1712 J. Opt. Soc. Am. A/Vol. 7, No. 9/September 1990 Boag et al.

Analysis of diffraction from doubly periodic arrays ofperfectly conducting bodies by using a patch-current model

Amir Boag, Yehuda Leviatan, and Alona Boag

Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel

Received November 29, 1989; accepted May 10, 1990

A novel solution is presented for the problem of three-dimensional electromagnetic scattering of a plane wave from adoubly periodic infinite array of perfectly conducting bodies. A set of fictitious spatially periodic and properlymodulated patches of magnetic current is used to simulate the scattered field. These patch currents are of dualpolarization and have complex amplitudes. The electromagnetic field radiated by each of the periodic patchcurrents is expressed as a double series of Floquet modes. The complex amplitudes of the fictitious patch currentsare adjusted to render the tangential electric field zero at a selected set of points on the surface of any of thescatterers. The procedure is simple to implement and is applicable to arrays composed of smooth but otherwisearbitrary perfectly conducting scatterers. Results are given and compared with an analytic approximation.

1. INTRODUCTION

The study of diffraction of a plane wave from periodic struc-tures is long standing. It has been motivated by academiccuriosity as well as many engineering applications. It is ofpractical importance in designing reflection and transmis-sion gratings often used as filters, broadband absorbers,polarizers, and frequency scanned reflectors. While singlyperiodic gratings have been treated extensively, doubly peri-odic gratings, being in general more difficult not only toanalyze but also to fabricate, have received considerably lessattention. One type of doubly periodic structure that hasbeen investigated by many researchers comprises infinitesi-mally thin planar doubly periodic screens in various configu-rations, often referred to as frequency-selective surfaces.1-3

Perfectly conducting screens of finite thickness consisting ofdoubly periodic arrays of apertures, known as inductivegrids, have also been extensively studied.45 While thestructures in Refs. 1-5 may be different, a common ingredi-ent of essentially all of them is that the fields in the variousregions can be represented by modal expansions relative tothe axis normal to the grating. These modal representa-tions are then matched by using appropriate boundary con-ditions, and the unknown modal coefficients are readily de-termined. In contrast to the above discussion, rigorousstudies of diffraction from doubly periodic arrays of finite-sized perfectly conducting scatterers, referred to as capaci-tive grids, have not been reported in the literature. In thislatter case the fields in the space gap between the scatterersforming the array cannot be represented in terms of analyti-cally known modal functions.5 Therefore even the methoddevised in Ref. 6 for the analysis of diffraction by doublyperiodic surfaces falls short, while differential method pro-cedures might be infeasible with present-time computerstorage and speed limitations.

In this paper we present a new method for analyzing three-dimensional electromagnetic scattering from doubly period-ic arrays. The technique is applicable to arrays composed ofperfectly conducting bodies of smooth, but otherwise arbi-trary, shapes. An example of an array is depicted in Fig. 1.

We follow the approach outlined in Ref. 7 for analyzingscattering by smooth homogeneous scatterers. The basicidea in Ref. 7 is as follows: Instead of employing surfaceintegral equations in solving for conventional electric andmagnetic surface currents, we solve for fictitious source cur-rents that lie a distance away from the surface. This ideahas been applied successfully to two-dimensional diffractionfrom gratings of cylinders,8 as well as to sinusoidal and ech-elette gratings.9-10 In Refs. 8-10, an expansion of periodicstrip currents is used for the unknown fictitious currentsthat simulate the periodic scattered field, and point match-ing is used for testing.

