Cross-field free electron laser instability for a tenuous electron beam

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Abstract
The free electron laser instability is investigated for a tenuous circulating electron beam propagating perpendicular to a uniform magnetic field B0êz and transverse wiggler field modeled by Bw sin k0yêx in planar geometry. Unlike the rippled‐field magnetron which operates at Brillouin flow, the present analysis assumes a low‐density electron beam with ω2p ≪Ω2c. Making use of a macroscopic cold‐fluid model for the electrons coupled with Maxwell’s equations for the fields, it is found that wave perturbations with ordinary‐mode polarization (δE∥B0 and δB⊥B0) amplify with characteristic maximum growth rate Im(δω)=ωp (Ωw/2ck0) and emission frequency ωr =(1+βE)γ2Ek0VE. Here, Ωw =eBw/γEmc, βE =VE/C, γE =(1−β2E)−1/2, and VE =−cE0/B0, where E0 is the applied electric field across the anode–cathode gap. Depending on the size of Ωw/ck0, the characteristic exponentiation time ω−1p(Ωw/2ck0)−1 for the cross‐field free electron laser instability can be relatively short in units of ω−1p.