Theoretical aspect of X-ray phase microscopy with transmission gratings

One of the advantages of X-ray grating interferometry is that it can work with spherical-wave illumination of X-rays. This means that the interferometry can be combined with an X-ray imaging microscopy to achieve a high spatial resolution. We have developed three types of X-ray phase imaging microscopies, which are based on a self-imaging phenomenon called the Talbot effect. The first type is an X-ray imaging microscopy just combined with X-ray Talbot interferometry, where two transmission gratings are used. The other two are novel X-ray phase imaging microscopies, where the self-image of a transmission grating is highly magnified and resolved by an image detector: one has been achieved by a single X-ray source and the other is what we call Lau-type X-ray phase imaging microscopy, where an array of mutually incoherent X-ray sources has been used. These two microscopies have overcome several problems of the Talbot-interferometer-type X-ray imaging microscopy, i.e., they provide a phase difference image (twin phase images separated by a specific distance) and hence have high sensitivities and high spatial resolutions. Quantitative phase imaging even for non-weak-phase objects has also been performed that are difficult to be achieved by the widely used Zernike phase-contrast microscopy. In this paper, we outline a theoretical description of the three types of the X-ray imaging microscopes using transmission gratings as well as the corresponding three types of X-ray projection microscopes.