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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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    $\begingroup$ I can't imagine that we are anywhere near degenerate density/pressure conditions in a free beam. By many orders of magnitude. I'd look for the primary constraint from space-charge effects or beam cooling. $\endgroup$ – dmckee Aug 23 '14 at 18:50
  • $\begingroup$ I have to agree with dmckee. It sounds like a perfectly semi-classical mechanism, no exclusion principle needed. I also don't see any way that the physical limit could ever be achieved for these accelerators. Magnetic fields around neutron stars have been estimated to be on the order of tens of millions of Tesla, probably even more. We are far, far away from technically achieving anything like those conditions in a beam line magnet, which means that nature can make far more capable x-ray lasers than we can. $\endgroup$ – CuriousOne Aug 23 '14 at 19:39
  • $\begingroup$ @dmckee Sure, I never recall all the most relevant tags. I thought about the space-charge effects and recalled all the cases where they are neglected. But then again, all these cases such as solids or white dwarfs are quasineutral. Okay, I guess I have my answer to the original question - Pauli exclusion would not be relevant. Do you care to write it out as an answer? I could maybe modify it to ask about the relevance of both charge and exclusion effects, so we can close the matter. $\endgroup$ – Void Aug 23 '14 at 19:43
  • $\begingroup$ I don't know what current beam densities are in electron machines, so it's hard to write a precise answer. It's just that achieving densities (for anything) where degeneracy comes into play is very difficult unless you just want to point at an atom. It is managed for BECs by making them very cold. $\endgroup$ – dmckee Aug 23 '14 at 19:47
  • $\begingroup$ @CuriousOne I don't see how the strength of stationary magnetic fields around neutron stars relates to lasers. There are some quite impressive energetic phenomena in nature far beyond our laboratory reach, but I do not recall any of them producing coherent light - which would be the case I would call a " natural laser". $\endgroup$ – Void Aug 23 '14 at 19:47
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Pauli found his principle first for electrons states in an atom. Call it the strong principle, for the lowest atomic orbitals and under "normal" temperature and other energetic circumstances it describes a natural law.

It was found out too that in metals the free electrons have a broad range of energy levels. Thats allow them to be in different distances from the bounded electrons and to use the free space in the metallic structur. Heating up such a system levels up the energies of the free electrons and made the niveau broader. In principle this is predicted in thermodynamics. The free electrons are chaotic particles which interact with other particles by electromagnetic radiation. All the system is in equilibrium (if not disturbed from outside), but each electron is perpetually changing his energy level.

Taking away from electrons thermal energy gives them less and less possibilities to exchange photons and the are less and less distinguishable. At least you can condensed matter.

Boundering electrons in a box with walls that have only one energy level reduces the thermal interaction for the electrons. This is the case in the artfully composed free-electron-laser. So we have to be careful with the use of the Pauli principle.

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