construction of an optical-carry adder

Construction of an optical-carry adder

J. Barry McManus and Roger S. Putnam

We have constructed a hybrid electronic/optical digital adder, based on a system suggested by B. Arazi [Proc.

IEEE 73, 162 (1985)], which uses an optical system for the carry-bits, giving fully parallel operation. Our

bulk-optic breadboard system has six acoustooptic modulators as optical switches, with a single He-Ne laser

source. It adds two 4-bit words at 10 MHz, with a 500-ns pipeline delay. The optical carry is based on a

multistage optical path, with light added or allowed to pass at each stage, according to whether carries are

generated or propagated at the corresponding column of the addition. For long addends, the system speed islimited by light loss through the carry path and by size effects.

1. Introduction

Hybrid electrooptic systems, which combine thestrengths of electronics and optics, may be useful forincreasing the speed of digital computing and signalprocessing. All-optical logic devices1 ' 2 may eventuallyfacilitate very high speed computation,3 but for thenear term, most logic functions will be best done withelectronics. We have been led to ask what can be donewith optics to help relieve particular speed bottlenecksin digital computing, rather than attempting to repli-cate the functions that can be done easily with elec-tronics. One important strategy for increasing thespeed of computing systems is to use parallel architec-tures, and optical techniques are often cited as a meansof providing increased parallelism.3'4 In digital addi-tion, the carry operation is the main source of delay inexecution. Recently, Arazi5 proposed a method forparallel digital addition which uses an optical systemfor carry-bit generation and propagation. In this pa-per, we report the demonstration of an optical-carryadder, which adds 4-bit words at a data rate of 10 MHz.This demonstration system uses the bulk optical tech-nology of acoustooptic modulators, beam splitters, anda He-Ne laser source. This system serves to clarifythe potential usefulness and limitations of the op-tical-carry adder. We will describe: the optical-carryadder concept, and how it can be extended to the

The authors are with Aerodyne Research, Inc., 45 Manning Road,

Billerica, Massachusetts 01821.Received 19 December 1986.0003-6935/87/081557-06$02.00/0.© 1987 Optical Society of America.

operations of subtraction and multiplication; the con-struction and performance of the demonstration op-tical-carry adder; how the system can be optimizedand what technology would be needed for an optical-carry adder to be competitive with all-electronic sys-tems.

II. Concept of the Optical-Carry Adder

In digital arithmetic, propagating carry bits fromcolumn to column is the primary feature which pro-longs operations. Addition tends to be a serial opera-tion, since the sum bit and carry bit at each columndepend on the carry bit from the previous column.The same can be said for subtraction, where borrowbits propagate from column to column. For simplic-ity, we will concentrate on addition in our initial dis-cussions. The serial nature of addition means thatmore time is required to add longer digital words.Significant increases in computation speed will resultfor systems that perform digital arithmetic in a fullyparallel fashion. In such a system, the computationtime would be independent of the number of bits.

The usual electronic solution to the serial nature ofaddition is to use carry-look-ahead (CLA) circuitry, 6

which allows parallel addition for a limited number ofbits. In the CLA, for each column added, all previouscolumns are checked simultaneously to determine if acarry is present. Only four logic gate delays are re-quired for execution of the addition, with two layers oflogic to determine the presence of carry bits in eachcolumn, and two layers of logic to form the sum of thetwo addend bits plus carry bit in each column.

