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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. Engineering, University of Dacca, 1961. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of E l e c t r i c a l Engineering We accept this thesis as conforming to the standards required from the candidates for the degree of Master of Applied Science Members of the Department of E l e c t r i c a l Engineering The University of B r i t i s h Columbia August 1965 In presenting th i s thes i s in p a r t i a l f u l f i lmen t of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r ee l y a va i l ab l e fo r reference and study. I fur ther agree that per -mission for extensive copying of t h i s thes i s for scho la r l y purposes may be granted by the Head of my Department or by h i s representat ives^ It is understood that copying or p u b l i -ca t ion of t h i s thes i s for f i n a n c i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT This thesis i s part of a f e a s i b i l i t y study concerned with the application of the Gabor Hologram to a proposed method of bandwidth compression of t e l e v i s i o n (video) signals obtainable by scanning a r e s t r i c t e d class of two tone pictures: the experimental work consists of producing and improving on the Gabor type Holo-grams and demonstrating the reconstruction. The proposed t e l e v i s i o n system w i l l employ two channels. One channel would be used to transmit the signal obtained by scanning the picture normally; the bandwidth allowed for this channel, though, would be much less than the normal bandwidth: the other channel of equally reduced bandwidth would be used to transmit the signal obtained, again by normal scanning, from the Hologram. At the receiver, optical superposition of the two pictures obtained from the two channels would be made. For a certain r e s t r i c t e d class of two tone pictures the di s t o r t i o n produced at the receiver by this method of picture transmission i s anticipated to be ne g l i g i b l e . The Fourier transform property of the Hologram i s speci-a l l y developed i n order to complement the two channel t e l e v i s i o n system. Considering the transform point of view i t has been shown theoretically.that the reconstruction lens suggested by Gabor i s not necessary for the two step imaging process and a lensless system has been developed. It i s demonstrated that better quality results can be obtained by using the modified process. i i In the experimental setup a positive lens necessary to widen the laser beam i s used. The effect of the noisy Airy rings created by this lens i s studied and a pinhole method of removing the noise has been suggested and i t s effectiveness demonstrated experimentally. F i n a l l y some examples of two step imaging using the modified lensless system are given,, The detailed account of the experimental work i s given to f a c i l i t a t e future work. i i i CONTENTS Page L i s t of I l l u s t r a t i o n s ............... .... v Acknowledgement . ................. v i i 1. Introduction ........... 1 2. Analysis of the Hologram 5 2*1 The Presnel Kirchhoff D i f f r a c t i o n Integral ... 5 2*2 Holograms as Spatial Fourier Transforms of the Object 8 2.3 Pr i n c i p l e of Lensless Reconstruction from a Hologram 12 2.4 Recording the Hologram: The Phase Problem ... 14 2.5 Frequency F i l t e r i n g with Hologram 17 3. Application of Holograms i n Television Transmis-3.1 Analysis 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 .. .. 31 4* Experimental Techniques and Results 32 4.1 The Elements of the System 32 4.2 The Technique of Producing the Hologram and Spatial F i l t e r i n g 40 5 * Conclusions . * . » * • . • • ' * * • * . * . . . • . . • * « o . o . « « . . . . . . . . . 49 Appendix A: Spatial F i l t e r i n g Using a Lens System ... 51 Appendix B: E f f e c t of Pinhole i n Removing the Airy Rings 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 Sym-bols Involved i n the D i f f r a c t i o n Integral ....... 7 2.2 Basic Arrangement for Spatial Spectral Analysis.. 15 2.3 Basis of Optical F i l t e r i n g 19 3.1 Quantization of Video Signal 22 3.2 Arrangement for Obtaining the Fourier Transform 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 25 3.4 Basic Scheme for Television Band Compression Using the Hologram . .....»•*«•••..........••..«.« 28 4.1 I l l u s t r a t i n g the Ef f e c t of Introducing a Conver-ging Lens i n the Path of a Coherent Light Beam... 34 4.2 Focussing of Coherent Light 35 4.3 Pinhole Used i n Removing the Airy Rings, 36 4.4 Experimental Arrangement for Illuminating the Object 37 4.5 Using a 35 mm Camera Back for Recording the Hologram ........................................ 39 4.6 Details of the Setup Used for Recording the HoXo^ i*stm » * # * * « * « o 9 * © « * » o » » * * * * * * » " » o » « a « « » * © « o o * * 40 4.7 Details of the Apparatus Used for Obtaining the Reconstuction ....... 42 4.8 Typical Object, i t s Hologram and Reconstruction from the Hologram 43 4.9 Photograph of Apparatus Used i n Performing the Experi 4.10 Holograms (Fresnel and Fraunhoffer) of Different types of Signals 45 4.11 Noisy Hologram and Reconstruction due to the Presence of the Airy Rings....... 46 v 4 . 1 2 Details of the F i l t e r i n g Setup . „ „ « . o . . . . . . . . . . . . . 4 7 4 . 1 3 E f f e c t of Spatial F i l t e r s on Some Typical S X ^ H £ t l S o o o o » o o a o » o o o o o e o o * o o o o o o « o o o o o o o o « « e « « « o o 4 8 A. l P r i n c i p l e of Spatial F i l t e r i n g Using Lenses ...... 51 B. l Amplitude Dist r i b u t i o n at the Focus of a Spherical Lens Illuminated with Laser Light ...... 53 B » 2 Removal of the Airy Pattern by the Pinhole ....... 54 v i ACKNOWLEDGEMENT The author would l i k e to thank Dr. M.P. Beddoes, the supervisor of this project, for the guidance and encourage-ment given during the period this work was carried out. Thanks are due to the staff members and the technicians of t h i s department for their help during the research. The author c o r d i a l l y acknowledges the illuminating discussions he had with Mr. P.J.W. Farr, the co—worker on the project, during the research. He i s greatly indebted to the Canadian Commonwealth Scholarship and Fellowship Committee for the award of a scholar-ship and to the National Research Council of Canada for f i n a n c i a l support f o r the research. Last but not the least the author would l i k e to express his sincere thanks to Miss H.A. Thomson for typing the masters for this thesis. l o INTRODUCTION 1 The process of two stage imaging using the d i f f r a c t i o n pattern of an object was discovered i n 1948 by Denis Gabor"*" of the Imperial Col lege , London, He suggested the name Hologram (gk. holos : whole ; gramma : writ ing) for the d i f f r a c t i o n pat-tern of a transparency because he bel ieved that th is contained the t o t a l information for reconstruct ing the object . Except for a "very few special cases the Gabor Hologram does not r e a l l y contain the whole information about the object . It misses out the phase information| yet under cer ta in circumstances i t can be used to reconstruct a reasonably good r e p l i c a of the object transparency. The name Hologram continues to be used for reconstruction using other methods. During the la s t two years several other types of Holograms have been produced and some of these contain the missing phase information. In th is thesis we s h a l l , however, use the word Hologram s p e c i f i c a l l y to describe the basic Hologram which Gabor described i n his paper"*". Any other type of Hologram w i l l be referred to with a sui tably qual i fy ing adjec t ive . The primary interes t of Gabor was the appl icat ion of the Hologram technique to e lectron microscopy. He had obtained some Holograms using a divergent e lectron beam and reconstructed an image from them by using v i s i b l e coherent l i g h t . The qual i ty of the reconstruct ion was poor par t ly because of the lack of a s u f f i c i e n t l y br ight source of highly coherent l i g h t and par t ly because of the system of lenses he had been using for obtaining the reconstruct ion . It has been demonstrated i n th is thesis that the system can be made completely lensless and a better qual i ty reconstruct ion can be obtained thereby. 