Holographic Data Reduction

K. A. Haines and D. B. Brumm

Very large bandwidths are required for the transmission of holographic data for systems such as TV. This

paper presents a technique in which the large bandwidths normally required are traded off for either in-

creased noise or decreased resolution in the image. The light radiated from the object is diffracted by an

intermediate dispersion medium and collected at the hologram aperture. Correct illumination of this

hologram provides an image beam that passes back through the intermediate medium and comes to focus

in the space originally occupied by the object. By proper selection of the dispersion medium, the holo-

gram aperture can be made extremely small, thus representing a large data reduction. The three di-

mensionality of the image and the original viewing angles are maintained. Included in the paper are ex-

perimental results that show reconstructed images after a data reduction of as much as 3600.

IntroductionEver since the resurgence of interest in coherent

optics, holographers have been in search of ways toapply holography to TV systems. One quickly realizesthat using straightforward holographic techniques forTV requires a transmission bandwidth that is greaterthan conventional TV by a factor of about 104. Thisis only one of the reasons for interest in bandwidthreduction techniques.

One could simply remove all but a narrow band ofspatial frequencies in the hologram by spatial filtering.While this reduces the bandwidth of the hologram, italso unfortunately degrades the three-dimensional prop-erties of the reconstructed image. For example, usingcollimated reference beams, this type of filtering re-stricts the viewing angle, which is defined as the angleover which the reconstructed image can be viewed, asshown in Fig. 1.

An alternative technique is one which involves asampling of the hologram. However, there are usuallydefinite drawbacks to such sampling systems. Forexample, most of these systems require the constructionof additional holograms if reconstruction of the sam-pling structure itself is to be avoided. Also, whilespace-spatial bandwidth product is reduced, the spatialbandwidth is not. Thus, one must still record andreimage on high-resolution relatively slow films.

Neither of these limitations exist with the techniquediscussed in this article.' This technique works in thefollowing way. As illustrated in Fig. 1, light from the

K. A. Haines is with the Holotron Corporation, Wilmington,Delaware; D. B. Brumm is with the Radar and Optics Labora-tory, The University of Michigan, Ann Arbor, Michigan.

Received 11 January 1968.

object is diffracted by a dispersion plane m(y) and fallsupon the hologram plane where it is recorded. Supposethat the hologram is constructed with a divergingreference beam from a point source, and suppose that itis then reconstructed with a beam originating from theopposite side of the hologram, but now converging tothe same point. This generates a conjugate wavefronttraveling from right to left, which is again diffracted bythe m(y) plane. If the dispersion function m(y) isstrictly a phase function, such as ground glass, for in-stance, a pseudoscopic image may be observed to theleft of the m(y) plane.2 The hologram aperture may bedecreased, thus decreasing the space-factor in the dataspace-spatial bandwidth product of the holographicdata. Alternatively, the distance D2 may be increased,thus decreasing the spatial bandwidth factor of thedata space-spatial bandwidth product. By the firsttechnique the size of the hologram may be made quitesmall. By the second, the film resolution requirementis reduced. The one-dimensional space-spatial band-width reduction of the system, that is, the data reduc-tion, is given approximately by the ratio R of the fol-lowing equation:

R - (2Xm/Di)(a/D,). (1)

2Xm is the object size and a is the hologram or collect-ing aperture size. This ratio is then the angle sub-tended by the object at the dispersion plane divided bythe angle subtended by the hologram at the dispersionplane. Since the space-spatial bandwidth product isreduced in two dimensions, the actual information re-duction ratio is R2.

