Coded Multiple Exposure Holograms

J. T. LaMacchia and D. L. White

Experiments have been performed in which 1000 holograms have been superimposed one upon the otherin the same area of photographic emulsion. Each hologram was formed with a uniquely coded referencebeam allowing reconstruction of only one of the superimposed holograms while the unaddressed hologramscontribute incoherent noise. Signal-to-noise ratios are calculated and measured as a function of thenumber of superimposed exposures. For the 1000-exposure hologram, the observed signal-to-noise ratiofor any one of the individual holograms was 10 dB.

1. Introduction

In the past, multiple exposure holograms have beenprepared in which different elements of the desired re-construction were exposed separately with the samereference beam.' Multiple exposure holograms werealso made in which both the reference beam and the ob-ject were changed.2 In the latter method, the differ-ent reference beams acted as codes with which indi-vidual reconstructions could be selected. By suchmethods a number of different images can be stored inthe same area of the photographic film. A convenientmethod for changing and coding the reference beam isthe use of a ground glass plate' whose position is changedbetween exposures.

Such methods are of specific interest if digital infor-mation is stored in the form of point sources. Formaximum usefulness of such an approach, the number ofsuperimposed holograms should be large. Any one re-constructed point is always surrounded by a halo ofnoise produced by cross correlation between any partic-ular reference beam code and the codes appropriate toall the other holograms recorded on the illuminated areaof the plate, as well as the noise produced by the auto-correlation of the diffuse reference. Accordingly thenumber of multiple exposures is limited by the achiev-able signal-to-noise ratio. In the past, fewer thantwenty superimposed holograms of continuous sub-jects have been obtained. In order to assess thetotal number that might be possible, a detailed under-standing of the degradation of the signal-to-noise ratiowith the number of superimposed exposures is required.The present paper presents such signal-to-noise calcula-tions which are followed by experimental data indicat-

The authors are with Bell Telephone Laboratories, Incor-porated, Murray Hill, New Jersey 07971.

Received 4 August 1967.

ing that for point sources over 1000 multiple exposuresare possible with a signal-to-noise ratio of 10 dB.

I. Signal/Noise Analysis

In this section we derive an expression for the signal-to-noise ratio when any one of the holograms is correctlyaddressed. Let the complex, scalar electric field of theith reference beam be given byfr(z) (x, y) in the plane ofthe photographic plate. The reference wave is codedby illuminating a piece of ground glass. Represent thesubject waveform at the plate for the ith hologram asfo(')(x, y). The amplitude transmittance of an exposedphotographic plate, in its linear region, is given by

T(x,y) = Const. I}-I'/, (1)

where I is the light intensity incident on the unexposedemulsion, r is the exposure time, and y is the slope of theH-D curve. In a sequence of N multiple exposures theindividual exposure time r() is made T/N and the inten-sities of the individual exposures are added by the emul-sion. Hence, the total intensity recorded by the emul-sion is given by the sum

NI = E I1(i),

i= 1N

I = E fr(i(X,y) + fo(i)(x,y):2.i= 1

The amplitude transmittance is thenN

T(x,y) = Const. { E (i)f0 (i)(x,y)i = 1

+ Pr) )(Xy)l 3>

(2)

(3)

which can be expanded4 in a Taylor series about2'fr(i) 2 + 3 fo(i) 12. Keeping only the first two termsof the expansion we have the result

January 1968 / Vol. 7, No. 1 / APPLIED OPTICS 91

SIGNALOUT

Fig. 1. Schematic diagram ofexperimental arrangement.

OBJECTPLANE

MIRROR

Const. T-rY'T~ x~y = N N + 1

4 E lfo(i)(X'Y)I' + Z fr"i)(XY)tZ' = 1 i =1 I

F N NX fr(i)(Xy)12 + E 1f,(i)(Xry)12

N

Ad o()(ZJ'l)fr *( )(XY)2i= 1

N 1- - E f0*(i)(xy)f'()(xy) (4)

2 i = 1

Since the terms preceding the square brackets in Eq. (4)appear identically in both signal and noise termsthroughout the subsequent development, we omit themhereafter.