2 Because of the lack of a s u f f i c i e n t l y bright source of coherent l i g h t the interes t i n Hologram technique died down u n t i l the advent of the laser i n I960. The invention of the laser revived the interes t i n th i s f i e l d and since then studies have been made into the various other p o s s i b i l i t i e s of the Hologram. A consider-able amount of work has been done since 1962 by Emmet Le i th and J u r i s Upatnieks in the Univers i ty of Michigan. They have extended the basic Gabor process to the reconstruction of continuous tone and three dimensional objects . They have developed the process as a new system, of lensless photography capable of producing the s t r i k -ing realism 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 recording the resultant d i f f r a c -t i o n pattern on a photographic p l a t e . The method developed by 2 L e i t h and Upatnieks d i f f e r s from this in that a reference beam obtained by s p l i t t i n g the o r i g i n a l beam is allowed to impinge on the d i f f r a c t i o n pattern. This resul t s i n a modulation of the d i f f r a c t i o n pattern according to the phase information carr ied by the Hologram forming beam. It i s possible to obtain a good qual i ty reconstruct ion of continuous tone objects by th is process. We sha l l refer to a Hologram made i n th i s way as a split-beam Holo-gram. For the purpose of th i s thesis we sha l l l i m i t our attent ion to the basic Hologram as described by Gabor. For such a Hologram, Gabor offered an explanation of the two step imaging process by considering the d i f f r a c t e d beam to consist of two par t s , the background wave and the secondary wave. He showed that i n the process of reconstruct ion from the Hologram i t i s possible 3 to get a wave part of which i s similar to the wave produced by the object. He argued that i t should, therefore, be possible to form an image of the o r i g i n a l by focussing the substitute wave issuing out form the Hologram with a spherical lens. It i s , however, possible to consider the whole process of two step in^#?iin:g; as consisting of two successive Fourier transformations i n a spatial coordinate system. Using this point of view i t i s possible to explain the two step imaging phenomenon more elegantly. This explanation, given i n Ch. 2, does not de-pend on the formation of a reconstructed image with a lens. Further, i t has been experimentally v e r i f i e d that i t i s possible to obtain a better reconstruction from a Gabor Hologram by making the system lensless. In this thesis we s h a l l be mainly interested i n the p o s s i b i l i t y of applying the Hologram technique to the bandwidth compression of a t e l e v i s i o n picture signal. A v i s u a l examination of the exaniples of the Holograms of simple patterns presented elsewhere i n t h i s thesis w i l l show that the intensity v a r i a t i o n i n the Hologram tends to be less sharp.than i n the O r i g i n a l . For example,, a straight edge representing a step function becomes a gradual and somewhat c y c l i c change i n i n t e n s i t y . We could, there-fore, i n f e r that the signal obtained by scanning the Hologram w i l l be less variable than the signal obtained by scanning the o r i g i n a l . This indicates that the spectrum of the Hologram w i l l occupy less bandwidth than the spectrum of the signal obtained from the o r i g i -nal* A scheme using two channels to achieve bandwidth reduction has been presented i n Ch. 3. I t has been shown that i t i s possible to obtain about f i v e to one reduction f o r a p a r t i c u l a r case. 4 A l t h o u g h t h e c a s e we d e a l t w i t h i s n o t a t y p i c a l e x a m p l e o f a v i d e o s i g n a l , y e t i t i n d i c a t e s v e r y c l e a r l y t h e p o s s i b i l i t y o f b a n d r e d u c t i o n f o r a c e r t a i n c l a s s o f s i g n a l s . We h a v e a l r e a d y s t a t e d t h a t t h e r e e x i s t s a F o u r i e r t r a n s f o r m r e l a t i o n s h i p b e t w e e n t h e H o l o g r a m a n d i t s o r i g i n a l . T h i s o p e n s u p t h e p o s s i b i l i t i e s o f a c h i e v i n g f i l t e r i n g o f s i g n a l s i n t h e s p a c e d o m a i n i n s t e a d o f i n t h e t i m e - f r e q u e n c y d o m a i n . T h i s i s o f i n t e r e s t b e c a u s e o f t h e f a c t t h a t t h e f r e q u e n c y c o m -p o n e n t s o f a p i c t u r e s i g n a l c a n be v i s u a l l y i d e n t i f i e d i n a H o l o g r a m a n d b y s e l e c t i v e t r a n s m i s s i o n o f t h e c o m p o n e n t s b a n d r e d u c t i o n c a n b e a c h i e v e d . T h i s c o u l d h a v e b e e n d o n e i n t h e t i m e d o m a i n a s w e l l b u t t h e a d v a n t a g e o f u s i n g t h e s p a c e d o m a i n i s t h a t t h e s e l e c t i o n o f c o m p o n e n t s b e c o m e s v e r y s i m p l e , b e i n g a m a t t e r o f m e r e l y p u t t i n g some m a s k s o n t h e H o l o g r a m s o t h a t t h e u n d e s i r e d c o m p o n e n t s a r e b l o c k e d o u t . The p r o c e s s o f f o r m i n g a H o l o g r a m a n d o b t a i n i n g a r e a s o n a b l y g o o d r e p l i c a o f t h e o r i g i n a l f r o m t h e H o l o g r a m r e q u i r e s a c a r e f u l a n d s p e c i a l t e c h n i q u e . C h a p t e r 4 g i v e s a d e t a i l e d a c c o u n t o f t h e e x p e r i m e n t a l t e c h n i q u e s . 2. ANALYSIS OF THE HOLOGRAM 5 We plan to investigate the f e a s i b i l i t y of using the Hologram for t e l e v i s i o n bandwidth compression. The p o s s i b i l i t y arises because the signal obtainable from scanning the Hologram seems to be less variable than that from scanning the o r i g i n a l . An exact knowledge of the equation of the Hologram i s necessary to predict the type of signal that w i l l be obtained when a given Hologram i s scanned* d i f f r a c t i o n integral which leads to the mathematical description of the Hologram. In la t e r sections a method of evaluating this integral i s shown and i t i s demonstrated that there exists a Fourier transform relationship between an object and i t s Holo-gram. The pri n c i p l e of reconstruction 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 discussed* 2.1 The Fresnel-Kirchhoff D i f f r a c t i o n Integral It i s possible to show that the Hologram produced by an aperture when illuminated by a point source of coherent and monchromatic l i g h t i s governed by the following integral formula, 3 generally known as the Fresnel-Kirchhoff d i f f r a c t i o n formula , This chapter commences by quoting the Fresnel-Kirchhoff 0(p) = - ^ JP(r+s) (cos a - cos y)dS (2.1) 6 where, 0 ( P ) = complex amplitude of l i g h t disturbance at a A e ^ r a point P i n the d i f f r a c t i o n pattern, complex l i g h t amplitude at a distance r from the point source, r = distance of point source from any point i n the aperture s = distance of the point P from the same point, in 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, 2ll P = — = phase s h i f t per unit length, jfy indicates the area of the aperture, and dS = elemental area. The angles a and P are the angles that the incident and the d i f f r a c t e d rays make with the normal to the plane of the aperture. Fig* 2.1 shows the arrangement of the system to obtain the dif f r a c -t i o n pattern and the distances 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 define a coordinate system (x, y, z) with the z di r e c t i o n perpendicular to the plane of the aperture. Let the coordinates of P, the point on the d i f f r a c t i o n pattern and P Q the point source be (x', y', z*) and ( X Q> y Q> Z q ) respectively. Suppose now that a two dimensional transparency i s placed i n the