Photorefractive adaptive resonanceneural network

Donald C. Wunsch 11, David J. Morris, Rick L. McGann, and Thomas P. Caudell

We describe a novel adaptive resonance theory (ART) device that is fully optical in the input-output

processing path. This device is based on holographic information processing in a photorefractivecrystal. This sets up an associative pattern retrieval in a resonating loop that uses angle-multiplexedreference beams for pattern classification. A reset mechanism is used to reject any given beam,permitting an ART search strategy. The design is similar to an existing nonlearning optical associativememory, but ours permits learning and makes use of information that the other device discards. It is asuitable response to the challenges of connectivity, learning, and reset presented by ART architectures.Furthermore, the design includes an efficient mechanism for area normalization of templates. It alsopermits the user to capitalize on the ability of ART networks to process large patterns. This new deviceis expected to offer higher information storage density than alternative ART implementations.

1. Introduction

Neural networks based on adaptive resonance theo-ry1-7 (ART) offer a number of implementation chal-lenges, the greatest of these being the resent mecha-nism, the massive interconnectivity requirements,and the requirement that these interconnections beadaptive. Previous research on holographic associa-tive memory8 has inspired us to create a relateddevice capable of learning and of normalizing memo-ries. We have been further encouraged by successfulimplementation of massive information storage in aphotorefractive crystal.9 Our research represents asynthesis of these results.

In this section we present the motivation for ourresearch by discussing ART and its associated chal-lenges for hardware implementation. In Section 2we describe our design, and in Section 3 we discussthe system layout in more detail. Section 4 de-scribes our (non-novel) laboratory progress towardrealization of a hardware implementation of an opti-cal ART network by means of incremental component-level hardware evaluations. Section 5 discusses fu-ture research, and Section 6 summarizes ourconclusions.

When this research was being performed, the authors were withthe Boeing Company, P.O. Box 24346, Seattle, Washington 98124-

0346. D.C. Wunsch is now with the Department of ElectricalEngineering, Texas Tech University, Lubbock, Texas 79409.

Received 9 June 1992.0003-6935/93/081399-09$05.00/0.c 1993 Optical Society of America.

ART is attractive because of its stable unsupervisedlearning properties and its ability to process largeinput-pattern fields. Recent successful applicationsof ART3 used input fields in excess of 107 nodes. Itappears that ART's scaling properties are limitedonly by hardware and software implementations.Thus we have significant motivation for the presentresearch. We present only a rough outline of theoperation of ART, which is covered extensively else-where.1-7 We merely want to point out some proper-ties of ART networks that are essential to theiroperation, yet present hardware challenges: resetand massive connectivity. An example illustratesthese points.

One type of ART network is depicted in severallayers in Fig. 1. The layers are as follow: R, therecognition layer; C, the comparison layer; I, theinput layer; V, the vigilance layer; Re, the reset layer.This grouping of layers is similar to that in Ref. 6 and,while it does not follow Carpenter's and Grossberg'soriginal description' exactly, it is functionally equiva-lent. Looking from left to right, we see the ART unitin action. First, the input is registered at the compar-ison layer and fed up to the recognition layer [Fig.1(a)]. In the second frame the recognition layer'swinner-takes-all property finds the node correspond-ing to the initial best guess [Fig. 1(b)]. This guess istested by replaying the winning node's previouslylearned template (weight values) onto the comparisonlayer. The input layer, which still contains theoriginal input, and the comparison layer, containingthe winning template, compare these two patterns by

10 March 1993 / Vol. 32, No. 8 / APPLIED OPTICS 1399

(b)

R O(

C SOS 000Y

I *0000* * * a a

(d)ART1: bottom-up pattern match-

ing [(a), (b)] is balanced by top-down feedback expectancy [(c), (d)].

sending competing signals to the vigilance node [Fig.1(c)]. The final frame shows an example of whathappens when the match is not good enough. Thevigilance node is here able to activate the reset layer.The reset layer suppresses only nodes at the outputthat have been recently active, and has no effect onthe rest [Fig. 1(d)]. In this example, only the priorwinner is affected. With this node removed, thenetwork reclassifies the pattern and continues to doso until it finds a good match. When a good match isfound, the winning node's activation is permitted tocontinue. The simultaneous activation of the inputnodes and a winning recognition node creates aresonance. This resonance is frustrated by the resetmechanism until a good match is found. The key isthat the pattern classification takes place in a feed-back loop and that learning does not set in untilresonance is permitted to persist. Until then, thereset mechanism permits a search for a better patternmatch, removes all classifications considered previ-ously, and suspends learning until the best answer isfound.

