1580 OPTICS LETTERS / Vol. 19, No. 19 / October 1, 1994

Statistical analysis of cross-talk noise and storage capacityin volume holographic memory

Xianmin Yi and Pochi Yeh

Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, California 93106

Claire Gu

Department of Electrical Engineering, Pennsylvania State University, University Park, Pennsylvania 16802

Received April 7, 1994

We present a statistical analysis of cross-talk noise and consider the cross-talk-limited storage capacity of volumeholographic memory. Random-walk approximation is employed to estimate the intrapage interpixel cross-talknoise and the signal-to-noise ratio. We obtain simple expressions of the storage capacity in terms of number ofbits per volume for a given signal-to-noise ratio or bit-error rate. The results indicate that the storage densityis reduced from the ultimate density of A 3 by a factor related to the signal-to-noise ratio.

Optical data storage in volume holographic mediahas been an important and exciting area of research.This research is driven mainly by the prospect ofparallel readout and an enormous storage capacityof -V/A 3 bits in volume V.1 In the spectral regimeof visible light, this translates into a storage densityof several terabits per cubic centimeter.

In practical applications, such as optical imageprocessing and pattern classification, the storagecapacity is limited by the cross-talk noise betweenholograms and pixels. Although some aspects ofthe cross-talk noise have been investigated,2 -6 ageneral theory of the storage capacity of the holo-graphic medium is not available. To illustrate this,we point out that the present state of the theory doesnot provide a definite answer for the storage densityat a given bit-error rate (BER). In this Letter wepresent a general theory of the storage capacity byconsidering the effect of the finite transverse size ofthe crystal on the storage capacity and by employinga statistical method to evaluate the cross-talk noise.We obtain, for what is to our knowledge the first time,expressions for the cross-talk-limited storage capac-ity in terms of the number of bits per unit volume ata given signal-to-noise ratio (SNR). The results arethen employed to compare angle multiplexing andwavelength multiplexing.

Referring to Fig. 1, we consider the recording andthe readout of a two-dimensional image in a holo-graphic medium. The two-dimensional image isone of several pages stored inside the holographicmedium by the use of angle or wavelength multiplex-ing, and it contains binary data. The interpixel crosstalk is dominated by the pixels within the same page.This becomes clear from the result of the analysis be-low. To study the intrapage interpixel cross-talknoise, we consider the imaging of an arbitrary inputobject given by

N2

U.(x, y) = E Am rectm=1

where 3 is the size of the pixel, S is the separation ofneighboring pixels, and m is shorthand notation forthe two integers m. and my that specify the addressof each of the pixels. Am can be either one or zero,depending on the on or off state of the correspondingpixel. By using the standard Fourier-optics analysiswe can obtain an expression for the output image.

It is important to note that the image is not anexact replica of the object; in other words, it suffersmany distortions. In this Letter we examine the ef-fect that is due to the finite size of the recordingmedium, and we neglect other noise sources. Eachsquare pixel of the object is transformed into a con-volution of a square and two sinc functions by virtueof diffraction. The diffraction-limited image of eachpixel thus consists of a main lobe and a series of side-lobes. The physical overlap of these images leads tointerpixel cross talk. At each of the grid points inthe output plane the optical amplitude consists of asignal amplitude and a noise amplitude. For thepurpose of clarity in describing the cross-talk noise,we assume that the size of each of the detectors ismuch smaller than the size of the main lobe of theimage of each pixel. Without loss of generality, letus examine the image amplitude of the input pixel atthe origin (0, 0). The signal amplitude and the noiseamplitude can be written as

U = Ao[f

N2

Un = Em.-

.0(0,)

dx sinc( a ) rect() (2)

Amf dx sinc( ) rect(X - M.)

xf dy sin ( y rect) (3)

x - mXS y - myS where D is the transverse size of the holographic1rect a , medium and f is the focal length of the lenses. Be-

a 1k '~ / cause the sine function decays as 1/x and each term(1) is a random variable that can be positive or negative,

0146-9592/94/191580-03$6.00/0 K 1994 Optical Society of America

October 1, 1994 / Vol. 19, No. 19 1 OPTICS LETTERS 1581

s

3 oCI I_a a O O CoEl zz T3000E

Object Plane (x0, y0)

y3

.1

Output Plane (x2,,

By evaluating the noise, we find that the interpixelcross-talk noise is critically dependent on the ratiobetween 8 and AfID. When the pixel size is an odd-integral multiple of the width of the sinc functionsidelobe, i.e., 3 = pAfID (where p = 1, 3, 5, 7, ...),the convolution sidelobes are highest and the SNRreaches its minimum. Because the absolute valuesof the amplitudes of the noises contributed from dif-ferent pixels decrease as 1/n and their sum diverges,the situation can be treated as a random-walk prob-lem. Using approximation (4) for the case of p = 1,we obtain the following approximate expression forthe minimum possible SNR:f f Ot f f a

