Wave diffraction by many superposed volume gratings Kun-Yii Tu and Theodor Tamir A multiple-scattering technique that was recently developed to evaluate wave diffraction by two superposed gratings is extended to situations in which there is an arbitrary number of gratings. In this approach the diffraction process can be represented in terms of a flow graph that serves as a template to construct algorithms for calculating the intensity of any diffracted order. We show that such calculations do not require a large computer memory if they are implemented byjudiciously tracking the relevant diffracted order throughout the flow paths. Using two types of typical grating structures as examples, we also investigate the effect of the relative grating phase on the diffraction efficiency. We thus find that the multiple-scattering analysis can readily identify those grating structures that are sensitive to the relative phase relationship. 1. Introduction A variety of applications in optics and acoustics, such as holographic interconnects, optical storage, and ultrasound imaging, have stimulated studies of wave diffraction by media whose dielectric properties are spatially modulated by several sinusoidal variations oriented along different directions. The pertinent configurations have usually been described in terms of superposed volume gratings that generate dif- fracted orders associated with multiple periodicities. The resulting diffracted field was obtained in the past by applying coupled-wave techniques to the corre- spondingboundary-valueproblem.", 2 However, these techniques require prohibitively large computer mem- ories if more than 2 gratings are involved, so that most of the analytical work reported so far has been restricted to only 2 superposed gratings. We have therefore recently developed 3 a different approach that may be readily applied to any number of superposed gratings. The key feature of our approach is to use a field representation that views the diffraction process as a well-defined sequence of multiply scattered waves. This wave sequence can be readily described in terms of a flow graph that provides a physically meaningful interpretation of the scattering process on the one hand and that directly serves as a template for developing effective computa- The authors are with the Department of Electrical Engineering, Polytechnic University, Brooklyn, New York 11201. Received 27 October 1992. 0003-6935/93/203654-07$06.00/0. t 1993 Optical Society of America. tional algorithms on the other hand. However, our previous research provided results for only 2 grat- ings; it did not discuss some of the important details that help to analyze and evaluate situations involving more than 2 gratings. We therefore address these issues here and illustrate the power of the multiple- scattering approach by examples involving many more than 2 superposed gratings. In particular, we consider an optical interconnect that involves 1 input and 12 outputs, which requires a total of 12 gratings, and a global interconnection between 3 inputs and 3 outputs, which requires a total of 9 gratings. In these examples, we specifically address the more difficult case involving gratings that are inclined at small angular separations between each other; this case is helpful in anticipating problems posed by the possible presence of large numbers of superposed gratings. Previously, the relative phases between the various superposed gratings were found to have a dramatic effect on the diffracted waves 4 and on the efficiencyof fan-out elements. 5 We therefore also examine here the effects of these phase relationships for the two examples of grating interconnections mentioned above. In this context we find that the diffraction efficiencyis sensitive to the phase differences between the gratings only if the gratings are recorded so that the grating wave vectors form closed loops. Further- more, we show that the multiple-scattering point of view provides clues to whether the grating phases may affect the diffraction results. We also justify and consolidate the mathematical basis of the 3654 APPLIED OPTICS / Vol. 32, No. 20 / 10 July 1993
Transcript
Wave diffractionby many superposed volume gratings
Kun-Yii Tu and Theodor Tamir
A multiple-scattering technique that was recently developed to evaluate wave diffraction by twosuperposed gratings is extended to situations in which there is an arbitrary number of gratings. In thisapproach the diffraction process can be represented in terms of a flow graph that serves as a template toconstruct algorithms for calculating the intensity of any diffracted order. We show that suchcalculations do not require a large computer memory if they are implemented by judiciously tracking therelevant diffracted order throughout the flow paths. Using two types of typical grating structures asexamples, we also investigate the effect of the relative grating phase on the diffraction efficiency. Wethus find that the multiple-scattering analysis can readily identify those grating structures that aresensitive to the relative phase relationship.
