h e v o l t a g e  grating  spatial  simple  gratings  there  from  sinusoidal fringe  grating w i l l  o n e beam t o t h e o t h e r  i s no v a r i a t i o n Therefore,  and f r i n g e p a t t e r n s ,  which i s t h e s h i f t f r i n g e s and g r a t i n g s  two w a y s :  a simple  nor-  required to  which  i n beam  ( F i g . 3.1).  i n e i t h e r the f r i n g e pattern w h e t h e r o r n o t we an e f f e c t i v e  would account  were p r e s e n t .  result  pattern  fy  e  coup-  ( e q . 3.8) i f t h e r e i s  <{> b e t w e e n t h e f r i n g e s a n d g r a t i n g  i n the Z direction.)  simple  defined  phase  means t h a t  3 that  sinusoidal holographic  and a t r a n s f e r of energy  "simple"  in  ( e q . 8.1)  upon a s i m p l e  non-zero  with  f o r the  the coupling. It  ling  the corresponding  o f t h e amount o f beam c o u p l i n g  with  values  8.3a i s a g r a p h o f t h e d e v e l o p m e n t o f t h e n o r m a l i z e d  recorder  development  lines  and hologram  the appropriate  (see F i g . 7.1), and i s r e p r e s e n t e d  2  chart  null  by a p p l y i n g  = 5.0 mW/cm ], t h e a n g l e s  r  the  was a c h i e v e d  8.2 w i t h  on  beams w e r e b a l a n c e d  p h a s e r e l a t i o n s h i p <J> b e t w e e n t h e f r i n g e p a t t e r n ing.  Points  as a f u n c t i o n o f t i m e .  of the coupled  [l (z=d)  'B*).  (Here, or the  are actually dealing  phase  shift  f o r the coupling c a n be o b t a i n e d  (J> c a n b e E  i f i n fact  from  t h e data  44  i)  direct  6  calculation  e  of  ii)  =  2 u  the  experimental  phase  3.8,  are  3.9  lines  vary  of  calculation The  which  The  identical from the  not  the  the  of  a full  and  by  the  2K  coefficient  angle  8  p  a and  b  inserted into  8.1. along  features of linear  with  the  note are  over the  with  different  sets  results  that  the  range of  same i n t e r c e p t a t E x p .  development of However,  despite  the  causes  piezoelectric  i n f i g u r e s 8.3  shown i n F i g . 8.3b  experiments.  examination  which  of  =  two  variation  observation,  0,  data  from  and  but  that the  also with  other  i n phase that a l l slopes the  means  i i above).  coupling  coupling  the  ( A p p e n d i x B)  data  and  three  only  ( i and  shows t h a t  voltage  p l o t t e d i n F i g . 8.4  have approximately  the  i s the  shifter  i s approximately  of  V:  and  runs.  exposure  voltage"  (degrees)  i s d e t e r m i n e d by  c a l c u l a t i o n s from the  These  lines  360  volts  and  7.1),  equations  with  1180  shift,  (Fig.  x  "nulling  2TT  where V phase  V — — V_  =  from the  coupling  be can  compared be  i t i s possible to  i t s irregular  reasons  can  f o r the  different  of  for  This of  the  entails  Fig.  8.5,  nominally  extract useful  behaviour.  time v a r i a t i o n  to that  information a  closer  c a l c u l a t e d phase  shifts.  8.2  Analysis  of Phase S h i f t  Figure strates  why  graphs  constant  values  several  apparent  fringe  8.4,  pattern  of  which  like the  Measurements shows t h a t  F i g . 8.6  the  g i v i n g the  phase  shift  v a r i e s with  predicted coupling  shift  are  not  r e a s o n s why  the  phase d i f f e r e n c e between the  v a r i e s as  the  hologram  observed  experimentally.  i s written.  for  time,  demon-  various  There grating  are and  45  FIGURE  8.4  Variation i)  of phase  Dashed  lines  shift  calculated  measured values ii)  Solid  lines  shifter.  with  f o r the  calculated  exposure using  for three  equations  diffraction directly  from  3.8,  runs: 3.9,  efficiency nulling  and  and  the  coupling,  voltage  of  PZT  R-BEAM 0  5  10  EXPOSURE  FIGURE Time  (J/cm ) 2  8.5  development  experiments. each  15  o f beam c o u p l i n g  T h e beam i n t e n s i t i e s  by o n e - h a l f t h e i r  sum.  for five  nominally  a r e n o r m a l i z e d by  identical dividing  From Young e t a l . ( 1 9 7 9 ) .  47  EXPOSURE  FIGURE  the  2  8.6  Beam i n t e n s i t i e s assumed  (J/cm )  values  calculated  from  o f the phase s h i f t  balk photovoltaic e f f e c t .  a computer model <j> ( c o n s t a n t )  From e l G u i b a l y  for different  associated (thesis).  with  48  8.2.1  Grating It  Bending  has  been  shown  gram g r a t i n g v a r i e s a l o n g illustrated  (1978) and  later  tal  1mm  t h i c k with  an  The  solid  lines  successive sponding  and  a  d e t e r m i n e d by by  shift  5.1)  that  during  degree  and  place the  the  these  mostly  first  by  and  virtual  depth  samples.  dashed  The  e  the  i s a measure of  the  i s the  shifter  same  holois  of  a  55  the  by  cryskV/cm. for  ten  corre-  dis-  coupling,.as  quantity  and  of  crystal  are  This PZT  field  lines  would produce the E.  the  This  case  i n t o the  which  i s mea-  a s s u m i n g no  other  place.  value  p y r o e l e c t r i c constant  of  Heating  during  1/20 the  a  first  ( i f any)  attempts  study-coupling before  the  the  of  crystal  crystal  minute of  by  by  met  PZT  with  one  several tenths  This  so  can of  therefore  Thermal  expansion  w r i t i n g by as  not  to  a  take  affect  in this  show  the  expansion would  initially  during  shifter  A)  expand i t along  experiments.  intervals  a d j u s t i n g the  (Appendix  0.5°C w i l l  success  regular  pyroelectric current  i l l u m i n a t i o n , and  present  lack of at  h e a t e d up  g r a t i n g wavelength.  i n the  e x p l a i n the  beam i n t e n s i t y  the  experiments  may  shifting.  (1979) f o r t h e  19°  $ ff  phase of  from a program developed  of phase with  i n Appendix  contraction  the  of  the  increases.  measurements showing the  about  reading  to  e l Guibaly  g r a t i n g which  take  obtained  where  using  (exposure)  photocurrent  centigrade.  x-direction  data  that  Expansion  From t h e (Fig.  shifts",  coupling  time  incidence  spaced time  phase  the  of  a l . , 1972)  as  by  variation  program  i n the  Thermal  modified  the  straight the  z-axis  which uses  angle  equally  nulling  variations  8.2.2  give  "effective  placement of  sured  the  i n F i g . 8.7,  Moharam  (Staebler et  only and  work i n  decreasing write  while  -160.0  -358.0  -356.0  -354.0  -352.0  -350.0  PHRSE S H I F T  -348.0  -3 46.0  -3 44.0  (DEC)  FIGURE 8.7 Time development of hologram c u r v a t u r e : 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 -  e f f e c t i v e displacement o f e q u i v a l e n t simple g r a t i n g  -342 0  -3 40 0  50  8.2.3  Bulk  Motion  As can  be  lent of  by  the  effect  by  fringe  shifting  ponent  i n the  Optical  Components the v a r i a t i o n  (x-axis) motion of  lenses i n the the  p a t t e r n from  motion p e r s i s t s  and  s l o p e s i n F i g . 