Analysis of cross talk in volume holographicinterconnections
Kun-Yii Tu, Hyuk Lee, and Theodor Tamir
We utilize a novel multiple-scattering formalism to study cross-talk effects in volume holographicinterconnections. Specifically, we explore the composition of that cross talk and evaluate its maximumexpected values as functions of various design parameters such as dynamic range, minimum separationbetween interconnections, and hologram size. Examples are given for canonic interconnections thatinvolve two superposed gratings, but the analysis and results are relevant to a broader class ofhigh-capacity interconnects.
I. IntroductionStudies of interconnections that use volume holo-grams have recently been stimulated by their poten-tial use in electronic-chip interconnects' and neuralnetworks.2 In such applications, interconnectingschemes with high densities and of compact sizes areessential. However, the capacity of interconnects thatutilize electrical wires is limited by interchannel crosstalk, which becomes severe as the interconnectiondensity increases. By contrast, optical interconnectsthat use freely propagating light beams generatemuch less cross talk than electrical wires becauselight waves do not interact directly with each other.Promising schemes for such optical interconnects usevolume holograms3 in which each individual intercon-nection is implemented by storing a sinusoidal grat-ing in a bulk crystal. The grating then acts as achannel that directs light from a specified input pixelto a specified output pixel by means of Bragg scatter-ing. For multiple interconnections, many such grat-ing channels are superposed in the same crystal andthe angular selectivity that is offered by Bragg diffrac-tion is relied upon to minimize cross talk.
The cross-talk behavior of volume holographic inter-connecting devices has been studied recently by usingcoupled-wave methods.4 5 However, these methodsprovide a rather poor description of the diffraction
When this research was performed the authors were with theDepartment of Electrical Engineering, Polytechnic University,Brooklyn, New York 11201. H. Lee is now with the Department ofElectrical Engineering, Seoul National University, Seoul, SouthKorea.
process, and their application to a large number ofinterconnects is complicated. We have therefore devel-oped a different approach6 that represents the dif-fracted field in terms of a well-defined sequence ofmultiple-scattered waves that may account for anynumber of superposed gratings. This wave sequencecan be interpreted readily in terms of a meaningfulqualitative physical picture of the scattering mecha-nism. Furthermore, the wave sequence can be de-scribed in terms of a flow graph that directly serves asa template for developing fast computational algo-rithms. These algorithms provide the intensity of anydesired diffracted order without the need for alsoevaluating orders that are not relevant to the applica-tion in question. Our wave formalism thus offers theadvantage of physical insight into the scatteringprocess because it can treat any number of gratings,on the one hand, and because, on the other hand, itsystematically and efficiently provides precise data onrelevant diffracted orders and related cross talk.
Here, we present a detailed analysis of cross-talkeffects in simple, basic forms of volume holographicinterconnects by applying the multiple-scattering ap-proach, which is outlined in Section II. In Section IIIwe identify two distinct varieties of cross talk that areproduced by a second grating being superposed on theone that establishes a desired interconnection. Thetwo varieties consist of spectral cross talk, which isdue to undesirable field components that belong tothe same spectral order as the desired signal, andspatial cross talk, which is produced by componentsthat may also reach the output point but belong toother spectral orders. Specifically, we examine crosstalk in interconnects of canonic types 1 2 (oneinput to two outputs) and 2 - 1 (two inputs to oneoutput), both of which involve only two superposed
gratings. However, we focus on small angular separa-tions between those two gratings so that the resultsare relevant to interconnects that contain manygratings for which those separations are necessarilysmall. The dynamic range of weighted interconnec-tions and the dependence of the two cross-talk variet-ies on the hologram thickness are presented in Sec-tion IV. Finally, in Section V we discuss the extensionof the multiple-scattering approach to situations thatinvolve more than two gratings, present data on thecomputer time that is required to obtain such results,and compare some of these data with those obtainedby using coupled-wave methods.
II. Multiple-Scattering Formulation of HolographicInterconnectionsA basic configuration 6 for high-capacity interconnec-tions that use volume holographic gratings is shownin Fig. 1. In order to guide a signal from an input pixeli to an output pixel o, a grating pattern is firstrecorded in a suitable crystal by interfering a beamincident from pixel i with another beam that isemitted by a source of o', where o' is the invertedimage of pixel o. After the grating is stored, the lightthat is introduced at pixel i is channeled by Braggscattering onto pixel o. Multiple interconnections canbe formed by exposing additional gratings sequen-tially or simultaneously. The latter procedure is moreefficient but it records undesirable additional crossgratings because of interference between light frompixels that are simultaneously active in the input or
INPUT
reference planes. However, cross gratings can beminimized by suitable arrangements, 7 so we shallignore their possible presence henceforth.
We specifically examine cross talk in elementaryinterconnections of types 1 2 (one input i feedingtwo outputs ol and 02) and 2 1 (two inputs i and i2feeding one output o), which can be implemented byonly two gratings in a two-dimensional configuration.The wave-vector diagrams of the two interconnectiontypes are indicated at the bottom of Fig. 1(b), whereK1 and K2 designate the two gratings with magni-tudes K1 = 2/d, and K2 = 2r/d2, respectively; thevectors kA, k1, k.2 and k., k 1, k 2 refer to input andoutput wave vectors, respectively. The transmittancebetween any two channels can be arbitrary, i.e.,weighted interconnections are also considered in thiswork.
We follow the approach developed in Ref. 6 anddescribe an elementary multiplexed volume hologramin Fig. 2 by means of two gratings inclined at an angleAX with respect to each other inside a dielectric layerof thickness z0. A plane wave with propagation vectork incident from the left produces diffracted wavesthat emerge to the right. The incident wave vector k.has a magnitude ko =W (Peo) ' 2 and denotes here thesingle incident vector k. for 1 -> 2 interconnects, oreither one of the two vectors kd and k-2 for 2 - 1interconnects. The diffracted orders are properlyidentified by the wave vectors
k,,., = k. + nK, + n2K2, with n, n2 = 0, ±1, ±2 ..... (1)
These vectors account for a finite number of homoge-OUTPUT neous propagating plane waves and an infinite dis-< crete spectrum of inhomogeneous nonpropagating
Cl (evanescent) plane waves. Only the former are usu-
x
o ft E £0
REFERENCE
(a)
1 - 2 INTERCONNECTION 2 1 INTERCONNECTION
(b)
Fig. 1. (a) Holographic interconnection scheme between an inputpoint i and an output point o. (b) Wave vectors for canonic inter-connects with two channels.