With increasing number of bits N in the addends,the CLA approach leads to a large number of gates(-N 2 ), each with many inputs (-N). Since the num-ber of inputs to a single logic gate islimited, for longdigital words (N > 8) the CLA circuitry must be modi-

15 April 1987 / Vol. 26, No. 8 / APPLIED OPTICS 1557

fied. The simplest approach to adding long digitalwords with CLA circuitry is to cascade 4-bit CLA ad-ders, so that 4 bits at a time are added in parallel.Commercially available7 4-bit CLA adders on a TTL-family chip can produce the final carry output in -5 ns,so that for N bits the addition delay is approximately(N/4) X 5 ns. A more sophisticated approach for longaddends is to use more layers of logic in the CLA tocombine the many inputs to each logic gate. In thisway the growth in execution delay is only logarithmic.For example, if the maximum number of inputs perlogic gate is 4, just three layers of logic can be used foran equivalent 32-input gate. More delay may be in-curred in this approach, however, since extra driverelements would be needed to drive the many logic gatesin parallel. Depending on the approach, the electronicCLA adders have a propagation delay that grows lin-early or sublinearly with the length of the digital wordsadded. The CLA circuitry has a great advantage overserial ripple-carry adders, but for long addends theCLA is not fully parallel and the propagation delayincreases.

The aim of the optical-carry adder (OCA) is to pro-duce a parallel adder with a propagation delay thatdoes not increase as the number of bits increases. TheOCA as presented by Arazi5 is shown schematically inFig. 1. It consists of two basic segments, the electroniclogic and the optical carry. The optical-carry segmentis a multistage optical path with a detector, modulator,and light source in each stage, and as many stages asbits in the addends. Some element, such as a beamsplitter, allows injecting and sampling the light at eachstage. Detected light indicates the presence of a carrybit at that stage. The electronic logic consists of anarray of half-adders (ADD and XOR gate in parallel),which drives the sources and modulators, plus finalXOR gates that produce the sums.

The operation of the OCA is essentially parallel.The input data pulses go to the half-adders, whoseoutputs set all the light sources and modulators atonce. The carry bits are generated as soon as the lightin the carry-line is detected. The sum bits are thenproduced in parallel with the final layer of electroniclogic. The electronic logic for each stage does not lookback at the earlier columns. The carry bits propagateoptically, along paths determined by the modulatorsand sources.

The delays involved in addition with this system aredue to the electronic logic, the process of light genera-tion and detection, and the optical propagation timealong the carry path. The electronic logic is only twolayers deep, even for long addends, so the electroniclogic delays are fixed (-2 ns). If the optical-carry lineis short enough, the optical propagation times can be asshort as desired (<1 ns). With fast modulators,sources, and detectors, the carry line delays should becomparable with those of the electronic logic. Thuswe expect that with an optical-carry adder, the delay inadding two 32-bit words could be as small as five gatedelays or -5 ns. This is significantly less than the-30-ns delay with all-electronic 32-bit adders. The

Optical Carry Line

(Ci) 45JC%) < 1 {19~ Sum6 i Inputs

Fig. 1. Schematic of optical-carry adder. Electronic input bits areAn, Bn. The optical-carry line has modulators (Mn), sources (In)which inject light, and detectors (D) which detect carry bits (C,).Electronic logic (AND, XOR gates) drives the sources and modulators

and produces the sum bits (Sn).

highest operating frequency of an OCA will be limitedby the electronic logic rise times. Therefore, the maxi-mum data rate for the OCA will be the same as forconventional electronic adders, provided the longerdelays in the conventional adder are handled in a pipe-lined fashion.

With small changes in the driving logic, the optical-carry adder can function as an optical-borrow subtrac-tor. If inverting gates are placed at the A inputs tothe half-adders' AND gates, and at the inputs to themodulators, the system calculates A-B, with the pres-ence of borrows signaled optically from column to col-umn. If we replace all the inverters that we use forsubtraction with XOR gates, we can convert the systemfrom an adder to a subtractor with a single switch (thatclamps all the second XOR inputs to logic-high).

