Complex waves on three-dimensional periodic arrays of lossy or lossless magnetodielectric spheres

The dispersion equation is derived and solved for the complex propagation constants of traveling waves (eigenmodes) supported by three-dimensional, infinite, periodic arrays of identical, electrically small, lossy or lossless, magnetodielectric spheres. Expressions are obtained for the effective permittivity and permeability of the arrays in frequency regions where the arrays can be regarded as continuous, homogeneous, isotropic media.