A general numerical framework for polarized radiative transfer in multidimensional graded-index media

The propagation of polarized light in graded-index (GRIN) media is widely encountered in applications such as remote sensing and medical imaging. In these media, the polarization ellipse described by the Stokes vector undergoes continuous rotation along curved ray paths, which complicates the modeling of polarized radiative transfer. Although the generalized vector radiative transfer equation (GVRTE) has been established to describe this phenomenon, it contains two complex source terms related to polarization rotation and ray torsion. As a result, no efficient numerical method has yet been developed to solve it. In this study, we introduce a rotation matrix to transform the local Frenet–Serret coordinate system of the curved ray into the global Cartesian coordinate system used in computation. This transformation allows the explicit derivation of the polarization rotation and torsion terms, thereby closing the GVRTE. To resolve the external collimated radiation term, a Runge–Kutta scheme is employed to determine the direction and intensity of the collimated Stokes vector within the Cartesian coordinate system. Based on the closed GVRTE derived above, a lattice Boltzmann method is constructed to discretize the equation in both time and space. Numerical results for Rayleigh-scattering GRIN media demonstrate the capability of the proposed framework to describe the coupled evolution of intensity and polarization state in multidimensional GRIN media. This work provides a deterministic numerical basis for future extensions to more complex scattering systems.