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    This study presents new mathematical formulas for 4π-spherical focusing, achieving ultra-high intensity electromagnetic fields. These findings advance high-intensity physics and particle production studies.

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    Area of Science:

    • Optics and Photonics
    • High-Intensity Physics
    • Quantum Electrodynamics

    Background:

    • 4π-spherical focusing is a technique to concentrate light with high numerical aperture.
    • Understanding focused electromagnetic fields is crucial for high-energy density physics and particle creation.

    Purpose of the Study:

    • To derive generalized mathematical formulas for focused electric and magnetic fields using 4π-spherical focusing.
    • To investigate the spatial distributions and intensity of these fields for different electromagnetic wave modes.
    • To analyze applications such as electron-positron pair production.

    Main Methods:

    • Investigated focused fields within the framework of diffraction optics.
    • Derived generalized mathematical formulas for spatial field distributions.
    • Analyzed transverse magnetic (TM) and transverse electric (TE) modes, with and without orbital angular momentum.
    • Calculated peak intensity for TM mode focusing.

    Main Results:

    • Developed singularity-free mathematical formulas for focused fields.
    • Demonstrated finite field strength and electromagnetic energy conditions.
    • Achieved peak intensity of 10^26 W/cm^2 for 100 PW laser power at 800 nm using TM mode.
    • Analyzed electron-positron pair production via Schwinger mechanism.

    Conclusions:

    • The derived formulas accurately describe 4π-spherical focusing.
    • TM mode 4π-spherical focusing yields the highest field strength and intensity.
    • The study provides a theoretical basis for high-intensity laser-matter interactions and particle physics.