The communication between the comparison layerand the recognition layer requires a massive numberof interconnects; furthermore, these connections mustbe capable of real-time adaptation. (Many neuralhardware implementations involve hard-wired connec-tion weights; clearly not an option for ART networks.)An ART unit of the type shown in Fig. 1 having Ninputs and M outputs requires the following types ofconnections: 2 MN adaptive, M2 + 2M inhibitory,5M excitatory, N inhibitory, 3N excitatory, and, ofcourse, N-input and M-output connections. This isa compelling argument for the importance of opticalimplementations.

The advantage of optics lies in the fact that imple-menting interconnects electronically is difficult be-cause of electromagnetic interference and the neces-sity of wires to carry the signals. A basic assumptionof the artificial neural systems paradigm is that the

importance of interconnects is dominant for a certainclass of problems. Psaltis and Abu-Mostafa'0 as-sessed the relationship of computing power to inter-connects, independent of the power of the individualprocessing elements. They suggest that an intercon-nection-dominated problem ". . .has the propertythat local decisions cannot be made until essentialinformation has been communicated from basicallythe entire input data. Thus useful computation canprogress only when all the input information hasbeen considered by the individual elements. For aparallel processor, this implies that all partial resultsneed to be globally communicated. It is this notionthat forms the basis for our conviction that communi-cation capability becomes the dominant factor. ... "10Holographic implementations offer superior commu-nication capability because of the degrees of freedominherent in the medium; it is this capability that weexploit in this implementation.

2. Optical Implementation

The design of the optical ART unit is shown in Fig. 2.It is a modification of the resonating loop reported bySoffer et al.,8 which is shown in Fig. 3. A keydifference between the figures is the nonlinear stor-age crystal of Fig. 2 in place of the hologram of Fig. 3.In our optical ART implementation the crystal isphotorefractive barium titanate (BaTiO3), which iscapable of recording multiple holograms in realtime.9" 1 It acts as part of a resonant cavity designedto converge on the correct images so that it behaves asan ART unit. Contrast this use of a crystal with Fig.3, in which a fixed hologram is used. The latterdesign permits the device to be capable of associativepattern retrieval, but not of learning. We also intro-duce two components between the storage crystal andthe right-hand phase-conjugate mirror (PCM1) in

Input Pattern

PCM 2

Reset Detector

Angle-MultiplexedReference Beams

#1 #2

Photorefractive- Memory

Crystal

l _ LReset SLM

(A7_ OutputImager

PCM 1Fig. 2. Photorefractive ART unit.

1400 APPLIED OPTICS / Vol. 32, No. 8 / 10 March 1993

R 000 R 0-0

f 000 Re 000 ReC G*e*CQQ O C ,*,000

4 O OlllOO TI *--000 I a000* u o a * * a

(a)

R o.o

f 000 ReC *Q0QooQ

I *0000

Eu * N

(c)Fig. 1. Network dynamics of

Stored ReferenceBeams

InputHologram

Lens 1

AhZ s

BS

X

Fig. 3. Holographic optical resonator: BS's, beam splitters.

Fig. 2; these are.a reset spatial light modulator (thereset SLM) and'a beam splitter to permit imaging ofthe angle-tultiplexed reference beams to identify theactive mode.

A brief review of the operation of the holographicresonator shown in Fig. 3 helps to explain the opticalART unit shown in Fig. 2. The holographic resona-tor is primed with a partial input pattern di, which isshown in the upper left of Fig. 3. This is reflected offthe second beam splitter toward the fixed, previouslyrecordezd hologram, which excites several referencebeams bi. These are retroreflected by a thresholdingPCM1 back toward the hologram, setting up a reso-nant loop between PCM1 and PCM2. The loop isbiased by the presence of the hologram and by theinjected signal di, and it suppresses all stored patterns(and their reference beams) except for the one mostclosely matching di. This causes a readout of thestored image di closest to the input di. The devicecan be considered as a pattern classifier by consider-ing the output reference beam bi to be an angularlymultiplexed classification code. Also, light that con-tains information from both di and ai is present at thepoint marked X in Fig. 3, but this information is notused. Reference 8 contains a more complete descrip-tion of the device illustrated in Fig. 3.

