Stimulated Emission in Free-Electron Lasers

I've been reading about free-electron lasers these days. The basic principle is clear:

In the undulator electrons spontaneousely emit synchrotron radiation. A self-attunement takes place, causing the electrons to be bunched into packages that are integer numbers of wavelengths apart. Thus the emitted radiation is coherent.

I wonder if this is only spontaneous emission (and coherence is purely due to the self-attunement) or if stimulated emission actually does play a role.

There is no stimulated emission in an FEL.

The SASE (Self-attuned spontaneous emission) process works as follows:

1. A bunch of electrons is accelerated.
2. The electrons pass a first undulator, creating synchrotron-source type light.
3. Light and electrons free-stream together, since both move at the speed of light, there is a back-reaction of the light on the electrons. This is the self-amplification phase: The (large-ish) electron bunch is cut up in several hundred "microbunches", which are compressed to length-scales on the order of microns (thus the name).
4. The micro-bunches pass a second undulator. Here, they also create synchrotron light, but the duration of the pulses is the length of the microbunch divided by the speed of light $$\Delta t \approx \frac{\mu m}{c} = \frac{10^{-6} m}{10^8 \frac{m}{s}} \sim 10^{-14} s.$$ This is how a FEL can create femtosecond timescale light bursts.
5. Since each microbunch emits its light coherently (note: this is a spatial coherence, not a phase coherence) the peak intensities are comparable to solid-state lasers, which is why these machines are called Free-Electron "lasers", even though there is no "lasing" in the conventional sense.

All of this is completely based on classical electrodynamics. There is no stimulated emission into a highly-populated bosonic state, as in conventional lasers.

What I describe above is the unseeded FEL process. One can replace step 2 with a conventional laser providing the intense light needed to induce microbunching.