Model Independent Description of Amplification and Saturation Using Green's Function

High-gain Free Electron Laser (FEL) is one of the many electron-beam instabilities that have a number of common features linking the shot noise, the amplification and the saturation. In this paper, we present a new, model-independent description of the interplay between these effects. We derive a simple formula for a maximum attainable gain before instability saturates. Application of this model-independent formula to FELs is compared with FEL theory and simulations. We apply the limitations resulting from these findings to FEL amplifiers used for seeded FELs and for Coherent electron Cooling.