Analysis of the sideband instability in the free-electron laser

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Abstract
A full linear stability analysis of the sideband instability in the free-electron laser is presented. All processes affecting the instability are identified and taken into account; these include, in addition to the usual nonparametric process leading to direct excitation of the Stokes and anti-Stokes processes, a parametric four-wave-mixing process in which the sideband modes are coupled through the pump mode, as well as further refractive terms, over and above that due to simple linear refraction, brought about by the presence of the potential wells created by the strong pump mode. The net result is an equation for the growth rate Γ that depends on all parameters describing these processes, and which is therefore more complicated than that usually quoted; moreover, the unstable mode is no longer ‘‘pure’’ Stokes, but is a mixed mode containing both sidebands.