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Dual phase grating based X-ray differential phase contrast imaging with source grating: theory and validation

Yongshuai Ge, Jianwei Chen, Peiping Zhu, Jun Yang, Shiwo Deng, Wei Shi, Kai Zhang, Jinchuan Guo, Huitao Zhang, Hairong Zheng, and Dong Liang

Open Access Open Access

Abstract

In this work, we developed a new theoretical framework using wave optics to explain the working mechanism of the grating based X-ray differential phase contrast imaging (XPCI) interferometer systems consist of more than one phase grating. Under the optical reversibility principle, the wave optics interpretation was simplified into the geometrical optics interpretation, in which the phase grating was treated as a thin lens. Moreover, it was derived that the period of an arrayed source, e.g., the period of a source grating, is always equal to the period of the diffraction fringe formed on the source plane. When a source grating is utilized, the theory indicated that it is better to keep the periods of the two phase gratings different to generate large period diffraction fringes. Experiments were performed to validate these theoretical findings.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Over the past two decades, the grating-based X-ray interferometry imaging, especially the Talbot-Lau imaging, has received a huge amount of research interests. In principle, three different images with unique contrast mechanism can be generated from the acquired same data. They are the absorption, the differential phase contrast (DPC), and the dark-field (DF) images. Usually, both the DPC and DF signals are considered as complimentary contrast information to the conventional absorption contrast information. Studies have shown that the DPC signal, which corresponds to the X-ray refraction information, may have advancements in providing superior contrast sensitivity for certain types of soft tissues [1–9]. Additionally, the DF signal, which corresponds to the small-angle-scattering (SAS) information, is particularly sensitive to certain fine structures such as microcalcifications inside the breast tissue [10–16].

Aiming at translating such a promising imaging technique into biomedical applications, a lot of efforts have been made to overcome the two major potential challenges encountered by the current Talbot-Lau imaging method. The first challenge is the prolonged data acquisition period due to the time consuming phase stepping procedure, and the second challenge is the reduced radiation dose efficiency due to the use of the post-object analyzer grating. During the past decade, different techniques [17–22] have been developed to shorten the data acquisition period. For instance, Zhu et al. suggested to extract the phase information with only two phase steps. Marschner et al. developed a fast DPC-CT imaging by integrating the helical scan with the phase stepping procedure. Miao et al. developed a novel DPC imaging method via a motionless phase stepping procedure using an electrically controlled focal spot moving technique. Ge et al. proposed a new analyzer grating design which integrates multiple phase stepping procedure together to achieve single-shot DPC imaging. The interference patterns can be directly recorded without the necessity of an analyzer $G_2$ grating if X-ray detectors having ultra-high spatial resolution are employed [23,24]. Moreover, the inverse Talbot-Lau geometry proposed by Momose et al. is also a promising approach to enlarge the periods of the interference patterns and thus ease their direct detection [25]. To realize the inverse Talbot-Lau imaging, recently, novel X-ray source embedded with periodic microstructured array anode has been developed and optimized [26,27]. Overall, these innovations are able to either significantly reduce the number of phase steps, or totally eliminating the mechanical phase stepping procedure to realize single-shot DPC imaging. Therefore, the total data acquisition time can be greatly reduced down to the similar level as of the conventional absorption imaging.

Alternatively, other studies have managed to replace the analyzer grating by one or two phase gratings [28–30]. Such new interferometer systems might potentially have better radiation dose efficiency, as have been experimentally demonstrated by Miao et al. with three nanometer phase gratings based far-field X-ray interferometer. Kagias et al. also have demonstrated the feasibility of using two phase gratings to realize DPC imaging. However, the requirement of micro-focus or mini-focus X-ray source in the above two pioneering experimental studies may still hurdle the wide applications of these new techniques. To meet the demand of high tube output flux, one solution is to add a source grating before an X-ray source with large focal spot size. This method is based on the Lau effect [31], and has already been widely used in the Talbot-Lau interferometer systems. Until now, this approach might be the most convenient solution to simultaneously overcome both the low beam flux problem and the low beam coherence problems.

So far, several theoretical explanations have already been developed for the dual phase grating XPCI systems [29,32]. To our best knowledge, unfortunately, there are no straightforward physical pictures to ease the interpretation of such a dual phase grating XPCI system. Therefore, this work attempted to provide a geometrical optics representation for the dual phase grating XPCI system based on wave optics. Moreover, we also hoped to derive a rigorous theoretical analyses on predicting the source grating period in a dual phase grating interferometer system.

The remains of this paper is organized as follows: the section II presents the theoretical analyses, the derived general representation of the diffraction fringe, the analyzer grating period, fringe visibility and period. In the section III, details of the experiments and results are presented. We discuss this work in section IV, and make a brief conclusion in section V.

2. Theory and results

In the theoretical discussions, assuming the dual phase grating based XPCI system is as follows: the X-ray source (arrayed or single slit) is positioned on the left, and the X-ray detector is positioned on the right. The default X-ray beam is monochromatic and propagates from the left to the right. The gratings are positioned between the source and detector. In particular, the distance between the source and the first phase grating (denoted as $G_1$) is $d_1$, the inter-space between the first phase grating and the second phase grating (denoted as $G_2$) is $d_2$, and the distance between the second grating and the detector (observation plane) is $d_3$, see Fig. 1.

 figure: Fig. 1.

Fig. 1. Illustration of the thin lens interpretation of a dual phase grating XPCI setups. The $G_1$ and $G_2$ are phase gratings. Herein, the solid arrows represent the forward imaging procedure, in which an arrayed source (in red color) positioned on the left hand side is assumed. Additionally, the backward imaging procedure is schemed by the dashed arrows, in which the arrayed source (in purple color) is assumed on the right hand side.

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The diffraction gratings are assumed having a form of periodic transmission function, denoted as $\textrm {T}(x)$. Its Fourier expansion is written as:

$$\textrm{T}(x)=\sum_{n=-\infty}^{n=\infty}a_{n}e^{\frac{i2\pi nx}{p}},$$
where $n$ represents the diffraction order, and $p$ denotes the grating period. Depending on the grating diffraction order $n$, the value of the coefficient $a_n$ varies. By default, the phase gratings used in this work have duty cycle of 0.5. In this paper, the duty cycle is defined as the fraction of the opening. For simplicity, this work only considers 1D gratings. Nevertheless, this theoretical analysis framework is also capable of dealing with 2D gratings.

