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

Publisher Website
Google Scholar
Abstract
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 B0=−B̂ cos k0zêx−B̂ sin k0zêy. 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 fb(z,p,t) =n0δ(Px)δ(Py)G(z,pz,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 G0( pz,t) =(2L)−1∫L−L dz G(z,pz,t) is assumed to have the rectangular form G0( pz,t) =[2Δ(t)]−1 for ‖pz−p0(t)‖ ≤Δ(t), and G0( pz,t) =0 for ‖pz−p0(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 p0(t), the momentum spread Δ(t), the amplifying wave spectrum ‖Hk(t)‖2, and the complex oscillation frequency ωk(t) +iγk(t). These coupled equations are solved numerically for a wide range of system parameters, assuming that the input power spectrum Pk(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 γk(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)mc2=[m2c4+e2B̂2/k20 +p20(t)c2]1/2, an increase in the momentum spread Δ(t), and a downshift of the growth rate γk(t) to lower k values. After sufficient time has elapsed, the growth rate γk has downshifted sufficiently far in k space so that the region where γk >0 no longer overlaps the region where the initial power spectrum Pk(t=0) is nonzero. Therefore, the wave spectrum saturates, and γ0(t) and Δ(t) approach their asymptotic values.