Fig. 1. Recording and readout geometry for angle multi-plexing (0 = ,-/2) and wavelength multiplexing (0 = 0).

the noise at a specific point detector contributed frompixels far away from it is negligible. It is thus legit-imate to evaluate the summation over all grid pointsfrom -- to +co as an approximation. Physically thismeans that intrapage cross talk between pixels im-poses a limitation only on pixel separation, not on thenumber of pixels. As we discuss below, the numberof pixels is limited by the interpage cross talk, whichis also related to the finite size of the holographicmedium.

Equation (3) directly gives us the noise amplitudeat a specific point detector generated from a giveninput pattern. We now examine the statistical be-havior of the noise at all the point detectors gener-ated from various input patterns. We note that fora specific input pattern at different point detectorsor for different input patterns at a specific point de-tector, Am has equal probability of being one or zerofor each m. Each of the terms in the summation ofEq. (3) has equal probability of being either positiveor negative. This leads to a mean value of zero, andthe statistical behavior of the noise is similar to thatof the random walk in probability theory. As thestep sizes are not equal, the correspondence is onlyan approximation.

Using Eqs. (2) and (3) and assuming SNR >> 1,we obtain a general expression for the SNR of theintrapage interpixel cross-talk:

( Af )2

D)SNRpi

2 f dx sinc( D x) rect x - nS)

According to the result, only pixels in the same row orcolumn as the pixel being considered contribute sig-nificantly to the noise. This is a direct result of therectangular-shaped crystal aperture. In the case ofother crystal geometries, the result should be slightlymodified. The SNR depends on the convolution ofthe sinc function and the rectangular function, whichin turn depends on 3D/Af. The convolution func-tion is sinclike, and the noise in the denominator ofapproximation (4) can be evaluated by a summationof all the sidelobes.

SNR~dd 3(2 (DS 2) (5)

When the pixel size is an even-integral multipleof the width of the sinc function sidelobe, i.e.,3 = qAfID (where q = 2, 4, 6, 8, ... ), the convolu-

tion sidelobes become lowest, and SNRpi reaches itsmaximum. Using approximation (4), we can esti-mate the maximum of the convolution sidelobes as[(2A 2 f 3)/(r 2 D3 S2 )](1/n 2 ). We note that the absolutevalues of the amplitudes of the noises at a specificpoint detector contributed from other pixels decreaseas 1/n2 . The sum converges quickly and can betaken as an upper limit of the total noise. In thiscase the noise can no longer be treated as a random-walk problem, and we have to redefine the SNR asthe ratio of the signal intensity to the upper limit ofthe intensity of the noise. Using Eqs. (2) and (3) forthe case of q = 2, we obtain the following expressionfor the maximum possible SNR:

(6)SNReven = ( 3 D 2 S2 24 A2f 2)

In the above discussion all the phases of the on pix-els are assumed to be identical. If the phases arerandom, the same result will be obtained.

According to approximation (5) and Eq. (6), we notethat the finite transverse size of the crystal leads tothe intrapage interpixel cross-talk noise, which im-poses a limitation on the pixel separation in eachhologram. It is known that the finite thickness ofthe crystal leads to an interpage cross-talk noise,which in turn limits the total number and the physi-cal size of the holograms that can be stored in thecrystal.3 4 Combining approximation (5) and Eq. (6)with those previous results, we can obtain the stor-age density in terms of the total number of bits perunit volume.

To obtain an expression for the cross-talk-limitedstorage density in wavelength multiplexing, we com-bine the above results with those obtained by pre-vious workers,4 including the frequency separationof adjacent holograms of Av = c/2t and the SNR ofSNRpa = 2f2/area, where c is the light velocity invacuum, t is the thickness of the crystal, f is thefocal length, and area is the area of the input plane(NS x NS). We note that the SNR is independent ofthe number of holograms stored in the crystal. For

YO Y'.

1582 OPTICS LETTERS / Vol. 19, No. 19 / October 1, 1994

160

140

IE

:S(D

120

100

80

60

40

20

01 10-2 10o

41 0- 1O-8 10 o o-12

Bit-error rate

Fig. 2. Storage density versus BER for both angle andwavelength multiplexing.

a wavelength variation from AO/2 to Ao, the numberof stored holograms is Npa = 2t/AO.

