# Principles of Different X-ray Phase-Contrast Imaging: A Review
## 2 Methods
### X-ray Holography  
Holography was introduced by [PERSON] in 1948 [5]. He coined \"holography\" from two Greek words \"holo\", meaning \"entire\" or \"complete\", and \"grafe\", denoting \"reading\". Compared with conventional absorption images that only record the intensity of the wave, holograms also contain phase information. Holography was first applied to electronic microscopes with coherent illumination. In holography, the object must be transparent and illuminated by a coherent wave so that the wave scattered by the object (object wave) and unscattered wave (reference wave) form interference patterns on the photographic plate, as shown in Figure 3a. When the photographic plate with interference patterns is illuminated by the reference wave only, it diffracts into the first \(\pm 1\) order, with the higher orders being negligible. Hence, a virtual image of the object appears at the position of the object with a real image at its conjugate position, as shown in Figure 3b. According to the two differentreference waves, namely, plane waves and point-source illumination, this method is called in-line and Fourier holography. Although holography was first realized on an electron microscope, electrons are not ideal for holographic imaging because they strongly interact with matter and create large phase shifts during scattering. In contrast, X-rays produce ideal holograms because their interaction with matter is much weaker. In 2002, [PERSON] et al. developed a prism X-ray interferometer [29]. Figure 4 shows that the X-ray deflected by the prism works as the reference wave and interferes with the beam that is refracted by the object. Figure 3: Sketch of (**a**) writing and (**b**) reading of in-line holography. The blue lines represent the reference wave, and the red lines represent the object wave. The green (solid) and yellow lines (dotted) represent the diffracted reference wave that forms the real (green) and virtual (yellow) images. The main progress in X-ray holography until 2010 was discussed in a review by [PERSON] et al. [30]. In 2007, [PERSON] et al. developed holography with extended reference by autocorrelation linear differential operation (HERALDO) on the basis of the work of [PERSON] [31; 32]. In 2010, [PERSON] et al. achieved a resolution of 110 nm with an integration time of 20 femtoseconds using extended references [33]. This method takes advantage of the sharp features of an extended reference object. The final image is retrieved by using the directional derivatives of the Fourier transformed hologram [34]. [PERSON] et al. achieved a resolution of 16 nm using this method in 2010 [35]. More details on the development and capabilities of X-ray holography are discussed in [PERSON] and [PERSON]'s article [34]. In this article, they revealed the achievement of imaging magnetic nanostructures. A spatial resolution of 10 nm was reached. In 2013, [PERSON] et al. implemented a deterministic method for solving the phase problem in hard X-ray in-line holography, which overcame the twin-image problem [36]. In Fourier transform holography (FTH), the image contrast is limited by the reference beam generated by a pinhole aperture. Increasing the pinhole diameter improves the signal but decreases the resolution [37]. In 2014, [PERSON] et al. decoupled the spatial resolution from image contrast using a reference beam generated by an FZP [37]. A spatial resolution of 46 nm was achieved. In 2018, [PERSON] et al. demonstrated high-fidelity imaging based on Fourier holography on free nanosamples with intense femtosecond X-ray pulses. Their method can be extended to a variety of references and samples. A lateral resolution below 20 nm was achieved [38]. In the same year, [PERSON] et al. imaged the insulator-metao transition in vanadium dioxide with element and polarization specificity using resonant soft X-ray holography. A resolution of 50 nm was achieved [39]. In the same year, [PERSON] et al. imaged the [Co/Ni]x4 multilayer magnetic structure within a nanocontact (NC) orthogonal pseudospin valve using HERALDO.