Analytic properties of the electromagnetic Green's function

TL;DR

This paper investigates the analytic properties of the electromagnetic Green's function expressed via the inverse Helmholtz operator, introducing a second frequency related to dielectric dispersion, and derives Kramers-Kronig relations connecting to eigenmode expansions.

Contribution

It introduces a second frequency as a new degree of freedom in analyzing the Green's function and extends its analytic properties and Kramers-Kronig relations to complex frequencies and wavevectors.

Findings

Green's function is analytic with respect to two complex frequencies with positive imaginary parts.

Derived Kramers-Kronig relations for the inverse Helmholtz operator and Green's function.

Extended analysis to characterize non-dispersive systems using the second frequency.

Abstract

The electromagnetic Green's function is expressed from the inverse Helmholtz operator, where a second frequency has been introduced as a new degree of freedom. The first frequency results from the frequency decomposition of the electromagnetic field while the second frequency is associated with the dispersion of the dielectric permittivity. Then, it is shown that the electromagnetic Green's function is analytic with respect to these two complex frequencies as soon as they have positive imaginary part. Such analytic properties are also extended to complex wavevectors. Next, Kramers-Kronig expressions for the inverse Helmholtz operator and the electromagnetic Green's function are derived. In addition, these Kramers-Kronig expressions are shown to correspond to the classical eigengenmodes expansion of the Green's function established in simple situations. Finally, the second frequency introduced as a new degree of freedom is exploited to characterize non-dispersive systems.