Here, we employ the basic strategy of Refs. 7-10 for facili-tating the analysis of three-dimensional scattering fromdoubly periodic arrays of isolated perfectly conducting scat-terers. We set up a simulated equivalent situation to theoriginal one in the region surrounding the scatterers. Thescattered field must be a source-free Maxwellian field satis-fying the radiation condition at z - , the periodicityconditions of the Floquet theorem," and the boundary con-dition on the surfaces of the scatterers. Instead of express-ing the scattered field as a conventional integral in terms ofthe physical surface currents, we simulate the actual field bythe fields of fictitious sources of yet unknown amplitudesthat lie a distance from the surface. Hence, in the simulatedequivalence the physical bodies are removed and the period-ic field that they scatter is simulated by the field of a set offictitious doubly periodic patches of currents satisfying theFloquet periodicity conditions and situated in the regionoriginally occupied by the scatterers. Each periodic patchcurrent lies in a plane parallel with the xy plane (the planespanned by the directions of the periodicity). All the patch-es are characterized by a common Fourier-transformablemagnetic-current density profile, which, for each periodicsource, is multiplied by an as yet undetermined constantcomplex amplitude. They are assumed to radiate in anunbounded homogeneous space filled with the same mediumas that surrounding the scatterers. Patches of electric cur-rent could be used as well. Patches of magnetic currentwere chosen because the electric field that they produce is

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Vol. 7, No. 9/September 1990/J. Opt. Soc. Am. A 1713

Unbounded Space (, E) magnitudes are specified by the respective periods. It isassumed that any inhomogeneity is confined between the z= -b and the z = b planes. The problem geometry togetherwith a relevant coordinate system is shown in Fig. 1. Itshould be noted that, according to our convention, the z axisis oriented downward. The medium surrounding the scat-terers is of permeability As and permittivity e. The mediumcan be dissipative; thus g and e are allowed to be complex.

A plane wave given by

Einc(r) = EinC exp(-jkinc r), (1)

Fig. 1. General problem of plane-wave scatteringperiodic grating of finite-sized scatterers.

from a doubly

easier to compute. Locating the sources some distance awayfrom the surface permits us to use periodic patch currentswith smooth current density profile that lie in planes parallelwith the xy plane spanned by the two directions of periodici-ty. This feature is attractive because it enables therepresentation of the field produced by each periodic cur-rent patch by uniformly convergent series of z-directed out-going and decaying plane waves known as Floquet modes.It follows that outside the grating region the total field radi-ated by the patches can also be represented analytically bymeans of these Floquet modes. Thus the fields can be deter-mined anywhere by summations of analytic terms. This is adesirable attribute as one avoids the surface integrationsassociated with the field computation at the three principalstages of the solution. The first stage is that of constructingmatrix equations for the problem, the second is that of test-ing the solution by checking the degree to which the bound-ary conditions are satisfied over a denser set of points on theboundaries, and the third is that of computing the scatteredfield and the reflection and transmission coefficients of vari-ous Floquet modes after the solution has been established.

The patch-current sources lying a distance away from theboundary surfaces produce a set of smooth field functions onthe surfaces that may be well suited for spanning the actualsmooth field on the boundaries. Furthermore, since we areactually using a basis of smooth field functions for repre-senting fields on the boundary, the boundary condition canbe enforced by a simple point-matching testing procedureand the unknown source amplitudes are readily determined.

The paper is organized in the following manner. Theproblem under study is specified in Section 2. The solutionis formulated in Section 3. Results of several numericalsimulations are presented in Section 4 and compared with ananalytic approximation in order to demonstrate the efficien-cy and accuracy of the proposed technique. Finally, a fewconclusions summarize the paper.

2. PROBLEM SPECIFICATION

Consider a doubly periodic array of scatterers. The array iscomposed of an infinite set of identical perfectly conductingscatterers arranged in a doubly periodic lattice. The latticeis described by two vectors d, and d2 lying in the xy plane.The vectors d, and d2 are referred to as lattice vectors. Theyare aligned with the two directions of periodicity, and their

with harmonic exp(jwt) time dependence assumed and sup-pressed, is incident on the grating. Here, kinc and E 0 c de-note, respectively, the wave vector and the amplitude of theincident field. Our objective is to determine the field scat-tered by the grating (Es, Hs) (i.e., the actual field minus theincident field). The field should be a source-free solution ofthe Maxwell equations and obey the Floquet periodicityconditions

Es(r + dp) = exp(-jkinc - dp)Es(r), p = 1, 2. (2)

In addition, (Es, Hs) should satisfy the boundary condition

n X ES = -h X Einc (3)

where S is the boundary of an arbitrary selected scattererand h is a unit vector that is normal to S.

