Traveling Waves on Three-Dimensional Periodic Arrays of Two Different Alternating Magnetodielectric Spheres

An exact
k
-beta (dispersion) equation to within the dipole scattering approximation has been obtained for a 3D array of two different alternating magnetodielectric spheres. The dispersion equation has the form of equating to zero the determinant of a system of four homogeneous equations in the normalized scattered field coefficients. Computationally efficient expressions are obtained for the coefficients of the homogeneous equation system as functions of the sphere radii, permittivities, and permeabilities, the free-space electrical separation distance of the array elements (
kd
), and the electrical separation distance (beta
d
) for the traveling wave supported by the array. For a given value of
kd
and an array of lossless scatterers the determinant equation can be solved for real beta
d
by a simple search procedure. For an array of lossy scatterers beta
d
is complex and a more difficult minimization in the complex plane is required to solve the determinant equation. The solution to the dispersion equation also yields values for the effective permittivity and permeability of the array regarded as a continuous medium. Computations were performed to investigate the performance of two-sphere arrays of lossless dielectric spheres, with the permittivities and radii of the two different dielectric spheres composing the array chosen so that the first magnetic dipole resonant frequency of one set of spheres equals the first electric dipole resonant frequency of the second set of spheres. Although it is shown that arrays composed of two different alternating purely dielectric spheres can behave as isotropic DNG media unlike arrays of identical dielectric spheres, the bandwidths are considerably narrower than those achievable with arrays of identical magnetodielectric spheres with appreciable permittivity and permeability close to each other. The practicality of using arrays of alternating two different purely dielectric spheres to fabricate DNG media depends on whether the narrow bandwidths are acceptable for the desired applications.