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A fully self-consistent theory of the free-electron laser is derived in the collective regime which includes all transverse variations in the wiggler field as well as the effects of a finite waveguide geometry. A set of coupled differential equations is found which describes an arbitrary radial beam profile, and a dispersion equation is obtained under the assumption of a thin monoenergetic beam which is solved numerically for the growth rates of the TE11 and TM11 modes in a cylindrical waveguide.
H. P. Freund andA. K. Ganguly
"Three-Dimensional Theory Of The Raman Free-Electron Laser", Proc. SPIE 0453, Free-Electron Generators of Coherent Radiation, (16 February 1984); https://doi.org/10.1117/12.947122
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H. P. Freund, A. K. Ganguly, "Three-Dimensional Theory Of The Raman Free-Electron Laser," Proc. SPIE 0453, Free-Electron Generators of Coherent Radiation, (16 February 1984); https://doi.org/10.1117/12.947122