The nonlinear analysis of self-field effects in free-electron lasers

A model of the self-fields associated with the charge density and current of the electron beam is incorporated into three-dimensional nonlinear formulations of the interaction in free-electron lasers for both planar and helical wiggler configurations. The model assumes the existence of a cylindrically symmetric electron beam with a flat-top density profile and a uniform axial velocity, and the self-electric and self-magnetic fields are determined from Poisson’s equation and Ampère’s law. Diamagnetic and paramagnetic effects due the electron beam interaction with the wiggler field are neglected; hence, the model breaks down when the wiggler-induced transverse displacement is comparable to the beam radius. The nonlinear formulations are based upon the arachne and wigglin codes, which represent slow-time-scale formulations for the evolution of the amplitudes and phases of a multimode superposition of vacuum waveguide modes. The electron dynamics in these codes are treated by means of the complete three-dimensional Lorentz force equations, and the representations for the self-fields are incorporated directly into this formulation. The results of the simulations are compared directly with an experiment at Lawrence Livermore National Laboratory based upon a planar wiggler and experiments at the Massachusetts Institute of Technology and the Naval Research Laboratory, which employed helical wigglers. These experiments employed intense electron beams with current densities of 200–1200 A/cm2 and comparable space-charge depressions of Δγself/γ0≊0.53%–0.78% across the beam. The simulations are in reasonable agreement with the experiments, and indicate that the self-fields tend to (1) reduce saturation efficiencies and (2) enhance beam spreading depending upon the magnitude of external beam focusing.