Space-variant holographic optical elements indichromated gelatin

Brian Robertson, Edward J. Restall, Mohammad R. Taghizadeh, and Andrew C. Walker

Spatially variant holographic optical elements in dichromated gelatin can implement many complicatedinterconnection patterns and networks in a straightforward manner. We demonstrate some prototype, highefficiency, point to point, off-axis interconnects for replay at the recording wavelength (514.5 nm). Designconsiderations and potential uses in the fields of optical computing and optical communications are exam-ined. Finally, we suggest a modified approach which allows for on-axis operation and replay in the nearinfrared.

I. IntroductionIn recent years there has been increasing interest in

free-space optical interconnection networks suitablefor use in communication and computing systems.These include several regular multistage architec-tures, for example, the perfect shuffle, half crossover,and Banyan and butterfly networks, all of which re-quire the spatial permutation of an array of lightbeams. To realize such space-variant networks(SVNETs), a variety of solutions relying on eitherclassical optics' or computer generated holograms2

(CGHs) have been proposed; i.e., prismatic retroreflec-tors can be used to make a half crossover network,3while a perfect shuffle may be implemented with sim-ple prisms and lenses.1 4 5 In addition these intercon-nection systems can be fabricated using a matched pairof space-variant holographic optical elements, whichessentially consists of two arrays of individual holo-graphic elements that can redirect collimated beams asrequired. Such an interconnect which performs a per-fect shuffle has recently been demonstrated by Hau-mann et al. at Erlangen.6

We describe a technique for fabricating multifacetholographic elements which provides a more compactand flexible solution than other current methods.These space-variant interconnects (SVIs) are similar

The authors are with Heriot-Watt University, Physics Depart-ment, Holography Group, Riccarton, Edinburgh EH14 4AS, Scot-land, U.K.

Received 10 September 1990.0003-6935/91/172368-08$05.00/0.© 1991 Optical Society of America.

to those proposed by Haumann et al. with the addi-tional benefit of providing point to point interconnec-tion by including focusing power in the elements.They can collimate, redirect, and focus down thebeams according to the required interconnect pattern.Furthermore, inclusion of an aperture imaging systemin the recording setup allows for the construction ofshort focal length elements with small facet sizes(<200 Am). This makes them directly compatiblewith current demonstration optical circuits.7

The advantages, requirements, and specific opticalsystem applications of SVIs are discussed in this paperwith particular emphasis on the role of the perfectshuffle. A number of prototype interconnects fabri-cated in dichromated gelatin (DCG) are described.These include several stages of a half crossover net-work, Banyan network, and a 10 X 10 stacked perfectshuffle.

II. Design ConsiderationsBefore discussing the SVIs it is useful to consider

briefly the specifications which a particular circuit, orsystem, impose on any interconnect intended for use inoptical communication or computing. Specifically,account must be taken of such parameters as fanout/fanin, efficiency, interconnect packing density, andwavelength of operation when designing any particularwiring scheme. It has been found that in general thetrade-off between these various factors must be as-sessed before a practical solution can be determined.We shall concentrate on holographic optical elements(HOEs) recorded in DCG, although most of the follow-ing comments equally apply to all recording materials.

Our first consideration relates to the type of opticalswitch, modulator, or detector with which the systemoperates. In general, to minimize both the optical

2368 APPLIED OPTICS / Vol. 30, No. 17 / 10 June 1991

power requirements and the response times these ac-tive elements must be as small as possible, which re-quires the use of tightly focused beams. In terms ofinterconnect design, any interconnection schememust, therefore, allow for point source to point sourceoperation as close to the diffraction limit as possible.In addition the type of active component employed inthe system will also determine the wavelength of oper-ation, which will most probably be in the near infrareddue to the availability of cheap and efficient lightsources in this region, i.e., semiconductor and solidstate lasers. Thus any space-variant HOE recordingscheme must be capable of producing good qualitycomponents for these wavelengths. Furthermore theymust be as efficient as possible to maximize the usageof laser power. This is probably one of the limitingfactors when scaling up an optical computing or com-munication switching system to any practical size, par-ticularly considering the low power limits of present,commercially available, single-mode diode lasers.Volume phase holographic materials, such as DCG,are, therefore, attractive media for recording SVIs giv-en that diffraction efficiencies of >90% and an absorp-tion of only a few percent are obtainable.

