Sample phase gradient and fringe phase shift in triple phase grating X-ray interferometry.

Triple phase grating X-ray interferometry is a promising new technique of grating based X-ray differential phase contrast imaging. Accurate retrieval of sample phase gradients from measured interference fringe shifts is a key task in X-ray interferometry. To fulfill this task in triple phase grating X-ray interferometry with monochromatic X-ray sources, the authors derived exact formulas relating sample phase gradient to fringe phase shift. These formulas not only provide a design optimization tool for triple phase grating interferometry, but also lay a foundation for quantitative phase contrast imaging.


Introduction
X-ray phase contrast imaging, which holds promising potential for medical imaging, can also be implemented by grating-based X-ray interferometry [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Currently most of the grating-based X-ray interferometry is implemented as Talbot-Lau interferometry, in which a single phase grating is employed as a beam splitter to split the X-ray into diffraction orders. The interference between the diffracted orders forms intensity fringes. High-contrast interference fringes can be formed at certain so-called Talbot-distances downstream [1][2][3][4]. To increase the grating interferometer's sensitivity, fine-pitch phase gratings of periods as small as a few micrometers or even sub-micrometers should be used. In this way the intensity fringe period is of comparable size as that of the phase grating period. Nevertheless, in medical imaging, it is only feasible to utilize hospital-grade imaging detectors, whose pixels are of a few tens of micrometers. To enable the fringe detection with common image detectors, one method is to use a fine-pitch absorbing grating placed at the entrance of the detector. This analyzer grating enables one to indirectly detect the fringe patterns through grating scanning [1][2][3][4]. By analyzing the fringes, a sample attenuation image, a phase gradient image, and a dark field image are reconstructed. However, the absorbing grating analyzer blocks more than half of the information-carrying X-ray, thereby greatly increases radiation dose in the imaging studies. This is a disadvantage of the Talbot-Lau X-ray interferometry for medical imaging applications, where radiation dose involved should be as low as reasonably achievable.
The recent emergence of dual and triple phase grating interferometry brings a solution for the dose inefficiency problems of the Talbot-Lau interferometry [17,18]. One of the unique advantages with the use of dual and triple phase gratings is that it enables the use of hospital-grade imaging detectors for resolving intensity fringes generated by the fine-pitch phase gratings, without the need of an absorbing grating [17,18]. In a previous work [19][20][21], based on a quantitative theory of the dual phase grating interferometry, we clarified the mechanism of the intensity fringe formation in dual phase grating interferometry. Indeed, each phase grating splits impinging X-ray into different diffraction orders. The diffracted waves interfere with one another, creating intensity fringes of different diffracted orders. A common imaging detector, through its pixel averaging effects, resolves only the beat patterns of large periodicities. Ge et al. [22] analyzed the formation mechanism of the dual-phase grating interferometry from the geometrical optics point of view. In this work, the understanding gained from the dual phase grating interferometry is extended to the triple phase grating interferometry. We found that the same fringe formation mechanism works for triple phase grating interferometry, as will be explained in Section 2. A rigorous analysis for triple phase grating interferometry is given in Appendix.