The optical-carry adder may also be useful in fastdigital multipliers. Digital multiplication can be per-formed in two steps8 9 ; first form all the (N 2) crossproducts of all the bits of two N-bit words, and secondadd together the appropriately grouped columns ofcross products. The first step can be done completelyin parallel with an array of AND gates. Since the col-umns have up to N elements, a fully parallel adderwhich performs the second step would be much morecomplex than for ordinary two-element addition. Onecan break up the N-element column additions intomanageable size by first adding the column elementstwo at a time, and then adding pairs of these results in astaged system of adders. One would need approxi-mately log2N stages and N - 1 adders to combine thecross products in an N-bit multiplication. Since theoptical-carry adder gives an advantage in propagationdelay in executing additions, that advantage would bemagnified where several stages of addition are used,such as in a multiplier.

Ill. Demonstration Optical-Carry Adder

To help clarify the issues concerning the perfor-mance and practicality of such a device, we have as-sembled a 4-bit optical-carry adder with commerciallyavailable components. We use bulk-optic technologyin our optical system, which consists of acoustoopticmodulators, beam splitters and a He-Ne laser source.We show a diagram of the system in Fig. 2.

1558 APPLIED OPTICS / Vol. 26, No. 8 / 15 April 1987

Fig. 2. Diagram of 4-bit demonstration optical-carry adder.

The optical-carry path contains four modulators inseries, so high modulator throughput is important.For this application we use acoustooptic modulators(AOMs) with the deflected beam as the modulator onstate. These AOMs (TeO2 crystals, manufactured byNewport E.O. Systems) have high deflection efficiency(-90%), reasonably fast rise times (20 ns), and TTLcompatible drive units. One drawback of an AOM forthis application is the switching delay caused by theslow acoustic propagation (4.2 X 105 cm/s for TeO 2).In our system, we operate with the laser at the acousticfocus for the greatest deflection efficiency, which re-sults in delays of -500 ns.

For the light sources, rather than use separatelypulsed lasers, we use a single 5-mW He-Ne laser withits output divided into beams that go through addi-tional AOMs. This allows us to use the same switchingwaveforms and timing with the sources as with thecarry path modulators. In the Arazi design, a 4-bitadder requires four sources, but we reduce that to threeby forming the final carry bit (C4) with an extra step ofelectronic logic, in which the last detector signal (C.) iscombined with the final column addend bits (A3,B3), asC4 = C, OR (A3 AND * B3 ). This saves one AOM anddoes not increase the propagation delay since the finalcarry is not used further.

The light in the carry path is sampled with 50% beamsplitters which address Si photodiodes (5-ns rise time).The detectors are followed by high speed amplifiers(600-MHz BW, 10OX gain) and fast voltage compara-tors with TTL outputs (9-ns delay). These detector/comparator modules provide TTL output pulses withinput optical power of 30 MtW, at switching speeds up to-15 MHz.

The electronic logic system is built with TTL chips,with typical propagation delays of 3 ns. Since theoptical-carry signals are delayed several hundrednanoseconds by the AOMs, we include matching de-lays in the electronic logic. Hence all the system de-lays are pipelined, and the clock frequency is not limit-ed by the acoustic delays. The logic delay elementsare shift registers, so the clock frequency must beadjusted so that an integral number of clock periods

equals the acoustic delay. When operating with anacoustic delay of 500 ns, the clock frequency is a multi-ple of 2 MHz.

We have operated the demonstration optical-carryadder up to a clock frequency of 10 MHz, and it dis-played the correct 5-bit sum of two 4-bit addends. Wehave observed the output with a set of five LEDs andwith an oscilloscope probe of the output channels.The input data pulses were derived from the systemclock. We selected the numerical values for the ad-dends with a set of switches. We had the option ofdropping out a fraction (1/2, 3/4, 7/8) of the data pulseswhile maintaining the same clock rate, to simulatevarying data streams (i.e., regular input words or allzeros), as well as to check that delayed logic pulsescorrectly aligned with the carry pulses. We operatedwith a carry delay of 500 ns, as determined by the pointof maximum deflection efficiency in the AOM. Thisdelay corresponded to five clock cycles at 10 MHz.