8.4,  a sideways  o f t h e m i r r o r and  the  such  shown b y  caused  Motion  of C r y s t a l  fringe the  optical  pattern.  Michelson  f o r days a f t e r  the  the  crystal  s y s t e m may  Direct  beam p a t h  due  t o the  residual  effect  the  above problems  of about result  of  creep  motion  shows t h a t  the p o s i t i o n  mechanical  equiva-  the  ( F i g . 7.1)  time  lOnm/min.  i n an  measurement o f  Interferometer adjustment  of phase with  of  of the  any  com-  com-  ponents. The 8.4  a non-zero  quantitatively also  straight cause a linear  grating change  before  i n the  r e g r e s s i o n of  8.4  the  from  PZT  phase  from  equations.  the  Fig. case  of  8.8  sjiCt^O) =  and  gives the  i s the  3.9  case,  and  result  the  two  o f u n c o n t r o l l a b l e and  wide v a r i a t i o n  i s due  in results  t o the phase  f o r the  shift  phase s h i f t  of  the  grating  voltage data  value  fy  0  =  5.0°  and  more r e l i a b l e  of  in Fig.  methods a g r e e  of the  dtp/dt = -10,  0,  the  shift,  two.  The  accurate i n the  a t t=0  5 and  to  pattern. lines  A  in  0.7°.  as  10  the  line the  line  to within ±  degrees  grating of  the  1.4°.  expected per  given  obtained  derivation  c o u p l i n g w h i c h w o u l d be -5,  constant.  chance  fringe  a  initially  (from which the  ±  geometry assumed  a  i t is  undergoing  of the  have had  measurement o f t h e phase  grating  in Fig.  c o u p l i n g t h a t f o l l o w s from  gets p r o g r e s s i v e l y l e s s  i s a graph  25°  the  e) l i n e s  (or exposure) i n s t e a d of remaining  the phase n u l l i n g  "simple"  In any  this  positions  i t i s a direct  3.8  f o r the  give the  above mentioned p r o c e s s e s  relative  shifter  the  gives a figure  the  were o b t a i n e d )  equations  departs  t=0  time  i s to  s l o p e was  i n F i g . 8.4,  increasing  back t o  Because by  This accounts  seen  change w i t h  Extrapolation  Because t h i s  unpredictable processes,  However, as  linear  Fig.  slope.  unpredictable.  8.5.  of  unit  for  the  time.  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 w h e r e <)) (  (deg/(J/cm )). 2  =  25°  and  f o r v a r i o u s v a l u e s o f <j>  52  The  crossover  experimental the phase strong  8.3  shown i n t h e run,  shift  indicating  due  enough t o  Calculation  first  to the  give the  two  t h a t the  bulk  ratio  joules/cm  2  impression  of  the  exciting  to calculate figure  of  the  voltaic  transport length, L ,  10.4°.  p  These c a l c u l a t i o n s  Q  5.0°.  *o  The  total  "  *p  =  tan"  occasionally  was  i n the  of  the  direction  From t h i s of  T h i s was  during  opposite  an  to  sometimes  Length  observed  shift  observed  reversed coupling.  beam i n t e n s i t i e s  phase  at a  <j> =  creep  of Photovoltaic Transport  arrived  using  was  photo-voltaic effect.  Young e t a l . _ (1979) u s e d measured  cases  diffraction a t an  exposure of  using equations they  efficiency  3.8  and  obtained a value  and  the  1.2  3.9,  and  f o r the  photo-  24nm.  can  now  phase  be  repeated  f o r the  shift  i s given  by  tan"  (E / E  present  set of  data  *D  +  (icL ) + p  1  1  D  v  )  where kxT' Ep  =  E  =  v  45  From t h i s  we  value  and  that obtained  drift  i n the  joules/cm  2  to the  manifests  phase  shift  restate  bulk itself  ± 5k  by  diffusion  V/cm  show t h a t L  in their To  due  can  i s the  p  =  2  equivalent  i s the v i r t u a l  13nm  ± 3nm.  The  field,  difference  Young e t a l . (1979) can  during the  time  taken  this  chapter,  field,  be  f o r the  between  e x p l a i n e d by  this a possible  exposure to reach  1.2  experiment. the  results,of  photovoltaic effect, i n the  h o l o g r a m as  L , p  has  the  electron  been measured as  a displacement  of the  transport length 13nm.  This  written grating  along  53  the +c-axis with respect to  t o the l i g h t f r i n g e p a t t e r n , which  beam c o u p l i n g b e t w e e n t h e t r a n s m i t t e d r e f e r e n c e  and  i n turn gives  subject  beams.  rise  54  9.  Studies  of  the  hologram w r i t i n g process  models  to p r e d i c t t h e i r  evaluation  of  time  of  the  development  the  i n t i m e and  diffraction  Though c o n s i d e r a b l e  success  has  in  no  success  computer models,  model p r e d i c t i o n s of culty  has  voltaic phase  shift  which  The through the  the  shift  been  a  use  phase  affected the  by  was  PZT  to  the  can  be  shifter  yield  during  and  least  of  several processes  which  later  used,  a phase  obtained,  which  corresponds  shift  due  to  generality of  s e v e r a l c r y s t a l s of  any  case,  measuring the  with  diffi-  the  photo-  determination  of  the  and  phase to  shift  with  the  to  calculations of  beam c o u p l i n g .  fringe pattern  the  This  does  It  has  not  variation is  apparatus used  best-fit phase  due  in  lines  shift  are  before  these  measuring  extrapolated i t has  been  disguise  the  original  value.  the  photovoltaic  effect  bulk  a transport this  directly  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  when t h e  value  In  some  f o u n d t h a t b o t h methods o f  the  make m e a s u r e m e n t s on  experimental  efficiency  i t was  same r e s u l t  establish-the  quantity  match e x p e r i m e n t  comparing t h i s  completely  and  former  such parameters as  w r i t i n g process.  v a r i a t i o n (at  the  an  to  observations  coupling.  Consequently,  g r a t i n g and  the  neither  linear,  attempts  been aimed a t  between the  quantity  beam  and  coupling.  measured d i f f r a c t i o n shift  and  crystals  around  i n p r e d i c t i n g the  coupling.  requires  crystal  conditions.  met  of  space c e n t r e  c a l c u l a t i o n of  this  phase  i n t e r c e p t gives  the  To  of  as  work h a s  i s nearly  This  i n the  rise  However, t h e  shift  LiNbOg  2.9°  a  the  which  experiments)  t=0.  of  constant  predicted.  to  gives  has  in ferroelectric  efficiency  been a c h i e v e d  development  length,  from the  to processes  the  the  present  shown t h a t  remain  such  been e n c o u n t e r e d  transport  SUMMARY  length  analysis  different  about  i t w o u l d be  doping  i t i s c l e a r that without  of  and  of  13nm. useful  under a  apparatus  For  to  variety  designed  to  55  hold the o p t i c a l i t will  components v e r y  still  f o r periods of s e v e r a l minutes or hour  n o t be p o s s i b l e t o m a t c h u p e x p e r i m e n t a l  predictions of the coupling.  