Sums of spherical waves for lattices, layers, and lines

Abstract

We consider the connections between sums of spherical wave functions over lattices, layers, and lines. The differences between sums over lattices and those over a doubly periodic constituent layer are expressed in terms of series with exponential convergence. Correspondingly, sums over the layer can be regarded as composed of a sum over a central line, and another sum over displaced lines exhibiting exponential convergence. We exhibit formulas which can be used to calculate accurately and efficiently sums of spherical waves over lattices, layers, and lines, which in turn may be used to construct quasiperiodic Green's functions for the Helmholtz equation, of use in scattering problems for layers and lines of spheres, and for finding the Bloch modes of lattices of spheres. We illustrate the numerical accuracy of our expressions. Show more