An electron beam traveling through a waveguide Raman uphyphen;converts a microwave traveling in the opposite direction, thus generating a coherent laser radiation. The frequency of the laser is tunable by the relativistic factor ggr; of the beam. A theory of this process is developed for an electron beam of arbitrary size. The pump and the laser eigenmodes are determined by the size of the waveguide, whereas the beam mode is confined to exist within the beam. The coupled eigenmode equations for the daughter waves are solved by firsthyphen;order perturbation technique. To simplify the problem, a nonmagnetized relativistic electron beam in which the electron density is parabolic with a maximum in the center (on the axis) is considered. An analytic expression for the growth rate of the scattered wave is obtained. It is found that the growth rate is roughly (a/b)1/2times the growth rate for stimulated Raman scattering in an infinite plasma with a uniform pump, where 2bis the separation of conducting planes andais the halfhyphen;width of beam mode. This is because the mode extent of the sideband wave extends to a width sim;b, whereas the region of parametric instability is sim;a. Our results show that for a given value ofb/a, the growth rate becomes maximum for a particular value of pump frequency. We plot our results for the growth rate of the TE1mode of the scattered wave as a function of the pump wave frequency.
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