We show that the entropy of the distinct connection patterns that are possible with multifacet optical interconnection architectures is approximately equal to the space-bandwidth product of the optical system

Brenner, K.-H.

Lohmann Adolf W.

Ozaktas Haldun, M.

Bilkent University Institutional Repository

418 J. Opt. Soc. Am. A/Vol. 10, No. 3/March 1993Interpretation of the space-bandwidth product as theentropy of distinct connection patterns inmultifacet optical interconnection architecturesHaldun M. OzaktasDepartment of Electrical Engineering, Bilkent University, 06533 Bilkent, Ankara, TurkeyK.-H. Brenner and Adolf W LohmannLehrstuhl Angewandte Optik, Physics Institute, University of Erlangen-Nirnberg, Staudstrasse 7/B2,W-8520 Erlangen, GermanyReceived July 22, 1992; revised manuscript received October 13, 1992; accepted October 16, 1992We show that the entropy of the distinct connection patterns that are possible with multifacet optical intercon-nection architectures is approximately equal to the space-bandwidth product of the optical system.1. INTRODUCTIONThe importance of the number of distinct connection pat-terns that can be wired up between N input and N outputnodes was emphasized by Keyes.",2 In the context of mi-croelectronic packaging, the input and output nodes maybe the terminals of logic gates on a chip or the pinouts ofchips on a multichip carrier. In the context of an opticalinterconnection system, the input nodes may be opticalsources and the output nodes optical detectors on the sur-face of electronic processing elements. The number Q ofdifferent ways in which the input nodes can be connectedto the output nodes is a measure of the flexibility affordedby that interconnection system. A designer confrontedwith the task of implementing a particular connectionpattern or circuit diagram has a much greater chance ofsuccess if Q is larger. Because Q can be extremely large,it is more convenient to work with its logarithm, fl =log2 Q, which can be interpreted as the information (en-tropy) required to specify a particular connection patternout of all the possible patterns. For instance, for a fullcrossbar interconnection network, where each input canbe tied to any set of outputs,N NQ 2 C(N, n) = (2N)N = 2N2,intoN2 ,since each of the N inputs has the choice of being con-nected to 0 c n - N output nodes in C(N, n) distinct ways{C(x,y) xV[y!(x - y)!]}. An alternative way of lookingat this is that there is a binary decision (to connect or notto connect) between each of the N2 possible input-outputpairs.Here we discuss the number of distinct connection pat-terns that are possible with a quite general class of free-space optical interconnection architectures, discussedpreviously by several authors.3 " The generic form of thearchitecture is depicted in Fig. 1, several other essentiallyequivalent forms being possible. The light originatingfrom a particular input channel is made to fall on anynumber of filter facets on the filter (Fourier) plane,through the use of appropriate phase elements correspond-ing to the desired positional shift in the Fourier plane.(Figure 1 shows the use of prisms as phase elements; if thelight is intended to fall on more than one filter facet, mul-tiple gratings should be used.) Each filter facet is as-sumed to be a phase element corresponding to a singledistinct displacement between the input and outputplanes. All input channels using the same filter facet willbe displaced by the same relative amount in the outputplane. If one's goal is to displace two input channels bydifferent magnitudes or directions in the output plane,different filter facets must be used. Hence this architec-ture allows partially space-variant mapping of the inputsto the outputs. 13The main result of the research presented in this paperis that the value of fl for this architecture is approxi-mately equal to the space-bandwidth product of the opti-cal system. This result provides a new interpretation ofthe space-bandwidth product, which is a measure of thethroughput and the cost of the system. In Section 2 wederive this result in a simple manner by ignoring edge ef-fects. In Section 3 we provide a more complicated deriva-tion that takes edge effects into account, which results ina slightly different but essentially identical result. Mostreaders will prefer to disregard this section, which hasbeen included for mathematical completeness.2. ANALYSISWe assume that each of the M filter facets corresponds toa distinct relative displacement between the input andoutput planes and that at least one input channel employseach filter facet. (If these conditions are not satisfied,the number of filter facets could be reduced without de-creasing the number of distinct connection patterns that0740-3232/93/030418-05$05.00 © 1993 Optical Society of AmericaOzaktas et al.Vol. 10, No. 3/March 1993/J. Opt. Soc. Am. A 419I nput Lens FOUFig. 