Light utilization in optical correlators

Joseph L. Horner

An analysis is made of the overall light efficiency in a coherent optical correlator. The results are appliedto a matched filter and an inverse filter. Kogelnik's coupled wave theory is applied to analyze the diffrac-tion efficiency of a filter recorded on a high-efficiency phase medium such as dichromated gelatin. Experi-mental results are presented for a matched filter and out-of-focus spatial filter, and the former is comparedto the theory with good agreement.

I. Introduction

The Vander Lugt optical correlatorl has proved to bea very useful and valuable addition to the collection ofoptical processing systems. It has been used in suchdiverse applications as classifying diatoms2 and in-specting welded seams.3 It is a powerful technique,because it can search an input scene for the desiredpattern without any mechanical motion and providesa simple way to fabricate the matched filter.

In a practical working system, it is desirable that theoverall utilization of the laser light source be as high aspossible. We can measure this as the ratio of light en-ergy in the correlation spot in the output plane to thelight energy at the input plane. A high throughput oflight is desirable in a practical working system to keepthe laser power requirements as small as possible, as, forexample, in a portable or airborne system, and toachieve a high signal-to-scattered light (noise) ratio.

We know that there are high diffraction efficiency(DE) recording media, such as dichromated gelatin(DCG), capable of near 100% DE. One might naivelywonder, as I did when I made a matched filter on thismedium, why an overall efficiency of only half of thisvalue was observed.

In a digital computer realization of these filters it isnot an important consideration, since the input energycan be increased without limit, and there is no suchthing as scattered light or detector noise to degrade thecorrelation spot. But in an optical analog processor,these are important considerations.

The author is with U.S. Rome Air Development Center, Solid StateSciences Division, Hanscom AFB, Massachusetts 01731.

Received 11 June 1982.

II. Discussion

In Vander Lugt's original correlator the filter wasformed holographically on a photosensitive plate, a B/Wholographic film such as 649F, for example. The lightthroughput of the system is determined by the effi-ciency of this filter. It is well known that a thin am-plitude hologram has a maximum DE of 6.25% and athick one even less, 3.7%. In practice one can typicallyachieve only a tenth of this (0.6%) because of the highdynamic range of the signal in the Fourier plane wherethe matched filter is recorded and the limited dynamicrange of typical films.

Following the work of Kogelnik, 4 we now realize thata thick phase-only hologram, or a volume hologram, canhave a theoretical DE of 100%, either in reflection ortransmission. Using a medium such as dichromatedgelatin (DCG) efficiencies of 80% are readily attainedin practice.

111. Analysis

It seems logical, then, to fabricate the Vander Lugtfilter on DCG. The question then arises: using a re-cording medium capable of 100% DE, can we achieve anoverall system efficiency, as defined in Sec. I, of100%?

An expression for the overall system efficiency 7 ofa correlator can be written for a 1-D case as

X = M Nf(x) * h(x) 2dxX IMf(x) 2dx

where f(x) is the input function, h (x) another function,N1M the DE of the recording medium, and the operator* indicates correlation.

A. Matched FilterLet us apply this to a very simple case, a 1-D slit of

width a, and let h(x) be the classical matched filter withuniform noise and unity normalization:

15 December 1982 / Vol. 21, No. 24 / APPLIED OPTICS 4511

(1)

f(x) = rect(x/a),

h(x) = f(x).

Putting these functions into Eq. (1),

7 = M

(2)

(3)

7.

S a/2 Irect(x/a) * rect(x/a)1 2dx_a/2

I rect(x/a) 2dx-a/2

= 2 /3 nm = 66.7%,

and for a 2-D aperture,

n = (2/3)2 nM = 44.4%,

at

2!(4)

(5)

when 71M 1.00. Thus even with a perfect medium thebest we can do is 44.4%.

B. Inverse Filter

Another type of Fourier plane spatial filter that canbe used and that will give a delta-function type of cor-relation peak in the output plane of a Vander Lugtcorrelator is the inverse filter, defined as

HW = 1F(w). (6)

In the case where f(x) is the rect(x/a) function, H(w) hasa number of singularities corresponding to the zeros inthe transform of rect(x/a), sinc(aw). In practice onehas to normalize and truncate the filter function,

H(w) = Fminj/F(C0)-

Equation (1) can be rewritten as

1 = '7Ak

/ b/2 IFmin -F(W) 2d

r-b/ 2 1 F(w) Ira/2I IF(x)2dx

_-a/2

= 71Mb IFmin12.

a

(7)

(8)

