  Synchrotron radiation emitted by electrons passing through an undulator
placed in a storage ring is decomposed in coherent modes. The case of ultimate
storage rings where the electron emittance is comparable to the emittance of
the photon fan is analyzed by means of the cross spectral density and the
coherent mode spectrum. The proposed method permits naturally the statistical
analysis and propagation of the cross spectral density along the beamline
optics. The coherence properties of the X-ray beam at any point of the beamline
are completely given in terms of the eigenvalues and coherent modes of the
cross spectral density.