To proceed, the standard Kirchhoff’s diffraction theory [33] is used to analyze the wave propagation procedure, i.e.,

$$\textrm{U}(x)=\frac{\textrm{U}_{0}e^{ikz}}{i\lambda z}\int\textrm{S}(\xi)e^{ik\left[\frac{(x-\xi)^{2}}{2z}\right]}d\xi,$$
where $\textrm {U}(x)$ denotes the X-ray field disturbance at location $x$, $\textrm {U}_{0}$ is the amplitude of the initial disturbance of the X-ray source, $\lambda$ denotes the wavelength of the assumed monochromatic X-ray beam, $z$ axis is along the wave propagation direction, and $\textrm {S}(x)$ represents the spatial X-ray source distribution. Take a single slit X-ray source as the example, we have
$$\textrm{S}(x)= \textrm{rect}\left(\frac{x}{\sigma}\right),$$
where $\sigma$ represents the width of the source, i.e., the opening of the slit. If a micro-focus X-ray tube is used, Eq. (3) should be adapted into a Gaussian function.

As shown in Eq. (2), the wave propagation procedure actually corresponds to an 1D convolution. After using the Eq. (2) repeatedly for multiple times, the X-ray field intensity at the detector plane can be derived (details can be found in the Appendix) and expressed as follows:

$$\begin{aligned} \textrm{I}_{3}(x_{3})=&\frac{\textrm{U}_{0}^{2}}{(d_{1}+d_{2}+d_{3})^{2}}\sum_{s=-\infty}^{s=\infty}\sum_{r=-\infty}^{r=\infty}C_{s}\left(\frac{d_{1}(d_{2}+d_{3})s+d_{1}d_{3}\frac{p_{1}}{p_{_{2}}}r}{d_{1}+d_{2}+d_{3}},\lambda, p_{1}, \phi_{1}\right) \\ &C_{r}\left(\frac{(d_{1}+d_{2})d_{3}r+d_{1}d_{3}\frac{p_{2}}{p_{1}}s}{d_{1}+d_{2}+d_{3}},\lambda, p_{2}, \phi_{2}\right)sinc\left(\frac{[d_{3}\frac{r}{p_{2}}+(d_{2}+d_{3})\frac{s}{p_{1}}]\sigma}{d_{1}+d_{2}+d_{3}}\right)\\ &e^{\frac{i2\pi[d_{3}\frac{r}{p_{2}}+(d_{2}+d_{3})\frac{s}{p_{1}}]x_{s}}{d_{1}+d_{2}+d_{3}}} sinc\left(\frac{[(d_{1}+d_{2})\frac{r}{p_{2}}+d_{1}\frac{s}{p_{1}}]p_{del}}{d_{1}+d_{2}+d_{3}}\right)e^{\frac{i2\pi[(d_{1}+d_{2})\frac{r}{p_{2}}+d_{1}\frac{s}{p_{1}}]x_{3}}{d_{1}+d_{2}+d_{3}}}, \end{aligned}$$
where $p_1$ and $p_2$ denote the period of $G_1$ and $G_2$, correspondingly. The $s$ and $r$ denote the difference of conjugated diffraction orders for each grating (see Eq. (31) in the Appendix), $\phi _{1}$ and $\phi _{2}$ represent the X-ray wave-front phase shifts induced on each phase grating, $x_s$ denotes the horizontal coordinate of the source plane, $x_3$ denotes the horizontal coordinate of the detector plane, and $p_{del}$ corresponds to the dimension of an individual detector element. According to the Nyquist-Shannon sampling theorem, in theory the minimum resolvable fringe period is at least two times of $p_{del}$. In this work, we only focused on discussing the $\pi -\pi$ phase grating based XPCI. Based on the results (see Eq. (32) and Eq. (33) in the Appendix) in previous literature [34,35], one has,
$$C_{s}\left(\frac{d_{1}(d_{2}+d_{3})s+d_{1}d_{3}\frac{p_{1}}{p_{_{2}}}r}{d_{1}+d_{2}+d_{3}},\lambda, p_{1},\pi\right)=-\frac{2}{\pi}sin\left(\frac{2\pi[d_{1}(d_{2}+d_{3})\frac{s}{p_{1}}+d_{1}d_{3}\frac{r}{p_{_{2}}}]\lambda}{(d_{1}+d_{2}+d_{3})p_{1}}\right),$$
and
$$C_{r}\left(\frac{(d_{1}+d_{2})d_{3}r+d_{1}d_{3}\frac{p_{2}}{p_{1}}s}{d_{1}+d_{2}+d_{3}},\lambda, p_{2},\pi\right)=-\frac{2}{\pi}sin\left(\frac{2\pi[(d_{1}+d_{2})d_{3}\frac{r}{p_{2}}+d_{1}d_{3}\frac{s}{p_{2}}]\lambda}{(d_{1}+d_{2}+d_{3})p_{2}}\right).$$
Substituting Eq. (5) and Eq. (6) back into Eq. (4), and only considering the smallest values of $s$ and $r$, i.e., $s=-2$ and $r=2$, or $s=2$ and $r=-2$, the following result can be derived:
$$\begin{aligned} &\textrm{I}_{3}(x_{3})=\frac{\textrm{U}_{0}^{2}}{(d_{1}+d_{2}+d_{3})^{2}}\bigg(1+\frac{8}{\pi^{2}}sin\left(\frac{4\pi d_{1}[(d_{2}+d_{3})-d_{3}\frac{p_{1}}{p_{_{2}}}]}{(d_{1}+d_{2}+d_{3})Z_{t,1}}\right)sin\left(\frac{4\pi d_{3}[(d_{1}+d_{2})-d_{1}\frac{p_{2}}{p_{1}}]}{(d_{1}+d_{2}+d_{3})Z_{t,2}}\right) \\ &sinc\left(\frac{2[d_{2}+d_{3}-d_{3}\frac{p_{1}}{p_{2}}]\sigma}{(d_{1}+d_{2}+d_{3})p_{1}}\right) e^{\frac{i4\pi[(d_{2}+d_{3})-d_{3}\frac{p_{1}}{p_{_{2}}}]x_s}{(d_{1}+d_{2}+d_{3})p_1}} sinc\left(\frac{2[(d_{1}+d_{2})-d_{1}\frac{p_{2}}{p_{1}}]p_{del}}{(d_{1}+d_{2}+d_{3})p_{2}}\right)e^{\frac{i4\pi[(d_{1}+d_{2})-d_{1}\frac{p_{2}}{p_{1}}]x_{3}}{(d_{1}+d_{2}+d_{3})p_{2}}}\bigg), \end{aligned}$$
where $Z_{t,1}=\frac {p_1^{2}}{\lambda }$, and $Z_{t,2}=\frac {p_2^{2}}{\lambda }$.

