Anisotropic representation for spatial dispersion in periodic metamaterial arrays

A rigorous anisotropic representation for 3D spatially dispersive periodic arrays is developed, beginning with the microscopic Maxwell equations, that yields causal, generalized, macroscopic permittivities and inverse permeabilities for the fundamental Floquet modes of the arrays. A significant feature of this formulation is that the generalized permittivities and permeabilities reduce to their conventional anisotropic-continuum definitions in terms of macroscopic averages at the low spatial and temporal frequencies. In addition, diamagnetic metamaterial arrays require no special considerations or modifications to accommodate their unusual characteristics. Analytic and numerical examples of dielectric-sphere and circular-cylinder arrays are given that confirm the theoretical results for the macroscopic permittivities and permeabilities of these arrays that exhibit electric and magnetic/diamagnetic macroscopic polarizations.