# Principles of Different X-ray Phase-Contrast Imaging: A Review
## 2 Methods
### Diffraction-Enhanced Imaging (DEI)  
Diffraction-enhanced imaging (DEI) derived from the contrast from absorption, refraction, and extinction was developed by [PERSON] et al. in 1997 [30]. As shown in Figure 13, this method is based on two crystals, which is also referred to as analyzer-based imaging. In this method, monochromatic X-rays are reflected by a crystal with an incident angle Bragg angle to the detector. By using a sample upstream of the crystal, which is usually termed as an analyzer, the incident beam to the analyzer is slightly refracted by the sample by \(\Delta\theta_{Z}\). In the DEI method, the analyzer is set to \(\theta=\pm\Delta\theta_{D}/2\), where \(\Delta\theta_{D}\) is the FWHM of the rocking curve. Thus, the intensity of the reflected beam is expressed as  
\[I_{B}=I_{R}R(\theta_{B}\pm\frac{\Delta\theta_{D}}{2}+\Delta\theta_{Z}), \tag{1}\]  
where \(R(\theta)\) is the analyzer reflectivity on the rocking curve. Because \(\Delta\theta_{Z}\) is very small, \(R(\theta_{0}+\Delta\theta_{Z})\) can be expressed as a two-term Taylor series approximation. \[R(\theta_{0}+\Delta\theta_{Z})=R(\theta_{0})+\frac{dR}{d\theta}(\theta_{0}) \Delta\theta_{Z} \tag{2}\]  
The intensities of the images taken at the low- (\(\theta_{L}\)) and high-angle (\(\theta_{R}\)) sides of the FWHM of the rocking curve are expressed as  
\[I_{L}=I_{R}(R(\theta_{L})+\frac{dR}{d\theta}(\theta_{L})\Delta\theta_{Z}) \tag{3}\]  
\[I_{H}=I_{R}(R(\theta_{H})+\frac{dR}{d\theta}(\theta_{H})\Delta\theta_{Z}) \tag{4}\]  
Figure 13: Sketch of diffraction-enhanced imaging (DEI) setup. Two crystals work as monochromator and analyzer. By solving these equations, the results are presented as follows:  
\[I_{R}=\frac{I_{L}(\frac{dR}{d\theta})(\theta_{H})-I_{H}(\frac{dR}{d\theta})( \theta_{L})}{R(\theta_{L})(\frac{dR}{d\theta})(\theta_{H})-R(\theta_{H})(\frac{ dR}{d\theta})(\theta_{L})} \tag{5}\]  
\[\Delta\theta_{Z}=\frac{I_{H}R(\theta_{L})-I_{L}R(\theta_{H})}{I_{L}(\frac{dR}{ d\theta})(\theta_{H})-I_{H}(\frac{dR}{d\theta})(\theta_{L})} \tag{6}\]  
In 2000, [PERSON] et al. applied this method to breast cancer specimens with a synchrotron source. Six out of seven cases showed enhanced visibility compared to digital radiology [101]. In 2000, [PERSON] et al. implemented the DEI method in CT [102]. In 2002, [PERSON] et al. used this method to image articular cartilage based on a synchrotron source. Both disarticulated and intact joints were visualized through their experiments. Gross cartilage defects, even at the early stages of development, could be visualized. A spatial resolution of 100 \(\upmu\)m was achieved [103]. In 2003, [PERSON] et al. applied a more general method to independently determine the refraction, absorption, and extinction of an object [104]. In 2007, [PERSON] et al. performed this method with a porcine eye as a sample based on a synchrotron source. The results clearly showed the cornea, iris, lens, retina, optic nerve, choroidal vasculature, and the ampulae of the vortex veins. A spatial resolution of approximately 100 \(\upmu\)m was achieved [105]. In 2009, [PERSON] et al. applied the DEI method in CT mode to human cancerous breast tissues yielding 14, 5, and 7 times higher CT numbers and 10, 9, and 6 times higher signal-to-noise ratios (SNRs) compared to conventional CT for masses, calcifications and spiculations, respectively. The DEI-CT mode yields a better SNR than conventional CT, even at ten times lower doses. This experiment was performed with a synchrotron source at a photon energy of 52 keV, which is similar to the K\(\upalpha\) lines of a tungsten anode X-ray tube, showcasing the potential of DEI-CT application using laboratory sources [106]. Several researchers applied this method using laboratory sources in 2009 [107; 108; 109; 110]. In 2014, [PERSON] et al. applied this method on a microfocus X-ray source [111]. In 2020, [PERSON] et al. improved the resolution by approximately a factor of 2 in analyzer-based imaging by cutting the monochromator collimator and thinning the Laue angle analyzer [112].