Time-Domain Simulations of the Nonlinear Maxwell Equations Using Operator-Exponential Methods

In this paper, we propose a Krylov-subspace-based operator-exponential method for time-domain simulations of the Maxwell equations with general nonlinear polarizations. This includes (classical) chi
⁽²⁾
- or chi
⁽³⁾
- nonlinearities and/or nonlinear coupled system dynamics. As an illustration, we compare the performance of our approach to certain well-known methods for the case of pulse self-steepening in a material with negative Kerr-nonlinearity. For this system, we also develop an appropriate analytical reference solution. In addition, we demonstrate that our approach allows to treat the complex nonlinear dynamics of various physical systems (classical and/or quantum) coupled to electromagnetic fields.