Another important parameter which must be takeninto account when designing any SVI is the intercon-nect packing density. This will be a major limitationin a large number of optical systems, particularly whenusing compact devices such as S-SEED, where spac-ings may be of the order of 10 ,um.8 By introducingimaging optics into the system, however, it is possibleto increase the channel separation up to the pointwhere recording individual facets would be feasible,i.e., 100-200 ,m in the case of DCG.

System architecture must also be considered whendesigning any interconnect scheme as it determinesboth the fanout/fanin requirements of the intercon-nect and also the flexibility required in the recordingsetup. First, we assume that a regular space-variantinterconnect must be designed to link up two P X Qarrays of optical switches with a fanin/fanout of F.This can be either in a stacked setup (P rows of Q bits)or in a folded interconnect (a 1 X PQ-bit input andoutput). The fanout/fanin requirements will deter-mine the overall efficiency of a DCG HOE as, general-ly, there exists a trade-off between replay fidelity andefficiency when F >1.9 This is characterized by a

-variation in the weightings of the fanned-out beamsand the appearance of higher diffraction orders.10 Inmany of the regular SVIs, however, only a one to onemapping or at most a fan-out/fan-in of two is required,both of which can be realized efficiently using a combi-nation of DCG SVIs and, if necessary, polarizing beamsplitters to provide lossless fanout.3 Using anythingother than polarization fanin/fanout will have addi-tional problems related to the interference effectswhich arise when several coherent beams overlap.8

111. Architectural ConsiderationsSpatially variant interconnects are most likely to

find their greatest use in communications networks

1El

2 3 4 5 6 7 8 9 10F] L1 I1 D1 C

1 6 2 7 3 8 4 9 5 10Fig. 1. Ten-facet 1-D illustration of a perfect shuffle.

and parallel processor architectures as part of the opti-cal link between either all-optical or hybrid optoelec-tronic switching devices.

Taking, for example, the perfect shuffle, it is appar-ent that in this case free space optical interconnectsoffer a significant advantage over electronic wiringsystems. The example in Fig. 1 is a simple 10-facet 1-D interconnect which could readily be fabricated withelectronics, albeit being susceptible to a certainamount of electromagnetic crosstalk. If, however, wedesire a 2-D interconnect to link logic arrays, the elec-tronic spaghetti required to perform such a link makesall-optical interconnects a far more desirable alterna-tive. Thus if, say, a 64 by 64 2-D interconnect isrequired, the electrical wiring becomes unmanageable.In addition to solving this problem, simple 2-D holo-graphic SVIs can provide wideband communicationpathways with very low time skew and negligible cros-stalk.

SV communications nets can incorporate many dif-ferent SVIs (usually separated by logic planes) toachieve a desired linkage. Any multiple of perfectshuffles, Banyans, half crossovers, butterflys, or ex-changes could be implemented as necessary. Mostnetworks, however, have an isomorphic perfect shuffleequivalent. Figure 2 shows how the eight-facet Ban-yan network achieves the same connectivity as a per-fect shuffle, as does any crossover net. Furthermorethese isomorphs introduce no additional componentcounts or latency into the nets.

Taking Fig. 2(a) as an example, the different linetypes emanating from each facet illustrate a fanout oftwo, most efficiently achieved by polarization division.The introduction of fanout into the perfect shuffle isnot strictly essential for this particular interconnect,but it has the advantage of avoiding a compare ex-change function at every logic plane (i.e., betweenneighboring facets/pixels) which would otherwise berequired to achieve full connectivity in log2N stages ofthe network (where N is the number of facets). Eachlogic plane in the network [Fig. 2(a)] serves two func-tions. The first is regenerative, so that each fanincomponent is fanned out by two and relayed so that

10 June 1991 / Vol. 30, No. 17 / APPLIED OPTICS 2369

Logic Iplanes.

a). Perfect Shuffle. b). Banyan.Fig. 2. Full interconnection of eight facets using two isomorphicnetworks: (a) the perfect shuffle and (b) the Banyan network. Ateach stage of connection, each facet has a fan-in and fan-out of two.

they are presented to the next interconnection stage assignals of the correct magnitude. This also limits theaccumulation of wavefront abberation, signal skew,and other desynchronizing effects, thereby improvingthe cascadability of the system. The second functionis to provide control: each plane having the potentialto be programmed to perform a simple Boolean func-tion on the fanned-in beams, thus providing some ex-ternally specified routing through the network. Fullconnectivity or broadcast capability is achieved by anOR function at each plane. The use of polarizationfanout permits a simple and efficient implementationof such a network, consisting of a beam splitting stage,two coupled SVIs (one performing the interconnect ofthe first line type and one performing the second), anda beam combining stage.