The operating frequency was limited by several fac-tors that became important near 10 MHz. The AOMrise times of 20 ns began to broaden the carry pulses.To some extent, this effect could be compensated forby trimming the logic pulse widths, but that wouldonly work up to -15 MHz. There was a variation inthe arrival time of the carry pulses according to thedifferent addends present. This addend-dependentjitter in the carry pulses was due to the path lengthvariation among the various carry paths, which couldbe as much as 1.8 m in our system, giving temporalvariations of 6 ns. The paths were relatively long inour system since the AOMs were mounted on base-plates 30 cm long, a length determined by the AOMfocusing optics.

The variations in carry pulse length and arrival timewere most apparent when both carry pulses and logicpulses were present at the final XOR gate, when the sumbit output should be completely canceled. This wouldbe the case if there was a carry from the previouscolumn and one addend input to the column wherecancellation should occur. Incomplete cancellationdue to errors in overlap of carry and logic pulses causedshort logic error spikes in the sum outputs. The errorspikes could be suppressed (to below logic-high levels)so as not to affect subsequent processing by increasingthe time constants of the output stage. This suppres-sion would only work if the error spikes were narrowwith respect to the total pulse width. At 10 MHz, theerror spikes were 10-20% of the total pulses, so we tookthat as the maximum operating frequency. The mini-mum light level in our system was 100 1iW, so themaximum frequency was well below the limit imposedby photon shot noise.

Our test system shows that it is relatively easy toconstruct an optical-carry adder that works at modestspeeds. The maximum frequency of our system islimited by the optical modulator rise times and by pathlength variations. The delay is dominated by the opti-cal modulator acoustic delay. The operation limits ofour system are far below what might be achieved byusing fast optical modulators in a compact, optimized


IN tN1 12 11

Fig. 3. Generalized carry path, with light sources (Ij), beam split-ters with reflectivities Rj, detectors (D), and lossy modulators (M).

design. We discuss some methods for building anOCA that would extend the performance of state-of-the-art electronic arithmetic in the following section.

IV. Technology for a Competitive Optical-Carry Adder

We will now consider some of the technology issuesthat will affect the ultimate applicability of the opti-cal-carry adder. The OCA has a potential advantageover electronic CLA adders in terms of reduced propa-gation delay for long digital words (16-32 bits). TheOCA would have the same maximum clock frequencyas all-electronic adders (provided the longer electronicadder delays were pipelined), and that frequencywould be limited by the electronic logic rise times. Inan ideal OCA, the optical system would have rise timesand delays less than those of the electronic logic, sovery fast light sources, modulators, and detector/com-parators, as well as a short carry path would be needed.

We use here the example of a 32-bit adder operatingat 100 MHz, with electronic logic rise times and delaysof 1 ns. To keep the addend-dependent jitter in theoptical carry to below 1 ns, the total carry path must be<30 cm long in air (or 14 cm in LiNbO 3 , 8.5 cm inGaAs). Thus, a 32-bit OCA could have only 0.25-0.9cm/stage in the carry path, with one modulator andone beam splitter per stage. This means that we needan optical technology that allows very compact de-vices. We will discuss below the use of integrated-optic technology for this application. At the sametime, we will discuss the optical modulator technologywhich must provide modulation bandwidths of at least1 GHz.

A serious limitation on the data rate is presented bythe integration time necessary for optical detection,especially with the large optical loss through a multi-stage path of modulators and beam splitters. To get agood system bit error rate of 10-9, one needs a signal-to-noise ratio of 12, and for this SNR at 100 MHz, oneneeds a power of 0.2-0.5 MtW (with a good Si photodi-ode).10 Thus if the source powers are 1 mW, one canafford a carry path loss of-33 dB. That would imply aloss budget of only 1 dB per modulator/beam splitterin a 32-bit adder. Even if the input powers increase to100 mW, the allowable loss per stage is <2 dB.

