Multiscale numerical method for nonlinear Maxwell equations

The aim of this work is to propose an efficient numerical approximation of high frequency pulses propagating in nonlinear dispersive optical media. We consider the nonlinear Maxwell's equations with instantaneous nonlinearity. We first derive a physically and asymptotically equivalent model that is semi-linear. Then, for a large class of semi-linear systems, we describe the solution in terms of profiles. These profiles are solution of a singular equation involving one more variable describing the phase of the solution. We introduce a discretization of this equation using finite differences in space and time and an appropriate Fourier basis (with few elements) for the phase. The main point is that accurate solution of the nonlinear Maxwell equation can be computed with a mesh size of order of the wave length. This approximation is asymptotic-preserving in the sense that a multi-scale expansion can be performed on the discrete solution and the result of this expansion is a discretization of the continuous limit. In order to improve the computational delay, computations are performed in a window moving at the group velocity of the pulse. The second harmonic generation is used as an example to illustrate the proposed methodology. However, the numerical method proposed for this benchmark study can be applied to many other cases of nonlinear optics with high frequency pulses.