Optical crossbar interconnections using variable grating mode devices

Optical crossbar interconnections using variable gratingmode devices

Ho-In Jeon and Alexander A. Sawchuk

In this paper we propose a system design for an optical crossbar interconnection network utilizing variablegrating mode liquid crystal devices (VGM LCDs) which are optical transducers capable of performingintensity-to-spatial-frequency conversion. The proposed system performs real-time, reconfigurable, andnonblocking, but nonbroadcasting, optical crossbar interconnections. The fundamental limitations and theperformance of the system are discussed.

1. Introduction

With advances in technology and the declining costof computer hardware, there have been several effortsto design computer architectures consisting of a largenumber of processors executing programs concurrent-ly. Such parallel computing architectures are roughlyclassified into single instruction-multiple data(SIMD) and multiple instruction-multiple data(MIMD) systems. These systems require a communi-cation network between processors and between pro-,cessors and memories. 1' 2 A single shared bus is gener-ally not sufficient, because in parallel processingsystems it is desirable to allow many processors to senddata to other processors and/or memories simulta-neously. The network design involves a trade-off be-tween factors such as cost; ease of reconfiguration;contention (the possibility that messages may collidein certain network configurations); bandwidth; andcomplexity of the control algorithm.3 The crossbarnetwork is known to be one of the most powerful inter-connection networks, in that it allows all processorsand/or memories to be dynamically interconnected inan arbitrary permutation without contention andwithout moving any existing interconnections. Thedrawback of a crossbar is that its complexity grows asO(N2 ). Many different types of multistage networkswith O(N logN) switches which provide parallel com-munications at a lower cost have been proposed, 4 how-ever, these networks suffer from contention problems.

The authors are with University of Southern California, Signal &Image Processing Institute, Los Angeles, California 90089.

Received 20 August 1986.0003-6935/87/020261-09$02.00/0.© 1987 Optical Society of America.

The use of optics for communication purposes hasbeen increasing because optical systems have potentialadvantages such as high space-bandwidth and time-bandwidth products, inherent parallelism, and low in-teraction between separate beams in a linear medium.This makes it attractive to design interconnection net-works with optical systems.

In recent years, there have been many efforts toimplement optical interconnection networks not onlyfor VLSI systems,5 but also for optical processors.6Several researchers have suggested and implementedvarious optical interconnection schemes. Some tech-niques are presented in Ref. 7; one is an optical digitalcomputer configured as a parallel switching network,others are essentially optical methods of performingmatrix-vector and matrix-matrix products. Severalfunctional interconnection schemes using optical gatesare described in Ref. 8. The implementation of bothspace-variant, space-invariant, and hybrid intercon-nection schemes using computer-generated subholo-grams is given in Refs. 6 and 9. An optical implemen-tation of the perfect shuffle exchange network usinglenses and Wollaston prisms is presented inRef. 10. Aconcept for optical crossbar networks using directionalcouplers is given in Ref. 11, although it provides only aplanar interconnection scheme which does not fullyexploit the advantages of 2-D optical processing sys-tems.

In this paper we propose a system design for opticalcrossbar interconnection networks using variable grat-ing mode (VGM) devices. These devices are opticaltransducers capable of performing an intensity-to-spatial-frequency conversion. The proposed systemperforms a real-time, reconfigurable, and nonblocking,but nonbroadcasting, optical crossbar interconnec-tion. In Sec. II, the operating principles and proper-ties of VGM devices are reviewed. Section III de-scribes how the proposed system works. The analysis

15 January 1987 / Vol. 26, No. 2 / APPLIED OPTICS 261

and fundamental limitations of the system are pre-sented in Sec. IV, and some new physical mechanismswhich can be used as interconnection elements aresuggested in the conclusion.

1. Variable Grating Mode Liquid Crystal Devices' 2,13

Variable grating mode liquid crystal devices (VGMLCDs) are a new class of optical transducers which canperform an intensity-to-spatial-frequency conversionover a 2-D image field. The VGM LCD primarilyconsists of a photoconductive layer in series with alayer of nematic liquid crystal mixture as shown in Fig.1(a). When a dc bias voltage above a threshold isapplied across the device to provide a voltage divisionbetween the two layers, a domain instability is formedin the liquid crystal layer as shown in Fig. 1(b). (Theword domain in this paper is reserved for the descrip-tion of the phenomenon of bright and dark stripeswhich appear when dc voltage is applied across thedevice. This phenomenon, which is caused by thevariation of the optical path length due to the periodicperturbation of the liquid crystal uniaxial index ellip-soid, is generally named domain structured instabilityin liquid crystals.) The width d of the domain period(line pair) is inversely proportional to the applied volt-age V according to

a Fig. 1. VGM cell: (a) schematic diagram of the VGM device con-(1) struction; (b) phase structure viewed through a polarizing micro-

scope.

where a is a constant that is dependent on the particu-lar liquid crystal material.

