ISSN:
1089-7666
Source:
AIP Digital Archive
Topics:
Physics
Notes:
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=−Bˆ cos k0zeˆx−Bˆ sin k0zeˆy. The analysis neglects longitudinal perturbations (δφ(approximately-equal-to)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+e2Bˆ2/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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.865415
