Multipole approach for analysis of passive and active metamaterials

Metamaterials are composites consisting of artificial metaatoms/metamolecules with typical sizes less than the respective wavelengths. A key point of the metamaterials, which differs them principally from the natural one (at least in a visible wavelength range) is the magnetic response on an external electric field. In this work we summarize the multipole expansion approach in order to describe analytically linear and nonlinear effective optical properties of passive metamaterials, and optical properties of metamaterials with gain. In order to validate our model, we applied the formalism for the split-ring resonator and the cut-wire structure and compare the results with rigorous numerical calculations. A new type of the second order nonlinearity - so called multipole nonlinearity - has been found to appear even for intrinsically linear metal, which the nano resonators consist of. The regular dynamics (transient dynamics and stabilized generation) and stochastic properties (linewidth of generation) of a nano laser have been described using the proposed model. By pumping of the appropriately positioned Quantum Dots (QDs) as a gain media, a magnetic response without optical losses can be obtained. Propagation of electromagnetic waves inside the metamaterials with QDs has been investigated, and dependence of a magnetic response on the gain has been analysed.