INFLUENCE OF THE TRAPPED-ELECTRON DISTRIBUTION ON THE SIDE-BAND INSTABILITY IN A HELICAL WIGGLER FREE-ELECTRON LASER

Use is made of the Vlasov-Maxwell equations to investigate detailed properties of the sideband instability for a helical wiggler free-electron laser with wiggler wavelength λm = 2π/k0 = const and normalized wiggler amplitude aw = eB̂w/mc 2k0 = const. The model describes the nonlinear evolution of a right circularly polarized primary electromagnetic wave with frequency ωs, wave number ks, and slowly varying amplitude âs (z,t) and phase δs (z,t) (eikonal approximation). The coupled Vlasov and field-evolution equations are analyzed in the ponderomotive frame ("primed" variables) moving with velocity vp = ωs/(ks + k0) relative to the laboratory. Detailed properties of the sideband instability are investigated for small-amplitude perturbations about a quasi-steady state characterized by an equilibrium electron distribution f0(γ0′) and a primary electromagnetic wave with constant amplitude â s0 = const (independent of z′ and t′) and slowly varying phase δs0(z'). A formal dispersion relation is derived for perturbations about a general equilibrium distribution f0(γ0′) that may include both trapped and untrapped electrons. For the case where only trapped electrons are present, the dispersion relation is reduced to a simple analytical form. Detailed properties of the sideband instability are investigated for the case where the trapped electrons uniformly populate the ponderomotive potential up to an energy γM′≤γ̂′+, where γ̂′+ is the energy at the separatrix. Analysis of the dispersion relation shows that the maximum energy of the trapped-electron population (γ′M) significantly affects detailed stability properties in the strong-pump and intermediate-pump regimes. © 1990 American Institute of Physics.