The optical ART unit of Fig. 2 was inspired by theSoffer et al. device, but it uses a BaTiO3 crystalinstead of a hologram to permit learning. Thissimple change is key in designing the device to behaveas an ART unit, but providing a reset mechanism toimplement ART correctly is also necessary. This isprovided by placing a detector in a position to recordthe overlap of the input pattern and the recordedtemplate (precisely the information that is discardedat location X of Fig. 3) and by controlling andmodulating the overall phase in the optically reso-nant loop. The reset detector of Fig. 2 is an integrat-ing photodetector. During initial setup, with a sin-gle template stored in the memory crystal and withthe same (matching) pattern input to the resonator,the phase of one of the two pump beams in PCM2 ofFig. 2 is adjusted12; this is done while the total powerat the reset detector is monitored to establish thephase at which maximum constructive interferencebetween the input pattern and the (matching) storedtemplate occurs. (This implementation of phase ad-justment requires that PCM2 be operated in thefour-wave-mixing mode, while the nonlinear modecompetition ncessitates that PCM1 in Fig. 2 be

operated in the thresholding photorefractive self-pumped mode.)

Subsequently, during pattern-search mode (whenone or more templates are stored in the memorycrystal and a new pattern is presented) the adjustablePCM2 pump-beam phase is left at the pre-establishedsetting, and the system is permitted to resonate untila single mode dominates. During this pattern-search mode the power used in the readout beams isinsufficient (for the time scales involved) to rewritethe memory crystal. When a single template isdominant in the resonator (as indicated by a constantsignal at the reset detector), the reset decision ismade. To make this decision, one sweeps the phaseof the pump to PCM2 through several cycles whilelooking for the presence of power modulation in eachpixel of the reset detector's field of view. Each pixelfor which an overlap between input and templateexists exhibits this modulation. The quality of thepattern match is determined from the number ofoverlap pixels detected by the reset detector.

Thus far we have explained the operation of thereset detector to measure the overlap of the inputpattern and template by the use of constructive ordestructive interference. This obtains the innerproduct between the input and the template. Asdiscussed in Refs. 1 and 2, however, we must normal-ized these measurements by input-pattern size.This can be accomplished in two ways. The simplerbut slower way is to shut off the pump beams to thefour-wave-mixing PCM2 (see Fig. 4). This is doneonly when the input pattern is first presented. Thereset detector then measures the total input patternintensity and saves this number electronically. Thepump beams are then switched back on, and the resetdetector reads the constructive interference patternrepresenting the overlap between the input and thetemplate. The reset decision is made from these twopieces of information. The faster way to do this is toadd an extra beam splitter and to bleed off a copy ofthe input pattern before injecting it into the resonator.This copy is then measured by an extra detector andis used for the reset decision.

I13PCM1

Fig. 4. Lab layout of the optical ART unit.

10 March 1993 / Vol. 32, No. 8 / APPLIED OPTICS 1401

If the reset detector's match measurement is toosmall, reset is indicated, and the steps describedbelow are performed. If the reset detector's matchmeasurement is above a user-determined threshold,reset if not triggered, and the system is permitted toresonate in its preferred mode, causing learning ofthe new pattern. This learning can be accelerated byincreasing the power in the readout beam for PCM2.

In the event that a reset condition is indicated, thefollowing actions take place. The optical systemviewing the reset SLM (off the beam splitter on theright-hand side of Fig. 2) is used to identify thelocation of maximum intensity at the reset SLMplane, which corresponds to the location of the refer-ence beam associated with the dominant systemmode (say, the kth reference beam). The (formerlyinactive) rest SLM is then activated to produce atten-uation at the known location at which the kth refer-ence beam focuses through the reset SLM. Thissignificantly reduces the gain of the kth system modeand permits continued pattern search, in which thekth mode is not permitted to compete. This processcontinues, with poorly matching modes excludedsequentially from competition by the reset SLM, untila dominant mode that sufficiently matches is found.If no such mode is found, the input pattern is storedas a totally new template by using a new referencebeam aligned along a direction previously unused.

The reset detector also can play a useful role innormalizing patterns by template size, which is arequirement of the ART neural network." 2 This isan important problem, not just for this specific neural-network implementation, but for holographic patternrecognizers in general. Some type of area-basednormalization of the templates needs to be performedin order for the pattern classification to be meaningful.However, the reset detector is in effect measuring thesize of the pattern that becomes the new recordedtemplate. Therefore, when reset is not indicatedand a new template is recorded, the constructiveinterference measurement is used to determine thedesired grating strength associated with that re-corded template. This is done by providing thedetector with enough electronic memory to store allthe template sizes up to the capacity of the system.This uses a small amount of memory, even for asystem of large capacity, because only one numbermust be stored for each template. This informationis used to modify the recording schedule for alltemplates. Small templates are recorded longer; bigones are recorded a shorter time. Alternatively,recording power can be increased by differentamounts, depending on template size. This makesthe recording schedule slightly more complicated, butthe complexity is compensated by improved perfor-mance of the system. A normalizing scheme isnecessary, and this one is useful because it capitalizeson system components that already exist.

