  X-ray imaging has conventionally relied upon attenuation to provide contrast.
In recent years, two complementary modalities have been added; phase contrast
and dark-field x-ray imaging, capturing weakly attenuating and sub-pixel sample
structures respectively. These three modalities can be accessed using a crystal
analyser, a grating interferometer or by looking at a directly-resolved grid,
grating or speckle pattern. Grating and grid-based methods extract a
differential phase signal by measuring how far a feature in the illumination
has been shifted transversely due to the presence of a sample. The dark-field
signal is extracted by measuring how the visibility of the structured
illumination is decreased, typically due to the presence of sub-pixel
structures in a sample. The strength of the dark-field signal may depend on the
grating period, the pixel size and the set-up distances, and additional
dark-field signal contributions may be seen as a result of strong phase effects
or other factors. In this paper we show that the finite-difference form of the
Fokker--Planck Equation can be applied to describe the drift (phase signal) and
diffusion (dark-field signal) of the periodic or structured illumination used
in phase contrast x-ray imaging with gratings, in order to better understand
any cross-talk between attenuation, phase and dark-field x-ray signals. In
future work, this mathematical description could be used as a basis for new
approaches to the inverse problem of recovering both phase and dark-field
information.