3. FORMULATION

A. Simulated Equivalent SituationWe now describe how the simulated equivalent situation tothe original one in the region surrounding the scatterers isset up. According to our general idea, in the simulatedequivalence that is shown in Fig. 2, the scattered field (Es,

Hs) is simulated by a field of a set of doubly periodic ficti-tious patches of magnetic current Mqi, q = 1, 2, i = 1, 2, ... ,N. These sources are located in the region occupied by thescatterers in the original situation and are treated as sources

Unbounded Homogeneous Space

(Einc ,Hinc)

kinc(gia)

(Es +Einc Hs +Hinc)

Periodic Patch Currents

/ Mathematical Boundary C

Fig. 2. Simulated equivalence for the region surrounding the scat-terers.

(EH")

\k kin'

Boag et al.

1714 J. Opt. Soc. Am. A/Vol. 7, No. 9/September 1990

radiating in an unbounded space filled with homogeneousmaterial that is identical to that surrounding the scatterersin the original situation. They have constant dimensionlessamplitudes Kqj that are yet to be determined. The electricfield Es at observation point r due to these sources is givenby

2 N

Es(r) = Z y KqjEqj(r),q=1 i=1

(4)

where Eqi describe the field due to a source Mqj of unitamplitude (Kq = 1). Obviously, since these periodic patch-es produce fields satisfying the Floquet periodicity condi-tions, the simulated scattered field [Eq. (4)] also satisfiesthem.

It is important to note that the location of the sources inthe simulated equivalence has not been specified yet. As faras the formulation is concerned, their location can be arbi-trary. The question of selecting source locations that aresuitable for a numerical solution is an important one. Fromthe numerous geometries considered in our earlier researchwith perfectly conducting and penetrable scatterers,7 -'0 wehave concluded that the sources should be placed on surfacesof a shape similar to that of the actual boundary. We willgive this issue further attention in Section 4.

B. Evaluation of the Unknown Amplitudes Kqj}By the construction, the simulated scattered field Es satis-fies the radiation and the periodicity conditions. Evidently,if a set of periodic patch currents Mqij could be found suchthat the boundary condition [Eq. (3)] was strictly satisfied,then Es would be the exact field scattered by the grating. Toobtain an approximate solution, the boundary condition isimposed at M selected points on the boundary S. Thisreduces the functional relation [Eq. (3)] to the matrix equa-tion

[Z]K = V, (5)

where

= [Z1 1 (6)ZI 1 22 6

is a 2N by 2M generalized impedance matrix,

[] (7)

is a 2N-element generalized unknown-current column vec-tor, and

V 2] (8)

is a 2M-element generalized voltage-source column vector.In Eq. (6), the matrices [Zpq] (p, q = 1, 2) denote M by Nmatrices whose (m, n) element is the tpm component of theelectric field at observation point r on S due to a patchcurrent Mqn of unit amplitude (Kqn = 1). Here, pm (p = 1, 2)are orthogonal unit vectors tangential to S at observationpoint rm on S. In Eq. (7), the vectors Kq (q = 1,2) denote N-element column vectors whose nth element is Kqn. Finally,in Eq. (8), the vectors V (p = 1, 2) denote M-element

column vectors whose mth element is the negative of the tpmcomponent of Einc at observation point r on S. Havingformulated the matrix equation [Eq. (5)], the unknown cur-rent vector can be found in a simple manner. If the bound-ary condition is imposed at M = N points on S, then theexact solution to Eq. (5) will be

K = [Zl-1 V. (9)

If, on the other hand, the boundary condition is forced at M> N points on S, then the solution, in a least-square sense,will be

K = [ZIt[Z1I-[Z]t V. (10)

This completes the solution of the matrix equation [Eq. (5)].Once the unknown current vector is derived, either from Eq.(9) or (10), one can readily proceed in evaluating an approxi-mate scattered field (Es, Hs) and, of course, any other field-related quantity of interest.