It should be evident that the viewing angle is limitedonly by the size of the dispersion plate and not by thesize of the hologram. In order to experimentallyverify this, an intermediate film plane was placed be-tween the object and dispersion planes, as shown in

June 1968 / Vol. 7, No. 6 / APPLIED OPTICS 1185

'A

IntermediatePlane

Objectf(n)

~~cin iI. I Viewing

LI Z:-- .,. Angle

I I --

Dispersion

Planem(y)

D

Hologram

Planeh(z)

0 TI a

-D2 A

Fig. 1. Dispersion plane system.

which can be rewritten, after successive Fourier trans-formations, as

92(Y) = { [i*(1y)n*(y) exp( jk y2)]* exp( -k2a2y2))

m(y) exp(kY 2)- (3)

Consider an on-axis point in a general object field.For this point g,(y) = exp[(-jk/2D1)y2]. With theexception of a phase function in the image plane, theimage functionf2 (x) can be expressed as

Fig. 1. Using a point object, a collimated referencebeam, and a random phase dispersion function, a seriesof photographs were taken, both at the intermediateplane and the point image plane, as the hologram aper-ture was decreased in size. The results are shown inFig. 2. Figure 2 (c) and (d) represents a hologramaperture reduction of about ten over those of Fig. 2(a) and (b). The photographs of Fig. 2 (a) and (c)were taken in the plane of the point image. Those ofFig. 2 (b) and (d) were in the intermediate plane. Itis evident that the light falls upon the point imagefrom the same range of angles in either case since thediameter of the intensity pattern at the intermediatefilm plane, as shown in Fig. 2 (b) and (d), is fixed.This indicates that the viewing angle in such a systemis indeed maintained as the space-spatial bandwidthproduct is decreased.

f2(x) = F(- )[F(__ ) exp(XD, x~~~i 2D12a2

(4)

where the function s is

-rn~ -jk 1 1s = (y) expI + D)-2 Di D

and F8 is the Fourier transform of s as given by

F,8 (u) = fs(x) exp(-j27rux)dx.

The reconstructed image can then be considered tobe a summation of such terms, each term representativeof a point in the object space.

In the following discussion, the characteristics of theimage resulting from the use of a ground glass dis-persion structure are derived and demonstrated. It isshown that such a structure causes the signal-to-noise

(a) (b)Analytical Description of the System

It is an impossibility to achieve a reasonable degreeof data reduction without sacrificing something, whetherit be signal-to-noise ratio, resolution of the recon-structed image, or viewing angle. In order to examinewhat happens to the image, it is convenient to firstpresent a general analytical description of the system.

The wavefront radiating from the object and fallingupon the dispersion plane in Fig. 1 is g(y), and that re-diffracted by the dispersion plane and forming an imageis 2(Y). For simplicity in the analysis, the hologramaperture is assumed to be Gaussian weighted with avariable standard deviation of a. Using the conven-tional simplification of the Fresnel diffraction formula,the wavefront at the hologram plane may be writtenas

h(z) = ft(y')m(y') exp -jk ('- )2dy' exp( 2)L2D2 Ij 2a' /(2)

Upon reillumination of the hologram, the conjugateimage wavefront is

(2(Y) = m(Y) J*(y')nz*(y') exp [2D ( - )2]dy,

ff1 2D 2 Iexp L-j (y - )2 exp - 2'/,12D, I ti2

(c) (d)

Fig. 2. Viewing angle retained with ground glass. (a) 10-cmhologram aperture; point source reconstruction. (b) 10-cmhologram aperture; image at intermediate plane. (c) l-emhologram aperture; point source reconstruction. (d) 1-cm

hologram aperture; image at intermediate plane.

1186 APPLIED OPTICS / Vol. 7, No. 6 / June 1968

or'

(a) (b)

Fig. 3. Ground glass dispersion function. (a) Large hologram

aperture. (b) Small hologram aperture.

ratio of the image to be reduced as the recorded data isreduced (hologram aperture is narrowed). Then adifferent type of random structure is described thatcauses the resolution of the image to deteriorate ratherthan the background noise to build up, as the data isreduced.

Example 1: Ground Glass Dispersion Function

Assume that the ground glass is a random phasefunction having no transmission amplitude variationsacross its surface. Let us also choose a model in whichthe spatial bandwidth of m(y) is extremely broad andin which the phase relationship between spectral com-ponents is random. Considering a point source objectlocated on axis, the characteristics of the reconstructedimage may be examined with Eq. (4).

