Partial coherence theory for x-ray phase contrast imaging technique with gratings

The partial coherence theory is applied to study the x-ray Talbot-Lau interferometry for phase contrast imaging with extended and polychromatic sources. Under a geometric condition for grating periods and distances between gratings, moiré fringes at the analyzer grating can be the same as the case of using one single slit. Although this geometric condition has been proposed before, we provide a quantitative explanation of the geometric condition compared to previous heuristic understanding. We then investigate the fringe visibility at the analyzer grating. For quasi-monochromatic sources, we derive an analytical formula and demonstrate that the visibility can reach 100% if the duty cycle of the source grating is less than an explicitly given bound. For polychromatic sources, we conduct numerical simulations to show that the visibility could reach near 100%. Both 1D- and 2D-grating systems are studied in this manuscript, respectively. Further work is to evaluate the theoretical results with physical experiments and establish design criteria for Talbot-Lau interferometers.