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    Area of Science:

    • Electromagnetism
    • Wave Optics
    • Mathematical Physics

    Background:

    • Cylindrical waves (CWs) reflection at planar interfaces is crucial in various optical phenomena.
    • Existing models often rely on spectral integrals for reflected CW analysis.
    • A comprehensive analytical solution for reflected CWs is needed.

    Purpose of the Study:

    • To develop an analytical solution for the reflection of cylindrical waves at planar interfaces.
    • To provide a new mathematical framework for analyzing reflected CWs.
    • To offer a physical interpretation of the reflected wave solution.

    Main Methods:

    • Spectral integral representation of the reflected CW.
    • Laurent series expansion of the Fresnel coefficient.
    • Analytical integration of the reflected-wave spectral integral.
    • Comparison with numerical quadrature methods.

    Main Results:

    • An exact analytical solution for the reflected cylindrical wave was derived.
    • The solution involves a combination of Bessel and Anger-Weber functions.
    • The analytical solution shows excellent agreement with numerical results.
    • A novel physical interpretation linking the solution to the image principle was established.

    Conclusions:

    • The developed analytical method provides an accurate and insightful approach to reflected cylindrical wave analysis.
    • The findings enhance the understanding of wave reflection phenomena at interfaces.
    • This work bridges mathematical solutions with physical principles in wave optics.