Holographic display of digital images by L. B. LESEM and P. M. HIRSCH IBM Scientific Center Houston, Texas and

J. A. JORDAN, JR. Rice University Houston, Texas

and transform it into a optical hologram, thereby allowing us to construct the three-dimensional image of the scatterer of the sound waves. The same procedures would translate a radar hologram into an optical one. These processes may also make possible magnification of microscopic images in the computer; the optical hologram of a microscopic object could be translated so as to yield a greatly magnified image. Such an image may well be free of the aberrations which so plague three-dimensional microscopy. The fulfillment of these ideas depends upon the solution of several problems which have inhibited the development of computer-simulated "holograms. Efficient computational techniques must be utilized in order that large-scale holograms may be generated in reasonable amounts of computer time. In this paper, we discuss first the mathematical representation of holography, then the computational techniques which we have used, and finally we exhibit some of the results of our efforts.

INTRODUCTION An optical hologram is a two-dImensional photographic plate which preserves· information about the wavefront of coherent light which is diffracted from an object 'and is incident upon the plate. A properly illuminated hologram yields a three-dimensional wave,front identical to that from the original object, and ~hus the observed image is an ~xact reconstruction of the object. The observed image has all of the usual optical propertie~ associated with real three-dimensional objects; e.g., parallax and perspective. The computer-simulation of the holographic process potentially provides a powerful tool to enhance the effectiveness of optical· holography. The construction of holograms in a computer provides a potential display device. For example, mathematical surfaces can be displayed, thus permitting design, to be visualized in three dimensions. In addition, computer generation of holograms can provide a new optical element (Le., lenses, apertures, filters, mirrors, photographic plates, etc.) without introducing the aberrations usually associated with such elements. Thus "perfect" systems can be examined, and experimental parameters can be easily varied. Also, a computer can perform operations, such as division, which are difficult or impossible, using real optical elements. Furthermore, computer holography can be used to filter the degrading effects of emulsion thickness and non-uniform illumination from optical holograms. The computer generation of holograms is part of a larger problem. Interpretation of holograms; i.e., the construction of digital images from holograms, con5titutes the "inverse" problem. Together, these two processes hold considerable promise. One can construct computer techniques which would take an acoustic hologram (the wavefront from a scattered sound wave)

Mathematical representation of the holographic process

Consider a monochromatic wave from a point source Po, which is incident upon an aperature A in a prime opaque screen. Suppose the screen is a distance r from Po and let the point P lie a distance s from the screen. Using Kirchoff's boundary conditions (Born and Wolf l ) one obtains the so called Kirchoff diffraction formula U(P)

Bff--A

[cos(n,r)

(1)

r+s cos(n,s)] dA

where B is constant which depends on the initial amplitude of the wave. In the limit of small angles and

41 From the collection of the Computer History Museum (www.computerhistory.org)

42

Fall Joint Computer Conference, ·1967

plane wave illumination this integral further can be simplified: U(a,b) -

B

f f

(2)

F(x-a, y-b) dxdy

A

where * F(x-a, y-b)

= ex p ~r 'k [(x-a)2 + (y-brl] 1~ _1

L 2z

J

The integral given in equation (2) can be extended by superposition to become T *.F = B

f f

T(x,y)F(x-a, y-b) dxdy

(3)

A

where the region A is a collection of apertures each with transmittance described by the complex function T(x,y). The integral in equation (3) is a convolution (T*F) of the function T(x,y), the image, with the function F(x-a, y-b) , the propagation function. If one adds a plane reference wave, ArelkOa, incident at a small angle () to the hologram plane to the diffracted wave front, the intensity of the total wave front is given by: H(a,b)

I T * F (a,b) + Arelkoa 12 = :T * F (a,b) 12 + A/ + (T * F)Are-lkOa + (T *F)* Are-Ikoa (4)

In the reconstruction process, the hologram whose plate darkening corresponds to the function H(a,b) is illuminated by a plane, coherent, monochromatic wave. Again, there is a convolution, but now (H(a,b)* F (a-x, b-y». The result of this convolution is three wave fronts: A wave with amplitude I T *F(a,b) 12 A/ is propagated in the direction of the illuminating wave. A wave with amplitude (T.F) * F is propagated at an angle () to the illuminating wave. This wave yields a real image, or a reconstruction, of T at a distance z. A wave front with amplitude (T.F).F is propagated at an angle -(} to the illuminating wave. This wave is identical with that which would be observed from an object located a distanc~ -z from the hologram, and thus yields a virtual image. Thus the first two terms in Eqn. 4 represent a central order image, the third term the real image' and the fourth term the virtual image. The term e-1koa acts as a shift operator and spatially separates the real image from the other two images; similarly the

+

"'We have recently modified this propagation function in order to spread the information over the whole of the hologram. This spreading is analogous to that produced by the diffuser plate used in optical holography. The numerical "diffuser" is achieved by multiplying F(x,y) by an appropriate real-valued function, e.g. l-e -C(X2 +y2) for some constant c. Our technique does not increase bandwidth needed to describe the object whereas the optical diffuser does.

