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    Area of Science:

    • Optics
    • Theoretical Physics
    • Applied Mathematics

    Background:

    • Light propagation follows Fermat's principle, leading to the optical Lagrange equation for determining light traces given a refractive index.
    • Designing optical systems often requires controlling light paths, which is challenging with fixed refractive indices.

    Purpose of the Study:

    • To solve the inverse problem of deriving refractive index distribution from a predetermined light trace in planar geometric optics.
    • To provide a theoretical framework for manipulating light rays in gradient-refractive-index (GRIN) media.

    Main Methods:

    • Formulated a dynamic equation model linking refractive index and light trace based on the optical Lagrange equation.
    • Employed the first integral method to solve the resulting partial differential equation for the refractive index distribution.
    • Utilized gradient-refractive-index (GRIN) medium distribution functions to control light paths.

    Main Results:

    • Successfully demonstrated a method to derive refractive index distribution from a given light trace.
    • Showcased the ability to control light trajectories and adjust rays by defining GRIN medium properties.
    • Illustrated the use of light traces to design geometric optical elements.

    Conclusions:

    • The proposed method offers a novel approach to designing ray dynamic systems by manipulating refractive index.
    • This work extends Lagrangian optics applications to gradient-refractive-index (GRIN) media for designing lenses, beam splitters, metasurfaces, and optical waveguides.
    • Provides theoretical guidance for precise ray manipulation in optical engineering.