Notice that the derived X-ray field intensity $\textrm {I}_{3}$ at the detector plane in Eq. (4) and Eq. (7) are in complex forms. Physically, only the real part of $\textrm {I}_{3}$, i.e., $\mathrm {Re}(\textrm {I}_{3})$, corresponds to the detectable beam intensity. For simplicity, we will use the general expression $\textrm {I}_{3}$ to represent $\mathrm {Re}(\textrm {I}_{3})$ in later discussions.

2.1 Geometrical interpretation

In previous studies [30,36], it is assumed that there has a virtual image plane (called as the $1^{st}$ image in this study) between $G_1$ grating and $G_2$ grating. Unfortunately, so far this assumption has not been rigorously demonstrated from theory. Herein, we would try to provide an intuitive interpretation of this virtual image plane based on our obtained intensity expression in Eq. (7).

First of all, suppose the $1^{st}$ image formed between $G_{1}$ and $G_{2}$ ensures that the geometrically magnified image of $G_{1}$ (when the X-ray source sits $d_1$ distance to the left) and $G_{2}$ (when the X-ray source sits $d_3$ distance to the right) are identical. Specifically, the distances between the $1^{st}$ image and the $G_1$ grating and $G_2$ grating are $r_1$ and $r_2$, respectively. Thus, having:

$$\frac{d_{1}+r_{1}}{d_{1}}p_{1}=\frac{d_{3}+r_{2}}{d_{3}}p_{2}.$$
The above assumption is made based on the principle of optical reversibility, which presupposes that the path of a light beam propagates through a series of optical media can be replaced by its opposite path if the light emits from the direction opposite to the original direction. In this study, there are two imaging procedures, the forward one and the backward one. In the forward imaging procedure, X-ray beam propagates from the left to the right, as illustrated by the solid lines in Fig. 1. In contrast, X-ray beam propagates from the right to the left in the backward procedure, as illustrated by the dashed lines in Fig. 1.

Upon Eq. (8), both $r_1$ and $r_2$ can be calculated as below:

$$r_{1} = \frac{d_{1}(d_{2}+d_{3}-d_{3}\frac{p_{1}}{p_{2}})}{d_{3}\frac{p_{1}}{p_{2}}+d_{1}},$$
$$r_{2} = \frac{d_{3}(d_{1}+d_{2}-d_{1}\frac{p_{2}}{p_{1}})}{d_{1}\frac{p_{2}}{p_{1}}+d_{3}}.$$
Immediately, the following relationships can be derived:
$$f_{1}:= \frac{d_{1}(d_{2}+d_{3}-d_{3}\frac{p_{1}}{p_{2}})}{(d_{1}+d_{2}+d_{3})} = \frac{r_{1}d_{1}}{r_{1}+d_{1}} = \frac{1}{\frac{1}{d_1}+\frac{1}{r_1}},$$
$$f_{2}:= \frac{d_{3}(d_{1}+d_{2}-d_{1}\frac{p_{2}}{p_{1}})}{(d_{1}+d_{2}+d_{3})} = \frac{r_{2}d_{3}}{r_{2}+d_{3}} = \frac{1}{\frac{1}{d_3}+\frac{1}{r_2}}.$$
Interestingly, results in Eq. (11) and Eq. (12) have the identical form as of the well known thin lens equation. Thus, the $G_1$ grating and $G_2$ grating can be considered as thin lens, $f_1$ and $f_2$ are their corresponding focal length. For the $G_1$ grating, $d_1$ is considered as the object distance, and $r_1$ is considered as the image distance. Similarly, for the $G_2$ grating, $r_2$ is considered as the object distance, and $d_3$ is the image distance, see Fig. 1.

2.2 Fringe visibility and period

Immediately, Eq. (7) can be further simplified into

$$\textrm{I}_{3}(x_{3}) =\frac{\textrm{U}_{0}^{2}}{(d_{1}+d_{2}+d_{3})^{2}}\bigg(1+\frac{8}{\pi^{2}}sin\left(\frac{4\pi f_{1}}{Z_{t,1}}\right)sin\left(\frac{4\pi f_{2}}{Z_{t,2}}\right)sinc\left(\frac{2f_{1}\sigma}{d_{1}p_{1}}\right)e^{\frac{i4\pi f_{1}x_{s}}{d_{1}p_{1}}}sinc\left(\frac{2f_{2}p_{del}}{d_{3}p_{2}}\right)e^{\frac{i4\pi f_{2}x_{3}}{d_{3}p_{2}}}\bigg).$$
From Eq. (13), the fringe visibility, denoted by $\epsilon$, is found to be:
$$\epsilon = \frac{8}{\pi^{2}}sin\left(\frac{4\pi f_{1}}{Z_{t,1}}\right)sin\left(\frac{4\pi f_{2}}{Z_{t,2}}\right)sinc\left(\frac{2f_{1}\sigma}{d_{1}p_{1}}\right)sinc\left(\frac{2f_{2}p_{del}}{d_{3}p_{2}}\right).$$
In Eq. (14), the first and second $sin$ functions are fast oscillating terms, the third and fourth $sinc$ functions are two slowly decaying terms (this is true especially for the main lobes). As a consequence, the optimal visibility is reached when the two $sin$ functions get to their maximum or minimum, namely,
$$\begin{aligned}f^{op}_1 & = \frac{2l_1+1}{8}Z_{t,1}=\frac{2l_1+1}{8}\frac{p_1^{2}}{\lambda},\;\;\; l_1\in\mathbb{Z} \end{aligned}$$
$$\begin{aligned}f^{op}_2 & = \frac{2l_2+1}{8}Z_{t,2}=\frac{2l_2+1}{8}\frac{p_2^{2}}{\lambda},\;\;\; l_2\in\mathbb{Z} \end{aligned}$$
Surprisingly, these optimal focal length $f^{op}_1$ and $f^{op}_2$ are exactly the self-image distance for a standard Talbot interferometer. Different $l_1$ and $l_2$ values will lead to varied dual phase grating XPCI setups. The system length achieves the most compact condition when $l_1$ and $l_2$ are equal to zeros. Based on Eq. (11) and Eq. (12), additionally, the optimal values for $r^{op}_1$ and $r^{op}_2$ can also be determined as follows:
$$\begin{aligned} r^{op}_1 & = \frac{f^{op}_1 d_1}{d_1-f^{op}_1}, \end{aligned}$$
$$\begin{aligned} r^{op}_2 & = \frac{f^{op}_2 d_3}{d_3-f^{op}_2}.\end{aligned}$$
From Eq. (13), the fringe period $p_{f}$ corresponding to the term $e^{\frac {i4\pi f_{2}x_{3}}{d_{3}p_{2}}}$ is equal to:
$$p_{f}=\frac{d_{3}p_{2}}{2f_{2}}=\frac{d_{1}+d_{2}+d_{3}}{2(\frac{d_{1}+d_{2}}{p_{2}}-\frac{d_{1}}{p_{1}})}.$$