The focus of this work is to understand, for triple phase grating interferometry, what are the attributes of an interferometry setup that determine the fringe phase shift ensued from sample refraction. This is an important task for interferometry for the following reasons. When a sample is introduced in an interferometer, sample refracts X-ray and distorts the original intensity fringe. Specifically, the local refraction angle α(x, y) is proportional to the sample's phase gradient ∂Φ s (x, y)/∂x [3,4]: where λ is the X-ray wavelength, and Φ s (x, y) is the sample's X-ray phase shift. Of note, Φ s (x, y) = −λr e ∫ ρ e (x, y, z) dz, where ρ e (x, y, z) is the sample electron density distribution and r e is the classical electron density which equals to 2.82E-15m. One important kind of fringe distortion is the localized fringe phase shifts Δϕ(x, y), which can be measured through phase stepping or Fourier analysis of intensity fringe pattern [1][2][3][4]. The map of the retrieved Δϕ(x, y) provides a differential phase contrast image of the sample. Furthermore, it is desirable to find out the functional relationship between the intensity fringe phase shift Δϕ and the sample phase gradient ∂Φ s (x, y)/∂x for two reasons. First, this functional relationship will enable one to retrieve a map of ∂Φ s (x, y)/∂x from the measured map of the fringe phase shifts Δϕ. The retrieved sample gradients ∂Φ s (x, y)/∂x from the angular projections can be used to reconstruct the 3D maps of the sample electron densities. Second, for a given projection, this functional relationship can be used to determine the ratio of |Δϕ/ 2πα|, which is called as the angular sensitivity of a interferometry setup in literature. The angular sensitivity of a setup tells the fringe phase shift generated by a unit refraction angle. This ratio depends not only on the configuration of the interferometer and phase grating periods, but also on the position of the sample in the ray path. Hence, the angular sensitivity provides a design optimization tool for interferometry.
The functional relationship between the fringe shift Δϕ and the sample phase gradient ∂Φ s (x, y)/∂x in the Talbot-Lau X-ray interferometry is well described in literature [23]. Recently we derived the corresponding functional relationship in dual phase grating X-ray interferometry [21]. In this work we set out to derive the formulas relating the fringe shift Δϕ and the sample phase gradient ∂Φ s (x, y)/∂x for triple phase grating interferometry. We hope this work will stimulate more studies on triple phase grating interferometry.

Methods
In order to find out the functional relationship between the fringe shift Δϕ and the sample phase gradient ∂Φ s (x, y)/∂x in triple phase grating interferometry, one should understand how the intensity fringe is formed. To start, we first give an intuitive explanation of the intensity fringe formation mechanism in the triple-phase grating interferometry. Figure 1 shows the geometrical configuration of a triple-phase grating interferometer. It consists an X-ray source S, three phase gratings G 1 , G 2 , and G 3 and an hospital-grade detector D. The periods of the three phase gratings are p 1 , p 2 , and p 3 respectively. As in the geometric configuration setup of the triple-phase grating interferometry( Fig. 1), R s denotes the sourceto-G 1 grating distance, R g12 , R g23 are the spacings of gratings G 1 , G 2 , and gratings G 2 , G 3 respectively, and R d marks the G 3 -to-detector distance. For convenience, we also denote L i the source-to-G i distance, i = 1, 2, 3, and L D the source-to-detector distance. That is L 1 = R s , L 2 = L 1 + R g12 , L 3 = L 2 + R g23 , and L D = L 3 + R d . The magnification factor of G i -todetector plane is M gi = L D /L i , i = 1, 2, 3. In addition, we denote the source-to-sample distance by L s .
In a previous work, based on the Wigner distribution formalism of wave diffraction, we developed a general theory of dual phase grating interferometry [19]. Using the same approach, we established a general theory for triple-phase grating interferometry. The derivation is outlined in the Appendix. According to this theory, the X-ray irradiance at the detector entrance is a sum of different diffracted orders, each of them is represented by an irradiance fringe pattern of the form exp [i2πx · (l/(M g1 p 1 ) + r/(M g2 p 2 ) + v/(M g3 p 3 ))], where l, r, and v are integers. Here, without going deeply into the theoretical derivation, we offer a heuristic physics picture of the fringe formation. First imagine what is the irradiance pattern in absence of the G 2 an G 3 gratings. Obviously, the irradiance pattern would be a sum of different diffraction orders generated by the G 1 phase grating alone. Each of the diffraction orders would be represented by exp [i2πx · l/(M g1 p 1 )], where l is an integer indexing the diffraction order. Similarly, in absence of the G 1 and G 3 gratings, the intensity pattern generated by the G 2 grating alone would be the sum of different diffraction orders, each of which is represented by exp [i2πx · r/(M g2 p 2 )], where r indexes the diffraction order generated by the G 2 phase grating alone. By the same argument, the G 3 grating alone would generate intensity patterns comprising diffracted orders of the form exp [i2πx · v/(M g3 p 3 )].
