Thin-film computing with the nonlinear interface

Vol. 6, No. 5/May 1989/J. Opt. Soc. Am. B 877

Thin-film computing with the nonlinear interface

Robert Cuykendall and Karlheinz Strobl

Center for Laser Science and Engineering, University of Iowa, Iowa City, Iowa 52242

Received July 18, 1988; accepted December 14, 1988

An optical integration technique using thin-film technology can be based on an interface with a diffusive Kerr-likenonlinearity. Idealized switching behavior is assumed in order to assess the ultimate worth of such an expedientapproach to optical computers if appropriate materials could be identified. This effort is justified by the results oftwo-dimensional simulations of the nonlinear interface carried out on the Cray X-MP/48 computer that show nearlywhole-beam switching with high contrast. The simplicity of the integration architecture is demonstrated bydesigning a thin-film half-adder.

INTRODUCTION

High-contrast switching at an interface between diffusivenonlinear media having opposite Kerr coefficients has beenreported' based on two-dimensional simulations of the opti-cal field redistribution effects. We recently found similarbehavior at the interface between linear and diffusive non-linear media. With polarization-coded inputs these inter-faces implement a symmetric self-controlled logic structuremore powerful than a NAND gate, which is both noise toler-ant and optically reversible.2 Such a switching device has,the additional advantage of computing the input signals at asurface, not while they are traveling through a bulk material.This is ideal for integration purposes since absorption lossescan be minimized.

Switching based on a nonlinear interface having one linearmaterial leads to an intriguing integration architecture, as-suming an idealized interface. It will be shown that bothcomputing and multiplexing elements can in principle beconstructed with thin films having the low-index linear ma-terial in common. This is a key feature in the proposed thin-film architecture since it avoids additional interfaces (beamsplitting) between different linear materials, which at highincident angles would cause nonnegligible reflections, great-ly complicating the circuits. Another feature is the multipleuse of component layers, leading to simple compact circuitsin two dimensions. We demonstrate the application ofthese ideas by designing a thin-film binary half-adder. Thiswork has been carried out in order to obtain an upper-limitindication of the potential worth of nonlinear inferface de-vices in optical computing.

NONLINEAR INTERFACE SIMULATION

Numerical computations of the behavior of an incident two-dimensional Gaussian beam at the interface between linearand nonlinear media have been carried out on a Cray X-MP/48 computer. A diffusive Kerr-like nonlinearity has beenassumed, relating the intensity of the beam I to the nonlin-ear mechanism density p through the one-dimensional dif-fusion equation

Do02 + Go -P = 0, (1)

where x is the distance from the interface. In this equationDo represents the diffusion coefficient, Go the generation-rate coefficient, and the recombination time constant forthe nonlinear mechanism. The quantity p may representthe density of free carriers, excited gas atoms, heat, etc. andis assumed proportional to the local nonlinear index of re-fraction:

n2NLI = ln2pP. (2)

Diffusion along the interface (z direction) was neglectedowing to the slow variation of the wave envelope in that'direction at high incident angles (Oo > 800).

The calculational technique employed3 is similar to thoseof Refs. 4 and 5. However, their analysis assumed a strictKerr nonlinearity with no diffusion of the nonlinear mecha-nism. Nondiffusive results obtained from the calculationswere previously found to agree exactly' with those obtainedby Tomlinson et al.

4 When carrier diffusion is modeled, theresults differ qualitatively owing to the nonlocal behavior ofthe nonlinearity. With diffusion, the index gradientchanges more slowly than the intensity gradient, simulatingmore the plane wave than diffusionless Gaussian behavior.,To study the switching from total internal reflection (TIR)to transparentization of the interface by a Gaussian-likebeam we selected two parameter sets: one for an incidentangle 0o = 870 (see Figs. 1 and 2) and one for 0 = 85°. Theoffset of the linear refractive index (A) was in the former case0.005 and in the latter 0.01, resulting therefore in a slightly'different ratio of incident angle to critical incident angle.All results'were calculated for a beam waist 2wo = 4, alinear refractive index nL = 1.5, and wavelength X = 0.3 im.In order to permit comparison between the nondiffusive and:the diffusive cases, we varied the product 2NL' until wefound a pair that showed the best switching behavior for one[Figs. 1(a) and 2(a)] and two [Figs. 1(b) and 2(b)] times theintensity. Note that in the particular case shown (not nec-'essarily the optimal parameter set, as discussed below) inputintensities 65 times greater were required in the diffusivecase to effect an index change of 0.005 in the nonlinearmedium. This greater intensity is required in order tochange the index of refraction over a broader area (becauseof diffusion) than the incident beam actually samples.