The reflectivities of the beam splitters in the carrypath can be adjusted to give the best utilization of theavailable optical power, which will yield the maximumsystem bandwidth. If we maximize the minimumpower level that reaches any of the detectors, over any,of the possible carry paths, we will have the maximumsystem optical detection bandwidth. The set of reflec-

Table 1. Optimum Reflectivitles and Minimum Light Level for All-EqualSet of Reflectivitles, as a Function of the Number of Bits N and Modulator

Transmission M

Optimum Minimum light levelsN reflectivities M = 1 M = 0.9 M = 0.8

4 0.5 6.2 X 10-2 4.6 X 10-2 3.2 X 10-28 0.25 1.1 X 10-2 5.3 X 10-3 2.3 X 10-3

16 0.125 2.4 X 10-3 5.0 X 10-4 8.5 X 10-532 0.0625 5.6 X 10-4 2.2 X 10-5 5.6 X 10-7

tivities for the beam splitters could refer equally wellto the intensity coupling coefficients for optical wave-guide couplers.

We calculate the optimum set of reflectivities, Rj, asa function of the number of bits N and the averagemodulator loss M for the generic system show in Fig. 3.The power D,., at the nth detector coming from the jthsource (of power Ij), is given by

n-iDni = IjMnjRnRi ]1 (1-R), j<n-1,

k=j+l

Dn ni = In_1RnRn1M

Dnn = In( - Rd

The simplest method of optimizing the reflectivitiesis to make them all equal, then adjust their values sothat the minimum Dnj is maximized. The minimumDnj is then for the carry path with n = N and j = 1, orthe path from the first source to the last detector (withall the source intensities equal). The minimum ismaximized with R = 2/N, so that the minimum poweris given by

DNl = MN_1 (2)2(- 2N2 , with I = 1.

We show some values of the minimum light level (orsystem throughput), for some values of N and M inTable I. The optimum R for a 4-bit system is 50%,which is what we used in the demonstration OCA.The throughput falls rapidly with increasing numberof bits and increasing loss. In a 32-bit system withoptimized all-equal reflectances (R = 1/16), the beamsplitter sampling of the light in the carry line accountsfor nearly all the allowable system loss of 33 dB, leavingvirtually no loss allowable to the modulators. In a 16-bit system, the beam splitter sampling loss is 26 dB,which leaves only 7-dB loss for sixteen modulators or0.44 dB each.

Having a set of reflectivities which are all the samedoes not give the best possible utilization of the opticalpower, since much more light would reach the detec-tors near the start of the carry line. We can increasethe minimum light level by using a parabolic distribu-tion of reflectivities, with the minimum Rj in the mid-dle of the carry path and high Rj on the ends, i.e.,

R1 =R0 +B [j (N+ 1)]2

We optimize the parabola coefficients, Ro and B, tomaximize the minimum Dnj for various number of bits


and modulator loss. The results are shown in Table II.Even though the throughput is better than with all-equal Rj, there is still a strong roll-off with the numberof bits and the modulator loss. In this case, for a 32-bitsystem, the sampling loss is now only 27 dB, but for asystem loss of 33 dB the allowable loss for each modu-lator can be only 0.25 dB.

We expect that the parabolic distribution of reflecti-vities gives nearly the optimum utilization of the opti-cal power. We have performed calculations of reflec-tivity optimization using Monte Carlo techniques, i.e.,where we repeatedly make small random changes inthe reflectivities, then recalculate the minimum lightlevel, and accept the changes if the minimum in-creases. Over many cycles, the reflectivity distribu-tions converge to sets of values that are well describedby the parabolic form shown above.

The bulk-optic technology that we use in the demon-stration OCA is unsuitable for a useful device thatexploits the advantage of the optical-carry concept.The AOMs that we use as carry path and source modu-lators have rise times that are too long and have rela-tively long switching delays. The large size of theAOMs leads to significant carry-bit jitter. Further, aserial arrangement of AOMs is difficult to align.These problems could be alleviated with guided-waveoptical technology.