C. Fields of Doubly Periodic Magnetic Patch-CurrentSourcesIn the simulated equivalence for the region surrounding thescatterers, the periodic scattered field is simulated by thefield of a set of 2N spatially periodic and properly modulatedfictitious patch-current sources placed outside that region.These patches lie in planes parallel with the xy plane. Theyare of dimensions s by s2 in the directions of the reciprocallattice vectors K = 2 X d2/Id1 X d21 and K2 = 2 X d/Id Xd21, respectively. It is assumed that si and S2 are sufficientlysmall compared with the dimensions of the bodies so thatthe patches can be completely enclosed inside the bodies.The current density of the ith periodic patch current Mqj (q= 1, 2, i = 1, 2,..., N) centered at a point r inside S isdescribed by

2

Mqj = tqjKqj(z - z)exp[j k'c (r-r)] J fPSP)p=1 n=-'

(11)

with {ipn = (r - ri- ndp) Kp/Kp and has a constant complexamplitude Kqi that is yet to be determined. Here, denotesthe Dirac delta function, z is the z component of r, and azqi

(q = 1, 2) are two unit vectors defining the directions of thesources centered at ri. The function f(-) in Eq. (11) is a real-valued window function of unit width characterized by acontinuous profile that is zero for all values of argumentoutside the interval (/2, /2) and of piecewise continuousderivative on that interval. Under these conditions f/sp)as a function of can be represented by a Fourier serieswhose convergence to fS/sp) on the period interval (-rl/Kp,

JrIKp) is absolute and uniform. It should be noted that theabove continuity requirements on f(-) are sufficient in orderto ensure uniform convergence of the Fourier series. How-ever, a smoother function f(-) should be preferred since itsFourier series converges faster. A specific choice forf(.) thathas been used in our numerical solution is

f() = 0.35875 + 0.48829 cos(27rt)+ 0.14128 cos(47rt) + 0.01168 cos(67rt), (12)

which is known in signal processing as the Blackman-Harriswindow.1 2 This window function and its Fourier transformare shown, respectively, in Figs. 3(a) and 3(b). As seen in

Boag et al.

Vol. 7, No. 9/September 1990/J. Opt. Soc. Am. A 1715

(16)kTmn = kT' + Ml 1 + nK2

and

kZmn = (k2 - kTmn kTmn)' 1 2, (17)

which are subject to the requirements Re(kzmn) 2 0 andIm(kzmn) • 0 for all m and n, which stem directly from theradiation condition at IzI Cow. Here, k is the intrinsic wavenumber in the surrounding medium, and k iTc and kTmn de-note the transversal to z components of the wave vectors ofthe incident field and of the mnth Floquet mode, respective-ly. Also, z is the unit vector in the z direction, and kZmn and-kZmn are, respectively, the z components of the wave vec-

0.50 tors of the z and -z traveling mnth Floquet modes. Thus, inEq. (14), k or k are used depending on whether z > zi or z

< zi, respectively. The coefficients

I C /2ap1 = fps I Vsp)exp(1bcp)d, p = 1, 2,1 e ZZ

J -s,,/2(18)

in Eq. (14) are the Fourier-series coefficients of the currentdensity profile in the two reciprocal lattice directions. Itshould be noted that the convergence rate of these Fourierseries will be affected by the choice of the patch dimensionss, and S2. Clearly, if sp is wider, the spectral spread of f sp)is narrower. As a direct consequence, the Fourier series willconverge faster. However, the ability of the sources to simu-late rapid field variations will obviously be less effective.

D. Field Representation in Terms of Floquet Modes0 50 100 150 An alternative representation for the scattered field Es,

which is valid for observation points r in the z < -b half-k space, results by substituting Eq. (14) into Eq. (13) and Eq.

