Efficient simulation of non-linear effects in 2D optical nanostructures to TM waves

We develop a theory to study stationary TM-type waves propagating in a nanostructured layer of 2D nonlinear optical metamaterial or plasmonic device. It is assumed that the layer is inhomogeneous and contains non-linear isotropic elemental materials with non-linearity and loss mechanisms, including both linear and non-linear losses. While modeling of the non-linear propagation of the TE-type scalar waves is straightforward, the TM-type waves within the standard E-field formulations of non-linear optics cannot be treated in a purely scalar H-field context since an implicit equation for the non-linear dielectric functions should be resolved otherwise. A new formulation, which is built on the solution of the implicit equation for the non-linear dielectric function, is proposed. We use a general cubic non-linearity to illustrate all of the important features of the proposed approach. The general solution for scalar H-field waves is validated versus our previously tested particular cases, and important differences are shown between those cases and the general solution. These details, for example, include the link between linear and nonlinear loss mechanisms, and connection between the linear and non-linear dielectric functions. The proposed approach is used for modeling a non-linear focusing device with optically controlled isotropic Kerrtype non-linearity; the simulation results prove the predicted functioning of the device. 2009 Elsevier B.V. All rights reserved.