Robust Fast Direct Integral Equation Solver for Quasi-Periodic Scattering Problems with a Large Number of Layers

We present a new boundary integral formulation for time-harmonic wave diffraction from two-dimensional structures with many layers of arbitrary periodic shape, such as multilayer dielectric gratings in TM polarization. Our scheme is robust at all scattering parameters, unlike the conventional quasi-periodic Green’s function method which fails whenever any of the layers approaches a Wood anomaly. We achieve this by a decomposition into near- and far-field contributions. The former uses the free-space Green’s function in a second-kind integral equation on one period of the material interfaces and their immediate left and right neighbors; the latter uses proxy point sources and small least-squares solves (Schur complements) to represent the remaining contribution from distant copies. By using high-order discretization on interfaces (including those with corners), the number of unknowns per layer is kept small. We achieve overall linear complexity in the number of layers, by direct solution of the resulting block tridiagonal system. For device characterization we present an efficient method to sweep over multiple incident angles, and show a 25× speedup over solving each angle independently. We solve the scattering from a 1000-layer structure with 3 × 105 unknowns to 9-digit accuracy in 2.5 minutes on a desktop workstation

Superalgebraically Convergent Smoothly-Windowed Lattice Sums for Doubly Periodic Green Functions in Three-Dimensional Space

This paper, Part I in a two-part series, presents (i) A simple and highly efficient algorithm for evaluation of quasi-periodic Green functions, as well as (ii) An associated boundary-integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three-dimensional space. Except for certain "Wood frequencies" at which the quasi-periodic Green function ceases to exist, the proposed approach, which is based on use of smooth windowing functions, gives rise to lattice sums which converge superalgebraically fast--that is, faster than any power of the number of terms used--in sharp contrast with the extremely slow convergence exhibited by the corresponding sums in absence of smooth windowing. (The Wood-frequency problem is treated in Part II.) A proof presented in this paper establishes rigorously the superalgebraic convergence of the windowed lattice sums. A variety of numerical results demonstrate the practical efficiency of the proposed approach

Three-dimensional quasi-periodic shifted Green function throughout the spectrum--including Wood anomalies

This work presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near "Wood anomaly frequencies". At these frequencies, one or more grazing Rayleigh waves exist, and the lattice sum for the quasi-periodic Green function ceases to exist. We present a modification of this sum by adding two types of terms to it. The first type adds linear combinations of "shifted" Green functions, ensuring that the spatial singularities introduced by these terms are located below the grating and therefore outside of the physical domain. With suitable coefficient choices these terms annihilate the growing contributions in the original lattice sum and yield algebraic convergence. Convergence of arbitrarily high order can be obtained by including sufficiently many shifts. The second type of added terms are quasi-periodic plane wave solutions of the Helmholtz equation which reinstate certain necessary grazing modes without leading to blow-up at Wood anomalies. Using the new quasi-periodic Green function, we establish, for the first time, that the Dirichlet problem of scattering by a smooth doubly periodic scattering surface at a Wood frequency is uniquely solvable. We also present an efficient high-order numerical method based on the this new Green function for the problem of scattering by doubly periodic three-dimensional surfaces at and around Wood frequencies. We believe this is the first solver in existence that is applicable to Wood-frequency doubly periodic scattering problems. We demonstrate the proposed approach by means of applications to problems of acoustic scattering by doubly periodic gratings at various frequencies, including frequencies away from, at, and near Wood anomalies