Self-Consistent Kinetic Description of the Free Electron Laser Instability in a Planar Magnetic Wiggler

The linearized Vlasov-Maxwell equations are used to investigate detailed free electron laser (FEL) stability properties for a tenuous relativistic electron beam propagating in the z direction through the planar wiggler magnetic field B0(x) = -Bw, cos k0zêx. Here, Bw = constant is the wiggler amplitude, and λ0 = 2π/k0 = constant is the wiggler wavelength. The theoretical model neglects longitudinal perturbations (δϕ = 0) and transverse spatial variations (∂/∂x = 0 = ∂/∂y). Moreover, the model is based on the Vlasov-Maxwell equations for the class of self-consistent beam distribution functions of the form fb(Z, p, t) = n,bδ(px) δ(Py) G(z, pz, t), where p = γmv is the mechanical momentum, and Py is the canonical momentum in the y direction. For low or moderate electron energy, there can be a sizable modulation of beam equilibrium properties by the wiggler field and a concomitant coupling of the kth Fourier component of the wave to the components k ± 2k0, k ± 4k0, ··· in the matrix dispersion equation. In the diagonal approximation, investigations of detailed stability behavior range from the regime of strong instability (monoenergetic electrons) to weak resonant growth (sufficiently large energy spread). In the limit of ultrarelativistic electrons and very low beam density, the kinetic dispersion relation is compared with the dispersion relation obtained from a linear analysis of the conventional Compton-regime FEL equations. Finally, assuming ultrarelativistic electrons and a sufficiently broad spectrum of amplifying waves, the quasi-linear kinetic equations appropriate to the planar wiggler configuration are presented.