aperture. Assume that the transmission c o e f f i c i e n t of the transparency i s t. In a s t r i c t mathematical sense the concept of a transmission c o e f f i c i e n t for a two dimensional transparency i s meaningless. However, for physical transparencies 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 thickness i s f i n i t e and a transmission 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, i . e . along the z d i r e c t i o n . Therefore t = t(x, y ) . This w i l l be very near-l y true i f the transparency i s s u f f i c i e n t l y t h i n . When the transparency i s placed i n the aperture the d i f f r a c t i o n pattern at P i s modified. The resultant pattern can be obtained from Eq. (2.l) by introducing t(x, y) i n the d i f f r a c -t i o n i ntegral and i s , 0 ( P ) = ~ 2 l JJt(x* y ) ^^ Ir^  (°OS a ~ Cos^)dS ^ (2.2) 2.2 Hologram; as a Spatial Fourier Transform of the Object : Evaluation of the D i f f r a c t i o n Integral and the Approximations  Involved A very important and interesting class of Hologram i s that i n which the Holograms are the spatial Fourier transforms of the corresponding object transparencies. This happens under certain special conditions when one has to make some of the distances involved i n Eq* (2.2) large and some of the angles small. We propose to investigate the exact conditions involved. We begin by applying the following constraints on Eq. (2.2), Cos a = - C o s f = Cos S , where £ i s a very small angle (2.3) ~rt = r ' " ^ ' = C o n s t a n t ( 2 o 4 ) 9 r' and s' as shown i n F i g . 2 . 1 r e p r e s e n t t h e d i s t a n c e s o f P Q and P from t h e o r i g i n 0 of t h e c o o r d i n a t e s y s t e m . E q . ( 2 . 3 ) and ( 2 . 4 ) a r e j u s t i f i e d i f t h e d i s t a n c e s o f t h e p o i n t s P q and P f r o m t h e t r a n s p a r e n c y a r e l a r g e b e c a u s e t h e n t h e v a r i a t i o n i n t h e t e r m (Cos a — Cosy) and l / ( r s ) w i l l be n e g l i g i b l e 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 f u n c t i o n exp j p ( r + s ) i n E q . ( 2 . 2 ) . I t s h o u l d be n o t e d t h a t a v e r y s l i g h t change i n ( r + s) p r o d u c e s a l a r g e change i n t h e t e r m exp j8 ( r + s ) . These two c o n d i t i o n s a l s o d i c t a t e t h a t t h e i l l u m i n a t i n g and t h e d i f f r a c t e d beam have a v e r y n a r r o w d i v e r g e n c e a n g l e • T h e r e f o r e , E q . ( 2 . 2 ) r e d u c e s t o , 0(F) = - j 4jfrf$JJ t U , y) exp [-jp(r + s)] dxdy A ( 2 . 5 ) The a p p e a r a n c e o f E q . ( 2 . 5 ) w o u l d l e a d one t o t h i n k i n t u i t i v e l y t h a t 0 ( P ) i s somehow r e l a t e d t o t h e two d i m e n s i o n a l s p a t i a l F o u r i e r t r a n s f o r m o f t ( x , y ) . I f we c o u l d s u i t a b l y m o d i f y t h e f u n c t i o n e x p J P ( r + s) i n E q . ( 2 . 5 ) i n t o t h e k e r n e l f u n c t i o n o f t h e s t a n d a r d two d i m e n s i o n a l F o u r i e r i n t e g r a l t h e n t h e p r o p o s i t i o n w o u l d be p r o v e d . ¥ e , t h e r e f o r e , have t o t r a n s f o r m t h e f u n c t i o n j p ( r + s ) i n t o t h e f o r m - j (<o x + to y ) . T h i s r e q u i r e s t h e e x p a n s i o n o f r and x y s i n t e r m s o f t h e r e c t a n g u l a r c o o r d i n a t e s . W i t h r e f e r e n c e t o F i g . 2 . 1 we c a n w r i t e , r = [ ( x 0 - x ) 2 + ( y o - y ) 2 + [<*2 + yl + zo' + * 2 + y 2 - 2 < v + y 0y> ] [ 2 2 2 T ^ r' + + y - 2 ( X Q X + y oy ) J T x 2 y 2 x o x + y o y 1 10 or r =• r x o x + y o y x ^ + y 2 f» + 2r T (x x + y y)' x o J oJ 2r ,3 • • 9 • (2.6) The series i n Eq, (2.6) converges as long as x/r' and y/r' are each less than unity. This, as we shall see l a t e r , w i l l always be the case. Simi l a r l y , s can be expressed as 2s x fx + y 1 y „ t + x 2 + v 2 (x'x + y ' y ) 2 ( 9 7 ) O c t ~ -> • • • • \ * ' { ) 2 s , J Therefore r + s = (r' + s') - (px + qy) + f(x, y ) , where P = A o (2.8) (2.9) (2.10) and, A f ( x , y) = \ i i o i (x x+y y) / l . 1 \ / 2, ,2\ o J oJ ' (wr + rr) (x +y ) - — ,3 (x'x+y ty)' a r 3 (2.11) Substiiuting Eq. (2.8) i n Eq* (2.5) we obtain, 11 0 ( P ) . _ j A C ^ y - " ' ) /7'rtU> y ).U*<*.yf| e - ^ ( p ^ ) d x d y 0 ( » x t » ) = K |/|t(x,y) exp ^ j p f (x,y)]J exp - j (ioxx + ayy)/j dxdy °* (2.12) Eq. (2.12) i s the analytic expression for the amplitude v a r i a t i o n i n a Hologram. In this equation, Ci>x = pp = spatial frequency i n radians/unit length i n the x di r e c t i o n (2.13) C0y = qP = spatial frequency i n radians/unit length i n y direction (2.14) K = — j A Co»W[,iP(r,+aO} = Constant (2.15) In the experiments described i n thi s thesis the system geometry was so arranged that a l l times we had X q = y Q = 0 . The expression fo 7r the spatial frequencies then become, «>x = 2TI ^ X r radians/unit length (2.16) toy = 2ft ^ ' g t - radians/unit length (2.17) Eq. (2.12) i s i n the form of a standard two variable Fourier i n -tegral transform. It i s important to note that i n the plane of the Hologram which we shall henceforth refer to as the spatial 4 frequency plane , the amplitude d i s t r i b u t i o n instead of being a direct Fourier transform of the transmission function t(x, y) i s modified by a phase factor exp jpf(x, y ) . For the Hologram to be the direct transform of the transmission function t(x,y) the aforesaid phase factor should be equal to u n i t y 0 This requires that f ( x , y) = 0. 12 The necessary condition for the Hologram to be a spatial Fourier transform of the transmission function i s then, r' and s' should be very large and maximum bounds of x and y be small. The factor P dictates the order of smallness of these factors. Since p may be quite large for the wavelengths of v i s i b l e l i g h t , t h i s condition becomes more meaningful i f we say Pf (x, y ) < 2% (2.18) It i s possible to simulate an i n f i n i t e l y large r' i n a f i n i t e distance by using a p a r a l l e l beam of l i g h t to illuminate the object transparency. The necessary condition as outlined 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 physical optics. 2.3 P r i n c i p l e of Lensless Reconstruction from a Hologram It i s interesting to note that i t i s possible to ob-t a i n a Hologram, which i s a Fourier transform of t(x, y), i n a large number of planes behind the object transparency as long as the condition outlined i n Eq. (2*18) i s approximately s a t i s f i e d . This i s i n contrast to the Holograms which are produced by using lenses'*. Such a Hologram i s produced by u t i l i z i n g the fact that the l i g h t amplitude distributions at the front and back focal plane of a spherical lens bear a Fourier transform r e l a t i o n to each other. Since i n such a system the transform r e l a t i o n i s not v a l i d for any other two planes i t i s necessary to position exactly the object transparency and the f i l m to record the Holo-gram at the two focal planes* The system which we have described i s inherently free from such a l i m i t a t i o n . 13 • Let us now assume that we have a transparency i n which 0(<*x» «y) has been recorded, no lens being used i n the process of recording.. This can be very simply done by placing a photo-sensit ive material i n the spa t ia l frequency plane. Since th is w i l l record the Fourier transform of t (x , y) we ought to be able to synthesize t (x , y) from th i s by taking an inverse Fourier transform- This could be accomplished by i l luminat ing the Hologram transparency with a l i g h t beam s imi lar to the one -producing i t . It might be argued that since we are using the same kind of beam as was U s e d i n producing the Hologram, the ker -nel function w i l l not change sign which is necessary for taking an inverse Fourier transform*.. This i s of l i t t l e s ignif icance since a change i n sign can be effected by properly ident i fy ing the coor-dinate axes during reconstruct ion . What i s important to note i s that we have taken two successive transforms to retr ieve the o r i g i n a l funct ion . We have, thus, achieved a completely lens-less system of imaging in two dimensions. The basic steps involved i n the two step process are i l l u s t r a t e d i n F i g . 