1. Introduction
A variety of applications in optics and acoustics, suchas holographic interconnects, optical storage, andultrasound imaging, have stimulated studies of wavediffraction by media whose dielectric properties arespatially modulated by several sinusoidal variationsoriented along different directions. The pertinentconfigurations have usually been described in termsof superposed volume gratings that generate dif-fracted orders associated with multiple periodicities.The resulting diffracted field was obtained in the pastby applying coupled-wave techniques to the corre-spondingboundary-valueproblem.", 2 However, thesetechniques require prohibitively large computer mem-ories if more than 2 gratings are involved, so thatmost of the analytical work reported so far has beenrestricted to only 2 superposed gratings.
We have therefore recently developed3 a differentapproach that may be readily applied to any numberof superposed gratings. The key feature of ourapproach is to use a field representation that viewsthe diffraction process as a well-defined sequence ofmultiply scattered waves. This wave sequence canbe readily described in terms of a flow graph thatprovides a physically meaningful interpretation of thescattering process on the one hand and that directlyserves as a template for developing effective computa-
The authors are with the Department of Electrical Engineering,Polytechnic University, Brooklyn, New York 11201.
Received 27 October 1992.0003-6935/93/203654-07$06.00/0.t 1993 Optical Society of America.
tional algorithms on the other hand. However, ourprevious research provided results for only 2 grat-ings; it did not discuss some of the important detailsthat help to analyze and evaluate situations involvingmore than 2 gratings. We therefore address theseissues here and illustrate the power of the multiple-scattering approach by examples involving manymore than 2 superposed gratings. In particular, weconsider an optical interconnect that involves 1 inputand 12 outputs, which requires a total of 12 gratings,and a global interconnection between 3 inputs and 3outputs, which requires a total of 9 gratings. Inthese examples, we specifically address the moredifficult case involving gratings that are inclined atsmall angular separations between each other; thiscase is helpful in anticipating problems posed by thepossible presence of large numbers of superposedgratings.
Previously, the relative phases between the varioussuperposed gratings were found to have a dramaticeffect on the diffracted waves4 and on the efficiency offan-out elements.5 We therefore also examine herethe effects of these phase relationships for the twoexamples of grating interconnections mentionedabove. In this context we find that the diffractionefficiency is sensitive to the phase differences betweenthe gratings only if the gratings are recorded so thatthe grating wave vectors form closed loops. Further-more, we show that the multiple-scattering point ofview provides clues to whether the grating phasesmay affect the diffraction results. We also justifyand consolidate the mathematical basis of the
Fig. 1. Geometry of wave diffraction by a thick dielectric layercontaining more than 2 superposed gratings.
multiple-scattering representation by presenting aproof of its general convergence in Appendix A.
2. Outline of the Multiple-Scattering Theory
To put our approach into proper perspective and toestablish notation, we first present a brief summaryof the multiple-scattering representation we haverecently developed3 for wave diffraction by two super-posed gratings and extend it to an arbitrary numberof such gratings. For this purpose we considerthe two-dimensional (a/y 0) geometry in Fig. 1involving a layer of large thickness z that con-tains many periodicities with arbitrary periodsd>(p = 1, 2, . . , N) aligned at various angular sepa-rations Af with respect to each other. A perpendicu-larly polarized (TE) plane wave having a wave vectorko = xuo + v0 is incident from the left. For simplic-ity all media are assumed to be lossless and isotropic,and the average dielectric constant of the modulatedlayer is taken to be equal to the exterior dielectricconstant E0 . The dielectric properties of the layer at
0 < z < z0 are then characterized by
E(r) = E0[1 + g(r)],
Ng(r) = M cos(K,, r +0)(1
R=1
Here r = xx + &z is the position vector, MR is themodulation amplitude, Kp = xW + V , is the gratingvector, and 0,, is the relative phase of the [uth grating.Apart from having considered only 2 gratings, Ref. 3ignored the possible presence of the phase term 0,which is specifically included now in Eq. (1). Theelectric field must then satisfy
(V2 + k2)E(r) = -k2g(r)E(r). (2)
We choose a solution in the iterative form
E(r) = E(0)(r) + ko f G(r, rj)g(rj)E(rj)drj, (3)
where E(°) refers to the incident plane wave andG(r, ri) is the free-space Green's function. Startingwith E(°) as a first trial field and using Eq. (3) as aniterative expression, we can systematically generatesuccessive terms representing higher-order scatteredcomponents. Omitting derivation details alreadygiven in Ref. 3, we then obtained the total scatteredfield as
E(r) = E.(m), (4)m=O v(m)
where m is the scattering level, v(m) represents theset of diffracted components at the mth level, andEn(m) is the scattered-field component belonging to thediffraction order n(m) = [nj(m), n 2 (m), . . . , nN(m)] Ev(m). Each index n(m) consists of a set of Nnumbers.Every n(m) component is generated by a flow process,as illustrated in Fig. 2 for a case involving only 2gratings. For situations havingN > 2, a correspond-ing flow graph can be generated similarly by using the
where (m) denotes the specific grating that causesscattering into the mth level.
