326 OPTICS LETTERS / Vol. 14, No. 6 / March 15, 1989

Continuous coherent erasure of dynamic holographicinterconnects in photorefractive crystals

Abdellatif Marrakchi

Bellcore, 331 Newman Springs Road, Red Bank, New Jersey 07701-7020

Received August 19, 1988; accepted December 20, 1988

The holographic double-exposure technique is applied to elementary gratings in photorefractive crystals. Theresult is an interconnection strength that is a continuous function of the phase shift between the two gratings. Infan-in and fan-out situations, where multiple phase gratings are required, it is possible to alter each interconnectionseparately without affecting the others.

The pioneering research of Van Heerden on holo-graphic associative memories and related storage ca-pacity1 has been an inspiration for the renewed inter-est in optical implementations of neural networks.More recently, with the development of powerfullearning algorithms2 and the possibility of real-timestorage and retrieval of information in photorefractivematerials,3 optical neurallike processors that can learnfrom experience are becoming an attractive alterna-tive for a whole class of problems that are not easilytractable with conventional serial computers. Someearly attempts to demonstrate the learning capabilityof holographic systems are under way, including a two-dimensional pattern classifier.4

At a more elementary level holographic gratings canbe used to implement interconnection links betweenindividual processing elements (or neurons) of twodistinct planes in a multilayered structure. The prin-cipal advantage of optics in this case is the large num-ber of independent interconnections that one can setup by recording multiple gratings.5 -7 One issue thatimpacts directly on the development of learning ma-chines based on real-time holography is the capabilityof continuously modifying a given interconnectionstrength (or weight) without affecting the others whenthe gratings share the same volume in the photorefrac-tive crystal.

Gabor et al. demonstrated that the superposition oftwo 7r-phase-shifted holograms can be used to performcoherent subtraction (or addition) of two-dimensionaltransparencies.8 This technique has also been suc-cessfully applied to the partial erasure of informationcontained in a transparency in the presence of otherholograms that are angularly multiplexed outside theBragg selectivity bandwidth of each other.9 In thisLetter we extend the principle of coherent erasure bydouble exposure of phase-shifted holograms to thecase of elementary gratings that implement real-timeoptical interconnections in photorefractive materials.The effect of continuously varying the phase shift onthe diffraction efficiency is quantified and shown to beapplicable to the simulation of synapses with pro-grammable variable weights, as would be required in alearning neural network. Issues that relate to fan-inand fan-out capabilities, which ultimately determine

the achievable level of parallelism and cascadability insuch processing architectures, are also addressed.

The proposed scheme of a holographic interconnectsystem is shown in Fig. 1. The purpose is to connect amatrix of sources in plane PIN with a matrix of detec-tors in plane POUT with fan-in and fan-out capability.To write the respective gratings that will form theoptical links, matrices of mutually coherent controlsources are needed in planes PC1 and PC2. All thebeams from PC1 represent the object wave in the tradi-tional sense of holography, and each beam from PC2represents the reference wave. Hence in this coherentsystem each interconnection set, defined by a specificconfiguration of Pc, and one reference beam from PC2,has to be recorded separately from all the others inorder to generate multiple optical links without appre-ciable cross talk. As an example, the dark areas in Fig.1 show a set of interconnections. Two gratings arewritten in the bulk of the photorefractive material byinterfering two sources from Pc, with one from PC2.The beam incident upon the crystal from PIN satisfiesthe Bragg condition and diffracts simultaneously fromthese two gratings, thereby connecting itself with twodifferent locations in the output plane. This situationwould correspond to a fan-out configuration.

The amplitude of the diffracted optical field is alinear function of the writing intensity modulationdepth to the first-order approximation. Consequent-ly the strength of a holographic interconnection couldbe controlled with the intensity of the recording beams

g S~~~~~~~~~~~~IN

P L PHOTO~~REFRACTIVEPC2 CRYSTAL

Fig. 1. Schematic of a holographic interconnect system.The photorefractive crystal is placed in the Fourier plane oflenses L. Beams from Pci and PC2 write the interconnectiongratings that link sources from PIN to detectors in POUT-The beam splitter BS separates the recording from the dif-fracted beams.

