Differential forms, metrics, and the reflectionless absorption of electromagnetic waves

Two classes of formulations are prevalent on the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves. In the first, additional degrees of freedom modify the curl and div operators of Maxwell's equations. This results in the so-called non-Maxwellian PML. The original Berenger formulation belongs to this class. The non-Maxwellian PML can be systematically derived by an analytic continuation of the coordinate space to a complex variables coordinate space (complex-space) which permits the extension of the PML to general geometries and media. In the second class of PML formulations, the additional degrees of freedom are entirely incorporated into modified constitutive tensors and the usual Maxwell's equations are recovered. This results in a Maxwellian PML. Interestingly enough, for all cases where the non-Maxwellian PML was derived, a Maxwellian PML was also later derived. This suggests a duality between the formulations and the possibility of a fundamental reason behind the existence of the Maxwellian PML. In this work, we review the PML concept using the language of differential forms to (i) explain the deeper reason allowing for the ubiquitous presence of the Maxwellian PML; (ii) to provide the general framework which unifies the various PML formulations; and (iii) to show that, in principle, many other classes (hybrid) of PML formulations can be derived in the frequency-domain. This is done by introducing a novel, geometrical interpretation of the PML in terms of a change on the metric of space and exploring the metric independence of Maxwell's equations unfolded by such a language.