When a write beam illuminates the cell, the inputintensity causes the voltage across the photoconduc-tive layer to decay and correspondingly enhances thevoltage across the liquid crystal layer. It follows thusthat the photoconductor implements an intensity-to-voltage conversion which changes the spatial frequen-cy of the liquid crystal layer according to Eq. (1). Thetransverse conductivity of the photoconductor is gen-erally low, so that the phase grating formed is highlylocal. If both processes are considered together, thedevice is seen to perform local intensity-to-spatial-frequency conversion. A collimated readout beam in-cident on the device at a wavelength where the photo-conductor is insensitive is angle-encoded within eachimage pixel by diffraction from each induced phasegrating and consequently can be steered in any 1-Ddirection orthogonal to the grating orientation. Thistype of process is shown schematically in Fig. 2. Inthis figure, an input consisting of two different bright-ness levels is impressed on the VGM device; two dis-tinct spatial regions with different grating spatial fre-quencies are created. If the VGM device isilluminated by a collimated readout beam at a wave-length where the photoconductor is insensitive, severaldistinct first-order diffraction patterns appear in theFourier transform plane of Fig. 2. By placing filters inthe Fourier transform plane to manipulate the ampli-tude of the diffracted components, an output imagewith a modified gray scale can be observed in theoutput plane. In this manner, the VGM LCD has been

Input VGMLiquid Crystal

Device

OutputFilterPlane

Fig. 2. Experimental setup for demonstration of intensity-to-posi-tion coding by means of an intensity-to-spatial-frequency conver-

sion in a VGM LCD.

used to implement arbitrary nonlinear point transfor-mations of image intensity.14'15

When designing an optical interconnection system,the operational properties of the VGM device must beconsidered. Some of these properties include the ac-cessible range of spatial frequencies; size of the diffrac-tion orders; the functional dependence of diffractionefficiency on the applied voltage; maximum diffrac-tion efficiency; response time; device uniformity; de-vice input sensitivity; and the polarization characteris-tics of the device.

The accessible range of spatial frequencies extendsfrom the threshold for grating formation to the onset ofdynamic scattering induced by high electric fields.The typical range of spatial frequencies is from 200 linepairs/mm to over 600 line pairs/mm.

The size of the diffraction orders is determined bytwo factors: diffraction from finite-sized pixel aper-tures and grating imperfections that cause local devi-

262 APPLIED OPTICS / Vol. 26, No. 2 15 January 1987

(a)

i I 100 m

(b)

ations from constant spatial frequency. The mostcommon type of grating imperfection is the joining orsplitting of grating lines as shown in Fig. 1(b). Thisgrating imperfection is not well understood, althoughit affects the accuracy of the optical computing and/orinterconnection systems. It is shown that, when abeam is incident on the imperfection, the spot size ofthe diffracted beam increases, which may increase un-expected crosstalk and thus reduce the number ofinterconnection elements. However, some experi-ments being carried out at the Hughes Research Lab-oratory have shown that the degree of grating imper-fections is proportional to the rate of change of appliedvoltage. 16 ,17

The diffraction efficiencies of the VGM LCD havenot been extensively modeled analytically, nor haveVGM LCDs been classified as thin or thick gratings.For simplicity of calculation, they are often regardedas thin gratings. In this paper, the experimental re-sults of the functional dependencies of diffraction effi-ciency on the applied voltage have been considered toevaluate the overall diffraction efficiency of the sys-tem. As can be seen in Fig. 3, the diffraction efficien-cies of the second-order components are -20% and arehigher than that of the first order. This phenomenonoccurs due to the peculiar nature of the birefringentphase grating formed by the VGM distortion.

