On the Derivation of Vector Radiative Transfer Equation for Polarized Radiative Transport in Graded Index Media

TL;DR

This paper derives a comprehensive vector radiative transfer equation for polarized light in graded index media, accounting for curved trajectories and geometrical effects on polarization transport, extending classic models.

Contribution

It introduces a complete derivation of the vector radiative transfer equation for graded index media, incorporating curved ray trajectories and polarization effects, which was not previously available.

Findings

Derived a generalized transfer equation valid for graded index media.

Presented multiple variants including Stokes parameters and coordinate system forms.

Extended classical equations to include geometrical and refractive index variations.

Abstract

Light transport in graded index media follows a curved trajectory determined by the Fermat's principle. Besides the effect of variation of the refractive index on the transport of radiative intensity, the curved ray trajectory will induce geometrical effects on the transport of polarization ellipse. This paper presents a complete derivation of vector radiative transfer equation for polarized radiation transport in absorption, emission and scattering graded index media. The derivation is based on the analysis of the conserved quantities for polarized light transport along curved trajectory and a novel approach. The obtained transfer equation can be considered as a generalization of the classic vector radiative transfer equation that is only valid for uniform refractive index media. Several variant forms of the transport equation are also presented, which include the form for Stokes parameters defined with a fixed reference and the Eulerian forms in the ray coordinate and in several common orthogonal coordinate systems.