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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 **B** ^{0}=−*B* _{ w } cos *k* _{0} *z* **e**̂_{ x }. Transverse spatial variations are neglected (∂/∂*x* =0=∂/∂*y*), and the case of an FEL oscillator (temporal growth) is considered. Assuming ultrarelativistic electrons and κ^{2}=*a* ^{2} _{ w }/(γ^{2} _{0}−1) ≪1, where *a* ^{2} _{ w } =*e* ^{2} *B* ^{2} _{ w } /*m* ^{2} *c* ^{4} *k* ^{2} _{0} and γ_{0} *m* *c* ^{2} is the electron energy, the kinetic dispersion relation is derived in the diagonal approximation for perturbations about general beam equilibrium distribution function *G* ^{+} _{0}(γ_{0}). Because of the wiggler modulation of the axial electron orbits, strong wave–particle interaction can occur for ω≊[*k*+*k* _{0}(1+2*l*)] β_{ F } *c*, where β_{ F } *c* 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 *l*th harmonic wave–particle coupling is proportional to *K* _{ l }(*b* _{1}) =[*J* _{ l } (*b* _{1})−*J* _{ l+1} (*b* _{1})]^{2}, where *b* _{1}=(*k*/8*k* _{0})κ^{2}. Assuming that *G* ^{+} _{0}(γ_{0}) is strongly peaked around γ_{0}=γ̂≫1, detailed *l*th 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*+*k* _{0}(1+2*l*)] Δ*v* _{ z } ‖≪1.

For monoenergetic electrons the characteristic maximum growth rate scales as [*K* _{ l } (*b*̂_{1})(1+2*l*)]^{1} ^{/} ^{3}, which exhibits a relatively weak dependence on harmonic number *l*. Here, *b*̂_{1}= 1/2 [*a* ^{2} _{ w }/(2+*a* ^{2} _{ w })] (1+2*l*). On the other hand, for weak resonant FEL instability, the growth rate scales as *K* _{ l } (*b*̂_{1})/(1+2*l*), which decreases rapidly for harmonic numbers *l*≥1.

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