The self-consistent pendulum picture of the free electron laser revised

We reconsider the one-particle dynamics of a Free Electron Laser, adopting the so-called universal scaling. By a fully hamiltonian treatment of the electron and radiation field variables, we show that the electron phase-plane is never that of a pendulum. Actually, besides an elliptic and a hyperbolic pendulum-like fixed point, an extra elliptic point is present at the same phase value as the hyperbolic one, for large values of the detuning parameter δ. On decreasing δ, these two points collapse, which implies a dramatic change in the electron orbit topology, at a value of the detuning parameter which coincides with the instability threshold for exponential gain in the many-electron system.