# Asymptotic Theory of the Ponderomotive Dynamics of an Electron Driven by a Relativistically Intense Focused Electromagnetic Envelope

## TL;DR

This paper develops an asymptotic theoretical framework to describe the relativistic ponderomotive effects on electrons driven by intense focused electromagnetic pulses, providing insights into electron scattering and energy distribution.

## Contribution

It introduces a rigorous asymptotic expansion method for modeling relativistic electron dynamics in intense laser fields, including phase-averaged solutions and scattering predictions.

## Key findings

- Predicts axially symmetric electron scattering patterns.
- Shows energy-angle dependence due to adiabatic invariance.
- Describes distribution of hot and cold electron scatter components.

## Abstract

A formalism for describing relativistic ponderomotive effects, which occur in the dynamics of an electron driven by a focused relativisticaly intense optical envelope, is established on the basis of a rigorous asymptotic expansion of the Newton and Maxwell equations in a small parameter proportional to the ratio of radiation wavelength to beam waist. The pertinent ground-state and first order solutions are generated as functions of the electron proper time with the help of the Krylov-Bogolyubov technique, the equations for the phase-averaged components of the ground state arising from the condition that the first-order solutions sustain non-secular behaviour. In the case of the scattering of a sparse electron ensemble by a relativistically intense laser pulse with an axially symmetric transverse distribution of amplitude, the resulting ponderomotive model further affords averaging over the random initial directions of the electron momenta and predicts axially symmetric electron scatter. Diagrams of the electron scatter directionality relative to the optical field propagation axis and energy spectra within selected angles are calculated from the compact ponderomotive model. The hot part of the scatter obeys a clear energy-angle dependence stemming from the adiabatic invariance inherent in the model, with smaller energies allocated to greater angular deviations from the field propagation axis, while the noise-level cold part of the scatter tends to spread almost uniformly over a wide range of angles. The allowed energy diapasons within specific angular ranges are only partially covered by the actual high-energy electron scatter.