Brightness of synchrotron radiation from wigglers

According to the literature, while calculating the brightness of synchrotron radiation from wigglers, one needs to account for the so-called 'depth-of-field' effects. In fact, the particle beam cross-section varies along the wiggler. It is usually stated that the effective photon source size increases accordingly, while the brightness is reduced. Here we claim that this is a misconception originating from an analysis of the wiggler source based on geometrical arguments, regarded as almost self-evident. According to electrodynamics, depth-of-field effects do not exist: we demonstrate this statement both theoretically and numerically, using a well-known first-principle computer code. This fact shows that under the usually accepted approximations, the description of the wiggler brightness turns out to be inconsistent even qualitatively. Therefore, there is a need for a well-defined procedure for computing the brightness from a wiggler source. We accomplish this task based on the use of a Wigner function formalism. We exemplify this formalism in simple limiting cases. We consider the problem of the calculation of the wiggler source size by means of numerical simulations alone, which play the same role of an experiment. We report a significant numerical disagreement between exact calculations and approximations currently used in the literature.