So far, the number set n(m) = [nj(m),n2 (m), . , nN(m)] has denoted a specific (partial) fieldcomponent that appears at the mth level and, as such,is a member of the larger set v(m). However, if werefer to Fig. 2, it is evident that each n(m) can bereached by following a unique path that starts at theinitial point n(0) = (0, 0, . . . , 0) and ends at thatparticular n(m). We can therefore use n(m) hence-forth to also denote this path. For each unique pathleading to some n(m), the corresponding field compo-nent En(m) is then given by
En(m) = An(m)Pn(rn) exp(ikn(m)* r), (6)
where the propagation vector of each scattered waveassociated with the number element n(m) is
kn(m) = XUn(m) + Wn(M), (7)
with
Un(m) = Uo + n(m)U + n2 (m)U2 + + nN(m)UN,
(8)
Wn(m) = [ko -Un ]1/2 . (9)
As implied by Eq. (6), the amplitude of En(m) requiresthe evaluation of the accumulated scattering strength
mn r kWM qz.
An(-) = TI an(q) = L(q) exp[ir(q)0o(q)], (10)q=1 q=1 4
0n(q)
where r(q) = 1 is the only nonzero element in thearray [n(q) - n(q - 1)]. It is important to note thatthe index n(m) in An(m) now specifies not only aparticular scattered-field component but also theunique path that led to it by following along theintermediate steps q = 1, 2, . . . , m in that path. Inaddition to the value of An(m), we must also find the
accumulated phase effects expressed by the phasecorrelation function 4Pn(m). For the same scatteringpath as was specified for An(.), we find that, at thenormalized thickness 4 = z/zo = 1,
'n(m) = L-lI i
m-is(S + iAn(r)) (11S + (A.(.) - n()]
q= [ +=1
(11)
where L-l(s) denotes the inverse Laplace transformin the s plane and phase mismatches are denoted by
An(q) = [Wn(q) - Vn(q)]ZO, (12)
with
Vn(q) = v + nl(q)Vl + n2(q)V2 + * + nN(q)VN, (13)
forq = 1,2,. . .,m.Finally, the field component of any particular dif-
fracted order (n1, n2 , . . , nN) is obtained by collectingtogether all the field components En(m) sharing thesame index n(m) = (nj, n2, .. , nu), which impliesthat all the scattered components (at any level)belonging to this spectral order n(m) must be identi-fied and summed up to construct the complete solu-tion. The field of any specific diffracted order canthen be written as
nln2,.,n = E Enln2-, nN'm=O v(m)
(14)
where the summations over m may be truncated atsome m = mo above which the higher-order scatteredcomponents become negligible.
