Long‐time quasilinear evolution of the free‐electron laser instability for a relativistic electron beam propagating through a helical magnetic wiggler

The long‐time quasilinear development of the free‐electron laser instability is investigated for a tenuous electron beam propagating in the z direction through a helical wiggler field B 0=−B̂ cos k 0 z ê x −B̂ sin k 0 z ê y . The analysis neglects longitudinal perturbations (δφ≂0) and is based on the nonlinear Vlasov–Maxwell equations for the class of beam distributions of the form f b (z,p,t) =n 0δ(P x )δ(P y )G(z,p z ,t), assuming ∂/∂x=0=∂/∂y. The long‐time quasilinear evolution of the system is investigated within the context of a simple ‘‘water‐bag’’ model in which the average distribution function G 0( p z ,t) =(2L)− 1∫ L −L d z G(z,p z ,t) is assumed to have the rectangular form G 0( p z ,t) =[2Δ(t)]− 1 for ‖p z −p 0(t)‖ ≤Δ(t), and G 0( p z ,t) =0 for ‖p z −p 0(t)‖ >Δ(t). Making use of the quasilinear kinetic equations, a coupled system of nonlinear equations is derived which describes the self‐consistent evolution of the mean electron momentum p 0(t), the momentum spread Δ(t), the amplifying wave spectrum ‖H k (t)‖2, and the complex oscillation frequency ω k (t) +iγ k (t). These coupled equations are solved numerically for a wide range of system parameters, assuming that the input power spectrum P k (t=0) is flat and nonzero for a finite range of wavenumber k that overlaps with the region of k space where the initial growth rate satisfies γ k (t=0) >0. To summarize the qualitative features of the quasilinear evolution, as the wave spectrum amplifies it is found that there is a concomitant decrease in the mean electron energy γ0(t)m c 2=[m 2 c 4+e 2 B̂2/k 2 0 +p 2 0(t)c 2]1 / 2, an increase in the momentum spread Δ(t), and a downshift of the growth rate γ k (t) to lower k values. After sufficient time has elapsed, the growth rate γ k has downshifted sufficiently far in k space so that the region where γ k >0 no longer overlaps the region where the initial power spectrum P k (t=0) is nonzero. Therefore, the wave spectrum saturates, and γ0(t) and Δ(t) approach their asymptotic values.