chapter  2
48 Pages

Metamaterials and homogenization of composites

The description of a metamaterial as a homogeneous medium involves averaging over the fluctuations of the electromagnetic fields at two levels. As explained in Section 1.3, the macroscopic Maxwell equations are obtained by averaging the rapidly fluctuating electromagnetic fields at atomic or molecular lengthscales over volumes that contain enough number of polarizable or magnetizable atoms/molecules. Within this framework, susceptibilities for bulk materials can be defined. In the case of metamaterials, the structural units of the metamaterial (see Chapter 3) are assumed to be sufficiently large on a molecular scale so that they can confidently be described by their bulk dielectric permittivity and magnetic permeability, and yet sufficiently small compared with the lengthscales over which the applied fields vary (typically a wavelength). Hence, only the fields due to the first few multipoles of the charge and current distributions induced in the structures contribute to the macroscopic polarization over lengthscales large compared to the metamaterial units. In other words, the fine structure of the charge and current distributions over the structural units is not discernible, but only a few averages such as the corresponding dipolar fields or (rarely) the quadrupolar fields can be resolved through the macroscopic polarization and magnetization. These average quantities determine the effective dielectric permittivity and the magnetic permeability tensors of the bulk metamaterial.