Homogenization of spatially dispersive metamaterial arrays in terms of generalized electric and magnetic polarizations

•A rigorous homogenization theory is developed for spatially dispersive electric and magnetic polarization in metamaterial arrays.•Reality and passivity conditions as well as reciprocity and causality relations are derived.•Causal constitutive parameters result for electric and magnetic or diamagnetic polarizations.•The theory is used to determine conditions for defining an electromagnetic continuum. An anisotropic homogenization theory for spatially dispersive periodic arrays is developed, based on the microscopic Maxwell equations, that yields causal, macroscopic permittivities, and inverse permeabilities for the fundamental Floquet modes of the arrays. (Macroscopic magnetoelectric coefficients are not required.) Reality conditions, reciprocity relations, passivity conditions, and causality relations are derived for these spatially dispersive macroscopic constitutive parameters. A significant feature of the formulation is that the macroscopic permittivities and permeabilities reduce to their anisotropic-continuum definitions in terms of ordinary 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. A numerical example of a 2D array comprised of circular–cylinder inclusions is given that confirms the theoretical results for the computed electric and magnetic or diamagnetic macroscopic polarizations.