2.2. Although we have developed the theory as outl ined above assuming that Fraunhofer d i f f r a c t i o n i s taking place, i t i s not ac tua l ly necessary to observe the Fraunhofer condit ion as given i n E q . (2.18) very s t r i c t l y i f the sole purpose i s two step imaging only . I f we do not observe the condit ion given by E q . (2.18) we get a d i f f r a c t i o n pattern that i s ca l l ed the Fresnel d i f f r a c t i o n . This corresponds to the case where f (x , y) defined by Eq . (2.1l) i s not n e g l i g i b l e . If such a case i s used a l l that happens i s that instead of gett ing back t (x , y) dur-ing the reconstruct ion process we get a function which is b a s i c a l l y 14 t(x, y) with a superimposed somewhat o s c i l l a t i n g modulation effect due to the exponential term i n which f (x, y) appears. The usefulness of such a reconstruction depends on the amount of allowable degrading effect due to noise that i s permissible. 2.4 Recording the Hologram : The Phase Problem We have so far t a c i t l y assumed that i t is possible to record the Hologram on a photographic f i l m . This i s true for an extremely limited number of special types of Hologram. A photo-graphic plate can record only positive functions as i t i s sensitive* to the intensity 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 . It i s , therefore, not possible to record f a i t h f u l l y the H 0logram of objects having either negative or com-plex values i n their spatial transforms using the simple means we have described. A way to get around this 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 into two parts, one part to form the Hologram (the signal beam) in the way we have described and the other part, i . e . the reference beam, i s allowed to shine in the plane of the Hologram. This produces a modulation of the intensity d i s t r i b u t i o n caused by the interference' of the reference and the signal beam in the plane of the Hologram according to the phase information carried by the Hologram forming beam. Such a Hologram known as the two-beam or split-beam type does not bear any resemblance what-soever to the o r i g i n a l object but i s able to reporduce high quality reconstruction of objects having complex transforms. The mathematical theory of the s p l i t beam Hologram i s yet to be 15 Hologram Object t-(x,y) T(co ,CO ) x* y a) Arrangement for Obtaining the Spat ia l Fourier Transform. Reconstruction Holograjii b) Arrangement for Obtaining the Inverse Fourier Transform. F i g . 2.2. Basic Arrangement for Spat ia l Spectral Ana lys i s . (P i s a point source of monochromatic and coherent l i g h t ) developed but the experimental technique connected with i t -has advanced quite far mainly due to the work of Leith and Upat-nieks who pioneered this technique,. This kind of Hologram is also capable of reconstructing three dimensional objects producing a genuine t h i r d dimensional effect »• We shall l i m i t our attention for the purpose of this thesis to the basic Hologram. A s p l i t beam Hologram although capable of giving better reconstruction does not show the Fourier transform relationship!because of the intensity modulation produced by the reference beam. Besides t h i s , so far as bandwidth compression i s concerned the s p l i t beam Hologram i s not suitable because of the extremely fine d e t a i l present i n such a Hologram which would require a band-width far 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 possible to obtain recog-nizable reconstruction from Holograms produced i n the way as shown i n F i g . 2.2a i f the objects consists mainly of transparent areas with a few dark patterns. Actually, as long as the.spatial transforms are such that they are either predominantly positive or predominantly negative a r e l a t i v e l y good reconstruction can be obtained. A t y p i c a l example i s that of a f l a t top pulse function either positive or negative. The spatial representation of a positive 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. It i s possible to obtain a reconstruc-t i o n from the Gabor Hologram of both ^hese pulses. By a Gabor Hologram we mean a Hologram which has been produced using the scheme shown on F i g . 2 . 2 a . For further det a i l s of the process 1.7, the reader i s referred to the section on experimental technique and experimental r e s u l t s . 2.5 Frequency F i l t e r i n g with Hologram We have seen that i n the process of forming a Hologram we perform a linear transformation of the object plane into a spatial frequency plane. This i s a two dimensional analog of the one dimensional time to frequency Fourier transformation. It i s possible to extend ideas from e l e c t r i c a l f i l t e r theory-to Hologram technique. The advantage of using the Hologram i s that i t i s very simple then to achieve elementary f i l t e r s capable of l i m i t i n g bandwidth of signals. Consider two two—dimensional spatial signals f^( x> y) and h(x, y ) . These are i n fact transparencies having transmission functions f. and h. 1 It i s desired to use h as the f i l t e r element to perform some desired operation on f^ and y i e l d f D ( x , y) as the output. In the space domain we have to perform a convolution for this purpose so that, f Q ( x , y) = JrJf±l<cr^) b(x -<r, y -/>,) dtf-d/=> (2.19) Since the process of convolution reduces to a simple m u l t i p l i c a -t i o n i n the frequency domain, the process can be simplified greatly by using Holograms instead of the signals themselves. Thus i n the spatial frequency domain employing the Holograms of f. and h we obtain l F (tt , tt ) = F.(tt , tt ) H(tt , tt ) (2.20) o x' y' l v x* y x* y 7 x ' 18 The function f Q ( x , y) can be obtained by taking an inverse transform of F^H. The steps involved are i l l u s t r a t e d i n Pig. 2.3. From E q . (2.20) we see that i t i s possible to modify the function f by suitably designing the function h(x, y) or H(» , «o ). As we have already said we could operate either in x y the plane of the object or i n the plane of the Hologram. Opera-tio n i n the plane of the transparency i s more complicated as i t involves the process of complex convolution involving spatial s h i f t s and also because of the fact that to evaluate the convo-lu t i o n i ntegral we have to go through a process of measuring the average l i g h t passed through the two transparencies f^(x, y) and h(x, y) placed back to back i n the object plane. In addition to this designing h(x, y) i s more complicated than H(« , <o ) for elementary f i l t e r s . Operation i n the spa t i a l frequency plane, that is- i n the plane of the Hologram i s more convenient and elegant. Suppose for example, we have a dark mask with a central opening of width 2x' by 2y'. If this 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 cut-off frequencies — 2-jx . X , — and i 2-rt , y ,— according to Eqs .< (2.16) A S A S and (2.17). It i s interesting to note that i t has two points, one i n the positive and the other i n the negative di r e c t i o n corres-ponding to the cut—off frequency. Similarly a dark spot placed on the optic axis i n the plane of the Hologram would mean the removal of the d.c. component of a sign a l . 19 SPATIAL FOURIER ANALYZER # 1 F. ( to , to ) H( « x , « ) # 2 F = F.H INVERSE F = F.H O 1 % FOURIER SYNTHESIZER^. # 3 F i g . 2.3. Basis of Optical F i l t e r i n g . 