The nonlinear interaction between an electron and multimode fields in an electromagnetically pumped free electron laser

We consider the motion of an electron under the influence of two counterpropagating multimode electromagnetic waves and a longitudinal electrostatic field. General equations of motion are presented from which a simple wavelength scaling condition is deduced for minimizing detrapping effects. An interaction coherence time or a maximum useful interaction length parameter is derived. The full equations of motion are numerically solved for a variety of external and initial conditions.