Lorentz algebraic approach in two and three dimensional polarization optics

Lorentz group and algebraic approach are significant and elegant language in optical spin angular momentum (SAM) and SAM-related polarization optics, which also has potential theoretical value in 3-D polarization optics and spin-orbital coupled vector-polarized beam (VPB). This article focuses on the Lorentz algebraic approach in 2-D and 3-D situations, where the first part includes a comprehensive analysis and review of 2-D polarization state (SoP) and polarized Lorentz transformations (PLT), in which the matrice form of generalized 2-D PLT and specific PLT in 2-D polarized devices will be naturally derived. In the second part, we will further explore the 3-D PLT theory and present a clear and convenient decomposed 3-D PLT model which exists in both generalized Jones matrices (GJM) and generalized Mueller matrices (GMM) representations, serving the transformation of 3-D coherency matrices and generalized Stokes vectors respectively. For the latter, the algebraic generators of decomposed 3-D sub-transformations (r-rotation, z-rotation and z-boost) will be defined and discussed for the first time, and the result show that they satisfy SO(3) and Lorentz symmetry respectively. The decomposed model presented can be used for 3-D polarized field simulation and calculation such as high-focused or axial-deviated beam, polarized ray-optics, and vector polarized beam.