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Mode Competition in the Quasioptical Gyrotron

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Memorandum rept.

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A set of equations describing the nonlinear multimode dynamics in the Quasioptical Gyrotron is derived. These equations, involving the slow amplitude and phase variation for each mode, result from an expansion of the nonlinear induced current up to fifth order in the wave amplitude. The interaction among various modes is mediated by coupling coefficients, of known analytic dependence on the normalized current I, the interaction length mu, and the frequency detunings Delta sub i corresponding to the competing frequencies Omega 1. The particular case when the modes form triads with frequencies Omega 1 Omega 3 - 2 Omega 2 approx 0 is examined in more detail. The equations are quite general and can be used to study mode competition, the existence of a final steady state, its stability, as well as its accessibility from given initial conditions. It is shown that when mubeta perp. 1, mu can be eliminated as an independent parameter. The control space is then reduced to a new normalized current I and the desynchronism parameters nui Delta sub i mu for the interacting frequencies. Each coupling coefficient G sub ij is written as G sub ij I S sub ij G sub ij nui, nuj, where the nonlinear filling factor S sub ij, carrying the information of the beam current spatial profile, can be computed independently. Therefore, it suffices to compute table of G sub ij as functions of nu1, nu2 and nu3 once to cover the parameter space. Results for a cold beam are presented here.

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  • Quantum Theory and Relativity

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