The free electron laser instability is investigated theoretically for a tenuous relativistic electron beam propagating parallel to a longitudinal wiggler field (B0+δB sin k0z)êz . The stability analysis is based on the linearized Vlasov–Maxwell equations for perturbations about the beam equilibrium f0b =(nb/2πp⊥)δ(p⊥−γbmV⊥) δ(pz−γbmVb). For (ωcb/k0Vb)(δB/B0) of order unity or larger, it is found that the transverse electron orbits experience a strong temporal modulation at harmonics of k0vz≂k0Vb. In the stability analysis, this orbit modulation is manifest by radiation amplification occurring near the simultaneous zeroes of ω=kc and ω−kVb−ωcb=nk0Vb for the general harmonic number n, which corresponds to an upshifted (for n≥1) wavenumber k̂n=(1−Vb/c)−1(ωcb/c+nk0Vb/c). For (ωcb/k0Vb)(δB/B0) >1, the instability is typically broadband with several harmonics excited. Moreover, depending on beam and field parameters, the characteristic maximum growth rate can be substantial with Im ω/ck0=(3/2)1/2 ×[(V2⊥/4c2)(ω2pb/c2k20) nJ2n(ωcbδB/k0VbB0)]1/3.
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Davidson et al. (1983) studied this question.
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