It is known that, in principle, an electron beam disturbs a spatial structure of the res- onator RF field. This effect is negligibly small when the resonator Q-factor is large enough and it becomes important as Q diminishes. In conventional gyrotrons the resonators are opened in an axial direction. Therefore, if such resonators are long enough and their diameter is relatively small, the electron beam first starts to affect the axial structure of the RF field that has been studied elsewhere [1,2]. In short resonators with a large diameter the influence of the electron beam on the transverse structure of the RF field may also be important. This effect is considered below. Consider a cylindrical resonator in which a thin annular electron beam is placed. Suppose that due to the effect of an electron beam the fields inside the beam (region 1) are different from those outside (region 2). In the vicinity of the beam these fields satisfy the boundary conditions , ,
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