3. Computational Algorithm
The formulation embodied in Eqs. (5)-(14) can bereadily translated into algorithms for evaluating thediffracted fields, but their implementation requiresthe clarification of several crucial computational as-pects. In this context, we recall that, in order tocalculate a specifically desired diffracted order, wemust identify all the scattering paths that lead toevery component (at all the levels m under consider-ation) belonging to this diffracted order. While manyroutines can be devised to determine these scatteringpaths, a natural procedure would use a parallel searchto track the paths; as an example, such a search wouldfollow the steps marked in Fig. 3, namely, from m = 1to m = 2, then m = 1 to m = 3, and so forth. Totrack the paths from level m to level m + 1, one mustensure that the scattering indices n(m) of all thecomponents at level m be retained in the computermemory. The identification of any diffracted orderat level m thus requires the determination of all thepaths that arrive from all preceding levels. Werecall3 that, for N gratings, the number of scatteringpoints at the mth level is (2N)m. Therefore a parallel-
Fig. 3. Parallel search for scattered components at the m = 2 level for a 2-grating case.
search algorithm would require a high memory capac-ity, especially when m and N are large.
To avoid such a memory cost, we developed analgorithm that searches and evaluates any desireddiffracted order n(m) by sequentially following thepaths leading to all these n(m), starting with the first(m = 1) level and ending at the last (m = mo) levelthat provides significantly large contributions. Toillustrate this type of search, we show in Fig. 4 thesearch for diffracted orders at the m = 2 level in a2-grating situation. In the figure, the encircled num-bers 1, 2,. . ., 10 indicate the sequence of the search.For example, consider the search for the diffractedorder n(m) = (0, 0). The search for any n(m) alwaysstarts at the topmost point (0, 0) and follows thebranch 1 to (1, 0) and then follows branch 2 to (2, 0).For our example, the search finds that (2, 0) ( 0),so that it ignores that end point and follows branch 3to (0, 0), which is now a desired end point. Thecontribution E() of that end point is then calculated,and its value is memorized. The search next exam-ines branches 4 and 5, both of which do not uncover adesired (0, 0) end point; therefore no calculationsneed be performed. It is important to note that, sofar, the path through branch 1 had to be retained sothat the search can return to the initial point (0, 0)and start a new search along branch 6, after which allthe path information through branch 1 can be dis-carded. The search then follows branch 7 and findsa new desired (0, 0) end point whose contribution E2)is calculated and added to the previous one (obtainedat the end of branch 3). The search subsequentlycontinues in the same fashion until all end points n(2)are examined. Of course, this discussion for theexample shown in Fig. 4 assumes that the precedinglevel m = 1 was already accounted for by an analo-gous previous search for that level. After complet-ing the task for the m = 2 level, the search may
continue for contributions at the m = 3 level, and soon. Following this search process from left to rightin the flow chart, we can thus sum all the scatteredfields that belong to the diffracted order of interestfrom level to level by using a minimal amount ofcomputer memory. This procedure can be easilymodified so as to permit intermediate diffractionorders of lesser importance to be easily bypassed, thusfurther minimizing the computational cost.
The determination of scattering paths recom-mended above belongs to a deterministic type ofapproach. Another procedure based on the MonteCarlo method was reported by Korpel and Bridge,6
but they applied it to more restricted situationsinvolving only a single grating. In our case, theprocedure illustrated in Fig. 4 was applied to theexamples discussed in Section 4, for which we haveevaluated the scattered-field components by using theinverse Laplace transform of Eq. (11). This trans-form operation was calculated by resorting to theMATLAB program7 with its impulse subroutine inthe control toolbox.
4. Numerical Results and Discussion
To demonstrate the power of this method, we presentbelow a set of numerical results for two types ofgrating configurations. The first consists of 12 grat-ings formed with 1 object wave and 12 referencewaves arranged in the geometrical relation shown inFig. 5(a). The angular separations A between thereference waves can be arbitrarily small, which im-plies that the grating wave vectors K, can be arbi-trarily close to each other. We assume that thesegratings are recorded in such a way that intermodula-tion (crossed) gratings that may be formed betweenany 2 of the 12 reference waves can be neglected.After the gratings are recorded, their readout can beperformed by injecting an incident wave so as to
1,0 -1 ,u U,I -,
2,0 1,1 1,-1 0, -2, -1,1 ,-I I
Fig. 4. Sequential search of scattered components belonging to the m = 2 level for a 2-grating case.