term e-1ko separates the virtual image from the other two.· Comp..utational techniques The computer, generation of holograms has been inhibited by the massive task involved in the straight forward numerical calculation of the convolution integrals of Eqn. 3. The first computer-generated holograms were reported by B. Brown and A. Lohmann2 in 1966. These holograms were of the "binary mask" type which uses only two grey levels to encode the information in Eqn. 4. In 1966 J. Waters8 extended this binary mask technique to three dimensions. Huang and Prasada6 suggest~d the use of transform techniques for holograms which, to first approximation, may be considered as Fourier transforms. In 1967, we7 made holograms with 32 grey levels using an array of 64 X 128 for the object. At the time of this writing we have 105 resolution points in our image. Since Eqn. 3 is a convolution integral, it may be readily evaluated using Fourier transform techniques. Indeed, the Fourier transform of H (a,b) is just the product of the Fourier transforms of T(x,y) and F(x-a, y-b): H(',1])

=

T(" 1])F(" 1])

(5)

The convolution can be computed by taking the in~ verse Fourier transform of Ha,1]). In evaluating T • Ii:digitally' we use finite Fourier transforms3 rather than infinite integrals. A mathematical flow diagram of the program for hologram construction is given in Figure 1. A major part of the program is that in which T * F is evaluated using a fast finite Fourier transform subroutine. In .our work T(x, y) is defined in basic "building blocks," arrays of 64 X 128 elements. These arrays are mapped onto the hologram plane using a propagation function F(x-a, y-b) which is defined over 128 X 512 elements. Conventional techniques for performing this map would require 2 29 machine operations (by a machine operation, we mean a complex multiply and add). For an array of N elements, the fast finite Fourier transform technique requires only N log2 N operations; i.e., 221 operations in our example. In the two-dimensional case, a further simplification can be made. The propagation function F(x, y) is separable; i.e., F(x, y) F 1 (x).F2 (y), where

=

*The first holograms, constructed by D. Gabor5 in 1948 lacked the reference beam. He showed that the terms (T"'F) and (T* F) '" were present on his photographic plates, but he could not spatially separate them. A. Lohmann first proposed the two-beam technique described here. Independently, E. Leith and J. Upatnieks discovered the two-beam technique and made the first usable holograms with a monochromatic laser source.

From the collection of the Computer History Museum (www.computerhistory.org)

Holographic Display of· Digital Images where B(r,s) == T(r,s)

PROPAGATION FUNCTION

IMAGE T(x,y)

F(o-x,b-v)

==0

F()I.BER TR~~ OF T(XtY) , T{l' .71)

1 FaRER TRANSFORM OF F(a-x,b-yh

IMAGE PROCESSING T{r .71)

Ftr ,n)

MULTIPLY TRANSFORMS 111.'7)·F('f.1J)

The finite Fourier coefficients B(j, k) and F(j, k) are given QY NF-l MF-l

. L2: B(r,s)exp(-2'7T1(--+--» .rj sk B(],k)= NF MF (7)

and NF-l MF-l

. ~~ rj sk F(],k)== L.,; L.,; F(r,s)exp(-2m( NF + MF-.» r==O 8==0

INVERSE TRANSFaN

,

T. F(a,b)

The inverse finite transform B(j,k).F(j,k) is defined as

INTERPOLATIO'J OF T-F(a,b)

A(n,m)==

sa UARE MAGNTUDE 2

IT"F + e ikdl IS HOLOGRAM

l SCALING AND PLOTTING OF HOLOGRAM

Figure I-Flow diagram for hologram construction 2

iky

Fl(x) == e and F 2 (y) ==~; . Thus the Fourier transform of F satisfies F(t, T}) == F 1 (t).F2 (1])' If the propagation function is an- NF x MF array, the savings in machine operatio'ns accomplished using the separability property is (NF.ML log2 NF.MF) (NF lo~ NF + MF log2 MF + NF.MF /2), a saving of 24 operations in our example. These savings are not fully achieved- however. The convolution which is obtained is not good over the NF x MF points of the propagation function, as can be demonstrated by the Helm-Sande theorem on finite convolutions.· If the object T(x,y) is defined over NF x MF points, where NF>NT and MF>MT, tben the finite convolution of the two is written as NF-l MF-l 2Z

W(n,m) ==

L L

r==O s==O n==O,I, ... , NF-l m==O,I, ... ,MF-l

the

product

B(r,s)F(n-r, m-s)

2: 2: B(j,k)F(j,k)exp(2m(-jn +

km

]

(8)

'=0 k==O

,J,

2

of

NF-l MF-l

J, ADD REFERENCE BEAM T4' F+e ikO'

ikx

r==O,l, ... NT-I s==O,I, ... MT-l r=N~ NT+l, . . . ,N~l s=MT, MT+l, ... , MF-l

r=O 8==0

I'

~

I

43

(6)

NF

MF

The Helm-Sande theore mstates that W (n,m)=A(n,m) only for the region NT