- z
Fig. 2. Basic geometry of a volume hologram that consists of twosuperposed gratings that implement an interconnection involvingtwo channels.
ally significant, and they are described by real propa-gation vectors kln2, which are generally differentfrom kln2, as discussed below. A few diffracted wavesare shown in Fig. 2 by arrows labeled with appropri-ate number pairs n = (n1, n2). For interconnectionpurposes, K and K2 are usually chosen so that allundesirable (n1 + n2 •4 0 or 1) higher orders are sup-pressed by rendering their fields evanescent in the zdirection. The index n = (0, 0) is usually reserved forthe incident field in a 1 2 interconnection, as shownin Fig. 2. The desirable (Bragg-scattered) signals arethen given by the diffracted orders n = (0, 1) or (1, 0),depending on the interconnection type. For a 2 -> 1interconnect, one of the two incident fields will also bedenoted here by n = (0, 0); however, the otherincident field as well as the Bragg-scattered orderswill then be associated with different number pairsn = (nl, n2) whose values are dictated by the intercon-nection function.
To minimize the effect of the boundaries at z = 0and z, we assume that the average permittivity inthe layer is equal to the exterior permittivity E,. Thehologram region is therefore characterized by theperiodic variation
where r is the position vector and Ml and M2 aremodulation amplitudes, which are much smaller thanunity in typical photorefractive crystals. For simplic-ity, we consider two-dimensional (ay - 0) situa-tions and perpendicularly polarized (TE) waves. Hencethe electric field is along y and its variation for theincident wave with n = (0, 0) can be taken as
E` = exp(iko 0 r), (3)
where a time dependence exp(-iot) is implied butsuppressed.
We have already stated in Ref. 6 that the field E ( isdiffracted in the periodic region by a multiple-scattering mechanism that can be effectively de-scribed by Feynman diagrams in the form of flowcharts. The chart for the case under discussion isshown in Fig. 3. This starts with the incident fieldE (", which is indicated by the point (0, 0) at the top ofthe diagram. This field is scattered by the K, or K2grating vectors, thus generating four new compo-
Fig. 3. Flow chart showing multiple scattering of a single inputwave by two gratings.
nents indicated by the set of number pairs v(1) = (1,0), (-1, 0), (0, 1), and (0, -1) at a first (m= 1)scattering level in Fig. 3. The sum of these fourcomponents constitutes a diffracted field El'), wherethe superscript identifies the pertinent scatteringlevel. Each one of these components undergoes ananalogous scattering process, thus generating the setv(2) of 16 number pairs shown at a second (m = 2)scattering level in Fig. 3; the sum of those 16 compo-nents is correspondingly denoted by El". This scatter-ing process continues and generates additional dif-fracted fields E 'm)(r) at every subsequent mthscattering level. In general, each field E(m) thus con-sists of 2m components that are identified by acorresponding set v(m) of number pairs. The com-plete field in the holographic layer is therefore in theform
E(r) = E I') (rm=0 (4)
where the field E(m) itself is given by the summation
E m(r) = >E(m,.v(.m) (5)
The subscript n (m) in En(m) denotes specific values ofn(m) = [nl(m), n2(m)] that must be summed up overthe entire set v(m) in order to comply with Eq. (5).However, we shall regard n (m) in a broader sense as afunctional term with a twofold connotation. On theone hand, n (m) identifies a specific scattered compo-nent at the mth level, as is implied in Eq. (5). On theother hand, n (m) may also refer to an entire path inthe flow chart by identifying the sequence of numberpairs from (0, 0) and ending at the specific wavecomponent n (m) = [nl(m), n2(m)] under consider-ation.
As an example of a particular path, n (m) = n (3) =(- 2, 1) appears in Fig. 3 as one out of 64 number pairsin the set v(3); for that case, n(m) = (-2, 1) alsodesignates that path (0, 0) (-1, 0) -> (-1, 1) (-2, 1) whenever the entire path, rather than its endpoint only, is intended. Hence every number pair andits corresponding path n (m) are characterized by asequence of intermediate steps n (q) where q = 0, 1,2,. . . m. In this example, q = 1 yields n (1) = (-1, 0),q = 2 yields n(2) = (-1, 1), etc. It can readily beshown6 that the sequence n (q) of number pairs isgenerated by the recursive relation
n(q + 1) = [n,(q + 1), n(q + 1)]
[nl(q) + 1, n2(q)]
[nl(q) - 1, n2(q)] with p.(q + 1) = 1
[nl(q), n2(q) + 1]
[nl(q), n2(q) - 1] with p.(q + 1) = 2
(6)
which is initiated with the values n(0) and n2(0) thatare prescribed by the number pair (nl, n2) of thespecific incident field under consideration. Thus forthe cases shown in Figs. 2 and 3, we have n(0) =n2 (0) = 0 for k incidence in a 1 - 2 interconnection.
In Eq. (6), the functional term pR(q) denotes thegrating associated with the qth scattering, i.e., [i(q) =
1 for scattering through K1, and pu(q) = 2 forscattering through -K2.
Using the above flow-chart concepts, we find6 thatan individual scattered component at the mth level isgiven by
En(.) = An(n)i.(m) exp[ikn(), r], (7)
where the functions An(m) and Tn(m) are described below.For this purpose, we first note that k(m) = k refersto the propagation vector of the diffracted ordern(m) = (n1, n2) under consideration. To clarify therelation between k11n2 and k n 2 we consider their xand z components, which are given by
(8) knln2
(9)
knn,. = un.,n2 + Vn,,.2S
kn,,n2 = Xlni,n2 + inj. 2,
where x and z are unit vectors in the respectivedirections. The x components of kn . and knn2 aretherefore equal. However, while the z component ofkfl, is the same as that given by Eq. (1), the zcomponent of kn1n2 satisfies
Wnln2 = (k2
o -un,,.,) (10)
The geometrical relationship expressed by Eqs.(8)-(10) is shown in Fig. 4 for several specific knn 2 andtheir corresponding wave vectors knln2. It is importantto observe that the magnitude of kn , is always equalto ko = o(,u[0 E0 )"
2 but the magnitude of kln 2 is gener-ally different from ko. Thus kln 2 may be a convenientquantity for identifying diffracted waves, but theiractual propagation vectors are knn 2. This holds forboth propagating and evanescent orders. However,unlike knn 2, which is always real, kln2 is complexfor evanescent orders because its component wjjn2along z is then imaginary. We also recall that the in-cident field for a 1 - 2 interconnect, as shown in Fig.4(a), is given by k,,O, but only one of the two incidentfields can be given by k,0 in a 2 - 1 interconnect.Thus, as shown in Fig. 4(b), the second incidencevector is given by k - for the case under consideration.
