Sums of spherical waves for lattices, layers, and lines

We consider the connections between sums of spherical wave functions over l attices, layers, and lines. The differences between sums over lattices and those over a doubly periodic constituent layer are expressed in terms of se ries 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 wh ich can be used to calculate accurately and efficiently sums of spherical w aves over lattices, layers, and lines, which in turn may be used to constru ct quasiperiodic Green's functions for the Helmholtz equation, of use in sc attering problems for layers and lines of spheres, and for finding the Bloc h modes of lattices of spheres. We illustrate the numerical accuracy of our expressions. (C) 2001 American Institute of Physics.