Figure 1(b) shows, in the absence of diffusion, beam

0740-3224/89/050877-07$02.00 © 1989 Optical Society of America

R. Cuykendall and K. Strobl

878 J. Opt. Soc. Am. B/Vol. 6, No. 5/May 1989 R. Cuykendall and K. Strobl

which an additional interference fringe switches through theinterface: with only a 4% increase in intensity, the TIR caseforms a transmitted self-focused channel, while the "trans-mit case" now forms two self-focused channels. We chose toshow the 0o = 87° results because they show best-pair behav-ior more typical of previously published results: Fig. 1(b)comprises a small glancing angle analog of the plot shown inFig. 9 of Ref. 4. Figure 2(b) shows notable reduction inbreakup into self-focused channels in the nonlinear trans-mission region for diffusive nonlinear media. It can also beseen [on comparing Fig. 1(a) with Fig. 2(a)] that self-focus-

60

30

X AXISINTENSITY

3.06

2 . 72

2.38

2.04

1 .70

1 .36

1.02

0.68

0.34

0.00

-30

Z AXIS

60

30

X AXIS

INTENSITY

2.75

2.45

2. 1 4

1 .83

(b)

Fig. 1. Numerical computations of intensity distributions for atwo-dimensional Gaussian beam incident at a nonlinear interface:(a) nondiffusive case at 00 = 870 with beam waist 2wo = 12 gm,wavelength X = 0.3 um, A = 0.005, nL = 1.5, and n2 NLI/A = 0.504, (b)n2NLI/A = 1.008. Note that the Kerr medium fills the negative-xhalf-space.

1 .53

1.22

0.92

0.61

0.31

breakup into two transmitted self-focused channels. Eachchannel appears to emanate from an interference fringecrossing the interface. Calculated intensity distributionsare shown only for the case 00 = 870 since qualitativelyequivalent switching behavior was found at 00 = 850. How-ever, since the latter case represented operation closer to thecritical incident angle for TIR, the nondiffusive best pairturns out to correspond to a single transmitted channel in-stead of two channels as in Fig. 1(b). The pair intensities inthis instance happen to be very close to the threshold at

0 .00

(b)

Fig. 2. Numerical computations of intensity distributions for atwo-dimensional Gaussian beam incident at a nonlinear interface:(a) diffusive case at 00 = 870 with diffusion length LD = 20 gm, beamwaist 2wo = 12 jim, wavelength X = 0.3 jim, A = 0.005, nL = 1.5, andn2NLI/A = 32.76, (b) nl2NLI/A = 65.52.

INTENS I TY

3 00

2.66

2.33

2.00 -

1.67 -

1.33 -

1.00 -

0.67

0 33 -

0. 00 -4 5

I NTENS I TY

6.39

5 .68

4.97

4 .26

3.55

2.84

2 13

1 .42

0.71

0.00

(a)

60

0

(a)

30

X AXIS

60

30XAXIS

_-_L -60-30

R. Cuykendall and K. StroblVol. 6, No. 5/May 1989/J. Opt. Soc. Am. B 879

ing of the reflected beam due to the nonlinear Goos-Hanchen effect discussed in Ref. 4 is substantially reduced.The reduction in self-focusing is attributed to the smearingof the nonlinear lens by the diffusion of p. Depending onthe actual materials selected, p may or may not diffuseacross the interface. However, diffusion across the interfaceboundary seems to reduce the chance that a surface wavewill be formed. 6 Hence in the diffusive results shown herethe nonlinear mechanism p has been permitted to diffuseinto the sourceless linear material in order to avoid furthercomplications owing to surface-wave formation.

