Perfectly Matched Layers implementation for E-H fields and complex wave envelope propagation in the SMILEI PIC code

The design of absorbing boundary conditions (ABC) in a numerical simulation is a challenging task. In the best cases, spurious reflections remain for some angles of incidence or at certain wavelengths. In the worst, ABC are not even possible for the set of equations and/or numerical schemes used in the simulation and reflections can not be avoided at all. Perfectly Matched Layers (PML) are layers of absorbing medium which can be added at the simulation edges in order to significantly damp both outgoing and reflected waves, thus effectively playing the role of an ABC. They are able to absorb waves and prevent reflections for all angles and frequencies at a modest computational cost. They increase the simulation accuracy and negate the need of oversizing the simulation usually imposed by ABC which normally leads to a waste of computational resources and power. In this paper, a uniform derivation of PML for finite-difference time-domain (FDTD) schemes and various geometries in Particle-In-Cell (PIC) codes is presented for Maxwell's equations and, for the first time, extended to the full envelope wave equation. An implementation of these methods in the open source PIC code SMILEI is proposed and benchmarked.