We now consider only the term in Eq. (4) which leadsto a virtual image, namely,

NTv(x,y) = E f0(i)(x'y)f*(i)(X'y). (5)

If the composite hologram is illuminated with one of thecoded reference waves, say f(J) (x, y), the virtual imagewavefront at the film is

fv(xy) = Ifr(i)(X'Y)l1fP)(X8)

N+ f 1)(xy) fo(i)(xy)fr*(i)(xy). (6)

The first term in Eq. (6) is referred to as the autocorrela-tion (AC) term. To facilitate the analysis we representthe diffuser as a collection of plane waves of equalamplitude and of random phase. Then the referenceelectric field in the hologram plane may be written sim-ply as

fri)(XY) = E Aexpi[kmz(i)x + kU)y + P.W], (7)= 1

where kmx and k, are the wave vectors in the x and ydirections, respectively, and the 4cm are the random

phases of the plane waves. The autocorrelation termbecomes

31 M

(fv)AC = MA2fo(')(x,y) + fo(i)(x,y) E E A2ail # n

X exp{i[kx,(i) - k(i)lx + i[km,/ - kzV/i)]Y

+ ij[4"(1) - n(D] }. (8)

The autocorrelation term reconstructs the signal fromthe ll coherent terms in Eq. (8) but also introduces anapproximately equal amount of noise power from theM(M - 1) incoherent terms. The cross correlationterm (CC) contributes only noise and may be written

N M M

(fv)cc = E fo"'(x,y) E E A2i Fj mn n

X exp{i[k1xi) - kn.(i)]x + [km.' 1 - k.,(W)] }. (9)

Since each of the (N - 1) cross correlation sources con-tains MI incoherent terms, the total noise contributedby the incorrectly addressed holograms is (N - 1) timesthe autocorrelation noise. Let us restrict our considera-tion only to point objects and assume further that theyare of equal intensity,

f,(i)(xy)12 = fo 2 for all 1 < J < N (10)

independent of x and y in the hologram plane. Thenthe incoherent addition of the AC and CC noise powersfrom a hologram aperture of area rh2 gives

PN = PAC + Pcc = 2oc(irh2)X {A4|f.1[M(M - 1) + M(N - 1)] }

=,~ (,och 2)A41f,|2V1IV. (113

We define the signal-to-noise ratio as the ratio of thepeak signal intensity in the reconstruction to the noiseintensity at the same point. The peak signal intensity5from the hologram is

Is = -eoC[(7rh')'/X'D']A4Ifo12M2 , (12)

where D is the distance from the hologram to the virtualimage plane.

The noise is distributed over an area of the imageplane determined by the extent of the correlation terms,Eqs. (8) and (9). If the diffuse screen subtends an

92 APPLIED OPTICS / Vol. 7, No. 1 / January 1968

BEAM

ARIABLET EXPANDING DIFFUSERVAIBE TELESCOPEATTENUATOR I______

649- FPLATE

RECONSTRUCTEDPOINT

6328He-NeLASER

MIRROR

_ IMAGEPLANE

>1 I A2

)LOGRAMPLANE

ANGLE RELATIVE TO SIGNAL DIRECTION

Fig. 2. Measured noise distribution in the virtual image planefrom a single exposure diffuse reference hologram.

Fig. 3. Photograph (4 X enlarged) of one virtual image re-

construction of signal and noise from ten-exposure hologram.

angle 0 at the hologram (see Fig. 1), then the noise willbe spread over a cone of angle 20. An exact calculationby Enloe6 shows that the noise is not uniform in theimage plane but is peaked at the reconstructed objectpoint. This is confirmed by our measurements for asingle exposure as shown in Fig. 2. From this figure, thepeak noise intensity can be expressed in terms of thetotal radiated noise power.

(7r/4)(20D)2 (13)

where KN is the peak-to-average ratio which we mea-sure. Therefore, we have for the signal-to-noise ratio

aS/i 7 = Ia/IN

- w'O'h'/KNNX'. (14)

According to Eq. (14), the signal-to-noise ratio willdecrease as the number of exposures increases. Themagnitude of S/N for a given number of exposures de-pends upon the relative angles subtended by the signaland the- noise. The larger 0, the more the noise is di-luted in the image plane. Similarly, the larger h, themore highly concentrated is the signal point image.

Ill. Experimental Procedure and Results

The experimental arrangement is shown in Fig. 1.The diffuse source 1 cm in diameter was placed 13 cmfrom the 649-F plate. The point source object origi-nated 20 cm from the plate. The signal-to-noise ratioswere measured by focusing the reconstructed image on a

pinhole 80 cm from the hologram and reading the inci-dent flux on a photomultiplier tube (PMT). An aper-ture of radius h between the hologram and lens con-trolled the size of the hologram contributing to the re-construction. A typical reconstruction from a ten-ex-posure hologram is shown in Fig. 3. The bright objectpoint appears surrounded by the auto and cross correla-tion noise, as described above.