The difference tetween kn, 2and kln 2 is conve-niently denoted by the wave-number mismatch
Pn 11n2 =Wn 112 -Vn 11n2s11
for which we show P2,-1 in Fig. 4(a) and p-l1 in Fig.4(b) as examples. We also observe that a phase-synchronism situation n = 0 occurs whenever aBragg condition is satisfied for the corresponding (n1,n2) diffraction order.
The first term in Eq. (7), using the quantities asdefined above, refers6 to the amplitude product
I I -
knlsn2 ()21knlsn2(b) 2-1
Fig. 4. Vector diagrams illustrating higher-order diffracted waves,with wave vectors kl,.2 shown on the left and actual propagationvectors k,,, shown on the right for (a) a 1 -> 2 interconnect and (b)a 2 - 1 interconnect. As drawn, (a) and (b) describe complemen-tary interconnections.
with individual amplitude factors
kOI2VI(q)Z 4
Wn(q)
where p1(q) = 1 or 2 was already defined in the contextof Eq. (6). The quantity An(m) thus measures the effec-tiveness of the scattering process in building up theamplitude of the field component at the point n(m) =(n1 , n2) under consideration, as recorded by the waveprogression along the path n(m), starting at n = (0, 0)and ending at the specific end point n = n(m).
The second term in Eq. (7) designates6 the phase-correlation function
whereY 2' denotes the inverse Laplace transform.Analogously to A,(m), the quantity 1Yn(m) measures thenet effect of accummulated phase mismatches thatare encountered by the wave progression along thepath n(m).
It is important to recognize that components belong-ing to any specific diffracted order (nl, n2), e.g., (1, 0)in Fig. 3, may appear at various levels m, as well as inseveral places at any specific level. Because the com-plete field En0 ,n2 comprises all the components that areidentified by a specific number pair (nl, n2), we have
Enln2= > 2Enln2 X (15)
where it is understood that, in the summation overany set v(m), we select only those components thatbelong to the diffracted order (n1 , n2) under consider-ation. For example, the diffracted order (1, 0) appearsin Fig. 3 after a first scattering (at level m = 1), butnot after a second scattering (at level m = 2), andthen reappears after a third scattering (at levelm = 3). In fact, only two (1, 0) entries are shown inFig. 3 at this last level (m = 3), but a total of ninesuch components then occur because other compo-nents in the group at level m = 2 scatter into a (1, 0)component at level m = 3. Ten separate El,0 compo-nents must then be combined, to an accuracy given bym = 3, to describe adequately the diffracted order (1,0) of the total field E(r) in Eq. (4).
Recall that the amplitude of the incident field in Eq.(3) is unity; the intensity of any diffracted order isgiven by the diffraction efficiency, which is defined as
wn,in2Tnn2 w 1Enl22
2. (16)
Diffracted fields that are produced by the presence ofa second grating will interfere with the desired fields.This effect expresses itself as cross talk, which can bequantified in terms of various 7nn as discussedfurther below. 2
III. Weighted Interconnections and Cross-TalkVarieties
In global weighted interconnections, each gratingwould ideally provide an independent channel with nospillover of signals from the other gratings. In termsof the intensities Ii and I,, at any input pixel i and anyoutput pixel o, respectively, the definition of anindependent interconnection can be expressed as
I, = t",J, (17)
where t is the desired transmittance (or ideal diffrac-tion efficiency) of that interconnection as long as onlythe grating that provides the channel i - o isaccounted for (all other gratings are assumed to beabsent). Cross talk can therefore be measured by thedifference between the light intensity I, which isdefined in Eq. (17), and the light intensity that is
actually received at the output pixel o in the presenceof other gratings. We then find that cross talk consistsof two different varieties, which are treated sepa-rately below.
A. Spectral Cross Talk
We first consider a 1 - 2 interconnection as shown inFigs. 1 and 4(a). This involves a single incident waveof the order n = (0, 0) which is diffracted by thegrating vectors K and K2 in a predetermined desiredratio into two separate outputs that are representedby the orders (1, 0) and (0, 1), respectively. The twointerconnecting channels are therefore (0, 0) -> (1, 0)and (0, 0) -* (0, 1), for which Bragg conditions aresatisfied so that
(18a)kWl = ko,
k = ko = koo + Ki,
k 2 0= ko,= ko+ K2,
(18b)
(18c)
with ko , k,,0, and kol having the same magnitude ko.Below we focus on the interconnection that is pro-vided by the grating vector K between the inputpixel, which is given by point (0, 0), and the firstoutput pixel, which is given by point (1, 0). It is thenclear from Eq. (15) or Fig. 3 that a second gratingvector K produces additional contributions to thesame (1, 0) diffracted order, all of which also appear inthe signal that is collected at the output pixel. Hencethe interconnection (0, 0) - (1, 0) ceases to beindependent and the contributions that are due to K2act as spectral cross talk. The term spectral is chosenhere because this cross-talk variety belongs to thesame spectral line that is intentionally sampled at theoutput pixel, so that it adds coherently to, but effec-tively distorts, the desired signal.
For a quantitative discussion of the spectral crosstalk, we assume (consistently with Fig. 4) that K andK2 are chosen so that only the orders satisfyingnl(q) + n(q) = 0 or 1 are propagating; all other ordersare evanescent and therefore negligible. The errorthat is introduced by disregarding the evanescentorders can be assessed readily by following Ref. 6, butthis error is exceedingly small if the thickness zo islarger than a few wavelengths, which is usually thecase. As an illustration, in Fig. 5 we show a flow chartup to the m = 5 level in which we retain only thosebranches that refer to scattering into propagatingorders. To highlight the diffracted order of interest,the (1, 0) diffraction points are placed inside rectangu-lar frames. Branches leading to Bragg (B-type) scatter-ing are denoted by solid lines and those leading tooff-Bragg (0-type) scattering by dashed lines. Toidentify specific paths on the flow chart, we label thepaths consisting of only B-type scatterings as B-typepaths and label all the others (involving one or more0-type scatterings) as 0-type paths. We can thenverify that the end points B1, B2, and B4 in Fig. 5identify paths that contribute to diffraction into thedesired (1, 0) order through scatterings by K only.
Fig. 5. Reduced flow chart (up to level m = 5) for the channelestablished by K1 in a 1 - 2 interconnect. Bragg and off-Braggtransitions are indicated by solid and dashed lines, respectively.
