Macroclump Model of the Nonlinear Evolution of the Sideband Instability in a Helical Wiggler Free Electron Laser

The nonlinear evolution of a helical wiggler free electron laser is investigated within the framework of a macroclump model for the trapped electrons. The model describes the nonlinear evolution of a right-circularly polarized electromagnetic wave with frequency w, and wavenumber k,, and slowly varying amplitude &,(z, t) and phase 8 ,(z, t) (eikonal approximation). The model further assumes that the trapped electrons can be treated as tightly bunched macroclumps that interact coherently with the radiation field. The analysis is carried out in the ponderomotive frame, which leads to a substantial simplification in both the analytical and numerical studies. As a first application, the nonlinear evolution of the primary signal is examined when 98' = 0 (no spatial variation of the wave amplitude and phase). The evolution equations are reduced to quadrature, and the maximum excursion of the wave amplitude a,, is calculated analytically. Subsequently, the nonlinear evolution of the sideband instability is investigated, making use of the equations describing the self-consistent evolution of the wave amplitude &, and phase 8,, which vary slowly with both space and time, together with the macroclump orbit equation. In the present analysis, the sideband signals are treated as perturbations (not necessarily small) about a constant-amplitude (&0 = const) primary electromagnetic wave with slowly varying phase 6'(z'). The coupled orbit and field equations are investigated analytically and numerically over a wide range of system parameters to determine detailed scaling properties of the sideband instability. The results of the present analysis suggest that free electron lasers operating with system parameters corresponding to the strong-pump regime [(f'/I, )'/4 > 1] are least vulnerable to the sideband instability. Moreover, the nonlinear evolution of the sideband instability is investigated numerically for system parameters corresponding to the Los Alamos free electron laser experiment. In several aspects, the numerical results are found to be in good qualitative agreement with the experimental results.