  We investigate the radiation from a bunch of relativistic electrons moving
along the cylindrical waveguide axis, assuming that the waveguide is partially
loaded by a medium with periodic dielectric permittivity and magnetic
permeability. The spectral distribution of the radiation energy flux through
the cross section of the waveguide is studied at large distances from the
medium. The analysis is based on the corresponding exact solution of Maxwell
equations for the case of a single electron moving along the waveguide axis.
The results of numerical calculations are presented in the special case of
layered medium consisting of a finite number of dielectric plates separated by
vacuum gaps. We show that under certain conditions on the problem parameters
the quasi-coherent Cherenkov radiation generated by the electron bunch inside
the plates is self-amplified at certain waveguide modes. A visual explanation
of this phenomenon is provided that reproduces the main features to rather good
accuracy.