0.1Hence Eqs. (15) and (16) yield
tio = I 1EBI + EB2 + EB4 12, (19)
where the equality implies that scattering levels thatare higher than m = 5 can be neglected, which isusually the case in such situations. The remainingpaths in Fig. 5 with end points B3, B5, B6, and B7and 01 through 07 include scattering by the secondgrating K2; we shall therefore refer to these paths assources of cross scattering. Because of interferencefrom cross scattering, the actual diffraction efficiencyfor the (1, 0) order is given by rlr0 rather than tl 0. Wethen find from Eqs. (15) and (16) that
W 7 7 12 (20)
defines the actual diffraction efficiency (or transmit-tance) for the (1, 0) order. Other actual efficienciesTnln can be derived analogously for different n =(n1, n2) orders. For the output channel that is denotedby (1, 0), it is then convenient to measure the effect ofspectral cross talk by
= -ITIO t,,01 .(21)Yi1,o - tlO
Interconnects with a large number of input-outputchannels necessarily require small values of Ak be-tween any two gratings, so that the dependence of 'Yl,0on A4 is important. We therefore vary A(\ by lettingK2 approach K, while keeping K, fixed and adjustingK2 continuously to satisfy the Bragg condition forscattering into the (0, 1) order, which corresponds todecreasing the separation between the two vectors k, 1and k 2 in Fig. 1(b). As an example, we show diffrac-tion efficiencies in Fig. 6 for the orders (1, 0) and (0, 1)as functions of the angular separation A. Here theorder (0, 1) is shown for comparison only. In bothcases (a) and in Fig. 6 we have M = M = M2, but,to illustrate the effect of the modulation index, themagnitude of M in case (a) is greater than that in case(b). We focus first on case (a) of Fig. 6 and recall that
0.00 1 0 3020
AO (rad)
Fig. 6. Variation of diffraction efficiencies t01 02 and Tnln2 versusangular separation A4\ for a 1 -*2 interconnect where K, ko = 1.33,00 = 41.680, zoId, = 3.25 x 104, M2 = 0 for tl0,, andM2 = M, for T1,0,with (a) Ml = 1.2 x 10-5 or (b) Ml = 8 x 10-6.
t 0 denotes the diffraction efficiency of the (1, 0) orderwhere K2 is absent, while Tl,0 and T0,l refer, respec-tively, to the diffraction efficiencies of the (1, 0) and(0, 1) orders where K2 is present.
If all the branches of a path n(m) satisfy synchro-nous (Bragg) conditions, i.e., Pn(q) = 0 for all q = 1, 2,... m, then Tn(m) reduces6 to im/Ml!. We also find6 thatthe magnitude of Tn(m) is a maximum under theseconditions and becomes smaller as any of its Pn(q)terms in Eq. (14) departs from zero. The decrease inI n(m) is determined by the number of off-Braggscatterings that are encountered along the path n(m)and by the degree of their mismatches pn(m). Thisimplies that 1Tn(J) < 1/i!, so that Eq. (7) leads tothe upper-bound limitation IEnm) I < IA n(m) I !- Wealso recall that Pn(q) measures the phase mismatch inthe n(q) branch. For the case discussed here, increas-ing the value of A(\ produces corresponding increasesin all Pn(q) except for those with n(q) = (0, 0), (1, 0), or(0, 1), as seen in Fig. 4(a). We then note that Tn(m) forE0 . with j = 1, 2, ... , 7 in Fig. 5 decreases as A41increases. As a result, we obtain two types of varia-tion for 1 ,0 in Fig. 6. For large values of A( > 20 ,rad,Tl0 saturates at 0.293 because the sum of E0 isnegligible compared with the sum of EB. The differ-ence between tl,0 and rl 0 in this range thus is due tothe B-type paths EB3, EB5, E,,, and EB7 mentionedabove. For values of Ak < 20 ,urad, on the other hand,the 0-type scattering paths account for larger ampli-tudes in the sum of E0 , so that T 0 reaches values thatare smaller than those in the range with larger A(\.
1722 APPLIED OPTICS I Vol. 31, No. 11 I 10 April 1992
I I I I I I
t(a)1 ,0
0,1
t (b)
1 0- 0,1
ru0
However, the difference between t and Tr,3 staysrelatively small because the first cross-scattered com-ponents appear only at the m = 3 level in themultiple-scattering representation (note that allI an(q) I < 1 for this case). For the same reason, we alsoexpect that the difference between t and T1,o de-creases when either M or (xn(q) decrease, as is seen inboth cases (a) and (b) in Fig. 6.
We also observe that rl'o and o,l are nearly equal inthe small Ali region that is considered in Fig. 6. Thishappens because the scattering processes for thoseorders are symmetric, i.e., the flow chart describingthe generation of the (0, 1) order is obtained from theflow chart in Fig. 5 that describes the generation ofthe (1, 0) order by interchanging all n, and n2 in thenumber pairs (n,, n2). Furthermore, for those twoflow charts we have p,,,,n2 Pn,, and wnl n2 wn2,nl sothat Ti,0 ToX
To discuss 2 - 1 interconnects, such as those inFig. 4(b), we recognize that their gratings may gener-ally be recorded differently from the gratings for 1 -o2 interconnects. However, a meaningful comparisonbetween the two cases can be made if the former areconstructed from the latter by just reversing vectorsas follows:
k2,1 = ki1-2; kl2-1
= kll-2; k 2 -1 = k2 , 2 . (22)
This means that the roles of input and output in thetwo interconnects are simply interchanged. The grat-ings themselves then remain physically the same, butthe directions of their vectors K, and K must bereversed as shown in Fig. 4.
The two inputs i and i in a 2 -> 1 interconnect canbe simultaneously active, but we shall focus on theinput k = k, because the second input k1 2 can beanalyzed in an analogous fashion. To clarify thisaspect, we note in Fig. 4(b) that the vector ko,, that isdirected into the point (0, 0) denotes a first inputwave, which is Bragg deflected by the grating vectorK into the point (1, 0); the latter point thus refers tothe diffracted order at the output of this 2 - 1interconnection. The second input wave has a wavevector that coincides with the vector k1,_1 of the firstinput so that it is Bragg scattered by the gratingvector K into the same output point (1, 0). Wetherefore get
,1= lk,2- + K - K2 = k0,0 + K - K2 = k l. (23)
If we then refer to Fig. 4 and restrict our discussion tothe interconnection channel (0, 0) (1, 0) that isspecified by the K, vector, we find that the scatteringcoefficients and phase mismatches of those 1 - 2 and2 -* 1 interconnections obey the one-to-one correspon-dence that is given by the matrix-pair equality
(n,, n,)'' = (1, 0) - (n, n) ".
