A Fast Summation Method for Oscillatory Lattice Sums

TL;DR

This paper introduces a novel fast summation algorithm for oscillatory lattice sums in wave scattering problems, providing rigorous analysis of Wood's anomalies with super-algebraic convergence and minimal computational time.

Contribution

A new summation method using Euler-Maclaurin and steepest descent techniques, offering improved efficiency and analysis of singularities in periodic wave scattering.

Findings

Super-algebraic convergence of the algorithm

Requires only milliseconds of CPU time

Provides rigorous analysis of Wood's anomalies

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.