1. Multifacet, partially space-variant optical interconnection arclnels in the input and output planes, and there are M filter facets. F(manner.are possible. The use of more than one filter facet withthe same displacement, or leaving a filter facet unused, issimply wasteful.) Since there are N output channels, atmost N distinct displacements are possible for each input(directed toward each of the N outputs), so that if eachfilter facet is to be distinct, M c N.[In fact, the set of possible displacements for a particu-lar input channel depends on its location in the inputplane. Although N distinct displacements are possible foreach input (directed toward each of the N outputs), a totalof =4N distinct displacements is possible when we con-sider all input channels. In the present derivation, we areassuming that all input and output channels can betreated uniformly regardless of their location in the inputor output plane. A more exact analysis is presented inSection 3.]For simplicity, the area and shape of the numerous in-put and output channels and those of the numerous filterfacets are assumed to be identical (as in Fig. 1). Let thearea of each input channel be denoted by a so that the areaof the input plane is Na (in square meters), and let thearea of each filter facet be denoted by a so that the area ofthe filter plane is Ma (in reciprocal square meters). Thespace-bandwidth product of the system (SB) is equal tothe product of the area of the input plane and the area ofthe filter plane, SB = NMaa. Also, the area of the inputchannels and the filter facets must satisfy aa - K, whereK is a coefficient of the order of unity, because the lightdistribution at the input channel and that at the filterfacet form a Fourier-transform pair. Thus we have 3NM s SB' SB/K. (1)We assume that the space-bandwidth product of the sys-tem is not underutilized, so that the above relation is sat-isfied with equality. Since M ' N, it also follows thatM ' NS and N Ž VB.We allow for arbitrary fan-out at the input plane andarbitrary fan-in at the output plane as well as the possibil-ity of some nodes not being involved in any connections.Lens Outputhitecture. f is the focal length of the lenses. There are N chan-'r simplicity we assume that these are arrayed in the Cartesian(As mentioned above, for fan-out to be established, theprisms in Fig. 1 must be replaced by multiple gratings.)Between 1 and N input channels may utilize the samefilter facet. All these channels will receive the same dis-placement in the output plane. i out of N channels mayutilize the same displacement in C(N, i) distinct ways.Summing over i, we obtainN>C(N, i) = 2N _ 1.Since the same multiplicity exists for each filter facet,there are (2 N - 1)M- 2NM distinct ways in which the Ninput channels can be mapped to the M displacements.(In fact, all the N input channels cannot utilize a givendisplacement. For instance, a channel at the rightmostedge of the input plane cannot utilize displacement vec-tors with positive components in the x direction becausethis would result in its being imaged outside the outputplane. The greater the magnitude of the displacement,the smaller the number of input channels that can utilizeit. As is mentioned above, we are ignoring such edge ef-fects in the present derivation.)Now we must determine the number of ways in which Mdistinct displacements can be chosen out of N possible dis-placements. This is simply C(N, M) since the particularorder in which the displacements are assigned to the filterfacets is not important. [If the displacements associatedwith two filter facets are interchanged, this can be com-pensated by adjusting the deflection components at the in-put plane that select the filter facet(s) that each inputchannel will use. Thus the two situations would not cor-respond to distinct connection patterns.]Therefore the number of distinct connection patternsthat can be implemented with M filter facets is given byQ = C(N, M)2m, (2)(1 = 10g 2 C(N, M) + NM = log2 C(N, M) + SB'. (3)The function C(N, M) will be approximately at its maxi-mum value when M N/2, rounded to the nearest in-Ozaktas et al.420 J. Opt. Soc. Am. A/Vol. 10, No. 3/March 1993teger. Using NM = SB', we find that M -/SB'12 andN - V2SB. SinceN2 C(NM) = 2NM-Oand thusNlog2 2 C(N,M) = N,M-Owe have log2 C(N, M) - N \/2SB', which is <<SB'because SB'>> 1. Since even the maximum value oflog2 C(N, M) << SB', we obtainQ SB', (4)which is our main result. In the simplest possible termsthis can be understood by noting that there is a binarychoice involved in whether we connect input channel i tofilter facet j, and since 1 i c N and 1 c j c M, thereare a total of NM = SB' binary decisions, which is noth-ing but the definition of Ql. To arrive at relation (4), itwas only necessary to show that the additional multiplic-ity contributed by the number of distinct ways in which asubset of the possible displacements are assigned to the Mfilter facets is insignificant.It is also interesting to investigate what fraction q of thetotal number of possible connection patterns 2N2 (whichcan all be realized with a full crossbar architecture) weare able to realize with this architecture. Dividing Q by2 , we obtain2NC(N, M) 1, (5)which is satisfied with equality when M = N. Takinglogarithms and remembering that the term