In this implementation the output of the neuralnetwork is the classification provided by the reference-beam identity, which can be read off by use of theoutput imaging optics viewing the reset SLM off the

beam splitter, as shown on the right-hand side of Fig.2. It is also possible to read out the stored templateinformation in a manner similar to that for thecomplete object output shown on the left in Fig. 3.In other words, the device can be used as either aheteroassociative or an autoassociative ART-basedmemory. Furthermore, it may be possible to replacethe angle-multiplexed reference beams with a thirdSLM. This could be used for associating input pat-terns with known output patterns for supervisedART-based learning.5 The image on the reset SLMwould be the inverse of the image on this thirdtraining SLM. We do not address the idea furtherhere, we simply mention the possibility.

The optical layout schematic of the device is shownin Fig. 4. For experimental verification of the de-sign, human observation of the reset signal andactivation of the reset SLM is acceptable. This roleculd ultimately be automated. The angle multiplex-ing of reference beams is carried out in a manneridentical to that described by Mok et al. 9 The detailsof the nonlinear competition between accessible modesin this type of device is described by Soffer et al.8

3. Experimental Setup

Three photorefractive BaTiO3 crystals are called forin the optical ART demonstrator implementationillustrated in Fig. 4. One is the memory crystal,used to record templates against which input pat-terns are classified. The other two are used forphase-conjugate feedback of laser light within theART resonator.

The memory crystal has two modes of operation,template storage and readout. The template storagemode calls for the use of relatively intense waves inthe reference beams and the object input wave(through the input SLM) in order to quickly create ormodify phase gratings within the photorefractivecrystal. The readout mode involves much lowerintensity waves in order to permit a search processbefore overwriting of the stored holograms occurs.In other words, the competitive pattern-matchingmechanism of ART can be realized at low power levelsto minimize erasure of templates recorded previously.Once the system has made a classification, a higherpower level is used to write the new updated templatequickly.'3 Over time, both the high-power writing ofnew templates and the integrated effects of low-powerreadout degrades templates learned previously. Thiscan be circumvented by rereferencing the storagecrystal with the template image(s). This could becompared with the refresh cycles necessary in adynamic random-access memory. Such rereferenc-ing could take place each time a new template iswritten after a classification search has terminated.However, such frequent rereferencing may not benecessary. It is possible to stagger power levels,9"14"15anticipating template erasure by writing earlier tem-plates at higher power and gradually decreasing thepower.

For the feedback PCM's, we employ four-wavemixing in PCM2 with controllable pump-wave power

1402 APPLIED OPTICS / Vol. 32, No. 8 / 10 March 1993

in a way such that adequate effective gain is estab-lished to overcome transmission and holographicefficiency losses within the resonator.' 6 PCM isoperated in the self-pumped mode so that the neces-sary thresholding mechanism is provided.

The reference beam used to record a particulartemplate must focus at the plane of the reset SLMafter propagation through the memory crystal.Each of the reference beams, which are angle multi-plexed at the memory crystal, focuses at a differentlocation on the reset SLM plane.

4. Adaptive-Resonance-Theory Memory-ComponentEvaluations

Here we consider implementation details necessaryfor optimal configuration of the BaTiO3 crystal thatserves as the holographic memory element. Some ofthese comments are of general relevance for phase-conjugate holography in neural-net implementations,particularly those involving BaTiO3. They are, how-ever, the result of our own incremental progresstowards implementation, and are not presented asoriginal contributions. (For other information inthis area, we recommend Refs. 16-19.) The centralpurpose of the experiments described below is todetermine the proper geometry and the feasibility ofmultiple-learned-template storage in the device.

A. Geometry and Grating Formation

To demonstrate the complete optical ART system, wefirst need to characterize the performance and todetermine the most effective beam and crystal geome-tries of the BaTiO3 memory crystal. The geometryused for these tests is defined in Fig. 5; k is thegrating spatial wave vector, c is the crystal c axis, a ismeasured inside the crystal, and y and v are mea-sured outside the crystal. All input polarizations areextraordinary (i.e., in the plane of the page). Ourcrystal is approximately 3 mm x 4 mm x 5 mm withthe crystal c axis normal to the 3 mm x 4 mm faces.Because BaTiO3 has an intrinsic phase shift of 900between the applied interference pattern and theresulting photorefractive phase grating, the processof recording holograms leads to two-beam coupling'6

in the crystal.In our application, beams 1 and 2 are the reference

and signal beams incident upon the memory crystal.