This derived fringe period $p_{f}$ of dual phase grating imaging system is exactly the same as of the results have been published in previous studies [29,32]. Since former studies were mainly focused on discussing the dual phase grating imaging with two identical $\pi$ phase gratings, i.e., $p_{1}=p_{2}=p$, therefore, the fringe period $p_{f}$ can be rewritten as:

$$p_{f}=\frac{(d_{1}+d_{2}+d_{3})p}{2d_{2}}.$$
We call such a system as symmetric dual phase grating interferometer system. In this special case, Usually, diffraction fringe with large enough period is desired to ease the detection in practice. As a result, decreasing the inter-grating distance $d_2$ is preferred in order to increase the fringe period $p_{f}$, provided that the phase grating period is fixed.

2.3 Spacing of X-ray sources

Similarly, a virtual period $p_0$ corresponding to term $e^{\frac {i4\pi f_{1}x_{s}}{d_{1}p_{1}}}$ can be defined as below:

$$p_{0}=\frac{d_{1}p_{1}}{2f_{1}}=\frac{d_{1}+d_{2}+d_{3}}{2(\frac{d_{2}+d_{3}}{p_{1}}-\frac{d_{3}}{p_{2}})}.$$
This equation has exactly the same form as of the Eq. (19). Since this virtual periodic pattern is positioned right on the source plane, therefore, it is reasonable to treat $p_0$ as the spacing between neighboring X-ray sources. In practice, such an X-ray source could be made by a group of individual emitters, or could be reached by adding a source grating before a normal X-ray tube with large focal spot size. As shown by the numerical results in Fig. 2, the fringe visibility increases if the source width or the duty cycle decreases. In particular, if the source width $\sigma$ is fixed, the fringe visibility periodically reaches its maximum as the source spacing $p_0$ increases, see the results in Fig. 2(a). Moreover, there only has one single optimal duty cycle corresponding to a certain source spacing $p_0$ to generate the highest fringe visibility, see the results in Fig. 2(b). Since smaller source width or duty cycle means better beam coherence, therefore, the arrayed X-ray source having smaller duty cycle is always recommended.

 figure: Fig. 2.

Fig. 2. Visibility distribution maps with respect to: (a) the source width $\sigma$ and the source spacing $p_0$, (b) the duty cycle and the source spacing $p_0$. The phase grating periods are assumed as $p_1=4.364\;\mu m$ and $p_2=4.640\;\mu m$. In image (a), the upper-left dark region above the dashed line corresponds to $\sigma >p_0$ and should be avoided. In image (b), the three dashed curves correspond to three different $\sigma$ values. In addition, these visibility maps have been normalized by their maximum values.

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By substituting Eq. (19) and Eq. (21) into Eq. (14), the expression of fringe visibility can be simplified into:

$$\epsilon = \frac{8}{\pi^{2}}sin\left(\frac{4\pi f_{1}}{Z_{t,1}}\right)sin\left(\frac{4\pi f_{2}}{Z_{t,2}}\right)sinc\left(\frac{\sigma}{p_{0}}\right)sinc\left(\frac{p_{del}}{p_{f}}\right).$$

2.4 Asymmetric dual phase grating system

Interestingly, if combining Eq. (19) and Eq. (21), $p_f$ and $p_0$ can be rewritten into the following forms

$$\begin{aligned} p_{f}=\frac{p_{0}p_{1}p_{2}}{2p_{0}(p_{1}-p_{2})+p_{1}p_{2}}, \end{aligned}$$
$$\begin{aligned} p_{0}=\frac{p_{f}p_{1}p_{2}}{2p_{f}(p_{2}-p_{1})+p_{1}p_{2}}. \end{aligned}$$
Notice that these expressions are only functions of the grating periods. According to the numerical results in Fig. 3, the $p_1 \le p_2$ relationship should always be guaranteed in order to generate diffraction fringes with large enough periods. This means the $G_1$ phase grating should have a smaller or equal period than the $G_2$ phase grating. Actually, when the source spacing $p_0$ is finite, for example, dozens of micrometers, $p_1 < p_2$ would definitely be a better choice to ensure large period of fringes. In contrast to the previously defined symmetric dual phase grating interferometer system, we call such a dual phase grating imaging system as asymmetric dual phase grating interferometer system. However, as the source spacing $p_0$ increases, the difference between $p_1$ and $p_2$ becomes negligible.

 figure: Fig. 3.

Fig. 3. Fringe period distribution maps obtained with respect to: (a) $p_1-p_2$ (when $p_2=4.640\;\mu m$) and $p_0$, (b) $p_2$ and $p_1$ (when $p_0=24\;\mu m$), and (c) $p_2$ and $p_1$ (when $p_0=60\;\mu m$). Plots in (b) and (c) correspond to the two circles marked on (a), respectively. As can be seen, the difference between $p_2$ and $p_1$ decreases as $p_0$ increases. Due to this reason, the bright band in plot (c) is less deviated from the diagonal direction than in plot (b).

