A particle in a laser field with fixed direction and infinite wavefront has analytic solutions in both the quantum and classical realms.
These solutions allow for the possibility of electromagnectic wavepackets modulated in the direction of motion.
The quantum solution of the Dirac equation was found by Volkov in the early days of quantum mechanics!
Volkov D M 1935 Z. Physik 94, 250
Contrary to expectations, the classical solutions appeared much later. The following book contains a very complete analysis of the classical case
Electrodynamics: A Modern Geometric Approach (Progress in Mathematical Physics) by William E. Baylis, Birkhäuser; Corrected edition (January 12, 2004)
The trajectories for linear and circular polarization can be seen in the following movies:
An interactive Mathematica cdf document can be downloaded from
The general theory is part of the lectures notes at
News: The case with linear polarization was published at the Wolfram Demonstrations Project:
Renan Cabrera"Classical Relativistic Particle in a Linearly Polarized Laser Field"
Wolfram Demonstrations Project
Published: September 15, 2017