r e s u l t s w i t h computer model  56  BIBLIOGRAPHY Cathey, W. Thomas: Optical Information Processing and Holography. Sons (1974). C a u l f i e l d , H.J. ( e d i t o r ) : Press (1979). Chanussot, G:  Handbook of Optical Holography (ch. 10).  Academic  F e r r o e l e c t r i c s 8, p. 671 (1974).  Channussot, F r i d k i n , Godefroy and Jannot: Chen, F.S.:  John Wiley and  App. Phys. L e t t . 31/ p. 3 (1977).  J . App. Phys. 40, p. 3389 (1969).  Chen, LaMacchia, Fraser:  App. Phys. L e t t . 13, p. 223 (1968).  Clark, Disalvo, Glass, Peterson:  J . Chem. Phys. 59, p. 6209 (1973).  C o l l i e r , Burckhardt and L i n : Optical Holography.  Academic Press, Inc. (1971).  Cornish, William D.: The Photorefractive Effect i n Lithium Niobate. t h e s i s , Dept. of Elec. Eng., U.B.C. (1975). Cornish, Moharam and Young: Dischler, Rauber:  J . Appl. Phys. 47, p. 1479  Ph.D.  (1976).  S o l i d State Comm. 17, p. 953 (1975).  e l Guibaly, Fayez H.F.: The Photorefractive Effect i n Lithium Niobate and i t s Applications. Ph.D. thesis, Dept. of Elec. 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Tamir, T. ( e d i t o r ) : Integrated Optics (2nd ed.) ( v o l . 7 of "Topics i n Applied Physics"). Springer-Verlag (1979). Turner, E.H.: von Baltz, R.:  Appl. Phys. Lett. 8, p. 303 (1966). Phys. Stat. S o l . (b) 89, p. 419 (1978).  von der Linde, Glass and Rogers:  Appl. Phys. L e t t . 25, p. 155 (1974).  von der Linde, Glass and Rogers:  Appl. Phys. L e t t . 26, p. 22 (1975a).  von der Linde, Glass:  Appl. Phys. 8y p. 85 (1975b).  von der Linde, Glass:  F e r r o e l e c t r i c s JL0, p. 5 (1976).  Young, Wong, Thewalt, Cornish:  Appl. Phys. L e t t . 2!4, 264 (1974).  Young, Moharam, E l Guibaly, Lun: J . Appl. Phys. 50(6), 4201 (1979).  58  APPENDIX A LiNbOg  A.l  Sources  Ohio.  The  crystals  Inc.,  former  of the melt  Table  s h o u l d be r e a l i z e d subject result  California,  can e x h i b i t  i n which  of processes,  five  Co., S o l o n ,  a poling from  electric  the melt.  o r congruent,  f o r each  where t h e  crystal  years o r so, these  not always  characteristics  Crystal  ( 4 9 . 0 m o l e % L i 0 2 v s . 48.6 m o l e %  the r e l e v a n t data  the past  by  a n d by Harshaw C h e m i c a l  stoichiometric  the latter  A - ll i s t s  prepared  i s r o t a t e d as i t i s withdrawn  i s either  t h a t over  t o a number  identical  work were c o m m e r c i a l l y  grown b y t h e C z o c h r a l s k i t e c h n i q u e ,  c o n t a i n s more L i t h a n  respectively).  A.2  i n this  i s a p p l i e d and the boule  composition  nally  used  Mountain View,  T h e y were  field  Data  of Crystals  The Technology  Crystal  used.  crystals  of a well recorded  LiO£ It  have  nature,  been  and as a  w h i c h may n o t b e r e p r o d u c i b l e i n a n o t h e r  nomi-  crystal.  Miscellaneous Properties Lithium  structure electric  niobate  (ABOg).  As a s t a b l e  d i p o l e moment e v e n  its  transition  its  moment.  hysteresis perature) .  temperature,  Below T loop  The c r y s t a l  can  be t h o u g h t  close packing  a distorted  a t room t e m p e r a t u r e ,  or Curie p o i n t  (T  (Nassau  with  with  o f an e l e c t r i c = 1470°  of polarization applied field  has a rhombohedral  of as being  crystal  i n t h e absence  as t h e e x t e r n a l l y  .5499, a = 5 5 ° 5 2 ' )  occupied  ferroelectric  , t h e degree  a =  gonal  i s a ferroelectric  field.  perovskite  i t e x h i b i t s an I f heated  above  k f o r LiNbOg) i t l o s e s  c a n be made t o v a r y  i s varied  (difficult  structure (point  group  3m  along  a  a t l e w temwith  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 .  composed  of sheets  the r e s u l t i n g  o f oxygen  i n approximately  octahedral interstices  b y L i , o n e - t h i r d b y Nb, a n d t h e r e m a i n i n g  hexa-  being one-third  o n e - t h i r d empty.  There a r e  It  59  two  separate  experiments relative  values  each  f o r t h e L i - 0 a n d Nb-0  was done w i t h  transparency  l i g h t o f X = 514.5nm, w h i c h  between  500nm a n d 2 5 ° C a r e n  distances.  =2.34  350nm a n d 5000nm. and n  _  =2.24.  The p r e s e n t  falls  set of  within the region of  The i n d i c e s  of refraction  The d i e l e c t r i c c o n s t a n t s  at  aree  „ r  o =  78  (perpendicular to c-axis)  pyroelectric  coefficient  coefficients  of.expansion  C. a l o n g  A.3  Electro-optic  crystal  field. is  yC/(m deg) a t 100°C.  2  2  o f 16.7 x 1 0 ~ ^ / d e g C . a l o n g  The  I t melts  a t 1260°C and has  the a-axis  and 2  x 10~^/deg  Behaviour  i s usually  s u f f i c i e n t to consider the permittivity  t o be c o n s t a n t s ,  However, i t i s t h e c a s e  a change  f o r some c r y s t a l s  i n the permittivity.  a change i n t h e i n d e x  coefficients  of  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  a small but detectable higher-order  cause to  i s 10~  = 32 ( p a r a l l e l t o c - a x i s ) .  r  the c-axis.  It a  and e  effect  whereby r e a d i l y  At o p t i c a l  of refraction,  (including  LiNbOg) t h a t attainable  frequencies this  and i s the b a s i s o f the  there fields  i s equivalent electro-optic  effect. The anisotropic  general  crystal D. i  where  e  =  tensor  expression  at optical  frequencies i s  ID  e e. .E. o 3  i s the tensor  relative  E t o the d i e l e c t r i c displacement pagation  with  magnetic  wave i n a c r y s t a l ,  propagate  permittivity  D.  respect to the crystal  f o r a given  ellipsoid,, called  f o r t h e d i e l e c t r i c p r o p e r t i e s o f an  Both  the optical  and d i r e c t i o n  the propagation  two waves o f d i f f e r e n t  Their refractive  indicatrix,  the applied electric  the p o l a r i z a t i o n  axes a f f e c t  and i n g e n e r a l  wave n o r m a l .  relating  indices  o f an  of proelectro-  velocities  may b e f o u n d  d e f i n e d by t h e f o l l o w i n g  field  may  by t h e  equation:  60  f—) n. ' l  =  v  1  ,  n.  =  1  w h e r e we h a v e u s e d t h e E i n s t e i n twice  i n t h e same t e r m  centre  perpendicular  ellipse The is  now  13k  k  second rank  r  "*  mn  symmetry and  the rest  elements  crystal  results  through i t s  of the i n d i c a t r i x ,  which  +  . ..1 '  £  x.x. i  =  1  i,j,k,£ =  1,2,3  j electro-optic  are inconvenient  to write  t o the following  coefficients,  out, contractions  conventions:  (ij)k  = 1,2,...,6 a n d m i s r e l a t e d  a uniaxial  i s cleaved  occurs  (i;j)(k£)  3-»-33, 4-»-23, 5*13, 6*12. is  k  rank t e n s o r s  Z  which  o f t h e axes o f t h e r e s u l t i n g  i n deformation  form a r e performed a c c o r d i n g  mk  any i n d e x  o f t h e two waves ( s e e F i g u r e A - 2 ) .  