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Video bandwidth compression using hologram technique Akhtar, Sayed Amin-u-Daulah 1965

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VIDEO BANDWIDTH COMPRESSION USING HOLOGRAM TECHNIQUE by SATED AMIN-u-DAULAH AKHTAR B.Sc. E n g i n e e r i n g , U n i v e r s i t y of Dacca, 1961. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of Electrical We accept t h i s  Engineering  t h e s i s as conforming to the  standards r e q u i r e d from the candidates f o r the degree of Master of A p p l i e d Science  Members of the Department of E l e c t r i c a l  Engineering  The U n i v e r s i t y of B r i t i s h August  1965  Columbia  In p r e s e n t i n g  fulfilment  of  the requirements f o r an advanced degree at the U n i v e r s i t y  of  British  Columbia,  available  this  thesis  I agree that  in p a r t i a l  the L i b r a r y s h a l l  f o r r e f e r e n c e and s t u d y .  make i t  I f u r t h e r agree that  m i s s i o n f o r e x t e n s i v e copying o f t h i s  thesis  for  freely per-  scholarly  purposes may be granted by the Head o f my Department o r by his  representatives^  cation of t h i s  thesis  without my w r i t t e n  It  i s understood that copying o r p u b l i -  for financial  permission.  Department The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  Columbia  gain s h a l l  not be allowed  ABSTRACT This t h e s i s i s p a r t of a f e a s i b i l i t y  study  concerned  w i t h the a p p l i c a t i o n of the Gabor Hologram to a proposed bandwidth compression  of t e l e v i s i o n  method of  (video) s i g n a l s obtainable by  scanning a r e s t r i c t e d c l a s s of two tone p i c t u r e s :  the experimental  work c o n s i s t s of producing and improving on the Gabor type grams and demonstrating The proposed One  Holo-  the r e c o n s t r u c t i o n . t e l e v i s i o n system w i l l  employ two channels.  channel would be used to transmit the s i g n a l obtained by  scanning the p i c t u r e normally; the bandwidth allowed f o r t h i s channel, though, would be much l e s s than the normal bandwidth: the other channel of e q u a l l y reduced bandwidth would be used to transmit the s i g n a l obtained, again by normal scanning, from the Hologram.  At the r e c e i v e r , o p t i c a l s u p e r p o s i t i o n of the two  p i c t u r e s obtained from the two channels would be made.  For a  c e r t a i n r e s t r i c t e d c l a s s of two tone p i c t u r e s the d i s t o r t i o n produced  at the r e c e i v e r by t h i s method of p i c t u r e t r a n s m i s s i o n  i s a n t i c i p a t e d to be n e g l i g i b l e . The F o u r i e r transform property of the Hologram i s s p e c i a l l y developed system.  i n order to complement the two channel  television  C o n s i d e r i n g the transform p o i n t of view i t has been  shown t h e o r e t i c a l l y . t h a t the r e c o n s t r u c t i o n lens suggested by Gabor i s not necessary f o r the two step imaging process and a l e n s l e s s system has been developed.  I t i s demonstrated that  b e t t e r q u a l i t y r e s u l t s can be obtained by u s i n g the modified process. ii  In the experimental  setup a p o s i t i v e lens  to widen the l a s e r beam i s used.  The  necessary  e f f e c t of the noisy A i r y  r i n g s created by t h i s lens i s s t u d i e d and a pinhole method of removing the noise has been suggested demonstrated  and i t s e f f e c t i v e n e s s  experimentally.  F i n a l l y some examples of two modified l e n s l e s s system are given,, experimental  step imaging u s i n g the  The d e t a i l e d account of the  work i s given to f a c i l i t a t e f u t u r e work.  iii  CONTENTS Page List  of I l l u s t r a t i o n s  Acknowledgement .  ...............  ....  .................  1.  Introduction  2.  A n a l y s i s of the Hologram  vii  ...........  1 5  2*1 The P r e s n e l K i r c h h o f f D i f f r a c t i o n I n t e g r a l  ...  2*2 Holograms as S p a t i a l F o u r i e r Transforms of the Object 2.3 P r i n c i p l e  5 8  of L e n s l e s s R e c o n s t r u c t i o n from  a Hologram  12  2.4 Recording the Hologram:  The Phase Problem  ...  2.5 Frequency F i l t e r i n g w i t h Hologram 3.  v  A p p l i c a t i o n of Holograms  14 17  i n T e l e v i s i o n Transmis-  3.1 A n a l y s i s of the Transmission of the Hologram of a Pulse  22  3.2 A p p l i c a b i l i t y of the process 4*  5*  .. ..  31  Experimental Techniques and R e s u l t s  32  4.1 4.2  32  The Elements of the System The Technique of Producing the Hologram Spatial F i l t e r i n g  and  40  Conclusions . * . » * • . • • ' * * • * . * . . . • . . • * « o . o . « « . . . . . . . . .  Appendix A:  S p a t i a l F i l t e r i n g U s i n g a Lens System  Appendix B:  E f f e c t of Pinhole i n Removing the A i r y  Rings  ...  49 51  53  iv  i  LIST OF ILLUSTRATIONS  2.1  I l l u s t r a t i n g the Coordinate System and the Symb o l s Involved i n the D i f f r a c t i o n I n t e g r a l .......  7  2.2  B a s i c Arrangement f o r S p a t i a l S p e c t r a l A n a l y s i s . .  15  2.3  B a s i s of O p t i c a l F i l t e r i n g  19  3.1  Q u a n t i z a t i o n of Video  22  3.2  Arrangement f o r Obtaining the F o u r i e r Transform  Signal  of an Aperture  •.  24  3.3  Hologram of an I n f i n i t e l y Long Narrow S l i t  3.4  Basic Scheme f o r T e l e v i s i o n Band Compression Using the Hologram . .....»•*«•••..........••..«.« I l l u s t r a t i n g the E f f e c t of I n t r o d u c i n g a Converging Lens i n the Path of a Coherent L i g h t Beam...  34  4.2  Focussing of Coherent L i g h t  35  4.3  Pinhole Used i n Removing  36  4.4  Experimental Object  4.1  25  the A i r y Rings,  28  Arrangement f o r I l l u m i n a t i n g the 37  4.5  Using a 35 mm Camera Back f o r Recording the Hologram ........................................  39  4.6  D e t a i l s of the Setup Used f o r Recording the HoXo^i*stm » * # * * « * « o 9 * © « * » o » » * * * * * * » " » o » « a « « » * © «  o o * *  40  4.7  D e t a i l s of the Apparatus Used f o r Obtaining the Reconstuction .......  42  4.8  T y p i c a l Object, i t s Hologram and R e c o n s t r u c t i o n from the Hologram  43  4.9  Photograph of Apparatus Used i n Performing the Experi  4.10 Holograms ( F r e s n e l and F r a u n h o f f e r ) of D i f f e r e n t types of S i g n a l s  45  4.11 Noisy Hologram and R e c o n s t r u c t i o n due to the Presence of the A i r y Rings.......  46  v  4 . 1 2 Details  of the F i l t e r i n g Setup . „ „ « . o . . . . . . . . . . . . .  47  4 . 1 3 E f f e c t of S p a t i a l F i l t e r s on Some T y p i c a l SX^H£tl  S  oooo»ooao»oooooeoo*oooooo«oooooooo««e«««oo  48  A. l  P r i n c i p l e of S p a t i a l F i l t e r i n g Using Lenses ......  51  B. l  Amplitude D i s t r i b u t i o n at the Focus of a S p h e r i c a l Lens I l l u m i n a t e d w i t h Laser L i g h t ...... Removal of the A i r y P a t t e r n by the Pinhole .......  53 54  B»2  vi  ACKNOWLEDGEMENT The the  author would l i k e to thank Dr. M.P. Beddoes,  s u p e r v i s o r of t h i s p r o j e c t , f o r the guidance and encourage-  ment given during the p e r i o d t h i s work was c a r r i e d out. Thanks are due to the s t a f f members and the t e c h n i c i a n s of t h i s department f o r t h e i r help during the r e s e a r c h . author c o r d i a l l y acknowledges the i l l u m i n a t i n g  The  d i s c u s s i o n s he  had with Mr. P.J.W. F a r r , the co—worker on the p r o j e c t , during the  research. He i s g r e a t l y indebted  S c h o l a r s h i p and F e l l o w s h i p  t o the Canadian Commonwealth  Committee f o r the award of a s c h o l a r -  ship and to the N a t i o n a l Research C o u n c i l of Canada f o r f i n a n c i a l support  f o r the r e s e a r c h . Last but not the l e a s t the author would l i k e to express  his  sincere thanks to Miss H.A. Thomson f o r t y p i n g the masters  for this  thesis.  1 lo The process  INTRODUCTION  of two stage imaging u s i n g the  diffraction  p a t t e r n of an object was d i s c o v e r e d i n 1948 by Denis Gabor"*" of the Imperial C o l l e g e , London, (gk. holos  : whole  He suggested the name Hologram  ; gramma : w r i t i n g ) f o r the d i f f r a c t i o n p a t -  t e r n of a transparency because he b e l i e v e d that t h i s the t o t a l  contained  i n f o r m a t i o n f o r r e c o n s t r u c t i n g the o b j e c t .  a "very few s p e c i a l  cases the Gabor Hologram does not r e a l l y c o n t a i n  the whole i n f o r m a t i o n about the o b j e c t .  I t misses out the  i n f o r m a t i o n | yet under c e r t a i n circumstances  it  The name Hologram continues  transparency.  to be used f o r r e c o n s t r u c t i o n using  During the l a s t two years  several  other types of  Holograms have been produced and some of these c o n t a i n the phase i n f o r m a t i o n .  In t h i s  word Hologram s p e c i f i c a l l y  t h e s i s we s h a l l , however, use  missing the  to describe the b a s i c Hologram which  Gabor d e s c r i b e d i n h i s paper"*".  Any other type of Hologram w i l l  be r e f e r r e d to w i t h a s u i t a b l y q u a l i f y i n g The primary i n t e r e s t the Hologram technique  phase  can be used to  r e c o n s t r u c t a reasonably good r e p l i c a of the object  other methods.  