Abstract
An analytical approach is presented to calculate the growth rate coefficient which describes the intensity evolution of a Cherenkov Free-Electron Laser (CFEL) operating in the linear high-gain regime. The effects of the energy spread on the growth rate of a CFEL are addressed by introducing a relaxation time in our analysis. The relaxation process is associated with a deviation of electron velocities from the average value due to an energy spread whatever its source. Therefore, the relaxation time can present a unified point of view to describe the gain deterioration caused by the inhomogeneous broadening of energy spread effects. In this study, the plasma fluid and Maxwell formulations are used to describe the dynamics of electrons. It is found that the growth rate coefficient is saturated at a maximum value when the interaction time is approximately five times the relaxation time. When the electron relaxation is neglected (i.e., the relaxation time is infinitely long), the gain expression derived by our approach agrees adequately with those reported in other studies. It is shown that our gain expression is generalized and simple enough to provide fast and reliable assessment of the performance of CFEL devices.