(b)Fig. 5. Wave-vector k diagrams for (a) recording and (b) readingout 12 superposed gratings KL, with j = 1, 2,..., 12. In (b) theletters A through L (A, B, C, K, and L are shown) denote wavevectors for the diffraction orders (1 0 0 0 0 0 0 0 0 0 0 0),(010000000000), ... , (O00000000001), respectively.The angular separations between references waves are given byA4)j forj = 1, 2 . . , 11.
coincide with one of the reference waves, as shown inFig. 5(b). Such a situation is relevant to a 12 1optical interconnect application. For small angularseparations A1\ between the gratings, as assumedhere, a total of 12 major diffracted orders[(lOOOOOOOoooo),(oloooooooooo),...,(O 0 0 0 0 0 0 0 0 0 0 1)] may appear. For brevity,we refer to these 12 orders by using the capital lettersA through L, respectively. The calculated diffractedintensities versus a uniform angular separation A4are shown in Fig. 6, in which we assume thatBragg-matched conditions in all 12 gratings are al-ways maintained. In the computation process, dif-
0.07 . . . . . .~ @*|e X **I*
0.06 A
~'0.05a)
iO0.04wD0r 0.03E
U~~~~~~~~~~
t2 0.02
0.01
0 0.5 1.0 1.5 2.0 2.5 3.0
A~~~~X lo -3
Fig. 6. Multiple-scattering results for 12 gratings arranged as inFig. 5(b), for zo/d = 500, Ki/ko = 0.4, and M = M = 1.65 x 10-4for all , with the angle of K being = r/2. The variation ofdiffraction efficiencies is shown for various orders A, B, ... , Lversus angular separation Ahi = A = const. for allj.
fracted orders that satisfy ni + n2 + * * + n12 • 0 or1 have been neglected; this is justified because theintensities of these diffracted orders are negligibleowing to their large Bragg mismatch. The numberQ of diffracted orders involved in the computation forsuch a 12-grating case is Q = 9,_Q = 145, Q = 937,. . . , if scattering levels are considered up to m = 1,m = 3, m = 5, .. ., respectively. If a coupled-waveapproachl,2 were used to solve such a situation, it mayrequire the computation of a 2Q x 2Q matrix, whichmay prove to be an impossible task for large Q.
The second case involves gratings formed by using3 reference waves and 3 incident waves, which re-quire a total of 9 gratings, as shown in Fig. 7(a).Such a structure is relevant to a global interconnectapplication between 3 inputs and 3 outputs. Fordistinction between the various gratings, the grat-ings' angular separations A4 are shown in the figuremuch larger than assumed. As in the previous case,we ignore the intermodulated gratings formed be-tween the reference and/or object waves. To probethe grating diffraction, we can inject 1 incident wave,as shown in Fig. 7(b), so that the wave coincides withthe first reference wave. This grating structureexhibits a configuration for which the grating wavevectors form a closed loop, whose relevance is dis-cussed below. The presence of such a closed loop,together with the symmetry of the grating structurewith respect to the z axis, causes many diffractedorders to have the same propagation vector; i.e.,different indices n(m) can lead to the same value of
- - -> object waves
- > reference waves
(a)
major output waves
orG
(b)Fig. 7. Wave-vector k diagrams for (a) recording and (b) readingout 9 superposed gratings. The angular separations between the3 object and 3 reference waves have the same value of Ali. In (b)the wave vector A, B, C, D, . . ., I represent the diffracted orders(100000000), (010000000), (001000000), ....(0 0 0 0 0 0 0 0 1), respectively. Some of the major orders appearin pairs belonging to the same diffraction orders; these pairs are Bwith D, C with G and F with H.
kn(m). For example, for the orders (1 0 0 0 0 0 0 0 0),
(1 -10100000),and(10 -1000100),theprop-agation vectors satisfy k 0 000 0 = k -=0 00000 k, o -100o100. The efficiencies of six major diffrac-tion orders are shown in Fig. 8, in which the phaseoi = for all .
