Multipole approach to metamaterials

An analytical description for plane-wave propagation in metamaterials is presented. It follows the usual approach for describing light propagation in homogeneous media on the basis of Maxwell's equations, although applied to a medium composed of metallic nanostructures. Here, as an example, these nanostructures are double (or cut) wires. In the present approach it is assumed that the carriers perform collective oscillations in a single wire. These oscillations are coupled to those in the adjacent wire; thus, the internal carrier dynamics may be described by a coupled-oscillator model. The multipole expansion technique is used to account for the electric and magnetic dipole as well as the electric quadrupole moments of these carrier oscillations within the nanostructure. It turns out that the symmetric normal mode is related to the electric dipole moment whereas the antisymmetric normal mode evokes simultaneously a magnetic dipole and an electric quadrupole moment. It is shown how effective permittivity and permeability can be derived from analytical expressions for the dispersion relation, the magnetization, and the electric displacement field. The results of the analytical model are compared with rigorous simulations of Maxwell's equations yielding the limitations and the domain of applicability of the proposed model.