Wave packet propagation by the Faber polynomial approximation in electrodynamics of passive media

Abstract

Maxwell's equations for propagation of electromagnetic waves in dispersive and absorptive (passive) media are represented in the form of the Schr\"odinger equation $i\partial \Psi/\partial t = {H}\Psi$, where ${H}$ is a linear differential operator (Hamiltonian) acting on a multi-dimensional vector $\Psi$ composed of the electromagnetic fields and auxiliary matter fields describing the medium response. In this representation, the initial value problem is solved by applying the fundamental solution $\exp(-itH)$ to the initial field configuration. The Faber polynomial approximation of the fundamental solution is used to develop a numerical algorithm for propagation of broad band wave packets in passive media. The action of the Hamiltonian on the wave function $\Psi$ is approximated by the Fourier grid pseudospectral method. The algorithm is global in time, meaning that the entire propagation can be carried out in just a few time steps. A typical time step is much larger than that in finite differencing schemes, $\Delta t_F \gg \|H\|^{-1}$. The accuracy and stability of the algorithm is analyzed. The Faber propagation method is compared with the Lanczos-Arnoldi propagation method with an example of scattering of broad band laser pulses on a periodic grating made of a dielectric whose dispersive properties are described by the Rocard-Powels-Debye model. The Faber algorithm is shown to be more efficient. The Courant limit for time stepping, $\Delta t_C \sim \|H\|^{-1}$, is exceeded at least in 3000 times in the Faber propagation scheme.