Propagation of ultroshort electromagnetic pulse in nonlinear metamaterials

The metamaterial, such as the negative-index material, has both dispersive permittivity and dispersive permeability. By incorporating the dispersive permeability into the nonlinear polarization, one can deal with the problem of nonlinear pulse propagation in metamaterial by using the same way as in ordinary material, or even direct borrowing the existing models for pulse propagation in ordinary materials. By doing so, we obtain a general three-dimensional nonlinear Schröinger equation suitable for few-cycle pulse propagation in metamaterial. This equation clearly demonstrates the role of dispersive permeability in pulse propagation: In the linear propagation aspect, its contribution is buried in the ordinary dispersive terms; while in the nonlinear propagation aspect, the dispersive permeability manifests itself as a nonlinear polarization dispersion, although it is a linear parameter. For Drude dispersive model, the dispersive permeability results in a negative self-steepening effect in the negative index region and negligible higher-order nonlinear dispersion terms. The titled direction of pulse due to the self-steepening effect in negative-index material is opposite to that in ordinary materials.