Grating strength and holding time both increase withdecreasing y because the diffusion of charges isfrustrated by the larger scale of the resulting grating.We selected y = 14° as a minimum y consistent withpractical mechanical and optical constraints. Thetheory for the photorefractive grating formation' inBaTiO3 predicts that, for a fixed y, the greatestgrating strength should be obtained with a 40°.In our geometry the crystal size and shape and theorientation of the c axis limit the external incidenceangle to roughly the interval 350 < < 145°, so thatrefraction limits the grating angle a to be consider-ably less than 400. In practice, we find that the bestperformance is obtained at 50°. A differentspecification of crystal dimensions to that permits abetter choice of , is clearly recommended for futureresearch in this area.

Grating-formation time and saturation strengthwere characterized by observing the onset and steady-state strengths for the two-beam coupling effect withinput beams of equal irradiance, 25 mW/cm2 at thecrystal. Illumination of the crystal interior by a150-W optical-fiber white-light illuminator was usedto erase stored gratings and to hold off formation ofgratings as a way to gain a starting time for thedetermination of grating formation rates. Figure 6shows the formation and saturation of gratings underthe selected geometry (i.e., with 3 = 500 and y 14°).The data plotted versus time in the figure are twophotodetector output traces representing the powerof beams 1 and 2, as well as a third trace that is thesum of the power of beams 1 and 2. Prior to event 1as shown in Fig. 6, the erasure illuminator was onand both detectors were blocked in order to setzero-signal base lines. At event 1 the detectors wereunblocked with the erasure illuminator left on inorder to prevent formation of the gratings; the powerof the two input beams is matched. At the time ofevent 2 the erasure illuminator is turned off, andformation of the gratings begins. The strength ofthe grating at any time is indicated by the powerimbalance between the initially balanced outputbeams. At steady state the ratio of the power inoutput-beam 2 to the power in output-beam 1 haschanged from unity to 8, which is consistent with asingle-beam scattering efficiency of 37.5%. The de-tector-sum trace remains near 100% throughout the

a)

3,00.

ID0co

80-

Memory Crystal

Fig. 5. Geometry for the two-beam-coupling grating-formationtests.

Event 1Time (seconds)

Fig. 6. Two-beam coupling ( = 500).

10 March 1993 / Vol. 32, No. 8 / APPLIED OPTICS 1403

500

grating formation, indicating that power is beingtransferred between the two beams with negligiblescattering into other directions. The time constantfor the grating formation at this power level is 7 s.

In the ART implementation a number of distinctgratings is formed, each with a signal-beam inputfrom the same (fixed) direction and a reference-beaminput along a distinctly different direction. In orderto ensure that multiple stored holograms have simi-lar formation time constants and decay rates, all thegratings need to be formed with nearly the samegeometry and spatial frequency magnitude Ik .

B. Competition with Self-Phase Conjugation

Since the two-beam coupling theory predicts that, forany angle x, the coupling strength for 1 = 90 - xshould be the same as for 13 = 900 + x, we attemptedto demonstrate the equivalent beam interaction at3 = 130°. The result of this experiment is shown in

Fig. 7. The two-beam coupling proceeds as in Fig. 6for several seconds immediately following event 2(the onset of grating formation). At event 3, how-ever, the interaction abruptly changes from that ofFig. 6, with both beams falling to 20% of the powerlevels observed between events 1 and 2. Concurrentwith event 3, the total power in both output beams(the upper trace) deviates sharply from 100%, alsosettling at 20% in the steady state. This deviationof the upper trace from 100% is indicative that someother process is competing with the two-beam cou-pling effect, stealing energy from the two outputbeams. Indeed, by observing the two input beams inthe retropropagating direction, we confirmed that theinput beams were undergoing phase conjugation.This competing nonlinear effect needs to be mini-mized in our optical ART memory implementation byselecting reference- and signal-beam geometries thatfrustrate the occurrence of the undesirable self-phaseconjugation.

Figure 8 plots the strength of the self-phase-conjugation response to a single input beam of extraor-dinary polarization as a function of the externalbeam-incidence angle 13', as illustrated in Fig. 9.For each input 13' the crystal was translated laterally

3 80%0'.

60%,

a0) 40%

20%qa0.

0%

Event 3

0 E 100Event 1

(1 + 2)

Beam 2200 i Beam 1

200 300 400 500

Time (seconds)

Fig. 7. Two-beam coupling frustrated by self-phase conjugation( = 130°).