Obviously a Banyan or crossover SVNET wouldhave an advantage over its perfect shuffle (PS) iso-morph because the second SVI in each stage is simply aone to one telecentric imaging interconnect. There is,however, one important distinction to be made: a PSnet simply replicates itself at each stage (unlike othernets), and so it can be folded in on itself to form aperfect shuffling machine (Fig. 3). This reduces thecomponent count and greatly simplifies interconnecttolerancing because only one interconnect is requiredno matter the number of facets. The additional over-head is the extra clocking control required to initiatecommunications across the net and to extract the datafrom the net after log2N iterations. The PS machinefinds its true niche within parallel processing ma-chines.

Novel electronic parallel processing architecturessuch as transputer based systems or data flow ma-chines' 2"3 are complex in computational processorability and low in processor connectivity. An alterna-tive is to have simpler processing capability but a farmore complex processor connectivity, e.g., neural net-

Fig. 3. Schematic illustration of a perfect shuffling machine. Dataare cloced through the interconnection machine (via lock-and-clocklogic planes-not shown) the required number of times to imple-

ment an algorithm or network scheme.

works or a RAM machine 4 (powerful Turing analog).As we outlined at the start of this section, a high degreeof connectivity is very difficult to achieve in electronicsbut not so with optical interconnects. Thus opticalparallel processors with simple processing ability butcomplex connectivity are becoming an increasingly in-teresting alternative to their electronic counterparts.

SVIs can play an important role in such processors,achieving the high degree of interconnection required(as already discussed, networks such as those in Fig. 2can provide full connectivity). Indeed there are manyexamples of computationally intensive algorithmswith spatially variant operations at their heart, such asbitonic merges and sorts, odd-even merging, polyno-mial evaluation, fast Fourier transform, and matrixtranspose.'5 Most of these are based on a perfectshuffle and would be very efficient if used in suchoptical parallel processors and can be adapted for oth-er interconnection patterns. The PS machine of Fig. 3could find its way into optical programmable logicarrays (PLAs) 16 or other novel optical architectures asan integral part of processors such as Hypercube basedconnection machine 7,18 or an optical version of a cellu-lar logic image processor (CLIP).19 Such an opticalCLIP is currently being developed in our departmentat Heriot-Watt.7 As its name suggests, it is a highlyparallel machine for processing images or performingalgorithms which are based on image manipulation.In the first case the architecture does not permit theprocessor to perform arithmetic as such, but with theaddition of a module containing a PS machine theconnectivity (between bits) required to carry out fullarithmetic is achieved. In addition to this arithmeticcapability, the PS machine would enable the opticalCLIP to perform functions which are difficult and slowto implement in electronics; such functions as image

2370 APPLIED OPTICS / Vol. 30, No. 17 / 10 June 1991

Inputarray.

Outputarray.

H2 P2Fig. 4. Schematic showing the implementation of a point to pointperfect shuffle off-axis interconnect. Plane P1 is the input array, P2is the output (interconnected array), HI is a perfect shuffling

SVHOE, and H2 is an inverse perfect shuffling SVHOE.

thresholding and arbitrary image shifting would bemade extremely efficient.

Whatever the architecture and whether the desiredapplication is optical computing or communicationsswitching networks it is clear that high quality spatial-ly variant optical interconnects will have an importantrole to play.

A. Recording of SVIs

Figure 4 illustrates the basic arrangement of a two-element SVI. The input signals (e.g., from an array ofoptical switches) are incident on the first space-varianthologram Hi which consists of an array of holographiclenses. Thus, if Hi is properly aligned, one focallength from the input plane P1, each of the signalbeams will become collimated. As the lenses haveslightly different reconstruction angles the collimatedbeams cross over to produce the required spatial per-mutation a distance L from Hi where a second mirrorimage or inverse function SVI H2 is placed. This will

produce concurrent spatial focusing of the beams atthe output plane P2.