a n d R.. are t h e l i n e a r and q u a d r a t i c i]k£  Because t h i r d and f o u r t h  w h e r e m,n  ink£  matrix)  A.l)  + Z . . E + R . ., „ E, E  2  *• i j  to  results  (eq.  that  I f the e l l i p s o i d  of polarization  o f an e l e c t r i c f i e l d  fnT  convention  t o t h e wave n o r m a l , t h e l e n g t h s  d e f i n e d by t h e equation  where z . . ijk  summation  i s t o be summed.  y i e l d the directions  presence  / e.. (e.. of diagonalized i i i i  We  t o i j and n t o k £ by t h e r u l e s  are interested  exhibiting  i n t h e s p e c i f i c case  the electro-optic  i n most o f t h e e l e c t r o - o p t i c  being  interdependant.  are (Turner  1966)  The m a t r i x  effect.  coefficients  o f LiNbOg,  Itshigh r  m  k  1+11, 2->-22,  being  form and numerical  which  degree o f zero,  values  of i t s  61  Dimens:.on (nnm) a b c  Crystal Number 1  1 5  (2) -,(3)  6  Doping  (mole %)  Composition of t h e Melt  3  20  b  undoped  congruent  10  10  a  0.1%  stoichiometric  (1)  a c c o r d i n g t o numbering  (2) (3)  broken - h a l f used was 7 x 3 x 20 mm. nominal - a c t u a l measurement i s 0.69 mm. FIGURE A-1:  optic  Iron  Polished Faces  of E l Guibaly  Table o f C r y s t a l  (1979).  Data  axis  FIGURE 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 uniaxial crystal.  FIGURE A-3 Application  of an e l e c t r i c f i e l d  Eg t o a l t e r t h e i n d i c e s  of refrac-  t i o n seen by a wave p r o p a g a t i n g in  t h e x^ d i r e c t i o n .  LiNb03  c rystal  ind i c e s no  axes  diagram axes  62  0  - r 22  0  The  22  2  direction  Figure the  n  0  The  indicatrix  +  0  22  If  field  n  (n  Equating  =  n  o  3.4  x  f o r the s p e c i a l  case  with the f i e l d  cm/Volt  1 0  28  x  "  30.8  x  "  the index  o f a wave p r o p a g a t i n g i n t h e  applied  i n the x^ d i r e c t i o n (see  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  2  o  - 22 2 r  ( V  we c o u l d  x 10"  c a n be u s e d t o o b t a i n  the i n d i c a t r i x here  l  n  2  E  r  +  1 + 2  +  r  1 3  E 3  33 33) 3 E  2  X  2  )  2  X  +  + 2(r  3  l  2  ( + 2  2  ( V  +  +  2  22 2  r  E  +  r  13  E  ) 2 X  3  2  (" 22 1 > 1 2 r  E  E )x x 1  3  X  =  1  1  (A.2)  = 0 and t h u s t h e above r e d u c e s t o  2  r  reduces t o  +  E )x x  13 3) l  consider f i e l d  indices,  8.6  are equal:  2  2  33  i s uniaxial  a crystal  o u r c a s e , E^ = E  we  LiNbOg  ( e q . 1) t h u s  ( V  «t2  0  :  that  + 2(r  For  33  0  =  ( V  r  0  =  n  22  0  12  axes  13  hZ  Because  principal  r  0  through  A-3).  13  0  change t o t h e a p p l i e d x  13  0  :  fact  r  E  E  3  x  2  (  +  n E  ~  as e f f e c t i n g  2  +  r  33  E  ) 3 x  3  changes  2  An  =  0  (A.3)  1  and A n  £  i n the r e f r a c t i v e  write  + An  ]  -  o'  the c o e f f i c i e n t s  2  E  x 3 1  +  2  0  of x ^  2  f n + An ) ~ >-e e  2  E_ x _  i  2  i  =  i n (3) and (4) and s o l v i n g  1  (A.4)  for An  0 /  w  e  g  e  t  63  An  and  similarly  simple  between  =  formula  r  13  E  3  2.  f o r the x-  An  This  O  V  =  2  coefficients  we  get  -  f o r the index  An a n d t h e a p p l i e d  field  change a l s o E for this  points  special  out the l i n e a r case.  relation  64  APPENDIX Piezoelectric  As reference upon  shown i n F i g u r e  beams was  altered  which the r e f e r e n c e  and  details  i)  Basic  Certain mechanical This  by a p p l y i n g  with  i s referred  matically,  develop  crystal o ( j , k Dk  =  the stress  applied  tion,  as  moment u p o n  field  of the e f f e c t  c a n be  to the  of stress.  represented The  state  the.faces  Mathe-  by a t h i r d rank of stress,  of nine  a., r e f e r s , }k  phe-  t h e shape o f t h e  effect".  by a s e c o n d rank t e n s o r  across  application  t o change  piezoelectric  a s p e c i f i c component  i n the j d i r e c t i o n  tensor  a, o f a  components,  f o r instance, to a  perpendicular  to the k  direc-  i n Figure B . l .  P^.  a l lnine  stress  i s given  thus-the  of a crystal  In the g e n e r a l  case,  components.  i s a vector quantity  each o f these  For instance,  three  of three  components  i n the 1 d i r e c t i o n ,  is a  the  comfunction  polariza-  by Pi  and  effect  specified  The p o l a r i z a t i o n ponents,  and  disc  e f f e c t " , and t h e complementary  and p o l a r i z a t i o n .  1 , 2 , 3 ) , where  force  An e x p l a n a t i o n  linearly proportional  o f an e l e c t r i c  t o as t h e "converse  i s completely  the object  to the p i e z o e l e c t r i c  mounted.  an e l e c t r i c  t h e moment b e i n g  the p i e z o e l e c t r i c  d . r e l a t i n g ijk  tion  was  between  Theory  nomenon, i . e . t h e a p p l i c a t i o n  of  a voltage  i s c a l l e d the " d i r e c t p i e z o e l e c t r i c  crystal,  Shifter  follow.  crystals  stress,  Phase  the phase r e l a t i o n s h i p  beam m i r r o r  of the device  Piezoelectric  7.1,  B  =  general P.  j.  =  d  0  j  (j,k=  k  expression d.  «  i l k  -j)  1,2,3)  f o r a l lthree  (B.l)  directions  of polarization i s (B.2)  65  FIGURE B.I The  stress  refers  to a force  direction dicular  tensor  across  t o the k  element  A  o ,  i n the j faces  perpen-  direction.  e-  FIGURE  B.2  Shape a n d d i m e n s i o n s o f t h e Vernitron piezoelectric  element.  66  where  d. represents i jk  metry  often renders  numerical  values  stress,  i  reverse that,  changed  impossible  (and  a  n  d  t o an e q u a l  of the crystal  i n t h e absence  s  o  t o apply w  e  c  a  n  o  n  l  a stress v  Equation  Furthermore, having  t h e number o f i n d e p e n d e n t  further  simplification  of the mathematical  the  first  a  single  digit  P^O of as i t i s  stress  I t i s conventional  d., . a s a r e s u l t  o f t h i s , and  l k j  i s reduced  from  27 t o 1 8 .  representation i s achieved  A by  reduc-  convention:  are replaced by  2 3 + 4  2 2 + 2  3 1 , 13 + 5  33  21,  + 3  1 2 + 6 t h e d ^ j where  we make e q u i v a l e n t c h a n g e s t o t h e s t r e s s l  Or  matrix:  i st o  according t o the rule:  c  In t h i s  tensile  i n e q . B.2 i s t h e f a c t  3 t o 2 according t o the following  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  consistency,  27  ^he e f f e c t  s u b s c r i p t r e m a i n s t h e same, b u t t h e s e c o n d a n d t h i r d  11+1  Factors  unstated  sum.  to setd . =  coefficients  t h e number o f s u b s c r i p t s f r o m  than  also applying the opposite  measure t h e i r  ing  stress,  sym-  I f we i g n o r e b o d y t o r q u e s ,  13k thus  fewer  t h e P^ i n e q . B.2 m u s t b e t h o u g h t  o^j without  c a n b e shown t o b e n e c e s s a r y )  so t h a t  crystal  a spontaneous p o l a r i z a t i o n  polarization.  