Except f o r  adjective.  of Gabor was the a p p l i c a t i o n of  to e l e c t r o n microscopy.  He had obtained  some Holograms u s i n g a divergent e l e c t r o n beam and r e c o n s t r u c t e d an image from them by using v i s i b l e of  light.  The q u a l i t y  the r e c o n s t r u c t i o n was poor p a r t l y because of the l a c k of a  sufficiently  b r i g h t source of h i g h l y coherent l i g h t and p a r t l y  because of the  system of lenses he had been using f o r o b t a i n i n g  the r e c o n s t r u c t i o n . the  coherent  I t has been demonstrated i n t h i s  system can be made completely  r e c o n s t r u c t i o n can be obtained  thesis  l e n s l e s s and a b e t t e r  thereby.  that  quality  2 Because  of the l a c k of a s u f f i c i e n t l y  coherent l i g h t the i n t e r e s t  i n Hologram technique  the advent of the l a s e r i n I960. the i n t e r e s t  in this  d i e d down u n t i l  The i n v e n t i o n of the l a s e r  f i e l d and since  i n t o the v a r i o u s other p o s s i b i l i t i e s  revived  then s t u d i e s have been made of the Hologram.  able amount of work has been done since  A consider-  1962 by Emmet L e i t h and  J u r i s Upatnieks i n the U n i v e r s i t y of M i c h i g a n . the b a s i c Gabor process  b r i g h t source of  They have  extended  to the r e c o n s t r u c t i o n of continuous tone and  three dimensional o b j e c t s .  They have developed the process  as a  new system, of l e n s l e s s photography capable of producing the  strik-  i n g r e a l i s m of t h i r d dimension. A Gabor Hologram i s produced by passing a coherent beam of l i g h t through a transparency and r e c o r d i n g the r e s u l t a n t  diffrac-  t i o n p a t t e r n on a photographic p l a t e . 2  The method developed by  L e i t h and Upatnieks  i n that a reference  differs  from t h i s  obtained by s p l i t t i n g the o r i g i n a l beam i s the d i f f r a c t i o n p a t t e r n .  This r e s u l t s  beam  allowed to impinge on  i n a modulation of  the  d i f f r a c t i o n p a t t e r n according to the phase i n f o r m a t i o n c a r r i e d by the Hologram forming beam.  It i s possible  to o b t a i n a good q u a l i t y  r e c o n s t r u c t i o n of continuous tone objects by t h i s p r o c e s s .  We  s h a l l r e f e r to a Hologram made i n t h i s way as a s p l i t - b e a m H o l o gram. For the purpose of t h i s attention  t h e s i s we s h a l l l i m i t our  to the b a s i c Hologram as d e s c r i b e d by Gabor.  For such  a Hologram, Gabor o f f e r e d an explanation of the two step imaging process by c o n s i d e r i n g the d i f f r a c t e d beam to c o n s i s t of two parts,  the background wave and the  i n the process  secondary wave.  He showed  of r e c o n s t r u c t i o n from the Hologram i t i s  that  possible  3 to get a wave part of which i s s i m i l a r to the wave produced by the  object.  He  argued that i t should,  t h e r e f o r e , be p o s s i b l e to  form an image of the o r i g i n a l by f o c u s s i n g the  s u b s t i t u t e wave  i s s u i n g out form the Hologram w i t h a s p h e r i c a l l e n s . I t i s , however, p o s s i b l e to consider of two  step in^#?iin g; as c o n s i s t i n g of two :  transformations  in a spatial  coordinate  of view i t i s p o s s i b l e to e x p l a i n the two more e l e g a n t l y .  This e x p l a n a t i o n ,  pend on the formation Further,  i t has  successive  system.  been experimentally  Fourier  Using t h i s  point  step imaging phenomenon  given i n Ch.  of a r e c o n s t r u c t e d  the whole process  2,  does not  de-  image with a l e n s .  v e r i f i e d that i t i s p o s s i b l e  to obtain a b e t t e r r e c o n s t r u c t i o n from a Gabor Hologram by making the  system l e n s l e s s . In t h i s t h e s i s we  s h a l l be mainly i n t e r e s t e d i n the  p o s s i b i l i t y of a p p l y i n g the Hologram technique to the bandwidth compression of a t e l e v i s i o n p i c t u r e s i g n a l . of the exaniples  of the Holograms of simple patterns  elsewhere i n t h i s t h e s i s w i l l i n the Hologram tends to be  and  l e s s sharp.than i n the O r i g i n a l .  be  s i g n a l obtained  l e s s v a r i a b l e than the  This i n d i c a t e s that the bandwidth than the nal*  For  a step f u n c t i o n becomes a  somewhat c y c l i c change i n i n t e n s i t y .  f o r e , i n f e r that the  presented  show that the i n t e n s i t y v a r i a t i o n  example,, a s t r a i g h t edge r e p r e s e n t i n g gradual  A v i s u a l examination  We  could,  there-  by scanning the Hologram w i l l  s i g n a l obtained  by  scanning the  original.  spectrum of the Hologram w i l l occupy l e s s  spectrum of the  A scheme u s i n g two  has been presented i n Ch.  s i g n a l obtained  from the  channels to achieve bandwidth 3.  to obtain about f i v e to one  origi-  reduction  I t has been shown t h a t i t i s p o s s i b l e r e d u c t i o n f o r a p a r t i c u l a r case.  4 Although  t h e case  video  signal,  band  reduction We  transform This in  opens  and be  by  f o r a  have  already  domain  selective  components masks  class  between  of  example  because  could  there  the Hologram of achieving  of using  becomes  that  c a n be v i s u a l l y  exists  a  done  the space  f i l t e r i n g  of  signals  domain.  the frequency identified band  i n the time  domain  Fourier  and i t s o r i g i n a l .  o f t h e components been  of  signals.  of the fact  have  of a  the possibility  instead of i n the time-frequency  transmission  This  clearly  stated that  the possibilities  the advantage  some  certain  of a p i c t u r e signal  achieved.  but of  up  i s not a typical  y e t i tindicates very  i s of interest  ponents  dealt with  relationship  the space  This  we  i s that  very  simple,  being  a matter  on t h e Hologram  so t h a t  the undesired  com-  i n a  Hologram  reduction can domain the  as  well  selection  of merely  putting  components  are  blocked out. The reasonably a  careful  account  process  good  of forming  replica  and s p e c i a l  a Hologram  of the original  technique.  of the experimental  from  Chapter  techniques.  4  and o b t a i n i n g the Hologram gives  a  a requires  detailed  5 2. ANALYSIS OF THE HOLOGRAM We plan to i n v e s t i g a t e the f e a s i b i l i t y of using the Hologram f o r t e l e v i s i o n bandwidth compression. a r i s e s because the s i g n a l obtainable  The p o s s i b i l i t y  from scanning the Hologram  seems to be l e s s v a r i a b l e than t h a t from scanning the o r i g i n a l . An exact knowledge of the equation of the Hologram i s necessary to p r e d i c t the type of s i g n a l that w i l l be obtained  when a given  Hologram i s scanned* This chapter commences by quoting  the F r e s n e l - K i r c h h o f f  d i f f r a c t i o n i n t e g r a l which leads to the mathematical d e s c r i p t i o n of the Hologram.  In l a t e r s e c t i o n s a method of e v a l u a t i n g  i n t e g r a l i s shown and i t i s demonstrated that there F o u r i e r transform gram.  this  exists a  r e l a t i o n s h i p between an object and i t s Holo-  The p r i n c i p l e of r e c o n s t r u c t i o n from a Hologram without a  lens i s then explained  and the a p p l i c a b i l i t y of such a Hologram  as a f i l t e r element i s d i s c u s s e d * 2.1  The F r e s n e l - K i r c h h o f f D i f f r a c t i o n I n t e g r a l I t i s p o s s i b l e to show t h a t the Hologram produced by  an aperture  when i l l u m i n a t e d by a p o i n t source of coherent and  monchromatic l i g h t i s governed by the f o l l o w i n g i n t e g r a l formula, 3 g e n e r a l l y known as the F r e s n e l - K i r c h h o f f d i f f r a c t i o n formula ,  0(p)  = -  ^  JP(r+s)  (cos a - cos y ) d S (2.1)  where,  0(P)  =  complex amplitude  6 of l i g h t disturbance at a  a p o i n t P i n the d i f f r a c t i o n p a t t e r n , Ae^  r  complex l i g h t amplitude the p o i n t  r  =  at a distance r from  source,  d i s t a n c e of p o i n t source from any p o i n t i n the aperture  s  =  d i s t a n c e of the p o i n t P from the same point, i n the aperture,  X  =  wavelength of the l i g h t used to produce the d i f f r a c t i o n pattern,  P  =  jfy and The  dS  =  phase s h i f t per u n i t l e n g t h ,  i n d i c a t e s the area of the aperture, =  elemental  area.  angles a and P are the angles t h a t the i n c i d e n t and the  diffracted Fig*  2ll —  2.1  rays make with the normal to the plane  of the aperture.  shows the arrangement of the system to o b t a i n the d i f f r a c -  t i o n p a t t e r n and the d i s t a n c e s and angles used i n the d i f f r a c t i o n formula are i l l u s t r a t e d . Let  us d e f i n e a coordinate system (x, y, z) with the  z d i r e c t i o n p e r p e n d i c u l a r to the plane  of the a p e r t u r e .  L e t the  coordinates of P , the p o i n t on the d i f f r a c t i o n p a t t e r n and P the p o i n t source be (x', y ' , z*) and ( > y > X  Q  Q  Suppose now t h a t a two dimensional p l a c e d i n the a p e r t u r e . of the transparency concept  q  ) respectively.  transparency i s  Assume t h a t the t r a n s m i s s i o n  is t.  coefficient  In a s t r i c t mathematical sense the  of a t r a n s m i s s i o n c o e f f i c i e n t  transparency  Z  Q  i s meaningless.  