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^{1}and Y. Z. Yin

^{2}

### Abstract

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 **B** _{0}=−*B*̂ cos *k* _{0} *z* **e**̂_{ x }−*B*̂ sin *k* _{0} *z* **e**̂_{ y }. 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 *f* _{ b }(*z*,**p**,*t*) =*n* _{0}δ(*P* _{ x })δ(*P* _{ y })*G*(*z*,*p* _{ z },*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 *G* _{0}( *p* _{ z },*t*) =(2*L*)^{−} ^{1}∫^{ L } _{−L } *d* *z* *G*(*z*,*p* _{ z },*t*) is assumed to have the rectangular form *G* _{0}( *p* _{ z },*t*) =[2Δ(*t*)]^{−} ^{1} for ‖*p* _{ z }−*p* _{0}(*t*)‖ ≤Δ(*t*), and *G* _{0}( *p* _{ z },*t*) =0 for ‖*p* _{ z }−*p* _{0}(*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 *p* _{0}(*t*), the momentum spread Δ(*t*), the amplifying wave spectrum ‖*H* _{ k }(*t*)‖^{2}, and the complex oscillation frequency ω_{ k }(*t*) +*i*γ_{ k }(*t*).

These coupled equations are solved numerically for a wide range of system parameters, assuming that the input power spectrum *P* _{ k }(*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 γ_{ k }(*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 γ_{0}(*t*)*m* *c* ^{2}=[*m* ^{2} *c* ^{4}+*e* ^{2} *B*̂^{2}/*k* ^{2} _{0} +*p* ^{2} _{0}(*t*)*c* ^{2}]^{1} ^{/} ^{2}, an increase in the momentum spread Δ(*t*), and a downshift of the growth rate γ_{ k }(*t*) to lower *k* values. After sufficient time has elapsed, the growth rate γ_{ k } has downshifted sufficiently far in *k* space so that the region where γ_{ k } >0 no longer overlaps the region where the initial power spectrum *P* _{ k }(*t*=0) is nonzero. Therefore, the wave spectrum saturates, and γ_{0}(*t*) and Δ(*t*) approach their asymptotic values.

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