Traveling Electromagnetic Waves on Linear Periodic Arrays of Lossless Penetrable Spheres

Abstract

Traveling electromagnetic waves on infinite linear periodic arrays of lossless penetrable spheres can be conveniently analyzed using the source scattering-matrix framework and vector spherical wave functions. It is assumed that either the spheres are sufficiently small, or the frequency such, that the sphere scattering can be treated using only electric and magnetic dipole vector spherical waves, the electric and magnetic dipoles being orthogonal to each other and to the array axis. The analysis simplifies because there is no cross-coupling of the modes in the scattering matrix equations. However, the electric and magnetic dipoles in the array are coupled through the fields scattered by the spheres. The assumption that a dipolar traveling wave along the array axis can be supported by the array of spheres yields a pair of equations for determining the traveling wave propagation constant as a function of the sphere size, inter-sphere separation distance, the sphere permittivity and permeability, and the free-space wave number. These equations are obtained by equating the electric (magnetic) field incident on any sphere of the array with the sum of the electric (magnetic) fields scattered from all the other spheres in the array. Both equations include a parameter equal to the ratio of the unknown normalized coefficients of the electric and magnetic dipole fields. By eliminating this parameter between the two equations, a single transcendental equation is obtained that can be easily solved numerically for the traveling wave propagation constant. Plots of the k - β diagram for different types and sizes of spheres are shown. Interestingly, for certain spheres and separations it is possible to have multiple traveling waves supported by the array. Backward traveling waves are also shown to exist in narrow frequency bands for arrays of spheres with suitable permittivity and permeability.