We believe that the apparent self-deflection of the trans-mitted beam back toward the interface [see Fig. 2(b)] iscaused primarily by modeling diffusion only in the z direc-tion and secondarily by the paraxial approximation, both ofwhich were necessary to keep run times within reasonablelimits. Since this is a cumulative numerical effect, the devi-ation from the true propagation direction increases as thebeam penetrates the nonlinear medium. This is consistentwith the observed behavior: the higher the transmittedintensity and the greater the incident angle, the shorter thelength scale where this self-bending is notable. The aboveexplanation is further supported by the significant reduc-tion in self-deflection in the nondiffusive case shown in Fig.1(b). This suggests that the only other form of related self-bending known to us, the self-deflection of beams withasymmetric beam profiles in nonlinear media,7 cannot be thecause for the observed self-bending. We know of no physi-cal effects in the nondiffusive case that would cause thetransmitted beam profile to be significantly less asymmetricthan in the diffusive case. For that reason, we think thatself-bending in reality will not occur on the length scale inwhich we are interested. If for any reason the input beam issufficiently asymmetric (e.g., half-Gaussian7) to cause self-deflection on the length scale of the interface, problems willarise in cascading multiple interfaces.

The calculation results indicate that nearly whole beamswitching at a nonlinear interface should be possible withreal (Gaussian) input beams. Moreover, the computationsvalidate the conceptualization of the nonlinear interface forlinearly or circularly polarized inputs, since by Refs. 8 and 9both the critical switching intensity I and the amplitudereflectivity r of an incident beam I are independent of theplane of polarization of the incident light for sufficientlyhigh incident angles 00. While these specific numerical ex-periments were carried out for a distance of 800 Aim along animaginary interface boundary in order to permit the stabil-ity of the computed solution beam to be observed, the actualdevice size is determined by incident angle, beam waist, andthe requirements for TIR, as discussed below.

Figure 2 shows nonlinear interface switching for specificratios of wavelength X to beam waist 2wo to diffusion lengthLD. Higher offsets (A > 0.01) in the diffusive case at 0 <85 required switching intensities outside the range wherethe computer program worked reliably. However, we canthink of no reason why the nonlinear interface should be-have differently at higher offsets. Limited effort was devot-ed to finding the optimal parameter set (o, A, w, LD, L,etc.), since actual transmission characteristics will differfrom the two-dimensional predictions anyway. With this inmind, we investigate the possibility of circuit integration(using the nonlinear interface as the single computing ele-ment), basing our calculations on the standard-model pa-

rameter values selected originally by Tomlinson et al. 4 : A =

0.02, 00 = 850, and nL = 1.5.

NONLINEAR THIN-FILM GATESince the computing of the input signals occurs at a surfaceand not while the signals are traveling through a bulk mate-rial, the nonlinear medium can be reduced to a thin film. Inthis case the absorption losses would be minimal. A sche-matic diagram of a thin-film realization of a nonlinear inter-face is shown in Fig. 3. The logic table defines the allowablecases, based on the assumption that a beam with intensity 1is reflected, while a beam with intensity 2 is transmittedthrough the diffusive nonlinear film. The thin-film gate canbe used, with some restriction (e.g., I, = 2 implies that I2 =0), both from the top (I) and from the bottom (12) side,permitting some limited polarization-independent multi-plexing with the computing element itself. Note that I isthe computing input, while the 2 input only reflects (multi-plexes).

The minimum thickness of the nonlinear thin film in Fig.3 follows from the requirement to guarantee TIR for the casewhen I = 2 = 1. The calculation below is shown in theplane-wave approximation, neglecting the intensity depen-dence of the index of refraction of the nonlinear film. [Amore careful analysis shows that these approximations in-crease the minimum thickness value only slightly (1%)].For TIR the refracted wave is therefore propagated onlyparallel to the surface and is attenuated exponentially be-yond the interface. The attenuation is described by theexponential factor

exp[-27r(2nLA - A2- nL2 cos2

0 )1 2 x/X], (3)

where nL is the index of refraction of the linear medium, nL -A is the index of refraction of the nonlinear film for negligi-ble intensities, and x is the depth of penetration. For A =0.02, 00 = 85, and nL = 1.5 at a depth of x = 2.5X, the electricfield is attenuated by more than a factor of 20. When thethickness of the nonlinear medium is chosen as 5, the TIRin the case I = 12 = 1 is therefore ensured. The minimumlength of the nonlinear medium, and therefore of the nonlin-ear thin-film gate, on the other hand, should be at least

I = 4wo/cos Oo + 5X tan 00, (4)

which is twice the projection of a beam with diameter 2w atthe nonlinear interface, plus the offset of the beam after ittravels through a 5A-thick layer of the nonlinear medium.The minimal width of the nonlinear film is twice the beamwaist: 4wo.

