Kinetic description of harmonic instabilities in a planar wiggler free-electron laser

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Abstract
The linearized Vlasov–Maxwell equations are used to investigate harmonic stability properties for a planar wiggler free‐electron laser (FEL). The analysis is carried out in the Compton regime for a tenuous electron beam propagating in the z direction through the constant‐amplitude planar wiggler magnetic field B0=−Bw cos k0zêx. Transverse spatial variations are neglected (∂/∂x =0=∂/∂y), and the case of an FEL oscillator (temporal growth) is considered. Assuming ultrarelativistic electrons and κ2=a2w/(γ20−1) ≪1, where a2w =e2B2w /m2c4k20 and γ0mc2 is the electron energy, the kinetic dispersion relation is derived in the diagonal approximation for perturbations about general beam equilibrium distribution function G+0(γ0). Because of the wiggler modulation of the axial electron orbits, strong wave–particle interaction can occur for ω≊[k+k0(1+2l)] βFc, where βFc is the axial velocity, ω and k are the wave oscillation frequency and wavenumber, respectively, and l=0, 1, 2, . . . are harmonic numbers corresponding to an upshift in frequency. The strength of the lth harmonic wave–particle coupling is proportional to Kl(b1) =[Jl (b1)−Jl+1 (b1)]2, where b1=(k/8k0)κ2. Assuming that G+0(γ0) is strongly peaked around γ0=γ̂≫1, detailed lth harmonic stability properties are investigated for (a) strong FEL instability corresponding to monoenergetic electrons (Δγ=0), and (b) weak resonant FEL instability corresponding to a sufficiently large energy spread that ‖Im ω/[k+k0(1+2l)] Δvz ‖≪1. For monoenergetic electrons the characteristic maximum growth rate scales as [Kl (b̂1)(1+2l)]1/3, which exhibits a relatively weak dependence on harmonic number l. Here, b̂1= 1/2 [a2w/(2+a2w)] (1+2l). On the other hand, for weak resonant FEL instability, the growth rate scales as Kl (b̂1)/(1+2l), which decreases rapidly for harmonic numbers l≥1.