Response times of current VGM devices are relative-ly slow. Work on measuring and improving the re-sponse time has been done at the Hughes ResearchLaboratory by Soffer et al. 17 The rise time for gratingformation from below to above threshold is typically ofthe order of 1 s, and the decay time is of the order of 50ms. The effects of thickness on the VGM responsetime were also tested. It is seen in general that thethinner the cell, the shorter the turn-on time (delaytime plus rise time). To improve the VGM response tobe compatible with TV frame rates or faster (of theorder of tens of milliseconds), Soffer et al. suggested anovel approach and some preliminary results forachieving a faster dynamic response by spoiling thelong range order of the periodic domains. In recentexperiments on these modified devices, the rise timehas been reduced to 600 200 ms and the decay timereduced to tens of milliseconds. Alternatives to theVGM device to improve the resonse time have beenexamined. One alternative device that preserves theessential feature (intensity-to-spatial-frequency con-version) uses the electrooptically induced birefringentchange in LC in wedge or prism-shaped devices. Somepreliminary experiments to examine these ideas havebeen performed.1 7

The input sensitivity of the VGM LCD, defined asthe input (writing) intensity per unit area per unitchange in grating spatial frequency, is determined byseveral factors. These include the slope of inducedgrating spatial frequency as a function of applied volt-age for the particular nematic liquid crystal mixture;the wavelength dependence of the photoconductivelayer photoconductivity; and the cell switching ratio(fractional increase in voltage across the liquid crystal

100

zZs

LI

z00- I

1001

U

- U

AAU AA

AAA-A

I I I I I I

* 2ND ORDER

* ST ORDER

aAA

I I I I I _

10 20 30 40 50 60APPLIED VOLTAGE (VDC)

70

Fig. 3. Diffraction efficiency as a function of applied voltage acrossa nematic liquid crystal mixture of phenylbenzoate (HRL2N40).The layer thickness was -6 Am, as defined by a perimeter Mylar

spacer.

layer from illumination at the threshold for gratingformation to saturation).

The uniformity of VGM LCD response depends onthe layer thickness, homogeneous mixing of the liquidcrystal material, and the as-deposited spatial depen-dence of photoconductive sensitivity. The nonunifor-mity of the device response characteristic can be acontributing factor in the establishment of the maxi-mum number of interconnection channels. The uni-formity can be improved significantly by improve-ments in the photoconductive layer depositionprocess.

The polarization-dependent properties of the VGMLCD are quite striking and provide an important fac-tor in designing a VGM crossbar interconnection net-work. When a dc bias voltage is applied across thenematic crystal layer, the orientation of the grating isconfigured such that the grating wave vector is perpen-dicular to the direction of unperturbed alignment,which is homogeneous and induced by unidirectionalrubbing or ion-beam milling. For all linear input po-larization angles, the even and odd diffraction ordersare found to be essentially linearly polarized. Asshown in Fig. 4, the even orders are linearly polarizedparallel to the domains comprising the VGM grating.For input polarization perpendicular to the domains,the even orders are found to be almost extinguishedand the odd diffraction orders are linearly polarizedwith a major axis that rotates counterclockwise withessentially constant intensity at the same rate as theinput polarization is rotated clockwise. This effect isthe same as that produced by a halfwave plate orientedat 450 to the grating wave vector. More physical andmathematical analyses of polarization-dependentproperties of the VGM LCD are given in Ref. 18.

III. Operating Principles of a 1 -D VGM Crossbar

In this section, a system which implements a real-time, reconfigurable, and nonblocking optical crossbar


-

network utilizing a VGM LCD is presented. Beforedescribing the operating principles of the system, it ishelpful to examine the diffraction properties of VGMphase gratings as shown in Fig. 5.

The VGM effect introduces a birefringent phasegrating in the nematic liquid crystal layer when a biasvoltage is applied. Thus the complex amplitudetransmittance function t(xy) of a VGM cell can bewritten as

t(x,y) = expL70(xy)], (2)

where

(xy) 27r[n(xy)-1] e(xy). (3)

Here e(x,y) is the crystal thickness and n(xy) is therefractive index. Periodic variations in e(x,y) andn(x,y) produce periodic variations in t(xy). For sim-plicity, we let the phase be

mrk(x,y) - sin(27rvx),2

PIN

VGM DOMAINORIENTATION

POIFFRACTED

T t -- *tl 2 34

1 2 3 4

2 3 4

l 2 3 4

Fig. 4. Polarization behavior of VGM diffracted orders, with illu-mination normal to the plane of the liquid crystal layer. The left-hand column indicates the input polarization associated with eachrow of output polarizations. The inset shows the corresponding

orientation of the VGM grating.

