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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=−B cos k0zex−B sin k0zey. 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 oscilla...

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