Iline,

11 12 I R I To 0 0 0o 1 0 11 0 1 01 1 1 1

2 0 0 2

ar medium non-l

Fig. 3. Nonlinear thin-film gate.

inear medium

850 55x12 T

880 J. Opt. Soc. Am. B/Vol. 6, No. 5/May 1989

THIN-FILM MULTIPLEXER

For simplicity we assume that the nonlinear thin-film gateuses linear polarization-coded input signals. The corre-sponding multiplexing element, which is necessary for thespatial separation and overlapping of the different signalchannels, is therefore a polarizer. The design of a thin-filmpolarizer matched to the nonlinear thin-film gate, and neces-sary for the polarization-sensitive wiring, is described below.A more detailed development will appear in a forthcomingpaper.'0

To calculate the characteristic values such as total reflec-tion and transmission of a given thin-film multilayer stack,we use the matrix method described by Macleod." A planewave' 2 with wavelength X traveling through a layer of thick-ness dr with an index of refraction n, and an angle 0 r mea-sured against the normal of the incident surface suffers aphase shift 6r, where

=r X nrdr cos Or = 7r/2 pi j = 2r. (5)

Parameter Xr0is the wavelength at which the rth layer acts asa X/4 wavelength stack, and gr is the relative thickness of therth layer. Note that layers are counted from entrance toexit, with the zero layer representing the entrance medium.

The first thin-film polarizer was designed by MacNeille'3

in 1946; he used three layers enclosed by two glass prisms.Since then, through improved film deposition techniques,the ability to use more layers (10-20) and computer-opti-mized thickness determination of the individual layers, thewavelength and angle of incidence region over which thethin-film polarizer maintains its performance have been sig-nificantly improved."",1 4 These polarizers, which are nowcommercially available, use an incident angle 300 ' 00 < 600,and most of them gain from the fact that the Brewster angle

Or- In order to design a polarizer at an angle 0o - 850, wecannot use the Brewster angle because there exists no knownmaterial that has an index of refraction >11 and a negligibleabsorption and that can be deposited as a film of controlledthickness. On the other hand, polarization-sensitive wiringin our integrated optical circuits needs only a very restricted

polarizer, one that has an extinction of roughly 3% for a"single frequency" and a fixed angle. Such a polarizer can

indeed be designed, and it takes only three layers to obtain(at least in theory) the desired performance.

The simplest thin-film polarizer is a symmetrical three-layer stack (HLH) formed by alternating thin linear filmswith high (H) and low (L) indices of refraction, which itself isenclosed by the same material that forms the middle layer.There exists today a large variety of materials that can bedeposited in the form of thin films,"1 with refractive-indexranges roughly from 1.25 to 2.6. Given a material, the indexstill depends somewhat on the wavelength as well as on thedeposition conditions and techniques. It is desirable to useas L material the same material as the linear medium in thenonlinear thin-film gate. A material with nL = 1.5, having anegligible absorption, and that can be deposited in any need-ed film thickness, is therefore assumed. The H materialshould have an index of refraction that is as high as possible

in order to approach the Brewster angle, giving a betterextinction coefficient. A material with nH = 2.35 is chosen,again with negligible absorption. This could be, for exam-

100%

0% -0.25 0.5 0.75 1 1.25 1.5

g2 /2

Fig. 4. Reflected intensity of a symmetric three-layer stackL(HLH)L for Oo = 850, nL = 1.5, nH = 2.35,g* = 0.8, andg2* = 0.071.

ple, ZnS, which is already being used extensively in thin-filmpolarizer production. Since the angle of incidence is deter-mined by the nonlinear thin-film gate, there are only twofree parameters, dl/X and d 2/X, which can be adjusted to getthe optimum polarizer.

