The entire concept of phase-contrast imaging using X-rays was first introduced by Bonse &
Hart in 1965 using a crystal interferometer setup . The setup is constructed of a single, highly perfect piece of crystal. Typically the crystal is a silicon crystal and it is cut into three crystal plates. The geometry of the setup is referred to as the Laue-Laue-Laue (LLL) interferometer setup. The first crystal plate is the beam splitter S, the second plate is the transmission element M, and the last plate is the analyzer crystal A [10, 57]. A. Momose introduced the use of the LLL interferometer setup in CT imaging to obtain tomographic phase-contrast images .
The working principle of the setup is as follows. When the incoming monochromatic X-ray beam satisfies the Bragg diffraction conditions, the splitter crystal divides the beam into two divergent X-ray beams. When the separate beams arrive at the transmission crystal, they are again both divided into two separate beams. From these four beams, the two convergent beams overlap at the analyzer crystal and there the beams are again divided into two and interference occurs.  An object in the beam path, between the transmission element and the analyzer crystal, causes a phase shift in the incoming beam. The amount of phase shift can be determined with the use of a phase shifter placed in the reference beam.
The beam incident on the first interferometer needs to be monochromatic and therefore a monochromator is placed before the splitter crystal. A typical LLL interferometer setup is depicted in Figure 6. 
Due to the short wavelength of X-rays, the interferometer setup imposes strict require-ments on the mechanical stability of the optical elerequire-ments. Good stability is achieved by using a monolithic crystal but this in turn limits the field of view (FOV) to sizes under what is needed for clinical applications.  In addition, as a monochromatic beam is required for imaging the feasibility of the setup is low and the demands of the setup are further increased . Due to these limitations of the crystal interferometer setup, grating-based differential phase-contrast (DPC) imaging has been introduced as an alternative interferometric method.
Figure 6: A typical LLL interferometer setup consisting of three plate crystals; the splitter, transmission element or mirror, and analyzer crystal. The monochromator is placed before the splitter crystal and the detector panel is placed after the analyzer crystal. The object to be images is placed in the beam path between the mirror and analyzer crystals causing a phase shift in the beam. The phase shift can be determined by a phase shifter placed in the reference beam. 
3.3.2 Grating Interferometry
DPC imaging is based on the use of gratings as optical elements to obtain phase contrast from the refraction of X-rays using a setup called the Talbot interferometer . The setup is named after the Talbot effect discovered by Henry Fox Talbot in 1836. He discovered that when a coherent wave propagated through a periodic structure, such as a grating, a self-image, also called the Moiré fringe pattern, of the structure is reconstructed behind the grating at distances
λ , (24)
where m = 1,2,3, . . ., p is the grating period, and dT is called the Talbot distance. In addition, the fractional Talbot effect was realized; a self-image of the phase grating is formed at distances
dm = (m−1/2)p2
4λ . (25)
Equation (25) can be expressed for any grating as dm = mp2
The contrast of the resulting self-image is at maximum for a phase grating when m is odd and for an amplitude or absorption grating, the maximum is achieved whenm is even. The coefficientη=1 for both amplitude and phase gratings that give a phase shift ofπ/2 andη=2 if the phase shift isπ. 
A grating-based interferometry setup using a coherent X-ray source is depicted in Figure 7 . The setup consists of an X-ray source, the sample to be imaged, a phase grating G1, an
absorption grating G2, and a CCD detector. The two gratings are separated by a distanced, wheredmust be a fractional Talbot distance so that maximum of the interference pattern is observed at G2. [14, 32, 33, 57] In theory, grating G1 can be an absorption or a phase grating but greater efficiency is achieved using a pure phase grating. Therefore, a phase grating with a π phase shift is a good choice. [14, 32] When the incident wavefront comes across the first grating G1, it is diffracted into the first two diffraction orders at each grating slit. The diffracted beams from neighboring slits interfere and generate a periodic intensity pattern in the plane of the absorption grating G2, which contains the refraction angle information. If an object is place between the X-ray source and the phase grating, it will cause refraction of the X-rays and as a result the interference pattern is modified compared to the intensity pattern created only by the grating. By comparing these interference patterns, the phase information of the sample can be obtained. [14, 32, 33, 52, 57]
Figure 7: A grating-based interferometer setup with an coherent X-ray source. A sample placed in front of the X-ray source causes distortions in the incident X-ray wavefront. The phase grating G1 causes diffraction in the wavefront, resulting in interference and therefore intensity modulations in the wavefront. The modulated wavefront is then analyzed by the absorption grating G2. The distance between the gratings G1 and G2 is d. 
Pfeiffer et al. introduced the use of an incoherent X-ray source in DPC imaging by proposing the use of a source grating G0 to create individually coherent but mutually inco-herent sources. A typical setup, called the Talbot-Laue interferometer, using a source grating is depicted in Figure 8. The source grating can contain a large number of apertures, all of which create sources of required coherence, and therefore conventional X-ray sources with source size of 1mm2 or more can be used. In order for the line sources created by the source grating to be beneficial for image-formation, the setup geometry should satisfy
p0 =p2· l
where p0 is the source grating period, p2 is the absorption grating period, l is the distance from the source grating to the phase grating, andd is the distance from the phase grating to the absorption grating. Final image resolution is given wd/l, which is determined by the source size, w.  Wang et al. studied the effect of a polychromatic X-ray source on the efficiency of the Talbot-Laue interferometer setup. The evaluation of the efficiency is interesting as the phase of the intensity oscillations is dependent on the wavelength of
used to assess the efficiency in the simulations. The results of the research showed that the visibility of the fringe pattern is not highly sensitive to the polychromaticity of the X-ray source even at higher fractional Talbot distances. 