f o r a two  dimensional  However, f o r p h y s i c a l t r a n s p a r e n c i e s  7  X t  F i g . 2.1. I l l u s t r a t i n g the Coordinate  System and the  Symbols Involved i n the D i f f r a c t i o n  Integral.  8 the t h i c k n e s s i s f i n i t e and a t r a n s m i s s i o n c o e f f i c i e n t does e x i s t . We assume that t does not vary across the transparency, the z d i r e c t i o n .  Therefore  t  l y true i f the transparency  =  t(x, y).  i s sufficiently  When the transparency  i . e . along  This w i l l be very nearthin.  i s placed i n the aperture the  d i f f r a c t i o n p a t t e r n at P i s m o d i f i e d .  The r e s u l t a n t p a t t e r n can  be obtained from Eq. ( 2 . l ) by i n t r o d u c i n g t ( x , y) i n the d i f f r a c t i o n i n t e g r a l and i s ,  JJ * ^^Ir^ ° ~ ^) t(x  0  (  P  )  ~ 2l  =  y)  (  OSa  Cos  ^  dS  (2.2)  2.2 Hologram; as a S p a t i a l F o u r i e r Transform of the Object : E v a l u a t i o n of the D i f f r a c t i o n I n t e g r a l and the Approximations Involved A very important  and i n t e r e s t i n g c l a s s of Hologram i s  that i n which the Holograms are the s p a t i a l F o u r i e r of the corresponding  object transparencies.  transforms  This happens  under c e r t a i n s p e c i a l c o n d i t i o n s when one has to make some of the d i s t a n c e s i n v o l v e d i n Eq* (2.2) l a r g e and some of the angles small.  We propose to i n v e s t i g a t e the exact c o n d i t i o n s i n v o l v e d . We begin by a p p l y i n g the f o l l o w i n g c o n s t r a i n t s on  Eq.  (2.2), Cos a  =  - Cosf  =  Cos S  , where £ i s a very small angle  ~rt  =  r'"^'  =  C  o  n  s  t  a  n  t  (2.3)  ( 2 o 4 )  9 a n d s' a s s h o w n i n F i g . 2 . 1 r e p r e s e n t  r' and  P from t h e o r i g i n  P  0 of t h e c o o r d i n a t e  and P from t h e transparency  the  points  the  v a r i a t i o n i n t h e term  negligible exp  i f the distances  are large  because  (Cos a — C o s y ) a n d l / ( r s ) w i l l  compared t o t h e v a r i a t i o n i n t h e o s c i l l a t i n g  j p ( r + s) i nEq.  slight exp  q  change  i n ( r + s) produces  j8 ( r + s ) .  illuminating  I t should  (2.2).  These  be n o t e d  a large  two c o n d i t i o n s  change  also  a n d t h e d i f f r a c t e d beam h a v e  Q  system.  ( 2 . 3 ) and ( 2 . 4 ) a r e j u s t i f i e d  Eq.  of P  the distances  that  a  of then  be function very  i n t h e term  dictate that the a very  narrow  divergence  angle• Therefore,  0(F)  =  - j  4  Eq. ( 2 . 2 ) reduces t o ,  jfrf$JJ  t U , y) exp [ - j p ( r + s ) ] dxdy (2.5)  A  The  appearance  that  0(P)  Fourier function the  of Eq. ( 2 . 5 ) would  lead  one t o t h i n k  i s somehow r e l a t e d t o t h e t w o d i m e n s i o n a l  transform  of t ( x , y ) .  I f we c o u l d  two d i m e n s i o n a l  Fourier  spatial  s u i t a b l y modify the  exp JP(r + s) i n Eq. ( 2 . 5 ) i n t o the kernel  standard  intuitively  function of  i n t e g r a l then the proposition  w o u l d be p r o v e d . ¥e, into  therefore,  t h e f o r m - j (<o x + to y ) . x  s  i n terms  Fig.  have t o t r a n s f o r m requires  j p ( r + s)  the expansion  o f r and  y  of the rectangular  2 . 1 we c a n w r i t e ,  This  the function  coordinates.  With  reference  to  10 r  =  [(x -x)  [<*  2  [  yl  +  ( y  2  0  2  +  +  z  o'  +  - y )  *  +  2  r'  o  2 +  + y  2  - < v  2  2  T  -  x  2  + o f»  y  0  T ^  2 + y  y y> ]  +  + y y)J  2(XQX  y  o  2  x  o  or x  r  =•  r  o  x  y  x  + y  ^  2r  +  x  +  y  o  1  y  (x x + y y)' o o ,3 2r  2  x  T  J  J  ••9 • (2.6)  The s e r i e s i n Eq, (2.6) converges as long as x / r ' and y / r ' are each l e s s than u n i t y .  T h i s , as we  shall  see l a t e r , w i l l  always  be the case. Similarly,  s can be expressed as  xx + y y f  x  1  „ t  +  + v O c t 2s  2  (x'x + y ' y )  2  ~  2s  2 ( 9  ->  • • • •  , J  7 )  \ * ' { )  Therefore r + s  =  ( r ' + s')  -  (px + qy)  + f(x, y),  (2.8)  where P  A =  o  (2.9) (2.10)  and, f ( x , y)  A =  \  i \ / 2, o ,2\ i / li . 1 (wr r r ) (x +y ) +  ( o+y o y)' — ,3 x  (x'x+y y)' r3  x  J  J  t  (2.11)  a  Substiiuting  Eq. (2.8) i n Eq*  (2.5) we  obtain,  0 ( P  )  .  _ j  0 (» t » )  A C ^ y - " ' ) /7'rtU> =  x  11 y )  .U*<*.yf|  K |/|t(x,y) exp ^ j p f (x,y)]J  exp  e  - ^ ( p ^ )  Ci> = pp x  f o r the amplitude v a r i a t i o n  In t h i s equation, =  s p a t i a l frequency i n r a d i a n s / u n i t length i n  =  (2.13)  s p a t i a l frequency i n r a d i a n s / u n i t length i n y direction  K  =  — j  A  Co  »W[,iP( + O} r,  a  In the experiments described  (2.14) =  Constant  fo r the s p a t i a l frequencies 7  «>  x  =  toy =  (2.15)  i n t h i s t h e s i s the system geometry  was so arranged that a l l times we had X expression  y  x  the x d i r e c t i o n C0y = qP  d  (2.12)  (2.12) i s the a n a l y t i c expression  i n a Hologram.  x  - j (io x + ayy)/j dxdy  °* Eq.  d  q  =  y  Q  =0.  The  then become,  r  r a d i a n s / u n i t length  (2.16)  2ft ^ ' g t -  r a d i a n s / u n i t length  (2.17)  2TI ^  X  E q . (2.12) i s i n the form of a standard two v a r i a b l e F o u r i e r i n t e g r a l transform.  I t i s important to note that i n the plane of  the Hologram which we s h a l l henceforth  r e f e r to as the s p a t i a l  4 frequency plane  , the amplitude d i s t r i b u t i o n i n s t e a d of being  a d i r e c t F o u r i e r transform i s modified  of the transmission  by a phase f a c t o r exp j p f ( x , y ) .  to be the d i r e c t transform  of the t r a n s m i s s i o n  f u n c t i o n t ( x , y) For the Hologram function  the a f o r e s a i d phase f a c t o r should be equal to u n i t y t h a t f ( x , y) = 0.  0  t(x,y)  This  requires  12 The  necessary c o n d i t i o n f o r the Hologram to be a s p a t i a l  transform  of the t r a n s m i s s i o n r ' and  x and y be  f u n c t i o n i s then,  s' should be very  small.  of these f a c t o r s .  The  large and maximum bounds of  f a c t o r P d i c t a t e s the order p may  Since  Fourier  of smallness  be quite large f o r the wavelengths  of v i s i b l e l i g h t , t h i s c o n d i t i o n becomes more meaningful i f we say P f (x, y ) < I t i s p o s s i b l e to simulate  2%  (2.18)  an i n f i n i t e l y large r ' i n a f i n i t e  d i s t a n c e by u s i n g a p a r a l l e l beam of l i g h t to i l l u m i n a t e object  the  transparency. The  necessary c o n d i t i o n as o u t l i n e d above i s e s s e n t i a l l y  what i s known as Fraurihofer d i f f r a c t i o n i n p h y s i c a l o p t i c s . 2.3  P r i n c i p l e of Lensless  Reconstruction  from a Hologram  I t i s i n t e r e s t i n g to note that i t i s p o s s i b l e to t a i n a Hologram, which i s a F o u r i e r transform  of t ( x , y ) , i n a large  number of planes behind the o b j e c t transparency as long as c o n d i t i o n o u t l i n e d i n Eq.  (2*18) i s approximately  the  the  satisfied.  This i s i n c o n t r a s t to the Holograms which are produced by lenses'*.  ob-  using  Such a Hologram i s produced by u t i l i z i n g the f a c t  that  l i g h t amplitude d i s t r i b u t i o n s at the f r o n t and back f o c a l plane  of a s p h e r i c a l lens bear a other.  Since  v a l i d f o r any  F o u r i e r transform  i n such a system the transform other two  r e l a t i o n i s not  planes i t i s necessary to p o s i t i o n  e x a c t l y the object transparency and gram at the two  r e l a t i o n to each  f o c a l planes*  The  the f i l m to record the system which we  i s i n h e r e n t l y f r e e from such a l i m i t a t i o n .  have  Holo-  described  13 • Let us now assume that we have a transparency i n which 0(<*» « y ) has been r e c o r d e d , no lens being used i n the process  of  x  recording..  This can be very simply done by p l a c i n g a photo-  s e n s i t i v e m a t e r i a l i n the s p a t i a l frequency p l a n e . w i l l r e c o r d the F o u r i e r transform of t ( x , to synthesize t ( x , transform-  Since  this  y) we ought to be able  y) from t h i s by t a k i n g an inverse  Fourier  This could be accomplished by i l l u m i n a t i n g the  Hologram transparency with a l i g h t beam s i m i l a r to the one producing i t .  I t might be argued that since we are using  the  same k i n d of beam as was U s e d i n producing the Hologram, the k e r n e l f u n c t i o n w i l l not change  sign which i s necessary  an inverse F o u r i e r transform*.. This i s  of l i t t l e  for taking  significance  since  a change i n s i g n can be e f f e c t e d by p r o p e r l y i d e n t i f y i n g the dinate axes during r e c o n s t r u c t i o n .  What i s  coor-  important to note  that we have taken two successive transforms to r e t r i e v e  the  original function.  lens-  We have, t h u s ,  achieved a completely  l e s s system of imaging i n two dimensions. i n the two step process  The b a s i c  are i l l u s t r a t e d i n F i g .  i s not a c t u a l l y necessary  to observe  steps i n v o l v e d  2.2.  Although we have developed the theory as above assuming that Fraunhofer d i f f r a c t i o n i s  outlined  taking place,  Eq.  (2.18) we get  it  the Fraunhofer c o n d i t i o n as  given i n E q . (2.18) very s t r i c t l y i f the sole purpose i s step imaging o n l y .  