Determination of k-β diagrams for arrays of penetrable spheres

In Shore et al. (2004) we have employed the source scattering-matrix formulation with vector spherical wave functions to investigate traveling electromagnetic waves on infinite linear periodic arrays of lossless electric dipoles (which can be spheres that scatter as electric dipoles) and lossless magnetodielectric spheres (which can scatter as both electric and magnetic dipoles), respectively. It is assumed that either the spheres are sufficiently small that only the dipole scattered fields can be excited, or the frequency is such that all the scattered multipole fields are negligible except the dipole fields. Accordingly, the sphere scattering is treated using only electric and magnetic dipole vector spherical waves, the dipoles being orthogonal to each other and to the array axis. Although our investigation was motivated in part by the recent theoretical demonstration by Holloway et al. (2003) that a doubly negative medium can be formed by embedding an array of magnetodielectric spheres in a background matrix, the results are equally applicable to arrays of lossless "metallic" nanospheres with negative relative permittivity and relative permeability equal to one. Accordingly, the purpose of this paper is to summarize the analyses given in detail in Shore et al. (2004), and to use them to determine the k-β diagrams for the traveling waves on linear arrays of magnetodielectric spheres and metallic nanospheres.