Lorentz algebraic approach in two- and three-dimensional polarization optics

Lorentz algebra is a significant and elegant language in 2-D SAM-related polarization optics, and it also holds potential theoretical value in 3-D polarization optics. This paper focuses on developing a decomposed generalized Mueller matrix (GMM) model for 3-D polarization transformations through a Lorentz algebraic approach. We first present a comprehensive analysis and review of the 2-D polarization state (SoP) and polarization transformations, introducing the necessary algebraic representations and approaches. Then, we further develop the 3-D transformation theory and present a convenient decomposed 3-D transformation model, which exists in both generalized Jones matrices (GJMs) and GMM representations. For GMM, the generator matrices of all sub-transformations (Er-rotation, Ez-rotation, and Ez-boost) are clearly defined and discussed for the first time, to our knowledge. And their correctness is verified from commutative relations and GMM simulations. Additionally, another simulation is presented to illustrate the potential application of decomposed GMM in non-paraxial beams and polarized ray-optics.