Statistical theory of high-gain free-electron laser saturation

We propose an approach, based on statistical mechanics, to predict the saturated state of a single-pass, high-gain free-electron laser. In analogy with the violent relaxation process in self-gravitating systems and in the Euler equation of two-dimensional turbulence, the initial relaxation of the laser can be described by the statistical mechanics of an associated Vlasov equation. The laser field intensity and the electron bunching parameter reach a quasistationary value which is well fitted by a Vlasov stationary state if the number of electrons $N$ is sufficiently large. Finite $N$ effects (granularity) finally drive the system to Boltzmann-Gibbs statistical equilibrium, but this occurs on times that are unphysical (i.e., excessively long undulators). All theoretical predictions are successfully tested by means of finite-$N$ numerical experiments.

Figure 1

Typical evolution of the radiation intensity using Eqs. (1, 2, 3); the detuning $\delta$ is set to $0$, the energy per electron $H∕N=0.2$ and $N={10}^4$ electrons are simulated. The inset presents averaged simulations on longer times for different values of $N:5\times {10}^3$ (curve 1), 400 (curve 2), and 100 (curve 3).

Figure 2

Comparison between theory (solid and long-dashed lines) and simulations (symbols) for a monoenergetic beam, varying the detuning $\delta$. The vertical dotted line, $\delta ={\delta }_c\simeq 1.9$, represents the transition from the low to the high-gain regime.

Figure 3

Comparison between theory (solid and long-dashed lines) and simulations (symbols) for a non-monoenergetic beam when the energy $\epsilon$, characterizing the velocity dispersion of the initial electron beam, is varied. The dotted lines represent the intensity and bunching predicted by the full statistical equilibrium given by Eqs. (16, 17), not very appropriate here, whereas the solid line and long-dashed lines refer to the Vlasov equilibrium defined by Eqs. (13, 14). The discrepancy between theory and numerical experiments is small over the whole range of explored energies.