16 / \

14

:R 12

10.1T 8

c 642

O0 l { l l l0 30 60 90 120 150 180

Input Beam Angle, 0' (Degrees)

Fig. 8. Input geometries permitting self-phase conjugation.

to find the maximum self-phase-conjugation strength.The plot of Fig. 8 shows that, in order to frustrate theonset of self-phase conjugation, we need 13' < 900.(This is useful not only because it permits analysis ofundesired competition with two-beam coupling, but-because it also provides the range of input geometriesthat permits phase conjugation as desired in PCM1 ofFig. 4.)

Considering the need to avoid self-phase conjuga-tion of the principal beams in the ART memorycrystal, we selected the geometry and beam assign-ments shown in Fig. 10. For the optical ART imple-mentation of the memory crystal there are twooperational modes: storage and search. In the stor-age mode the memory crystal is illuminated by areference beam and a signal beam simultaneously.In the selected geometry ( 50) neither the signalnor the reference beam can self-conjugate during thestorage mode. In the search mode the crystal isilluminated by a signal beam in the absence of anyreference beam. The hologram forms a virtual im-age of any reference beam associated with a near-matching stored template. The reference beam(s)generated in the memory crystal propagate out of thecrystal and through the reset SLM. They are re-flected by PCM1 and propagate back into the crystal,where they scatter from the stored hologram(s) toproduce backward propagating beam(s) along thesignal beam path. For the memory-crystal geometrywe selected and with respect to the self-phase-conjugation geometry illustrated in Fig. 9, the inputsignal beam is along a path with 3' 43, the inputreference beam is along a path with 13' 57°, theretropropagating signal path has angle 1' 137, and

Input Beam

Phase-Conjugate Beam

PCMI

Fig. 9. Self-phase-conjugation geometry.

1404 APPLIED OPTICS / Vol. 32, No. 8 / 10 March 1993

k

Not usedReference

Memory Crystal

Fig. 10. Selected memory-crystal geometry.

the retropropagating reference path has angle 13'1230. In this geometry the retropropagating ver-sions of the signal and reference beams both have 13'angles that may permit self-phase conjugation. Inthe event that self-phase conjugation of the retroprop-agating beams interferes with the optical ART imple-mentation, we attempt to reduce the effect by avariety of means, including translation of the crystalto avoid the total internal reflection at the crystal-airinterfaces that can provide optical feedback requiredfor self-pumped phase conjugation.18

C. Grating Decay Rate

Experiments were performed to determine the rate atwhich gratings, formed in the geometry shown in Fig.5, decay during the process of readout. Figure 11shows the grating scattering efficiency (i.e., the grat-ing strength) as a function of time under four differ-ent illumination conditions. The four curves corre-spond to the decay in the presence of a single readoutbeam of four different powers: 1.7 mW and 600, 170,and 20 jiW; the highest power is equal to the power ineach of the two beams during the grating formationprocess shown in Fig. 6. In the 20-puW case thedecay rate is converging on the intrinsic (zero-illumination) decay rate of the grating. The 1.7-mWcurve indicates an exponential decay time constant of- 175 s. For the grating decay caused by the forma-

tion of other gratings (requiring input of two 1.7-mWbeams) the decay time constant would be - 88 s.

The grating scattering efficiency at zero time foreach curve in Fig. 11 is 3 7.5%. This representsthe maximum (saturation) grating strength achiev-

40%-

35%-

> 30%-C

'3 25%

t, 20%

o 15%

o 1 0%

5% -

0% -

able with our selected geometry, which plays a role indetermining the counterbalancing gain required ofthe phase-conjugating feedback mirror to permitresonance in the optical ART implementation.

D. Storage of Multiple Matched Gratings

Given the grating-formation and grating-decay timeconstants, Mok et al.9 provide the means for determin-ing exposure-time sequences appropriate for the for-mation of multiple holograms with equal strength atthe end of the storage process. We see no reasonwhy LiNbO3 , their choice of crystal, could not also beused in this device. This would provide the largetemplate storage capacity they demonstrated, but atthe price of requiring higher power levels. Usingtheir technique, we analyzed the storage properties ofBaTiO3. Figure 12 shows the maximum matched-grating scattering efficiency that can be achieved as afunction of the total number of gratings stored, whenthis method, with our formation and decay timeconstants of 7 and 88 s, respectively, is used. ForFig. 12 we assume a 37.5% saturation scatteringefficiency (consistent with both the two-beam cou-pling strength at saturation and the grating scatter-ing efficiency measurements), and we assume thatnegligible time passes between formation of the se-quential gratings.

