Revisiting the Fokker-Planck approach and its relation to the sample’s real-space correlation function in single-grid X-ray dark-field imaging

This paper revisits the Fokker-Planck (FP) generalization of the transport-of-intensity equation (TIE) to describe diffusive paraxial X-ray imaging and its relation to capturing small scale structural information of the sample described by its real space autocorrelation function. We show how the diffusion coefficient in FP is related to the real-space correlation function of the sample and to the autocorrelation length of the setup and support this with experimental results. Experiments were carried out on a polychromatic, non-coherent laboratory setup proving the FP framework for that too. We demonstrate that by eliminating the effect of beam hardening, we can get some quantitative dark-field results in such setup. The results provide context for several recent findings e.g. regarding the behavior of dark field with propagation distance. We also discuss the relation to the small-angle X-ray scattering function.