As is rigorously shown in Appendix, the X-ray irradiance at the detector entrance is a result of cross-modulation between the fringe patterns generated by each of the three phase gratings G 1 , G 2 and G 3 , respectively. As a result the irradiance pattern at the detector entrance is a weighted sum of different diffracted orders, each of which is represented by a product: where the triple integers (l, r, v) represent the diffracted orders in the triple-phase grating interferometry. The closed form equation of the weight, i.e., the magnitude of the diffracted order is Q(l, r, v) defined in Eq. (15) in the Appendix. For a given diffracted order (l, r, v), Eq. (2) shows that its period is given by Eq.
(3) presents a general formula of fringe period for any diffracted order (l, r, v) in the triple phase grating interferometry.
As is mentioned in the Introduction, a necessary condition for high-sensitivity interferometry is to use fine period phase gratings of micrometer or sub-micrometers. Nevertheless, in the context of grating based phase contrast imaging, it is desirable to use a hospital-grade imaging detector with pixels of tens micrometers. Hence, as long as the detector pixel size p D ≫ p 1 , p 2 , p 3 , most of the fine fringe patterns are rendered to a constant background.
However, with proper setup, some of the fringe patterns can have much broad fringe periods and these broad fringe patterns can then be resolved by the detector. These broad fringe patterns are indeed the beat patterns corresponding to diffraction orders with a specific order, say (l, r, v). To determine which (l, r, v)-order could correspond to broad fringes, one needs to investigate Eq. (3) to find a suitable geometric setup, i.e. appropriate M g1 , M g2 , M g3 , and the order (l, r, v). In Section 3.2, we give an example on how to find these broad period fringe patterns. Once we have understood the fringe formation mechanism, we are ready to tackle the problem of fringe phase shift caused by sample refraction.
Before investigating the fringe phase shift caused by sample refraction in triple-phase grating interferometry, let's consider a single-phase grating interferometer setup as is shown in Figs. 2(a) and (b). As can be seen from Fig. 2(a), when sample is down stream from the phase grating G, the refracted ray causes a lateral fringe shift of Δx = α · (L d − L s ), where L d is the source-to-detector distance, L s is the source-to-sample distance. Since the period of the l-th order intensity fringe is M g p/l, where M g = L d /L g is the magnification factor from grating to detector, and p is the grating period, and a lateral fringe shift of one period is equivalent to a fringe phase shift of 2π. Therefore, for the l-th order fringe, the phase shift caused by the sample refraction is: However, if the sample is upstream of the grating G, the refracted ray makes an effective angle α eff to the locally unperturbed ray, as is shown in Fig. 2(b). The effective angle differs from the sample refraction angle α. Referring Fig. 2(b), note that β ≈ tan(β) = ξ/L g , and α ≈ tan(α) = ξ/(L g − L s ). One has α eff = α − β ≈ α · (1 − (L g − L s )/L g ) = α · (L s /L g ). Hence the sample refraction results in a lateral shift of the grating associated fringe by Δx = α eff · (L d − L g ) = α · (L s /L g ) · (L d − L g ). So for the l-th order intensity fringe, the phase shift caused by the sample refraction is Therefore combining Eq. (4) and Eq. (5), the phase shift can be written as Here min{a, b} denotes the smaller number of a, and b.

Fringe phase shifts caused by sample refraction
Now consider a triple-phase grating interferometer as is shown in Fig. 1, in which the gratings are marked as G i with i = 1, 2, 3 and various interval lengths are indicated. As is explained in Section 2, the interference fringes in triple-phase grating interferometry are consist of various diffraction orders. As is shown in Eq. (2), each of the orders is indexed by three integers (l, r, v) and represented by a product of three exponentials, which represent the intensity modulation generated by each individual phase grating. This being so, the samplegenerated phase shift for a fringe of order (l, r, v) will be a sum of the three fringe phase shifts associated with each individual phase grating. Therefore, we have the samplegenerated fringe phase shift as: where λ is X-ray wavelength.