Figure 4 shows the variation of the calculated reflectedintensity with the relative thickness of the middle layer (r =2) for a polarizer that is the "best" compromise (g,*) amongthe different competing factors for the application at hand:a polarizer that behaves as a vertical mirror (vertical polar-ized beams reflected, R, = 97%, horizontal beams transmit-ted) with a symmetric reduction in performance for smalldeviations from the chosen parameter set: 00, nr, dr, and X.Choosing different g, and g2 values, one could get an R, <

98.8%, but small deviations have a larger effect on the perfor-mance. For the chosen gl* and g2* (shown in Fig. 4), itfollows from Eq. (5) that d, = X/9 and d2 = X/7.4. Thereforethe best micro v mirror is only X/2.8 thick.

THIN-FILM HALF-ADDER

By combining the nonlinear thin-film computing and multi-plexing elements designed above, an integration architec-ture is illustrated by designing a binary half-adder. A 1-bithalf-adder performs modulo 2 addition of two binary digits,Ai and Bi, and outputs the sum Ai @1 Bi and carry AiBi.Figure 5 is a schematic diagram of a half-adder for whichonly two'5 nonlinear thin-film gates (computing primitives)have been used, and the multiplexing is done with v and hmirrors. Because of the intrinsic difference of h- and v-polarized beams (only h-polarized beams have a Brewsterangle), thin-film polarizer performance is far better for vmirrors.

Three different thin-film realizations of a half-adder areshown in Fig. 6. The pictures are drawn to scale (except forthe thickness of the nonlinear medium) for an incident angle00 = 850. The y dimension has been enlarged by a factor of11 for improved visualization. The solid lines show thecomputing beams, while the dashed lines show additionalunavoidable signal channels (duplication and inversion ofthe v input signal) characteristic of the thin-film logic. The

half-adder version in Fig. 6(a) requires the smallest amountof space. The version in Fig. 6(b) is roughly 5% longer than



that in Fig. 6(a) but can be constructed using the least A h

number of layers. It is also the most flexible design: (1) it is B h

transparent to an h beam traveling from VI to V 3 (or vice M

versa) and can therefore be used to communicate with cir- 850

cuits in planes above or below the actual half-adder and (2) Asubstitution of the polarizer V1 or V 3 with a conventionalmirror M permits redirectioning of some of the inputs andoutputs. Note that the two h mirrors in Fig. 5 have been M 2

replaced in Figs. 6(a) and 6(b) by the conventional mirrors IM, and M2 and the v mirror V2. This also reduces the 1 ABh (A'B+AB)hminimum volume required for these designs by a factor of 2. (a)The third half-adder, shown in Fig. 6(c), is twice as high asthe others and needs an additional pump beam, but it has L

the definite advantage of a continuous rather than inter- H 7 /9

rupted nonlinear thin film and is therefore easiest to manu- X / 9

facture. The design of Fig. 6(c) is independent of the non- A h L

linear film thickness, while the designs of Figs. 6(a) and 6(b) Bv h B'h

have a small such dependence. Figure 7 shows the half- M 1 - 5

adder in Fig. 6(c), emphasizing the layered structure, whichcomprises 15 layers. Note that the minimum-layer design[Fig. 6(b) with V 3 and V 5 replaced by a conventional mirror i X B,MI requires only 12 layers.

To estimate the theoretical minimum volume require-ment for a thin-film half-adder, we consider here only the I 2case shown in Fig. 6(a). The other cases follow in a straight- " ABh (A'B+AB h

forward manner. The minimum distance between the two (b)nonlinear thin-film gates in Fig. 6(a) is again the length ofthe nonlinear thin film defined by Eq. (4). Thus is the 1 h

characteristic minimum length scale for this half-adder real- v / B'hization. The smallest half-adder has then a length Lha = 31 A h /0

and a height Hha = I/tan 00 + 5X + 2X/2.8, where the thick- / ness of the v mirrorsV and V3 has been included. /

All three dimensions of the half-adder depend linearly on (AB+AB') v

A~~~~~ h Ba B h AB

Bh A'Bv 'hA'BhV \fis A B', \

AB Bv Fig. 6. Thin-film half-adder: (a) smallest volume, (b) most flexi-

A B'V ABA)h/ 4 ble, and (c) simplest to manufacture.

