A Systematic Study of Perfectly Matched Absorbers

Abstract

In the numerical modeling of electromagnetic radiation and/or scattering problems by finite methods, the computational domain is usually truncated by an artificial boundary on which suitable absorbing boundary conditions (ABCs) are imposed. An alternative approach to mesh truncation, which was introduced by Berenger for finite difference time domain (FDTD) implementation, employs a region which is designed to absorb plane waves whose frequency and incident angle are arbitrary. Since the plane waves are transmitted into the region without any reflection, the region is called a perfectly matched layer (PML). In this chapter, our main objective is to systematically derive the partial differential equations satisfied within the PML media. It is demonstrated that both Maxwellian PMLs (the anisotropic and the bianisotropic PML media) as well as non-Maxwellian PMLs (Berenger PML) can be realized by assuming a field decay behavior within the PML. Causality and reciprocity issues, and their implications in the proper design and operation of perfectly matched absorbers, are also discussed. It is shown that if the constitutive parameters of the PML medium satisfy the Kramers-Kronig relations, the medium is causal and does not exhibit a singular behavior at lower frequencies. This, in turn, enables us to extend the PML concept to the static case. The reciprocity concept is also important in the design of PMLs, since the PML medium occupies a bounded domain in mesh truncation applications. The medium must be reciprocal and, as a consequence, the decay behavior of the waves traveling in the longitudinal direction must be identical to that of the waves reflected from the terminating boundary and traveling in the opposite direction. Some examples of causal/non-causal and reciprocal/non-reciprocal PMLs are given in the work to illustrate these behaviors.