Theoretical analysis of the coherence properties of a grating

In this work we aim to clarify theoretically the spatial coherence properties of the x-ray beam in the focal plane after interaction with a variable line space (VLS) grating. Assuming that the VLS grating is oriented horizontally, we are interested in the coherence properties of the beam in the vertical dispersion direction. We first consider a fully spatially coherent x-ray beam illuminating the grating. We show that the spatial coherence properties in the focal plane depend on the bandwidth of the incoming radiation. Being fully spatially coherent up to the VLS grating, the spatial coherence properties degrade in the focal plane of the VLS grating. We attribute this to coupling of the spatial and frequency components in the focal plane of the grating. Next, we examine partially coherent x-ray beams incident on a VLS grating. We assume that this radiation is of the Gaussian Schell-model type and obtain an analytical expression for the spatial coherence properties of the beam in the focal plane of the VLS grating for such a field. Next, we consider a monochromator setting that is provided by installing slits in the focal plane of the VLS grating and examine the degree of coherence in this case. Finally, we evaluate the degree of coherence at different apertures of the exit slits, assuming coherent illumination of the grating and a bandwidth of $1.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$.