To the best of our knowledge, this is the first study that derived out the mathematical relationship between the interference fringe shift and the sample phase gradient for triple phase grating interferometry. Equation (7) shows that the sample generated fringe phase shift is proportional to the sample phase gradient with a proportion constant ξ(l, r, v). Note that ξ bears rich physics with it. First, the ratio |ξ| / λ is equal to the angular sensitivity |Δϕ/ (2πα)| [21,23], which is a measure of fringe phase shift generated per unit sample-refraction angle. Hence Eq. (7) can be used as a guiding tool for optimizing angular sensitivity. Obviously, Eq. (7) dictates that ξ = 0 when sample is placed on source plane or detector entrance. Moreover, Eq. (7) implies that for a given geometric configuration, ξ is a piecewise linear function of the source-to-sample distance L s with breakpoints L s = 0, L 1 , L 2 , L 3 , and L D . So when sample position changes within two adjacent breakpoints the angular sensitivity varies monotonically. Therefore, the maximum of |ξ| will be achieved only at one of the breakpoints. In other words, the maximum of |ξ| will occur at one of the grating planes. This feature revealed by Eq. (7) suggests a practical way to optimize angular sensitivity. In order to find the maximal value of |ξ|, using Eq. (7), one only needs to compare the calculated sensitivity values for sample being placed at each of the three phase grating planes and selects the largest one. Second, the proportional constant ξ is also called the auto-correlation length in literature [4,5,18]. This quantity ξ determines the sensitivity of a setup for detecting small angle X-ray scattering from the fine structures of the sample. But this topic is out of the scope of this study [4,5,18].
To see how to use the general formula of Eq. (7), we divided possible sample positions into four intervals. They are (i) the interval between source and G 1 grating, (ii) that between G 1 and G 2 gratings, (iii) that between G 2 and G 3 grating, and finally, (iv) that between G 3 grating and the detector entrance. We applied Eq. (7) in each of the four intervals and found the corresponding interference fringe phase shifts as follows.
In interval (i) the fringe shift is given by: In interval (ii) the fringe shift is: Δϕ ii (l, r, v) = λ ∂Φ s (x, y) ∂x ⋅ L s l M s − 1 M g 1 p 1 + r M g 2 − 1 M g 2 p 2 + v M g 3 − 1 M g 3 p 3 .
In interval (iii) the fringe shift is: In interval (iv) the fringe shift is: The M s in Eqs. (9)(10)(11) is the magnification from sample plane to detector entrance M s = L D / L s .

Predictions on fringe period and angular sensitivity of a triple-phase grating interferometry setup
The performance of a triple-phase grating interferometer is governed by several aspects: (i) if the generated fringe period is large enough to be detected by a common detector; (ii) if the visibility of the fringe pattern is high enough to tolerate noise; (iii) with a given triple-phase grating interferometer, where should we place the sample to get higher angular sensitivity?
Specifically, assume we employ three π-phase gratings of design energy E D = 20keV, and X-ray beam is monochromatic [17]. The periods for the three gratings are p 1 = p 3 = 2p 2 = p = 1μm. According to Eq. (3) in Section 2, the fringe period for diffracted order (l, r, v) is To select a fringe for detection, one should make sure that the fringe has adequate intensity modulation, and thereby good visibility. As is shown in the Appendix, the magnitude of the diffracted order of (l, r, v) is given by Q(l, r, v) defined in Eq. (15) in the Appendix, which is inversely proportional to l, r, and v. In order to attain a high intensity modulation and thus a high visibility in beat fringe patterns, we should choose |l|, |r| and |v| as small as possible.
