Phase space distribution of brilliance of undulator sources

As an optical source, an undulator can be described by its distribution of brilliance in phase space (angle and position in the transverse plane). This is a convolution of single-particle diffraction pattern and electron beam distribution. Approximate peak brilliance and phase space widths are given. By analogy between brilliance and Wigner function, the depth-of-field and diffraction widths are shown to be essentially the same thing. The peak brilliance is shown to have a broad maximum as a function of the beam β function (β ≅ L/2π) but is practically independent of β at short wavelengths