physically  In p r a c t i c e ,  B.2 i m p l i e s t h a t i f a  and o p p o s i t e  o f any s t r e s s ,  i n the total  moduli.  interdependent,  the sign of polarization.  change e f f e c t e d  °ji'  many o f t h e  n e e d be c o n s i d e r e d .  i n t h e case  present a  s  t h e 27 p i e z o e l e c t r i c  °6  °5  0  a,, 4  o  b  2  °5  °4  i . e . a. . ,k 3  j = 4,5,6. tensor  + a.  3  For  toget  ,  j =  1,...,6  °3  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  moduli take  t h e form o f t h e  67  '12 A  l  etc. l  B.2 i s now d.  we n o t e t h a t  stress field  i n the converse  f o r the strain  e  e  l l  e  _ 31  e  e  and  then  relating  relating  o f e q . B.2 i s  by e f f e c t i n g  the following  substi-  2 2  32  e  13  e  23  e  33_  e  'z<  l  l  -v  e  2 e  3-  writing  The " V e r n i t r o n " The  was a p o l e d ,  i  which  Lead  Zirconium phase  thickness-expanding dimensions  coefficient  relates  = 1,2,3  j =  1,2,...,6  1  piezoelectric  (PZT) w i t h  piezoelectric 20%,  a r e t h e same a s t h o s e  Thus t h e a n a l o g u e  i n matrix notation  12  ID  titanate  effect.  the coefficients  i s used,  E. i  d. . E.  ii)  effect  effect  components:  e  21  1,2,...,6  ( N y e , p . 115) t h a t  i n the direct  t o o c a n be e x p r e s s e d  tution  j =  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  d. .. ijk  This  = 1,2,3  i t c a n be shown  and p o l a r i z a t i o n and s t r a i n  i  3  the experiments dealt  and  36—I  written  .a.  !D  In  16  *26  d  relation  l  21  U 31  The  13  Piezoelectric  s h i f t i n g element sintered  ceramic  and c o n v e n t i o n a l  given  the s t r a i n  Titanate  i n the Vernitron  i n the 3 d i r e c t i o n  Element  used i n these disc  of lead  experiments  zirconium  a x e s a s shown i n F i g . spec sheet  B.2.  The  i s d ^ = 285 pm/V  to the applied  voltage  i n the  ±  68  (  _J  i O  c E E  LO  IT)  o-  LU  o c  I  (U-  i  <1>  v C-  C  FIGURE B - 3  Response of PZT t o an a p p l i e d  voltage:  The PZT element was i n s e r t e d ; i n t o • one arm' of;., a Michelson interferometer,  and the v o l t a g e  • •!  required • > . _  t o d i s p l a c e the f r i n g e s by one p e r i o d aas measured. :  69  3 direction. is  d  3  3  = 219 pm/V A  thereby of  The v a l u e ± 2%  lower  study  on t h e t i m e  initiates  ( F i g . B.3).  i n the ceramic  i s somewhat i n c o n v e n i e n t  step  the voltage  measuring where.  through  required t o step  solution  i s , a change  immediately  Interferometer  the voltage  o n t h e PZT a n d  exponential  as, during  i n voltage  across  approach  t o t h e new  the r e l a t i v e l y  r a n i n t o problems with  ceramic  dimension.  l o n g time r e q u i r e d t o  range, t h e hologram b e i n g  writing of the gradually s h i f t i n g  of reducing  the  a s s u m i n g i t s new d i m e n s i o n s , b u t  the required voltage  due t o t h e c o n t i n u e d  obvious  That  an a p p r o x i m a t e l y  This  The  l a b using a Michelson  t h e c o u p l i n g o f t h e w r i t i n g beams i s s e t b y t h e r e l a x a t i o n t i m e  does n o t r e s u l t  changes  i n this  ( F i g . B.2).  limit  t h e PZT c e r a m i c  rather  obtained  t h e beam i n t e n s i t y  to inhibit  thermoelectric effects,  fringe writing  as mentioned  studied  pattern. during else-  70  APPENDIX Michelson (PZT  In t h i s the  PZT  phase  shifter  bench.  Figure  is  less  than  by  the  form  a  sage o f  at  any  shifter,  pattern.  of  fixed  the  the  Setting variation  of  isolation  o f f e r e d by  first  clearly  was to  second  source  c a u s e d by the  decrease minute  or  i n the so  and  the  by  on  the  screen.  I f one  the  voltage  on  will  provide  a means o f  (see  bench.  Appendix up  on  the  the  |D^-D I  the  two  2  beams  mirrors  of  Three  by  the  moving i n the was  types  formed  (M^,  will  M)  will  2  cause  path  by  was  conducting  split  minimized  the  and  the  length over  and  i s mounted  counting the  the  pas-  piezoelec-  bench  and  by  degree of  trans-  and  speed of  path  causing  the  variations in  tubing.  a  The  length  bench with  increase  a b o u t one  days t o  slam-  at night.  differential  a slow monotonic  a p e r i o d of  The  machinery) being  experiments  covering  a  the  mechanical  were n o t i c e d ' .  beams i n c a r d b o a r d  with  observing  o f t e n accompanied by  beam p a t h s  c o n s i s t e d of  decaying  mirrors  calibrating  of v a r i a t i o n  b e n c h , and  further isolating length  the  a means t o m e a s u r e t h e  random v a r i a t i o n s i n t h e  gradually  If  |Dj-D^  optical  was  differential  geometry.  B).  noise  i n path  optical  shifter  minimized  This  the  the  distance  calibrate  of  laser,  was  and  variation  device  This  lengths.  greenhouse of  point  componts  a i r currents  phase path  glass type  of  stability  to b u i l d i n g v i b r a t i o n (occupants,  optical  etc.  of  used both to  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  interferometer  due  was  back t o g e t h e r  fringe p o s i t i o n provides  the  ming doors,  the  screen  the  the  V a r i a t i o n i n the  shifter  the  vibration)  mechanical  length  brought  then varying  f r i n g e s on  constant  mitted  and  the  diagram of  coherence  (BS)  bench  Interferometer  c h e c k on  schematic  f r i n g e s t o move a s  PZT  tric  to  temporal  diffraction  destructive a  the  Interferometer  calibration,  Michelson  and  C.l is a  beam s p l i t t e r  pattern  on  work a  C  a  plexi-  The  third  or  fringe  per  f r i n g e per  few  FIGURE  C.1  Schematic of Michelson i n t e r f e r o m e t e r bench v i b r a t i o n  as used t o  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  detect  element.  72  hours.  This  attributed properly by  was  to a  immediately gradual  aligned.  waiting  a few  This  evident  every  r e l a x a t i o n of was  days a f t e r  t r e a t e d by alignment  the  time the mirror  holders  redesigning before  e q u i p m e n t was  the  t a k i n g any  after  PZT  the  holder  serious  s e t up, optics and  also  and  was  were simply  measurements.  