# Principles of Different X-ray Phase-Contrast Imaging: A Review
## 2 Methods
### Coherent Diffraction Imaging  
In 1952, [PERSON] suggested that [PERSON]'s theorem could be extended to determine the crystal structure by measuring the diffraction intensities at a half-integral of \(h\), which is midway between the Bragg peaks [8]. This lensless method called coherent diffraction imaging (CDI) was then introduced into noncrystalline specimens, and high-resolution phase images were obtained [59]. The progress and applications of the method have been thoroughly discussed in the reviews of [PERSON] [30] in 2010 and [PERSON] in 2011 [60]. In the present review, we briefly introduce the principle of phase-retrieval algorithms based on the iterative method of [PERSON] [61]. We discuss the progress and applications after 2011. The experimental setup for the CDI method is shown in Figure 7. In the CDI method, after illumination by a coherent X-ray source, the diffraction patterns of an object with transmission function \(f(x)\) are recorded by the detector as scattered intensities. The Fourier transform is then applied to the scattered intensities for \(F(u)\) such that the object  
Figure 6: Sketch of the twin zone plates under [PERSON]’s X-ray DIC microscope. is synthesized by the inverse Fourier transform of \(F(u)\). A random guess of phase \(\varphi(u)\) is applied to \(F(u)\) for the inverse Fourier transform to retrieve the object's phase. Image \(f^{\prime}\ {}_{1}(x)\) is formed. The object has finite support, meaning the outside of the support should be zero. Because the resulting \(f^{\prime}\ {}_{1}(x)\) would not be zero outside the support, \(f^{\prime}\ {}_{2}(x)\) is set to \(f^{\prime}\ {}_{1}(x)\) inside the support and zero outside. Then, \(F_{2}(u)\) is obtained by applying the Fourier transform onto \(f_{2}(x)\). Since \(F_{2}(u)\) is not equal to the measured scattered intensities, \(F^{\prime}\ {}_{2}(u)\) is set to \(|F(u)|\exp(\mathrm{i}\varphi_{2}(u))\), and \(f^{\prime}\ {}_{2}(x)\) is obtained using its inverse Fourier transform. This procedure is repeated multiple times, as shown in Figure 8.  
In 2013, [PERSON] et al. observed fully hydrated and unstained biological specimens and achieved a resolution better than 25 nm using coherent X-ray diffraction microscopy [62]. In the same year, [PERSON] analyzed the size distribution and electron density projection of particles with the CDI method using focused hard X-ray free-electron laser pulses [63]. In 2016, [PERSON] et al. performed an in situ spatiotemporal study on an electrodeposition process in a sealed wet environment and provided an experimental demonstration of the theoretical prediction of Turing-Hopf electrochemical pattern formation [64]. In 2016, [PERSON] et al. used the CDI method, which imaged the atomic motion of laser-excited molecular iodine with temporal and spatial resolutions of 30 fs and 0.03 nm, respectively, due to the interference between the nonstationary excitation and stationary initial charge distribution [65]. In the same year, [PERSON] et al. demonstrated a formalism of 3D CDI by scanning the energy of the incident X-ray beam instead of moving the sample, which dramatically improved the temporal resolution of this method [66]. In 2004, [PERSON] and [PERSON] presented a new iterative phase-retrieval algorithm called ptychography [67; 68]. This method is a combination of scanning transmission X-ray microscopy (STXM) and the CDI method. It measures multiple diffraction patterns by scanning the coherent X-ray probe over an extended specimen instead of illuminating  
Figure 8: Flow of the iterative procedure of the CDI method. Figure 7: Sketch of the coherent diffraction imaging (CDI) setup. a finite specimen with a spread source in the CDI method. It exploits a convolution between the Fourier transform of a finite illumination function and the diffraction pattern of the object in a Fourier space. The principle, milestone progress, and application of ptychography were thoroughly discussed in a review by [PERSON] in 2018 [PERSON] (2018). In 2019, [PERSON] et al.