Propagation of electromagnetic waves in periodic lattices of spheres: Green’s function and lattice sums

Abstract

We have recently exhibited expressions for Green’s functions for dynamic scattering problems for gratings and two-dimensional arrays, expressed in terms of lattice sums. We have also discussed efficient techniques to evaluate these sums and how their use in Green’s function forms leads naturally to Rayleigh identities for scattering problems. These Rayleigh identities express connections between regular parts of wave solutions near a particular scatterer and irregular parts of the solution summed over all other scatterers in a system. Here we discuss these ideas and techniques in the context of the problem of the scattering of a scalar wave by a regular lattice of spheres. We discuss expressions for lattice sums which can be integrated arbitrarily-many times to accelerate convergence, a computationally efficient Green’s function form, and the appropriate Rayleigh identity for the problem. We also discuss the long-wavelength limit and obtain the Maxwell-Garnett formula for lattices of perfectly conducting spheres.