1 interconnects that have the same grating configura-tion and that obey Eqs. (22)-(24) as complementaryinterconnects. If we then compare the scattered fieldsobtained through Eq. (7) and examine their flowcharts (up to m = 5), we find by induction that
1,0 12 1,0 2wo,0 1- (25)
which yields T1 oi2 = ri02-1 if Eq. (16) is also accountedfor. Hence the spectral cross-talk behavior given byTi, 0 is the same in complementary interconnects andtherefore requires no separate discussion for the 2 1 case. However, the spectral cross-talk intensity in a2 - 1 interconnection will be generally different fromthat in a corresponding 1 - 2 interconnection if thegratings are not the same, i.e., T1, 1 2
T if theinterconnections are not complementary.
Diffraction efficiencies for the (1, 0) order in a 2 - 1interconnect are shown in Fig. 7. Although the (0, 1)order is also shown in Fig. 7, it relates to cross talk ofthe spatial variety, which will be discussed below. Thedecrease of Tlo as A - 0 occurs here for the samereason that T1, decreases in Fig. 6. Similarly, if wecompare cases (a) and (b) in Fig. 7, we note that Tl,t,,0, and the gap 0t1 - T1,0 decrease as the modulationindex M = Ml = M2 decreases. The effect of thesevariations on the spectral cross talk is illustrated inFig. 8, which shows curves for constant values of yl0in a typical pair of complementary interconnects. Thediffracted order (1, 0) was calculated as a function of
0.3
aC0.2- 0.2
0.._
cl
0.1
0.00 5 10 15 20
(24)
For example, Fig. 4 shows that p2, in the 1 - 2 caseis equal top- 1 in the 2 - 1 case.
We shall henceforth refer to pairs of 1 -2 and 2 ->
AO (irad)
Fig. 7. Variation of diffraction efficiencies t, 2 and T,n, versusangular separation A for a 2 - 1 interconnect where K k, = 1.33,0, = 41.68-, zold = 6.5 x 104, M2 = 0 fort l, 0, and M = Ml for ,,with (a) Ml = 6 x 10-' or (b) Ml = 3 x 106.
Fig. 8. Variation of angular separation Ali versus ideal efficiencyt1, for fixed spectral cross talk yio = 0.5%, 1%, and 2% (solid lines),and spatial cross talk Xi,, = 0.5%, 1%, and 2% (dashed lines). Herez0 = 6.5 x 104d, is fixed and t1,, is varied by changingM, = M, = M,K1lk, = 1.33, and 00 = 41.68°, and K, is adjusted to maintain aBragg condition at all A+.
tlo by varying Ml = M2 = M only; the hologramthickness z0 was held constant.
To examine the behavior of spectral cross talk, weconsider the curve for yl 0 = 1% as an example. Theregion tl,0 > 0.029 is dominated by scatterings of theB type, which cause more than 1% spectral cross talkeven at large values of Ack (for which 0-type contribu-tions are very small). In the region 0.015 < t <0.029, contributions of the B type decrease to lessthan 1%, but 0-type contributions increase as Adkdecreases. We then see that Ack decreases as t decreases if the value 8l 0 = 1% is maintained. As t1 0decreases below 0.015, the total cross talk caused bythe combined effect of B- and 0-type scatterings isless than 1% at AXk = 0. This threshold appearsbecause cross-scattered components that contributeto the (1, 0) diffracted order start two levels higherthan the components that are scattered by the firstgrating. Analogously to the yi 0 = 1% curve, all otherylo curves start at a specific threshold value of tl,0 andrise slowly at first but then become almost vertical atsufficiently high values of AO).
We thus find for both 1 -*2 and 2 -> 1 interconnec-tions that, for Ack above a minimal value, the spectralcross talk is practically independent of the angularseparation between the gratings.
B. Spatial Cross Talk
The incident and diffracted fields considered heretheoretically can be focused by perfectly collimating
lenses into corresponding dimensionless points. Asshown in Fig. 4 for both 1 -*2 and 2 -* 1 interconnec-tions, an incident wave diffracts not only into thedesirable output waves but also generates unwanteddiffracted components, which are oriented into theclose vicinity of the desired output pixel. Theseunwanted fields include all nl + n2 = 1 orders, withthe exception of the desired (1, 0) and (0, 1) orders for1 - 2 interconnects, and the (1, 0) order for 2 - 1interconnects. Because the detection aperture at theoutput pixel cannot be focused into an ideal point butextends over a finite effective spatial region, suchunwanted diffracted orders may also register andthus cause cross talk. This effect is denoted here asspatial cross talk, which evidently consists of undesir-able components with propagation vectors kn n thatare different from the desired ki,0 and/or k l vectors.For the interconnection channel established by Kl,we can measure spatial cross talk by
1Xo = t
1,0 u(0.,02)(26)
where u (nl, n2) is the set of all diffracted orders(n1, n2) that propagate in a direction other than thatof the desired output signal. The spatial cross talkXn,,% for other channels n = (ni, n2 ) (1, 0) canbe analogously defined.
For a 1 -* 2 interconnect, the angular separationbetween the two desired output beams is given by 8,as shown in Fig. 4(a), and the two closest unwanteddiffracted orders (-1, 2) and (2, -1) occur at angles5, and 8 away from the desired orders (0, 1) and(1, 0), respectively. For a complementary 2 1interconnect, the closest unwanted orders are (2, -1)and (0, 1), which are shown in Fig. 4(b) to be sepa-rated by 8. and 8+ from the desired (1, 0) order,respectively. On the basis of the grating equations (9)and (10), we find for small Adk that those angularparameters are all closely equal and satisfy 3+ 8-8 = 2Alv. If the aperture at the output pixel is smallerthan 2Ack, the detection of unwanted diffracted ordersis suppressed and spatial cross talk is then elimi-nated. However, such a scheme for preventing spatialcross talk is practical only in cases involving twochannels, as discussed here. If more than two grat-ings are used, the corresponding grating equationsshow that the angular separation between a desireddiffracted order and certain of the unwanted orderscan be much smaller than 2Ack rather than closelyequal to it. We shall therefore determine conditionsfor minimizing spatial cross talk by assuming themore realistic possibility that all of it can be detected.
