A simple theory of gain and saturation in FEL oscillators

We develop a simple theory of gain and saturation in FEL oscillators, using a Maxwell–Lorentz approach. Our analysis includes inhomogeneous broadening effects. We derive a simple expression for the FEL gain as a function of: (a) optical intensity, (b) energy detuning, and (c) energy-spread. The analytic results are discussed and are shown to be in good agreement with one-dimensional, time-independent, numerical simulations.