A free-electron laser with an electron ring and electromagnetic pumping wave

A free-electron laser with an electron ring and electromagnetic pumping wave is studied. The drifting and rotating electron ring of a large radius comes from the magnetic cusp and goes through a cylindrical waveguide interacting with the TE mode of the waveguide. By making use of the Vlasov distribution theory and a three-dimensional wave equation, the dispersion relation of the scattered wave in the Compton region is deduced. In the numerical analysis, the growth rate and radiation frequency of the scattered wave as a function of the guided magnetic field, electron beam energy, electron rotating ratio, and radial position of the electron ring are discussed.