# Principles of Different X-ray Phase-Contrast Imaging: A Review
## 2 Methods
### Differential X-ray Microscopy  
Differential interference contrast (DIC) imaging was first introduced into visible-light microscopy by [PERSON] [6]. Figure 5 shows that the plane wave produced by the first polarizer is split into ordinary (O) and extraordinary (E) rays by the first prism. The separation of these two beams is smaller than the resolution of the system. Then, the O- and E-lights pass through different areas of the specimen, resulting in different phase shifts. After the objective, the O- and E-lights are recombined by the second prism and form elliptically polarized light because of the different wavefronts. Finally, the light enters the second polarizer, which is orthogonal to the first polarizer and creates DIC imaging. In 2001, [PERSON] et al. achieved X-ray DIC imaging by inserting a pair of zone plates upstream of the sample [7]. These two zone plates were set parallel with the lateral displacement (orthogonal to the optical axis) smaller than the resolution of the system. The distance between these two zone plates along the optical axis was small compared with that between the zone plates and the detector, as shown in Figure 6. The incident plane wave split into a zeroth-order plane wave and higher order spherical waves after the first zone plate (ZP1). These waves then split into zeroth and higher-order harmonics through the second zone plate (ZP2). Only the combinations of the zeroth-order of ZP1 and the first order of ZP2 and vice versa contributed to the image while appropriate apertures blocked the others. These two waves interfered with each other and formed diffraction patterns because of the different wave paths. The presence of a specimen changed the diffraction patterns. Later, in 2002, a 320-nm resolution was achieved by their group [50]. In the same year, [PERSON] et al. introduced DIC imaging into both full-field imaging and scanning X-ray microscopy with a 200-nm resolution with the fabrication of a novel FZP doublet [51]. All these experiments were performed at the European Synchrotron Radiation Facility. Figure 5: Sketch of the differential interference contrast (DIC) microscope. Similar to X-ray holography, the DIC method also requires highly coherent and brilliant sources. That constrains the application of this method to sources other than synchrotron sources, X-ray free-electron lasers, or laboratory sources in the extreme-ultraviolet wavelength region [52]. In 2005, [PERSON] et al. introduced a liquid-nitrogen jet laser-plasma source at a water window into this method and achieved a resolution of approximately 321 nm [52]. In 2006, [PERSON] et al. combined an overlaid binary grating and a zone plate through a logically exclusive OR operation and directly presented a magnetic phase contribution [53]. In 2007, [PERSON] et al. demonstrated this method using a compact soft X-ray microscope. A resolution of 30 nm was achieved [54]. In 2008, [PERSON] et al. separated the amplitude and phase information by applying a structured aperture [55]. In 2012, [PERSON] et al. designed single-optical-element optics for this method based on modified photon sieves (MPSs), providing better resolution than the Fresnel zone plate-based DIC, and higher image contrast than normal photon sieves. A resolution below 45 nm was achieved [56]. In 2015, [PERSON] et al. proposed differential interference contrast digital in-line holography by using a single optical element to split the incident light into a plane wave and a converging spherical wave. A two-dimensional DIC effect was generated simultaneously. This method is capable of in situ metrology [57]. In 2018, [PERSON] et al. demonstrated a DIC method for cells using hard X-ray holography and knife-edge filtering in the spatial frequency domain. Good quality phase-contrast images of polystyrene beads and human HeLa cells were obtained [58]. With the new fabrication technique of FZP, the X-ray DIC method can easily achieve a resolution of tens of nanometers. The invention of the FZP doublet reduces the difficulty of alignment.