  In this work, we develop a new systematic and self-consistent approach to
homogenize arbitrary non-magnetic periodic metamaterials. The proposed method
does not rely on the solution of an eigenvalue problem and can fully
characterize the effects of frequency dispersion, magneto-electric coupling,
and spatial dispersion, even in frequency band-gaps or when the materials are
lossy. We formulate a homogenization problem to characterize a generic
microstructured artificial material, and demonstrate that it is equivalent to
an integral-differential system. We prove that this complex system can be
reduced to a standard integral equation and solved using standard methods. To
illustrate the application of the proposed method, we homogenize several
important metamaterial configurations involving split ring resonators and
metallic wires.
