Theoretical explanation of spatial coherence and source size characterization by X-ray grating interferometry

The unprecedented spatial coherence of modern diffraction-limited storage rings (DLSRs) enables advanced X-ray experimental techniques. A precise beam characterization will be very helpful to exploit these techniques. Although X-ray grating interferometry (XGI) is widely used for this purpose, our previous work showed that the XGI-measured visibility width is significantly smaller than the coherence length (defined by the width of the local complex coherence function) for highly coherent beams [Lu et al. (2025), Opt. Lett. 50, 6746-6749]. To clarify what XGI actually probes, we present here a theoretical analysis. We show that, under the assumption of a global quadratic phase profile for the incident beam, XGI measures the normalized integrated cross-spectral density (ICSD), which represents a global correlation function of the beam. Based on this, we derive an analytical formula for exact coherence length measurement for ideal Gaussian Schell-model beams. Despite this difference from the coherence length, XGI yields reliable source size estimates. To explain this interesting fact, we use the ICSD to show that XGI can measure the source size within an acceptable error, even for DLSRs, provided the cross-spectral density of the source satisfies the Schell-model condition in mean and difference coordinates.