Since all phase gratings employed are π-gratings, the first nonzero term of the modulation coefficients C, as is defined in Eq. (18) of the Appendix, is |l| = |r| = |v| = 2. Hence to form a beat pattern as explained in Section 2, a choice of l = −r = v = 2 is preferred. In this way, the fringe period of the diffracted order of (2, −2, 2) is reduced to p fr = L D 2 R g 23 − R g 12 ⋅ p . (13) So as long as one sets |R g23 − R g12 | ≪ L D , the total interferometry distance, the fringe period is much larger than the phase grating periods. Large fringe period is an attribute that are seeking for. In the simulation, setting L D = 2m, and (R s , R g12 , R g23 , R d ) = (0.5, 0.5, 0.51, 0.49) meters respectively, we will expect that the fringe period is p fr = 100μm, which is well resolvable by a common hospital-grade detector.
To affirm the assertion above, we perform numerical simulation through wave propagation. In the simulation we assume a 20keV point X-ray source and a detector of period p D = 20μm. The three phase gratings are π gratings at design energy of 20keV and periods p 1 = p 3 = 2p 2 = 1μm [24]. As is stated in the last paragraph, with the setup of L D = 2m, and (R s , R g12 , R g23 , R d ) = (0.5, 0.5, 0.51, 0.49) meters, the beating fringe period is p fr = 100μm. So one fringe period will occupies 5 pitches in the detector. Figure 3 shows the intensity fringe pattern, which confirms the theoretical results. Several fringe patterns of different X-ray energies are shown in Fig. 3. As photon energy changes, the fringe diffraction order may either be unchanged, or be changed to the l = 1 diffraction order from l = 2 for the triple π grating setups. Hence, the fringe period is either unchanged or is doubled, as is shown by Eq. (12), or by the general formula of Eq. (3). Meanwhile, the magnitude of fringe modulation varies significantly with photon energy. Under this setup, the fringe visibility, defined as V = (max(I) − min(I))/(max(I) + min(I)), is as high as V = 0.46 as is shown from numerical simulation in Fig. 3 for X-ray energy E D = 20 keV. Now with the given setup, we turn on to find the optimal sample position so that the angular sensitivity is maximized. First, let us look at what Eq. (7) predicts. Recall from Eq. (7) that the factor ξ is a piecewise linear function of sample position L s . |ξ| is maximized when sample is at one of the grating plane, since ξ = 0, when L s = 0 or L D . By setting L s = L 1 , L 2 , or L 3 , one finds the corresponding values of ξ are −(5.00E3)λ, −(1.01E6)λ, and (4.90E3)λ respectively. Here λ is the X-ray wavelength, which equals to 6.2E-11m for a 20keV X-ray source. A plot of ξ/λ with respect to L s is shown in Fig. 4, where L s spans from L 1 /2 to (L 3 + L D )/2. Thus Eq. (7) predicts that L s = L 2 is an optimal sample position where the interferometer's angular sensitivity is maximized.

Simulations to validate the theory
To validate the Eq. (7) in Section 3, we perform a numerical simulation with a sphere of diameter 5mm placed half way between the second and third gratings with L s = 1.255m. The sphere is filled with a hypothetical tissue, which has the same chemical composition as breast adipose tissue but with lower mass density. In this way the retrieved fringe phase shifts are less than π. We employed ray-tracing to get the projected sample attenuation A 2 (x, y), and the phase map of the sample. From the sample phase map we computed the sample phase gradients. Using Eq. (7) and the sample phase gradients we compute the theoretical map of the fringe shift Δϕ(x, y) caused by sample refraction. The theoretical map of the sample attenuation A 2 (x, y) and the map of the fringe phase shift Δϕ(x, y) are shown in Figs. 6(a) and 6(b) respectively. We then employed Fresnel propagation down stream through G 1 , G 2 , the sample sphere, the grating G 3 and finally to the detector plane to get the projected image. The simulated fringe patterns at the detector, with and without samples, are shown in Fig. 5. The fringe phase shift Δϕ(x, y) and the attenuation map A 2 (x, y) were retrieved with the Fourier fringe analysis method developed in [14][15][16]25], which we call the FT-retrieval for short. The maps of the retrieved attenuation and the fringe phase shift are shown in Figs. 6(b) and 6(e) respectively. As comparison, profiles of the attenuation A 2 (x) and the fringe phase shift Δϕ(x) along the central line across grating are shown in Figs. 6(c) and 6(f) respectively. The red-dash curves represent the theoretical values, while the blue-solid curves are the retrieved values from the intensity fringes. The good match between the theoretical and FT-retrieved values of Δϕ(x) validates the fringe phase shift formula of Eq. (7). The The oscillations in Fig. 6(c) is due to the FT-phase retrieval errors, which results from the rapidly varying attenuation at the sphere boundary. To avoid the infinity values of the phase gradient at the edges of the sphere, in the simulation we slightly modified the sphere boundary to render the boundary values finite.