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By far, we have established a rigorous theoretical framework to predict all the key parameters needed in a new dual phase grating interferometer system. The flowchart in Fig. 4 lists one example to perform such parameter estimations. Herein, we assumed the already known parameters are the wavelength of the mean X-ray beam energy, the source grating period $p_0$, the phase grating periods $p_1$ and $p_2$. Following the listed step-by-step calculations, the $d_1$, $d_2$, $d_3$, $r_1$, $r_2$, and the diffraction fringe period $p_f$ were all be estimated. Notice that the system key parameters estimated in this way are all optimal ones that ensure the highest fringe visibility.

 figure: Fig. 4.

Fig. 4. Flowchart to estimate the key parameters of the dual phase grating interferometer system used in our experiments. In this example, the mean beam energy and grating parameters are predetermined. The system geometry and fringe period are parameters to be calculated.

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3. Experiments and results

3.1 Experimental settings

Experimental validations were performed on an in-house XPCI bench. The system includes a rotating-anode Tungsten target X-ray tube (Varex G242, Varex Imaging Corporation, UT, USA). It was operated at 55.00 kV (mean energy of 28.00 keV) with continuous fluoroscopy mode with 0.40 $mm$ nominal focal spot. The X-ray tube current was set at 12.5 mA, with a 20.00 second exposure period for each phase step. The X-ray CCD detector (OnSemi KAI-16000, XIMEA GmbH, Germany) has a native element dimension of 7.40 $\mu m$ and an effective imaging area of 36.00 $mm$ by 24.00 $mm$. The absorption grating $G_0$ has a period of 24.00 $\mu m$, with a duty cycle of 0.35. On this special $G_0$ grating, there are four round active areas with diameter of 2.00 $cm$. The first phase grating $G_1$ has a period of 4.364 $\mu m$, with a duty cycle of 0.50, and the second phase grating $G_2$ has a period of 4.640 $\mu m$, with a duty cycle of 0.50. These grating specifications were provided by the manufacturer (Microworks GmbH, Germany). With these available gratings in our lab, the optimal system imaging setup were calculated as the following: $d_1= 527.90 \, mm$, $d_2 = 108.93 \, mm$, and $d_3 = 1623.43 \, mm$. Moreover, the distance between the X-ray source and $G_0$ was fixed at 400.00 $mm$. In this forward imaging system, the $G_0$ grating is utilized as the source grating.

Keep the same grating setup in the above forward system, another experiment was performed by replacing the Varex G242 X-ray tube by a flat panel detector with native element dimension of 194.00 $\mu m$ (3030Dx, Varex Imaging Corporation, UT, USA), and replacing the XIMEA detector by a micro-focus X-ray tube (L9421-02, Hamamatsu Photonics, Japan) with $7.00 \mu m$ focal spot size. The micro-focus tube was also operated at 55.00 kV (mean energy of 28.00 keV). This setup is used to perform the backward imaging experiments. This time, the $G_0$ grating was used as an analyzer grating. We would like to mention that if the optical reversibility assumption is true, then the detected fringes in this backward setup should have very large period due to the Moiré effect between the diffraction fringe and the $G_0$ grating.

Finally, to demonstrate that the estimated inter grating distance $d_2 = 527.90\;mm$ between $G_1$ and $G_2$ indeed was the optimal value, the distance $d_2$ was scanned over the range of $d_{2}\pm 8.00\;mm$. Both the fringe visibility and period were compared and plotted. Finally, a PMMA rod with diameter of $10.00\;mm$ was imaged.

3.2 Experimental results

Quantitative comparison results of the fringe period and fringe visibility between the theoretical calculations and experimental measurements under different $d_2$ are shown in Fig. 5. Overall, the experimental results agreed well with the theoretical calculations. In particular, plots in Figs. 5(a) and 5(b) correspond to the forward imaging system, and plot in Fig. 5(c) corresponds to the backward imaging system. As the $d_2$ deviated from its optimal position, i.e., $\Delta d_{2}\ne 0.00\;mm$, the detected fringe visibility of both the forward and backward imaging systems were decreased. In other words, the fringe visibility obtained from the two systems reach to their maximum at the same $\Delta d_{2}=0.00\;mm$ position. Such an experimental observation clearly demonstrated the correctness of the assumed optical reversibility for dual phase grating XPCI. Notice that the theoretical calculations were performed using polychromatic X-ray beams, whose spectra were estimated according to real experimental settings. As the distance $d_2$ decreased, the fringe period was gradually increased. When the forward imaging system setup was operated at its optimal visibility condition, the measured fringe period was about $70.00\;\mu m$, which is very close to the theoretically estimated $69.42\;\mu m$. Since the detected fringe period from the backward imaging system setup was considered as infinite (large enough), so we did not plot its variations. The experimentally acquired diffraction fringes of the two system setups are presented in Figs. 5(d) and 5(e), correspondingly.

 figure: Fig. 5.

Fig. 5. Quantitative comparison results of the fringe period and fringe visibility between the experimental measurements (dots or error bars) and the theoretical calculations (solid lines). (a), (b) and (d) correspond to fringe period, fringe visibility and the fringe image measured from the forward imaging system, (c) and (e) are the measured fringe visibility and fringe image from the backward imaging system, The theoretical calculations were performed using polychromatic X-ray beams. The error bars in plots (b) and (c) were estimated from ten independent measurements on different region-of-interests (ROIs). The images in (d) and (e) have dimensions of 200 pixels by 300 pixels. The effective pixel dimensions are 7.40 $\mu m$ and 194.00 $\mu m$ in (d) and (e), correspondingly. On image (e), the four individual regions containing fringes are the active areas on our $G_0$ grating.

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The imaging results of the PMMA rod are presented in Fig. 6. The experiments were performed on the forward dual phase grating XPCI interferometer with source grating. For illustration purpose, the original images were $10\times 10$ binned, resulting an effective pixel dimension of 74.00 $\mu m$. In addition, line profiles of each contrast mechanism were depicted as well. Clearly, the asymmetric dual phase grating XPCI system with source grating designed by our proposed theory indeed can generate three unique contrast mechanisms.

 figure: Fig. 6.

Fig. 6. Imaging results of the PMMA rod obtained from the forward system setup with $G_0$ grating. Images in (a)-(c) correspond to the absorption, DPC, and DF contrast. Plots in (d)-(f) represent the vertically averaged line profiles of images (a)-(c), correspondingly. The effective image pixel dimension is 74.00 $\mu m$. The scale bar denotes 10.00 $mm$.