To explore the variation of Xl01 first we again recall6
that Tn(m) is maximum at Ad = 0 and that unwanteddiffraction orders involve only 0-type scattering. Wethen conclude from Eqs. (7)-(16) and (26) that nfor (nl, n2) belonging to u(n, n2) peaks at AX = 0.This argument disregards the possible presence ofdestructive interference between the various compo-nents in Eq. (15); such interference can be neglected
because only the first scattered order is usuallydominant. Second, we recognize that the highestvalues of Tnln are provided by the closest diffractedorders (2, -') and (-1, 2) for 1 - 2 interconnects,and by (0, 1) and (2, - 1) for 2 -l 1 interconnects, asseen in Figs. 4(a) and 4(b), respectively; these ordersoccur at the lower scattering levels that are shown inFig. 3. Quantitatively, the intensities of the (2, -1)and (-1, 2) orders are much smaller than the inten-sity of the (0, 1) order, but the (0, 1) order itself mayreach the same magnitude as the desired (1, 0) order,as shown in Fig. 7. As an example, for the 1 - 2interconnection covered in Fig. 8, both r, l and T-i2are smaller than 0 and they can then be neglectedin comparison with t = 0.0297 for 81,0 = 1% andValo1 = 0.1726; in fact the ratios T2,-l/Tl,0 and T-1,2/Tl,0become even smaller as t decreases. A similarsituation holds also for the (2, - 1) order in a 2 - 1interconnection, but the (0, 1) order may then bequite appreciable. We shall therefore address prima-rily the 2 - 1 case in the discussion below.
To illustrate the above considerations, we show inFig. 8 the spatial cross talk in a 2 1 interconnect byplotting fixed values Xl, = 0.5%, 1%, and 2% withdashed lines. The variation of tl0 is covered only up tothe maximum efficiency of spectral cross-talk linesshowing values Yj,0 = 0.5%, 1%, and 2%, respectively,because spatial cross talk at the higher values of t, 0 isobscured by the larger values of spectral cross talk.We observe that the lines of constant X1,0 are almosthorizontal and thus only slightly vary with t. Thisbehavior can be explained by noting that, for small t,0(or I aIo l) and for a value of A that is not too close tozero, the (0, 1) diffracted order is generated mostly atthe m = 1 level, so that Eqs. (7)-(14) yield
To -0, I-0,-sinc-0--jI' (27)Wo [ i 2
where all higher-order (m > 1) scatterings have beenneglected. Similarly, if ( 10 is small enough, the effi-ciency t 0 can also be approximated by using only them = 1 component, which yields
t1,0 =W0,0 1, 0 (28)
Therefore the spatial cross talk that is defined in Eq.(26) is approximated by
Xio w 2a, sinc I -I' (29)
which is not a function of t1,0and therefore plots as ahorizontal line in Fig. 8. We therefore conclude thatspatial cross talk is essentially independent of thetransmittance t. Because t is proportional to M., itfollows that spatial cross talk is independent of M,and M2.
The above behavior is in contrast to that obtainedfor the spectral cross talk which, for greater than
a certain minimum value, was independent of Ackrather than t. As a result, if an interconnection is tolimit total cross talk (which is well approximated bythe sum of ylo + rlo) to below a maximum permissiblevalue, its design points must be contained in a regionthat is given roughly by a vertical stripe in Fig. 8. Asan example, to assure that the total cross talk is below1%, this region is given by Ac > 21 lirad and t <0.028 for the interconnect illustrated in Fig. 8. Thissimplified design rule holds well for values of Ack thatlie beyond the first dip in Fig. 7, but for smaller valuesof Adk the lines of constant Xl,0 may deviate apprecia-bly from the horizontal.
It is important to recall that the above consider-ations were developed for 2 - 1 interconnects only.For 1 -l 2 interconnects, the role of spatial cross talkcan be ignored because the relevant values of y ,n2 arethen considerably smaller than Yn,2 even as Adk - 0.Design criteria for - 2 interconnects are thereforedictated only by spectral cross-talk considerations,i.e., they can be based solely on curves such as thosefor yi,( in Fig. 8.
IV. Dependence of Cross Talk on Various Parameters
The results shown in Figs. 7 and 8 clarify well thegeneral behavior of cross talk and some of the limita-tions it may impose, but they provide only marginalinsight into the parametric aspects that are requiredfor design purposes. We therefore discuss below theeffects that are produced by the more relevant physi-cal variables.
A. Effect of Modulation Indices M, and M,
The terms M, and M2 appear only in Eq. (13), so thatEq. (19) suggests that t 0 generally increases (but notlinearly) with increasing values of M. Hence theeffect of varying M, (and analogously M2) wouldaccount for a variation that is similar to that shown inFig. 8. We shall therefore not elaborate on this aspectany further.
B. Effect of Crystal Thickness zo
In Fig. 8 the behavior of cross talk is illustrated for afixed value of the crystal thickness z which is givenby = zdl = 6.5 x 104. For other values of z0, weexpect that the curves in Fig. 8 will occupy differentpositions. To examine the effect of z0, we focus on the1% lines for both Ylo and Xlo and reproduce them for
= 6.5 x 104 in Fig. 9 but also add corresponding 1%lines for /2 and 2. The spectral cross-talk curves(shown solid) exhibit the same threshold and maxi-mum values of t1,0. Inside the range of t that isbounded by these threshold and maximum values,the lines move up with respect to Adk as z decreases.For the range t 0 < 0.03, i.e., for ~yj0 < 1%, the spatialcross-talk lines (shown dashed, for 2 -l interconnec-tions only) also move up as z decreases. We thusdeduce that, in order to limit cross talk to a value thatis smaller than a given maximum level, e.g., tomaintain it below the X1,0 = 1% line in Fig. 9, it ispossible to lower the Xlo line by increasing z0 and thus
Fig. 9. Variation of angular separation Ad versus ideal efficiencyt, 0 for spectral cross talk 5l = 1% (solid lines) in both 1 -> 2 and2 -> 1 cases and spatial cross talk Xlo = 1% (dashed lines) in the 2 ->1 case. The parameters and their variations are the same as in Fig.8 except that now three different values of z0 are considered, witht = zold, = 6.5 x 104.
work with smaller values of A+k. However, note thatthe maximum value of t,,0 (_ 0.03 in Fig. 9) is notaffected by changing z0 because it is determined by'Ylo
To clarify the behavior of the curves in Fig. 9, weshow rl, andTo l in Fig. 10 as functions of Aid for twodifferent thicknesses z0 but for the same value of Mz(with M = M = M2 . We recall that tl,0 is given only byB-type scatterings in Eq. (19), so that its value isdetermined by the product Mz0 only. Thus keepingMz0 fixed is equivalent to fixing tl,0. Hence, as seen inFig. 10, the efficiencies T1,0 and ro i become equal ati\ = 0, and rl,, saturates to values that are almostequal for different z. As already discussed in thecontext of Fig. 7, this happens because B-type scatter-ings are dominant at both Adk = 0 and large Ack, butnot in between. Second, the dashed curve for smallerz0 has the same appearance as the solid curve forlarger z0, except that the former is stretched out alongthe horizontal axis more than the latter. This hap-pens because, as z0 decreases, larger values of Adk arerequired to provide equivalent effects, as expected.
