As I understand it, a free electron laser can basically be pictured as a synchrotron light source with an undulator which by the particular setup causes the electrons to self-attune so that they produce light which is both coherent and monochromatic (as opposed to the usual synchrotron which produces monochromatic but incoherent light). The process of self-attuning can be also replaced by attuning the electrons by an external laser.
Skipping the details, the resonance caused by the right set of parameters causes something which is called "microbunching" in the electron beam, i.e., an effect which can be classically pictured as all the electrons clumping around points in the beam with a distance of one optical wavelength in between. Same spatial points means same undulation, so they oscillate throughout the beam with the same phase causing coherence of emitted light. (I have just read about it, this might be a slightly naive description.)
All the electrons at one place and with one velocity? Pauli exclusion principle comes quickly to mind.
It seems to me that at least for small wavelengths (say ångströms $\to$ X-rays) the bunching has to be done at the cost of a noticeable velocity dispersion of the electrons due to the Pauli exclusion principle. This would mean a dispersion of the produced wavelengths and possibly a tendency to further dilute the bunched electron packets.
Also, a larger required brightness of the output means more electrons and once again a wider peak in the phase space to fit the electrons into. This would in a very hand-waivish manner suggest that there is a certain constraining relation between the wavelength/monochromaticity/coherence/brightness of the outcome.
So the question is: Does such a fundamental constraint of a free electron laser exist? And is it relevant for the current state of art or the near future?
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