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    The modal method with coordinate transformation provides a general analytical solution for 1D grating diffraction using a T-matrix. This approach clarifies the conditions under which Rayleigh expansion is valid in transformed media.

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    Area of Science:

    • Electromagnetics
    • Optics
    • Computational Physics

    Background:

    • Electromagnetic grating diffraction is crucial for optical devices.
    • Analytical solutions are sought for complex grating structures.
    • The modal method and Rayleigh expansion are common analytical techniques.

    Purpose of the Study:

    • To develop a general analytical solution for 1D grating diffraction.
    • To investigate the relationship between modal expansion and Rayleigh expansion validity.
    • To introduce a T-matrix formulation for grating analysis.

    Main Methods:

    • Implementing the modal method.
    • Utilizing curvilinear coordinate transformation.
    • Deriving a T-matrix solution.

    Main Results:

    • A general analytical T-matrix solution for 1D grating diffraction was obtained.
    • The validity of Rayleigh expansion was shown to be dependent on modal expansion in transformed media.
    • The coordinate transformation method provides a rigorous framework.

    Conclusions:

    • The modal method combined with coordinate transformation offers a powerful tool for grating analysis.
    • This work establishes a clear criterion for the applicability of Rayleigh expansion.
    • The T-matrix method provides a comprehensive solution for electromagnetic grating diffraction problems.