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A linear theory for a free electron laser(FEL) with a three dimensional coaxial wiggler and an axial magnetic field is considered. A coaxial waveguide is partially filled with a relativistic and annular electron beam. Variation of the wiggler in the radial direction is taken into account, Parametric decay of the wiggler pump wave, in the beam frame, into a space-charge wave and an electric-magnetic (EH) waveguide mode is analyzed. With relativistic treatment of the electron oscillations in the wiggler field, a nonlinear wave equation is derived which describes the interaction between the wiggler field and the excited waves. To analyze the spatial amplification of the excited waves the relativistic fluid model is employed. A formula for the spatial growth of the excited eigenmodes is derived with the lab-frame temporal growth taken to be zero. The electromagnetic effects of the space-charge wave as well as the effects of the longitudinal components of the backscattered wave are considered. The boundry effects of the annular electron beam and the waveguid wall effects are retained.
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