\ l~v > the beam waist 2wo. Gaussian beams (TEMO), the kind ofSum=(A8B+AB')h beam that we are dealing with, have the following beam

waist dependence:Carry=ABh W(z) = W0 [1 + (Z/ZR)2]/2,

Pv' h PQ0v'P'

0h

ZR :=7rWO 2n/A, (6)nonlinear interface zZ(

where z is the distance from the focus, 2wo is the minimumPo v'PQ h beam waist, and n is the index of refraction of the medium

A/h through which the beam is traveling. Only over a distancevertical polarization mirror IzI << ZR can a Gaussian beam be approximated by a parallel

v,h v beam.v For the operation of the thin-film half-adder, it is neces-

sary that the intensity incident upon the individual nonlin-hI horizontal polarization mirror H o p ear thin-film gates be independent of the path through

v,h h which the light beam reaches the gates. That means that theFig. 5. Nonlinear interface half-adder with v and h mirrors. longest path that is allowed between the individual gates in


882 J. Opt. Soc. Am. B/Vol. 6, No. 5/May 1989

Linear medium with low index of refraction

Linear medium with high index of refraction

Nonlinear medium

Mirror

Fig. 7. Schematic layout for thin-film half-adder corresponding toFig. 6(c).

the circuit has to be <<2ZR. This criterion limits the tight-ness of the focusing, and therefore the minimal size of theoptical circuits, unless the beam travels in a waveguide fromone element to the other or soliton pulses can be used. Thelatter option requires that the L medium be nonlinear andwill be discussed in a future paper. (By controlling thediffusion coefficient rate for a given thickness of nonlinearmedium, one should actually be able to control the self-focusing in order to improve cascadability of a single thin-film gate by minimizing beam expansion after the beamleaves the nonlinear material.)

The longest path between the two nonlinear thin-filmgates in the half-adder [Fig. 6(a)] is

Lmax = 21/sin 0 0 21. (7)

An intensity attenuation of 10% corresponds to a beam di-ameter change of 4.9% and to a distance IzI = O.3 2ZR from thefocus. Using this as a criterion to calculate the minimumbeam waist, we obtain from Eqs. (6) and (7) the relation

0.327rwo 2nL0.3 2 ZR = = Lmax/2 46wo + 57X. (8)

The solution of this relation is wo 32X. Inserting this valueinto Eq. (4), we obtain for the minimum volume of a thin-film half-adder

with the same amount of manufacturing steps hundreds ofhalf-adders can be produced. The number is limited onlyby the film extension in the third dimension and the mini-mum distance between two adjacent adders necessary toavoid channel interference due to diffraction. The mini-mum separation is therefore about twice the beam waist.This indicates the high potential that this kind of opticalcircuit has for parallel calculation. The problem of deposit-ing strips of different materials on the same horizontal planehas to be solved only in one direction and requires an accura-cy of only tens of micrometers, which is within the presentstate of the art. Ion- or laser-enhanced chemical-vapor de-position, or similar techniques, could be used to produce thedesired structure.

Only four materials are necessary to build the thin-filmhalf-adders: a nonlinear material (NL), the correspondinglinear material (L), another linear material (H) with an in-dex of refraction as high as possible, and a material for themirrors. The mirrors M could be a simple thin-film alumi-num or gold coating, depending on the wavelength of lightused. The combination of NL and L forms the nonlinearthin-film gate that computes and allows some polarization-independent multiplexing, while the combination of L andH forms the v mirror for polarization-sensitive wiring (mul-tiplexing). Having the L material in common avoids addi-tional interfaces, and therefore beam splitting, making sim-pler circuits possible. This integration architecture permitshigh flexibility in the circuit design since every element canbe used at least twice, like the V2 polarizer or the right-handnonlinear film in Fig. 6(b). Some circuit elements can alsobe used from the back side for another computing circuit,minimizing the necessary total number of computing andmultiplexing elements.

