physics of photons and needs of SR experiments

On the topic of #3 from my last post, I try to learn a bit of how to understand photons. I posted a question on Stack Exchange here. I referenced a paper by KJ Kim on the Wigner function approach to synchrotron radiation, and asked the question:

When one represents radiation via a Wigner function, is this really quantum mechanics?

I got some links to papers about Wigner functions and about photons. I think that quantum optics has a lot to say on this topic, but unfortunately I seem to have lost my copy of Mandel and Wolf. I was referred to papers on the wave function of a photon, such as this by Iwo Bialynicki-Birula. The wave function defined by Bialynicki-Birula is referenced here in a 2011 New Journal of Physics article by Saldanha and Monken. In this article, they extend the photon wave function approach to "to include the interaction of photons with non-absorptive continuous media."

I was interested in this question because I wanted to understand undulator radiation better. The question is what one should calculate? How should one represent the radiation such that it covers the properties used by the experiments with synchrotron radiation? In "Undulators, Wigglers and Their Applications", edited by Elleaume and Onuki, the brilliance is identified with the Wigner function and is computed for undulators, wigglers and bending magnets. Its a dense book, but contains an up-do-date perspective on these topics. Regarding codes, the synchrotron radiation may be computed in the near field with SRW. But this just computes different components of the Electric field for a given electron beam source and undulator construction. What to do with the output, and are there still open foundational questions?

Once one knows the radiation fields, one can propagate them, and probably ray optics is sufficient for most purposes. The Shadow code may be used for this purpose, and allows one to enter lenses, mirrors and other optical components in the x-ray beamline.

So, this is sort of the the landscape, as far as books, theoretical frameworks, and software for synchrotron radiation. Certainly I'm biased to that used at my institute, but I think it covers a good amount.

For some practical questions, I think one can look at how some different electron beam parameters affect the photon beam. For example, the electron beam energy spread, or the tilt angle of the electron beam. Next, the question is, for the experiments one does with the radiation, what are really the important parameters. Brightness is important, but it seems to be a stand-in for a more detailed case by case examination. Does one need large numbers of photons (flux?), does one want a round beam? Are the coherence properties of the x-rays important? In the latter case, it appears that the Wigner function does not tell all, but in faction one needs to compute the mutual intensity (the argument of the integral in the definition of the Wigner function).