Two-dimensional, phase modulated lattice sums with application to the Helmholtz Green’s function

Abstract

A class of two-dimensional phase modulated lattice sums in which the denominator is an indefinite quadratic polynomial Q is expressed in terms of a single, exponentially convergent series of elementary functions. This expression provides an extremely efficient method for the computation of the quasi-periodic Green’s function for the Helmholtz equation that arises in a number of physical contexts when studying wave propagation through a doubly periodic medium. For a class of sums in which Q is positive definite, our new result can be used to generate representations in terms of θ-functions which are significant generalisations of known results.

An FFT-based Algorithm for Efficient Computation of Green's Functions for the Helmholtz and Maxwell's Equations in Periodic Domains

“…In these two papers, Linton gave the conclusions that, if only a few values are needed, the Ewald's method is the most efficient method, while if a large number of values are needed, the lattice sums method is better. There are many papers concerning the lattice sums method and the Ewald's method; see [9,30,23,5,20,28,1,11,21,26] for the 2D case and [18,6,11,21,26,20,28,20,15,25,17] for the 3D case. There are also some other methods considering the numerical evaluation of the Green's functions; see, e.g., [2,22,3,4].…”