(b) (13) into Eq. (4), by using the inequality

Fig. 3. (a) Plot of the window function f(t) given by Eq. (4).(b) Fourier transform of the window function f(Q) depicted in (a).

Fig. 3(b), the Fourier transform of the Blackman-Harriswindow is characterized by a very low sidelobe level (<-92dB).

The periodic patch currents are treated as source currentsradiating in an unbounded space filled with homogeneousmaterial that is identical to that composing the region sur-rounding the bodies. The electric field Eqi at an observationpoint r due to the periodic patch-current source Mqi of unitamplitude centered at ri can be derived from

Eqi(r) = -V X Fqi(r), (13)

where Fqi is the electric vector potential represented as aseries of Floquet modes:

Fqj(r) = aqi E > 2 A exp[-jk' (r - ri)]. (14)m=- n=-O 1 m

In Eq. (14), kin is the wave vector of the mnth Floquet modedefined by

(15)kMn = kTmn + kZmn,

with

Z < Zi i, (19)

and subsequently by interchanging the order of the summa-tions involved. One obtains

E- = Z Z Em .exp(-jkn r),m=-X n=--

where the space-harmonic expansion coefficients are

alma2n 2 XE = 2k- Kqjk-n X tqj exp (jk- n - ri) .

Zmn q1 i

(20)

(21)

Equation (20) with E n given by Eq. (21) constitutes a fieldrepresentation for the scattered field Es in the z < -b half-space in terms of an infinite discrete set of Floquet modes.We refer to this field as the reflected field.

Similarly, in the half-space below the lowest point of thescatterers an alternative representation in terms of Floquetmodes for the total field E, valid for observation points r inthe z > b half-space, is obtained by using the inequality

Z > Zi V i. (22)

The result is

(23)E+ = E I Emexp(-jk * r),m=-= n=-X

f(X)1.0

0.8

0.6

0.4

0.2

0.0-0.50 -0.25 0.00 0.25

x(a)

f (k)If (0) [dBI0-

-30

-60

-90

-120

-150-150 -100 -50

Boag et al.

1716 J. Opt. Soc. Am. A/Vol. 7, No. 9/September 1990

where

V = 5mObnOEic

alma2n 2 N

2kZ n n Kq=k1 =1 exp(jkm - ri). (24)

We refer to this field as the transmitted field. Note that thetransmitted mode of the zeroth order is a superposition ofscattered and incident fields. The field representations[Eqs. (20) and (23)] are also referred to as the space-harmon-ic representation for the reflected and transmitted fields.The coefficients E n given by Eq. (21) and E +n given by Eq.(24) are referred to as the space-harmonic expansion coeffi-cients for the reflected and transmitted fields, respectively.They are the parameters of interest in problems involvingscattering from periodic structures.

4. NUMERICAL RESULTS

As a numerical study case, we consider the problem of dif-fraction from a doubly periodic orthogonal array comprisingperfectly conducting spheres. Some representative compu-tations obtained with a computer program developed byusing the preceding formulation are given and comparedwith an analytic approximation. Unfortunately, we werenot able to find appropriate data in the literature for com-parison with our results.

As a first step, we define the accuracy criteria that will aidus in evaluating the numerical results. In our solution weforce the boundary condition to be obeyed only at a finitenumber of selected points on the boundary. Naturally, onecan question the behavior of the fields on the boundarybetween the matching points because it can, in general, bequite different from what is required by the boundary condi-tion, thereby rendering the results inaccurate. Toward thisend, we carry out a study of the convergence of the boundarycondition error AEb, defined by

1bc= x (Es + Einc)IAEb = h X sI Ei nc l on S.Wnlnax

(25)

This boundary-condition check certainly does not totallyvalidate the solution, but it can immediately indicate faultyresults.

