Coherence and Wigner Function and diffraction limit

Single electrons emit radiation, but the total result comes from all the electrons.
One may compute the Wigner function or equivalently the mutual coherence function resulting
from a single electron. One may then compute the convolution with the electron beam distribution. When the electron beam size is smaller than the intrinsic size of the x-ray beam, this is the so-called diffraction limit. What happens here at very small emittance?
The paper by Geloni et. al. examines this issue.
The recent paper by Bazarov (on arxiv) also explores this issue and has computations of the Wigner function out of undulators.

Important properties of radiation

It seems that there exist two separate cultures in the field of synchrotron radiation and light sources. There is the "machine" side and the "experiments" side. The machine side mostly thinks about electrons, and the experiments side mostly thinks about photons. The devices like the wigglers and undulators sit in between.
As someone coming from the machine side, I'm trying to learn about the important properties of the photons or electromagnetic fields. I'm reading about undulators, wigglers, and bending magnets and their radiation.

So. One should calculate the power radiated. One should understand polarization. One should understand coherence. Is the Wigner function important? What is computed from SRW? How can one interface this with a ray tracing code?

Then there are the standard things people compute and use to compare sources:
brightness (or brilliance) and flux.

As emittance decreases for storage ring light sources, one claimed benefit is in the coherence of the resulting light. And indeed, the claimed benefit for FEL based light sources is also with respect to coherence (among other properties).
So I am trying to understand partial coherence, how one describes it, and how it may be influenced by decreased electron beam sizes. To start, here is a recent review paper by Nugent which reviews x-ray science and covers properties of coherence, including the Wigner function description:

Stack exchange questions

In an effort to enlist outside help and expand the connections to outside fields, I have been asking some questions on the different Stack Exchange sites. These are the questions that I think are some of the areas that could use development and injection of ideas:

programming:

Suggestions for getting an open source electron beam tracking code going

physics:

Interpretation of Wigner function in optics

mathematics:

Stability region of a polynomial map

Computational Science:

Uses of Power Series Maps

Electrons through wigglers to Photons: Codes

The x-rays in a synchrotron are produced by high energy electrons traversing magnetic fields.
The main devices that produce the magnetic fields are undulators or wigglers.

To model this process, therefore, first one needs to model the electrons in the ring, and to have a good model of the fields in the ring. This is the purpose of a tracking code. I focus on the code AT, since I have experience, and the elements representing the physics are easily accesible and available, so that one knows where to look if there are problems or questions.

To really know all the fields in the ring that act back on the electron beam, one must do measurements and modelling of the elements. For the insertion devices, there is the Mathematica based code Radia, which allows one to build an insertion device and compute the fields. One can also compute the kick map which will act on the electrons and can reduce the dynamic aperture and lifetime, particularly for narrow pole-width insertion devices.

Next, one needs to compute the radiation that will result from the electrons. For this, the code SRW is quite accurate and state-of the art. Next, one should model how this radiation can progress down the beam-line. For this purpose, the code Shadow is well developed.

In modeling both the electron beam dynamics and the radiation effects from undulators, one needs to have a model for the magnetic field of a given undulator. The field may be computed using a code such as Radia. But there are also representations of the field in terms of harmonics such as the Halbach representation.
Here's a description of the symplectic integration routine for an analytical representation from Forest.
One may also represent it via a kick map.
The latter two representations serve the purpose of the tracking of electrons for the purpose of tracking to determine dynamic aperture. Given the electron orbit, one can compute the radiation emitted. This can be done with e.g. SRW

electron and photon beam sizes

Once one knows the electron beam properties, then one can calculate the radiation properties.
What are the basics? What photon beam is produced by a given electron beam as it goes through a bending magnet or undulator?
One computes the brightness via the Wigner function. Does this give the beam sizes also? The single electron brightness needs to be convolved with the beam distribution.

The basic result is that the photon beam sizes ($\Sigma_{x,y}$) and divergences ($\Sigma_{x',y'}$) come from the electron beam sizes ($\sigma_{x,yb}$) and divergence ($\sigma_{x',y'b}$) and the single electron radiation size ($\sigma_{x,ye}$) and divergence ($\sigma_{x',y'e}$) added in quadrature.