Extending this to a 2-D slit,

b n (F 2 . (9)

To evaluate this, let F(min) = 1/30, i1M = 1.00, and b =10a;

=0.01%. (10)

The inverse filter is not a very energy efficient deviceand results in degrading the noise performance of thesystem.5 The reason is that the filter has high attenu-ation in that portion of the spectrum where the signalis highest, namely around the dc and low-frequencyregion, and amplifies where the signal is least, namely,in the vicinity of the zeros of the input sinc function.This has been verified experimentally.6

IV. Recording Consideration

In recording a hologram, the K ratio is defined as

K = Iref/Isig, (11)

0

1

K Ratio

Fig. 1. Plot of theoretical diffraction efficiency vs K ratio (K =

Ief/Isig). Circles are experimental points, from Ref. 7, Fig. 25,p. 54.

(12)I = Aref exp(iaex) + AsigI2

= IAsig2(K + 1) {1 + K cos xl (13)

If DCG behaved like a linear photographic film, the DEwould be proportioned to

( 2VW2\K +J

(14)

However, there is no reason to suppose DCG behavesthis way. Instead we must go to Kogelnik's4 treatmentof a volume phase hologram. Kogelnik's expression forDE [Eq. (43)] is

'2k = sin2 (V2 + 42)1/2/(1 + 42/v2 )

wherev = 7rnld/X(CRCS)1/2 ,

= Od/2Cs,

(15a)

(15b)

(15c)

and ni = index modulation in the medium, d is themedium thickness, X the wavelength, CR and Cs geo-metric factors associated with the beam geometry, and0 a dephasing factor. If we work at the Bragg angle, thevalue of t is zero, and Eq. (15a) reduces to

t = sin2 (v). (16)

To evaluate this, we will use the experimental dataof Colburn et al.7 and assume that nj varies as themodulation factor in Eq. (13). From Ref. 7, d = 17 um,X = 0.488 ptm, (CS CR)1 12 1.00, giving

v = 109nj. (17)

At K = 1, the experimentally determined value of - is0.84, requiring that n1 = 0.011. Putting these into Eq.(16), we find,

where Iref and Isig refer to the intensities of the referenceand signal beams, respectively. Consider two inter-fering beams producing a hologram on a square lawdetector. The intensity pattern is

'7k = in2(1.16 M . (18)

A plot of this is shown in Fig. 1 together with anotherexperimental point at K = 50. The agreement is rea-sonably good.

4512 APPLIED OPTICS / Vol. 21, No. 24 / 15 December 1982

A problem arises in trying to record the Fouriertransform of the slit pattern in a single exposure witha single reference beam and is shown in Fig. 2. If we fixthe reference level at 50% of the dc peak, the K value forthis region is 2.0, corresponding to an Nk of 0.79, fromEq. (18). However, for the first side lobe peak, the Kvalue is 0.5/0.047 = 10.6, giving an 7k of 0.37. What thismeans is that the side lobes will not be effective incontributing to the correlation spot. If we make thereference beam comparable to the side lobes, the overallenergy efficiency suffers, because the K ratio of the dcspot will be large, giving a small 7k, and the central spot,where most of the energy is, will not contribute appre-ciably to the correlation spot. Manipulating the K ratioin the various spectral regions was the basis of Casa-sent's work in selective spectral correlation.8

V. Experiment

A. VanderLugt Matched FilterTo keep the K ratio near unity in all regions of the

spectrum, a VanderLugt filter was made using twodifferent reference beam intensity levels. The DCGwas prepared from 649F holographic film following therecipe of Colburn et al.7 9 The exposing wavelength was0.488,um, with an intensity level of -150 mJ.

To prevent double exposure, two masks were made.The first was made of chromium film on glass, the samesize as the dc spot in the Fourier plane. Chromium wasselected to absorb the high-intensity level withoutburning. The second mask consisted of a contact printof the first mask on a 649 photographic plate. A pre-cision holder was devised to hold the two masks, in se-quence, in exact registration with the unexposed DCGplate. The DCG VanderLugt filter was then exposedin two steps. First the chrome-on-glass mask was po-sitioned to block the dc spot, the reference beam in-tensity was adjusted to equal the first side lobe peakintensity, -and an exposure was made. Then this maskwas removed, and the second mask (the one on 649 film)

was put into position. The intensity level of the refer-ence beam was adjusted to be 50% of the peak dc spotintensity, and an exposure was made. In this way theK ratio of both the dc spot and the side lobes was nearunity. The results will be presented and discussed inthe next section.