73  APPENDIX D Applications  The larger  field  storage  o f holograms  o f modern o p t i c a l  o f Holography  i nferroelectric  technology  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 the  interfacing  reconstruction  o f holography  o f three-dimensional  wave c o m p o n e n t s  t h a t have p a s s e d  w o u l d make u s e o f b o t h the  relative  Further of  immunity  of optical  qualities  resolution, high  processing that  data  c o m p u t e r memory  Desirable high  the high  and reduced  35mm s l i d e s  mixed with  studied.  because  image b r i g h t n e s s ,  a reference  techniques  (Tamir,  will  t o mix  p . 309).  This  frequencies and  interference.  be d i s c u s s e d a n d an example  displays  35mm  durability,  upon u s i n g  i s formed by a l e n s  (e.g.  slides) simplified  (Clay; i n C a u l f i e l d ) focussed-image  on a r e c o r d i n g  holomedium  beam.  have been  c a n come u p w i t h  many p o s s i b l e u s e s  n o t t o be p r a c t i c a l use l i e s  slow t o adopt t h i s  when  i n the f i e l d new t e c h n i q u e ,  for  three-  carefully of artistic presumably  nature.  of the classical  i n t e r f e r o m e t r y setups  interferometry, with  (Brandt;  fibres  s t o r e d image  improved  t h e major p o t e n t i a l  of i t s technical  holographic  I t h a s b e e n shown  d i s p l a y s , many o f them a p p e a r  All  ties  techniques  One e x a m p l e o f  given.  c a n be c o n s i d e r a b l y  but a r t i s t s  holographic  long  develop-  involves the real-time  rate attainable a t optical  cost of copies.  I t seems t h a t  endeavour,  images u s i n g  i n two-dimensional  Though t h e i m a g i n a t i o n dimensional  information handling.  signals t o electromagnetic  grams, where a n image o f t h e o b j e c t and  i s part of the  c u r r e n t l y undergoing r a p i d  new d e v i c e s  through  applications o f holographic  a holographic  are  t o other  crystals  i nCaulfield).  the latter  These  allowing  have  their  analogs i n  g r e a t l y expanded  i n c l u d e t h e u s e o f much l a r g e r  capabili-  aperture.  74  application for  to  multiple  studies  exposure  Pattern optical  data  location  of  form of  and  character  processing.  between an  the  reference  techniques  a reference  correlation performs  i n v o l v i n g random o r  The  correlation  by  the  complex  pattern  involves  and  niques  alter  use  computer  of  success  Caulfield). transverse difficult to  of  see  field)  microscope. taking  two  offer  bulk  at  and  later  Time v a r y i n g  to  optical  and  of  unusual  they  a an  research  has  image a t  low  and  can  single,  geometrical  be  coding,  Optical may  tech-  include  Ch.  the  9).  moved t o o t h e r (Cox;  field, This  i t is can  be  magnification with  during  high often overcome (and  good  conventional detailed  by  construction.  designed  exiting  be  a  areas,  in  t o have a  growth) can  them  trans-  e l e c t r o n microscope  magnification  crystal  and  hologram).  (Catheg;  l a r g e volume.  (HOE)  input  interpreting aerial  microscopy  depth of  system  Fourier  etc.  i s designed  the  the  of  multidimensional  to  ability,  enhance  superposing  other  of  images,  in itself to  of  a  degree  common  conjugate  relate  holographic  (e.g.  element  i n t o any  provide  study  i t at high  processes  holograms  holographic  ability  viewing  as  microscope  throughout  most  applied  and/or  extraction, efficient  handling  expense of  The  manipulation  invented  with  making a hologram  e n t e r i n g wavefront  the  the  detail  a  the  of  presence  i s , in effect,  television  holographic  a conventional  fine  by  of  been a c h i e v e d  successive A  single,  has  originally  magnification  some e x t e n t  depth  the  Because  to  content  generated holograms,  Although  limited  x-rays,  instance).  Fourier transforms  information  graphics  potential  examining the  function.  (which  ( g e n e r a l l y , the  computer  was  i m a g e by  i s s t o r e d as  function  information  Holography images.  latter  and  c u r r e n t l y main areas  determine the  input  m u l t i p l y i n g the  maps, m e d i c a l ,  higher  i s to  a reference  image enhancement,  r e c o g n i t i o n and  photographs  here  i n an  and  reference  Image p r o c e s s i n g signals)  aim  f u n c t i o n s , where t h e  wavefronts,  (in vibration studies, for r e c o g n i t i o n are  pattern input  diffuse  to  transform  wavefront.  c o n f i g u r a t i o n s or  They  special  any  75  spectral large  but this  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  amount o f a b e r r a t i o n t o t h e s y s t e m  extent use  characteristics,  that  "one s h o u l d  conventional  resort  (Close; i n Caulfield).  e l e m e n t s w h i c h must  conform t o an u n u s u a l  very  ( i nthe s p i r i t  of  narrow s p e c t r a l  and coma) t o such  an  t o t h e u s e o f a HOE o n l y when i t i s i m p o s s i b l e t o  lenses and m i r r o r s "  l a r g e elements  (e.g. astigmatism  reflectivity  shape such  as a helmet v i s o r ,  of a Fresnel lens),  must be u s e d  Examples a r e design of  a n d where l a r g e  (as i n a i r c r a f t  surfaces  "heads up"  displays). The niques (2-D  c o n s i d e r a b l e number o f o t h e r  include:  spectroscopy,  optical  Finally,  and  devices. second  i n this  regard 1980).  i s less  four broad  time  severe  rapidly  accumulated  data  description  A  high  potentially  t o be d e s c r i b e d may  represent  of optical  o f an o p t i c a l  devices.  subpico-  data  on  that the  c a p a c i t y , and between a c c e s s  time and  devices  than  the others  storage  (similar  and f a s t  random a c c e s s  a solution  listed.  to archival  but cap-  (for rapid  storage  read-write-erase  holographic  t o these  One  ( l a r g e , amounts o f  recording storage  later),  capcity, fast  faster,  I t i s seen  storage  fast  t o be r e a d  memories).  type  on t h e p o t e n t i a l  c a t e g o r i e s o f memory; a r c h i v a l  occasionally altered),  f o r larger,  atten-  has l e d t o t h e development o f  to holographic  read-mostly  recording  measurements.  D l a n d D2 g i v e p e r f o r m a n c e  and storage  (as i n most computer  the  Figures  methods  ( e . g . f o r image  The n e e d  to capitalize  f o r holographic  occasionally accessed),  able o f being  storage  contouring  tech-  which has r e c e i v e d , most  storage.  