Polarized Line Formation in Multi-dimensional Media. IV. A Fourier Decomposition Technique To Formulate The Transfer Equation With Angle-Dependent Partial Frequency Redistribution

TL;DR

This paper introduces a Fourier decomposition method for solving the polarized radiative transfer equation in multi-dimensional media, accounting for angle-dependent partial frequency redistribution and Hanle effect, improving modeling of solar atmospheric polarization.

Contribution

It generalizes Fourier series expansion of angle-dependent PRD functions to multi-D media, enabling simplified transfer equations for polarized light with angle dependence.

Findings

Formulated multi-D polarized RT equation with angle-dependent PRD.

Showed Fourier components become azimuthally independent or dependent as appropriate.

Derived transfer equations for Fourier components suitable for iterative solution methods.

Abstract

To explain the linear polarization observed in spatially resolved structures in the solar atmosphere, the solution of polarized radiative transfer (RT) equation in multi-dimensional (multi-D) geometries is essential. For strong resonance lines partial frequency redistribution (PRD) effects also become important. In a series of papers we have been investigating the nature of Stokes profiles formed in multi-D media including PRD in line scattering. For numerical simplicity so far we restricted our attention to the particular case of PRD functions which are averaged over all the incident and scattered directions. In this paper we formulate the polarized RT equation in multi-D media that takes into account Hanle effect with angle-dependent PRD functions. We generalize here to the multi-D case, the method of Fourier series expansion of angle-dependent PRD functions originally developed for RT in 1D geometry. We show that the Stokes source vector $\bm{S}=(S_I,S_Q,S_U)^T$ and the Stokes vector $\bm{I}=(I,Q,U)^T$ can be expanded in terms of infinite sets of components ${\tilde{\bm{\mathcal{S}}}}^{(k)}$, ${\tilde{\bm{\mathcal{I}}}}^{(k)}$ respectively, $k\in[0,+\infty)$. We show that the components ${\tilde{\bm{\mathcal{S}}}}^{(k)}$ become independent of the azimuthal angle ($\varphi$) of the scattered ray, whereas the components ${\tilde{\bm{\mathcal{I}}}}^{(k)}$ remain dependent on $\varphi$ due to the nature of RT in multi-D geometry. We also establish that ${\tilde{\bm{\mathcal{S}}}}^{(k)}$ and ${\tilde{\bm{\mathcal{I}}}}^{(k)}$ satisfy a simple transfer equation, which can be solved by any iterative method like an Approximate Lambda Iteration (ALI) or a Biconjugate-Gradient type projection method provided we truncate the Fourier series to have a finite number of terms.