is  two  I f we do not observe the c o n d i t i o n given by a d i f f r a c t i o n p a t t e r n that i s  Fresnel d i f f r a c t i o n .  c a l l e d the  This corresponds to the case where f ( x ,  defined by E q . ( 2 . 1 l ) i s not n e g l i g i b l e .  If  such a case  used a l l that happens i s that i n s t e a d of g e t t i n g back t ( x , i n g the r e c o n s t r u c t i o n process we get  a f u n c t i o n which i s  y)  is y)  dur-  basically  14 t ( x , y) w i t h a superimposed somewhat o s c i l l a t i n g modulation e f f e c t due to the exponential  term i n which f (x, y) appears.  The u s e f u l n e s s of such a r e c o n s t r u c t i o n depends on the amount of allowable degrading 2.4 Recording  e f f e c t due to noise that i s p e r m i s s i b l e .  the Hologram : The Phase Problem  We have so f a r t a c i t l y assumed that i t i s p o s s i b l e to r e c o r d the Hologram on a photographic extremely l i m i t e d number of s p e c i a l  film.  types  graphic p l a t e can r e c o r d only p o s i t i v e  This i s true f o r an  of Hologram.  A photo-  f u n c t i o n s as i t i s  sensitive* to the i n t e n s i t y d i s t r i b u t i o n and not to the amplitude d i s t r i b u t i o n of l i g h t .  I t i s , t h e r e f o r e , not p o s s i b l e to r e c o r d  f a i t h f u l l y the H logram of objects having 0  plex values  i n t h e i r s p a t i a l transforms  we have d e s c r i b e d .  either  negative  or com-  u s i n g the simple means  A way to get around t h i s i s to use a r e f e r -  ence beam obtained by s p l i t t i n g up the Hologram forming beam i n t o two p a r t s , one part to form the Hologram (the s i g n a l beam) i n the way we have d e s c r i b e d and the other p a r t , i . e . the reference beam, i s allowed  to shine i n the plane  of the Hologram.  This produces a modulation of the i n t e n s i t y d i s t r i b u t i o n  caused  by the interference' of the reference and the s i g n a l beam i n the plane  of the Hologram according to the phase i n f o r m a t i o n  by the Hologram forming beam.  carried  Such a Hologram known as the  two-beam or split-beam type does not bear any resemblance whatsoever to the o r i g i n a l object but i s able to reporduce high quality  r e c o n s t r u c t i o n of o b j e c t s having  complex  transforms.  The mathematical theory of the s p l i t beam Hologram i s y e t to be  15 Hologram Object  t-(x,y) T(co ,CO ) x* y a) Arrangement f o r Obtaining the  Spatial  Fourier  Transform. Reconstruction Holograjii  b) Arrangement f o r Obtaining the  Inverse F o u r i e r  Transform.  Fig.  2.2.  Basic Arrangement f o r S p a t i a l S p e c t r a l  Analysis.  (P  and  is  a point  coherent  source of monochromatic  light)  developed but the experimental technique connected with i t has advanced q u i t e f a r mainly due to the work of L e i t h and Upatnieks  who  pioneered t h i s technique,.  This k i n d of Hologram is  also  capable of r e c o n s t r u c t i n g three dimensional o b j e c t s producing a genuine for  t h i r d dimensional e f f e c t  the purpose  »• We  s h a l l l i m i t our a t t e n t i o n  of t h i s t h e s i s to the b a s i c Hologram.  A  split  beam Hologram although capable of g i v i n g b e t t e r r e c o n s t r u c t i o n does not show the F o u r i e r transform r e l a t i o n s h i p ! b e c a u s e of the i n t e n s i t y modulation produced by the reference beam. t h i s , so f a r as bandwidth compression  i s concerned  the  beam Hologram i s not s u i t a b l e because of the extremely  Besides  split fine  d e t a i l present i n such a Hologram which would r e q u i r e a bandwidth f a r i n excess of the present t e l e v i s i o n  system.  Gabor"*" has shown that i t i s p o s s i b l e to o b t a i n recogn i z a b l e r e c o n s t r u c t i o n from Holograms produced shown i n F i g . 2.2a  i n the way  i f the o b j e c t s c o n s i s t s mainly of transparent  areas w i t h a few dark p a t t e r n s .  A c t u a l l y , as long as t h e . s p a t i a l  transforms are such that they are e i t h e r predominantly or  positive  predominantly negative a r e l a t i v e l y good r e c o n s t r u c t i o n can  be obtained.  A t y p i c a l example i s that of a f l a t top pulse  f u n c t i o n e i t h e r p o s i t i v e or n e g a t i v e . of  as  The  spatial representation  a p o s i t i v e pulse i s a s l i t on dark background whereas a  negative pulse can be represented as a narrow dark band. on. a transparent background.  I t i s p o s s i b l e to o b t a i n a r e c o n s t r u c -  t i o n from the Gabor Hologram of both ^hese p u l s e s .  By a Gabor  Hologram we mean a Hologram which has been produced using the scheme shown on F i g . 2 . 2 a .  For f u r t h e r d e t a i l s of the process  1.7, the reader i s r e f e r r e d to the s e c t i o n on experimental technique and experimental 2.5 Frequency We  results.  F i l t e r i n g with Hologram have seen that i n the process of forming a Hologram  we perform a l i n e a r t r a n s f o r m a t i o n of the o b j e c t plane i n t o a s p a t i a l frequency plane.  This i s a two dimensional analog of the  one dimensional time to frequency F o u r i e r t r a n s f o r m a t i o n . i s p o s s i b l e to extend ideas from e l e c t r i c a l f i l t e r Hologram technique.  theory-to  The advantage of u s i n g the Hologram i s  that i t i s very simple then to achieve elementary of  It  filters  capable  l i m i t i n g bandwidth of s i g n a l s . Consider two  and h(x, y ) .  spatial signals f^( > x  y)  These are i n f a c t t r a n s p a r e n c i e s having t r a n s m i s s i o n  f u n c t i o n s f . and 1  It  two—dimensional  h.  i s d e s i r e d to use h as the f i l t e r  element to  perform  some d e s i r e d o p e r a t i o n on f ^ and y i e l d f ( x , y) as the  output.  In the space domain we have to perform a c o n v o l u t i o n  for  D  t h i s purpose f (x, Q  so t h a t , y)  J J ± < ^) r f  =  l  cr  b(x -<r, y -/>,)  dtf-d/=> (2.19)  Since the process of c o n v o l u t i o n reduces to a simple t i o n i n the frequency domain, the process can be g r e a t l y by u s i n g Holograms i n s t e a d of the s i g n a l s Thus i n the s p a t i a l frequency domain employing f. l  multiplica-  simplified themselves.  the Holograms of  and h we o b t a i n F  o  (tt , tt ) x' y'  =  F.(tt , tt ) l x* y v  H(tt , tt ) x* y 7  x  (2.20) '  18  The f u n c t i o n f ( x , y) can be obtained by t a k i n g an inverse Q  transform of F^H. Pig.  The steps i n v o l v e d are i l l u s t r a t e d i n  2.3. From E q . (2.20) we see t h a t i t i s p o s s i b l e to modify  the f u n c t i o n f H(»  x  , «o ). y  by s u i t a b l y d e s i g n i n g the f u n c t i o n h(x, y) or  As we have already s a i d we could operate e i t h e r i n  the plane of the o b j e c t or i n the plane of the Hologram.  Opera-  t i o n i n the plane of the transparency i s more complicated as i t i n v o l v e s the process of complex c o n v o l u t i o n i n v o l v i n g  spatial  s h i f t s and a l s o because of the f a c t t h a t to evaluate the convol u t i o n i n t e g r a l we have to go through a process of measuring the average  l i g h t passed through the two t r a n s p a r e n c i e s f ^ ( x , y)  and h(x, y) p l a c e d back to back i n the o b j e c t plane. to  In a d d i t i o n  t h i s d e s i g n i n g h(x, y) i s more complicated than H(«  elementary  , <o ) f o r  filters. Operation i n the s p a t i a l frequency plane, t h a t is- i n the  plane of the Hologram i s more convenient and e l e g a n t . for  Suppose  example, we have a dark mask w i t h a c e n t r a l opening of  width 2x' by 2y'.  I f t h i s mask i s placed i n the Hologram plane  then i t w i l l correspond to a low pass f i l t e r having the c u t - o f f f r e q u e n c i e s — 2-jx . , — X  and  i 2-rt  A S  and  (2.17).  , ,— y  according to Eqs . < (2.16)  A S  I t i s i n t e r e s t i n g to note t h a t i t has two p o i n t s , one  i n the p o s i t i v e and the other i n the negative d i r e c t i o n c o r r e s ponding  to the c u t — o f f  frequency.  S i m i l a r l y a dark spot p l a c e d on the o p t i c axis i n the plane of the Hologram would mean the removal of the d.c. component of  a signal.  19  SPATIAL FOURIER ANALYZER # 1  F  F. ( to , to  1  H( « , « x  # 2  = F.H O  )  %  INVERSE FOURIER SYNTHESIZER^. # 3  F i g . 2.3. Basis of O p t i c a l  Filtering.  )  F  = F.H  20  It  i s p o s s i b l e to o b t a i n r e c o n s t r u c t i o n from a  Hologram which has been subjected to frequency  filtering.  The  r e s u l t a n t i s a d i s t o r t e d r e p l i c a of the o r i g i n a l , the d i s t o r t i o n depending pn the type of f i l t e r being used. of and  f i l t e r i n g are shown i n the s e c t i o n on experimental  technique  results. Besides these elementary  to  T y p i c a l examples  forns of f i l t e r s  synthesize more complicated f i l t e r s  Hologram.  i t i s possible  i n the plane of the  One such p o s s i b i l i t y i s to design a f i l t e r f o r  maximizing the s i g n a l to noise r a t i o .  The design of such a 4  f i l t e r has been d e s c r i b e d by Vander Lugt  and r e q u i r e s the  use of the split—beam technique d e s c r i b e d e a r l i e r .  21 3. APPLICATION OP HOLOGRAMS IN TELEVISION TRANSMISSION The  f e a s i b i l i t y of applying the Hologram to band  reduced t e l e v i s i o n t r a n s m i s s i o n i s i n v e s t i g a t e d i n t h i s The  chapter.  