Modal formulation of Kirchhoff's law for reciprocal structures

Abstract

Kirchhoff's law provides a relation between the incident fields a structure can absorb and those it thermally emits. It is generally formulated as a relation between the angle-dependent emissivity and absorptivity of a structure. In this letter, we propose to extend the definition of the absorptivity and emissivity to account for interference and coherence effects. These new definitions are used to derive a modal form of Kirchhoff's law for reciprocal structures, in which "absorptive" and "emissive" modes can be paired. We show that the formulation of Kirchhoff's law strongly depends on the basis used to express the fields. Three different formulations are proposed, which are related to three popular bases: plane waves characterized by their direction of propagation, plane waves characterized by their transverse wave vector, and the spatial distribution of the tangential electric and magnetic fields along a closed surface that contains the structure.