and

A nonlinear analysis of Free-Electron Laser amplifiers in three-dimensions is described. A set of coupled nonlinear differential equations is derived which governs the self- consistent evolution of either the TE or TM waveguide modes and the trajectories of an ensemble of electrons subject to the radiation and wiggler fields. In particular, we consider magnetostatic field configurations consisting of either (1) a planar polarized wiggler in conjunction with a rectangular waveguide or (2) a helical wiggler and axial guide field combination in conjunction with a cylindrical waveguide. Both self-field and space-charge effects are neglected in the formulation, and the analysis is valid for the high-gain Compton (sometimes called the strong pump) regime. The analysis is multi-mode in the sense that an arbitrary number of either TE or TM modes may be included. The analysis is in good agreement with Free-Electron Lasers experiments at Lawrence Livermore National Laboratory and the Massachusetts Institute of Technology which employ each of these configurations. 1