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A three-dimensional theory of the stimulated Raman scattering mechanism for a coaxial free-electron laser (FEL) is developed. The relativistic mass correction is included in the analysis and the boundary effects through the radial dependencies of the wave field quantities are taken into account. The configuration consists of a coaxial cylindrical waveguide with annular electron beam. Parametric decay of the wiggler pump wave, in the beam frame, into a space-charge wave and an electric-magnetic waveguide mode is analyzed in three dimensions. A nonlinear wave equation for the three-wave interaction is used to obtain a formula for the spatial growth rate of the excited eigenmodes. Numerical analysis is conducted to study the growth rate and required relativistic factor. Growth rate is found to be substantially larger than in the case of one-dimensional coaxial wiggler. Significantly higher growth rate is also obtained compared to a similar FEL with a helical wiggler instead of the coaxial wiggler.
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