m)rn=4

(4)

where the parameter m represents the peak-to-peakexcursion of the phase delay and the spatial frequencyv is determined by the magnitude of applied voltage.Combining Eqs. (2) and (4) and assuming a squareVGM cell of dimensions r by r, the windowed complextransmittance function t(x,y) becomes

t(x,y) = exp - sin(2irvx)] rect (-) rect (5)

If the VGM cell is located in the front focal plane of apositive lens with a focal length f and is illuminated bya unit-amplitude, normally incident coherent planewave, the field distribution U(xo,yo) at the Fouriertransform plane is given by19

Fig. 5. Diffraction property of the VGM phase grating.

sume that pvf is large enough, we can omit cross termsto obtain the intensity distribution I(xo,yo) at the Fou-rier transform plane given by

I(XOYO) () > (2 () svXf) sinc{

(7)

From Eq. (7) we see that there are components at manydiffraction orders. The peak intensity at the pthorder is

U(x0,y0) = 1 Fjt(x,y))1 Xo _Yo

= -F {exp [i 2 sin(27rvx) rect ) rect (T)} xo y,ixf I 1i I (1) (Yg~fX=Xj'f'=Yj

(2)(fx -V, fY))2 1

X sinc (rfx) sinc(-fy)Il XO YO

fX=Tf, f Y=f

= E J ( 2 ) sincji(fx-pv)) sinc{rfyjl x0 y

jAf pie P ( 2 )X =in {xf (Xo-pan} si°},fy=

= ~~~~ ~sinc (x -T v~) sinc {~}

where we denote the Fourier transform by F and usethe identity

exp [i 2 sin(2rvx) = E J (2) exp(j2irpvx),

and Jp is a Bessel function of the first kind, order p.Here p determines the diffraction orders. If we as-

P ( 2 )

X2f2

and the spacing between each order is vXf.The VGM crossbar system is shown in Fig. 6. There

are 2K + 1 electrical input lines, each driving a sepa-

264 APPLIED OPTICS / Vol. 26, No. 2 / 15 January 1987

T2=- 7 J,

Af P=-.

-i LD-LLDK,

- LD,

-O

IK/ VK

Fig. 6. Schematic diagram of a 1-D VGM crossbar.

t(Xy) = exp[ sin27rv(x - ir) rectX ( )rect(-)

(9)

where N = 2K + 1. Utilizing Eq. (6), we obtain theamplitude distribution at the Fourier transform plane

U(X0'Y0) = xf Ft(xy)}1 Xo Yf o

2 K r=\ 2r °*, j XT ~ § sinc -r( -p sinc Xfi=-K p=-- ~ If / j

* exp(-j2ipvij) exp (-j2iriTx0/Xf). (10)

Omitting the multiplicative complex phase factors, weget

2 K X~

U(X0 y 0) = E Z 2 (i=K p=--

rate laser diode denoted by LDi, -K,...,i,...,+K.The light from each laser diode separately illuminatesone element in a 1-D array of 2K + 1 electricallycontrolled VGM cells (or subcells) as shown in Fig. 6.Any beam spreading or collimating optics is omittedfor clarity in Fig. 6. Each VGM cell of the 1-D arrayconsists of a layer of nematic liquid crystal mixtureplaced between two transparent electrodes and eachcell has vertical dimension r. There are no photocon-ductive layers in the structure. Thus the spatial fre-quency vi of the ith cell is controlled by the appliedvoltage Vi. The Fourier transform lens forms the dif-fraction pattern of the VGM array in the back focalplane, where there is a mask to eliminate unwanteddiffraction orders. A detector array in the back focalplane detects output signals from positive and nega-tive diffraction orders and electrically combines themto produce an output in 2K + 1 parallel channels.Thus the VGM cells serve as an electrically controlla-ble beam steering array to direct the input signal lightto one of several spatially separated output channels.Because the center of the ith VGM cell is displaced byi-, the complex transmittance function t(xy) of theith VGM cell is given by

ti(x,y) = exp[i 2 sin2irv(x - ir)] rect ( i) rect (Y) (8)

and the overall amplitude transmittance functiont(x,y) of the array of N VGM cells becomes

X sinc (X - Xfvip)} sinc {Y} (11)

