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    Area of Science:

    • Optics and Photonics
    • Plasma Physics
    • High-Intensity Laser Physics

    Background:

    • Conventional focusing optics have limitations in achieving ultra-high laser intensities.
    • Relativistic flying mirrors offer a novel approach to laser focusing.
    • The 4π-spherical focusing scheme is crucial for high-intensity applications.

    Purpose of the Study:

    • To develop a theoretical formalism for laser field configuration focused by a 4π-spherical scheme.
    • To analyze the impact of phase errors on the focused electromagnetic field.
    • To provide mathematical expressions for femtosecond laser pulses and continuous waves.

    Main Methods:

    • Theoretical development of a focusing formalism.
    • Application of the 4π-spherical focusing scheme.
    • Mathematical derivation of focused field expressions using spherical Bessel functions and harmonics.

    Main Results:

    • A theoretical framework describing 4π-spherical focusing with phase errors was established.
    • The focused field configuration was expressed as a linear combination of spherical Bessel functions and spherical harmonics.
    • Three-dimensional intensity distributions near focus were visualized for fields with phase errors.

    Conclusions:

    • Relativistic flying mirrors can achieve laser intensities exceeding conventional methods.
    • The developed theory accurately describes focused fields, including those with phase errors.
    • The findings are applicable to high-power femtosecond laser systems and continuous wave scenarios.