Nonlinear electromagnetic metamaterials: Aspects on mathematical modeling and physical phenomena

Abstract

Electromagnetic metamaterials with simultaneously negative effective dielectric permittivity ε and magnetic permeability μ (double-negative or left-handed or negative-index metamaterials) have a long history going back to the seminal paper of Veselago in 1968. Such metamaterials exhibit unusual and remarkable effects, like, e.g., the reversal of Snell's law, the support of backward-propagating waves, and the possibility of obtaining a perfect lens. Nonlinear left-handed metamaterials are also very useful in tunable structures with intensity-controlled transmission and in switching the material properties from left- to right-handed and back. In this paper, we present aspects on mathematical modeling and physical phenomena governing electromagnetic wave propagation in nonlinear metamaterials. The metamaterials under consideration are described by a Drude-Lorentz frequency model of the linear parts and a Kerr-type behavior of the nonlinear parts of ε and μ, respectively. We show that in the left-handed band of the metamaterial wave propagation is governed by a higher-order nonlinear Schrödinger (NLS) equation, and derive analytically ultra-short bright or dark solitons solutions. Then, we investigate wave propagation in the frequency band gaps, i.e., in the frequency regimes where ε and μ have different signs, and, hence, linear waves are evanescent. In these band gaps localization of waves is possible if a nonlinearity is induced in the metamaterial medium. We derive a dissipative short-pulse equation (DSPE) governing ultrashort pulses that may be formed in the band gaps and discuss its solitons solutions.