X-ray dark-field via spectral propagation-based imaging

Dark-field X-ray imaging is a novel modality which visualises scattering from unresolved microstructure. Current dark-field imaging techniques typically require precision optics in a stable environment. Propagation-based imaging (PBI) is an optics-free phase-contrast imaging technique that can be used to recover phase information by modelling the propagation of a diffracted wavefield. Based on the Fokker--Planck equation of X-ray imaging, we propose a dual-energy PBI approach to capture phase and dark-field effects. The equation is solved under conditions of a single-material sample with spatially slowly-varying dark-field signal, together with an a priori dark-field spectral dependence. We use single-grid dark-field imaging to fit a power law to the dark-field spectral dependence, and successfully apply the PBI dark-field retrieval algorithm to simulated and experimental dual-energy data.