We thus conclude from Fig. 10 that, in general,fixed values of y, oc t1 0 - Tio = constant (e.g., for1%) move toward higher values of Ac as z0 decreases.This explains why the solid (spectral) curves move upin Fig. 9 as z0 decreases. Similarly, fixed values of
0.3
>1
cC
i 0.2
.2a)
D 0.1
0.00 5 10 15 20
A (rad)Fig. 10. Variation of diffraction efficiency Tv,,2 versus angularseparation Ad for complementary interconnects. Here M, = M, =M, KIk, = 1.33, and 0, = 41.680 as in Fig. 8. Two values of z. areplotted, with C = zId, = 6.5 x 104, but MzId, = 0.39 is fixed in allthe curves shown. The T,,1 curves hold only for the 2 - 1 case andthe tlo curve refers to M2 = 0.
Xl o I Tol = constant also move toward larger values ofAd as z0 decreases, so that the dashed (spatial) linesalso move up in Fig. 9 as z0 decreases.
C. Effect of Dynamic Range
So far, we have assumed for simplicity that Ml =M = M. In practice, however, interconnects of theanalog type where M1• M2 may be needed, as in thecase of neural nets. The presence of cross talk willthen affect the allowable relationship between M, andM2. This will determine an effective dynamic rangethat can be conveniently measured in terms of a ratior = M2 M. To explore the effect of varying r in 1 -l 2interconnects, we show the variation of T1i0 and Toj forseveral r in Fig. 11. We observe that T, 1 ;l 0 as r 1;also r0, - 0 and rTI, - t 0 as r -> 0. These effects areexpected because larger cross talk is produced bygratings with larger values of M, or M2.
The situation for 2 -l 1 interconnects is illustratedin Fig. 12, which shows that the variation of rlro is thesame as that of 1 -*2 interconnects in Fig. 11. On theother hand, r01, behaves quite differently in that it isnow oscillatory rather than nearly constant. In partic-ular, for large Ack we find that T7,l is generally lessthan Tl0; however, for r > 1, oi can exceed To atsufficiently small values of A. We also note that theextrema of the curves in Figs. 11 and 12 are alignedalong the same values of A. This happens because
1726 APPLIED OPTICS I Vol. 31, No. 11 10 April 1992
0.8
cJ0
aCD
az
0
C.)CU
0
0.6
0.4
0.2
0.00
Fig. 11. Variationseparation A for 141.68°, z Id, = 6.5 xand 2.5.
0.3
C)c
C)
._
(V
C0
C)
0
0.2
0.1
0.00
5 10 15AO (grad)
of diffraction efficiency T, versus ang- 2 interconnects, where KlIko = 1.33, e104, M1 = 4 x 10-6, and r = MAIM, = 0.5,
5 1 0
9
I.10
1
15 20
the variations of both T,, and T, 0 with Ack are deter-mined only by Tn(m), which does not depend on r, i.e.,the effect of r is only to scale effectively the curves forto or T1 along the vertical axis.
After understanding the variation of the diffractionefficiencies T1,0 and Tojl, we illustrate the typical behav-ior of cross talk in Fig. 13 by again using y10 = X1, =1% curves for 0.1 < r < 10. We then see that largerratios r confine permissible t to smaller values ifspectral cross talk must be limited below yl,, = 1%.On the other hand, similarly limiting spatial crosstalk to X, to some maximum amount requires thatthe minimum value of A be increased as r increases.As an example, tj 0 should be < 2 x 10' and Adk shouldbe 2 240 prad for the total cross talk not to exceed 1%for the situation shown in Fig. 13. Hence largerdynamic ranges r require larger values of Adk andsmaller values of t 0 in order to maintain total crosstalk below a desired limit value.
V. Extensions of Scattering Formalism andComputational Considerations
i So far, we have focused on l- 2 and 2 12o interconnects that use TE-mode incidence in the
presence of only two superposed sinusoidal gratingsin two-dimensional geometries. We therefore outline
[ar below the extension of the multiple-scattering formu-0 lation to situations that are mathematically more.0, complex. However, the discussion is restricted to
two-dimensional geometries because three-dimen-sional problems are considerably more complicated.We also present here representative figures for thecomputer times that are required to calculate data bymeans of programs based on the multiple-scattering
1 00
(a
10
0° ,10110-4 I0-3 10-2 10-1
t1, 0
AO grad)
Fig. 12. Variation of diffraction efficiency T,,,,, versus angularseparation A for 2 - 1 interconnects, where Klk = 1.33, 0 =41.68°, zold, = 6.5 104, M1 = 3 x 10', and r = M2A!1 = 0.5, 1.0,and 2.0.
Fig. 13. Variation of angular separation A versus ideal efficiencytl,0 for spectral cross talk y,,o = 1% (solid lines) in both the 1 - 2and 2 1 cases, and for spatial cross talk Xo = 1% (dashed lines) inthe 2 -- 1 case. The grating parameters and their variation are thesame as in Fig. 8 except that now r = MAIM 1 = 0.1, 0.5, 1, 2, and 10.
10 April 1992 I Vol. 31, No. 11 I APPLIED OPTICS 1727
I I I I I
0,1 for r=2.5
1,0 for r=0.5
1,o 0 1 for r=1 .0
for r=2.5- - - - - _ '/ - - - - - - - - -
…............. .........................<X0, for r=0.5
_____________ _______ .~0,
I I ,
--- r=0.5
- r=1.0....... r=2.0
'1,0
.,
..
..
. . . . .
-
approach, and compare these figures with those ob-tained if coupled-wave methods are used instead.
A. Scattering in the Presence of More Than Two GratingsThe extension of the multiple-scattering formalismfrom a 1 - 2 to a 1 -> N > 2 interconnection requiresa number of gratings that is equal to the desirednumber of outputs N. The situation where N > 2 canbe obtained in a rather straightforward manner bysimply examining the construction of the flow chartin Fig. 3. It follows that, instead of using only p. = 1 or2, we must then take V. = 1, 2, 3, . . , N in all quanti-ties involving the integers n,,, such as kn,2 N = k3 +n1K, + n2K2 + ... + nNKN, En n n etc. As a result,sequences of number pairs n(qt tor9 = 2 are replacedby sequences of number triplets, quadruplets, etc. forN = 3, 4, etc., respectively.