Because we combine the +1 and -1 order diffractedbeams incident on a particular pair of detector ele-ments, we need a spatial filter which can block beamsfrom all other orders without blocking any of the +1and -1 order beams having spatial frequencies fromXfV-K to XfvK. Suppose that the range of spatial fre-quencies that can be formed in VGM devices is

Vmin < Vi < Vmax (12)

for -K,... ,i,... ,K, then the smallest distance of thesecond-order beam from the origin in the Fouriertransform plane is given by 2 Xfvmin as shown in Fig. 7.To avoid overlap between the maximum distance ofthe first order and the minimum distance of the secondorder, we must have

2XAfmin > 'fAmax,

or

VmaxPmin > P2 (13)

Thus, for example, if we use a Vmax of 600 cycles/mm,Vmin must be >300 cycles/mm, the vi terms are confinedto the region of 300 cycles/mm < vi < 600 cycles/mm,and the detectors in the array must be placed betweenXfvmin and Afvmax. If we place the spatial filter de-signed in these conditions in the Fourier transform

Fig. 7. Intensity distribution in the Fourier transformplane of a 1-D VGM crossbar.


I I

�_ f f__�

S. (x

r r x r 2i2 2 2 2 2Fig.8. Nidentical rectangular pulses with pulse widthr modulated

by N different frequencies.

plane, only the +1 and -1 order terms of Eq. (11) arepassed and the filtered complex amplitude field is

jUf ( 2 ) sinc {(X 0 - Xfvdl siac { °}M i=_K ITJ XV f

+T2 Im\ K T ) ~_jf- -2 E sinc{ (X + fv) sinc{-

Jxf \/i=-K (x0f + sinc

(14)

Again, if we assume that r is large enough to omit thecross terms, the intensity distribution in the Fouriertransform plane can be written as

I(xo,yo) = IU(xoyo)12

IT2} 1 ( 2 ) IEX {t(°f )) {A= {A } J 1 () sinc f (X0 - Xfvi)} sinc 2 Y}

+f { 2 ( 1-))

X E sinc 2 T (x0 + fvi)} sinc 2fTY} * (15)

This intensity distribution consists of N = 2K + 1 sinc2functions which overlap each other in the Fouriertransform plane as shown in Fig. 7. If the VGM cellsize r is small, we have many input channels, althoughdiffraction in the Fourier plane will cause crosstalk toadjacent output channels. On the other hand, if islarge, it reduces the maximum number of input chan-nels because the overall dimensions of the array andthe optical system are fixed. Thus, there must be atrade-off between the maximum number of intercon-nection elements and the signal-to-noise ratio due tothe crosstalk. A discussion of this problem and other

fundamental limitations such as reconfiguration time,diffraction efficiency, broadcasting, etc. is given in Sec.IV.

IV. System Analysis and Fundamental Limitations

One of the most important factors which must betaken into account in analyzing an optical interconnec-tion network is the maximum number of interconnec-tion elements that can provide a tolerable signal-to-noise ratio. The optimization problem involving thenumber of interconnection elements, the signal-to-noise ratio, and crosstalk is discussed in this section.Because the proposed system is reconfigurable, thereconfiguration time is another important factor forassessing the system performance. The overall dif-fraction efficiency of the system is also presented inthis section.

We model the system as a type of frequency-divi-sion-multiplexing system which is bandlimited be-tween fvmin and fvmax- We want to transmit theamplitude of N spatially disjoint rectangular pulsessi(x), each having pulse width r as shown in Fig. 8.Figure 9 shows the spectrum of the multiplexed signal.Utilizing ideal bandpass filters as shown in Fig. 9, wecan demultiplex the N signals. However, the spec-trum of each rectangular pulse is not bandlimited and,as the number of signals increases, the degree of cros-stalk will increase. This consequently lowers the sig-nal-to-noise ratio, where we define the noise as thetotal leakage of interfering terms in a given specificchannel due to the sinc2 functions arising from othersignals. We want to find the maximum number N inthe sense that all channels have some minimum speci-fied signal-to-noise ratio.

