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    Area of Science:

    • Optics and Photonics
    • Laser Physics
    • Quantum Optics

    Background:

    • Geometric modes in lasers are crucial for various applications.
    • Understanding vortex structures is key to controlling laser beam properties.
    • Previous studies have explored different laser modes, but circularly geometric modes require specific investigation.

    Purpose of the Study:

    • To theoretically demonstrate the circularly geometric mode using the inhomogeneous Helmholtz equation.
    • To analyze the factors determining the vortex structures of these modes.
    • To experimentally generate and verify the vortex structures of circularly geometric modes.

    Main Methods:

    • Theoretical analysis solving the inhomogeneous Helmholtz equation with pump distribution.
    • Numerical calculations of transverse lasing modes and cavity degeneracy.
    • Experimental generation using a selectively pumped solid-state laser and a π/2 mode converter.
    • Interferometric analysis using a Mach-Zehnder interferometer to probe vortex structures.

    Main Results:

    • The circularly geometric mode can be theoretically solved from the inhomogeneous Helmholtz equation.
    • Vortex structures are determined by the minimum order, total number, and degeneracy of transverse lasing modes.
    • Experimentally generated circularly geometric modes exhibit vortex structures consistent with theoretical predictions.
    • Interference patterns confirm the predicted vortex structures, validating the theoretical analysis.

    Conclusions:

    • The theoretical framework accurately describes the generation and properties of circularly geometric modes.
    • Experimental validation confirms the ability to generate and analyze these modes in solid-state lasers.
    • The findings provide a foundation for controlling and utilizing laser vortex structures in optical systems.