Amplification of undulator radiation in a waveguide

The amplification of stimulated undulator emission from electrons is theoretically investigated, in cases where phase velocity is different from the speed of light. The cases of normal and anomalous Doppler effects are considered for a real free-electron-laser system where the undulator field is excited in a waveguide or a stripline. It is shown that the difference between the phase velocity and the speed of light can lead to a significant change in the conditions of excitation and characteristics of undulator radiation, making them worse in the case of a smooth waveguide, and improving them in the case of diaphragmmed structures. Expressions for the gain coefficient and emission line width are found and analyzed. It is pointed out that changes in the phase velocity of the waves along the undulator can increase the efficiency of the system.