p o s s i b i l i t y of a c h i e v i n g band compression by u s i n g Holograms '.  a r i s e s because the i n t e n s i t y d i s t r i b u t i o n i n a Hologram, as be  seen from the examples presented  than i n the o r i g i n a l .  The  i n Ch. 4,  can  i s more smooth  sharp v a r i a t i o n i n the o r i g i n a l , f o r  example, a t h i n b l a c k l i n e on a white background gives a Hologram that spreads out over a l a r g e r area and has gradual  change i n i n t e n s i t y .  a somewhat c y c l i c  and  This i s true f o r a l l p i c t u r e s which  are b a s i c a l l y i n the form of l i n e drawings..  Since the  intensity  d i s t r i b u t i o n becomes smooth i n the Hologram i t w i l l give r i s e to a slowly v a r y i n g video where by slowly we the video b a s i s we  s i g n a l when scanned by a t e l e v i s i o n camera,  mean t h a t the rate of v a r i a t i o n i s slower than  s i g n a l obtained by scanning  the  could then p r e d i c t that the video  o r i g i n a l . On an s i g n a l obtained  the Hologram would p o s s i b l y cover l e s s bandwidth than the f o r such o b j e c t s .  intuitive from original  This statement i s not true f o r c e r t a i n types 7  of objects and Hologram methods. of a three dimensional  For example , the  transmission  o b j e c t by the split-beam Hologram  technique  r e q u i r e s enormous bandwidth. I t i s p o s s i b l e to represent  a two  dimensional  where changes i n i n t e n s i t y l e v e l s along the scanning are very few,  by a t r a i n of r e c t a n g u l a r p u l s e s .  picture,  direction  I f the changes  are f a r apart each pulse can be t r e a t e d s e p a r a t e l y .  The  analysis  of the t r a n s m i s s i o n of a pulse r e p r e s e n t i n g such a p i c t u r e , is  22 given  i n Sec.  3.1.  On the b a s i s of t h i s a n a l y s i s a two  system of band reduced t r a n s m i s s i o n F i n a l l y , i n Sec.  3.2  a p p l i c a t i o n of the process i s 3.1  Analysis The  channel  i s proposed.  an account of the  possible  given.  of the Transmission of the Hologram of a Pulse p o t e n t i a l c o m p r e s s i b i l i t y of t e l e v i s i o n p i c t u r e  s i g n a l depends upon i t s excessive  redundancy.  "Most simply  the redundancy a r i s e s i n a s i g n a l , such as the t e l e v i s i o n s i g n a l , by v i r t u e of the f a c t that i f c e r t a i n parts of the g known the of a few  other parts may  s i g n a l are  be guessed" , i . e . , most p i c t u r e s  consist  sharp t r a n s i t i o n s of i n t e n s i t y with the r e s t of*the  being more or l e s s even.  As a r e s u l t the t e l e v i s i o n  area  picture  can be quantized i n s e v e r a l d i s c r e t e steps as shown i n F i g .  3.1b.  I t i s p o s s i b l e to transmit  by  t r a n s m i t t i n g these  a reasonably good q u a l i t y p i c t u r e  pulses.  In t h i s s e c t i o n we of the  s h a l l t r e a t a very s p e c i a l c l a s s  s i g n a l s taken form those represented by F i g . 3.1b.  some p i c t u r e s as i n l i n e drawings, the pulses  a) F i g . 3.1  representing  b) Quantization  of a Video S i g n a l  a) O r i g i n a l S i g n a l b) Quantized S i g n a l  In the  23 t r a n s i t i o n i n l e v e l occur s u f f i c i e n t l y f a r apart i n time so that the t r a n s m i s s i o n of each pulse can be t r e a t e d s e p a r a t e l y .  We  -could t h e r e f o r e study the t r a n s m i s s i o n of a s i n g l e pulse as a representative We  case.  p l a n to transmit the Hologram of a f l a t  topped  pulse i n s t e a d of the pulse i t s e l f over the t e l e v i s i o n The  s p a t i a l r e p r e s e n t a t i o n of such a pulse i s e i t h e r a narrow  slit  on a dark background or a dark band on a transparent background* We  can begin with the case of a r e c t a n g u l a r aperture  having dimensions 2a and 2b. -narrow s l i t by l e t t i n g one  This can be converted  sions and the coordinate  into a  of the dimensions become very l a r g e and  the other becoming very s m a l l .  We  channel.  The  aperture, the d i f f e r e n t dimen-  system are i l l u s t r a t e d i n F i g .  3.2.  then have, +a 0(P)  +b  K I  =  J  -a  t ( x , y) exp  j ^ - j (cc^x + « y ) ] y  dxdy  -b (3.1)  Eq.  (2.12) f o r 0f (x, y)  (3.1) f o l l o w s from Eq.  e s s e n t i a l l y a case of Fraunhofer For an aperture, we t(x, Therefore we  y)  sin extending  end up w i t h the one  Constant  (3.1),  aco  s i n b«  =  y i n f i n i t e l y along y d i r e c t i o n  dimensional sin  0(P)  C  This i s  have,  x For a narrow s l i t  0.  diffraction.  =  o b t a i n form Eq.  =  we  F o u r i e r transform,  aid X  ,  (3.3)  24  F i g . 3.2* Arrangement f o r O b t a i n i n g the F o u r i e r Transform of an a p e r t u r e . (Note t h a t as b tends to i n f i n i t y and a tends to zero ve o b t a i n the F o u r i e r transform of a f l a t topped p u l s e ) .  which i s a w e l l known f u n c t i o n having the shape as given i n P i g . 3.3.  0(P)  =  0(« ) x  co or x' x  P i g . 3.3 Hologram of an I n f i n i t e l y Long Narrow S l i t Now, form Eq. (2.16), _2JL co = A s' x -x being the x-coordinate of any p o i n t i n the Hologram plane. T  Therefore,  22L 2Ll sin a X s'  0(P)  a  _  where,  g  A  c  X  s'  S i n pgx'  (3.4)  a  Assume now that the Hologram i s scanned along the x d i r e c t i o n w i t h a spot v e l o c i t y v, so t h a t , x' =  v t , t being the time.  (3.5)  Therefore, the time s i g n a l obtained by scanning the Hologram can be w r i t t e n as,  26  = c  0(t) where,  =  0  .  .  = c  s^ir* Pgvt  S i n <o, t 1  W-j^t  '  P gv  (3.6)  _ 0(t)  ,  a  S i n co t -r-i—  =  =  S i nfi>,t - r - l — (3.7)  2 E  Q  where,  2 E  = Sin  The time f u n c t i o n  0(t) =  2 E  ttt —  i n Eq. (3.7) i s the  i n v e r s e F o u r i e r transform of the f o l l o w i n g E(tt)  =  E  function,  ,  (3.8)  This can be shown as f o l l o w s ,  e(t)  =  j  E(tt) (exp jwt) dtt =  ^1  j  E exp jttt dtt  ^ 1 Sin tt,t  =  = 2 E  0(t)  (3.9)  Therefore, the s i g n a l obtained by scanning the Hologram has a frequency spectrum  of width tt^ r a d i a n s / s e c .  T h e r e f o r e , the  bandwidth i s r  l  ~  2-rt  2it  _  ~  X  ~  Xs'  u.iu;  A v e r y i n t e r e s t i n g p r o p e r t y of the Hologram i s apparent from Eq. (3.10). i.e,  the narrower  As the width of the s l i t  'a' decreases,  the pulse becomes the l e s s e r becomes the  frequency band requirements* a zero width spectrum  As a—»-0, f ^ — » • 0 g i v i n g r i s e to  f o r a zero width  slit.  27 The main reason that a, wide band i s r e q u i r e d f o r the t r a n s m i s s i o n of a t e l e v i s i o n p i c t u r e i s that to reproduce  sharp  edges i n a p i c t u r e we must make the system r i s e time short enough which means the use of a wide bandwidth.  I f we  convert a p i c t u r e  i n t o i t s Hologram then the sharp edges i n the o r i g i n a l give r i s e to a p a t t e r n which i s spread out i n the Hologram thereby making the frequency range  (corresponding to the sharp edge) of the  s i g n a l obtained by scanning the Hologram s m a l l .  On the other  hand, a broad v a r i a t i o n i n the o r i g i n a l tend to give r i s e to f i n e r v a r i a t i o n s i n the Hologram and t h e r e f o r e the bandwidth would be more i f we  scan the Hologram i n s t e a d of the o r i g i n a l  f o r t h i s c l a s s of s i g n a l s . A p o s s i b i l i t y of saving the channel width, t h e r e f o r e , a r i s e s i f we  can p r o p e r l y handle the two b a s i c c l a s s e s of s i g n a l s .  The f o l l o w i n g scheme could p o s s i b l y be  used.  Let us have two channels f o r p i c t u r e t r a n s m i s s i o n as shown i n P i g . 3.4.  In both the channels we  introduce low pass  f i l t e r s the pass band of which can be v a r i e d at w i l l .  Channel  #1 p r i m a r i l y t r a n s m i t s the broad v a r i a t i o n s i n the s i g n a l whereas channel #2 p r i m a r i l y t r a n s m i t s the f i n e r d e t a i l s i n a p i c t u r e . At the r e c e i v i n g end the Hologram could be r e c o n s t r u c t e d to r e t r i e v e the o r i g i n a l , the broader v a r i a t i o n s of which have been removed by the f i l t e r . c o u l d be superimposed good q u a l i t y  The two p i c t u r e s i g n a l s thus obtained i n the r e c e i v e r to o b t a i n a comparatively  picture.  A t y p i c a l order of magnitude c a l c u l a t i o n w i l l be usef u l to get an estimate of the band requirements f o r the two n e l s as shown i n P i g . 3.4.  We  s h a l l again use the case of a  chan-  VIDEO  OBJECT  LOW  PASS RECEIVER  TRANSDUCER  t(x,y)  FILTER  SIGNAL  C H A N N E L  1  #1  OPTICAL  SUPERIMPOSED  FOURIER  DISPLAY  ANALYZER  C H A N N E L  HOLOGRAM  VIDEO  #2  LOW  RECEIVER  TRANSDUCER x  SIGNAL  x' y'  Fig.  OPTICAL  PASS  3 » 4 » ' B a s i c Scheme  FILTER  f o r T e l e v i s i o n Band  FOURIER SYNTHESIZER  Compression  Using  the Hologram  ro co  29  typical  square topped pulse because  of the ease and elegance  w i t h which i t can be handled mathematically. Assume,  -  Scanning rate  =  L  V i d t h of frame =  =  525 x 30  =  -  15,750 l i n e s / s e c  that p o r t i o n of the Hologram which i s being scanned  Therefore,  v  =  V  =  W L  (3.11)  Prom Eq. (3.10) *1 For  X  _  X  a t y p i c a l case,  which  ¥  =  5 cm  X  =  6.328 x 1 0 ~ cm (He-Ne gas l a s e r  =  1.25 x 1 0  5  light)  gives, f  Let  "  us suppose  that the f i l t e r  9  x g i n the second channel i n F i g . 