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    Area of Science:

    • Optics and Photonics
    • Materials Science
    • Applied Mathematics

    Background:

    • Metasurfaces offer unique light manipulation capabilities.
    • Studying metasurfaces in non-flat geometries presents significant challenges.
    • Existing models often struggle with complex surface curvatures.

    Purpose of the Study:

    • To develop a mathematical approach for metasurfaces in non-flat geometries.
    • To establish analytical conditions for phase discontinuities on curved surfaces.
    • To provide a unified framework for both near and far field analyses.

    Main Methods:

    • Formulation of a vector Snell's law adapted for abrupt interface discontinuities.
    • Derivation of analytical conditions relating surface curvature to refracted directions.
    • Inclusion of both near-field and far-field electromagnetic considerations.

    Main Results:

    • Established analytical conditions that guarantee the existence of phase discontinuities.
    • Demonstrated the applicability of the vector Snell's law to curved metasurfaces.
    • Provided a comprehensive mathematical framework for non-flat metasurface design.

    Conclusions:

    • The proposed mathematical approach enables precise study and design of metasurfaces in non-flat geometries.
    • The derived conditions offer a pathway to engineer phase discontinuities for novel optical functionalities.
    • This work advances the understanding and application of metasurfaces beyond planar configurations.