Electromagnetic source transformations and scalarization in stratified gyrotropic media

TL;DR

This paper demonstrates the equivalence of different scalarization methods for Maxwell's equations in stratified gyrotropic media, extending previous approaches and clarifying the relationships between source transformations, gauge choices, and potential formulations.

Contribution

It unifies and extends scalarization techniques for Maxwell's equations in gyrotropic media, showing their equivalence and clarifying the role of gauge and source transformations.

Findings

Scalarization methods are equivalent in gyrotropic media.

Vector potential approach can be extended with proper gauge conditions.

Scalar Hertz potentials are invariant under source transformations.

Abstract

It is known that with restrictions on the type of the constitutive equations, Maxwell's equations in non-uniform media can sometimes be reduced to two 2nd order differential equations for 2 scalar quantities only. These results have previously been obtained in two quite different ways. Either by a ``scalarization of the sources'', where the relevant scalar quantities are essentially vector potential components and where the derivation was limited to isotropic media; or alternatively by using the ``scalar Hertz potentials'', and this method has been applied to more general media. In this paper it is shown that both methods are equivalent for gyrotropic media. We show that the scalarization can be obtained by a combination of transformations between electric and magnetic sources and gauge transformations. It is shown that the method based on the vector potential, which previously used a non-traditional definition of the vector potentials, can also be obtained using the traditional definition provided a proper gauge condition is applied and this method is then extended from isotropic to gyrotropic media. It is shown that the 2 basic scalar Hertz potentials occurring in the second method are invariant under the source scalarization transformations of the first method and therefore are the natural potentials for obtaining scalarization. Finally it is shown that both methods are also equivalent with a much older third method based on Hertz vectors.