the recent  i n u s e as compared  between a c c e s s  memory c o s t  f o r data  i n an attempt  (Marcatelli,  devices  trade-off  list  note  size  the application data  speeds;  generation  and p a r t i c l e  digital  memories  memories  We  gate  various  of  we d e a l w i t h  expensive  holographic  data  processing)  i n a p p l i e d 'research; less  can  a t high  maps o f 3-D o b j e c t s ) ; m u l t i p l e i m a g e  or p a r a l l e l  tion  especially  applications of holographic  memory  memory o f  v a r i e d needs, o r  76  occupy a unique position i n a h i e r a r c h i c a l set of solutions. considerations ( H i l l , 1976)  I n i t i a l design  indicated that holographic memories would have  these f i v e basic features: i)  they would be Fourier transform holograms (as opposed to d i r e c t images) i n order to protect against l o c a l i z e d loss of data due to medium imperfections or surface dust;  ii)  they would be i n a 2-D page format, as the 3-D  imaging  capability  of holograms offers no advantages; iii)  the information would be i n 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 b i t 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  presented i n F i g . D.3.  storage system using a f e r r o e l e c t r i c c r y s t a l i s  The Bragg angle s e l e c t i v i t y of such a medium i s used to  enable the superposition of multiple holograms (up to 500), so the address of a b i t i s XY$.  The main o p t i c a l components are the l i g h t source, beam deflectors,  page composer, recording medium, and detector array. The l i g h t source should provide intense (~1 Watt), collimated, coherent l i g h t pulsed or gated at about 1 MHz the spectrum to take advantage of the A and wavelength.  -  3  and i n the blue or green part of  relationship between storage density  The argon-ion gas laser meets these requirements,  although i t  displays a low conversion e f f i c i e n c y of e l e c t r i c a l power to o p t i c a l power. Fast and accurate beam deflectors are needed to p o s i t i o n the laser beam for reading, writing and erasing operations.  Acousto-optic and e l e c t r o -  optic deflectors are the main candidates, with the l a t t e r being f a s t e r but suffering, l i g h t loss through surfaces and having a high cost.  Because  77  electro-optic often be  coefficients  cascaded  of  an  form  composer ratio, large  data  array of  beam t o  are  0.8  input  a data  m  crystal  effect.  shut"  to  at the  frame  have  electro-optic 2  deflection  m  which  array.  angles,  interfere  The  full  i s the  composer, which  with  resolution,  per  c o n s t r u c t e d by  and  page.  RCA  may  and  the  reference  of t h i s  page  contrast  sufficiently  A u s e f u l but  (Labrunie  than  consists  a high  a  l e a d lanthanum  i s faster  and  requirements  page a d d r e s s i n g  l a r g e number o f b i t s been  are  deflector  o b j e c t beam p a t h  speed, h i g h  alternate possibilities data  large deviations they  achieved.  detector  a high  composer has  (PLZT) b l o c k  give  binary apertures  t o accomodate a  Among a number o f  A digital  i s l o c a t e d i n the  u n i f o r m i t y , and  page  to produce  deflectors  device  matrix  small  ys. have been  that i t exhibit  aperture  titanate  of  "open o r  stability,  liquid  cascading  times  The  too  t o i n c r e a s e the  c o n s t r u c t e d by  random a c c e s s  are  the  slow  e t a l . , 1974).  zirconate  liquid  crystal  determines  to a  version. The extent  how  of  ideal  the  recording material i s a  flexible  rewrite  structed  capability.  data  photodiode  complete  a high  available high To  the  light,  long  no  an  must e x h i b i t  random a c c e s s  and  system w i l l  and  be.  lifetime  used  electrical one  high  periods  or  and  signal two  The  these  the  addressing  readout  by  technology  characteristics efficiency  readout,  to  be  an  erase-  requirements.  of  switches able  words. to  and  high  holographically recon-  would c o n s i s t  and  large  nonvolatile storage,  a l l of  convert  detectivity  to allow  t o a l l words.  to  The  large diffraction  material f u l f i l l s  or p h o t o t r a n s i s t o r and  energy f o r b r i e f  component  density storage, nondestructive  date  pattern into  entire  sensitivity  array of photodetectors  These photodiodes optical  of  to provide  The  efficient  medium a r e  make maximum u s e resolution  and  central  one  f o r each b i t .  to store The  sensing  incident  a r r a y must  allow  construct large defect-  78  free  arrays of this  type  already exists  i n t h e form  of silicon-diode-array  camera  tubes. Although order  of X  - 2  ,  the theoretical  a n d f o r 3-D  upon t h e system by o t h e r aperture  effects  number o f b i t s addresses larity  p u t a lower  t h e number optical  nonmechanical  d u c e d b y 3M i n 1 9 7 4 . rates,  factors  limit  d e n s i t y f o r 2-D  (n/X)^, p r a c t i c a l  decrease  this  on s p o t  deflector  p e r p a g e , a n d t h e medium  Several merical  of the order  p e r p a g e , beam  a l l limit  storage  limit.  size,  resolution  memories a r e c o m m e r i c a l l y memory was  I t i s a 50 M b i t  and uses a s e t of separate  imposed  For instance,  optical  limits  r e c o r d i n g range,  image c r o s s t a l k  o f XY and granu-  superposed. available.  data  The f i r s t  processor  system capable  2-D h o l o g r a p h i c  limits the  t h e number  the Megafetch data  read-only  i s of the  limitations  detector noise  o f h o l o g r a m s t h a t c a n be  optical  systems  plates.  com-  intro-  o f 15 Mbaud  data  79  — i  r  MAGNETIC TAPE  10'  « HOLOSCAN  I  I  I  OPTICAL BIT BY BIT » I3M 1360  GRUMMAN MASSTAPE » 3AMPEX TEM ' O UNICOW  MOVABLE HEAD DISKS 10-  FIXED HEAD DISKS DRUMS CCD  10"  CORES  © MEGAFETCH HOLOGRAPHIC  MOS  10"  BIPOLAR  io-  J  10'  L  J  10"  108  J  L  10'  10'  L 10  STORAGE CAPACITY (bits)  FIGURE D - l ( a ) A comparison o f v a r i o u s memory types i n terms o f access time and cost per b i t of stored data.  (from C a u l f i e l d )  80  T  r  MAGNETIC TAPE  10* OPTICAL BIT BY BIT  MOVABLE HEAD DISKS FIXED HEAD; DISKS  1CT  DRUMS  CCD  10-  CORES  HOLOGRAPHIC 10~  MOS  BIPOLAR  J  10"  io-  10"  L  l  io-  10"  i  l  t -1  to  10  10'  MEMORY COST (cents/bit]  FIGURE D - l ( b ) A comparison o f v a r i o u s memory t y p e s i n terms o f a c c e s s time and  storage c a p a c i t y ,  (from C a u l f i e l d )  POLARIZATION SENSITIVE  COUIMATING lt"S  HIGK-EFflCIENCY GRATING  FIGURE D - 2 A h o l o g r a p h i c o p t i c a l memory system u s i n g t h r e e - d i m e n s i o n a l (volume) storage (from C a u l f i e l d )  82  APPENDIX  A number o f d a t a work.  Among t h e s e  was  E  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  the hologram w r i t i n g  and r e a d / e r a s e  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 c a s e  shift.  