Metamaterials form a widely studied class of subwavelength-period photonic crystals that might be considered artificial material. Currently, several widely used computational homogenization techniques allow obtaining the effective material parameters of composite media. Nevertheless, most of them have their limits of applicability. Some of them are physically baseless, which often results in misinterpretation of the provided results.
In this study, we propose the microscopical approach for the extraction of effective optical constants. To do that, we reformulate the mathematically ill-posed problem of scattering by an infinite photonic crystal in terms of the corresponding currents injection. This allows us to obtain effective permittivity as a function of arbitrary frequency and wavevector. In particular, we demonstrate that the approach successfully copes with a crystal of optically-dense particles supporting magnetic dipole Mie resonance and correctly describes the contribution of the resonance to both permittivity and permeability.