An additional criterion that must be satisfied in the loss-less case is the conservation of the power flow along the zaxis. Hence, it is important to check if the sum of thereflected and transmitted powers equals the incident power.To that effect, we examine the power conservation error AP,defined by

pinc - Pm n P E Z |

=p mn mn (26)pine

p _ kzmn 12mn k 1E (28)

are the power flows per unit area of the mnth Floquet modesin the -z direction for reflected modes and in the +z direc-tion for transmitted modes. Here, q is the intrinsic imped-ance of the medium. The power-conservation check by nomeans totally validates the solution. However, we surmisethat a solution of a relatively large number of unknowns,which satisfies not only the boundary conditions but also thepower conservation law within some acceptably low error, isnot likely to be defective.

We consider a plane wave traveling in a lossless mediumincident upon a grating of scatterers at some angle (inc, 4,inc).The incident field is given by Eq. (2) with

kinc = k(sin Zfinc cos Zinc k + sin Zinc sin inc 5 + cos ffic 2)

(29)

In the following example, we consider a grating composedof spherical scatterers of radius r = 0.2X and periods d1 = d2= 0.8X, A being the wavelength of the incident wave. As

0

-20

CS

0

Cs

0.1 0.3 0.5

r r

0.7

Fig. 4. Plots of max(AEb,) and APversus r/r for the case of a planewave normally incident (inc = inc = 0) upon an orthogonal arraywith periods d = d2 = 0.8X composed of perfectly conductingspheres of radius r = 0.2X obtained with N = 44 patch-currentsources per sphere.

-20

-40-

..

xr

-60

where the primed sums indicate that the summations aretaken over only the propagating modes. In Eq. (26),

Pinc =pin = z` 1E incJ2

kvj (27)

-8010

'AP

max(^EC)

30 50 70

is the power flow per unit area in the z direction of theincident field and

IT

Fig. 5. Plots of max(AEb,) and AP versus N for the case of Fig. 4with the sources situated on a sphere of radius r = 0.2r.

.... 11.111-1.1.1

Boag et al.

,

Boag et al. ~~~~~~~~~~~~~~Vol. 7, No. 9/September 1990/J. Opt. Soc. Am. A 1717

Table 1. Efficiencies P j~ /pinc, Power-Conservation Error, and Boundary-Condition Error for the Case of aTM-Polarized Plane Wave of Wavelength Xa

0 inc 00 150 300 450 600 750

p: 1 0 /pinc ()- 5.47 5.23 5.35 4.67 4.55p~o/pdnc () 85.6145 43.14 59.80 65.28 71.14 63.47

pt 1 0 /pinc %)- 20.55 20.12 19.82 21.75 23.90

P-/pinc (% 14.3855 30.84 14.85 9.55 2.44 8.08

AP' (% 0.8 X i0-5 0.3 X 10-2 0.9 X 10-3 0.1 X 10-3 0.2 X 10-2 0.4 X 10-2

MaX(AEbN) (% 0.02 0.08 0.08 0.05 0.04 0.03

a Plane wave of wavelength X incident in the xz plane ((pin, = 00 at various angles oin upon a grating with periods di d2 0.8X composed of perfectly conducting

Spheres of radius r = 0.2X obtained with N = 54 patch-current sources situated on spheres of radius rt = 0.2r.

mentioned above, previous studies 7-10 have shown that it is

advantageous to place the sources on surfaces similar in

shape to the actual boundary. With this in mind, thesources Mqi are placed on a concentric sphere of radius r81, r5

<r. We use 2N patches of dimensionss S~ 2 =0. 135X andspace them evenly on the sphere. The boundary conditionsare matched atM =N points. Plots of max(AEb~)and A.?asa function of rs at normal incidence for N = 44 are depictedin Fig. 4. Observe that rs 0.2r seems to be the optimalsource location. Both the power-conservation error and themaximum-boundary-condition error increase as one moves

away from this optimum. This increase is rapid if thesources approach the center (rs -~ 0), while it is relativelygradual as the sources approach the boundary of the body.Plots of max(AEC and AP as a function of Nfor 75 = 0. 2r areshown in Fig. 5. The approximately exponential decay of

the errors with increasing number of sources observed in Fig.5 clearly demonstrates the fast convergence of the procedureat hand. Table 1 presents the efficiencies p~n/pinc in the

various spectral orders for several incident angles. Alsoshown are the power conservation error AP' and the maxi-mum-boundary-condition error max(AEb,).