Although it is possible to fabricate a single SVIwhich can achieve this spatial permutation, the beamswill arrive at the next logic plane with slightly differentangles. A two-element interconnect, on the otherhand, allows for all the beams to come out along paral-lel axes. Such a scheme is, therefore, capable of beingused with a telecentric imaging setup, leading to easieralignment and better performance of the overall opti-cal system. It is also more compatible with anglesensitive switching devices, such as those based onFabry-Perot etalons, waveguide structures, and fibers.

To minimize crosstalk effects we must first ensurethat the input beams do not overfill the aperture of anindividual pixel. Second, it is necessary to take intoaccount the diffraction of the beams between HI andH2. This will be helped if the input corresponds to anarray of Gaussian beams, for example, as generated bya Dammann grating.2021 Section V will cover morefully the limitations that this places on the design of aSVI.

Another factor which must be to taken into consid-eration is the polarization dependence of the holo-grams. This mainly arises from the off-axis geometryintroducing different Fresnel transmission coeffi-cients for each polarization. (The polarization depen-dence of the grating is by comparison negligible. 10)Thus, if the replay angle is 300, the transmission coeffi-cients of the s- and p-polarization components will be94.2 and 97.5%, respectively. One solution to compen-sate for these differences is to use suitable opticalcoatings. Whether these polarization differencescause any difficulties depends on the actual systemtolerances. However, we show in Sec. VI how to avoidsuch problems by going to an on-axis system.

B. Recording SchemeA recording setup capable of accurately producing

the type of space-variant interconnects illustrated in

0HE

_-* fh -

Fig. 5. Recording geometry required to fabricatethe DCG holographic interconnects. M1,2,3, mir-rors; LI, FO,R, lenses; A1,2, apertures; BIS, beam

splitter; and H, holographic plane.

10 June 1991 / Vol. 30, No. 17 / APPLIED OPTICS 2371

f I / Aperture (A2).Fig. 6. Effect of displacing a collimating lens F0 a distance p.

Fig. 4 is shown in Fig. 5. Essentially it is identical tothe step and repeat exposure systems previously usedto record high quality lenslet arrays,9 with the excep-tion that the object arm collimating lens F0 is mountedon an x/y stepper motor. The reference beam is de-rived from a spatially filtered and collimated planewave. Lens Li produces a point source a distance fh infront of the holographic plane H, while aperture Al ispositioned so as to be precisely imaged at H. Similarlythe object arm consists of a spatially filtered and colli-mated beam which illuminates the aperture A2. A 4-Flens system then images this aperture at the holo-graphic plane, accurately overlapping with the objectbeam. Each facet of the SVI, therefore, has well de-fined edges, allowing the maximum possible intercon-nect density to be achieved.

A variation in the replay angle (AO) is obtained bytranslating collimating lens F0 a distance p normal tothe object beam optic axis. This causes a slight angu-lar displacement of the collimated beam with respectto the aperture (Fig. 6), which is related to the lensmovement by the following relations:

tan(AO) = p/fo, (1)

where fo is the focal length of Fo. If a precise 4-Fimaging system is used, the wavefront present at A2will be reproduced at the holographic plane. Thus atranslation of Fo will cause a change in the object beamangle and hence in the replay angle of the reconstruct-ed wavefront. If there is no translation, the HOEfacet will reconstruct at the standard replay angle 0.To relate this to an actual spatial permutation, consid-er Fig. 7. If the two interconnecting HOEs are sepa-rated by a distance L and each facet has dimensions ofd by d, to link facet n to facet n + m requires an angulardisplacement of 0 + Alm, which can be derived usingthe relation

tan(O + AOm) = h+md (2)L

Substituting this into Eq. (1), we find that the distancewe are required to translate the collimating lens isgiven by

Pm = f tan [tan(h +md) m 0]. (3)

Thus to record a SVI with this system we simplytranslate F0 to give the required object beam angle and

Fig. 7. Schematic of the interconnection angle AO required to con-nect facet n to facet n + m in an interconnection distance L.

expose a facet. By successively stepping the holo-graphic plate vertically and/or horizontally and re-peating the process, a complex 2-D space-variant in-terconnect can be realized. The advantage of thistechnique lies in the accuracy to which apertures A2and Al can be imaged and overlapped at H. Problemscan, however, arise when trying to record large replayangle variations. First, the maximum variation in re-cording angle Atmax, which can be realized during fab-rication, is limited by the numerical aperture of therecording optics. Second, as larger angles are ap-proached, aberrations become more significant.These two factors impose an upper limit on AOmax ofonly a few degrees. This in turn restricts the mini-mum allowable hologram separation L for any particu-lar SVI scheme.

