  The (spectral) brightness for partially transverse coherent sources as
Synchrotron Radiation (SR) and Free-Electron Laser (FEL) sources can be defined
as the maximum of the Wigner distribution. Then, the brightness includes
information on both coherence and wavefront characteristics of the radiation
field. For undulator sources, it is customary to approximate the
single-electron electric field at resonance with a Gaussian beam, leading to
great simplifications. Attempts to account for the modified spatial and angular
profile of the undulator radiation in the presence of detuning due to energy
spread currently build on the simplified brightness expression derived under
the assumption of Gaussian beams. The influence of energy spread on undulator
radiation properties is becoming important in view of diffraction-limited rings
with ultralow emittance coming on-line. Here we discuss the effects of energy
spread on the brightness of undulator radiation at resonance, as well as
relevant relations with coherence properties.
