Multimode simulations without particles in the quasi-opticalgyrotron

A set of coupled nonlinear differential equations, involving the slow amplitude and phase variation for each mode, is used to simulate the multimode dynamics in the quasi-optical gyrotron. The interaction among various modes is mediated by coupling coefficients of known analytic dependence on the normalized current I, the interaction length μ, and the frequency detunings Δ_i corresponding to the competing frequencies ω_i. The equations include all the possible resonant combinations of up to four different frequencies, ω_i-ω_j+ω _k-ω_l≃0, among a set of N participating modes, keeping terms up to fifth order in the wave amplitudes. The formalism is quite general and can be used to study mode competition, the existence of a final steady state and its stability, and its accessibility from given initial conditions. It is shown that when μ/β_⊥≫1, μ can be eliminated as an independent parameter. The control space is then reduced to a new normalized current I and the desynchronism parameters ν_i=Δ_iμ for the interacting frequencies. Numerical simulations for cold beams of various cross sections demonstrate that ν_i is the most important parameter for the system behavior. Overmoding is not determined by the frequency separation δω among the cavity modes per se, but by the separation among the corresponding desynchronism parameters