A three-dimensional finite-difference time-domain algorithm based on the recursive convolution approach for propagation of electromagnetic waves in nonlinear dispersive media

We present for the first time a successful formulation of a three-dimensional finite-difference time-domain algorithm that is based on the recursive convolution approach and is used to evaluate the propagation of electromagnetic waves in nonlinear dispersive media. We treat in particular the case where the nonlinear polarization term depends only on the product of the square of the electric field and the third-order electric susceptibility function. However, the approach is general and we can easily extend the formulation to the product of any power of the electric field with the nth-order electric field susceptibility function. We find that, in contrast to the usual formulation for linear dispersive materials which uses a simple linear relationship between the next-time-step electric field and the previous-time-step electric field, the formulation for nonlinear dispersive materials with a third-order susceptibility function results in coupled nonlinear cubic equations which relate the next-time-step electric field vector to the previous-time-step electric field vector. Consequently, the coupled nonlinear cubic equations must be solved at each time step to advance the electric field vector.