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4. Discussion

In this work, we have proposed a new theoretical explanation framework for dual phase grating XPCI using wave optics. First, we found that the diffraction imaging mechanism can be explained by the geometrical optics for the first time. Within this simplified representation, we demonstrated that the X-ray phase gratings can be treated as thin lenses. Second, by exploring the optical reversibility principle, our theory provides a new scheme to predict the period of the arrayed source. Finally, we showed that all the key parameters, like the inter-grating distances, the diffraction fringe period, the phase grating periods, and so on, can all be estimated using this new theoretical framework. Experiments were performed to validate these theoretical findings. Results demonstrated that this developed theoretical analyses framework is precise, and can be used to explain and design such a dual phase grating XPCI system.

Based on our theory, we found that the period of the arrayed source $p_0$ and the period of the diffraction fringe $p_f$ can be treated as two paired counterparts. Without explicitly specifying the X-ray beam propagation direction, it is kind of hard to define their physical meanings, see Eq. (4) and Eq. (7). However, as long as the X-ray beam propagation direction is selected, their meaning would be defined automatically. For example, in the forward imaging system, $p_0$ denotes the period of the arrayed source, and $p_f$ denotes the period of the diffraction fringe. However, in the backward imaging system, their meanings are interchanged. We believe such observation implies exactly the inherent optical symmetry character of the dual phase grating XPCI. Thus, for the first time we are able to explain the wave optics based dual phase grating imaging theory into a geometrical representation form, which is similar to the classical thin lens imaging theory. Within this new theoretical representation, the two phase gratings can be treated as two thin convex lens. As a result, an image of the X-ray source (arrayed or single) is formed at the $r_1$ distance downstream of the $G_1$ grating, and it is imaged again by the $G_2$ grating to form the final image (diffraction fringe) on the detector plane. In order to generate the optimal diffraction fringe visibility, our theory shows that the effective optical lengths of $G_1$ and $G_2$ gratings should equal to their own Talbot self-image distances. If the inter-distance between the $G_1$ and $G_2$ gratings does not satisfy the derived optimal conditions, see Eq. (17) and Eq. (18), the detected fringe visibility will be decreased. When the first-order Talbot distances are selected, the dual phase grating based XPCI system will have the most compact geometry.

We would like to emphasize that the discussed optical reversibility and symmetry are mainly for the source grating period $p_0$ and diffraction fringe period $p_f$. In other words, if the arrayed source has a period of $p_0$, the forward imaging procedure would generate diffraction fringe with period of $p_f$. As a contrary, when the arrayed source has a period of $p_f$, the backward imaging procedure would generate diffraction fringe with period of $p_0$. This is to say $p_f$ is equal to the period of the formed image of the source on the detector plane, and $p_0$ is equal to the period of the formed fringe on the source plane. However, the beam intensity, the imaging field-of-view (FOV), and the fringe visibility may not always satisfy such optical reversibility and symmetry assumption. Take the fringe visibility as an example, the obtained fringes in the forward and backward systems may be different if the duty cycles of the sources used in the two imaging procedures are different, see Eq. (22).

In principle, the periods of the two phase gratings can be any values if there is no source grating. The two phase gratings could be identical, or they could have totally different periods. However, our theory shows that when a source grating is added, the period of the formed diffraction fringe from identical $G_1$ and $G_2$ grating combination is always equal to the period of the source grating, i.e., $p_0=p_f$. In practice, as shown in Fig. 2, it is better to keep the source grating opening small ($<15\;\mu m$) to guarantee high enough fringe visibility. Additionally, it is also better to keep the period of a source grating small (between $20\;\mu m$ to $50\;\mu m$) to keep a fairly high X-ray beam usage efficiency. As a result, the identical $G_1$ and $G_2$ combination would not be recommended. As a contrary, we think the asymmetric dual phase grating interferometer should be a better choice if a source grating is used. Specifically, the period of $G_1$ grating is slightly smaller than the period of $G_2$ grating, assuming that the X-ray beam propagates through the $G_1$ grating first and the $G_2$ grating afterwards. With the asymmetric system design, the period of the diffraction fringe becomes larger than the source grating period. As a consequence, the asymmetric grating design would be more favorable in practical applications.

For our experiments, the measured fringe visibility was not very high (around $5.00\%$ to $6.50\%$). We thought this might due to the low spatial coherence of the X-ray beam. By increasing the beam coherence or reducing the phase grating periods, the fringe visibility could be enhanced. In addition, the polychromatic X-ray beam used in experiments may also degrade the fringe visibility. We noticed a similar work [37] published recently also tried to estimate the period of the source grating using purely geometric configurations. Despite of the same conclusions, however, our theoretical analyses shown in this work were based on wave optics.

This study has several limitations: First, we only focused on discussing the $\pi -\pi$ dual phase grating system, and results for the $\frac {\pi }{2}-\frac {\pi }{2}$ dual phase grating system were not presented. However, results for the $\frac {\pi }{2}-\frac {\pi }{2}$ system should be able to be derived readily, and similar conclusions can also be obtained. Second, during the theoretical analyses, we ignored the X-ray detector non-idealities and assumed ideal response. Under such circumstances, the signal integration was performed within one single detector element, as shown in Eq. (35). However, in practice, the detector response may not be perfect and the real response function needs to be considered for rigorous analyses. Third, no quantitative results were obtained for the system sensitivity in current theoretical framework, and this question would be investigated in future work. Finally, the total length of our current experiment setup was over $2.00\;m$. This is because of our used large phase grating periods. We believe by properly reducing their periods, it is able to shorten the tube-to-detector distance down to 1.00 $m$.

In future, we would like to extend this newly developed dual phase grating XPCI theoretical analysis framework into the triple phase grating based XPCI system. We would like to perform comparisons with the already published theoretical results in literature [29]. Additionally, we are also interested in making a rigorous analyses of the radiation dose performance between the new phase grating based XPCI system and the conventional Talbot-Lau system.

5. Conclusion

In conclusion, a new theoretical explanation framework for dual phase grating XPCI systems using pure wave optics was developed. Based on the optical reversibility principle, the dual phase grating XPCI procedure can be explained by the geometrical thin lens imaging theory. The period of the arrayed source is equal to the period of the diffraction fringe formed on the source plane. When a source grating is used, it is recommended to make the periods of the two phase gratings unequal.

