Into the third dimension: stochastic measurements of Stokes parameters within the Poincar\'e sphere

TL;DR

This paper introduces a statistical framework for analyzing Stokes parameters in spherical coordinates, unifying circular and linear polarization treatment, with new distributions and estimators relevant for polarimetry in astrophysics.

Contribution

It presents a novel three-dimensional statistical approach to Stokes parameters, including new distributions and estimators, enhancing polarization analysis in astrophysical observations.

Findings

Circular and linear polarization are statistically intertwined.

New distributions and estimators for 3D polarization analysis.

Discussion on pitfalls of lower-dimensional analysis.

Abstract

Inspired by recent use of polarimetry to study the Cosmic Microwave Background and extragalatic supernovae, a foray into the statistical properties of Stokes parameters expressed in spherical coordinates is began, allowing circular polarization and linear polarization to be treated in a unified manner. The use of spherical coordinates is quite necessary as it permits a Stokes polarization state to be expressed in terms of the customary polarization angles and degree of polarization usually needed for human interpretation. As shall be demonstrated, circular and linear polarization are not statistically independent quantities but intertwined in a way that is especially important, for instance, at low signal-to-noise. New distributions, classical estimators, and marginalizations are presented for this "three-dimensional" polarization problem including a generalization of the Rice distribution. The paper concludes with discussion regarding the potential pitfalls of a lower dimensional analysis.