A time-domain finite element method for Kerr and Raman type nonlinear hyperbolic metamaterials with application for enhanced third-harmonic generation

In this paper, we derive a time-dependent Maxwell's equation model to simulate electromagnetic wave propagation in nonlinear hyperbolic metamaterials. We approximate both permittivity and permeability by the Drude-Lorentz model and consider the third-order nonlinear polarization for this model. We propose a semi-implicit time-domain finite element scheme, and establish the stability of this numerical scheme. This model and our proposed numerical method can characterize both the linear and nonlinear properties of materials and aid in designing nonlinear hyperbolic metamaterials to enhance the high harmonic generation. Extensive numerical results confirm the optimal convergence rate of our numerical scheme and showcase the enhancement of high harmonic generation in two-dimensional nonlinear multilayer hyperbolic metamaterials. This paper is the first one on developing and analyzing a time-domain finite element method to simulate the electromagnetic wave interaction with nonlinear hyperbolic metamaterials.