IV. ResultsUsing such an experimental setup, we have recorded

many prototype SVIs, including several stages of a halfcrossover network, a Banyan network, and a 10 by 10perfect shuffle interconnect. All holograms were re-corded and replayed at 514.5 nm using DCG derivedfrom Kodak 649K plates. Preprocessing and develop-ment techniques were based on those of Chang andLeonard. 2 2

Figure 8 shows the masked input array to a 10 by 10stacked inverse perfect shuffle and the resulting out-put of the interconnect. In this particular case thefacet size was 500 /Am, the standard replay angle 0 was300, the interconnection distance was 5 cm, and theSVI replay focal length fh was 2 cm at 514.5 nm. Theefficiency of each SVI was measured to be 90%, givingthe overall interconnect an 81% efficiency. As can beseen from the figure, there is a slight positioning error.This is principally due to inaccuracies in the position-ing of the holographic plate by the stepper motor sys-tem and can be easily avoided by using a better transla-tion stage. An important consideration when makinginterconnects in this way is that the beam in the objectarm of Fig. 5 must be greatly expanded to use only theuniform part of an expanded Gaussian beam. If this is

2372 APPLIED OPTICS / Vol. 30, No. 17 / 10 June 1991

a L01

n-1n

n+1

N

a). Input Image. b). Output Image.

not the case, we can see from Fig. 6 that the amplitudeprofile across the aperture will vary with AO. This inturn introduces a varying K ratio (irradiance ratiobetween reference and object arms) during the holo-graphic recording. This is undesirable because, as-suming a limited available index modulation, the re-duced modulation depth will limit the maximumachievable efficiency.

V. Scaling ConsiderationsThere are several important issues which must be

considered when attempting to scale such an intercon-nect to smaller dimensions and different operatingwavelengths, including diffractive effects, which canlead to undesirable crosstalk, wavelength-shift-in-duced aberrations, and alignment tolerances. Thesecan lead to several restrictions being imposed on theinterconnect beyond which either fabrication or opera-tion is impractical.

The first restriction relates to the loss of powercaused by overfilling each input pixel. First, assumethat all the beams incident on H1 (Fig. 4) have thesame /e 2 Gaussian spot diameter of OH1. As the in-tensity profile of a Gaussian beam extends to infinity, afraction of the incident power will spill over into adja-cent pixels. This power will not necessarily be dif-fracted along the interconnect path if the Bragg condi-tion is not correctly satisfied. Instead it remains in thezeroth order, missing the second SVI. While this re-duces the possibility of any crosstalk occurring frompixel overfill, the spot size should nonetheless be keptwell within the dimensions of the pixel to optimizeoverall efficiency. A good restriction to follow is thatWH1 < d/3. Taking into account the square packingdensity of the SVI this corresponds theoretically toover 99% of the input power being diffracted into thecorrect channel.

The effect of diffraction while passing between H1and H2 must also be taken into account when design-ing any SVI as it can fundamentally limit the perfor-mance of the interconnect. In particular, when deal-ing with high packing densities and large hologramseparations, the array of beams at H2 can easily overfillthe pixels, again giving rise to possible crosstalk prob-lems. As long as wH1 < d/3, and no aperturing of the

Fig. 8. (a) The input mask to the inverse perfectshuffle interconnect and (b) the output from the

interconnect.

input Gaussian beam occurs, paraxial ray analysis andGaussian beam propagation theory can be used to pre-dict these diffractive effects23 and hence minimizethem through proper SVI design. Obviously the opti-mum solution corresponds to one in which a beamwaist is relayed halfway between the two space-variantHOEs. The beam radius at H2 WH2 is then equal toWH1. Theoretical prediction for the position of theinput array which achieves this geometry requires anaccurate knowledge of the lenslets focal lengthf. Thismay differ slightly from fh due to the nature of therecording beams, i.e., f = fh only if on recording aGaussian object beam waist sits fh in front of the holo-gram and a Gaussian reference beam waist fh behindthe holographic plane. This is complicated, however,by the fact that both recording arms contain apertureduniform irradiance beams-an arrangement requiringthe use of numerical computation for a complete anal-ysis. Fortunately, designing and operating the SVIsystem described in Sec. IV do not require a preciseknowledge of f. Finding the correct displacement canbe achieved in practice during the setting up stage byfine tuning the separation between P and H1 until thecorrect beam diameter at H2 is found.