5.3  passes only up to 100 k c / s . Then, g  = a  ~ 100 x 10  f  J  s" = 1.25 x 10*  Q  1.25 x 1 0  ' =  . ..-4 0.8 x 10 a  cm  y  a _  max s*  —4  S l i t s of width l e s s than 0.8 x 10 passed through t h i s channel.  x s' can t h e r e f o r e be  For Holograms of wider s l i t s we  need a wider band. The l i m i t of r e s o l u t i o n R expressed as the r a t i o of the  maximum r e s o l v a b l e s l i t to the frame width i s then,  30  a  2  R = —JS* ¥  s' F = ^—  where,  2  =  g  _ W  5  =  s' =  ¥  2 e  F ,  max ; *  1 0 0 typically  Therefore, a 1 0 0 kc/s bandwidth gives a r e s o l u t i o n l i m i t f o r the Hologram as R =  x  2  018  x  1 0 ~  4  x  100  =  0.016  This f i g u r e sets the l i m i t on the l a r g e s t p i c t u r e p o i n t that could be t r a n s m i t t e d by the second channel.  A l l finer  details  of the o r i g i n a l p i c t u r e could be t r a n s m i t t e d w i t h i n the 1 0 0 kc/s bandwidth. P i c t u r e p o i n t s i n the o r i g i n a l which are l a r g e r than those d i c t a t e d by R above could be sent by conventional means by a narrow band system.  The bandwidth r e q u i r e d to transmit  p i c t u r e p o i n t s determined by R = 0 . 0 1 6 by conventional means can be c a l c u l a t e d as f o l l o w s . For  R  =  0.016,  Number of b i t s per l i n e  =  Numbers of l i n e s / s e c  15,750  =  Q 016  =  62.5  Therefore, number of b i t s per second 10  =  15,750  x  62.5  = 9 8 5  3  With a conventional p i c t u r e q u a l i t y a bandwidth of 9 8 5 kc/s should do.  Therefore u s i n g a t o t a l bandwidth of 1 . 0 8 5 Mc/s we  should be able to send a comparatively This compares f a v o r a b l y with t e l e v i s i o n system.  good q u a l i t y p i c t u r e .  5 Mc/s bandwidth of a standard  x  31 3.2 A p p l i c a b i l i t y of the Process The  Case which we have developed i s v a l i d f o r a s i n g l e  f l a t top p u l s e .  I t should be r e a l i z e d that t h i s i s not an  example of a t y p i c a l t e l e v i s i o n p i c t u r e s i g n a l , not even of the c l a s s of the o b j e c t s we have d i s c u s s e d i n the preamble. i f i n a l i n e drawing the consecutive the l i n e of scanning  However,  p i c t u r e elements along  are p l a c e d s u f f i c i e n t l y f a r apart then we  could consider them as a s e r i e s of separate  pulses each of which  could be t r e a t e d s e p a r a t e l y meaning thereby  that the spectrum  (Hologram) of one remains e s s e n t i a l l y separate  from the other.  This i s n e a r l y true because the spectrum of a pulse f u n c t i o n decays down to a n e g l i g i b l e value very q u i c k l y , i n about four of f i v e c y c l e s of r e p e t i t i o n . A p o s s i b l e use f o r such a system could be the i n s t a n t aneous t r a n s m i s s i o n of two tone p i c t u r e s l i k e documents, machine drawings, e t c . over telephone existence these  trunk l i n e s .  Such a need i s i n  days and t h i s system could be f r u i t f u l l y used.  A review of other p o s s i b l e f i e l d s where the band—reduced t r a n s m i s s i o n of two tone p i c t u r e i s necessary  i s given i n paper by  9 Cherry, Kubba, e t a l . I t should be noted t h a t the p o s s i b i l i t y of two channel t e l e v i s i o n t r a n s m i s s i o n u s i n g Holograms f o r bandwidth compression i s only at the suggestion  stage:  the scheme has not been made.  experimental  v e r i f i c a t i o n of  4. EXPERIMENTAL TECHNIQUES AND  RESULTS  I t i s d e s i r a b l e to have a review of the experimental techniques because there are a c e r t a i n number of steps and precautions to be taken to make the system of two even s a t i s f a c t o r y .  step imaging  Ve s h a l l give a d e t a i l e d account of these  i n t h i s chapter. In Sec. 4.1  an account of the main elements i n v o l v e d  i n producing the Gabor Hologram i s given. pinhole  I t i s shown that a  stop i s necessary to remove the excess noise e f f e c t  produced by the A i r y r i n g s formed by the s p h e r i c a l In Sec. 4.2 system  i s explained.  included.  lens.  the method of u s i n g the elements  of the  D e t a i l e d measurements of the setup are  R e s u l t s u s i n g the m o d i f i e d Gabor method are presented.  Some examples of s p a t i a l f i l t e r i n g , having b e a r i n g on t e l e v i s i o n band compression 4.1  are i n c l u d e d .  The Elements of the System The Source:  The f i r s t n e c e s s i t y i s f o r a source of  monochromatic and coherent l i g h t .  The n a t u r a l choice would be  a continuous wave l a s e r because these are the best p o s s i b l e  sources  of s u f f i c i e n t l y b r i g h t monochromatic and coherent l i g h t at "the present time.  I t i s p r e f e r a b l e to use a l a s e r with output i n  the blue-green end of the v i s i b l e  spectrum  as most photographic  m a t e r i a l s have t h e i r maximum s e n s i t i v i t y i n that r e g i o n . the time t h i s work was was  not very common.  At  being c a r r i e d out, t h i s k i n d of l a s e r The l a s e r used i n our experiments  Spectra—Physics Model 130 helium—neon gas l a s e r .  is a  This emits an  33  almost p a r a l l e l beam of r e d l i g h t a t 6 3 2 8 1 2 . 5 mm at the e x i t a p e r t u r e . rated.  with a diameter of  The. output power i s 0 . 2  The beam was adjusted to operate i n the T E M  a uniphase  wavefront.  q o  milliwatts mode g i v i n g  (see P i g * 4.1a).  The D i v e r g i n g System;  As has been noted i n the l a s t  paragraph,  the output beam of the l a s e r i s only 2 . 5 mm i n  diameter.  To be able to i l l u m i n a t e a s u f f i c i e n t area of- the  o b j e c t transparency  ( t y p i c a l l y about a centimeter square) i t i s  necessary tb widen t h i s beam.  This can be done by p l a c i n g a  short f o c a l l e n g t h p o s i t i v e lens i n the path of the beam.  In  our work we used a f s 2 . 2 / 5 5 mm Asahi Autotakumar camera l e n s . The use of a lens i n t h i s way r e s u l t s i n the formation of the p a t t e r n c a l l e d the A i r y of b r i g h t l i g h t  rings* I t c o n s i s t s of a c e n t r a l  surrounded  patch  by a s e r i e s of r i n g s , the i n t e n s i t y  as a f u n c t i o n of r a d i u s r being g i v e n b y ^ ,  Kr) where,  and  f 2J (Sf r ) * | n  =  I„  X  J  2  f  (if')  d  =  diameter  f  =  f o c a l l e n g t h of the l e n s ,  X  =  wave l e n g t h ,  i s the f i r s t  In Eq. (4.1) I  q  (4.1)  of the beam,  order B e s s e l f u n c t i o n .  r e p r e s e n t s the i n t e n s i t y at the centre of the  spot and i s r e l a t e d to the power i n the beam i n the f o l l o w i n g way,  34  a).  Spectra-Physics  b). Airy  Model 130 gas l a s e r operating  rings produced by a  c ) » Removal of the A i r y  55 mm Asahi Autotakumar  by 220 micron  lens when placed i n  pinhole  beam as shown i n  F i g . 4.1.  the  Fig.  a).  Illustrating  the E f f e c t  Capacity of a Pinhole Rings.  mo de  q o  rings  diameter  as shown i n  4.3.  of Introducing a Converging  Lens i n the Path of a Coherent L i g h t Beam and the  Airy  in T E M  Filtering  i n Removing the Noise Produced by the  35 where,  A  =  area of the beam c r o s s - s e c t i o n  P  =  power i n the l i g h t beam.  The  photograph i n F i g . 4.1b shows the e f f e c t of i n t r o d u c i n g  the  55 mm f o c a l length lens i n the path of the l a s e r beam from  the Model 130 l a s e r .  F i g . 4.2 Focussing The The  of Coherent L i g h t  Hologram formed i n such a beam i s very  noisy.  r e c o n s t r u c t i o n i s also very poor as can be seen from the  t y p i c a l examples ( F i g . 4.11)attached with t h i s t h e s i s .  It is  necessary to remove these A i r y r i n g s to get a c l e a n Hologram. This can be done by i n t r o d u c i n g a pinhole of the l e n s .  E f f e c t of the pinhole  ( F i g . 4.1c) can be explained  by c o n s i d e r i n g the pinhole  i n the frequency plane.  the pinhole  i s i n c l u d e d i n the appendix B. Pinhole:  as a low  Further d i s c u s s i o n about  For a 55 mm f o c a l length l e n s , l i g h t  of wavelength 6328 A, and a beam diameter of 2.5 mm, P  point  on removing the A i r y r i n g s  pass f i l t e r  The  around the f o c a l  —  2  j ^  =  27.8 microns  36  220 microns  Pig. the  4.3.  Pinhole used i n removing the A i r y d i s c produced by  spherical lens.  View shown i s  m a g n i f i c a t i o n 33,100 X . are p a r t i c l e s  a microphotograph with area  The small grey specs i n s i d e  the  hole  of s i l v e r which the f i x e r f a i l e d to remove from  the f i l m s u r f a c e .  Apparently these microscopic s i l v e r  do not produce any v i s i b l e  degrading  particles  effect.  F i l m used to record and reduce the pinhole to proper size  (220 microns) was I l f o r d F o r m o l i t h developed  developer  and followed by f i x i n g i n  Industrafix.  in Kodalith  37 As has been shown i n the appendix i t i s not necessary to use pinhole of e x a c t l y t h i s s i z e to get s a t i s f a c t o r y r e s u l t s . used a pinhole of 220 microns i n diameter. way  The most  of making a pinhole of t h i s s i z e appears to be  reduction.  