Résumé
It has been noted that the usual approaches for homogenization (e.g. Bruggeman) could lead to singularities when considering composites comprising materials with positive and negative permittivities. This situation is specifically encountered in the field of metamaterials, in particular in the visible range. The question then arises to unerstand if these resonances are just artifacts or possess, on the contrary, a clear physical meaning. In this work, we study the homogeneous properties of a bidimensional structure that is made of a periodic set of metallic wires embedded in a dielectric host medium. The structure is considered in a region of wavelengths that are much larger than the period of the structure. The work comprises a theoretical part where we develop a two-scale approach to the homogenization of the structure. As it is the case for the common physical approach, it leads to an effective permittivity with strong (electrostatic) resonances as well. The theoretical results, and the presence of resonances, are confirmed by numerical computations based on a rigorous modal approach. In the numerical results we consider specifically the case of silver and gold nanowires, described by a dispersive negative permittivity. We show that the main parameters for the onset of resonances is the optical filling ratio of the structure. Keeping in mind the possibility of performing experiments, it is far easier to keep the geometrical filling ratio constant and to consider strongly dispersive materials in the range of wavelengths considered. Here, besides the crucial fact that they are widely used in nanotechnology, silver and gold nanowires comply with our needs in the visible region of the spectrum.