The width of the correlation function in Eq. (4) de-pends mainly on the frequency spread of the functionF,(x/XD,) and has little dependence on the size of thehologram aperture. That is, the resolution of the pointimage depends on the extent of the m(y) function.However, the intensity of the correlation peak is re-

lated to a, the size of the hologram aperture. Owing

to the random nature of F, the correlation function for

off-axis points is relatively constant. As a becomes

smaller, this correlation becomes larger. This implies

that the background noise in the image function is uni-

form and that it increases with decreasing aperture.

In the limit, as a goes to zero, the term in the square

braces is a delta function, and f2(x) - F,(x/XDI); that

is, the image function f2(x) is entirely noise.An experimental demonstration of these effects for a

ground glass dispersion function is shown in the image

plane photographs of Fig. 3. Figure 3 (a) is a photo-

graph of the image using a relatively large hologram

aperture. The two-dimensional data reduction ratio

R2 was ten. In Fig. 3 (b) the hologram aperture di-

ameter was decreased so that the data reduction ratio

was 400. From these figures it appears that when re-

ducing the hologram aperture the 'resolution of the

image is altered very little. However, there is a marked

increase in the noise background as predicted.

Example 2: A Resolution ReducingDispersion Function

In many cases it is not of interest to trade off back-ground noise in the image field for a data reduction.It may be more desirable to select a dispersion mediumthat causes the image resolution to decrease as a func-tion of data reduction, that is, as the hologram aperturedecreases, but which contributes very little to back-ground noise buildup.

A dispersion medium that does exactly this is onewhich is again a purely phase function. The dispersionfunction can then be written as m(y) = exp [j0(y)],where 0(y) is a random function of y. However, thereare two essential restrictions on the statistics of thisrandom function. First, it is essential that the firstderivative of the phase 0(y) have a standard deviationthat is large enough so that light from all parts of theobject field is diverted by the entire dispersion planeinto the relatively narrow hologram aperture. Thatis, it is essential that

kT 2mDi D (5)

where a- is the standard deviation of the phase rate 6,and 2 Xm and 2 Yn are the dimensions of the object anddispersion functions, respectively. When this require-ment is satisfied, the large viewing angle is maintained.Second, it is essential that the second derivative of thephase is small, so small in fact that 0 can be consideredconstant within a resolution element of the recon-structed wavefront. This resolution element size isXD2 /a. The second restriction describes a function forwhich the phase relationship between all spectral com-ponents is not entirely random, and for that reason thediscussion of Example 1 does not apply. Using thedispersion function having these special characteristicsresults in a conjugate image wavefront g2(y) that is theconjugate of the object wavefront, except for an over-laying mask. When this mask is considered as a dif-fraction medium, its spatial frequencies may be directlyrelated to a point spread function in the image plane.For example, a component whose spatial frequency is fgives rise to an image point location error of 8, wherea = Dxf.

The equation that relates the autocorrelation of themask function to the autocorrelation of 0 is, from theAppendix,

[R( - R(T)2] (

where 32 = (k2 a2 /D22) + U2. Rn(r) is the autocor-

relation function of the mask function M and R(r) isthe autocorrelation function of 0.

From this equation, it is evident that the width ofthe autocorrelation function of R3 1 (r) decreases as thesize of the hologram aperture a decreases. The spatialfrequencies of the image spread increase, thus contrib-uting to a poorer resolution. Similarly, as the cor-relation function for 0 becomes broader, that is, as 0becomes less rapidly varying, the image point spread

June 1968 / Vol. 7, No. 6 / APPLIED OPTICS 1187

(a) (b) (c)Fig. 4. Resolution reducing dispersion function. (a) Data

reduction ratio R = 140; (b) R' = .500; (c) R = 3600.

function narrows. There are practical limits to thebreadth of R(r), however, since a point is reached atwhich the image will appear excessively spotty whenviewed by eye.

