  This paper describes a theory of transverse coherence properties of Undulator
Radiation. Our study is of very practical relevance, because it yields specific
predictions of Undulator Radiation cross-spectral density in various parts of
the beamline. On the contrary, usual estimations of coherence properties assume
that the undulator source is quasi-homogeneous, like thermal sources, and rely
on the application of van Cittert-Zernike (VCZ) theorem, in its original or
generalized form, for calculating transverse coherence length in the far-field
approximation. The VCZ theorem is derived in the frame of Statistical Optics
using a number of restrictive assumptions: in particular, the quasi-homogeneous
assumption is demonstrated to be inaccurate in many practical situations
regarding undulator sources. We propose a technique to calculate the
cross-spectral density from undulator sources in the most general case. Also,
we find the region of applicability of the quasi-homogeneous model and we
present an analytical expression for the cross-spectral density which is valid
up to the exit of the undulator. For the case of more general undulator
sources, simple formulas for the transverse coherence length, interpolated from
numerical calculations and suitable for beamline design applications are found.
Finally, using a simple vertical slit, we show how transverse coherence
properties of an X-ray beam can be manipulated to obtain a larger coherent
spot-size on a sample. This invention was devised almost entirely on the basis
of theoretical ideas developed throughout this paper.