It is interesting to observe in this particular gratingstructure that the grating phases 0, may have aprofound impact on the diffraction efficiencies. Bytaking a random phase distribution for the gratings,as illustrated in Fig. 9, we observe a dramatic differ-ence in the diffraction efficiency as compared with thecase shown in Fig. 8. However, for the gratings ofthe 12 - 1 interconnect case shown in Figs. 5 and 6,the magnitudes of the diffraction efficiencies are notaffected if the phases Op. are changed to any othervalues.
This behavior can be easily understood if themultiple-scattering paths are traced, as clarified inFig. 10. In Fig. 10(a), for example, a component ofthe diffracted order (1 0 0 0 0 0 0 0 0 0 0 0) is pro-duced by the path ( 0 0 0 0 0 0 0 0 0 0 0)
(100000000000)-(10000 -1000000)(1 0 0 0 0 0 0 0 0 0 0 0); to reach the last element in
this path, scatterings by both K6 and -K 6 at levelsm = 2 and m = 3 must occur, which are accompaniedby phase changes r(q)op(q) that effectively cancel eachother. The net phase term that is thus left is causedby scattering from the K1 grating vector, which doesnot affect the diffraction efficiency. Thus in the caseof 12 - 1 gratings, all the paths that lead to ahigher-order level contribution to a first-order dif-fracted order involve branches that must be scatteredback and forth by the same grating, thus cancelingthe phase produced by such scattering pairs (for anyarbitrary phase). However, in the case of 3 - 3gratings, the grating vectors form a closed loop, inwhich case a higher-order level can be reached with-out having to retrace the sequence of the scatteringgratings. In this case, phase cancellation does not
0.20
8)aa,
0
i!0
0
0.16
0.12
0.08
0.04
0.000.0 0.5 1.0 1.5 2.0 2.5 3.0
AO x 10 3
Fig. 9. Multiple-scattering results for 9 gratings arranged as inFig. 7(b). The variation of diffraction efficiency is shown versusangular separation Ali [given in Fig. 7(a)]. The parameters are thesame as those in Fig. 8 except that the grating phases for the wavevectors A, B, C, D, E, F, G, H, and I are given by a set of randomnumbers (2.0485, 0.47229, 2.1405, 1.2121, 1.2181, 1.5700, 0.4635,1.8447, and 2.6565, respectively).
generally occur, and the diffraction efficiency there-fore becomes a function of the relative phase distribu-tion of the gratings. As an example for a 3 - 3grating, we see in Fig. 10(b) that a scattering path isgiven by (000000000) -- (001000000) -(0 0 1 0 0 0 0 0 -1) - (0 0 1 0 0 0 1 0 -1) [or(1 0 0 0 0 0 0 0 0)]. This path involves scatteringsby the K3, -K9 , and K7 grating vectors, whose phaseterms do not cancel. Hence the diffraction efficiencyis expected to be different if the grating phases aremodified.
The convergence of the multiscattering approachdescribed here was verified numerically by the ac-cepted procedure of successively obtaining resultsthat accounted for an increasing number m of scatter-ing levels. In all cases considered by us, this proce-dure indicated a reasonably rapid converging process.
(1,0,0,0,0,0,0,0,0,0,0,0) (output wave)
0.14
0.12
0a.U
r
u
0Co
0X
0.10
0.08
0.06
0.04
0.02
0.00
0.0 0.5 1.0 1.5 2.0 2.5 3.0
AOl x 10- 3
Fig. 8. Multiple-scattering results for 9 gratings arranged as inFig. 7(b). Here zo/di = 500, K1/ko = 0.4, and M, = M = 1.65 10-4 for p. = 1, 2,.. ., 9, with the angle of K1 being ' = w/2. Thevariation of diffraction efficiencies is shown for the A, B. . . , Forders versus angular separation Ad [given in Fig. 7(a)]. For thiscase the grating phases 0. are all set equal to zero for all p.
