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    This study introduces a novel decomposed generalized Mueller matrix (GMM) model for 3-D polarization transformations using Lorentz algebra. This approach simplifies complex optical systems and has potential applications in advanced optics.

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    Area of Science:

    • Optics and Photonics
    • Mathematical Physics
    • Polarization Optics

    Background:

    • Lorentz algebra is a powerful tool in 2-D polarization optics.
    • Extending these algebraic methods to 3-D polarization optics offers significant theoretical potential.
    • Generalized Mueller matrices (GMMs) are essential for describing polarization transformations.

    Purpose of the Study:

    • To develop a decomposed generalized Mueller matrix (GMM) model for 3-D polarization transformations.
    • To utilize a Lorentz algebraic approach for modeling these transformations.
    • To explore the application of this model in non-paraxial beams and polarized ray-optics.

    Main Methods:

    • Comprehensive analysis and review of 2-D polarization state (SoP) and transformations using algebraic representations.
    • Development of 3-D transformation theory and a decomposed 3-D transformation model.
    • Definition and discussion of generator matrices for sub-transformations (rotations and boosts) within the GMM framework.

    Main Results:

    • A convenient decomposed 3-D transformation model is presented in both generalized Jones matrices (GJMs) and GMM representations.
    • Generator matrices for r→-rotation, z→-rotation, and z→-boost sub-transformations are defined and discussed for the first time.
    • The correctness of the GMM model is verified through commutative relations and simulations.

    Conclusions:

    • The Lorentz algebraic approach provides an elegant framework for 3-D polarization optics.
    • The decomposed GMM model offers a simplified and effective method for analyzing 3-D polarization transformations.
    • The model shows potential for applications in non-paraxial beams and polarized ray-optics.