The results in Fig. 12 can be used to determine theminimum gain required of the phase-conjugatingfeedback mirror in the optical ART implementation.For example, the figure shows that, if seven gratingsare to be stored, the maximum grating scatteringefficiency is 20%. The two-pass efficiency is 4%for the memory crystal, which, combined with thedemonstrated 20% reflectivity for PCM1, leads to around-trip efficiency of 0.8%, which must be coun-terbalanced by a gain in PCM2 of - 125.

We are encouraged by the high capacity theoreti-cally possible with volume holography and by theactual results achieved by others in this area ofresearch.9 In the short run we expect that theattainable capacity of the system will be limited bythe number of holograms stored, which should be inthe hundreds, if not thousands, of memories. In thelong run, as the ability to store multiple holograms in

35

5 30.G

.' 25

0.2 10

5

02 3 4 5 6 7 8 9 10 11 12

Number of Matched Gratings

Fig. 12. Maximum scattering efficiency versus the number ofmatched gratings.

10 March 1993 / Vol. 32, No. 8 / APPLIED OPTICS 1405

0 100 200 300 400 500 600 700 800 900 1000

Time (seconds)

Fig. 11. Grating decay rates.

photorefractive crystals improves, we expect to belimited by the reset SLM or by the reference beamsteering mechanism. All of these are generous limi-tations, however, and promise much better storagecapacity than alternative ART neural-network imple-mentations. 2 0-2 4

We recently became aware of Ref. 18, in whichChiou describes a strong anisotropy in cross talk inmultiplexed volume holograms for both self-pumpedphotorefractive PCM's and two-wave mixing at theFourier plane. The increased cross talk reported forangle multiplexing orthogonal to the nominal plane ofincidence has implications on the storage capacity ofthe optical ART neural network. Chiou shows thatthis increased cross talk may be mitigated for certainclasses of input objects by photorefractive mixing atimage planes rather than at Fourier planes. Thisissue requires further study.

5. Future Work

The next stage in our preparations for implementa-tion of the optical ART is the demonstration of aphase-conjugating feedback mirror with gain charac-teristics suitable for operation with the memorycrystal described above. Further experiments willinvolve determining the geometry and the thresholdfor the reset detector and demonstrating the ability todefeat a selected template in favor of another. Thetemplate selection process can then itself be verified,followed by construction of a full prototype.

6. Conclusions

The optical ART unit is capable of processing largepatterns and potentially has a large template capacity.The device's capacity should ultimately approach thecapacity of holographic storage systems, which poten-tially greatly exceed electronic storage capabilities.9"10Furthermore, the device is all-optical in the informa-tion-processing path; the reset detector's electronicsare never used sequentially when an input patternhas already been learned or when it matches anexisting template sufficiently. ART has been shownto be useful with extremely large input fields, such asthose expected in high-resolution images.3 The highstorage density of this device should enable it tooutperform alternative ART implementations,20-24

which do not offer the same capability for processingthe large number of pixels in a high-resolution image.(These implementations are more likely to be usedwith applications in which speed is more importantthan the capability to handle large numbers of pixels.)Finally, we have also shown how the architectureprovides a convenient means for normalizing thememory templates based on template area, an obser-vation that may be applicable to more general classesof holographic information-retrieval systems.

This work was funded by The Boeing Company,Computer Services Research and Technology Group.We appreciate fruitful discussions with R. AaronFalk, Gail Carpenter, Robert J. Marks II, Steve

Rogers, and Jonathan Kane. We also appreciateseveral helpful remarks by the reviewers.

References and Notes1. G. A. Carpenter and S. Grossberg, "A massively parallel

architecture for a self-organizing neural pattern recognitionmachine," Comput. Vis. Graph. Image Process. 37, 54-115(1987).

2. S. Grossberg, "Competitive learning: from interactive activa-tion to adaptive resonance," Cogn. Sci. 11, 23-63 (1987).

3. T. P. Caudell, S. D. G. Smith, G. C. Johnson, and D. C. WunschII,, "An application of neural networks to group technology,"in Applications of Artificial Neural Networks II, S. K. Rogers,ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1469, 612-621(1991).

4. A. M. Waxman, M. Seibert, R. Cunningham, and J. Wu,"Neural analog diffusion-enhancement layer and spatio-temporal grouping in early vision," in Advances in NeuralInformation Processing Systems I, David Touretzky, ed. (Kauf-mann, New York, 1989), pp. 289-296.

5. G. Carpenter, S. Grossberg, and J. Marshall, "ARTMAP: aself-organizing neural network architecture for fast super-vised learning and pattern recognition," in Proceedings of theInternational Joint Conference on Neural Networks, (Instituteof Electrical and Electronics Engineers, New York, 1991), Vol.1, pp. 863-868.

