Homogenization of metamaterial surfaces and slabs: the crossed wire mesh canonical problem

The objective of this paper is to discuss the differences between the homogenization of a bulk metamaterial and the homogenization of a thin metamaterial slab (complex surface). To this end, we study the canonical problem of propagation of electromagnetic waves in a mesh of crossed metallic wires embedded in a dielectric slab. We prove that it may not be appropriate to characterize the complex surface with the same effective parameters as the bulk metamaterial and the usual boundary conditions. We develop a modified homogenization procedure that is suitable to homogenize the complex surface using the bulk material effective parameters. The results of our analytical model are compared with full wave numerical results, yielding an excellent agreement for long wavelengths. Our formulation is valid for slabs of arbitrary thickness. We hope that this work may contribute to a more profound understanding of the interaction of electromagnetic waves with the emerging metamaterials.