As a comparison test, a single-exposure VanderLugtfilter was made with the reference beam intensity op-timized at10

Iref = V/Idc Ilst lobe,

which for the case of sinc(aw) is

Iref = 1.0 0.047 = 0.22

(19)

or 4.7:1 compared to the dc spot peak intensity. Theseresults will be discussed in the next section.

B. Out-of-Focus FilterIt has been suggested by some authors11 12 that one

way to mitigate the dynamic range problem in trying torecord a Fourier transform plane signal is to move theholographic plate to a position out of the Fourier planeby a small amount Af, where f is the focal length of theFourier transform lens. This was experimentally testedfor values of Af/f of 5 and 10%. The focal length f was500 mm in all cases, and only single exposures weremade on the DCG plate. The trade-off for this is thatthe correlation signal is no longer space invariant. 1 3

The results of this experiment will be reported in thenext section.

V. Results

The results of the above experiments are presentedin Table I. First, a word about how the raw data werenormalized.

To minimize the statistical spread in the raw data,which occurs in any photographic process, each exper-iment was repeated six times, and the results were av-eraged. There is another problem with DCG. If abatch of plates is prepared, the measured DE slowlydecreases over a period of a week or two, even if the

Fig. 2. Plot of sinc2 function showing the difficulty of recording itwith a single reference beam.

15 December 1982 / Vol. 21, No. 24 / APPLIED OPTICS 4513

(20)

Table I

Regular Two-exposurematched matched Out-of-focus

DE Filter (%) (%) 5% 10%

Measured 58 47.2 62% 69.3%Calculated 44.4 44.4

prepared plates are refrigerated. To remove thisvariation, a test plate was made at the beginning of eachexperiment, exposing the plate to two equal intensitybeams and then measuring the resultant DE.

The formula used for correcting the raw data was1 6

X7 = - E iRD/TP,6 \=i

where a7 is the corrected DE reported in the table, 17RD

is the raw data, and 77TP is the DE of the test plate.The results in Table I show good agreement between

theory and experiment.The single-exposure filters showed higher DE than

the two-exposure ones, probably because of nonlinearityin the DE vs n, curve. Theoretical values of 7 for theout-of-focus experiment were not computed due to thecomplications of such a calculation.

VI. Conclusion

I have attempted to define a measure of the usefulefficiency of a coherent optical correlator because thisis important in a practical working system and does notshow up in a computer simulation study. I have ex-perimentally checked the calculations in the case of theclassical matched filter and found good agreement.

We see that even if we use a recording medium ca-pable of 100% DE, such as DCG, at best we can obtainonly 4/9 of this in the optical matched filter correlator.

However, this is almost 3 orders of magnitude betterthan using the usual amplitude-only holographic filterfabricated on 649F film. We also see that while theout-of-focus scheme does significantly increase theoverall correlator efficiency, the sacrifice in the space-invariant property probably does not justify this tech-

(17) nique.

References1. A. Vander Lugt, IEEE Trans. Inf. Theory II-10, 139 (1964).2. S. P. Almeida and J. K. Eu, Appl. Opt. 15, 510 (1976).3. J. M. Sandrik and R. F. Wagner, Appl. Opt. 20, 2795 (1981).

4. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).5. J. M. Wozencraft and I. M. Jacobs, Principles of Communication

Engineering (Wiley, New York, 1965), p. 242-244.6. J. L. Horner, J. Opt. Soc. Am. 59, 553 (1969).7. W. S. Colburn, R. G. Zech, and L. M. Ralston, "Holographic

Optical Elements," Tech. Rep. AFAL-TR-72-409 (1973).8. D. Casasent and A. Furman, Appl. Opt. 16, 1662 (1977).

9. It was discovered that it is important that the plates be relativelyfresh, even though they have been frozen in the meantime. Usingplates a year and a half old, even though they had been frozen,a DE of only 40% was obtained. Using fresh plate, DEs of 70%

were readily achieved.10. The function O7K is symmetric in the sense that flK(K = n) = VK(K

= 1/n). To make the K ratio of the dc spot and first lobe equalKdc = Iref/Idc = Klobe = Ilobe/Iref l giving Iref = '/dcIobe

11. B. Goldman, Optik 34, 254 (1971).12. K. Chalasinska-Macukow and T. Syoplik, Appl. Opt. 18, 1436

(1979).13. A. Vander Lugt, Appl. Opt 6, 1221 (1967).

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4514 APPLIED OPTICS / Vol. 21, No. 24 / 15 December 1982