Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging

Simple analytical expressions are derived for the spatial resolution, contrast and signal-to-noise in X-ray projection images of a generic phase edge. The obtained expressions take into account the maximum phase shift generated by the sample and the sharpness of the edge, as well as such parameters of the imaging set-up as the wavelength spectrum and the size of the incoherent source, the source-to-object and object-to-detector distances and the detector resolution. Different asymptotic behavior of the expressions in the cases of large and small Fresnel numbers is demonstrated. The analytical expressions are compared with the results of numerical simulations using Kirchhoff diffraction theory, as well as with experimental X-ray measurements.