A solution of the vector problem of diffraction of a plane electromagnetic wave by a finite cone

We have elaborated the linear and nonlinear theories of monoenergetic electron beam instability in a hybrid ring structure. Both the vacuum and plasma modes are possible. The type a realized mode depends on the beam parameters, ring waveguide geometry, and plasma density. In the vacuum mode, the instability is dissipative by nature, which is connected with the existence of SCW energy losses by excitation of a fast SCW. In the plasma mode, the beam instability (auto modulation) is stimulated by the fields induced in plasma. When the bunches formed in this way are propagating along the diffraction grating that consists of the system of periodic metallic rings, the bunches are emitting EMW. If the EMW radiative condition becomes broken, the excitation process comes to the end. The transformation efficiency increases when relativistic beams are used. Therefore, the usage of ring structure is especially efficient in the SHF band (in particular, for mm wavelengths)