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Lie algebraic approach to GRIN optimization.

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    A Lie algebraic approach offers a systematic way to calculate aberration coefficients for light rays in gradient-index (GRIN) lenses. This method provides explicit formulas for optimization, avoiding repetitive ray tracing for improved lens design.

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    Area of Science:

    • Optics and Photonics
    • Mathematical Physics

    Background:

    • Aberration coefficients are crucial for understanding and correcting optical system performance.
    • Ray propagation in inhomogeneous media, particularly gradient-index (GRIN) lenses, presents unique challenges for traditional optical design methods.

    Purpose of the Study:

    • To present a systematic Lie algebraic approach for determining aberration coefficients in GRIN lenses.
    • To derive analytical expressions for aberration coefficients applicable to arbitrary rotationally symmetric GRIN lenses.
    • To demonstrate the utility of these formulas in lens optimization routines.

    Main Methods:

    • Utilizing Lie algebraic methods to model ray propagation through optical systems.
    • Deriving explicit analytical expressions for aberration coefficients.
    • Applying optimization routines that leverage the derived analytical formulas.

    Main Results:

    • Analytical expressions for aberration coefficients of rotationally symmetric GRIN lenses were successfully derived.
    • The developed formulas facilitate optimization without the need for iterative ray tracing.
    • Optimization of a radial lens recovered a known classical solution, validating the approach.
    • A GRIN lens with axial inhomogeneity was also optimized, showcasing the method's versatility.

    Conclusions:

    • The Lie algebraic approach provides an efficient and systematic framework for calculating aberration coefficients in GRIN lenses.
    • The derived analytical expressions are valuable for advanced optical design and optimization, particularly for complex GRIN systems.
    • This method enhances the speed and accuracy of optical system design by reducing computational overhead.