Integrated-optic technology is attractive for this ap-plication since one can build a compact chip that hasintegrated modulators, beam splitters (waveguide cou-plers), and possibly sources and detectors. Most ofthe integrated-optic devices that have been describedin the literature are made with lithium niobate(LiNbO3).11 2 This material has a high electroopticcoefficient and low loss waveguides can be readilyformed in it. Electrooptic waveguide modulators inlithium niobate have been demonstrated with band-widths in excess of 15 GHz.1 3 One can envision asingle-chip OCA in LiNbO3 with detectors and sources(laser diodes) connected to the edges via optical fibers.There are two drawbacks to using lithium niobate-based integrated optics for an OCA, and these are theissues of loss and size. We estimate that there wouldbe a loss of at least 6 dB per stage, due to propagation,bends, and coupling effects.12 That amount of loss ismuch higher than what could be tolerated in an adderfor long digital words. A single stage of such a devicemay require 4-6 cm of propagation length, due to thevery gradual bends that are used to minimize loss, aswell as the length necessary for the electrooptic modu-lation to occur. Therefore, this device would not be ascompact as desired.

An alternative to the passive material, lithium nio-bate, for an integrated-optic OCA, is to use an activematerial, like gallium aluminum arsenide (GaA-lAs).11,1 2"14 In an active material, laser diodes, detec-tors, waveguides, and even electronics can be com-bined on a single integrated wafer. The mostimportant advantage that an active material offers forthe OCA is the possibility of incorporating traveling-wave laser amplifiers (TWLA) in the structure. Dis-

Table 11. Optimum Reflectivities and Minimum Light Level for ParabolicDistribution of Reflectivities with the'Optimized Parabolic Coefficients Ro

and B

MinimumOptimum light

N M RO B level

4 1.0 0.40 0.18 1.8 X 10 -i0.9 0.38 0.20 1.5 X 1 0-l0.8 0.35 0.23 1.3 X 10 -l

8 1.0 0.20 0.049 4.6 X 10-20.9 0.16 0.052 2.8 X 10-20.8 0.13 0.058 1.6 X 10-2

16 1.0 0.093 0.0060 8.9 X 10-30.9 0.055 0.0062 2.9 X 10-30.8 0.027 0.0069 6.7 X 10-4

32 1.0 0.046 0.0006 2.0 X 10-30.9 0.014 0.0007 1.6 X 10-40.8 0.0025 0.0008 5.2 X 10-6

crete TWLAs have demonstrated a single-pass opticalgain of more than 25 dB, with an electrical modulationbandwidth in excess of 10 GHz.15 The issue of lossbecomes much less important with optical amplifiersin the carry path. In addition, the amplifiers could beused as the carry-path modulators, so that separateelectrooptic modulators would be unnecessary. Thistype of device could be quite compact since the func-tions of modulator and amplifier are combined, andthe TWLA itself is rather short (300 Mm). If a com-pact waveguide tap is used, the length of the carry pathcould be consistent with the requirements of a 32-bitadder operating with a 100-MHz data rate (0.25 cm perstage). It would be difficult to reduce the size of eachstage (with modulator and waveguide tap) to the 0.025cm that would be appropriate for a 1-GHz data rate.The size of the optical components thus limits themaximum data rate in an optical-carry adder.

V. Conclusions

We have demonstrated the optical-carry adder con-cept in a 4-bit adder constructed with commerciallyavailable components. This has shown that it is rela-tively straightforward to construct an adder based onthe scheme outlined by Arazi5 and has helped clarifythe system limits. The maximum data rate for ourdevice (10 MHz) is limited by several factors, includ-ing modulator rise time (-20 ns), addend-dependentjitter in carry-bit arrival time due to the size of oursystem (-6 ns), and electronic delays (4-8 ns). Ourdemonstration adder has pipelined delays of 500 ns,which is mostly due to the acoustic propagation time inthe acoustooptic modulators. This delay is not inher-ent to the optical-carry adder concept, but reflects thechoice of a modulator technology that allows high opti-cal throughput and convenience of operation.