As the wavelength varies from AO/2 to AO, the ratio3D/Af will also change accordingly by a factor of 2.Based on the above discussion, the intrapage inter-pixel cross-talk noise will vary with the wavelength.The cross-talk-limited storage density p, in terms ofthe number of bits per volume, is thus between thefollowing limits:

3qr 2 1 1pA(min) = 2 (SNRoddSNRpa) A0

3

PA(mx)=3 1 (7PANm x) = [SNRp.(SNRe.00 )YJ2] A0

3

Note that the SNR values defined here are in termsof the ratio between the signal intensity and the vari-ance of the noise amplitude. To translate this SNRto the measurable quantity of the BER, we apply thecentral-limit theorem to the noise amplitude and ob-tain a Gaussian probability density function. As aresult, the intensity probability-density function isan exponentiallike distribution. From this intensitydistribution, we can calculate the probability of error,i.e., the BER. As discussed above, the intrapage in-terpixel cross talk for aD/Af = even integer is nottreated as a random walk problem, and SNRaven isthe ratio of the signal intensity and the upper limitof the noise intensity. So SNRVn can be chosen to bequite small, for instance, 4. Then the threshold leftfor the interpage cross-talk noise can be one fourthof the signal intensity. To achieve a BER of 10-9we need a SNR of 150, i.e., SNRpa = 150. UsingSNRodd = 150, SNReven = 4 for a BER of 10-9, we havepA(min) = 7 X 10-4 /AO3 and pA(max) = 1.0 x 10-2/A0

3.Because of the variation in wavelength, the max-imum storage density, which represents an upperlimit, cannot be achieved in practice. We note thatthe storage density is -2 to 3 orders of magnitudelower than 1/Ao3.

In the case of angle multiplexing, the interpagecross-talk noise is given by SNRpa = (2tf)1(AdyNh),

according to Ref. 3, where dy is the size of the out-put plane in the y direction and Nh is the number ofholograms stored in the crystal. Such an interpagecross talk gives a limitation on only one dimensionof the output plane. The other dimension is limitedby the paraxial approximation. Let a = dr/f, wheredo is the size of the output plane in the x direc-tion. Using approximation (5) and Eq. (6), we obtainthe following expression for the lower limit and theupper limit of the cross-talk-limited storage densityin terms of the number of bits per volume:

37T2 a 1p 0(min) -~4 (SNRoddSNRpa) A3

p 0(max) = 3 a As (8)2 SNRpa(SNReven) 1 12 A' 8

Again using SNRpa = 150, SNRodd = 150, andSNReven = 4 for a BER of 10-9 and a = 0.2, we ob-tain a storage density of po(max) = 1.0 x 10-3/A 3

and po(min) = 6.6 x 10-5 /A3. Using A = Ao/2 asan example, we obtain po(max) = 8.0 X 10-3/AO3

and po(min) = 5 x 10-4/Ao3. In angle multiplex-ing we can adjust A and f to satisfy the condition forminimum intrapage interpixel cross talk. Thus themaximum storage density is practically achievable.

Figure 2 illustrates the range of storage density asa function of the BER for both angle and wavelengthmultiplexing. We note that there is a significantarea of overlap between these two schemes of opti-cal storage.

In conclusion, we present a statistical theory of thecross-talk-limited storage capacity of both angle- andwavelength-multiplexed holographic memory. Weobtain, for what is to our knowledge the first time,expressions for the number of bits per unit volumeat a given BER. It is found that the storage densitycan be significantly lower than the ultimate stor-age density of 1/A3 by 2 to 3 orders of magnitudebecause of a desired BER of 10-9. Also, we havemade comparisons between angle- and wavelength-multiplexing schemes.

Claire Gu acknowledges support by the NationalScience Foundation under Young Investigator pro-gram ESC-9358318. The work at the University ofCalifornia, Santa Barbara, is supported by the U.S.Air Force Office of Scientific Research and the U.S.Office of Naval Research.

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Mok, J. Opt. Soc. Am. A 9, 1978 (1992).4. K Curtis, C. Gu, and D. Psaltis, Opt. Lett. 18, 1001

(1993).5. G. A. Rakuljic, V. Leyva, and A. Yariv, Opt. Lett. 17,

1471 (1992).6. A. Yariv, Opt. Lett. 18, 652 (1993).

pA(max)

AV /p 6 (max)

_ p 0l(niin) *-. -