Two-dimensional Vlasov treatment of free-electron laser sidebands

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Abstract
The Kroll–Morton–Rosenbluth equations [IEEE J. Quantum Electron. Q E ‐ 1 7, 1436 (1981)] for a helical‐wiggler free‐electron laser are generalized to treat an electron beam with a prescribed radial density profile and an equilibrium distribution function that is an arbitrary function of the longitudinal action J. The principal approximation is the assumption that betatron frequencies of beam particles are low compared with typical synchrotron frequencies. Vlasov equilibria for finite‐amplitude primary waves with time‐varying phase are calculated for several distribution functions. Using these equilibria, radial eigenvalue equations for the frequency and growth rate of small‐amplitude sidebands are derived and solved numerically. The radial mode structure is found to have no appreciable effect on sideband growth when the beam radius is large compared with [2ks min(Ω0, dφ0/dz)]−1/2, where ks and φ0 are the wavenumber and phase of the primary wave and Ω0 is the maximum synchrotron ‘‘frequency’’ in z of trapped electrons. In these effectively one‐dimensional cases, the dispersion relation depends only on the distribution function and on a dimensionless density parameter η̄=kwa2wω2b/(c2γ3rΩ30i), where kw is the wiggler wavenumber, aw=eAw/(mc2) is the dimensionless wiggler vector potential, ωb is the maximum plasma frequency of the beam, and γr is the Lorentz factor for resonant particles. Both the upper and lower sidebands for a deeply trapped distribution (J≊0) have a maximum growth rate of (31/2/2)(η̄2/2)1/3 for η̄≪1 and (31/2/2)(η̄/2)1/3 for η̄≫1, and distributions with a spread in J invariably show slower sideband growth. For beams with a smaller radius, the radial density variation causes a further reduction in the peak sideband growth rate and narrows the spectrum of unstable modes.