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.