Asymptotic theory of ponderomotive dynamics of an electron in the field of a focused relativistically intense electromagnetic envelope

Based on a formal asymptotic solution of Maxwell's equations for a field propagating in a vacuum and of the relativistic Newton equation for an electron driven by the corresponding Lorentz force, we construct a description of the relativistic ponderomotive dynamics of an electron in the field of an intense focused electromagnetic envelope. A small parameter for the asymptotic expansion is proportional to the ratio of the radiation wavelength to the focal spot radius. Using the obtained averaged model of ponderomotive dynamics, electron beam scattering patterns are plotted with regard to an angle relative to the electromagnetic field propagation axis for laser pulses with a Gaussian transverse intensity distribution, as well as the energy spectra of scattered electrons corresponding to individual ranges of this angle. Scattered particles are absent in some ranges of the polar angle.