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Synchrotron radiation emitted by electrons passing through an undulatorplaced in a storage ring is decomposed in coherent modes. The case of ultimatestorage rings where the electron emittance is comparable to the emittance ofthe photon fan is analyzed by means of the cross spectral density and thecoherent mode spectrum. The proposed method permits naturally the statisticalanalysis and propagation of the cross spectral density along the beamlineoptics. The coherence properties of the X-ray beam at any point of the beamlineare completely given in terms of the eigenvalues and coherent modes of thecross spectral density.

Coherent modes of X-ray beams emitted by undulators in new storage rings