6. T. W. Ryan and C. L. Winter, "Variations on adaptive reso-nance," in Proceedings of the First International Conferenceon Neural Networks, M. Caudill and C. Butler, eds. (Instituteof Electrical and Electronics Engineers, New York, 1987), pp.767-776.

7. B. Moore, "ART1 and pattern clustering," in Proceedings ofthe 1988 Connectionist Summer School at Carnegie-MellonUniversity, C. Touretsky and G. Hinton, eds. (Kaufmann, NewYork, 1989).

8. B. H. Soffer, G. J. Dunning, Y. Owechko, and E. Marom,"Associative holographic memory with feedback using phase-conjugate mirrors," Opt. Lett. 11, 118-120 (1986).

9. F. H. Mok, M. C. Tackitt, and H. M. Stoll, "Storage of 500high-resolution holograms in a LiNbO3 crystal," Opt. Lett. 16,605-607 (1991).

10. D. Psaltis and Y. S. Abu-Mostafa, "Theoretical investigation ofparallelism and connectivity in optical computers," in OpticalProcessing: Interim Report, Rep. UDR-TR-85-148 (Universi-ty of Dayton Research Institute, Dayton, Ohio, 1985), pp.9.1-9.13.

11. A. Yariv, Optical Electronics (Holt, Rinehart & Winston, NewYork, 1985), pp. 499-525.

12. J. Feinberg, "Optical phase conjugation in photorefractivematerials," in Optical Phase Conjugation, R. A. Fisher, ed.(Academic, Orlando, Fla., 1983), p. 421. In our ART imple-mentation, phase adjustment and modulation are performedon the reading beam of the photorefractive four-wave PCM.In this way, one can affect the phase of the phase-conjugatebeam without directly modifying the phase of the photorefrac-tive grating that is responsible for the phase conjugation.

13. This corresponds to the fast-learning mode of ART, as de-scribed in Ref. 1. Slow learning in ART has also beenanalyzed by G. L. Heileman, M. Georgiopoulos, and C. Abdal-lah, in "A dynamical adaptive resonance architecture," IEEETrans. Neural Net. (to be published). This slow learningcould be implemented in our device by operating always at thelower power level.

14. D. Psaltis, "Optoelectronic implementations of neural net-works," in Tutorial Notes, International Joint Conference onNeural Networks (Institute of Electrical and Electronics Engi-neers, New York, 1990), p. 46.

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15. D. Psaltis, D. Brady, and K. Wagner, "Adaptive optical net-works using photorefractive crystals," Appl. Opt. 27, 1752-1759 (1988).

16. J. Feinberg, "Optical phase conjugation in photorefractivematerials," in Optical Phase Conjugation, R. A. Fisher, ed.(Academic, Orlando, Fla., 1983), pp. 417-443.

17. M. Cronin-Golomb, B. Fischer, J. 0. White, and A. Yariv,"Theory and applications of four-wave mixing in photorefrac-tive media," IEEE J. Quantum Electron. QE-20 (January1984).

18. A. Chiou, "Anisotropic cross talk in an optical interconnectionby using a self-pumped phase-conjugate mirror at the Fourierplane," Opt. Lett. 17, 1018-1020 (1992).

19. B. Fischer, J. 0. White, M. Cronin-Golomb, and A. Yariv,"Nonlinear vectorial two-beam coupling and forward four-wave mixing in photorefractive materials," Opt. Lett. 11,239-241 (1986).

20. A. Rao, "VLSI implementation of neural classifiers," NeuralComputat. 2, 35-43 (1989).

21. D. C. Wunsch II, T. P. Caudell, D. Capps, and R. A. Falk, "Anoptoelectronic adaptive resonance unit," in Proceedings of theInternational Joint Conference on Neural Networks (Instituteof Electrical and Electronics Engineers, New York, 1991), Vol.I, pp. 541-549.

22. T. P. Caudell and D. C. Wunsch II, "A hybrid optoelectronicART1 neural processor," in Proceedings of the InternationalJoint Conference on Neural Networks (Institute of Electricaland Electronics Engineers, New York, 1991), Vol. II, p. 926.

23. J. Kane andM. Paquin, "POPART: partial optical implemen-tation of adaptive resonance theory 2," IEEE Trans. NeuralNet. (to be published).

24. S. W. Tsay and R. W. Newcomb, "VLSI implementation ofARTi memories," IEEE Trans. Neural Net. 2,214-221 (1991).

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