From Eq. (6) it is evident that the noise, which maybe attributed to the higher frequencies of R1M(T), doesnot build up as a is decreased. These desired char-acteristics are demonstrated by the experimental resultsshown in Fig. 4. This series of photographs was againtaken in the image plane. The object was a ring sur-rounding a cross. For the photograph of Fig. 4 (a),the space-spatial bandwidth reduction ratio R2 was140, for Fig. 4 (b) it was 500, and for Fig. 4 (c) it was3600. Observe that the resolution does actually de-crease in the manner predicted while the noise back-ground is altered very little.

In the actual experiment, the dispersion medium wasan aluminized random surface on glass, reflecting lightinto the hologram aperture rather than transmitting it.This was a convenient method for obtaining the largevariation of diffraction angles across the surface.

ConclusionsThe dispe-sion function of the second example pro-

vided an image resolution loss without a decrease insignal-to-noise ratio. An additional benefit of thattype of dispersion function is that the critical realign-ment requirement during the reconstruction process isgreatly alleviated over that for the ground glass system.A lateral misalignment on the order of the width of thecorrelation peak of R(r) will provide at least a blurryimage.

It is desirable to use as a dispersion function a sur-face that varies slowly. However, an excessivelyslowly varying surface will cause the image to appearspotty, the size of these spots being related to the widthof the correlation slope function R(r). A compromisesolution for the slope function must be chosen.

With the dispersion plane systems discussed so far,reconstruction of the conjugate images has beenachieved. While this simplified the experimental work,the system is not limited to pseudoscopic images.However, one must use a different though related dis-persion function on reconstruction than that used in thehologram construction step in order to reconstruct non-pseudoscopic images. An example of a system that re-constructs an orthoscopic image is one for which theactual image of the dispersion plane is focused by atelescope of unit magnification into a space that is

readily accessible. In this focused plane is placed theconjugate of the dispersion function. For the mirrordispersion function system used to obtain the results ofFig. 4, this conjugate mirror is simply the imprint ofthe original surface.

Finally, it should be mentioned that the dispersionfunction need not be confined to a planar function.It is entirely possible to carry out this process with afunction having a depth or axial variation. An ex-ample of a system using a dispersion function in twoplanes is one in which the first plane is an array of smalldiameter lenses that focuses the object onto a groundglass plate in the second plane. This entire assemblymay then be considered as the function m(y). Usingsuch a system yielded results for which the resolutiondeteriorated with decreasing aperture. With thisspecific dispersion function, there may be some diffi-culty in avoiding focus on the lens structure itself whenthe image is viewed by eye. However, it does illus-trate the use of a three-dimensional dispersion function.Others have obtained reasonably good results in theirwork on imaging using this type of composite lensstructure.3

This work was performed at the Radar and Optics LaboratoryInstitute of Science and Technology, The University of Michigan,and was sponsored under contracts from the United States AirForce Avionics Laboratory, Wright-Patterson Air Force Baseand from NASA. This paper was presented at the Fall Meetingof the Optical Society in Detroit, October 1967 [J. Opt. Soc.Amer., 57, 1412A (1967)].

AppendixIn this appendix, an equation is developed that re-

lates the correlation function of the overlaying mask tothe statistics of the dispersion function m(y).

If the second derivative of the phase of the dispersionfunction is so small that the first derivative may be con-sidered constant within a resolution element of the re-constructed dispersion plane, then

°(Y) -D X f, ~(u)du for Y'- < 2Y, - Y ~~~~a (A-i)

= [O(nJ') - (y)]/(Y - Y).

Using this approximation in Eq. (3), and assumingan on-axis point object, one obtains

92(Y) exp(jky2/2D1)M(d), (A-2)

where

I(6) = f exp jk (y" - 2y'y)-2 Y2] exp(jy'0)dy'.

The first exponential term in this equation is simplythe point image reconstruction function. The functionM is the mask function, and it is this portion of theabove equation whose spectral density is examined inthe following.

The problem of analyzing this spectral density, oralternatively the corresponding autocorrelation func-tion, can be solved by using the transform method of

1188 APPLIED OPTICS / Vol. 7, No. 6 / June 1968

- -

nonlinear systems,4 where is assumed to be the Gauss-ian input to the system and M(0) is the output.

