Shifting curves based on the detector integration effect for x-ray phase contrast imaging

In theory, we find that the actual function of the analyzer grating in the Talbot-Lau interferometer is segmenting the self-images of the phase grating and choosing integral areas, which make sure that each period of self-images in one detector pixel contributes the same signal to the detector. Furthermore, in the case of the lack of an analyzer grating, the shifting curves are still existent in theory as long as the number of fringes is non-integral in a detector pixel, which is a sufficient condition for creating shifting curve. The sufficient condition is available for not only the Talbot-Lau interferometer and the inverse geometry of Talbot-Lau interferometer, but also the x-ray phase contrast imaging system based on geometrical optics. In practical applications, we propose a method to improve the performances of the existing systems by employing the sufficient condition. This method can shorten the system length, is applicable to large period gratings, and can use the detectors with large pixels and large field of view. In addition, the experimental arrangement can be simplified due to the lack of an analyzer grating. In order to improve detection sensitivity and resolution, we also give an optimal fringe period. We believe that the theory and method proposed here is a step forward for x-ray phase contrast imaging.