Discussion and conclusions
The functional relationship between fringe phase shifts and sample phase gradients, as is derived in Eq. (7), enables the retrieval of sample phase gradient in triple phase grating interferometry. Obviously, it lays a foundation for quantitative differential phase contrast imaging. Moreover, the volumetric images of the sample electron densities can be reconstructed from the sample gradients retrieved in different angular projections. On the other hand, for implementing triple phase grating interferometry, Eq. (7) is especially useful. In fact, the angular sensitivity of an interferometry setup is an indicator of its imaging sensitivity. The angular sensitivity of an interferometer depends not only on the phase shifts and periods of the three phase gratings, but also on the geometric configuration of the interferometer and the sample position. There is a pressing need of a design optimization tool to predict how sensitive a specific setup is. The functional relationship derived in Eq. (7) enables one to predict the angular sensitivity of an interferometer setup with triple phase gratings. In this work, as an example, we studied the angular sensitivities of three π phase grating setups. As is shown in Fig. 4, this interferometer demonstrates large variations in the angular sensitivity for different sample placements. Hence Eq. (7) provides a useful tool for system optimization. We stress that, our formula of Eq. (7) is applicable to any triple phase grating setups. For example, one may adopt setups with π/2-π-π/2 phase gratings [17]. By using Eq. (9), we found that, in the π/2-π-π/2 setups with identical grating period of 1μm, the angular sensitivity peaks as well when the sample is on the G 2 grating plane, but the sensitivity is different. This is because the beat fringe patterns of the three π grating setups attain a diffraction orders (l, r, v) = (2, −2, 2) and its complex conjugate, but the beat fringe patterns of π/2-π-π/2 setups are of diffraction orders (l, r, v) = (1, −2, 1) and its complex conjugate. So Eq. (7) dictates that the angular sensitivity of the π/2-π-π/2 setups are half that of the three π phase grating setups. Due to the space limitation, we are unable to discuss more details about the various implementations of triple phase grating interferometry.
Compared to Talbot-Lau interferometry, a decisive advantage of triple phase grating setups with fine grating periods lies in that the fringe period can be directly resolved by common imaging detectors. This is good for medical imaging applications, as is explained in the introduction. Of course, the dual phase grating interferometry can be implemented with common imaging detectors as well. As for angular sensitivities with dual phase grating setups, we derived a corresponding formula in a previous work [18,21]. In that work we showed that, for dual phase grating setups, the optimal sample placement is on the first grating plane. Apparently, this conclusion is different from what we found in Section 3 for setups with triple phase gratings. Compared on the basis of the same grating periods and total system length, an optimal triple phase grating setup can achieve few times higher in angular sensitivity than that with dual phase grating setups. With triple phase grating setups, there are more pathways for wave interference and beat patterns formation, as compared to the dual phase grating setups. In addition, in dual phase grating interferometry the spacing between the two phase gratings are very narrow for allowing large fringe period [19]. In the triple phase grating interferometry the space between gratings are usually ample and convenient for sample placement. In addition to fringe visibility, the average intensity of the fringe patterns also determines the quality of the sample phase gradient images. Exactly speaking, it is the ratio of the 1st-order to the 0th-order Fourier components of the fringe patterns that determines image quality [26]. As a disadvantage of triple phase grating interferometry, note that its fringe visibility is generally lower than that of the Talbot-Lau and the dual phase grating interferometry. This is because the intensity modulation is essentially a product of the modulation contributed by each of the three phase gratings, as is shown in Appendix. Another limitation of this work lies in that we did not include polychromatic X-ray in our derivation. Polychromatic X-ray can distort fringe phase shifts by spectral averaging and beam hardening [27][28][29][30]. More research is needed on these issues.