0146-9592/89/060326-03$2.00/0 © 1989 Optical Society of America

March 15, 1989 / Vol. 14, No. 6 / OPTICS LETTERS 327

by placing amplitude modulators in the plane PC,.The main drawback of such a technique, however,derives from the dependence of the photorefractiveresponse time on the modulation depth itself. Henceone would need a fairly accurate tracking and controlof each grating exposure for any set of interconnec-tions, leading to a complex and impractical system.

The double-exposure technique is a two-stage pro-cess in which a phase shift is induced on one of thecontrol beams between recordings. The conventionalways of inducing this phase shift are either to electro-optically phase modulate the beam or to reflect it off apiezoelectrically driven mirror. In the following ex-periments we use a mirror mounted on a stack ofpiezoelectric ceramics. In the scheme proposed inFig. 1 a matrix of such modulators would be placed inthe control plane Pc,.

When two coherent beams are allowed to interferewithin the bulk of a photorefractive material the resul-tant intensity distribution is mapped onto a corre-sponding refractive index modulation owing to chargegeneration, transport, and trapping. If the writingbeams are plane waves and the recording is assumed tobe in the linear regime, the resultant phase grating isdescribed by

An(x) = An, cos(KGx), (1)

in which An, is the saturation refractive index modula-tion and KG is the grating wave vector. In the double-exposure technique a second grating that is phaseshifted with respect to the first grating is recorded inthe same material and at the same spatial frequency.Hence if we assume an identical modulation at satura-tion, the composite grating is described by

An(x) = AnS[cos(KGx) + cos(KGX + 4)], (2)

which can be rewritten as

An(x) = 2An, cos(4/2)cos(KGx + 4/2), (3)

in which '1 is the phase shift between the two gratings.An analysis of Eq. (3) reveals that the spatial frequen-cy of the composite grating is not altered, that thediffracted beam will suffer a phase variation, and,most importantly, that the overall modulation depthof the grating is multiplied by cos(4/2). Consequent-ly, since the diffraction efficiency is proportional tothe square of this modulation depth (to the first-orderapproximation), the diffracted intensity will vary ascos2(b/2), thereby yielding a variable interconnectionbetween the readout source and the detector plane.

In our experiment the recording medium is a singlecrystal of photorefractive bismuth silicon oxide(BiU2 SiO2 0, or BSO). A holographic grating with afringe spacing of 1.8 ,m is formed in the bulk of thismaterial by interfering two plane waves originatingfrom an argon laser operated at a wavelength of 514nm. One of the plane waves is reflected off a piezomirror driven by a square-wave voltage, with a longperiod compared with the material's response time formeasurement purposes. The phase difference be-tween the two recorded gratings is dictated by thevoltage-induced displacement of the piezo mirror.The composite grating is read out in real time with a

He-Ne laser incident at the Bragg angle, although forphase matching over a wider spatial bandwidth itwould be preferable to use the same wavelength as forwriting. In this particular experiment the photore-fractive recording is performed in the diffusion re-gime, i.e., no external field is applied to the crystal.

Figure 2 shows oscilloscope traces of the diffractedintensity () and the relative phase () between thetwo gratings for three sets of experimental values. Asexpected, when the second grating is superimposedonto the first one the diffracted intensity decreases byan amount that is related to the magnitude of thephase shift between the two recordings. In Figs. 2(a)to 2(c) the applied voltage to the piezo mirror inducesa shift that is less than, equal to, and larger than r,respectively. Consequently the diffracted intensityat the minimum point decreases to a value that isequal to zero for 4I = 7r [Fig. 2(b)] before increasingagain. Since the writing beams are continuously on,the diffraction efficiency increases to its saturationvalue after erasure. Even though there are two com-peting mechanisms, i.e., recording of the second grat-ing and erasure of the first grating, it appears that thecoherent erasure process has a response time that ismuch faster than that of the space charge field build-up.10 Intuitively this could be explained by the moreeffective erasure with an adequate light distribution.Under normal circumstances grating decay is inducedby uniform illumination throughout the whole crystal.However, for an identical total intensity, if we choose adistribution that perfectly matches the phase gratingbut is spatially shifted the trapped charges are in fact