20 It i s possible to obtain reconstruction from a Hologram which has been subjected to frequency f i l t e r i n g . The resultant i s a distorted 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. Typical examples of f i l t e r i n g are shown i n the section on experimental technique and r e s u l t s . Besides these elementary forns of f i l t e r s i t i s possible to synthesize more complicated f i l t e r s i n the plane of the Hologram. One such p o s s i b i l i t y i s to design a f i l t e r for maximizing the signal to noise r a t i o . The design of such a 4 f i l t e r has been described by Vander Lugt and requires the use of the split—beam technique described 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 transmission i s investigated i n this chapter. The p o s s i b i l i t y of achieving band compression by using Holograms '. arises because the intensity d i s t r i b u t i o n in a Hologram, as can be seen from the examples presented i n Ch. 4, i s more smooth than in the o r i g i n a l . The sharp v a r i a t i o n i n the o r i g i n a l , for example, a thin black li n e on a white background gives a Hologram that spreads out over a larger area and has a somewhat c y c l i c and gradual change i n i n t e n s i t y . This i s true for a l l pictures which are b a s i c a l l y i n the form of line 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 varying video signal when scanned by a t e l e v i s i o n camera, where by slowly we mean that the rate of v a r i a t i o n i s slower than the video signal obtained by scanning the o r i g i n a l . On an i n t u i t i v e basis we could then predict that the video signal obtained from the Hologram would possibly cover less bandwidth than the o r i g i n a l for such objects. This statement i s not true for certain types 7 of objects and Hologram methods. For example , the transmission of a three dimensional object by the split-beam Hologram technique requires enormous bandwidth. It i s possible to represent a two dimensional picture, where changes in in t e n s i t y levels along the scanning d i r e c t i o n are very few, by a t r a i n of rectangular pulses. If the changes are far apart each pulse can be treated separately. The analysis of the transmission of a pulse representing such a picture, is 22 given i n Sec. 3.1. On the basis of this analysis a two channel system of band reduced transmission i s proposed. F i n a l l y , i n Sec. 3.2 an account of the possible application of the process i s given. 3.1 Analysis of the Transmission of the Hologram of a Pulse The potential compressibility of t e l e v i s i o n picture signal depends upon i t s excessive redundancy. "Most simply the redundancy arises i n a signal, such as the t e l e v i s i o n signal, by virtue of the fact that i f certain parts of the signal are g known the other parts may be guessed" , i . e . , most pictures consist of a few sharp transitions of intensity with the rest of*the area being more or less even. As a r e s u l t the t e l e v i s i o n picture can be quantized i n several discrete steps as shown i n F i g . 3.1b. It i s possible to transmit a reasonably good quality picture by transmitting these pulses. In t h i s section we s h a l l treat a very special class of the signals taken form those represented by F i g . 3.1b. In some pictures as i n li n e drawings, the pulses representing the a) b) F i g . 3.1 Quantization of a Video Signal a) Original Signal b) Quantized Signal 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 far apart i n time so that the transmission of each pulse can be treated separately. We -could therefore study the transmission of a single pulse as a representative case. We plan to transmit the Hologram of a f l a t topped pulse instead of the pulse i t s e l f over the t e l e v i s i o n channel. The spatial representation of such a pulse i s either a narrow s l i t on a dark background or a dark band on a transparent background* We can begin with the case of a rectangular aperture having dimensions 2a and 2b. This can be converted into a -narrow s l i t by l e t t i n g one of the dimensions become very large and the other becoming very small. The aperture, the di f f e r e n t dimen-sions and the coordinate system are i l l u s t r a t e d i n F i g . 3.2. We then have, +a +b 0(P) = K I J t(x, y) exp j ^ - j (cc^x + « y y ) ] dxdy -a -b (3.1) Eq. (3.1) follows from Eq. (2.12) for 0f (x, y) = 0. This i s e s s e n t i a l l y a case of Fraunhofer d i f f r a c t i o n . For an aperture, we have, t(x, y) = Constant Therefore we obtain form Eq. (3.1), sin aco sin b« x y For a narrow s l i t extending i n f i n i t e l y along y direction we end up with the one dimensional Fourier transform, sin aid 0(P) = C X , (3.3) 24 F i g . 3.2* Arrangement for Obtaining the Fourier Transform of an aperture. (Note that as b tends to i n f i n i t y and a tends to zero ve obtain the Fourier transform of a f l a t topped pulse). which i s a well known function having the shape as given i n Pig. 3.3. 0(P) = 0(«x) co or x' x Pig. 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), co = x _2JL A s' -xT being the x-coordinate of any point i n the Hologram plane. Therefore, 0(P) sin a 22L 2Ll X s' a X s' _ c Sin pgx' (3.4) where, g A a Assume now that the Hologram i s scanned along the x direction with a spot v e l o c i t y v, so that, x' = vt, t being the time. (3.5) Therefore, the time signal obtained by scanning the Hologram can be written as, 26 0 . a , . Sin <o, t 0(t) = c s^ir* = c 1 Pgvt W-j^ t ' where, = P gv (3.6) _ Sin co t Sin fi>, t 0(t) = - r - i — = 2 E - r - l — (3.7) Q where, 2 E = Sin ttt The time function 0(t) = 2 E — i n Eq. (3.7) i s the inverse Fourier transform of the following function, E(tt) = E , (3.8) This can be shown as follows, e(t) = j E(tt) (exp jwt) dtt = j E exp jttt dtt ^ 1 ^ 1 Sin tt,t = 2 E = 0(t) (3.9) Therefore, the signal obtained by scanning the Hologram has a frequency spectrum of width tt^ radians/sec. Therefore, the bandwidth i s r l ~ 2-rt _ 2it ~ X ~ Xs' u . i u ; A very interesting property of the Hologram i s apparent from Eq. (3.10). As the width of the s l i t 'a' decreases, i . e , the narrower the pulse becomes the lesser becomes the frequency band requirements* As a—»-0, f ^— » • 0 giving rise to a zero width spectrum for a zero width s l i t . 27 The main reason that a, wide band i s required for the transmission of a t e l e v i s i o n picture i s that to reproduce sharp edges i n a picture we must make the system r i s e time short enough which means the use of a wide bandwidth. If we convert a picture into 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 pattern which i s spread out i n the Hologram thereby making the frequency range (corresponding to the sharp edge) of the signal obtained by scanning the Hologram small. 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 rise to f i n e r variations i n the Hologram and therefore the bandwidth would be more i f we scan the Hologram instead of the o r i g i n a l for t h i s class of signals. A p o s s i b i l i t y of saving the channel width, therefore, arises i f we can properly handle the two basic classes of signals. The following scheme could possibly be used. Let us have two channels for picture transmission as shown i n Pig. 3.4. In both the channels we introduce low pass f i l t e r s the pass band of which can be varied at w i l l . Channel #1 primarily transmits the broad variations i n the signal whereas channel #2 primarily transmits the f i n e r d e t a i l s i n a picture. At the receiving end the Hologram could be reconstructed to retrieve the o r i g i n a l , the broader variations of which have been removed by the f i l t e r . The two picture signals thus obtained could be superimposed i n the receiver to obtain a comparatively good quality picture. A t y p i c a l order of magnitude calculation w i l l be use-f u l to get an estimate of the band requirements for the two chan-nels as shown i n Pig. 3.4. We s h a l l again use the case of a O B J E C T t ( x , y ) 1 O P T I C A L F O U R I E R A N A L Y Z E R HOLOGRAM x x ' y ' TRANSDUCER V I D E O S I G N A L LOW P A S S F I L T E R C H A N N E L #1 C H A N N E L #2 TRANSDUCER V I D E O S I G N A L LOW P A S S F I L T E R R E C E I V E R R E C E I V E R SUPERIMPOSED D I S P L A Y O P T I C A L F O U R I E R S Y N T H E S I Z E R F i g . 