I am not sure to fully answer the question but this exist.

http://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis

Regards

Romain

Thanks Romain- its a helpful wikipedia page.

Interesting to see detM=n1/n2 (indices of refraction at start and end). In the electron optics case its always detM=1 for traversing any magnetic fields (neglecting radiation).

Anyway, if you start in vacuum and end in vacuum you still have detM=1.

So we could talk about the x-ray beam line in the same way we talk about an electron transfer line- discuss matching, invariants, etc.

I'll keep reading.

Transport of electrons in a beam line is quite similar to transport of light in a focusing system. The Rayleigh range corresponds to the Twiss beta function... Read this reference ... Dattoli et al. Il Nuovo Cimento D, March 1992, Volume 14, Issue 3, pp 271-278 Towards a wave theory of charged-beam propagation.

Thanks Dr. Giannessi. After looking at your paper and others I start to have a better understanding of this. The next question is to understand what properties are required/desired by the experiments. Typically we characterize the source of the photons by its brightness. But this is not something that is directly relevant to the experiments. For example, at ESRF, we have both high and low beta sections, which can give the same brightness (emittance is the same), but lead to different sizes of photon beams in the hutch. Suppose we have the matrix M that characterizes the beamline, together with a certain amount of flexibility in tuning it (change the position or focal length of a lens, shift a mirror, etc.) I'd like to understand what properties we want to optimize in the hutch. Can we describe the beam envelope in terms of beta functions and emittance as we do with an electron beam. Then, would we say we want to minimize the emittance of the photon beam at the hutch? How do we talk about this, and how can we talk about the overlap between the electron beam and the photon beam?

In general, describing a photon beam with rays corresponds to the case of geometrical optics, i.e. each photon propagates along a straight line in vacuum. So you want to make transition from a wave description to geometrical optics and wonder when you can do that. There is very close analogy in quantum physics. Full quantum description is a wave description and classical description is the same as geometrical optics when each particle propagates along the line. In order to make transition from one description to another you need a system occupying high energy levels (Bohr-Sommerfeld quantization rule). These states are characterized by the property that uncertainty condition for them is much larger than the fundamental limit of \hbar / 2. Uncertainty condition can be interpreted as conventional emittance. So, in order for classical approach to work you want emittance to be much larger than quantum limit. Then you can describe quantum system with classical equations. (There is additional factor of N - number of electrons because they are fermions, but this factor would not be for bosons, i.e. photons).

It is the same for radiation. In order to describe electro-magnetic wave with geometrical optics you want your radiation to occupy large area in the phase space, much larger than uncertainty limit. Uncertainty limit is a fully coherent wave. So, you want your radiation to be incoherent. In this case you can break it into rays "photons" which do not interfere with each other. Then you can describe each photon with a position and direction of motion which are independent parameters unlike radiation front which is described with a function of \vec{r} rather then 2 vectors \vec{r} and \vec{r^\prime}.

That's the logic of when you can describe radiation as an ensemble of non-interacting photons.To do that properly mathematically, you need to construct spectral intensity - Fourier transform of the 2-point autocorrelation function other the short scale. This spectral intensity has \vec{r} and \vec{k} as independent variables, and it can be considered as the photon distribution in the phase space. If the area of this distribution is larger than \sqrt{ }>\lambda / (4\pi) then the radiation is incoherent and interference effects can be typically ignored. After that all the conventional accelerator methods can be applied including matrix transform and emittance concepts. BTW, spectral intensity for waves is exactly the same as Wigner distribution in quantum mechanics. Typically, description with Wigner distribution works well for uncorrelated electrons.

Now back to your question of minimizing photon emittance. As I said before, this description works only for incoherent light. Such a light is generated by electrons which contribute to the light incoherently (light generated by one electron does not affect emission properties of other electrons). Then the overall light distribution function is equal to the convolution of the single-particle spectral intensity and electron distribution. In general, you want to maximize this convolution to get the brightest photon beam.

You may want to check of the following reference and Sec.6 in particular. Kwang-Je has more or less the same logic as I described above.

K.-J. Kim, "Characteristics of Synchrotron Radiation," AIP Conference Proceedings, 184, 565 (AIP, 1989)