If we denote by XfAv the bandwidth of the bandpassfilter, N becomes

N f= fVmax - fvmin BXfAv Av

(16)

where we let XfVmax - XfVmin be the overall bandwidthXfB of the system. The signal power P8 at the ithdetector is given by

{r2 Ji ( 2Av,

PS= - Xf J' J"ftli A,) sinc f j~j -XfVil dv, (17)

which integrates the intensity distribution in the Fou-

Fig. 9. Spectrum of the multiplexedsignal.


r

S' (XJ

rier transform plane due to si(x). The noise power PNat the ith detector is

p = 2 2 sinc2 (v-Xfv 1)} dv,

jvi

(18)

and represents the summation of the power in all theother signal terms in the region from

Xf (vi- 2A) to Xf (vi + )

By changing the variables of Eq. (17) to 1/(f)(v - Xfvi) = v', we get

A (v - Afvi) = rv',

A dv = dv',

(19)

(20)

and the upper and lower limits of the integration givenin Eq. (17) become

A {xf (vi + 2- Xfvi} =AV

AXf i 2 ) Afi -2A/Ix (i ~v- AV - Xf} = - AV

respectively. Since v' is a dummy variable, Eq. (17)can be written as

4J (m) ^

P f 2 sinc 2 (Tv)dv

X\f f: ( )X 21

2

XfJ B

2iN

where we have used Eq. (16).Referring to Fig. 10, we can further simplify Eq. (18)

because the N sinc2 functions are identical and sym-metric. Here we consider the case for N = 4. Thenoise power at the third detector is given by the sum-mation of areas:

area(fgkj) + area(egki) + area(fgkh),

where area(fgkj) represents the leakage effect of thefirst channel, area(egki) is the second channel, andarea(fgkh) is the fourth channel. Because we have theidentities

area(fgki) = area(abdc),area(eghi) = area(cdge),area(fgkh) = area(hkml),

we obtain the noise power of the third detector by thesummation of two areas:

area(abge) + area(hkml).

This means that we can calculate the noise power atthe ith detector without summing the power in all theother signal terms in the region from

Xf (Pi - 2-) to Xf (vi + 2A)V

Fig. 10. Diagram for calculation of the noise power of the ithdetector.

i.e., the noise power at the ith detector can be ex-pressed as the sum of two integrations of the function

sinc2 {Tf (x- f}from Xfvmin to

At (i-A2P) and f(vi + AP)

to Xfvmax:

r Xf I vv

=N sin Jx 2/ (v - Xfvi dv

+ fx sinc2(If (v-Xfv)} dv]- (22)

Again, by changing variables as expressed in Eqs. (19)and (20), the upper and lower limits of the integrationsgiven in Eq. (22) become

AXf i 2 ) A/i -2 = 2N

WV $fvmin- Xfvil = Vmin -Pi

I 1Xfvmax- fvi = Vmax -Vi

b At {At~Ix (vi + AP \-fi = AV = 2N Xf Iv2) 2 22Nrespectively. Thus Eq. (22) becomesV2(M) B.2

PN 2 f 2N sinc (rv)dv + B sinc2( r)d* (23)

Since vi can be expressed by

: V~~~~i = Pmn + (i- )/A + A V

= vmin + (i-)v2) 2

VMin I

the lower limit of the first integration of Eq. (23) be-comes


Vmin -{min + (i - 2) } = - (i ) Nand the upper limit of the second integration becomes

Vmax - vmin + (i _ 1) B} = B - - = (N -i + 1) B .

Finally, the noise power at the ith detector is given by

IV sin_ 2v(d I\2Nsic\21 = A ) [ Dsinc v^dv + sinc2(Tv)d]

(24)

From Eqs. (21) and (24), we obtain the signal-to-noiseratio at the ith detector by

BJ2N sinc2 (Tv)dvI__

2NPS

(N-i+ ~JN_ sinc2 (rv) dv + ( )N sinc2(rv)dv

Note that r is given by - = D/N, where D is the overallsize of the VGM array. Substituting this into Eq. (25)we obtain

B

2UN sinc 2 ( v) dv . (6ps B: sinc2 (Pv'dv (N

N Jsi (N v) dv + 2N sinc2 D ) d

Intuitively, Eq. (26) should be a monotonically de-creasing function of N. Because the sinc2 functioncannot be integrated in closed form, it is mathemati-cally difficult to find an analytic expression for Eq.(26). For this reason, in Fig. 11 we give a numericalcalculation for the maximum number of interconnec-tion elements arbitrarily assuming a minimum SNR of1. We perform this calculation for the detector ele-ment located in the center of the N elements; thiselement is subject to the maximum crosstalk and pro-vides a worst-case estimate. Thus, for example, if D =70 mm, mi = 400 cycles/mm, max = 800 cycles/mm,we obtain the maximum number of interconnectionelements as N = 127. As shown in Fig. 11, the maxi-mum number of interconnection elements increases asthe minimum spatial frequency of the VGM deviceincreases and the overall size D of the array increases.

