Influence of intense equilibrium self-fields on the cyclotron maser instability in high-current gyrotrons

Abstract

The linearized Vlasov–Maxwell equations are used to investigate the influence of intense equilibrium self-fields on the cyclotron maser instability. A uniform density (n̂b) electron beam propagates parallel to an applied axial magnetic field B0êz with average axial velocity βbc. The particle trajectories are calculated including the influence of the radial self-electric field and the azimuthal self-magnetic field. Moreover, the linearized Vlasov–Maxwell equations are analyzed for body-wave perturbations localized to the beam interior, assuming electromagnetic perturbations about the equilibrium distribution function f0b=(n̂b/2pπ⊥) ×δ(p⊥−γbmV⊥) ×δ(pz−γbmβbc). Near the beam axis (ω2pbr2/c2≪1), it is found that the transverse electron motion is biharmonic, with oscillatory components at the frequencies ω+b and ω−b defined by ω±b =(ωcb/2) ×{1±[1−(2ω2pb/ω2cb) ×(1−β2b)]1/2}. Similarly, the electromagnetic dispersion relation for waves propagating parallel to B0êz exhibits two types of resonance conditions: a high-frequency resonance (HFR) corresponding to ω−kβbc=ω+b, and a low-frequency resonance (LFR) corresponding to ω−kβbc=ω−b. Both the HFR branch and the LFR branch exhibit instability, with detailed stability properties depending on the value of the self-field parameter s=ω2pb/ω2cb. Moreover, the LFR branch is entirely caused by self-field effects, whereas the HFR branch represents a generalization of the conventional cyclotron maser mode to include self-field effects. The full dispersion relation is analyzed numerically, and the real oscillation frequency ωr=Re ω and growth rate ωi=Im ω are calculated for both types of modes over a wide range of system parameters s, β⊥, βb, and kc/ωcb. Analytic estimates are made of the cyclotron maser growth properties in circumstances where β2⊥γ2z/2 is treated as a small parameter. [Here, γz=(1−β2b)−1/2. ] It is found that the maximum growth rate is given by ωi=(2γ2z)−1 ×[s(2β2⊥γ4z−s)]1/2ωcb, which occurs for wavenumber k=km=γ2zβbωcb/c. As the beam density (s) is increased, the growth rate ωi increases to the maximum value ωmaxi =γ2zβ2⊥ωcb/2 for beam density s=sm=β2⊥γ4z. As s is increased beyond sm, the growth rate ωi decreases to zero for s=s0=2β2⊥γ4z. Similarly, the instability bandwidth Δk =(2γzωcb/c)[γ2z−β−1⊥ ×(s/2)1/2]1/2 approaches zero as s approaches s0.