3 » 4 » ' B a s i c Scheme f o r T e l e v i s i o n B a n d C o m p r e s s i o n U s i n g t h e H o l o g r a m ro co 29 t y p i c a l square topped pulse because of the ease and elegance with which i t can be handled mathematically. Assume, - -Scanning rate = L = 525 x 30 = 15,750 lines/sec Vidth of frame = that portion of the Hologram which i s being scanned = V Therefore, v = W L (3.11) Prom Eq. (3.10) *1 " X _ X For a t y p i c a l case, ¥ = 5 cm X = 6.328 x 10~ 5 cm (He-Ne gas laser l i g h t ) which gives, f = 1.25 x 10 9 x g Let us suppose that the f i l t e r i n the second channel in F i g . 5.3 passes only up to 100 kc/s. Then, ~ f 100 x 10 J . a . . - 4 g = s" = Q ' = 0.8 x 10 cm a 1.25 x 10* 1.25 x 10 y a _ max s * —4 S l i t s of width less than 0.8 x 10 x s' can therefore be passed through th i s channel. For Holograms of wider s l i t s we need a wider band. The l i m i t of resolution R expressed as the ratio of the maximum resolvable s l i t to the frame width i s then, 3 0 2 a 2 g s' R = —JS* 5- = _ W = 2 F , ¥ ¥ emax ; * s' where, F = ^— = 1 0 0 t y p i c a l l y Therefore, a 1 0 0 kc/s bandwidth gives a resolution l i m i t for the Hologram as R = 2 x 0 1 8 x 1 0 ~ 4 x 1 0 0 = 0 . 0 1 6 This figure sets the l i m i t on the largest picture point that could be transmitted by the second channel. A l l f i n e r d e t a i l s of the o r i g i n a l picture could be transmitted within the 1 0 0 kc/s bandwidth. Picture points i n the or i g i n a l which are larger than those dictated by R above could be sent by conventional means by a narrow band system. The bandwidth required to transmit picture points determined by R = 0 . 0 1 6 by conventional means can be calculated as follows. For R = 0 . 0 1 6 , Number of b i t s per line = Q 016 = 6 2 . 5 Numbers of lines/sec = 1 5 , 7 5 0 Therefore, number of bi t s per second = 1 5 , 7 5 0 x 6 2 . 5 = 9 8 5 x 1 0 3 With a conventional picture quality a bandwidth of 9 8 5 kc/s should do. Therefore using a t o t a l bandwidth of 1 . 0 8 5 Mc/s we should be able to send a comparatively good quality picture. This compares favorably with 5 Mc/s bandwidth of a standard t e l e v i s i o n system. 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 for a single f l a t top pulse. I t should be realized that this i s not an example of a t y p i c a l t e l e v i s i o n picture signal, not even of the class of the objects we have discussed in the preamble. However, i f i n a li n e drawing the consecutive picture elements along the l i n e of scanning are placed s u f f i c i e n t l y far apart then we could consider them as a series of separate pulses each of which could be treated separately 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 nearly true because the spectrum of a pulse function decays down to a negligible value very quickly, i n about four of five cycles of r e p e t i t i o n . A possible use for such a system could be the instant-aneous transmission of two tone pictures l i k e documents, machine drawings, etc. over telephone trunk l i n e s . Such a need i s i n existence these days and thi s system could be f r u i t f u l l y used. A review of other possible f i e l d s where the band—reduced trans-mission of two tone picture i s necessary i s given i n paper by 9 Cherry, Kubba, et a l . It should be noted that 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 transmission using Holograms for bandwidth compression i s only at the suggestion stage: experimental v e r i f i c a t i o n of the scheme has not been made. 4. EXPERIMENTAL TECHNIQUES AND RESULTS It i s desirable to have a review of the experimental techniques because there are a certain number of steps and precautions to be taken to make the system of two step imaging even sat i s f a c t o r y . Ve s h a l l give a detailed account of these i n this chapter. In Sec. 4.1 an account of the main elements involved i n producing the Gabor Hologram i s given. It i s shown that a pinhole stop i s necessary to remove the excess noise effect produced by the Airy rings formed by the spherical lens. In Sec. 4.2 the method of using the elements of the system i s explained. Detailed measurements of the setup are included. Results using the modified Gabor method are presented. Some examples of spatial f i l t e r i n g , having bearing on t e l e v i s i o n band compression are included. 4.1 The Elements of the System The Source: The f i r s t necessity i s for a source of monochromatic and coherent l i g h t . The natural choice would be a continuous wave laser because these are the best possible sources of s u f f i c i e n t l y bright monochromatic and coherent l i g h t at "the present time. It i s preferable to use a laser with output i n the blue-green end of the v i s i b l e spectrum as most photographic materials have their maximum s e n s i t i v i t y i n that region. At the time this work was being carried out, this kind of laser was not very common. The laser used i n our experiments i s a Spectra—Physics Model 130 helium—neon gas laser. This emits an 33 almost p a r a l l e l beam of red l i g h t at 63281 with a diameter of 2.5 mm at the exit aperture. The. output power i s 0 .2 milliwatts rated. The beam was adjusted to operate i n the T E M q o mode giving a uniphase wavefront. (see Pig* 4.1a). The Diverging System; As has been noted i n the l a s t paragraph, the output beam of the laser i s only 2 .5 mm i n diameter. To be able to illuminate a s u f f i c i e n t area of- the object transparency ( t y p i c a l l y about a centimeter square) i t i s necessary tb widen this beam. This can be done by placing a short f o c a l length positive lens i n the path of the beam. In our work we used a f s 2 . 2 / 5 5 mm Asahi Autotakumar camera lens. The use of a lens i n this way results in the formation of the pattern c a l l e d the Airy rings* I t consists of a central patch of bright l i g h t surrounded by a series of rings, the intensity as a function of radius r being given b y ^ , f 2J n ( S f r ) * | 2 K r ) = I„ X J f (4.1) ( i f ' ) where, d = diameter of the beam, f = focal length of the lens, X = wave length, and i s the f i r s t order Bessel function. In Eq. (4.1) I q represents the int e n s i t y at the centre of the spot and i s related to the power i n the beam i n the following way, 34 a ) . Spectra-Physics Model 130 gas laser operating i n T E M q o mo de b ) . A i r y rings produced by a 55 mm Asahi Autotakumar lens when placed i n the beam as shown i n a ) . c ) » Removal of the Airy rings by 220 micron diameter pinhole as shown i n F i g . 4 .3 . F i g . 4 .1 . I l l u s t r a t i n g the Ef fec t of Introducing a Converging Lens i n the Path of a Coherent Light Beam and the F i l t e r i n g Capacity of a Pinhole in Removing the Noise Produced by the A i r y Rings. where, A = area of the beam cross-section 35 P = power i n the l i g h t beam. The photograph i n F i g . 4.1b shows the effect of introducing the 55 mm focal length lens i n the path of the laser beam from the Model 130 laser. F i g . 4.2 Focussing of Coherent Light The Hologram formed i n such a beam i s very noisy. The reconstruction i s also very poor as can be seen from the ty p i c a l examples (Fig. 4.11)attached with t h i s thesis. It i s necessary to remove these Airy rings to get a clean Hologram. This can be done by introducing a pinhole around the focal point of the lens. E f f e c t of the pinhole on removing the Airy rings (Fig. 4.1c) can be explained by considering the pinhole as a low pass f i l t e r in the frequency plane. Further discussion about the pinhole i s included i n the appendix B. The Pinhole: For a 55 mm focal length lens, l i g h t of wavelength 6328 A, and a beam diameter of 2.5 mm, P — 2 j ^ = 27.8 microns 36 220 -microns P i g . 4 .3 . Pinhole used i n removing the Airy disc produced by the spherical lens . View shown i s a microphotograph with area magnification 33,100 X. The small grey specs inside the hole are par t i c l e s of s i l v e r which the f ixer f a i l e d to remove from the f i lm surface. Apparently these microscopic s i l v e r par t i c l e s do not produce any v i s i b l e degrading e f fect . Fi lm used to record and reduce the pinhole to proper size (220 microns) was I l f o r d Formolith developed i n Kodal i th developer and followed by f i x i n g i n Industraf ix . 