The reconfiguration time of the VGM crossbar de-pends essentially on the response time of the VGMdevice used. The response time of existing VGM de-vices is of the order of hundreds of milliseconds, aresponse time which is very long compared to that ofLED arrays. Thus, the reconfiguration time of theVGM crossbar is of this order. However, this longreconfiguration time of the VGM crossbar may betolerated in applications where large blocks of data aretransferred, such as in graphics or image processing.Recent work to improve the VGM response time mayreduce the reconfiguration time in the future.

The overall diffraction efficiency of the system canbe crudely estimated using previous VGM experimen-

70D=80

20 260 300 340 380MIN. SPATIAL FREQ)UENCY (CYCLES/mm )

Fig. 11. Maximum number of interconnection elements with re-spect to the minimum spatial frequency of the VGM device and the

overall size D of the array.

tal measurements and the diagram shown in Fig. 3. Itcan be seen from Eq. (15) that the intensity distribu-tion of +1 and -1 order beams is symmetric withrespect to the origin of the x0 axis. Thus, if we electri-cally combine the +1 and -1 order beams in the detec-tor as shown in Fig. 6, the diffraction efficiency of thesystem can be improved by a factor of 2 given by 0.08 X2 = 0.16 = 16%, where the approximate value of 0.08was obtained from Fig. 3. If we use the second-orderbeam, we can get the diffraction efficiency up to 0.2 X 2=0.4 = 40%.

The VGM crossbar has a shortcoming in that it doesnot perform broadcast (one-to-many) interconnec-tions. This property can be explained by comparingthe operating principles of VGM devices with those ofacoustooptic cells. Acoustooptic devices may havecomposite spatial frequencies making up the propa-gating phase grating produced according to the electri-cal signal. On the other hand, the spatial frequenciesof the phase grating formed in VGM devices areuniquely determined by the applied dc bias voltageand do not propagate. In other words, the phase grat-ing is not of a composite form but of a fundamentalfrequency. Thus, this VGM crossbar cannot be di-rectly used where broadcasting is required.

V. Discussions and Conclusions

Advances in technology and the declining cost ofcomputer hardware have encouraged the design ofcomputer architectures consisting of a large number ofprocessors executing programs concurrently. Suchparallel computing architectures require communica-tion networks between processors, and between pro-cessors and memories, which allow many processors tosend data to other processors and/or memories simul-taneously. Optical techniques for these communica-tion purposes provide some potential advantages.Many techniques for implementing optical intercon-nection schemes have been reported in the literature.7

Real-time reconfigurable optical interconnectionsystems are the most desirable for existing applica-tions. Some of the possible schemes to implement


- forl

these use two acoustooptic Bragg cell arrays which cansteer the input beam two-dimensionally. Anotherscheme is to use real-time holography and/or comput-er-generated holograms. Yet other schemes utilizespatial light modulators. Techniques for implement-ing real-time optical interconnection networks utiliz-ing directional couplers and matrix-vector and ma-trix-matrix processors have been reported. In thispaper we have described a real-time, reconfigurable,and nonblocking, but nonbroadcasting, optical cross-bar interconnection network exploiting the propertiesof VGM LCDs. The fundamental limitations and ananalysis of the performance of the system are present-ed.

This research is supported by DARPA/ARO Con-tract DAAG29-84-K-0066.

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EThe History of Science Society has announced a programof grants to unaffiliated scholars-people trained in the his-tory of science who are either unemployed, unaffiliated withany institution making use of that training, or employed ei-ther part-time or without prospects of continuation or re-newal. With the support of the Rockefeller Foundation, sup-plemented by the C. Doris Hellman Pepper Memorial Fundand the Culpeper Foundation, awards of up to $1,000 will bemade to facilitate research or travel to prospective job inter-views. Applicants must have a Ph.D in the history of scienceor a closely related field. For information write Joseph W.Dauben, HSS Coordinator of Programs, Dept. of History,Herbert H. Lehman College, CUNY, Bedford Park Blvd.West, Bronx, NY 10468.


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Optical crossbar interconnections using variable grating mode devices