In conclusion, in this work, by using intuitive analysis of the beat pattern formation and sample refraction in triple phase grating interferometry, the authors derived exact formulas relating sample phase gradients to fringe phase shifts. These formulas not only provide a design optimization tool for triple phase grating interferometry, but also lay a foundation for quantitative phase contrast imaging.
NB In the manuscript review process, we were informed by a reviewer that there is a recent preprint [31], which also investigates the angular sensitivity of triple phase grating interferometry. Different from our work, the core method adopted by this preprint is based on the thin lens imaging theory developed for the dual phase grating interferometer [22]. The preprint does not present any explicit formula of angular sensitivity for triple grating interferometry.

Funding
National Institutes of Health (1R01CA193378).
The constant I in in Eq. (14) is the intensity at the first phase grating G 1 's entrance, and μ in is the complex coherence degree, which is defined as μ in (Δ) = ∫ I src (s) ⋅ exp [i2πsΔ] ds ∫ I src (s) ds , (19) where I src is the X-ray source intensity distribution.
Due to the space limitation, we present an outline of the proof. We use the Wigner function formalism to derive a general form of the equation. The Wigner function of an X-ray wavefront at z = s 1 is defined as the Fourier transform of the mutual intensity of the wavefront J s1 (x 1 , x 2 ; y) [32][33][34]: As the wave propagates, the Wigner function evolves from z = s 1 to z = s 2 in the following way [32][33][34]: W s 2 − 0 (x, u; y) = W s 1 x − λ s 2 − s 1 ⋅ u, u; y .
Repeating the process of Wigner function evolution from G 2 through G 3 downstream to the detector's entrance, we get the mutual intensity, after a tedious computation, at the detector entrance: where W = (w 1 , w 2 , w 3 ), where W = (w 1 , w 2 , w 3 ), Since S i is a periodic function of period p i , i = 1, 2, 3 (see Eqs. (23), (30), and (32)), following the same arguments presented in the Appendix of [35], one can show that for real numbers v and s, S i (v, s) = Ψ i (v, s) ⋅ Ξ 1/p i (v), where Ξ q (r) = ∑ n ∈ ℤ δ(r − nq), δ is the Dirac delta function, and when v = n/p i , n is integer, Ψ i can be analytically expressed in the form of Eq. (18). Substitute Eq. (36) back to Eq. (34), after a tedious computation, one finishes the proof of the assertion. Schematic of an X-ray triple-phase grating interferometer with a micro-spot X-ray source.  Numerical simulation on fringe period of beating fringe patterns. In the simulation, a point X-ray source is assumed and a 20μm detector is employed. The phase gratings G 1 , G 2 , and G 3 are π gratings at design energy E D = 20keV with periods p 1 = p 3 = 2p 2 = 1μm. The geometric setup for the interferometry is L D = 2m, and (Rs, Rg 12 , Rg 23 , R d ) = (0.5, 0.5, 0.51, 0.49) meters respectively. To compare the influence of X-ray energy to fringe visibility, multiple X-ray sources (E = 15, 17.5, 20, 22.5 and 30 keV respectively) are simulated. As can be seen, the visibility is maximized when X-ray energy equals the phase grating design energy E D = 20 keV for this geometric setup. The change of the signed angular sensitivity (ξ/λ) with respect to the sample position.  Simulated fringe patterns at the detector, with (Fig. 5(a)), and without the object (Fig. 5(b)).  Simulation results for validation of Eq. (9). In this simulation, we employ the same interferometry setup as in Fig. 3. A 5mm diameter sphere filled with 100% adipose tissue is placed half way between the second and the third gratings with L s = 1.