It should be emphasized that, for any calculated quantityof interest, one should examine the numerical convergenceby comparing the results for an increasing number of sourcesand match points. If the computed results are sufficientlyclose, it can be assumed that a satisfactory accuracy has been

achieved. To validate the solution, one should at the sametime check the behavior of the error in the boundary condi-tion between the matching points and examine the decreasein the power-conservation error. These checks are easilyexecuted with summations of analytic terms. As explainedearlier, neither of these checks totally validates the result.However, they can undoubtedly serve as useful tools for

testing whether the result is trustworthy. One can also,without appreciable difficulty, consider additional sourcelocations, each providing a check against the others. It maybe added that, even for choices that are less than optimal in

source location, the solution converges. The rate of conver-gence might be sensitive to the location of the sources,though usually not heavily.

Figure 6 shows plots of the 0 component of the surface-induced current J given by

resembles that obtained for the case of scattering from asingle sphere designated as di= d2=

The scattering mechanism from the doubly periodic grat-ing can be formally represented as an infinite series of multi-

ple scattering among the individual scatterers composingthe grating.'3 In this way, the scattering characteristics ofthe infinite grating can be expressed in terms of those of thesingle scatterer. An analysis that neglects the mutual inter-actions among the scatterers is presented for the two-dimen-sional case in the Appendix of Ref. 8. This is a first-orderapproximation that is expected to be accurate only if theperiod is large compared with the wavelength in the sur-rounding medium. It should be pointed out that, if a graz-ing mode is excited, this first-order approximation is oftennot satisfactory. The presence of a grazing mode stronglyenhances the mutual interaction among the scatterers. Itcan be shown, further, by analogy with Ref. 8 that the effi-ciencies of the propagating Floquet modes p:,,/pinc (with theexception of the zeroth-order transmitted mode) are relatedto the respective values of the scattering cross section o-(O, 0)of the single scatterer at (0 In, ,), where 0 In and ck In are themode angles. This relation can be expressed as follows:

(31)pin co Oiclcs O0 I~n(kldl X d21)

Figure 7 shows a plot of ra(O, 0)/[Icos 0I(kdld 2)2] obtained by

using the exact eigenvalue solution for the single sphericalscatterer of radius r = 0.2X as a function of 0 in the xz plane

1J.1/IHncI2.5

2.0

1.5

1.0

0.5

0.0

- d,1 ,=0.8X----d12=1.52.

--- d 1,2 =2.5k

. . . dl,2 =-

J = At X (Hinc + H5 ) on S (30)

versus 0 in the xz plane for various values of periods d, = d2 .

Note that, for di = d2= 2.5X, the surface current closely

0 45 90 135 180

0 (eg)

Fig. 6. Magnitude of induced electric surface current Jo versus 0for gratings of various periods and for the single-scatterer case.

Boag et al.

1718 J. Opt. Soc. Am. A/Vol. 7, No. 9/September 1990

0.15

0.10:

0.05-X x

0.00

Fig. 7. Vaat discrete7ra(O, OVO[cIof normal ii= d2= 2.5X0.2X.

for the norectangulasymmetry,degon .the left-hamodes wit.that the coed discrete

5. DISCI

The technianalysis ofperfectly cod is that tIboundary rivable fielican be conThe generathogonal giing scatter

have been demonstrated. It has also been shown that in thelimiting case of widely spaced spherical scatterers the nu-merical solution agrees well with an approximate analyticsolution.

Y. Leviatan's present address is the Department of Elec-trical Engineering and Computer Science, George Washing-ton University, Washington, D.C. 20052.

