Asymmetric Perfectly Matched Layer for the Absorption of Waves

The perfectly matched layer (PML) has become a standard for comparison in the techniques that have been developed to close the system of Maxwell equations (more generally, wave equations) when simulating an open system. The original Berenger PML formulation relies on a split version of Maxwell equations, with numerical electric and magnetic conductivities. We present here an extension of this formulation, which introduces counterparts of the electric and magnetic conductivities affecting the term which is spatially differentiated in the equations. The phase velocity along each direction is also multiplied by an additional coefficient. We show that under certain constraints on the additional numerical coefficients, this ‘medium’ does not generate any reflection at any angle or any frequency and is thus a perfectly matched layer. Technically it is a superset of Berengerʹs PML to which it reduces for a specific set of parameters, and like it, it is anisotropic. However, unlike the PML, it introduces some asymmetry in the absorption rate and is therefore labeled an APML, for asymmetric perfectly matched layer. We present here the numerical considerations that have led us to introduce such a medium as well as its theory. Several finite-difference numerical implementations are derived (in one, two, and three dimensions) and the performance of the APML is contrasted with that of the PML in one and two dimensions. Using plane wave analysis, we show that our APML implementations lead to higher absorption rates than the considered PML implementations. Although we have considered in this paper the finite-difference discretization of Maxwell-like equations only, the APML system of equations may be used with other discretization schemes, such as finite elements, and may be applied to other equations, for applications beyond electromagnetics.