We  s t a r t e d with a b l a c k c i r c u l a r spot  j i n c h i n diameter.  This was  reduced  a b l a c k spot 220 microns i n  dia.  gave us the d e s i r e d p i n h o l e .  The  I l f o r d Pormolith developed i s an extremely  a  We  convenient  photographic approximately  i n s i z e i n two  steps to  A contact p r i n t from t h i s f i l m used i n the process  i n Kodak K o d a l i t h developer.  was  This  h i g h c o n t r a s t f i l m with very good r e s o l u t i o n  and i s normally used f o r r e c o r d i n g two hole obtained i n t h i s way  tone o b j e c t s .  The  shows under the microscipe  (400  to have a r e g u l a r c i r c u l a r shape. transparent though.  The hole i s not  pinX)  perfectly  In the photomicrograph s e v e r a l b l a c k spots  r e p r e s e n t i n g minute s i l v e r p a r t i c l e s can be seen.  However, these  do not seem to s e r i o u s l y hamper the q u a l i t y of p i n h o l e . t o r y r e s u l t s have been obtained with t h i s p i n h o l e . i l l u m i n a t i n g scheme i s t h e r e f o r e as shown i n P i g .  Satisfac-  The b a s i c 4.4. Coll. Lens  CW LASER hole Diverging Lens a)  F i g . 4*4.  Experimental  a). Fresnel D i f f r a c t i o n  ect  Div. Lens  Pinhole  Obj ect  b)  Arrangement f o r I l l u m i n a t i n g the b ) . Fraunhofer  Object,  Diffraction  The Recording Medium: need a photographic medium. of t h i s medium.  38 For r e c o r d i n g the Hologram we  There are two b a s i c  requirements  The r e s o l u t i o n l i m i t i n lines/mm should be  h i g h so that f i n e d e t a i l s i n the d i f f r a c t i o n l i n e s are properlyrecorded.  Secondly, the f i l m speed should be reasonably high  so that exposure  time can be made r e l a t i v e l y s h o r t .  The short  exposure  time i s necessary to prevent the e f f e c t of j a r r i n g the  system.  Besides these two b a s i c requirements the f i l m must be  red s e n s i t i v e because  of the type of l a s e r being used.  Several types of l o c a l l y and e a s i l y a v a i l a b l e were t r i e d out.  Best r e s u l t s were obtained from Kodak High  C o n t r a s t Copy, type M 402 f i l m .  This i s a pan f i l m ,  w e l l i n t o the red and has a h i g h r e s o l u t i o n f i g u r e . ten r a t i n g i s ASA  films  64.  The  tungs-  I t has been p o s s i b l e to use an exposure  short as l / l 5 seconds with t h i s f i l m . f i l m i s e i g h t minutes  sensitive  as  Development time f o r t h i s  at 70°F i n Kodak D 76 f o l l o w e d by a 3  minute f i x i n g i n I n d u s t r a f i x .  Workers"'""*' i n t h i s f i e l d have  suggested the use of s p e c t r o s c o p i c p l a t e s l i k e the Kodak 649  F.  This f i l m has an extremely h i g h r e s o l u t i o n f i g u r e of 2000 lines/mm but the exposure  index i s extremely small being only ASA 0.0016  f o r white l i g h t and even smaller f o r red l i g h t . it  Besides t h i s  i s g e n e r a l l y necessary to place a s p e c i a l order to o b t a i n t h i s  f i l m , s e v e r a l months d e l i v e r y time being u s u a l . The type M 402 f i l m comes i n 35 mm t h i s we have used the Asahi Pentax 35 mm t a k i n g lens removed.  s i z e and to h o l d  camera w i t h i t s p i c t u r e  This has a f o c a l plane s h u t t e r which  used to time the exposures.  was  The viewing pentaprism i s very  h e l p f u l i n a d j u s t i n g the p o s i t i o n of the Hologram on the f i l m .  39  Timing mech. ( f o c a l plane s h u t t e r )  F i g . 4.5. Using a 35 mm the Hologram.  Camera Back f o r Recording  40 Since the viewing m i r r o r f l i p s up during the exposure we get a f a i t h f u l r e c o r d i n g of the Hologram on the f i l m , no other surfaces being present  optical  i n the path of the l i g h t beam during the  a c t u a l exposure. 4.2 The Technique of Producing  the Hologram and S p a t i a l F i l t e r i n g  We have already d e s c r i b e d the elements i n v o l v e d i n the system to produce the Hologram and to o b t a i n a r e c o n s t r u c t i o n from i t .  A j u d i c i o u s use of these  the system work s a t i s f a c t o r i l y . necessary  elements i s necessary  To produce the Hologram i t i s  to l i m i t the distances' i n v o l v e d w i t h i n c e r t a i n bounds  so t h a t the r e c o n s t r u c t i o n process a length. i n F i g 4.6.  does not r e q u i r e too great  A t y p i c a l setup i n v o l v i n g the d i s t a n c e s used i s shown In t h i s setup f ( x , y)  £cf : Eq. ( 2 . 1 l ) J  n e g l i g i b l e and hence the phase modulating disturbance Unless  to make  otherwise  i s not i s present.  s p e c i f i c a l l y mentioned t h i s setup w i l l be used  f o r a l l experiments.  Model 130 gas l a s e r A  B  C  D  F i g . 4.6. D e t a i l s of the Setup Used f o r Recording A  :  the Hologram  D i v e r g i n g l e n s , f : 2.2/55 mm Asahi Autotakumar, s i x element.  B  :  P i n h o l e , 220 microns d i a .  C  :  Object, area about h a l f a centimeter  square.  D  :  35 mm  camera back to r e c o r d Hologram.  high c o n t r a s t copy, Type M 402. a centimeter  F i l m used:  Hologram area:  bench as shown i n F i g . 4.9  was  used to hold up the  T y p i c a l exposure w i t h the Spectra-Physics l a s e r turned  about  square.  A l l equipment should be r i g i d l y mounted.  gas  41 Kodak  An  optical  equipment. Model  130  on to i t s maximum b r i l l i a n c e i s l / l 5 second.  A t y p i c a l example of the Hologram and the r e c o n s t r u c t i o n from i t i s shown i n F i g . 4.8. involved.  The  Note the m a g n i f i c a t i o n f a c t o r s  r e c o n s t r u c t i o n i s l a r g e r than the o r i g i n a l  cause of the f a c t that a d i v e r g e n t beam was  used.  The  be-  exact  setup and the d i s t a n c e s i n v o l v e d in. the r e c o n s t r u c t i o n process i s not the same as the p o s i t i o n at which i t was was  necessary  taken.  This  to o b t a i n a r e c o n s t r u c t i o n w i t h i n the length of  the o p t i c a l bench. p l a c i n g C i n F i g 4.7  It  i s p o s s i b l e to get a r e c o n s t r u c t i o n by  at 52.5  cm form the p i n h o l e .  The  recon-  s t r u c t i o n i s then f u r t h e r away and bigger i n s i z e ,  i Some more examples of Holograms of step and f u n c t i o n s ai*e g i v e n i n F i g . 4.10. p o s i t i v e Holograms. can be c l e a r l y  The  pulse  A l l examples shown are  s p a t i a l frequencies i n these Holograms  identified.  A  :  D i v e r g i n g l e n s , f : 2.2/55 mm Asahi Autotakumar, s i x element.  B  :  P i n h o l e , 220 microns d i a .  C  :  Hologram  D  :  R e c o n s t r u c t i o n , recorded on P o l a r o i d Type 55 P/N  F i g . 4.7. D e t a i l s of the Apparatus f o r Obtaining Reconstruction.  film  43  m i l , ^ ! ^ . ' ; • ; Mr 1  in  1—  a) O r i g i n a l , 32 X m a g n i f i c a t i o n .  b) Hologram, 11.6  c) Lensless r e c o n s t r u c t i o n by  d) Gabor r e c o n s t r u c t i o n using  X magnification.  modified method, a c t u a l  r e c o n s t r u c t i n g l e n s , 6.4 X  size.  magnification.  F i g . 4.8. T y p i c a l Object, i t s Hologram and Reconstruction the Hologram. (Note m a g n i f i c a t i o n f a c t o r s ) .  from  Fig. 4 . 9 .  Photograph of Apparatus Used i n Performing the Experiments.  45  ) Hologram  [pf(x,  y)  =T oJ  of a negative p u l s e .  ) Hologram  [pf(x,  y)  =  Holograms Types of  |^"pf(x, y)  £  O-J  £  o]  of a negative p u l s e .  oJ  of a p o s i t i v e p u l s e .  F i g . 4.10.  b) Hologram  d) Hologram of a step  (Fresnel Signals.  [ p f ( x , y) function.  and Fraunhofer)  of  Different  46 The e f f e c t  of the pinhole i n removing the noise from the Hologram  as w e l l as the r e c o n s t r u c t i o n i s the i l l u s t r a t i o n s  F i g . 4.11.  c l e a r l y demonstrated by comparing  i n F i g . 4.11 w i t h those of F i g 4 . 8 .  Noisy Hologram and R e c o n s t r u c t i o n due to Presence of the A i r y R i n g s .  the  47 For s p a t i a l f i l t e r i n g i t i s p o s s i b l e to use two d i f f e r e n t setups. One i s the scheme described  i n Sec. 2.5.  This has the disadvan-  tage of r e c o r d i n g the Hologram on the f i l m p r e v i o u s l y . scheme i s to use a one step process as described This scheme i s i n p r i n c i p l e e q u i v a l e n t Sec.  A  simpler  i n Appendix A.  to the one described i n  2.5 but gives a b e t t e r r e s u l t since the problems o u t l i n e d  i n Sec. 2.4 are avoided. The  f o l l o w i n g setup was used f o r f i l t e r i n g .  The r e s u l t s  are i n d i c a t e d i n F i g . 4.13.  A  B  C  D  E  F  l e n s , f : 2.2/55 mm Asahi  G  A  :  Diverging  Autotakumar  B  :  Pinhole,  C  :  C o l l i m a t i n g l e n s , f : 2.8/40 mm K i l f i t t - M a k r o - K i l a r E  D  :  Object transparency, about h a l f a centimeter square  E  :  Hologram forming le'ns, f s 5.6/135 mm  220 micron d i a .  Schneider-Kreuznach-  Componan F  :  F i l t e r element placed at the f o c a l p o i n t of the lens E  G  :  Imaging l e n s , f : 2.5/10 i n c h  H  .  F i l t e r e d image recorded Fig.  on P o l a r o i d Type 55 P/N  4.12. D e t a i l s of the F i l t e r i n g Apparatus.  48  ) Removal of d . c . component 0.025 cm negative s t e p .  from  b) Removal of high frequencies from a 0.025 cm step.  