In this paper we have demonstrated that thin-film opticalcomputing circuits, at least in principle, can be constructedbased on the idealized behavior of the nonlinear interface.Further research is necessary16 to determine the actualswitching characteristics of the nonlinear thin-film gate inorder to assess beam regeneration and refocusing require-ments in real circuits. It should also be noted that realisticcomputing circuits would need to operate with pulses ratherthan cw signals. Any deviation from a rectangular-pulsetime envelope would degrade switching behavior even for anideal nonlinear interface, thus limiting the clock rate. Sincethe thin-film architectural approach works out so well in thecase of an ideal interface having one linear material, webelieve that a high potential may exist for the nonlinearinterface in optical computation and for extending theseideas to other interface configurations (both media nonlin-ear) and signal types (soliton).

Vha = Lha X Hha X 4wo = (4587 X 138 X 128)X3= 10-5 cm 3 ,

(9) ACKNOWLEDGMENTS

where an intermediate X = 0.5 ,m has been used to obtainthe last result.

DISCUSSION

One important feature of this thin-film architecture is thatthe layers have no restriction in the third dimension, so that

The authors thank H. A. Macleod and D. R. Andersen forhelpful discussions. Calculations were performed at theNational Center for Supercomputing Applications at theUniversity of Illinois, supported by a National ScienceFoundation block grant to the University of Iowa.

R. Cuykendall is also with the Department of Electricaland Computer Engineering, University of Iowa.

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REFERENCES AND NOTES

1. R. Cuykendall, "Three-port reversible logic," Appl. Opt. 27,1772 (1988).

2. R. Cuykendall and D. R. Andersen, "Reversible computing: all-optical implementation of interaction and Priese gates," Opt.Commun. 62, 232 (1987); "Reversible optical computing cir-cuits," Opt. Lett. 12, 542 (1987).

3. D. R. Andersen, R. Cuykendall, and J. Regan, "SLAM-vector-ized calculation of refraction and reflection for a Gaussian beamat a nonlinear interface in the presence of a diffusive Kerr-likenonlinearity," Comput. Phys. Commun. 48, 255 (1988).

4. W. J. Tomlinson, J. P. Gordon, P. W. Smith, and A. E. Kaplan,"Reflection of a Gaussian beam at a nonlinear interface," Appl.Opt. 21, 2041 (1982).

5. D. Marcuse, "Reflection of a Gaussian beam from a nonlinearinterface," Appl. Opt. 19, 3130 (1980).

6. P. Varatharajah, A. Aceves, J. V. Moloney, D. R. Heatley, and E.M. Wright, "Stationary nonlinear surface waves and their sta-bility in diffusive Kerr media," Opt. Lett. 13, 690 (1988).

7. G. A. Swartzlander, Jr., and A. E. Kaplan, "Self-deflection oflaser beams in a thin nonlinear film," J. Opt. Soc. Am. B 5, 765(1988).

8. A. E. Kaplan, "Theory of hysteresis reflection and refraction of

light by a boundary of a nonlinear medium," Sov. Phys. JETP45, 896 (1977).

9. P. W. Smith, W. J. Tomlinson, P J. Maloney, and J. P. Her-mann, "Experimental studies of a nonlinear interface," IEEE J.Quantum Electron. QE-17, 340 (1981).

10. K. H. Strobl and R. Cuykendall, "Thin-film optical circuits,"submitted to Appl. Opt.

11. H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Hilger,Bristol, UK, 1986).

12. Since the multilayer stack is composed of only linear materials,and the incident beam is roughly parallel, the plane-wave de-scription of the underlying problem is adequate.

13. S. M. MacNeille, U.S. Patent 2,403,731 (1946).14. R. P. Netterfield, "Practical thin-film polarizing beam-split-

ters," Opt. Acta 24, 69 (1977).15. Note that a transistor-based half-adder needs on the order of 16

computing primitives (transistors).16. We are now conducting an experiment to study the behavior of

the nonlinear interface. Preliminary results [K. H. Strobl andR. Cuykendall, "Gaussian-beam reflection at a nonlinear inter-face," submitted to Opt. Lett.] show much higher switchingcontrast than obtained in a previous experiment reported by P.W. Smith and W. J. Tomlinson, "Nonlinear optical interfaces:switching behavior," IEEE J. Quantum Electron. QE-20, 30(1984).


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Thin-film computing with the nonlinear interface