A Fast Summation Method for Oscillatory Lattice Sums
This method offers a fast and rigorous solution for computing lattice sums in periodic wave scattering, which is important for applications like grating design and photonics.
The paper presents a new fast summation method for oscillatory lattice sums in wave scattering problems, achieving super-algebraic convergence in milliseconds of CPU time and providing a rigorous analysis of Wood's anomalies.
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.