37 As has been shown i n the appendix i t i s not necessary to use a pinhole of exactly this size to get satisfactory results. We used a pinhole of 220 microns i n diameter. The most convenient way of making a pinhole of this size appears to be photographic reduction. We started with a black c i r c u l a r spot approximately j inch i n diameter. This was reduced in size in two steps to a black spot 220 microns i n d i a . A contact print from this gave us the desired pinhole. The f i l m used i n the process was I l f o r d Pormolith developed in Kodak Kodalith developer. This i s an extremely high contrast f i l m with very good resolution and i s normally used for recording two tone objects. The pin-hole obtained i n this way shows under the microscipe (400 X) to have a regular c i r c u l a r shape. The hole i s not perfectly transparent though. In the photomicrograph several black spots representing 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 seriously hamper the quality of pinhole. S a t i s f a c -tory results have been obtained with this pinhole. The basic illuminating scheme i s therefore as shown in Pig. 4.4. C o l l . Lens hole Diverging Lens ect CW LASER Div. Pin-Lens hole Obj ect a) b) F i g . 4*4. Experimental Arrangement for Illuminating the Object, a). Fresnel D i f f r a c t i o n b). Fraunhofer D i f f r a c t i o n 38 The Recording Medium: For recording the Hologram we need a photographic medium. There are two basic requirements of t h i s medium. The resolution l i m i t i n lines/mm should be high so that fine d e t a i l s i n the d i f f r a c t i o n lines are properly-recorded. 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 short. The short exposure time i s necessary to prevent the effect of j a r r i n g the system. Besides these two basic requirements the f i l m must be red sensitive because of the type of laser being used. Several types of l o c a l l y and easily available films were t r i e d out. Best results were obtained from Kodak High Contrast Copy, type M 402 f i l m . This i s a pan f i l m , sensitive well into the red and has a high resolution figure. The tungs-ten rating i s ASA 64. It has been possible to use an exposure as short as l / l 5 seconds with this f i l m . Development time for this f i l m i s eight minutes at 70°F i n Kodak D 76 followed by a 3 minute f i x i n g i n Industrafix. Workers"'""*' i n th i s f i e l d have suggested the use of spectroscopic plates l i k e the Kodak 649 F. This f i l m has an extremely high resolution figure of 2000 lines/mm but the exposure index i s extremely small being only ASA 0.0016 for white l i g h t and even smaller for red l i g h t . Besides this i t i s generally necessary to place a special order to obtain this f i l m , several months delivery time being usual. The type M 402 f i l m comes i n 35 mm size and to hold th i s we have used the Asahi Pentax 35 mm camera with i t s picture taking lens removed. This has a focal plane shutter which was used to time the exposures. The viewing pentaprism i s very helpful i n adjusting the position of the Hologram on the f i l m . 39 Timing mech. (focal plane shutter) F i g . 4.5. Using a 35 mm Camera Back for Recording the Hologram. 40 Since the viewing mirror f l i p s up during the exposure we get a f a i t h f u l recording of the Hologram on the f i l m , no other optical surfaces being present i n the path of the l i g h t beam during the actual exposure. 4.2 The Technique of Producing the Hologram and Spatial F i l t e r i n g We have already described the elements involved i n the system to produce the Hologram and to obtain a reconstruction from i t . A judicious use of these elements i s necessary to make the system work s a t i s f a c t o r i l y . To produce the Hologram i t i s necessary to l i m i t the distances' involved within certain bounds so that the reconstruction process does not require too great a length. A t y p i c a l setup involving the distances used i s shown in F i g 4.6. In thi s setup f ( x , y) £cf : Eq. (2.1l)J i s not negligible and hence the phase modulating disturbance i s present. Unless otherwise s p e c i f i c a l l y mentioned th i s setup w i l l be used for a l l experiments. Model 130 gas laser A B C D F i g . 4.6. Details of the Setup Used for Recording the Hologram A : Diverging lens, f : 2.2/55 mm Asahi Autotakumar, six element. B : Pinhole, 220 microns d i a . C : Object, area about half a centimeter square. 41 D : 35 mm camera back to record Hologram. Film used: Kodak high contrast copy, Type M 402. Hologram area: about a centimeter square. A l l equipment should be r i g i d l y mounted. An optical bench as shown i n F i g . 4.9 was used to hold up the equipment. Typical exposure with the Spectra-Physics Model 130 gas laser turned 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 reconstruction from i t i s shown i n F i g . 4.8. Note the magnification factors involved. The reconstruction i s larger than the o r i g i n a l be-cause of the fact that a divergent beam was used. The exact setup and the distances involved in. the reconstruction process i s not the same as the position at which i t was taken. This was necessary to obtain a reconstruction within the length of the optical bench. It i s possible to get a reconstruction by placing C i n F i g 4.7 at 52.5 cm form the pinhole. The recon-struction i s then further away and bigger i n si z e , i Some more examples of Holograms of step and pulse functions ai*e given i n F i g . 4.10. A l l examples shown are positive Holograms. The spatial frequencies i n these Holograms can be c l e a r l y i d e n t i f i e d . A : Diverging lens, f : 2.2/55 mm Asahi Autotakumar, six element. B : Pinhole, 220 microns dia. C : Hologram D : Reconstruction, recorded on Polaroid Type 55 P/N f i l m F i g . 4.7. Details of the Apparatus for Obtaining Reconstruction. 43 1— m i l , 1 ^ ! ^ . ' ; • ; Mr i n a) Original, 32 X magnification. b) Hologram, 11.6 X magnification. c) Lensless reconstruction by d) Gabor reconstruction using modified method, actual reconstructing lens, 6.4 X si z e . magnification. F i g . 4.8. Typical Object, i t s Hologram and Reconstruction from the Hologram. (Note magnification f a c t o r s ) . F i g . 4 . 9 . Photograph of Apparatus Used i n Performing the Experiments. 45 ) Hologram [pf(x , y) =T oJ b) Hologram |^"pf(x, y) £ O-J of a negative pulse. of a negative pulse. ) Hologram [ p f ( x , y) = oJ d) Hologram [ p f ( x , y) £ o] of a pos i t ive pulse. of a step funct ion . F i g . 4.10. Holograms (Fresnel and Fraunhofer) of Dif ferent Types of S ignals . 46 The effect of the pinhole i n removing the noise from the Hologram as well as the reconstruction i s c l e a r l y demonstrated by comparing the i l l u s t r a t i o n s i n F i g . 4.11 with those of F i g 4.8. F i g . 4.11. Noisy Hologram and Reconstruction due to the Presence of the A i r y Rings. 47 For spatial f i l t e r i n g i t i s possible to use two dif f e r e n t setups. One i s the scheme described i n Sec. 2.5. This has the disadvan-tage of recording the Hologram on the f i l m previously. A simpler scheme i s to use a one step process as described i n Appendix A. This scheme i s i n pr i n c i p l e equivalent to the one described i n Sec. 2.5 but gives a better result since the problems outlined in Sec. 2.4 are avoided. The following setup was used for f i l t e r i n g . The results are indicated i n F i g . 4.13. A B C D E F G A : Diverging lens, f : 2.2/55 mm Asahi Autotakumar B : Pinhole, 220 micron d i a . C : Collimating lens, 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 half a centimeter square E : Hologram forming le'ns, f s 5.6/135 mm Schneider-Kreuznach-Componan F : F i l t e r element placed at the focal point of the lens E G : Imaging lens, f : 2.5/10 inch H . F i l t e r e d image recorded on Polaroid Type 55 P/N F i g . 4.12. Details of the F i l t e r i n g Apparatus. 