Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect

The recently developed fast Fourier factorization method resolves linear Maxwell equations in a truncated Fourier basis using correct factorization rules. In nonlinear optics, Maxwell equations present a discontinuous product of two simultaneously discontinuous functions for which no rule of factorization applies. Using an iterative method which avoids such a type of factorization, we extend the fast Fourier factorization method to nonlinear optics. We demonstrate the good convergence of the method by studying deep metallic gratings with grooves filled with a nonlinear material, illuminated in TM polarization with a high intensity plane wave.