As an example, for three (N = 3) superposed grat-ings, each wave component is scattered into sixdifferent orders at every scattering level (instead offour such orders for N = 2). This implies that Eq. (6)is replaced by
which can be initiated with n(0) = (0, 0, 0) if thesingle input wave has an incident wave vector ko,0 .We thus find for N = 3 that 6 (instead of 4')scattered components appear at every mth level. Byinduction, we deduce for N superposed gratings that2N new components appear at every scattering point,so that the mth scattering level includes (2N)m
components.To generalize 2 -l 1 interconnections to larger Q
1 situations, where Q > 2 denotes the number ofinputs, we first recall that a complete analysis of the2 - 1 case required two flow charts. The first of these(shown in Fig. 3) is generated by Eq. (6) with aninitial value n = (0, 0); the second one is similar but isgenerated with a different initial value by Eq. (6), e.g.,n = (1, - 1) for the case corresponding to that of Fig.4(b). Because Q - 1 interconnects require Q separategratings, each input wave corresponds to a differentinitial value n in Eq. (6), which generates (2Q)'components at the mth scattering level for a total ofQ(2Q)' components if all inputs are active. Forgeneral Q -* N interconnects, it follows that thenumber of scattered components that is generated atevery mth level by all the inputs is Q(2QN)Y, whichequals N(2N 2
)m if Q = N. For such an N - Ninterconnection, the total number of scattered compo-nents that is summed up over all levels up to andincluding m is N[(2N2 )m+l - 1](2N - l)-1, which iswell approximated by 2mN 2m+ for large m or N > 2.
The number of components therefore increases rap-idly with N, but the progression of the scatteredwaves remains systematic. The evaluation of alldiffracted orders thus can be handled adequately by acomputer program that expands on the schemeadopted for the simpler N = 2 case.
B. Nonsinusoidal GratingsTo extend the formalism to nonsinusoidal gratings,we can replace g(r) of Eq. (3) by the more generalform
N
g(r) = E g(r),=1
(31)
where p = 1, 2, 3, . . , N refers to specific gratings, asbefore. Each of these gratings now consists of aperiodic modulation given by the Fourier series
In terms of flow charts, Eq. (32) implies that everygrating scatters waves through vectors given by +K,,+2K,,, +3K,,, etc., rather than through ±KA only.The additional scattered components then can beaccounted for by suitably augmenting the flow chartsand the formulation given in Eqs. (6)-(16).
C. TM-Mode IncidenceThe TM-mode (parallel polarization) situation can beaddressed by noting that, as discussed elsewhere,8 thepertinent wave equation involves a function g(r) thatincludes not only the function g(r) of Eq. (3) but alsoits derivatives. However, g(r) has the same periodic-ity as g(r) and generally includes higher-order har-monics in a manner similar to that shown in Eq. (31).Hence this case can be treated by applying theapproach already outlined in Subsection V.B.
D. Computational Considerations
All the quantitative data presented in this paper wereobtained on a 25-MHz 386 personal computer with aprogram based on MATLAB software. Most resultswere calculated by retaining up to m = 5 scatteringlevels. To evaluate 50 separate points for the dif-fracted order (1, 0) in the presence of two gratings, atypical run takes 17 s if the paths are restricted ton,(q) + n2 (q) = 0 and branches. If all paths areaccounted for, the same run takes 2.3 min. For threesuperposed gratings, on the other hand, an analogoustypical run that yields 50 points for the diffractedorder (1, 0, 0) takes 1 min for correspondingly re-stricted paths, and 6.9 min if all paths are included.As expected, calculation times increase rapidly withthe number of superposed gratings.
The multiple-scattering approach is considerablyfaster compared with coupled-mode methods if, as isusually the case, only selected diffracted orders needto be evaluated. This advantage may not hold if manyhigher orders must also be evaluated; in fact, it
appears that both techniques need about the sameamount of time to calculate results if the samenumber of higher-order components is retained. How-ever, even in that case, the multiple-scattering ap-proach requires considerably less memory than thecoupled-mode techniques; in fact, the effectiveness ofthe latter may be severely limited by memory require-ments if high accuracies impose the retention of alarge number of higher diffracted orders.
VI. Concluding RemarksWe have examined the mechanism that causes crosstalk in volume holographic interconnections by apply-ing a multiple-scattering analysis to canonic 1 - 2and 2 1 interconnects. Although these canonicunits require only two superposed gratings, theyserve as basic elements for clarifying diffraction ef-fects in configurations that involve a larger number ofsuch gratings. This approach has revealed the pres-ence of cross talk of the spectral type, which is causedby unwanted higher-order scatterings that belong tothe same order as the desired diffracted order, as wellas cross talk of the spatial variety, which is caused byscattered components that belong to different spec-tral orders. We have specifically examined the effectsof varying the angular separation A between thegratings, the modulation depth M., the diffractionefficiency t 2, the crystal thickness z, and the dy-namic range r in the canonic interconnects underconsideration.
Both types of cross talk may be detected at theinterconnection output, but they behave quite differ-ently in terms of the physical parameters involved.Specifically, spectral cross talk is independent of Adkwhereas spatial cross talk is independent of M . As aresult, different restrictions are needed for the grat-ing parameters if the total cross talk must be keptbelow a desired limit. To achieve such a limit, intercon-nects should be designed so that the separation Adkbetween any two gratings is larger than a minimumvalue and diffraction efficiencies t,,n2, are kept belowcertain maximum values. These conditions becomemore restrictive as the dynamic range r of the inter-connecting channels is increased.
Most of the results presented here were derived byapplying analytical and numerical considerations toholographic interconnects that involve TE-mode inci-dence in the presence of only two superposed sinusoi-dal gratings in two-dimensional geometries. How-ever, we focus on small values of Ac\, and the resultswe obtained for the canonic two-grating intercon-nects are indicative of those expected in configura-tions that involve many superposed gratings. Further-
more, we have outlined the extension of the rigorousmultiple-scattering formalism to a larger number ofgratings, as well as to nonsinusoidal modulations andto TM-mode incidence. An analytical examination ofthese extensions suggests that most qualitative as-pects and many of the quantitative considerationsdeveloped here for two gratings also apply to configu-rations that involve a larger number of gratings intwo-dimensional geometries. In particular, we expectthat the minimum value of A that is needed to limitcross talk will not change appreciably if the number ofgratings increases. On the other hand, we expect thatdiffraction efficiencies may have to be lowered (bydecreasing either M, or z0) to maintain cross talkbelow a prescribed limit as the number of gratings isincreased. However, the application of such consider-ations may have to be modified considerably forvolume holographic interconnections containing grat-ings that are superposed in three-dimensional pat-terns.
This work was supported by the National ScienceFoundation and by the New York State Science andTechnology Foundation under its Centers for Ad-vanced Technology Programs. We express our appre-ciation to the reviewers whose constructive com-ments enabled us to introduce revisions that haveconsiderably improved this paper.
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