Appendix

Using Eq. (1) and Eq. (2), the X-ray wave field disturbance right before the $G_1$ grating is obtained as following:

$$\textrm{U}_{1}(x_{1}, y_{1})=\frac{\textrm{U}_{0}e^{ikd_{1}}}{d_{1}}e^{ik\left[\frac{(x_{1}-x_{s})^{2}}{2d_{1}}+\frac{(y_{1}-y_{s})^{2}}{2d_{1}}\right]},$$
where $(x_s, y_s)$ denotes the source coordinates. Afterwards, the X-ray field interacts with $G_1$ grating via
$$\textrm{U}_{1}^{'}(x_{1}, y_{1})=\textrm{U}_{1}(x_{1}, y_{1})\textrm{T}_{1}(x_{1};p_1).$$
Then, the X-ray field reaches the $G_2$ grating with field disturbance of
$$\textrm{U}_{2}(x_{2}, y_{2})=\frac{\textrm{U}_{0}e^{ik(d_{1}+d_{2})}}{d_{1}+d_{2}}\sum_{n=-\infty}^{n=\infty}a_{n}e^{\frac{i2\pi n(-d_{1}n\pi+kp_{1}x_{s})}{kp_{1}^{2}}}e^{ik\left[\frac{[x_{2}-(x_{s}-\frac{2\pi d_{1}n}{kp_{1}})]^{2}}{2(d_{1}+d_{2})}+\frac{(y_{2}-y_{s})^{2}}{2(d_{1}+d_{2})}\right]}.$$
Similarly, X-ray field starts to interact with $G_2$ grating,
$$\textrm{U}_{2}^{'}(x_{2}, y_{2})=\textrm{U}_{2}(x_{2}, y_{2})\textrm{T}_{2}(x_{2};p_2).$$
Finally, the X-ray wave arrives at the detector plane with field disturbance written as below:
$$\begin{aligned} \textrm{U}_{3}(x_{3}, y_{3}) = &\frac{\textrm{U}_{0}e^{ik(d_{1}+d_{2}+d_{3})}}{d_{1}+d_{2}+d_{3}}\sum_{n=-\infty}^{n=\infty}\sum_{m=-\infty}^{m=\infty}a_{n}b_{m}e^{\frac{i2\pi n(-d_{1}n\pi+kp_{1}x_{s})}{kp_{1}^{2}}}e^{\frac{i2\pi m[-(d_{1}+d_{2})m\pi+kp_{2}(x_{s}-\frac{2\pi d_{1}n}{kp_{1}})]}{kp_{2}^{2}}}\\ &e^{ik\left[\frac{[x_{3}-(x_{s}-\frac{2\pi d_{1}n}{kp_{1}}-\frac{2\pi(d_{1}+d_{2})m}{kp_{2}})]^{2}}{2(d_{1}+d_{2}+d_{3})}+\frac{(y_{3}-y_{s})^{2}}{2(d_{1}+d_{2}+d_{3})}\right]}. \end{aligned}$$
Therefore, the beam intensity can be expressed as:
$$\begin{aligned} \textrm{I}_{3}(x_{3}, y_{3}) = &\frac{\textrm{U}_{0}^{2}}{(d_{1}+d_{2}+d_{3})^{2}}\sum_{n=-\infty}^{n=\infty}\sum_{n^{'}=-\infty}^{n_{'}=\infty}\sum_{m=-\infty}^{m=\infty}\sum_{m^{'}=-\infty}^{m^{'}=\infty}a_{n}a_{n^{'}}^{*}b_{m}b_{m^{'}}^{*}\\ &e^{-\frac{i\pi[(d_{1}+d_{2})d_{3}(m^{2}-m^{'2})p_{1}^{2}+2d_{1}d_{3}(mn-m^{'}n^{'})p_{1}p_{2}+d_{1}(d_{2}+d_{3})(n^{2}-n^{'2})p_{2}{}^{2}]\lambda}{(d_{1}+d_{2}+d_{3})p_{1}^{2}p_{2}{}^{2}}}\\ &e^{\frac{i2\pi[d_{3}(m-m^{'})p_{1}+(d_{2}+d_{3})(n-n^{'})p_{2}]x_{s}}{(d_{1}+d_{2}+d_{3})p_{1}p_{2}}}e^{\frac{i2\pi[(d_{1}+d_{2})(m-m^{'})p_{1}+d_{1}(n-n^{'})p_{2}]x_{3}}{(d_{1}+d_{2}+d_{3})p_{1}p_{2}}}. \end{aligned}$$
Let $n=n^{'}+s, m=m^{'}+r$, Eq. (30) can be simplified into
$$\begin{aligned} \textrm{I}_{3}(x_{3}, y_{3}) = &\frac{\textrm{U}_{0}^{2}}{(d_{1}+d_{2}+d_{3})^{2}}\sum_{n^{'}+s=-\infty}^{n^{'}+s=\infty}\sum_{n^{'}=-\infty}^{n^{'}=\infty}\sum_{m^{'}+r=-\infty}^{m^{'}+r=\infty}\sum_{m^{'}=-\infty}^{m'=\infty}a_{n^{'}+s}a_{n^{'}}^{*}b_{m'+r}b_{m^{'}}^{*}e^{-\frac{i2\pi s(2n+s)\lambda}{\frac{2(d_{1}+d_{2}+d_{3})}{d_{1}(d_{2}+d_{3})+d_{1}d_{3}\frac{p_{1}}{p_{2}}\frac{r}{s}}p_{1}^{2}}}\\ &e^{-\frac{i2\pi r(2m+r)\lambda}{\frac{2(d_{1}+d_{2}+d_{3})}{(d_{1}+d_{2})d_{3}+d_{1}d_{3}\frac{p_{2}}{p_{1}}\frac{s}{r}}p_{2}^{2}}}e^{-\frac{i2\pi[(d_{2}+d_{3})\frac{s}{p_{1}}+d_{3}\frac{r}{p_{2}}]x_{s}}{(d_{1}+d_{2}+d_{3})}}e^{-\frac{i2\pi[(d_{1}+d_{2})\frac{r}{p_{2}}+d_{1}\frac{s}{p_{1}}]x_{3}}{(d_{1}+d_{2}+d_{3})}}. \end{aligned}$$
Be aware that index $s$ and $r$ represent the difference between the conjugated diffraction orders for the first and second grating, respectively. They are not equal to the grating diffraction orders. Using equation [34,35,38]
$$\sum_{l=-\infty}^{l=\infty} C_{l}(d,\lambda, p_{g},\Delta\phi_{g})=\sum_{l=-\infty}^{l=\infty}\sum_{n=-\infty}^{n=\infty}g_{l+n}g_{n}^{*}e^{\frac{-i2\pi l(l+2n)\lambda d}{2p_{g}^{2}}},$$
where
$$\begin{aligned} & C_{l}(d,\lambda, p_{g},\Delta\phi_{g})=\left\{ \begin{array}{cc} 1 & l=0\\ -(1-cos\Delta\phi_{g})\cdot(-1)^{\lfloor4k\lambda d/p_{g}^{2}\rfloor}\cdot\frac{sin(4k^{2}\pi\lambda d/p_{g}^{2})}{k\pi} & l=2k\neq0\\ -i2sin\Delta\phi_{g}\cdot\frac{sin(4\pi\lambda d/p_{g}^{2}\cdot(k+1/2)^{2})}{\pi(2k+1)} & l=2k+1 \end{array}\right\}, \end{aligned}$$
the Eq. (31) can be further simplified as following,
$$\begin{aligned} \textrm{I}_{3}(x_{3}, y_{3}) = &\frac{\textrm{U}_{0}^{2}}{(d_{1}+d_{2}+d_{3})^{2}}\sum_{s=-\infty}^{s=\infty}\sum_{r=-\infty}^{r=\infty}C_{s}(\frac{d_{1}(d_{2}+d_{3})+d_{1}d_{3}\frac{p_{1}}{p_{_{2}}}\frac{r}{s}}{d_{1}+d_{2}+d_{3}},\lambda, p_{1}, \phi_{1}) \\ &C_{r}(\frac{(d_{1}+d_{2})d_{3}+d_{1}d_{3}\frac{p_{2}}{p_{1}}\frac{s}{r}}{d_{1}+d_{2}+d_{3}},\lambda, p_{2}, \phi_{2}) e^{\frac{i2\pi[d_{3}\frac{r}{p_{2}}+(d_{2}+d_{3})\frac{s}{p_{1}}]x_{s}}{d_{1}+d_{2}+d_{3}}} e^{\frac{i2\pi[(d_{1}+d_{2})\frac{r}{p_{2}}+d_{1}\frac{s}{p_{1}}]x_{3}}{d_{1}+d_{2}+d_{3}}}. \end{aligned}$$
where $\phi _{1}$ and $\phi _{2}$ denotes the phase shift on $G_1$ and $G_2$ grating correspondingly.

