Self-consistent Vlasov description of the free electron laser instability

Abstract

A self-consistent description of the free electron laser instability is developed for a relativistic electron beam with uniform density propagating through a helical wiggler field B0=−B̂ cosk0zêx−B̂ sink0zêy. The analysis is carried out for the class of solutions to the Vlasov–Maxwell equations described by fb(z,p,t)=n0δ(Px)δ(Py)G(z,pz,t), where Px and Py are the exact canonical momenta invariants perpendicular to the beam propagation direction. The linearized Vlasov–Maxwell equations lead to an exact matrix dispersion relation which is valid for perturbations about general beam equilibrium G0(Pz) and which includes coupling to arbitrary harmonic number (n) of the fundamental wiggler wavenumber k0. No à priori restriction is made to low beam density (as measured by ω2P/c2k20) or small wiggler amplitude (as measured by ω̂c/ck0=eB̂/γ̄mc2k0). Moreover, no assumption is made that any off-diagonal elements in the matrix dispersion relation are negligibly small. A detailed numerical analysis of the exact dispersion relation is presented for the case of a cold electron beam described by G0(Pz) =δ(pz−p0). It is shown that the instability bandwidth increases rapidly with increasing wiggler amplitude ω̂c/ck0. Moreover, except for very modest values of wiggler amplitude, it is shown that the growth rate calculated from an approximate version of the dispersion relation can be in substantial error for large values of (k+nk0)/k0. Preliminary estimates of the influence of beam thermal effects are also presented.