T h i s was  Figure  8.7  (solid Two  (1)  slightly lines),  "ZERO"  i s used  and used  programs a r e given  t o o b t a i n the dashed  of the s t r a i g h t  line  lines  grating  as t h e c o r r e s p o n d i n g c u r v e d  "EXPCUP"  i s used  t o generate  of coupling curves  initial  phase  but constant  work t o o b t a i n t h e d a t a f o r  i n F i g u r e 8.7, w h i c h that  would produce  shows t h e  t h e same  o f F i g u r e 8.8, w h i c h  would a r i s e  r u n shown  curvature.  here:  f o r t h e case o f a  variation  i n t h e phase  h o l d i n g the c o u p l i n g and d i f f r a c t i o n  the experimental  phase  grating.  the curves  (|> f o l l o w e d b y a t i m e  ("SIMDAT" i s a f i l e from  which  w r i t t e n by  development o f t h e g r a t i n g  coupling  sort  o f a non-zero  i n this  showing the time  of the remaining  displacement  (2)  altered  program  i n this  i n Figure  8.2.)  shows t h e non-zero  (d<j>/dt data  dds£0).  measured  1 2 3 4 5 6 7 8 9 10 11 12 13 1-1 15 1G 17 18 19 20 21 22 23 24 25 26 27 23 29 30 31 32 33 34 35 36  37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60  c  C C C C C C C C C C C C C C  1  20  2 10  FILE:  "ZERO"  ' Z E R O ' INPUTS SELECTED DATA FROM ' H 0 L 0 3 ' (10 TIME S A M P L E S , 12 Z - S E G M E N T S ) DESCRIBING THE PHASE (PHIHG) AND AMPLITUDE (AMP) OF THE BENT HOLOGRAM GRATING, AND MOVES THE GRATING OVER TO A POSITION STRADDLING THE FRINGE PATTERN SO THAT THE COUPLING IS NULLED ( I . E . , THE E X I T I N G R AND S BEAMS ARE SET E Q U A L . ) AMPCON IS A CONSTANT CONVERTING THE F I E L D AMPLITUDE OUTPUT FROM ' H 0 L 0 3 ' TO INDEX VARIATION AMPLITUDE FOR USE IN THE SUBROUTINE ' T H R U ' . 'DO 10' >> TIME STEPS REAL T I M E ( 1 0 ) . A M P ( 1 2 , 1 0 ) . P H I H G ( 1 2 , 1 0 ) , AMPC0N=5.2E-6 CALL MODEL(AMPCON,TIME,PHIHG,AMP) WRITE(5,1) FORMAT(' TIME ' . ' E F F E C T I V E S H I F T ' ) DO 10 1=1,10 DO 20 J = 1 . 1 2 AMPV(J)~AMP(J,I) PHIHGV(.J) = PHIHG( J , I ) CONTINUE CALL EOUAL(PHIHGV,AMPV.PHIEO) P H I E Q 0 = P H I E Q * 4 5 . / A T A N ( 1 .0) W R I T E ( 5 , 2 ) TI M E ( I ) , P H I E O D FORMAT(' ' , F 4 . 2 , F 1 2 . 2 ) CONTINUE STOP END  AMPV(12).  PHIHGV(12)  c+ + * + * + * + **-r + *t* + *** + + + ********** + * + *t + ** + + ** + ** + + SUBROUTINE C C C C C C  MODEL(AMPCON,TIME,PHIHG,AMP)  MODEL READS THE STORED OUTPUT FROM H0L03 IN TWO 12X11 MATRICES ( A P . F ) WHICH GIVE INTERNAL FIELD AMPLITUDE AND PHASE S H I F T , AND M U L T I P L I E S THEM BY CONSTANTS "CONV" TO CONVERT THE PHASE TO RADIANS AND "AMPCON" TO CONVERT THE F I E L D AMPLITUDE TO AN INDEX AMPLITUDE. "AMPCON" MUST BE GIVEN IN THE CALL STATEMENT.  51 10  20  REAL T I M E ( 1 0 ) , P H I H G ( 1 2 , 1 0 ) , A M P ( 1 2 . 1 0 ) DO 10 I - 1 , 10 READ(5,51) T I M E ( I ) , ( P H I H G ( J , I ) . d = 1 . 1 2 ) F0RMAT(F7.3,12(F8.3)) CONTINUE DO 20 1=1,10 READ(5,51) T I M E ( I ) , ( A M P ( d , I ) , J * 1 . 1 2 ) CONTINUE CONV=ATAN(1.)/45. DO 30 I 1 , 10 DO 40 J=1 ,12 • AMP(J,I)=AMP(j,I)*AMPCON P H I H G ( U , I ) = P H I H G ( J , I ) *CONV CONTINUE 3  40  00 w  61 62 63 64 65 67 G8 69 70 71 72 73 74 75 76 77 78 79. 80 81 82 83 84" 85 86 87 88 89 90 91 92  g3  94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120  30  CONTINUE RETURN END  SUBROUTINE C C C C C C  EQUAL(PHIHG,AMP,PHIEO)  EQUAL DOES A SEARCH FOR THE PHASE ANGLE "PHIEQ" WHICH WOULD RESULT IN NO COUPLING IF THE HOLOGRAM GRATING WAS A SIMPLE LINEAR O N E . I . E . , IT IS THE " E F F E C T I V E " PHASE SHIFT OF THE GRATING MEASURED EXPERIMENTALLY WHEN THE PZT VOLTAGE IS ADJUSTED FOR ZERO COUPLING. "DO 10" >>SEARCH S T E P S ; 'DO 20" >> S H I F T S GRATING BY PHIEQ  20  10 11  REAL P H I H G ( 1 2 ) , A M P ( 1 2 ) , P H I H G T ( 1 2 ) PHIGAP = ( P H I H G ( 1 2 ) - P H I H G ( 1 ) ) / 2 . PHIEQ=PHIHG(1)+PHIGAP DO 10 1 = 1,13 DO 20 0 = 1 . 1 2 PHIHGT (<J) =PHIHG( J ) - PHI EQ CONTINUE CALL T H R U ( P H I H G T , A M P , R I N T , SI NT) PHIGAP=PHIGAP/2. I F ( R I N T . L T . 1 . 0 ) PHIEQ=PHIEQ-PHIGAP I F ( R I N T . G T . 1 . 0 ) PHIEQ=PHIEQ+PHIGAP I F ( A B S ( R I N T - 1 . 0 ) . L E . 0 . 0 0 0 1 ) GO TO 11 CONTINUE RETURN END  £t*************+++*********+*+***********+***** SUBROUTINE T H R U ( P H I H G , A M P , R I N T , S I N T ) C C C C  "THRU" SENDS TWO BEAMS OF EQUAL DEGREES TO THE NORMAL THRU THE I N T E N S I T I E S AT THE OUTPUT SIDE I S 5 1 5 . 5 NM AND LAYER THICKNESS  AND UNIT STRENGTH AT + / - 19 12 LAYERS AND G I V E S THEIR (RINT,SINT). THE WAVELENGTH IS D Z . ( S E E MOHARAM P74 FOR MATH)  REAL A M P ( 1 2 ) , P H I H G ( 1 2 ) COMPLEX R . S . R S T O R E . C Z T . S Z T WL=514.5E-7 DZ=.068/12. PI=4 . * A T A N ( 1 . 0 ) THETA=19.*PI/180.  10  R=CMPLX(1.,0.) S=CMPLX(1.,0.) DO 10 1=1,12 RSTORE =R C=PI*AMP(I)/WL/COS(THETA) CZT=COS(C*DZ) SZT=SIN(C+DZ) R=R*CZT-(0.,1.)+S*SZT* & CMPLX(COS(PHIHG(I)),-SIN(PHIHG(I))) S = S * C Z T - ( 0 . , 1. )*RSTORE*SZT* & CMPLX(COS(PHIHG(I)),SIN(PHIHG(I))) CONTINUE  121 RINT=REAL(R*CONdG(R)) 122 SINT=REAL(S*CONUG(S)) 123 RETURN 124 END 125 of F i l e 1 2 c* 3 C F I L E : " E X P C U P " >> TO REDUCE 'SIMDAT' PARAMETERS. 4 c* 5 REAL R C U P ( 8 ) , S C U P ( 8 ) , R D I F ( 8 ) , S D I F ( 8 ) o CALL INPUT(RCUP,SCUP,RDIF,SDIF) 7 8 CALL ANDAT(SDIF.RCUP) 9 CALL S I M D A T ( S D I F ) 10 STOP 11 END 12 13 14 SUBROUTINE INPUT(RCUP,SCUP,RDIF,SDIF) 15 1G c INPUT READS DATA FROM F I L E 'SIMDAT'. NORMALIZES THE 17 c FIGURES, AND PUTS THEM INTO THE PROPER VECTORS. 18 19 REAL R C U P ( 8 ) , S C U P ( 8 ) , R D I F ( 8 ) , S D I F ( 8 ) 20 READ(5.51 ) ( R C U P ( I ) ,1 = 1 . 8 ) , ( S C U P ( I ) , 1 = 1 , 8 ) , 21 1 ( R D I F ( I ) , 1 = 1 , 8 ) , ( S D I F ( I ) , 1 = 1,8) 22 51 FORMAT ( 8 F 5 . 1 ) 23 DO 10 1=1,8 24 RCUP(I)=RCUP(I)*2./(RCUP(I)+SCUP(I)) 25 SDIF(I)=SDIF(I)/(SDIF(I)+RDIF(I)) 26 10 CONTINUE 27 WRITE ( 6 , 1) ( S D I F ( I ) , I = 1 , 8 ) , ( R C U P ( I ) , I = 1,-8) 28 1 FORMAT( ' ','SDIF= ' . 8 F 7 . 3 . / . ' RCUP= '.8F7.3) 29 RETURN 30 END 31 32 33 34 SUBROUTINE ANDAT(SDIF,RCUP) 35 36 C "ANDAT" CALCULATES THE PHASE SHIFT FROM THE NORMALIZED 37 C ' COUPLING AND DIFFRACTION EFFICIENCY FOR EACH DATA POINT. 38 39 REAL S D I F ( 8 ) , R C U P ( 8 ) , P H I ( 8 ) 40 DO 10 1=2,8 41 PHI(I)=45./ATAN(1.0)*ARSIN((RCUP(I)- 1.)/SIN(2.*ARSIN(SORT(SDIF(I))))) 42 10 CONTINUE 43 WRITE(6, 1) ( P H I ( I ) , I = 2 , 8 ) 44 1 FORMAT (' ',/,'PHI= ****.** '.8F7.2) 45 RETURN 46 END 47 48 49 50 SUBROUTINE S I M D A T ( S D I F ) 51 52 c "SIMDAT" STARTS WITH THE OBSERVED VALUES FOR THE 53 c D I F F R A C T I O N E F F I C I E N C Y AND CALCULATES THE COUPLING 54 c FOR VARIOUS VALUES OF P H I ( T = 0 ) AND D/DT ( P H I )  03  

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