Rapidly convergent quasi-periodic Green functions for scattering by arrays of cylinders—including Wood anomalies

Abstract

This paper presents a full-spectrum Green-function methodology (which is valid, in particular, at and around Wood-anomaly frequencies) for evaluation of scattering by periodic arrays of cylinders of arbitrary cross section-with application to wire gratings, particle arrays and reflectarrays and, indeed, general arrays of conducting or dielectric bounded obstacles under both transverse electric and transverse magnetic polarized illumination. The proposed method, which, for definiteness, is demonstrated here for arrays of perfectly conducting particles under transverse electric polarization, is based on the use of the shifted Green-function method introduced in a recent contribution (Bruno & Delourme 2014 , 262-290 (doi:10.1016/j.jcp.2013.12.047)). A certain infinite term arises at Wood anomalies for the cylinder-array problems considered here that is not present in the previous rough-surface case. As shown in this paper, these infinite terms can be treated via an application of ideas related to the Woodbury-Sherman-Morrison formulae. The resulting approach, which is applicable to general arrays of obstacles even at and around Wood-anomaly frequencies, exhibits fast convergence and high accuracies. For example, a few hundreds of milliseconds suffice for the proposed approach to evaluate solutions throughout the resonance region (wavelengths comparable to the period and cylinder sizes) with full single-precision accuracy.

On the evaluation of quasi-periodic Green functions and wave-scattering at and around Rayleigh-Wood anomalies

“…Wave-scattering by periodic media, including RW anomalous configurations, at which the quasiperiodic Green function ceases to exist, has continued to attract significant attention in the fields of optics [17,22,33,34,35,36,39,45,50] and computational electromagnetism [3,8,4,9,10,31,14,26,42,39,18]. Classical boundary integral equations methods [43,49,52] have relied on the quasi-periodic Green function (denoted throughout this work as G q κ ), which is defined in terms of a slowly converging infinite series (equation (27)).…”

“…Since it introduces new spatial singularities, this technique was first applied to problems where the domain boundary coincides with the graph of a periodic function. The contribution [4] introduced a slightly different use of the shifted Green function from its original inception which allows for application to more general domains.…”

Stable, high-order computation of impedance–impedance operators for three-dimensional layered medium simulations

“…Paramount among these interfacial methods are those based upon integral equations (IEs) [17,18], but these face difficulties. Most have been addressed in recent years through (i) the use of sophisticated quadrature rules to deliver HOS accuracy; (ii) the design of preconditioned iterative solvers with suitable acceleration [19]; and (iii) new strategies to avoid periodizing the Green function [20][21][22][23][24][25][26][27]. Consequently, they are a compelling alternative (see, for example, the survey article of [18] for more details); however, two properties render them noncompetitive for the parametrized problems we consider compared with the methods we advocate here.…”

An FFT-based Algorithm for Efficient Computation of Green's Functions for the Helmholtz and Maxwell's Equations in Periodic Domains

“…There are also some other methods considering the numerical evaluation of the Green's functions; see, e.g., [2,22,3,4]. In particular, the cases that are near or at the Wood anomalies, which are very difficult to dealt with, have been considered in [22,3,4]. For very large k's (> 10 5 ), Kurkcu and Reitich introduced the NA method in 2D domains, which comes from the integral representations (see [10]).…”