Free electron laser saturation: exact solutions and logistic equation

Models attempting an analytical description of free-electron laser (FEL) devices have been proposed in the past. They provided interesting results, leading either to a deeper understanding of the FEL dynamics and to semi-analytical formulae, useful for the preliminary design of self amplified spontaneous emission and oscillator FELs. Most of these models work well until the level of mild saturation. In this paper, we comment on the so-called logistic model and a more recent analysis describing the FEL evolution in terms of Jacobi elliptic functions. Both models are shown to be suited to describe the evolution from the low signal to the onset of saturation. We attempt therefore an extension of these theoretical formulations using a delayed logistic model, capable of including characteristic features like the post saturation power oscillations.