With this method, the autocorrelation RM(r) of thenonlinear system output is

RM(r) fF(wi) expQ;w2doiFw)eXp(_0r2 22) dw2f} 2 ) b (2)

exp[-R(T)w0w21, (A-3)

where F(cwl) is the Fourier transform of M. That is,

F(wl) = fM(d) exp(-j~w,)dO

[jk - k'a2 = exp (C12 - 2D 2 W12j.

Inserting this function into Eq. (A-3) gives

RM(r) f exp[ -=o(RW2 + d-) -1 2-]dI

fexp[ 2 ( - y2)- W2 2 - dW2

where 2

grating,= (k 2a2/D2

2) - (jk/d) + a.2.

(A-4)

After inte-

Rm(T) a2/[14 - R(T)2]2

X exp - k2 Id-R2 2T? + + ___ _)

Xex 2d 4 - R 3 + R--F+ (A-5)

If there is to be any appreciable data reduction,either D2 >> Ymax or a <<Ymax. With these in-equalities, the exponential term may be neglected.With the exception of a weighting function of y acrossthe mask function,

RM [r] {a'/ [4 - R(r)'] }I (A-6)

In an approximate manner, we can relate the stand-ard deviation of the function R(r) to the imagepoint spread. For example, if the width of Rm(r) is T,the image point spread is approximately AX = XD,/T.

References1. K. A. Haines and D. B. Brumm, Proc. IEEE 55,1512 (1967).2. E. N. Leith and J. Upatnieks, J. Opt. Soc. Amer. 56, 523

(1966).3. R. V. Pole, Appl. Phys. Lett. 10, 20 (1967).4. W. B. Davenport and W. L. Root, Random Signals and Noise

(McGraw-Hill Book Co., Inc., New York, 1958), Chap. 13.

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1 Editorial Foreword-Kurt E. Shuler

3 Inversion Mechanisms in Gas Lasers-W. R. Bennett, Jr.

31 Measurement of Excited State Relaxation Rates-W. R. Bennett, Jr.,P. J. KindImann, and G. N. Mercer

55 Properties of Optical Cavity Modes-A. G. Fox

56 Collison Lasers-Gordon Gould

65 Charge Transfer as a Possible Laser Pumping Mechanism-J. WilliamMcGowan and R. F. Stebbings

70 High-Resolution Tuned-Laser Spectroscope-H. J. Gerritsen and M. E.Heller

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105 Inverted Population Distributions Produced by Chemical Reactions-H. P. Broida

106 Vibrational-Rotational Population Inversion-J. C. Polanyi

128 Molecular Beam Studies of Internal Excitation of Reaction Products-D. R. Herschbach

145 Measurement of Energy Transfer in Molecular Collisions-A. B.Callear

171 Laser Possibilities of Chemically Excited Molecules Formed withAtomic Species-T. T. Kikuchi and H. P. Broida

179 Flame Laser: Model and Some Preliminary Experimental Results-R. Bleekrode and W. C. Nieuwpoort

201 Population Inversions Produced by Chemical Depletion of GroundStates-Michael L. Seman

181 Explosion Flame Emission-John A. Howe

187 Exploratory Research on Population Inversions in Gaseous Explo-sions-I. Wieder, R. R. Nieman, and A. P. Rodgers

193 Atomic and Molecular Fluorescence Excited by Photodissociation-R. N. Zare and D. R. Herschbach

201 Photodissociation of Thallium Bromide and Cesium Bromide-W. T.Walter and S. M. Jarrett

205 Laser Action in Rare Earth Chelates-A. Lempicki, H. Samelson, andC. Brecher

214 Mechanism of Energy Transfer in Some Rare-Earth Chelates-M. L.Bhaumik and M. A. Al-Sayed

1190 APPLIED OPTICS / Vol. 7, No. 6 / June 1968

2: Chemical Lasers 1965