involvingC(N, M) is negligible,log2 = N(M-N) = SB' - N2 0 (6)which is satisfied with equality when M = N. For agiven space-bandwidth product, the number of distinctconnection patterns that are possible with our architec-ture is fixed and cannot be increased by increasing eitherN or M. By increasing N, the number of potential connec-tion patterns 2N2 can be increased, but since the numberof connection patterns that can be implemented with ourarchitecture is fixed, the ratio -q will be decreased.Since we assume that each filter facet corresponds to adistinct displacement and is utilized for at least one con-nection, Q(M) is the number of distinct connection pat-terns that can be implemented with exactly M filterfacets; the connection patterns that can be implementedwith less than M filter facets are not included. For agiven value of N, the space-bandwidth product determinesthe maximum number of filter facets M that we can af-ford to have (SB/N). The total number of distinct connec-tion patterns that can be implemented with at most M =SB'/N filter facets isME 2Ni 2 NM = 2SD'The geometric series is dominated by and almost equal toits largest term. This means that almost all the connec-tion patterns that can be implemented with at mostM filter facets require exactly M filter facets. A corollaryis that almost all the 2N2 connection patterns that are pos-sible with a full crossbar require that M = N facets beused to be implemented with the architecture under con-sideration. The number of connection patterns for whicha smaller number of facets are sufficient is negligible incomparison.3. CONSIDERATION OF EDGE EFFECTSIn this section we repeat the above derivation, taking intoaccount edge effects.Remember that we assume that the input and outputplanes consist of Cartesian arrays of \N/ X VN channels.As discussed above, there are at most N distinct displace-ments that are possible for each input. However, when allinput channels are considered, the total number of dis-tinct displacement vectors possible is (2VNN - 1)2 4N.(This is the number of distinct vectors that can be drawnbetween pairs of points in a X \N/- array.) A giveninput cannot utilize =3N out of these =4N displacementssince they would image the input outside of the arrayof \N/ X \N/i output channels.First we establish an upper bound for l. We can chooseM displacements out of the =4N possible in C(4N, M) dis-tinct ways. In Section 2 we calculated that the inputchannels can be mapped to a particular set of M displace-ments in 2 NM ways, assuming that any number of inputchannels between 1 and N could utilize each displacement.However, since in fact all N input channels cannot utilizea given displacement, this was an overestimate. Thus weare led to the upper boundQ log2 C(4N,M) + NM = log2 C(4N,M) + SB' SB',(8)where in the last step a similar argument as used in arriv-ing at relation (4) is invoked.Now we derive a lower bound for l. Let (i, j) denotethe horizontal and vertical components of a particular dis-placement vector, which must satisfy-(NN-- 1) i s (N - 1) , (9)-(VN - 1) 'j '(VN - 1) . (10)Without loss of generality, assume for the moment thati, j Ž 0. Let (k, 1) denote the coordinates of a particularinput channel such that 1 5 k _ \/N and 1 1 ' VN.If the values of k or 1 are large, this input may not be ableto utilize the given displacement because doing so wouldcause the displacement to be imaged outside the arrayof output channels. More precisely, only those inputssatisfying1 k ' NlN - i, (11)1 _ I N__ ' - j (12)can utilize the displacement with components (i, j). Themaximum number of inputs that can utilize this particu-lar displacement is (VN - IiI)(VN - Ill), which is exactlythe number of inputs contained in the rectangle bound byrelations (11) and (12), and the absolute values will alsoOzaktas et al.Vol. 10, No. 3/March 1993/J. Opt. Soc. Am. A 421ensure validity for negative values of i andj. Any numberof inputs between 1 and (N - lil)(\N - il) may utilizethis particular displacement, in [2('V-Ni')(V'N-1j' - 1] dis-tinct ways.Now assume that we have chosen a particular set of Mdistinct displacements out of the =4N possible displace-ments. The total number of distinct ways in which theinputs can be mapped to these M displacements is then(13)The right-hand side can be calculated as_ _i = [ -2-1)2 2E (VN - i1) = IMN - 2 M41 2Thus, using MN = SB', we obtainQ -(/I - M/4 +1/4)2.where the product is taken over all the M displacements.