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    Area of Science:

    • Optics and Photonics
    • Quantum Mechanics
    • Mathematical Physics

    Background:

    • The Wigner distribution function (WDF) is a key tool for analyzing quantum states and optical beams in phase space.
    • Bessel-Gauss (BG) beams are a class of optical beams with unique propagation characteristics, combining properties of Bessel beams and Gaussian beams.
    • Understanding the WDF of BG beams is crucial for applications in optical imaging, laser physics, and quantum information.

    Purpose of the Study:

    • To derive and present three novel, equivalent mathematical expressions for the Wigner distribution function (WDF) of a Bessel-Gauss (BG) beam.
    • To explore the mathematical structure of the WDF for BG beams, revealing connections to Laguerre-Gauss functions, Bessel functions, and Fourier series.
    • To analyze the phase-space properties, symmetries, and limiting cases of the WDF for BG beams.

    Main Methods:

    • Derivation of three distinct mathematical representations for the WDF of a BG beam.
    • Utilizing Laguerre-Gauss functions in double summations for the first expression.
    • Employing modified Bessel functions in a single summation for the second expression.
    • Developing a compact integral representation for the third expression, linking the WDF to Fourier series coefficients.

    Main Results:

    • Three equivalent expressions for the Wigner distribution function (WDF) of a Bessel-Gauss (BG) beam were successfully derived.
    • The first expression involves double summations of Laguerre-Gauss functions, while the second uses a single summation of modified Bessel functions.
    • The third expression is a compact integral representation showing the WDF of a BG beam is proportional to the mth coefficient of a complex Fourier series.

    Conclusions:

    • The derived expressions provide versatile tools for analyzing the WDF of BG beams.
    • The WDF of BG beams can be expressed in terms of fundamental mathematical constructs like Laguerre-Gauss functions, Bessel functions, and Fourier series.
    • Further analysis of symmetries and phase-space quadratic forms (Hamiltonian, Lagrangian, orbital angular momentum) deepens the understanding of BG beam characteristics.