A Convolution-Free Mixed Finite-Element Time-Domain Method for General Nonlinear Dispersive Media

In this paper, a mixed finite-element time-domain (FETD) method is presented for the simulation of electrically complex materials, including general combinations of linear dispersion, instantaneous nonlinearity, and dispersive nonlinearity. Using both edge and face elements, the presented method offers greater geometric flexibility than existing finite-difference time-domain (FDTD) implementations, and in contrast to existing nonlinear FETD methods, also incorporates both linear and nonlinear material dispersions. Dielectric nonlinearity is incorporated into the Crank-Nicolson mixed FETD formulation via a straightforward Newton-Raphson approach, for which the associated Jacobian is derived. Moreover, the dispersion is modeled via the Möbius z transform method, yielding a simpler more general algorithm. The method's accuracy and convergence are verified, and its capability demonstrated via the simulation of several nonlinear phenomena, including temporal and spatial solitons in two spatial dimensions.