Multimode nonlinear analysis of free-electron laser amplifiers in three dimensions

The nonlinear evolution of a free-electron laser (FEL) amplifier is investigated for a configuration in which an electron beam propagates through an overmoded rectangular waveguide in the presence of a planar wiggler with parabolically tapered pole pieces. The analysis is fully three dimensional and describes the evolution of an arbitrary number of resonant TE and/or TM modes of the rectangular guide as well as the trajectories of an ensemble of electrons. Numerical simulations are conducted for parameters consistent with the 35-GHz amplifier experiment performed by Orzechowski and co-workers [Phys. Rev. Lett. 54, 889 (1985); 57, 2172 (1986)], in which the ${\mathrm{TE}}_{01}$, ${\mathrm{TE}}_{21}$, and ${\mathrm{TM}}_{21}$ modes were observed. The theory is found to be in good agreement with the experiment. Surprisingly, comparison with a single-mode analysis shows that the enhancement of the efficiency of the ${\mathrm{TE}}_{01}$ mode obtained by means of a tapered wiggler is significantly greater (as well as being in substantial agreement with the experiment) when the ${\mathrm{TE}}_{21}$ and ${\mathrm{TM}}_{21}$ modes are included in the simulation.