Magnifi-  c a t i o n 25 t i m e s .  c a t i o n 25 t i m e s .  ) Square wave imaged  d) E f f e c t  through  of low pass f i l t e r on  system shown i n F i g . 4 . 1 2 .  the  No f i l t e r  fication  13.5  used.  Magnification  same square wave. Magni13.5 times.  times.  Fig.  Magnifi-  4.13. Effect Typical  of S p a t i a l F i l t e r s on Some Signals.  49  5. CONCLUSIONS The F o u r i e r transform  property of a Gabor Hologram  which i s u s e f u l f o r studying bandwidth reduced t e l e v i s i o n t r a n s mission i s developed i n Ch. 2* and  (2.18).  The key formulae are Eqs. (2.12)  A l o g i c a l e x p l a n a t i o n , based on the transform  property, of the two step imaging process  i s developed i n Sec. 2.3.  I t i s then demonstrated that the r e c o n s t r u c t i n g lens suggested by Gabor i s unnecessary and s u p e r i o r q u a l i t y r e c o n s t r u c t i o n can be obtained by omitting; t h i s .  The frequency  which a l s o f o l l o w s from the transform  filtering  property  property of the modified  Hologram i s d i s c u s s e d i n Sec. 2.5. The  f e a s i b i l i t y of a p p l y i n g the Hologram  techniques  to t e l e v i s i o n t r a n s m i s s i o n a t reduced bandwidth i s s t u d i e d i n Ch. 3.  A block diagram of a proposed two channel system i s  given i n F i g . 3.4.and estimates  of the p o t e n t i a l saving i n band-  width f o r a s p e c i a l case are given i n Sec. 3.2.  The system has  not been e x p e r i m e n t a l l y v e r i f i e d and p o s s i b i l i t y of f u t u r e work exists i n this  direction.  Experimentally,  a number of techniques  have been  d i s c o v e r e d or developed f o r Hologram work, and these  are d e s c r i b e d  i n Ch. 4. The use of a p o s i t i v e lens to diverge the narrow l a s e r beam (2.5 mm d i a ) to cover an area of 5mm x 5mm i s necessary. The n o i s y A i r y r i n g s created by t h i s lens have been removed by using a pinhole.  This technique  e x p l a n a t i o n of the process  i s d e s c r i b e d i n Sec. 4.1 and an  i s given i n the Appendix.  Typical  50 r e s u l t s of the Gabor method of r e c o n s t r u c t i o n of two tone n a l s are given i n Sec. 4.2.  origi-  The r e s u l t s , taken as a whole, i n d i  cate that the Hologram method of r e c o n s t r u c t i o n has been  improve  over t h a t achieved by Gabor and, i t i s thought, to the l e v e l at which experiments place.  of the two  channel t e l e v i s i o n system can take  51 APPENDIX A SPATIAL FILTERING USING A LENS SYSTEM 12 I t has been demonstrated  that i f an o b j e c t transpar-  ency i s p l a c e d at One f o c a l plane of a s p h e r i c a l lens and i l l u m i nated with a p a r a l l e l coherent beam then the F o u r i e r transform, i . e . the Hologram, i s d i s p l a y e d at the other f o c a l plane.  This  p r i n c i p l e can be u t i l i z e d f o r performing s p a t i a l f i l t e r i n g i n one  step.  This has the advantage that i t i s not necessary to  r e c o r d the Hologram on a f i l m thereby e l i m i n a t i n g the t r o u b l e s i n d i c a t e d i n Sec. 2.4.  By p l a c i n g another lens (which i n e f f e c t  takes the i n v e r s e F o u r i e r transform) behind the Hologram plane it  i s p o s s i b l e to produce  an image of the o r i g i n a l  transparency  i n one s t e p . The setup a f i l t e r  setup i s i l l u s t r a t e d i n F i g . A . l .  In such a  element as d e s c r i b e d i n Sec. 2.5 i s i n t r o d u c e d  Imaging lens Laser  Obj ect transp  Filter element (Hologram plane)  Image  F i g . A . l . P r i n c i p l e of S p a t i a l F i l t e r i n g Using Lenses.  52 i n the Hologram plane. the image plane.  This w i l l r e s u l t i n a f i l t e r e d image i n  A b l a c k spot placed at the f i l t e r  render an image w i t h the d.c. component removed. aperture p l a c e d i n the f i l t e t f plane w i l l filter  will  S i m i l a r l y an  simulate a low pass  and the image obtained would be l a c k i n g iri the high  frequency plane.  plane  component.  Other f i l t e r s  can also be achieved i n t h i s  53 APPENDIX B EPPECT OF PINHOLE IN REMOVING THE AIRY RINGS When a beam of p a r a l l e l l a s e r l i g h t i s passed through a converging  lens a p a t t e r n known as the A i r y rings i s formed  (see F i g . 4.1b).  The amplitude d i s t r i b u t i o n i n t h i s  when observed at the f o c a l p o i n t of the lens i s  pattern  10  A , V*> 2  A  =  A  d  =  diameter of the l e n s ,  f  =  f o c a l l e n g t h of the l e a s ,  X  =  wavelength of the l a s e r l i g h t .  (B.l)  where,  and  F i g , B . l * . Amplitude D i s t r i b u t i o n a t the Focus of a S p h e r i c a l Lens I l l u m i n a t e d with Laser L i g h t . Now i f t h i s l i g h t i s used to i l l u m i n a t e a pinhole p l a c e d at the focus of the l e n s , then the d i f f r a c t i o n p a t t e r n produced by the p i n h o l e can be expressed as ,  54  JACoso" 2X  U(P)  exp  J Cos^ . 2X o  J (x) x  (  x  /  )  /  / / exp j p ( r + s) ds  JjOaw)  x  K  2 J, l ±  j p ( r + s) ds  *  paw  (B.2)  '  where,  P  =  2-n;/X ,  and  w  =  r a d i u s v e c t o r of a p o i n t i n the d i f f r a c t i o n  a  = r a d i u s of p i n h o l e ,  pattern. Assume now that the s i z e of the pinhole i s such t h a t  x  =  n,  i . e . i n other words i t passes only the c e n t r a l hump of the A curve A  =  shown i n F i g . B . l .  Therefore, i f r i s greater  than x,  0 .  The expression f o r U(P) would then be the product shown i n F i g * B . 2 a) and b ) .  of the two curves  The r e s u l t i s  in Fig. B.2 c).  the curve  J-^paw) Paw  A  M  U ( P ) = A.M  a,  b )  F i g . B.2. Removal of the A i r y P a t t e r n by the P i n h o l e . This shows that the annoying r i n g s are removed and we get a smooth v a r i a t i o n i n the i l l u m i n a t i o n . This i s confirmed  55 by the photograph i n F i g . 4.1  c).  To get the s i z e of the pinhole we  set x  =  %.  Therefore,  or, r  =  (Xf)/d  =  13.9  A pinhole of 27.8 the 55 mm  =  (6328 x 10~  x 5.5)/(2.5 x 1 0 )  cm  _1  microns.  micron diameter placed at the f o c a l plane  of  lens would then remove the A i r y ring's. It  i s extremely  d i f f i c u l t to produce a w e l l d e f i n e d  c i r c u l a r pinhole of t h i s s i z e * it  8  A bigger pinhole can be used i f  i s placed s l i g h t l y o f f the f o c a l plane of the s p h e r i c a l l e n s ,  the requirement  being that the r a d i u s should be  pinhole blocks anything more than the f i r s t  such that the  zero of the J ^ ( x ) / x  function. In e f f e c t the pinhole can be considered as a low pass f i l t e r p l a c e d at t h e , s p a t i a l frequency The A i r y r i n g s can be approximately frequency  plane  (cf : Sec.  considered as the  2.5).  spatial  d i s t r i b u t i o n of a pulse the low f r e q u e n c i e s of which  are being only passed by the p i n h o l e .  56 REFERENCES 1*  Gabor, D., "Microscopy by Reconstructed Wavefronts." Proc* Roy* Soc*. A* 197> 1949, pp. 454-486.  2*  L e i t h , E . N* and Upatnieks, J . , "Wavefront R e c o n s t r u c t i o n with Continuous Tone O b j e c t s , " J . Opt. Soc. Am., v o l * 53, Dec. 1963, pp* 1377-1381.  3«  Born, M. and Wolf. E., P r i n c i p l e s of O p t i c s , 2nd ed* New York: Pergamon P r e s s , Inc., 1964, pp. 378-382.  4.  Vander Lugt, A. " S i g n a l D e t e c t i o n by Complex S p a t i a l F i l t e r i n g , " IEBETrans• on I n f o r m a t i o n Theory, v o l . IT - 10, A p r i l 1964. Number 2, pp. 139-145.  5.  Cheatham, T* P* and Kohlenberg, A., " O p t i c a l F i l t e r s — T h e i r Equivalence to and D i f f e r e n c e . f r o m E l e c t r i c a l Networks:," 1954 IRE N a t l Conv. Record, p t . 4. pp* 6-12.  r  f  6*  L e i t h , E . N. and Upatnieks, J . , "Wavefront H e c o n s t r u c t i o n w i t h D i f f u s e d I l l u m i n a t i o n and Three Dimensional O b j e c t s , " J . Opt* Soc* Am** v o l . 54. Nov. 1964,' pp..' 1295-1301*  7.  L e i t h , E . N. and Upatnieks, J . , "Photography by L a s e r * " S c i e n t i f i c American* June 1965, v o l . 212, pp. 24-35*  8*  Cherry, C. and G o u r i e t , G. G*, "Some P o s s i b i l i t i e s f o r the Compression of T e l e v i s i o n S i g n a l s by Recoding," Proc* IEE, v o l . 100, Jan* 1953, pp. 9-18.  9*  Cherry, C*, Kubba, M. H., Pearson, D. E* and Barton, M* P., "An Experimental Study of the P o s s i b l e Bandwidth Compress i o n of V i s u a l Image S i g n a l s , " P r o c . IEEE, Nov. 1963, pp. 1507-1517.  10*  O l i v e r , B. M*, "Some' P o t e n t i a l i t i e s of O p t i c a l Proc* IRE. Feb* 19.62, pp. 135-141.  11.  Holeman, J * , "The 3-D Hologram Process and Image R e c o g n i t i o n by S p a t i a l F i l t e r i n g , ? G . E * Advanced Technology L a b o r a t o r i e s T e c h n i c a l Information S e r i e s P u b l i c a t i o n No. 64GL159* Date: Sept* 9» 1964.  Masers."  1  12.  Rhodes J r * . J * E * . " A n a l y s i s and Synthesis of O p t i c a l Images," Am. J * of Phys.. v o l . 21, May 1953, pp. 337-343.  13*  Beddoes, M* P* and Akfrtan S« A., "Some Techniques f o r P r o ducing Holograms w i t h L a s e r s , " A paper accepted f o r p r e s e n t a t i o n i n October 1965 a t the Canadian E l e c t r o n i c Conference sponsored by the Canadian branch of the IEEE.  

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