A Fast Summation Method for Oscillatory Lattice Sums

Published

March 6, 2017

Author(s)

Ryan Denlinger, Leslie Greengard, Zydrunas Gimbutas, Vladimir Rokhlin

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.

Citation

Journal of Mathematical Physics

Volume

58

Issue

023511

Pub Type

Journals

Download Paper

https://doi.org/10.1063/1.4976499

Local Download

Keywords

Lattice sums, Euler-Maclaurin formula, Wood’s anomalies

Citation

Denlinger, R. , Greengard, L. , Gimbutas, Z. and Rokhlin, V. (2017), A Fast Summation Method for Oscillatory Lattice Sums, Journal of Mathematical Physics, [online], https://doi.org/10.1063/1.4976499, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=921601 (Accessed October 9, 2025)