A Three-Dimensional Finite-Difference Time-Domain Formulation for Nonlinear Materials With the Frequency-Dependent Electronic Conductivity and Polarization

We present a three dimensional finite difference time domain (FDTD) algorithm that is used to evaluate the propagation of electromagnetic waves in conductive and dispersive materials that exhibit the frequency dependent electric conductivity and polarization. We consider a case where the electric conductivity has the linear property, specifically through the first order (linear) electric conductivity function, and the electric polarization has both linear and nonlinear properties, specifically through the first order (linear) and third order (nonlinear) electric susceptibility functions. The resulting FDTD algorithm shows that the nonlinear dispersive material with a third order susceptibility function results in coupled nonlinear cubic equations for the three components of the electric field vector, relating the next time step electric field vector to the previous time step electric field vector. This contrasts the usual algorithm of the linear conductive and dispersive material, which has a simple linear relationship between the next time step electric field and the previous time step electric field. Consequently, the coupled nonlinear cubic equations must be solved at each time step to simulate the behavior of the electric field vector in the nonlinear dispersive material that contains both frequency dependent electric conductivity and polarization.