Abstract
Plasma fluid and Maxwell equations are used to derive an analytical expression for the high-gain of a Cherenkov Free-Electron Laser (FEL). The evolution of the gain from the linear to the non-linear regime is investigated. The gain expression is then generalized to predict the linear and non-linear effects at any stage of interaction, permitting us to identify the criteria of optimal design of Cherenkov FELs. In this study, the spatial dependence of the characterizing parameters of electron beam velocity and density, such as the average value and modulation amplitude, are considered. Also, the phase shift between the bunched electrons and the radiation field is taken into account. In the linear limit, it is shown that our gain expression agrees satisfactorily with those obtained in previous studies.