Generalized Jones-Mueller calculus for optical anisotropic media.

Area of Science:

• Optics and Photonics
• Mathematical Physics
• Polarization Optics

Background:

• Existing 3-D matrix calculi (generalized Jones matrix calculus - GJM, and Mueller matrix calculus - GMM) are incomplete and lack a unified framework.
• Current GJM models 3-D anisotropic interactions in a paraxial-like manner, neglecting vectorial light path influence and defining differential GJM (dGJM) as fixed.
• The independence of GJM and GMM hinders a comprehensive understanding of 3-D polarized transformations.

Purpose of the Study:

• To establish a global, bi-directional connection between GJM and GMM for a unified 3-D polarization framework.
• To develop a pure-matrix approach for modeling 3-D anisotropic interactions along arbitrary light paths.
• To refine the theoretical foundation of 3-D polarization optics for non-paraxial applications.

Main Methods:

• Introduction of a Lorentz-like algebra to establish a double-covering homomorphism between SL(3,C) and the Lorentz-like group (LLG).
• Development of a pure-matrix approach that incorporates the vectorial light path for 3-D anisotropic interaction.
• Theoretical exploration and establishment of the global connection between GJM and GMM.

Main Results:

• A global and bi-directional mapping between GJM and GMM is established via the Lorentz-like algebra and LLG.
• A novel pure-matrix approach enables polarization modeling along arbitrary light paths, overcoming paraxial limitations.
• The proposed theories provide a refined framework for 3-D polarization optics.

Conclusions:

• The established homomorphism provides a unified theoretical basis for 3-D polarization matrix calculus.
• The new modeling approach accounts for vectorial light path effects in 3-D anisotropic media.
• This work lays the foundation for advanced non-paraxial polarization applications.