Multipole metamaterials

This article investigates the multipole expansion approach in order to describe analytically linear and nonlinear optical properties of metamaterials. It is shown that especially the consideration of electric quadrupole multipoles (not only the magnetic dipole one) is required by basic principles and provides the possibility to adequately analyse an effective magnetic material response. To motivate the appearance of such multipole terms, numerical methods to calculate the near field radiation patterns as well as far field properties are used. The dispersion relation of a single metamaterial layer and 3D infinitely expanded bulk metamaterials are derived as well. In order to validate the model for particular cases, the formalism for the split-ring resonator and the cut-wire structure is applied. In principle, the formalism is not restricted by any special metaatom geometry. Only the intrinsic electron dynamics in the respective metaatom must be known.