Repositioning the beam waist in this fashion alsorequires some degree of aberration analysis as the ho-logram is shifted slightly away from its intended recon-struction geometry. Generally this causes only a veryslight change in curvature, so that the replay wave-fronts are still more or less correct. In addition, aseach n to n + m interconnect mapping has a slightlydifferent path length, there must be slight variations inWH2 and the output focal lengths. Because the differ-ence in path length is so small, however, typically onlya fraction of a percent, this effect will be of no signifi-cance. Take, for example, a 16 X 16 stacked perfectshuffle of 200-,gm channel separation, 2.5-mm focallength, and an interconnect separation of 5 cm. Thevariation in path length for this particular system cor-responds to <20 Am, which should cause no practicalproblems.

Fabricational inaccuracies and misalignment of H1and H2 with respect to the optic axis will lead todeterioration in the interconnect efficiency, distortionof the output array, and increased levels of crosstalk.

10 June 1991 / Vol. 30, No. 17 / APPLIED OPTICS 2373

Inputarray.

Outputarray.

P1 H1 H2 H3 H4 P2Fig. 9. Schematic implementation of a four-element on-axis wave-length-shiftable point to point crossover interconnect. Plane P1 isthe input array, P2 is the output array, H1 and H4 are both lensletarrays for collimating the light, and H2 and H3 are both similar

redirecting planar grating interconnects.

Recent studies24 have shown that, as the dimensions ofthe system are scaled down, the tolerances on position-al and recording accuracy become more severe. In-deed, even with a facet size of 500 Am, some difficultyin providing adequate alignment was found. This wasmainly due to the off-axis nature of the SVI whichmade it difficult to define the optic axis. Thus, toimplement practical SVIs where facet sizes are <200Am, the interconnect packaging must be able to pro-vide accurate alignment capabilities.

The final scaling consideration relates to operatingthe DCG SVI at a wavelength different from that it wasrecorded at. If it is wavelength shifted to a near infra-red wavelength the SVIs would become impossible tomake because each facet would require a differentplane wave recording angle to ensure Bragg matchingat the replay wavelength. Our solution to this is tocreate a four-element space-variant interconnect.This involves separating the focusing and directingoperations into two separate HOEs: a lenslet arrayand a corresponding array of plane grating elementswhich are sealed together to reduce reflection losses(Fig. 9). This has the additional benefit of making theinterconnect on-axis. It is then much simpler to buildin a wavelength shift into the separate elements; pla-nar gratings are simple, and we have already demon-strated the recording of high quality lenslet arrays at514 nm for replay at 850 and 1064 nm.9 In addition,the close cascaded nature of the holograms2 425 willreduce the sensitivity of the interconnect to changes inreplay wavelength,24 and, as all the beams travel moreor less normally to the HOE surfaces, the final SVNETwill be less polarization dependent.

VI. Conclusions

With our prototype SVIs we have demonstrated theability to implement complicated optical interconnec-tion patterns with important applications in the fieldsof optical computing and communications. Relativelysimple point to point interconnections are possiblewith high efficiency and low component count. Arraysizes are the main limitation of this approach, because

of the finite aperture of the translated lens (F0 in Figs.5 and 6). We anticipate solving this problem by mov-ing to a fiber/aperture based recording system insteadof the translatable imaging optics.

We are grateful for the many fruitful discussionswith J. S. Snowdon, S. Bowman, F. A. P. Tooley, and J.Turunen on the relevance of this work and would liketo thank I. R. Redmond for his invaluable advice. B.Robertson and E. J. Restall acknowledge SERC Casestudentships with Pilkingtons plc. and British Aero-space plc., respectively.

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