by L. B. LESEM and P. M. HIRSCH
IBM Scientific Center Houston, Texas and
J. A. JORDAN, JR.
Rice University Houston, Texas
INTRODUCTION
An optical hologram is a two-dImensional photo-graphic 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-dimen-sional 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 dis-play 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 as-sociated with such elements. Thus "perfect" systems can be examined, and experimental parameters can be easily varied. Also, a computer can perform opera-tions, 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, con-5titutes 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)
41
and transform it into a optical hologram, thereby allow-ing 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 magnifica-tion 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.
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 Wolfl) one obtains the so called Kirchoff diffraction formula
U(P)
B f f - - -
[cos(n,r)A r+s cos(n,s)] dA
(1)
where B is constant which depends on the initial amplitude of the wave. In the limit of small angles and
42 Fall Joint Computer Conference, ·1967 with transmittance described by the complex function T(x,y). The integral in equation (3) is a convolution
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)* 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 or-der 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 forward numerical calculation of the convolution inte-grals of Eqn. 3. The first computer-generated holo-grams 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 informa-tion in Eqn. 4. In 1966 J. Waters8 extended this binary mask technique to three dimensions.
Huang and Prasada6 suggest~d the use of trans-form techniques for holograms which, to first approxi-mation, 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 ele-ments. Conventional techniques for performing this map would require 229 machine operations (by a ma-chine operation, we mean a complex multiply and add). For an array of N elements, the fast finite Four-ier transform technique requires only N log2 N opera-tions; i.e., 221 operations in our example.
In the two-dimensional case, a further simplifica-tion can be made. The propagasimplifica-tion funcsimplifica-tion F(x, y) 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.
IMAGE
Figure I-Flow diagram for hologram construction
2 2
ikx iky
Fl(x) == e2Z and F2(y) ==~; . Thus the Fourier trans-form of F satisfies F(t, T}) == F1(t).F2(1])' If the propagation function is an- NF x MF array, the sav-ings in machine operatio'ns accomplished using the separability property is (NF.ML log2 NF.MF) the finite convolution of the two is written as
NF-l MF-l
W(n,m) ==
L L
B(r,s)F(n-r, m-s) (6) r==O s==On==O,I, ... , NF-l m==O,I, ... ,MF-l
Holographic Display of· Digital Images 43 where
A hologram require~ resolution and frequency con-tent four times greater than that needed to define the alterna-tively obtained' good results by interpolating the
con-vo~ution. We insert, by parabolic interpolation, three pomts between each pair computed in our convolution.
Thus the interpolated convolution is made four times as large in negligible computer time. The interpolation requires! approximately 3N operations compared to the N log2 N operations needed for the Fourier ap-proach.
44 Fall Joint Computer Conference, 1967
Figure 2-An image of the Greek letter lambda that was read into the computer
Holographic Display of Digital Images 45
Figure 3-A digital hologram plotted on an IBM 2995, Model 2
46 Fall Joint Computer Conference, 1967 Experimental verification of computer-generated
holography
The test of a computer-generated hologram is whether the H(a,b) defined in Eqn. 4 can be translated to a photographic plate, which in turn can be illuminated by coherent light to form an image. An image is first digitally read in as the A in Figure 2. We then com-pute H(a,b) and plot it using the IBM 2995 Model 2 photographic plotter, which has 32 grey levels and plots on a 40" X 60" piece of'film with a spot size of 10 mils., Figure 3. We then photographically reduce this film to a 35 mm slide which is then illuminated with a collimated laser beam. Figures 4-6 exhibit the laser reconstructions which we have achieved from some of our computer-generated holograms. The image arrays contained 384 X 256 elements and the final hologram array contained 1530 X 256 elements. The physical dimensions of the reconstructed images are I" X l.5'!
Note the virtual; central and real images
CONCLUSIONS
We have shown that the computer generation of holo-grams is feasible technique for the display of digitally.
defined data. Major savings in computer time can be achieved using the fast finite· Fmirier transf().rm· algor-ithm and by taking advantage of the separability of the propagation function.· Further improvements can be achieved using interpolation.
Figure 5-The real image of an optical reconstruction of the Jetter "B"
Figure 6-The real image of an optical reconstruction. of the lambda in Figure 2
ACKNOWLEDGMENT
We wish to acknowledge the efforts of Mr. D. E.
Richards who did the bulk of the programming for the computations reported herein. His work in pro-ducing efficient programs has greatly expedited our work.
REFERENCES
M BORN E WOLF Principles of optics
Pergamon Press New York 1959 2 B R BROWN A W LOHMANN
Complex spatial filtering with binary masks Appl Optics vol 5 p 967 1966 3 J W COOLEY J W TUKEY
An algorithm for the machine calculation of complex Fourier series
Mathematics of Computation vol 19 p 297 1965 4 H D HELMS
Fast convolutions using the fast Fourier transform
Holographic Display of Digital Images
4.
7Unpublished Technical Memorandum Bell Telephone Laboratories Inc 1966
5 D GABOR
Microscopy by reconstructed wave/ronts Proc Roy Soc vol A 197 p 454 1949 6 T S HUANG B PRASADA
Considerations on the generation and processing 0/
holograms by digital computers
MIT Res Lab of Elect Quar Prog Rep no 81 p 199 1966
7 L B LESEM P M HIRSCH J A JORDAN JR Computer generation and reconstruction 0/ holograms Proceedings Polytechnic Institute of Brooklyn Sympos-ium on Modern Optics 1967
8 J P WATERS
Holographic image synthesis utilizing theoretical methods Applied Physics Letters vol 9 no. 11 pp 405-407 1966