Intensity patterns of a focused electromagnetic spherical wave with aberration

The laser pulse focused by a relativistic flying parabolic mirror can exceed the laser intensity focused by conventional physical focusing optics. Depending on the Lorentz γ-factor, the focal length of the relativistic flying mirror in the boosted frame of reference becomes much shorter than the incident beam size. The 4π-spherical focusing scheme is applied to describe such a focused field configuration. In this paper, a theoretical formalism has been developed to describe the field configuration focused by the 4π-spherical focusing scheme with an arbitrary phase error of an incident electromagnetic wave. The focused field configuration is described by the linear combination of the product of the spherical Bessel function and the spherical harmonics, resulting in the same expression as the multipole radiation. The mathematical expression showing the focused field for the femtosecond laser pulse, as well as the continuous wave, has been derived for the application to the femtosecond high-power laser. We show the three-dimensional intensity distribution near focus for the 4π-spherically focused electromagnetic field with phase error.