We have shown how the ultimate performance of anOCA is limited by light loss and by size effects. Even ifthe carry path reflectivities are optimized to make thebest use of the available optical power, the light loss issevere in a multistage OCA. Small optical throughputincreases the integration time necessary for optical


detection, thereby limiting the maximum data rate.The most direct answer to the problem of loss appearsto be to use traveling-wave semiconductor laser ampli-fiers as the carry path modulators, such that the carrysignals are regenerated at each stage. An OCA withamplifiers might be fabricated on a single wafer ofGaAlAs using the techniques of optoelectronic integra-tion. Since the maximum variation in the arrival timeof the carry bits depends on the length of the carrypath, it is important to make the carry path as short aspossible (preferably <20 cm).

The optical-carry adder offers the potential advan-tage of a shorter propagation delay than is possiblewith an all-electronic adder, when adding long digitalwords (16-32 bits). The maximum data frequencywould be limited by the speed of the electronic logic inthe system. An OCA should be able to add 32-bitwords at a data rate of 100 MHz, with a propagationdelay of 5-10 ns, compared with 30-40 ns for electronicadders.

The authors are pleased to acknowledge the supportfor this research provided by the Rome Air Develop-ment Center, Griffiss Air Force Base, NY, under con-tract F30602-84-C-0130. We are grateful to H. JohnCaulfield for helpful discussions during the initiationof this project.

This paper is based on one presented at the OSAAnnual Meeting, October 1986, paper ML5.

References

1. S. D. Smith et al., "Nonlinear Optical Circuit Elements as LogicGates for Optical Computers: The First Digital Optical Cir-cuits," Opt. Eng. 24, 569 (1985).

2. H. M. Gibbs, Optical Bistability: Controlling Light with Light(Academic, Orlando, 1985).

3. A. A. Sawchuck and T. C. Strand, "Digital Optical Computing,"Proc. IEEE 72, 758 (1984).

4. A. Huang, "Architectural Considerations Involved in the Designof an Optical Digital Computer," Proc. IEEE 72, 780 (1984).

5. B. Arazi, "An Electro-Optical Adder," Proc. IEEE 73, 162(1985).

6. M. M. Mano, Digital Design (Prentice-Hall, Englewood Cliffs,NJ, 1984).

7. Motorola Schottky TTL Data Book (1983).8. V. Chandran, T. F. Krile, and J. F. Walkup, "Optical Techniques

for Real-Time Binary Multiplication," Appl. Opt. 25, 2272(1986).

9. S. Israel, S. C. Gustafson, and E. S. Cooley, "AsynchronousIntegrated Optical Multiply Accumulate with Sideways Sum-mer," Appl. Opt. 25, 2284 (1986).

10. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, &Winston, New York, 1976), Chap. 10.

11. R. C. Alferness, "Guided Wave Devices for Optical Communica-tions," IEEE J. Quantum Electron. QE-17, 946 (1981).

12. S. E. Miller, "Integrated Optics: A Technology with a Future,"Proc. Soc. Photo-Opt. Instrum. Eng. 517, 2 (1984).

13. C. M. Gee, G. D. Thurmond, and G W. Yen, " 17-GHz BandwidthElectro-Optic Modulator," Appl. Phys. Lett. 4, 998 (1983).

14. E. Garmire, "A Comparison of LiNbO3 and III-V Semiconduc-tor Technologies for Integrated Optics," Proc. Soc. Photo-Opt.Instrum. Eng. 460, 6 (1984).

15. J. C. Simon, "Semiconductor Laser Amplifier for Single ModeOptical Fiber Communications," J. Opt. Commun. 4,51 (1983).

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construction of an optical-carry adder

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