Lattice Sums Accommodating Multiple Sublattices for Solutions of the Helmholtz Equation in Two and Three Dimensions

TL;DR

This paper derives exponentially convergent series for lattice sums involving multiple sublattices in 2D and 3D Helmholtz problems, improving computational efficiency for electromagnetic scattering in complex lattice structures.

Contribution

The authors present novel Ewald-based series expressions for multi-sublattice lattice sums in 2D and 3D Helmholtz solutions, extending prior methods to more complex lattice configurations.

Findings

Derived exponentially convergent series for lattice sums

Validated results through comparison with direct computations

Enhanced computational efficiency for multi-sublattice systems

Abstract

The evaluation of the interaction between objects arranged on a lattice requires the computation of lattice sums. A scenario frequently encountered are systems governed by the Helmholtz equation in the context of electromagnetic scattering in an array of particles forming a metamaterial, a metasurface, or a photonic crystal. While the convergence of direct lattice sums for such translation coefficients is notoriously slow, the application of Ewald's method converts the direct sums into exponentially convergent series. We present a derivation of such series for the 2D and 3D solutions of the Helmholtz equation, namely spherical and cylindrical solutions. When compared to prior research, our novel expressions are especially aimed at computing the lattice sums for several interacting sublattices in 1D lattices (chains), 2D lattices (gratings), and 3D lattices. We verify our results by comparison with the direct computation of the lattice sums.