Analytical modelling of linear and nonlinear properties of metamaterials based on multipole expansion

A simple analytical model has been developed within the scopes of the macroscopic Maxwell's equations. In the framework of this model the dispersion relation for plane waves has been calculated for the case of Cut-Wire (CW) and Split Ring Resonator (SRR) geometries. The dispersion relation has been compared with rigorous numerical calculations. A possible way to introduce the electric and magnetic material parameters has been suggested. Validity criteria and applicability limitations of the developed model are discussed. A new type of nonlinearity specific for the metamaterials - Multipole Nonlinearity - is identified based on the developed model, wheras the second harmonic generation (SHG) process is considered in detail.