It is shown that the Wigner distribution provides a proper framework in which the propagation and imaging characteristics of synchrotron radiation can be analyzed, taking fully into account the effect of diffraction and electron beam emittance. The Wigner distribution can be interpreted as a phase-space density of photon flux, i.e., the brightness, and transforms through a general optical medium in the same way as in the case of collection of geometric rays. The brightness due to a collection of electrons can be calculated by a simple convolution of the brightness of a single electron with the electron phase space distribution function. Expressions for the brightness of bending magnets, wigglers and undulators are given.
"A New Formulation Of Synchrotron Radiation Optics Using The Wigner Distribution", Proc. SPIE 0582, Insertion Devices for Synchrotron Sources, (5 May 1986); doi: 10.1117/12.950906; https://doi.org/10.1117/12.950906