For ESRF, these are available here.
So, actually, the beam sizes depend on the frequency- which harmonic for an undulator.
Here is the paper by Tanaka and Kitamura describing the impact of energy spread on the photon beam sizes.

The horizontal beam size and divergence are given by:
${\sigma }_x=\sqrt{{\epsilon }_x{\beta }_x+{\eta }_x^2{\sigma }_{\delta }^2}$
${\sigma }_{xʹ}=\sqrt{\frac{{\epsilon }_x}{{\beta }_x}+{\eta }_{xʹ}^2{\sigma }_{\delta }^2}$

To understand this, we need to understand the emittances and the Twiss parameters and the dispersion.

The same equations apply to the bunch length and energy spread. The longitudinal beta function is given to first order by
${\beta }_z=\frac{C{\alpha }_c}{{\mu }_s}$
with ${\alpha }_c$ the momentum compaction factor and ${\mu }_s$ equal to $2pi{\nu }_s$.

For ESRF these are available here.
references:
article by M. Sands.

electron tracking and non-linear dynamics

So, the basic mechanism of the synchrotron light source involve highly relativistic electrons moving through magnetic fields. What happens?
Well, we have dipole magnets that bend the electrons, and basically create the circular trajectory. Then we have quadrupole magnets for focusing, sextupole magnets for chromatic effects and other stability issues. To better understand, and control the electrons, we also need to model them.
Since there are lots of magnets, one needs a code to do this. Here, there's some nice aspects and some messes.

Basically, the equations of motion are given by the Lorentz force law. Newton's laws give a differential equation that needs to be integrated to get the orbit. So one needs a code that can integrate through the magnets. There are conserved quantities, and the motion is symplectic. So one needs a symplectic integrator in order to not lose precision over time.

A symplectic integrator comes in different orders. To get good results, one usually needs at least fourth order. One needs this together with flexible ways to define the magnetic fields. One code that can do this is the Matlab based code called AT- Accelerator Toolbox. An attempt at an open source collaboration is here. The underlying integrators are coded in C, which can be compiled as Mex files, and the definition of the fields to create the ring is done with Matlab.

Non-linear dynamics is mainly relevant for injection efficiency and Touschek lifetime.
One may also want to optimize parameters to improve the non-linear dynamics. One example is the tune shift with amplitude.
For the case of sextupoles distributed around a synchrotron, the general result is

Beam Lifetime

The electron beam loses electrons over time. The lifetime $\tau$ is defined as
$\frac{1}{\tau}=\frac{1}{I}\frac{dI}{dt}$

The lifetime itself may be divided into two pieces, a vacuum lifetime and a Touschek lifetime with the total lifetime given by:
$\frac{1}{\tau}=\frac{1}{\tau_v}+\frac{1}{\tau_T}
The vacuum lifetime is result of the scattering process of electrons off of the gas molecules in the beam pipe. The Touschek lifetime is the result of the scattering of electrons off of each other.
An expression valid for most synchrotron lights sources is:
$\tau_T^{-1}=\frac{r_e^2 I_b}{8\pi e\gamma^3\sigma_s}\oint_C\frac{F\left(\left[\frac{\delta_{\rm acc}(s)}{\gamma\sigma_{x^{\prime}}(s)}\right]^2\right)}{\sigma_x(s)\sigma_{x^{\prime}}(s)\sigma_z(s)\delta_{acc}^2}ds,$

The lifetime scales as
$\tau \approx \frac{\sigma_s\sqrt{\epsilon_z}\delta_{acc}^3}{I_b}$

The bunch length may be measured with a streak camera. Its value depends on the current and is given by solving the Haissinski equation, below the microwave instability.
The vertical emittance may be measured.
The energy acceptance is challenging. It may be given by the RF acceptance or by the dynamic acceptance. Computing this quantity is the work of a tracking code.

In AT, one uses the function atcalc_TouschekPM.