This article presents full-spectrum, well-conditioned, Green-function methodologies for evaluation of scattering by general periodic structures, which remains applicable on a set of challenging singular configurations, usually called Rayleigh-Wood (RW) anomalies (at which the quasi-periodic Green function ceases to exist), where most existing quasi-periodic solvers break down. After reviewing a variety of existing fast-converging numerical procedures commonly used to compute the classical quasi-periodic Green-function, the present work explores the difficulties they present around RW-anomalies and introduces the concept of hybrid “spatial/spectral” representations. Such expressions allow both the modification of existing methods to obtain convergence at RW-anomalies as well as the application of a slight generalization of the Woodbury-Sherman-Morrison formulae together with a limiting procedure to bypass the singularities. (Although, for definiteness, the overall approach is applied to the scalar (acoustic) wave-scattering problem in the frequency domain, the approach can be extended in a straightforward manner to the harmonic Maxwell's and elasticity equations.) Ultimately, this thorough study of RW-anomalies yields fast and highly-accurate solvers, which are demonstrated with a variety of simulations of wave-scattering phenomena by arrays of particles, crossed impenetrable and penetrable diffraction gratings and other related structures. In particular, the methods developed in this article can be used to “upgrade” classical approaches, resulting in algorithms that are applicable throughout the spectrum, and it provides new methods for cases where previous approaches are either costly or fail altogether. In particular, it is suggested that the proposed shifted Green function approach may provide the only viable alternative for treatment of three-dimensional high-frequency configurations with either one or two directions of periodicity. A variety of computational examples are presented which demonstrate the flexibility of the overall approach.

本文介绍了用于评估一般周期结构的散射的全谱，条件良好的格林函数方法，该方法仍适用于一组具有挑战性的奇异配置，通常称为瑞利伍德（RW）异常（准奇异结构）。定期格林函数不再存在），大多数现有的准周期求解器会崩溃。在回顾了通常用于计算经典准周期格林函数的各种现有快速收敛数值程序之后，本工作探讨了它们围绕RW异常出现的困难，并介绍了“空间/光谱”混合表示的概念。这样的表达式既允许修改现有方法以在RW异常处获得收敛，也可以对Woodbury-Sherman-Morrison公式进行略微推广，并提供绕过奇异性的限制程序。（尽管为了明确起见，将整体方法应用于频域中的标量（声）波散射问题，但可以直接将其扩展到谐波麦克斯韦方程和弹性方程。）最终，对这一过程的深入研究RW异常产生快速且高精度的求解器，并通过各种粒子阵列，交叉不可穿透和可穿透的衍射光栅以及其他相关结构对波散射现象进行了多种模拟，从而证明了这一点。尤其是，本文开发的方法可用于“升级”经典方法，从而产生适用于整个频谱的算法，并且它为以前的方法成本高昂或完全失败的情况提供了新方法。特别地，建议所提出的移位格林函数方法可以为治疗具有一个或两个周期性方向的三维高频配置提供唯一可行的选择。给出了各种计算示例，这些示例演示了整体方法的灵活性。建议提出的移位格林函数方法可能为治疗具有一个或两个周期性方向的三维高频配置提供唯一可行的选择。给出了各种计算示例，这些示例演示了整体方法的灵活性。建议提出的移位格林函数方法可能为治疗具有一个或两个周期性方向的三维高频配置提供唯一可行的选择。给出了各种计算示例，这些示例演示了整体方法的灵活性。