Lawrence Berkeley National Laboratory

The focused field and its intensity distribution achieved by the 4π-spherical focusing scheme are investigated within the framework of diffraction optics. Generalized mathematical formulas describing the spatial distributions of the focused electric and magnetic fields are derived for the transverse magnetic and transverse electric mode electromagnetic waves with and without the orbital angular momentum attribute. The mathematical formula obtained shows no singularity in the field in the focal region and satisfies the finite field strength and electromagnetic energy conditions. The 4π-spherical focusing of the transverse magnetic mode electromagnetic wave provides the highest field strength at the focus and the peak intensity reaches 10²⁶ W/cm² for the laser power of 100 PW at 800 nm wavelength. As an example of using the mathematical formula, the electron-positron pair production via the Schwinger mechanism is analyzed and compared with previous results.