REFERENCESI 1. J. P. Montgomery, "Scattering by an infinite periodic array of

| ........ thin conductors on a dielectric sheet," IEEE Trans. Antennas0 45 90 135 180 Propag. AP-23, 70-75 (1975).2. C. C. Chen, "Transmission through a conducting screen perfo-

0 (deg) rated periodically with apertures," IEEE Trans. Microwave* * * n c Theory Tech. MTT-18, 627-632 (1970).lues of reflection and transmission efficiencies p/pnc 3. T. Cwik and R. Mittra, "The cascade connection of planarmode angles (asterisks) and the corresponding plot of periodic surfaces and lossy dielectric layers to form an arbitrary

os 01(kdid2 )2] versus 0 for P = 0 (solid curve) for the case periodic screen," IEEE Trans. Antennas Propag. AP-35, 1397-aicidence (inc = inc = 0) upon a grating with periods di 1405 (1987).composed of perfectly conducting spheres of radius r = 4. C. C. Chen, "Transmission of microwave through perforated flat

plates of finite thickness," IEEE Trans. Microwave TheoryTech. MTT-21, 1-6 (1973).

5. G. H. Derrick, R. C. McPhedan, and L. C. Botten, "Theory ofmal incidence of the x-polarized plane wave on a crossed gratings," in Electromagnetic Theory of Gratings, R.r grating defined by d = d2 = 2.5X. Owing to Petit, ed. (Springer-Verlag, New York, 1980), Chap. 7.it is sufficient to show the interval from 0 to 180 6. G. H. Derrick, R. C. McPhedan, D. Maystre, and M. Nevibre,

Also shown are computed efficiencies appearing in "Crossed gratings: a theory and its applications," Appl. Phys.~d s o s o w n a e c o m u t e d ffi c i e c i e s p p e a r n g i n1 8 , 3 9 -5 2 (1 9 7 9 ).nd side of relation (31) for various propagating 7. Y. Leviatan, A. Boag, and A. Boag, "Generalized formulationsIh the wave vectors lying in the xz plane. Note for electromagnetic scattering from perfectly conducting andntinuous curve nearly coincides with the calculat- homogeneous material bodies-theory and numerical solu-values, tion," IEEE Trans. Antennas Propag. 36, 1722-1734 (1988).8. A. Boag, Y. Leviatan, and A. Boag, "Analysis of two-dimension-

al electromagnetic scattering from periodic grating of cylindersusing a hybrid current model," Radio Sci. 23, 612-624 (1988).

USSION 9. A. Boag, Y. Leviatan, and A. Boag, "Analysis of two-dimension-al electromagnetic scattering from non-planar periodic surfaces,que presented in this paper provides a complete using a strip current model," IEEE Trans. Antennas Propag. 37,

diffraction by doubly periodic arrays of smooth 1437-1446 (1989).onducting bodies. The main feature of the meth- 10. A. Boag, Y. Leviatan, and A. Boag, "Analysis of diffraction fromhenducti bodes. loted mai fetro the meth-l echelette gratings, using a strip-current model," J. Opt. Soc.lie patch currents located away from the physical Am. A 6, 543-549 (1989).produce a set of smooth periodic analytically de- 11. A. Hessel, "General characteristics of traveling-wave antennas,"ds on that boundary. Thus boundary conditions in Antenna Theory, R. E. Collin and F. J. Zucker, eds.veniently imposed in the point-matching sense. (McGraw-Hill, New York, 1969), Chap. 19.d method has been applied in this paper to or- 12. F. J. Harris, "On the use of windows for harmonic analysis withthe discrete Fourier transform," Proc. IEEE 66, 51-83 (1978).atings composed of spherical perfectly conduct- 13. V. Twersky, "Multiple scattering by double-periodic planar ar-ers. The numerical convergence and accuracy ray of obstacles," J. Math. Phys. 16, 633-643 (1975).

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