Efficient computation of the 2D Green's function for 1D periodic layered structures using the Ewald method

We present an alternative direct procedure for applying the Ewald approach to obtain the Green's function for an array of line sources with 1D periodicity. The approach is the 2D analog of that of Wilton, Jackson and Champagne (see URSI Int. Sym. on Electrom. Theory, Thessaloniki, Greece, May 1998). Furthermore, we derive an algorithm for choosing the Ewald splitting parameter E that extends the efficiency of the method when the periodicity is somewhat larger than a wavelength. The case of periodic 2D multilayered media is also treated analogously done by Wilton et al. for the 3D case. In particular, the dyadic Green's function formalism of Michalski (1990), which yields mixed potential integral equations for layered media, is combined with the Ewald method.


I. Introduction
In applying numerical full wave methods to periodic structures, fast and accurate means to evaluate the periodic Green's function are often needed. The free space Green's function for three dimensional (3D) problems with 2D periodicity has been efficiently accelerated via the Ewald method in [l], and has been extended for the evaluation of the Green's function in multilayered media in [2]. In (31 the extension of the Ewald method to 2D problems with 1D periodicity was given for the case of coplanar source and observation points. An extension of the approach to the non coplanar case was also briefly summarized in [3], in which a formula for non coplanar source and observation was obtained by integrating in closed form the results in [l]. Here, we present an alternative direct procedure for applying the Ewald approach to obtain the Green's function for an array of line sources with 1D periodicity. 'The approach is the 2D analog of that of 111. Furthermore, we derive an algorithm for choosing the Ewald splitting parameter E that extends the efficiency of the method when the Periodicity is somewhat larger than a wavelength. The case of periodic 2D niultilayered media is also treated analogously as in [2] for the 3D case. In particular, the dyadic Green's fnnction formalism of [4], which yields mixed potential integral equations for layered media, is combined with the Ewald method. An analogous formulation, to be reported in the future, has also been successfully applied to accelerate the Green's function for a periodic linear array of point sources (1D periodicity).

Green's Function Transformation
Consider the element-by-element superposition of the fields radiated by an infinite array of progressively phased line sources: where kSo = ksinBo is the phase gradient along the z direction with equivalent scan angle 00 and R, = [ (~-r ' ) ' + ( z -z ' -n d )~]~/~ is the distance between the observation point r = (z, 2 ) and the nth source point r; = (z'+nd, 2 ' ) (see Fig.1). For simplicity the homogeneous medium is supposed to have small losses; hence k = k, + j k t = IkleJmr, ki < 0, & < 0. By integrating in y' the fundamental identity for the Ewald transformation for an array of poiut sources One notes that the series GI does not decay expoiientially, while the series G2 has Gaussian convergeiice because Re(s) > E on tlie path of integration.
Transformation of C2(r,r'). Efficieiit evaluation of Gz is based on the evaliiatio~~ of the

Field Representation for Multi-Layered Periodic Media
For siniplicity, we deal here only with electric currents and their radiated magnetic vector potential dyads and scalar potentials. We choose the representation of "Formulation C" of [4]. There, the scattered electric field E,(r) produced by a periodic current J(r') is represented as  pp.194,195], all the voltages and currents for all combinations of polarizations and unit source types may he expressed in terms of these two characteristic scalar Green's functions and 1;heir derivatives. Extraction of the Asymptotic Form of Series. In (9), for numerical convenience terms L?fm(z,z'), asymptotic for large q to the terms of(z,z') of the spectral dyadic Green's function, are subtracted term by term from the original series and their sum is added back as a separate series. Removing the asymptotic form of each term from the spectral sum 196 1 Fig. 3. (a) Array of line sources in a multilayer dielectric environment. The direct array of line sources and the "first" two asympt,otic (for large 9 ) array-images are used in the asymptotic acceleration, making the first 9 slim in ($1) rapidly convergent (b) Test case: two strip gratings in a dielectric layer. (c) Current on top and bottom strip grat,ings, solved by the method of moments using the periodic Green's function of Sec.11. accelerates its convergence. The reinaiiiiiig sum in (9) may then be accelerated via the Ewald nietbod. Eacli of these terms, in turn, is an asymptotic form of the various transmission line voltages and curreiits, aiid consists of up to three terms of the form rpamexp [-j(k,,z+k,,lz-I' -zpl)]/kz,,; e = 1 , 2 , 3 . For the first term, rYm = 1 and zg = 0, and the term is readily recognized as t,he direct contribution from the original array of line sources radiating in a homogeneous inedinni. For the remainirig terms, the constant I'rm represents the asymptotic layer boundary reflectioii coefficient for large q and with a = e or h. These terms represent reflections appearing to arise from (quasi-)images located at z'+zg, in layers above or below the origiiial source layer as shown in Fig.3a. All three terms result in series having the spectral form of liomogeneous medium periodic Green's functions; the Ewald method may thus he used to sum them. The potentials K* and P, may be treated the same way. Numerical Example: Convergence. The test problem shown in Figure 3b consists of two periodic strip gratings (period d = 1.3m) on top (2 = 0) and in the middle ( z = -1m) of a dielectric layer (relative dielectric permeability t = 2.3), which resides in the region -2m< z < 0. Each strip is Im wide. The periodic structure is illuminated from above hy a plane wave with unitary inagiletic field H directed along y, and frequency equal to 100MHz.
Tlie current 011 both top and bottom strips, shown in Fig.3c, is evaluated by the method of moments defined on a single array cell, using the periodic Green's function developed in tlie previous sections. The solution is compared with a reference solution constructed with the corresponding three diniensional (3D) problem of an infinite 2D array of metallic patches with dinieiisioiis 1 x 1 in, and periodicities d, = 1.3m and d, = lm. In the 2D problem the evaluatioii of tlie Green's function requires only a few terms to achieve accuracy to four significant digits: tlie first spectral sum in (9) is over indices -5 < q < +5, while the second spectral sum iii (9) transformed according to the Ewald method, as explained in the previous sections, is over iudices from -1 to + l . The bottom strip current is larger than the top one since the bottom strip is ,506 wavelengths long.