48 ) Removal of d . c . component from 0.025 cm negative step. Magni f i -cation 25 times. ) Square wave imaged through system shown in F i g . 4.12. No f i l t e r used. Magnif icat ion 13.5 times. b) Removal of high frequencies from a 0.025 cm step. M a g n i f i -cat ion 25 times. d) Ef fec t of low pass f i l t e r on the same square wave. Magni-f i c a t i o n 13.5 times. F i g . 4.13. Ef fect of Spat ia l F i l t e r s on Some Typical S ignals . 49 5. CONCLUSIONS The Fourier transform property of a Gabor Hologram which i s useful for studying bandwidth reduced t e l e v i s i o n trans-mission i s developed i n Ch. 2* The key formulae are Eqs. (2.12) and (2.18). A l o g i c a l explanation, based on the transform property, of the two step imaging process i s developed i n Sec. 2.3. It i s then demonstrated that the reconstructing lens suggested by Gabor i s unnecessary and superior quality reconstruction can be obtained by omitting; t h i s . The frequency f i l t e r i n g property which also follows from the transform property of the modified Hologram i s discussed in Sec. 2.5. The f e a s i b i l i t y of applying the Hologram techniques to t e l e v i s i o n transmission at reduced bandwidth i s studied 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 potential saving in band-width for a special case are given i n Sec. 3.2. The system has not been experimentally v e r i f i e d and p o s s i b i l i t y of future work exists i n thi s d i r e c t i o n . Experimentally, a number of techniques have been discovered or developed for Hologram work, and these are described in Ch. 4. The use of a positive lens to diverge the narrow laser beam (2.5 mm dia) to cover an area of 5mm x 5mm i s necessary. The noisy Airy rings created by thi s lens have been removed by using a pinhole. This technique i s described i n Sec. 4.1 and an explanation of the process i s given i n the Appendix. Typical 50 results of the Gabor method of reconstruction of two tone o r i g i -nals are given i n Sec. 4.2. The r e s u l t s , taken as a whole, indi cate that the Hologram method of reconstruction has been improve over that achieved by Gabor and, i t i s thought, to the level at which experiments of the two channel t e l e v i s i o n system can take place. 51 APPENDIX A SPATIAL FILTERING USING A LENS SYSTEM 12 It has been demonstrated that i f an object transpar-ency i s placed at One focal plane of a spherical lens and illu m i -nated with a p a r a l l e l coherent beam then the Fourier transform, i . e . the Hologram, i s displayed at the other focal plane. This princip l e can be u t i l i z e d for performing spatial 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 record the Hologram on a f i l m thereby eliminating the troubles indicated i n Sec. 2.4. By placing another lens (which i n effect takes the inverse Fourier transform) behind the Hologram plane i t i s possible to produce an image of the o r i g i n a l transparency in one step. The setup i s i l l u s t r a t e d i n F i g . A . l . In such a setup a f i l t e r element as described i n Sec. 2.5 i s introduced Laser Obj ect transp F i l t e r element (Hologram plane) Imaging lens Image Fi g . A . l . Principle of Spatial F i l t e r i n g Using Lenses. 52 i n the Hologram 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 the image plane. A black spot placed at the f i l t e r plane w i l l render an image with the d.c. component removed. Similarly an aperture placed i n the f i l t e t f plane w i l l simulate a low pass f i l t e r and the image obtained would be lacking iri the high frequency component. Other f i l t e r s can also be achieved i n this plane. 53 APPENDIX B EPPECT OF PINHOLE IN REMOVING THE AIRY RINGS When a beam of p a r a l l e l laser l i g h t i s passed through a converging lens a pattern known as the Airy 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 th i s pattern when observed at the focal point of the lens i s 10 where, and A = A A , 2V*> d = diameter of the lens, f = focal length of the leas, X = wavelength of the laser l i g h t . (B.l) F i g , B.l*. Amplitude D i s t r i b u t i o n at the Focus of a Spherical Lens Illuminated with Laser Light. Now i f thi s l i g h t i s used to illuminate a pinhole placed at the focus of the lens, then the d i f f r a c t i o n pattern produced by the pinhole can be expressed as , 5 4 U(P) JACoso" 2X exp jp(r + s) ds J Cos^ . 2X o 2 J, K l ( x ) / / ± / / exp jp(r + s) ds J x(x) JjOaw) (B.2) x * paw ' where, P = 2-n;/X , a = radius of pinhole, and w = radius vector of a point i n the d i f f r a c t i o n pattern. Assume now that the size of the pinhole i s such that x = n, i . e . i n other words i t passes only the central hump of the A curve shown i n F i g . B . l . Therefore, i f r i s greater than x, A = 0 . The expression for U(P) would then be the product of the two curves shown i n Fig* B . 2 a) and b). The result i s the curve i n F i g . B . 2 c ) . J-^paw) A Paw M U(P) = A.M a, b ) F i g . B.2. Removal of the Airy Pattern by the Pinhole. This shows that the annoying rings are removed and we get a smooth v a r i a t i o n i n the illumination. This i s confirmed 55 by the photograph i n F i g . 4.1 c ) . To get the size of the pinhole we set x = %. Therefore, or, r = (Xf)/d = (6328 x 10~ 8 x 5.5)/(2.5 x 10 _ 1) cm = 13.9 microns. A pinhole of 27.8 micron diameter placed at the focal plane of the 55 mm lens would then remove the Airy ring's. It i s extremely d i f f i c u l t to produce a well defined c i r c u l a r pinhole of this size* A bigger pinhole can be used i f i t i s placed s l i g h t l y off the focal plane of the spherical lens, the requirement being that the radius should be such that the pinhole blocks anything more than the f i r s t zero of the J^(x)/x function. In effect the pinhole can be considered as a low pass f i l t e r placed at the,spatial frequency plane (cf : Sec. 2.5). The Airy rings can be approximately considered as the spatial frequency d i s t r i b u t i o n of a pulse the low frequencies of which are being only passed by the pinhole. 56 REFERENCES 1* Gabor, D., "Microscopy by Reconstructed Wavefronts." Proc* Roy* Soc*. A* 197> 1949, pp. 454-486. 2* Leith, E. N* and Upatnieks, J., "Wavefront Reconstruction with Continuous Tone Objects," J . Opt. Soc. Am., v o l * 53, Dec. 1963, pp* 1377-1381. 3« Born, M. and Wolf. E., Principles of Optics, 2nd ed* New York: Pergamon Press, Inc., 1964, pp. 378-382. 4. Vander Lugt, A. r "Signal Detection by Complex Spatial F i l t e r i n g , " IEBETrans• on Information Theory, v o l . IT - 10, A p r i l 1964. Number 2, pp. 139-145. 5. Cheatham, T* P* and Kohlenberg, A., "Optical F i l t e r s — Their Equivalence to and Difference.from E l e c t r i c a l Networks:," 1954 IRE N a t f l Conv. Record, pt. 4. pp* 6-12. 6* Leith, E. N. and Upatnieks, J . , "Wavefront Heconstruction with Diffused Illumination and Three Dimensional Objects," J . Opt* Soc* Am** v o l . 54. Nov. 1964,' pp..' 1295-1301* 7. Leith, E. N. and Upatnieks, J., "Photography by Laser*" S c i e n t i f i c American* June 1965, v o l . 212, pp. 24-35* 8* Cherry, C. and Gouriet, G. G*, "Some P o s s i b i l i t i e s f o r the Compression of Television Signals 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 Possible Bandwidth Compres-sion of Visual Image Signals," Proc. IEEE, Nov. 1963, pp. 1507-1517. 10* Oliver, B. M*, "Some' P o t e n t i a l i t i e s of Optical Masers." Proc* IRE. Feb* 19.62, pp. 135-141. 11. Holeman, J * , "The 3-D Hologram Process and Image Recognition by Spatial Filtering, 1? G.E* Advanced Technology Laboratories  Technical Information Series Publication No. 64GL159* Date: Sept* 9» 1964. 12. Rhodes J r * . J* E*. "Analysis and Synthesis of Optical Images," Am. J * of Phys.. v o l . 21, May 1953, pp. 337-343. 13* Beddoes, M* P* and Akfrtan S« A., "Some Techniques for Pro-ducing Holograms with Lasers," A paper accepted for presentation i n October 1965 at the Canadian Electronic Conference sponsored by the Canadian branch of the IEEE. 

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