A fast summation method for oscillatory lattice sums

Abstract

We present a fast summation method for lattice sums of the type which arise when solving wave scattering problems with periodic boundary conditions. While there are a variety of effective algorithms in the literature for such calculations, the approach presented here is new and leads to a rigorous analysis of Wood's anomalies. These arise when illuminating a grating at specific combinations of the angle of incidence and the frequency of the wave, for which the lattice sums diverge. They were discovered by Wood in 1902 as singularities in the spectral response. The primary tools in our approach are the Euler-Maclaurin formula and a steepest descent argument. The resulting algorithm has super-algebraic convergence and requires only milliseconds of CPU time.

Periodic Fast Multipole Method

“…Second, there is no essential obstacle to extending the scheme to treat oscillatory problems (such as the Helmholtz or Maxwell equations) in two and three dimensions. The various sums and integrals, however, must be treated with more care, as they are conditionally convergent, permit "quasi-periodic" boundary conditions and are subject to resonances (Wood anomalies) [2,3,12,13,17,34]. Third, the scheme can be coupled with integral equation methods and the fast multipole method to solve periodic boundary value problems when the unit cell contains inclusions of complicated shape.…”

“…Of particular note is [45] which extends a three-dimensional version of the kernel-independent FMM library [33] to permit the imposition of periodicity on the unit cube in one, two or three directions. (See [7,12,13,17,27,30,34,37,39] for further discussion and references, largely in the context of the Poisson, Helmholtz and Maxwell equations.) Unfortunately, lattice sumbased approaches are less efficient when the unit cell has high aspect ratio, as illustrated for a doubly periodic problem in Figure 2.…”

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

“…A number of methodologies have been developed which, for configurations away from RW anomalies, can be used to evaluate the quasi-periodic Green function efficiently and accurately-including lattice sums [18,27,28], Laplace-type integral representation [6,7,23,24,52], the Ewald summation method [2,12,20,27,40] and, recently, the Windowed Green function (WGF) method [3,9]. (In fact the WGF method yields algebraic high-order convergence even at RW anomalies when used in conjunction with the shifted Green function [3,8,9]).…”

“…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)).…”

Spectrally-accurate numerical method for acoustic scattering from doubly-periodic 3D multilayered media

“…However, in this approach, the quasi-periodic Green's function suffers from slow convergence and becomes impractical for 3D problems. There are many methods to remedy the slow convergence such as the Ewald summation method [45,46,47], spatial-spectral splitting [48], or a lattice sum [49,50,51]. Moreover, the quasi-periodic Green's function does not exist at Wood anomalies.…”