This article describes a complete theory of spatial coherence for undulatorradiation sources. Current estimations of coherence properties often assumethat undulator sources are quasi-homogeneous, like thermal sources, and rely onthe application of the van Cittert-Zernike theorem for calculating the degreeof transverse coherence. Such assumption is not adequate when treating thirdgeneration light sources, because the vertical(geometrical) emittance of theelectron beam is comparable or even much smaller than the radiation wavelengthin a very wide spectral interval that spans over four orders of magnitude (from0.1 Angstrom up to 10^3 Angstrom). Sometimes, the so-called Gaussian-Schellmodel, that is widely used in statistical optics in the description ofpartially-coherent sources, is applied as an alternative to thequasi-homogeneous model. However, as we will demonstrate, this model fails toproperly describe coherent properties of X-ray beams from non-homogeneousundulator sources. As a result, a more rigorous analysis is required. Wepropose a technique, based on statistical optics and Fourier optics, toexplicitly calculate the cross-spectral density of an undulator source in themost general case, at any position after the undulator. Our theory, that makesconsistent use of dimensionless analysis, allows relatively easy treatment andphysical understanding of many asymptotes of the parameter space, together withtheir region of applicability. Particular emphasis is given to the asymptoticsituation when the horizontal emittance is much larger than the radiationwavelength, and the vertical emittance is arbitrary. This case is practicallyrelevant for third generation synchrotron radiation sources.
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