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    Area of Science:

    • Optics
    • Computational Physics
    • Mathematical Physics

    Background:

    • Ray tracing in gradient-index (GRIN) media is complex due to spatial variations in refractive index.
    • Existing methods often require specific solutions for different GRIN field configurations.
    • Accurate ray tracing is crucial for designing optical systems and understanding light propagation.

    Purpose of the Study:

    • To propose a transformation method based on optical Hamilton equations for 3D ray tracing in axially inhomogeneous GRIN media with cylindrical symmetry.
    • To simplify the ray tracing process by transforming physical space into a virtual space.
    • To develop a unified expression for ray tracing applicable to various GRIN fields.

    Main Methods:

    • A canonical transformation is applied to convert optical conjugate variable pairs from physical space to a virtual space.
    • The ray trace in the virtual space is solved using a uniform expression.
    • The solution is then inversely transformed back into the physical space.
    • A Hamilton principal function (S function) is utilized, providing both the real ray trace and its conjugate trace.

    Main Results:

    • The proposed method offers a simplified and unified approach to 3D ray tracing in complex GRIN media.
    • The S function simultaneously yields the real ray trace and its conjugate trace (direction cosines).
    • The conjugate trace can be independently derived from the S function, offering insights into propagation direction.
    • The S function is also applicable to 2D cases with axial inhomogeneity or cylindrical symmetry.

    Conclusions:

    • The transformation method based on optical Hamilton equations provides an efficient and versatile tool for 3D ray tracing in GRIN media.
    • The S function offers a powerful way to analyze ray propagation, providing both real and conjugate traces.
    • The method's applicability to 2D special cases enhances its utility for various optical system analyses.