The noise was measured by slightly detuning theground glass from its reconstructing position. Then thesignal was tuned in by carefully repositioning the dif-fuser. The required replacement precision is a functionof the resolution of the diffuser aperture. That is, if thefar field pattern of the diffuser is coarse, the replacementtolerance is relaxed. The allowable displacement of thediffuser from its original position is of order X/0 andamounted to about 15 u in our experiment.

The measurements were made with 5-mm and 9-mmapertures at the hologram and apertures ranging from100 A to 500 ,4 at the PMT. K was measured from aplate having only one exposure and was found to be five.(See Fig. 2.)

Figure 4 shows the measured S/N ratios that havebeen corrected for the finite aperture at the detector.This correction ranged from 0.8 dB to 12.7 dB, depend-ing upon the aperture required in any particular mea-surement to obtain sufficient output from the PMT.For the 1024-exposure hologram, the correction factorwas 12.7 dB. The theoretical prediction is calculatedfrom Eq. (14).

IV. Conclusion

The results compare favorably with the predictedS/N. The fluctuation in the data is primarily due tothe statistical variation in the low background noiselevel. The results indicate that as many as 105 superim-posed holograms may be obtained when the subject is anarray of separated point sources. While the use ofpoint subjects may be unacceptable in applications re-quiring storage of complex images, point subjects areideally suited to digital information storage. The re-sults also indicate that the practical limitation in storingmany data arrays may not be one of optical signal-to-noise ratio, but rather one of available intensity. Sincethe signal power diffracted by the hologram is dividedequally among all of the stored data points, the power

NUMBE OF SUEIPSDXOUE

3 0-

I 10 100 1000

NUMBER O SUPERIMPOSED EXPOSURES

Fig. 4. Plot of measured S/IN corrected for system geometry as afunction of the number of multiple exposures.

January 1968 / Vol. 7, No. 1 / APPLIED OPTICS 93

available to any one point is greatly diminished.Therefore, electrical noise in the detector rather thanoptical noise in the reconstruction is likely to determinethe over-all S/N.

The authors thank R. J. Collier, H. W. Kogelnik, andC. B. Burckhardt for their helpful discussions. Theyalso thank J. R. Merchant for his fine technical assis-tance and E. T. Doherty for describing his unpublishedresults on multiple exposure holograms.

References1. G. W. Stroke, F. H. Westervelt, and R. G. Zech, Proc. IEEE

55, 109 (1967).2. M. Marchant and D. Knight, Opt. Acta 14, 199 (1967). D.

Gabor, Nature 208, 422 (1965).3. R. J. Collier and K. S. Pennington, Appl. Opt. 6, 1091 (1967).4. G. W. Stroke, An Introduction to Coherent Optics and Holog-

raphly (Academic Press Inc., New York, 1966), p. 107.5. Al. Born and E. Wolf, Principles of Optics (Pergamon Press,

Inc., New York, 1965), p. 396.6. L. H. Enloe, Bell System Tech. J. 46, 1479 (1967).

Soviet Journal of Optical Technology

Publishes, in English translation, articles and brief communications describing original work,as well as reviews of progress in optical science and technology. It reports on new instru-ments for astronomy, spectroscopy and spectrophotometry, electrooptics and optoelectron-ics, geodesy, photometry and colorimetry, microscopy, refractometry and nephelometry,sensitometry, optical measurement and control, as well as photography (still and moviecameras, projectors). In addition, the journal presents articles in the areas of optical design,technology of optical manufacturing, as well as properties of optical glasses and crystals.

Editor in Chief: A. L. Nikitin

Vice Editor in Chief: A. A. Lebedev

Scientific Editor: E. P. Semenov

Managing Editor: R. B. Mullov

Howard A. Robinson Scientific Translation Editor

Annual Subscription rates:

(One volume a year)

OSA members:

U.S.A. and possessions, Canada, Mexico

Elsewhere $13.50

Nonmembers:

U.S.A. and possessions, Canada, Mexico

Elsewhere $26.00

Order from Subscription Handling, American Institute of Physics, 335 East 45th Street, NewYork,N.Y. 10017.

Published by the American Institute of Physics for the Optical Society of America.

94 APPLIED OPTICS / Vol. 7, No. 1 / January 1968

$12.50

$25.00