DIFFRACTIONEFFICIENCY

PHASE (b e 0-A

(a)

(b)

03 > 2 > |

712 13 ;Il

(c)

Fig. 2. Oscilloscope traces of the diffracted intensity andthe phase shift between the recorded gratings. The magni-tude of the shift increases from (a) to (c).

328 OPTICS LETTERS / Vol. 14, No. 6 / March 15,1989

1. 0

0. 8

~0.6 4

0. 2

0. 00 100 200 300 400

Phas. Shift

Fig. 3. Normalized diffraction efficiency as a function ofthe phase shift in units of degrees between the two gratingsrecorded with a double-exposure technique.

T1-

112

Fig. 4. Oscilloscope traces of the diffracted intensity in afan-out situation and of the phase shift of one of the writingbeams.

more effectively reexcited during erasure, owing inpart to a larger local optical power density. In addi-tion, contrary to a technique that relies on varying theintensity of the writing beams, the erasure time con-stant is independent of the phase shift between thetwo gratings, leading to an identical exposure time forall recordings and hence to a simpler system.

The confirmation of the cos2() behavior of the dif-fracted intensity is illustrated in Fig. 3. Here thenormalized diffraction efficiency after erasure, i.e., theminimum intensity in Fig. 2, is plotted as a function ofthe phase shift between the two gratings. Theory andexperiment are in good agreement. A variable andcontrollable interconnection strength is thus possiblewith this technique. Obviously linearity is still anissue, although this problem can be alleviated by hav-ing an adequate nonlinear relationship between theapplied voltage and the phase shift induced by themodulator.

In a practical interconnection scheme many gratingswill share the same volume in the photorefractive crys-tal. It is thus necessary to find out whether erasure ofone grating will affect the other gratings. In a fan-outsituation one source from plane PIN is connected toseveral detectors in plane POUT; in a fan-in situationseveral sources are connected to one detector. In bothcases multiple phase gratings have to be recorded inthe material. In one experiment illustrating fan out,two gratings with an angular separation of 4 mrad arerecorded in the BSO crystal in the diffusion regime.

Since the Bragg selectivity for the He-Ne beam is notcritical with this small angular separation, two wavesare diffracted. During the recording one of the writ-ing beams is phase shifted while the efficiency is con-tinuously monitored. The angle of incidence of thereadout beam is such that an equal intensity is dif-fracted along both directions. Figure 4 shows oscillo-scope traces of the diffracted intensity in each beamand the relative phase shift. In this particular experi-ment erasure of one grating does not affect the othergrating. For larger angular separations between thewriting beams it has been reported,9 and we have con-firmed, that the different gratings do not affect eachother either. However, whenever several inputsources have to be directed to the same output, coher-ent diffraction might become a problem if the relativephases of the different diffracted beams affect theoverall intensity through either constructive or de-structive interference."1 One way to alleviate thisproblem would be to assign known randomly distrib-uted phases to the interconnection gratings.

In summary, it is shown that the double-exposuretechnique with a variable phase shift between the tworecorded gratings yields a continuously graded inter-connection strength between two spatially separatedplanes. The nonlinear relationship between theweight of this optical link and the phase shift is de-scribed by a cos2(Ql) function, as experimentally veri-fied. When several holographic gratings are recordedto simulate fan in and fan out, it is possible to modifyone interconnection weight without significantly af-fecting the other gratings. Combined with the largestorage capacity available with a holographic record-ing, this double-exposure technique could be suitablefor the optical implementation of learning neural net-works with continuously variable weights.

I acknowledge the critical review of this manuscriptby S. F. Habiby, G. A. Hayward, W. M. Hubbard, S.Jain, E. G. Paek, and A. C. VonLehmen.

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