## 3 Crystal-Based X-ray Interferometer  
A crystal-based XI is an amplitude-division interferometer that uses the XRD of crystal wafers and/or blocks to divide, reflect, and recombine X-rays. Various crystal-based XIs with different optical configurations, such as the triple Laue (LLL) interferometer (Bonse-Hart type) using triple Laue-case XRD (Figure 2a) [2], the triple Bragg (BBB) interferometers using triple Bragg-case XRD (Figure 2b) [15; 16], and crystal-based XIs combining the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **Detecting** & **Density** & **Dynamic Range** & **Spatial** & **Typical** \\
**Physical Value** & **Resolution** & **of Density** & **Resolution** & **Measurement** \\  & & & & & **Time for CT** \\ \hline Crystal-based & & & & & \\ X-ray & & & & & \\ interferometry & \(\cos\) (\(p\)) & High & Narrow & \(\sim\)30 \(\mu\)m & \(\sim\)3 h \\ (CXI) & & & & & \\ Diffraction- & & & & & \\ enhanced imaging & \(\frac{\partial p}{\partial x}\) & Middle & Wide & \(\sim\)10 \(\mu\)m & \(<\)30 min \\  & & & & & \(<\)1 min for white \\ Grating-based & & & & & synchrotron \\ X-ray & & & & & radiation \\ interferometry & \(\cos(\frac{\partial p}{\partial x})\) & Low & Middle & \(\sim\)10 \(\mu\)m & 2 h for mono. \\ (GXI) & & & & & synchrotron \\ Propagation-based & & & & & radiation \\ imaging (PI) & \(\
abla^{2}p\) & Low & Wide & \(\sim\)3 \(\mu\)m & \(\sim\)1 h \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main specifications of four major PCXI methods at SAGA Light Source (synchrotron radiation facility of Saga prefecture in Japan) [7]. Laue- and Bragg-cases of XRD (Figure 2c) [17], have been devised, as shown in Figure 2. Furthermore, another type of crystal-based XI has been developed that uses simultaneous coplanar and non-planar X-ray diffractions for high-resolution imaging and suppresses beam blurring [18; 19]. It diffracts X-rays simultaneously by satisfying multiple diffraction conditions on different lattice planes. Among these interferometers, the LLL interferometer and its separated type [20] are currently used for practical PCXI. As shown in Figure 2a, the incident X-rays are divided into two beams (object and reference) split by the Laue-case XRD at the first wafer (S: splitter), and the two beams are reflected at the second wafer (M: mirror) and re-combined at the third wafer (A: analyzer), forming two interference beams. If a sample is placed in the object beam path, the interference beam intensities are changed by the wave superposition, so the phase shift caused by the sample can be detected on the basis of the intensity changes. The optical configuration is the same as that of the Mach-Zehnder interferometer used in the visible light region. However, the XRD conditions of each wafer have to be satisfied simultaneously for crystal-based XI operation, so a crystal-based XI is generally fabricated from a silicon ingot monolithically. In addition, since the temporal and spatial coherence lengths of X-rays, including SR, are several micrometers, the displacement \(\Delta x\) of the object and reference beam paths on A must be less than several micrometers for high interference. In other words, it is important to fabricate a crystal-based XI so that the differences between A-M and M-S distances are less than a few micrometers by using an ultra-precision crystal-forming technique. The intensity \(I\) of the interference X-rays emitted from the A wafer of the interferometer is given by  
\[I=I_{1}+I_{2}+2\gamma\sqrt{I_{1}I_{2}}\cos\phi \tag{1}\]  
where \(I_{1}\) and \(I_{2}\) are the intensities of the object and reference beams, \(\gamma\) is the degree of coherence, and \(\phi\) is the phase difference between the object and reference beams. If the sample is placed in the object beam path, Equation (1) can be rewritten as  
\[I^{\prime}=(I_{1}-\Delta I_{1})+I_{2}+2\gamma^{\prime}\sqrt{(I_{1}-\Delta I_{ 1})I_{2}}\cos(\phi+\Delta p) \tag{2}\]  
by using intensity change \(\Delta I\), phase shift \(\Delta p\) by sample, and decreased \(\gamma^{\prime}\).