(This step corresponds to taking the Mth power of 2N - 1in Section 2.) The total number of distinct connectionpatterns is then(14)Q = i - 1],where the sum is over the C(4N, M) different possible setsof M displacements. [This step corresponds to multiply-ing by C(N, M) in Section 2.]Because the last expression is difficult to evaluate, wemust satisfy ourselves with a lower bound. Any term inthe above summation is a lower bound for Q; hence wechoose a term that is both easy to evaluate and as large aspossible, one that corresponds to a set of displacementsthat are symmetric and as short as possible {since agreater number of inputs can utilize shorter displace-ments [relations (11) and (12)]}. More specifically, wechoose the set of M displacement vectors with components(i, J), satisfying-(- 1)/2 cic (MI-- 1)/2, (15)-(VM - 1)/2 cj c (VM - 1)/2. (16)Thus we can now explicitly express the product and writea lower bound for Q,Q ' H H [2 (v-ij)(V-ijj) - 1], (17)jiwhere the ranges of the products are as given by relations(15) and (16). Taking logarithms,Li Ž > log2 [2(Vi)(VN J I) - 1]. (18)j iIf we ignore the -1 in the square brackets, relation (18)is clearly an excellent approximation unless Iii and 1ijare close to lN/N. Iil and ill will attain their maximumvalues \N/N - 1 only when M attains its maximum value(2VN - 1)2 4N. Even in this case the overall error in-curred by ignoring the -1 in the square brackets will bequite small because the terms that are most affected arefew in number and small in magnitude compared with theterms for which ignoring the -1 is an excellent approxi-mation, which are many in number and large in magni-tude. Thusj (N - ij)(VN-il). (19)j iSince M c (2VN - 1)2 < 4N, we have N > M/4; henceSB' = MN > M2/4 and M < 2S. Noting that theright-hand side of relation (21) is a decreasing functionof M,(V/IS _ M/4 + 1/4)2 > (IB - 2V;i-/4 + 1/4)2> SBV/4, (22)so thata >SB'4(23)Combining our lower and upper bounds, we haveSB74 5 Li • SB'. Hence, apart from a numerical factorof the order of unity, Li - SB'.4. CONCLUSIONWe have shown that the value of Li for the quite generalclass of multifacet partially space-variant free-space opti-cal interconnection architectures described in this paperis approximately equal to the space-bandwidth product ofthe optical system. Thus we have provided a new inter-pretation of the space-bandwidth product.Li is a measure of the flexibility of a particular inter-connection architecture in being able to accommodate agiven connection pattern or circuit diagram. The space-bandwidth product is a measure of the throughput of anoptical system as limited by physical considerations and isalso closely related to the cost of the system. By showingthat these two quantities are essentially equivalent, wehave provided a bridge between physical concepts andarchitectural-circuit concepts.Finally we note that, although we have assumed that weare dealing with a fixed connection pattern, the systemcould be dynamically reconfigured if active deflectors areemployed at the inputs and outputs.ACKNOWLEDGMENTH. M. Ozaktas acknowledges the support of the Alexandervon Humboldt Foundation through a postdoctoral re-search fellowship.REFERENCES1. R. W Keyes, "Communication in computation," Int. J. Theor.Phys. 21, 263-273 (1982).2. R. W Keyes, The Physics of VLSI Systems (Addison-Wesley,Reading, Mass., 1987).3. E. Bradley, P. K. L. Yu, and A. R. Johnston, "System issuesrelating to laser diode requirements for VLSI holographic op-tical interconnects," Opt. Eng. 28, 201-211 (1989).(20)(21)Ozaktas et al.= V M__Nfj [2(N1N-1'1)(VN---Ijl) 11,422 J. Opt. Soc. Am. A/Vol. 10, No. 3/March 19934. G. E. Lohman and A. W Lohmann, "Optical interconnec-tion network utilizing diffraction gratings," Opt. Eng. 27,893-900 (1988).5. R. K. Kostuk, J. W Goodman, and L. Hesselink, "Opticalimaging applied to microelectronic chip-to-chip interconnec-tions," Appl. Opt. 24, 2851-2858 (1985).6. R. K. Kostuk, "Multiple grating volume reflection hologramswith application to optical interconnects," Ph.D. dissertation(Stanford University, Stanford, Calif., 1986).7. M. R. Feldman, C. C. Guest, T. J. Drabik, and S. C. Esener,"Comparison between electrical and free space optical inter-connects for fine grain processor arrays based on intercon-nect density capabilities," Appl. Opt. 28, 3820-3829 (1989).8. M. R. Feldman and C. C. Guest, "Interconnect density capa-bilities of computer generated holograms for optical intercon-nection of very large scale integrated circuits," Appl. Opt. 28,3134-3137 (1989).9. H. M. Ozaktas and J. W Goodman, "Lower bound for the com-munication volume required for an optically interconnectedarray of points," J. Opt. Soc. Am. A 7, 2100-2106 (1990).10. H. M. Ozaktas, Y Amitai, and J. W Goodman, "Comparisonof system size for some optical interconnection architecturesand the folded multi-facet architecture," Opt. Commun. 82,225-228 (1991).11. H. M. Ozaktas, Y. Amitai, and J. W Goodman, "A threedimensional optical interconnection architecture with mini-mal growth rate of system size," Opt. Commun. 85, 1-4(1991).12. H. M. Ozaktas, 'A physical approach to communication limitsin computation," Ph.D. dissertation (Stanford University,Stanford, Calif., 1991).13. G. E. Lohman and K.-H. Brenner, "Space-variance in opticalcomputing systems," Optik (Stuttgart) 89, 123-134 (1992).Ozaktas et al.

Interpretation of the space-bandwidth product as the entropy of distinct connection patterns in multifacet optical interconnection architectures

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Interpretation of the space-bandwidth product as the entropy of distinct connection patterns in multifacet optical interconnection architectures

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