Factorizations for 3-rotations and polarization of the light in Mueller-Stokes an Jones formalisms

TL;DR

This paper derives formulas for all 2- and 3-element factorizations of SU(2) and SO(3) groups, enabling comprehensive parameterizations of polarization transformations in light optics.

Contribution

It provides explicit formulas for all possible 2- and 3-element factorizations of SU(2) and SO(3), facilitating complete polarization rotation analysis.

Findings

Six 2-element factorizations define all Euler-type angles.

Six 3-element factorizations cover all parameterizations by three angles.

Relations enable decomposition of pure polarization rotators into elementary components.

Abstract

Formulas describing all 2-element and 3-element factorizations of arbitrary element of the groups SU(2) and SO(3,R) are derived. Six 2-element factorizations, $ (U_{2}U_{3}U'_{2}), (U_{3}U_{2}U'_{3}), (U_{3}U_{1}U'_{3}), (U_{1}U_{3}U'_{1}), (U_{1}U_{2}U'_{1}), (U_{2}U_{1}U'_{2})$, provide all possible way to define Euler type angles; and six 3-element ones, $ (U_{1}U_{2}U_{3}), (U_{1}U_{3}U'_{2}), (U_{2}U_{3}U_{1}), (U_{2}U_{1}U_{3}), (U_{3}U_{1}U_{2}), (U_{3}U_{2}U_{1})$ provide all possible ways to parameterize the unitary and orthogonal groups by three elementary angles. In thecontext the light polarization formalism of Stokes-Mueller vectors and Jones spinors, relations produced give a base to resolve arbitrary pure polarization rotators into all possible sets of elementary rotators of two or three constituents.