Figure 8: A grating-based interferometer setup using an incoherent X-ray source. The setup contains the X-ray source, a source grating G0, a phase grating G1, an analyzer grating G2, the sample to be imaged, which is placed between gratings G0and G1, as well as the detector placed behind grating G2. Image modified from .
Phase-stepping is performed to separate the phase information of the signal from ab-sorption information and other contributions, such as imperfections of the grating . The total phase shift could also be obtained straight from the intensity profile by integration but phase-stepping provides a more precise way of extracting only the phase information .
Phase stepping is performed by moving one of the gratings, typically G2, in the direction of its period (x in Figure 8) over one period of the grating. For every point of the phase-stepping scan an image is taken and from these, the average intensity of the period for each pixel can be found. From these averages, the oscillations phase φcan be determined for all pixels. The phasesφof the intensity oscillations in each pixel are related to the phase profile Φ of the wavefront, the wavelength λ, the period of the absorption grating period p2, and the decrement of the real part of the refractive indexδ by
Asφ contains only phase information, the phase of the object can be obtained from Equa-tion (28) by integraEqua-tion. This phase-stepping procedure is done both with and without the sample in place to obtain the phase information of the sample. [33, 52] A total of three steps are required for the phase-stepping scan to obtain φ for a sinusoidal intensity oscillation, although typically more steps are acquired [28, 33, 52]. This is because the intensity curve has three different components: amplitude, phase and a constant . Some drawbacks of the phase-stepping process include its time-consuming nature and the fact that multiple exposures have to be done to obtain the phase-stepping curve . These limit the imple-mentation of the technique especially into clinical use but the technique has been widely used in research [3, 17, 27, 28, 43, 46].
The 3D distribution of the X-ray refractive index can be obtained by tomographic
re-constructing of Φ. This requires a sufficient number of projections at various angles around the sample. The average signal for each pixel can also be obtained from the phase-stepping scan, which corresponds to the signal obtained from conventional radiographic imaging, and therefore the attenuation coefficient can also be calculated. As a result, one phase-imaging protocol yields both phase and attenuation information.  This was exploited by Herzen etal., who studied materials with similar attenuation coefficients and refractive indices using a self-built liquid phantom system. They discovered that these materials can be best distin-guished from each other by simultaneous measurement of both quantities.  This could be of use also in clinical settings in certain applications.
Zhu et al. proposed a new approach, which eliminates the need for the phase-stepping procedure. The approach was tested at a synchrotron facility but can also be implemented on a laboratory source by using a source grating to create an array of slit sources. The new approach is based on the similarities of the crystal analyzer method and grating inter-ferometry. Both of the techniques measure the refraction angle signals, which contain the phase information of the sample. The phase information can be obtained by adjusting the gratings to a position where the intensity curve is linear. This approach reduces the amount of projections required for the reconstruction of the final image and Zhu etal. obtained en-hanced contrast compared to the phase-stepping protocol. The use of the new method, named reverse-projection method, eliminated ring artifacts and horizontal stripe artifacts that were present in the images of the same object obtained using the phase-stepping method. 
In order to obtain tomographic images with high quality using grating interferometry the setup needs to be carefully designed. This includes but is not limited to the precise placement of the gratings with respect to one another, the grating fabrication process, and the stability of the system . The quality of the gratings is essential for the setup and largely depends on the process by which they are made with . In addition, the gratings are required to have the correct ratio of periods in order for the setup to produce valuable information. For an incoming plane wave, the period of the analyzer grating p2 is given by the period of the phase gratingp1 by p2 =p1/2 and for an incoming spherical wave with radius L, the period is given by p2=p1/2·L/(L−d) [17, 32, 52].
Silicon is a popular choice for the gratings used in DPC imaging [17, 27, 52]. The phase grating should induce a phase shift of π onto the passing X-rays and the analyzer grating should be highly absorbing . The silicon grating structures can be manufactured by a process including electron-beam lithography, and wet chemical etching into silicon. Addi-tionally, for the absorption and source gratings electroplating of gold is required [17, 27, 52].
The source grating does not require as precise manufacturing as the dimensions of the grating are larger than that of the phase grating and absorption grating . The fabrication of the phase and absorption gratings requires extreme precision as the period of the phase grating is under 10µm and the period of the absorption grating needs to be half of that of the phase grating. In addition, electroplating of the gold into the absorption grating rarely results in complete filling of the grooves.  David etal. have been involved in the development of the grating interferometer technique and have also provided a detailed study on the grating fabrication process .
As with all novel techniques, grating-based interferometry also has limitations for the introduction of the technique into clinical use. Many of the studies using grating interferom-etry have been done on synchrotron sources, which are not appropriate for clinical use. 
The use of the source grating has shown promising results in transferring the technique to be used with polychromatic X-ray sources . Additionally, acquisition and post-processing times of phase-contrast images are long. This is mainly due to the fact that these issues have
not been of primary concern to researchers and could significantly be reduced with research and the development of suitable algorithms and protocols. Imaging of human tissues in-vivo also presents issues as structures of interest are inside different tissues that are of varying densities. Additionally, the introduction of phase-imaging into clinical settings requires the assessment of radiation dose imparted on the subject of interest.