In particular, the results shown in Figs. 6, 8, and 9were obtained by retaining scattering levels up to onlym = 7, which yielded an accuracy higher than thatshown by the scales in those figures. As alreadydiscussed above, our algorithms used a minimalamount of memory; therefore no memory problemswere encountered. Thus for scattering levels m <10 we estimate that a memory of 1 Mbyte is morethan sufficient for any number of gratings. On theother hand, the computational time may becomeunduly long for cases requiring the inclusion of highscattering levels. Most importantly, however, forarbitrary angles A4i we have now presented resultsfor a significantly larger number of gratings forwhich, to the best of our knowledge, no other studieshave been reported so far.
5. Conclusions
By using a multiple-scattering representation of thescattered fields, we have presented a technique forevaluating diffraction effects produced by media thatcontain an arbitrary number of superposed gratings.This approach has an advantage in that it usesmeaningful flow graphs to systematically constructeffective algorithms for evaluating the intensity ofany desired diffracted order. In particular, we havepresented a systematic procedure for tracking thesegraphs so that calculations can be implemented byminimizing both computer memory and computationtimes. Furthermore, we have shown that the grapharchitecture can be examined in conjunction with thewave-vector configuration of the gratings so as toreveal whether the relative phase relationship be-tween the gratings affects the efficiency of any dif-fracted order.
We have obtained and discussed results for situa-tions involving as many as 12 gratings having smallangular separations. Such results may not be easilyobtained by using coupled-wave methods becausetheir application would require a prohibitively largecomputer memory. However, our formulation isrestricted to two-dimensional situations, and there-fore the results may not be immediately applicable tomore realistic three-dimensional cases. On the otherhand, this formulation holds for arbitrarily small angularseparations Ali between any 2 gratings. Hence, to theextent that a large number of gratings can be accommo-dated in a two-dimensional situation by using arbitrarilysmall values of A, the present approach provides apowerful analytical tool for further work.
Appendix A: Convergence of the Multiple-ScatteringSolution
Because the multiple-scattering solution in Eq. (4) isin the form of a series summation, it is appropriate toprove its convergence. For this purpose, we use theinequality
E(m)= En(m) < IEmI = i JE.(fl. (Al)v(m) >(m)
The maximum value of I can be obtained if themaxima of IAn(m) I and I nm) I are known. From Eq.
(10) the magnitude of An(m) is determined by Wn(q),
whose value increases as the diffraction order movesaway from a Bragg-matched condition. Hence amaximum value ma of all IUan(q)l must occur overall the possible scattering paths. (We assume herethat we exclude the possible presence of Rayleigh-type anomalies, for which Wn(q) = 0.) Thus the maxi-mum of IAn(m) is (max)m. For T'n(m), on the otherhand, Eq. (11) indicates that the maximum of I Tnm Iis given by 1/m!, as was discussed in Ref. 3. Usingthe maxima of both IAn(m) I and I 1 we can evalu-ate the possible maximum of IE(m) in inequality (Al)by taking into account the total number (2N)(m) ofscattering paths at each mth scattering level. Wethen obtain
IEW~m. = (2Namx)mIE~)Imo, i!
(A2)
We can then verify the convergence of the seriessummation in Eq. (4) by using the d'Alembert testinvolving the ratio
I E(m+') Ima
I E(m) Im
(2NcxmaX)m+l
(m + 1)!
(2Ncimax)m
m!
2Ntmax
+ 1 (A3)
which implies that, for fixed values of N and amax,there exists a minimal value of m for which the aboveratio becomes less than one. This condition satisfiesthe d'Alembert convergence test. However, this proofholds for the convergence of E(m) in inequality (Al), inwhich we retain forward-scattering components only.This proof may need to be modified if backward-scattered components are also included.
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.
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