Now, we take the source size and the detector pixel size into consideration. Assuming the source has a slit shape and is defined by Eq. (3), an ideal X-ray detector with element dimension of $p_{del}$ is utilized, thus, Eq. (34) becomes

$$\begin{aligned} \textrm{I}_{3}(x_{3}, y_{3}) = &\frac{\textrm{U}_{0}^{2}}{(d_{1}+d_{2}+d_{3})^{2}}\sum_{s=-\infty}^{s=\infty}\sum_{r=-\infty}^{r=\infty}C_{s}(\frac{d_{1}(d_{2}+d_{3})s+d_{1}d_{3}\frac{p_{1}}{p_{_{2}}}r}{d_{1}+d_{2}+d_{3}},\lambda, p_{1}, \phi_{1}) \\ &C_{r}(\frac{(d_{1}+d_{2})d_{3}r+d_{1}d_{3}\frac{p_{2}}{p_{1}}s}{d_{1}+d_{2}+d_{3}},\lambda, p_{2}, \phi_{2}) \frac{1}{\sigma}\int_{x_{s}-\frac{\sigma}{2}}^{x_{s}+\frac{\sigma}{2}}e^{-\frac{i2\pi[(d_{2}+d_{3})\frac{s}{p_{1}}+d_{3}\frac{r}{p_{2}}]t}{(d_{1}+d_{2}+d_{3})}}dt\\ &\frac{1}{p_{del}}\int_{x_{3}-\frac{p_{del}}{2}}^{x_{3}+\frac{p_{del}}{2}}e^{-\frac{i2\pi[(d_{1}+d_{2})\frac{r}{p_{2}}+d_{1}\frac{s}{p_{1}}]v}{(d_{1}+d_{2}+d_{3})}}dv, \end{aligned}$$
namely,
$$\begin{aligned} \textrm{I}_{3}(x_{3}, y_{3}) = &\frac{\textrm{U}_{0}^{2}}{(d_{1}+d_{2}+d_{3})^{2}}\sum_{s=-\infty}^{s=\infty}\sum_{r=-\infty}^{r=\infty}C_{s}(\frac{d_{1}(d_{2}+d_{3})+d_{1}d_{3}\frac{p_{1}}{p_{_{2}}}\frac{r}{s}}{d_{1}+d_{2}+d_{3}},\lambda, p_{1}, \phi_{1}) \\ &C_{r}(\frac{(d_{1}+d_{2})d_{3}+d_{1}d_{3}\frac{p_{2}}{p_{1}}\frac{s}{r}}{d_{1}+d_{2}+d_{3}},\lambda, p_{2}, \phi_{2})sinc(\frac{[d_{3}\frac{r}{p_{2}}+(d_{2}+d_{3})\frac{s}{p_{1}}]\sigma}{d_{1}+d_{2}+d_{3}})\\ &e^{\frac{i2\pi[d_{3}\frac{r}{p_{2}}+(d_{2}+d_{3})\frac{s}{p_{1}}]x_{s}}{d_{1}+d_{2}+d_{3}}} sinc(\frac{[(d_{1}+d_{2})\frac{r}{p_{2}}+d_{1}\frac{s}{p_{1}}]p_{del}}{d_{1}+d_{2}+d_{3}})e^{\frac{i2\pi[(d_{1}+d_{2})\frac{r}{p_{2}}+d_{1}\frac{s}{p_{1}}]x_{3}}{d_{1}+d_{2}+d_{3}}}. \end{aligned}$$

Funding

National Natural Science Foundation of China (11804356, 11535015, 11674232, 61671311); Chinese Academy of Sciences Key Laboratory of Health Informatics (2011DP173015);National Key Research and Development Program of China (2016YFA0400900).

Acknowledgments

The authors would like to thank Dr. Zhicheng Li at Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, for his support during experiments, and Mr. Peizhen Liu at Shenzhen University for his assistance during data processing.

Disclosures

Y. Ge and J. Chen have made equal contributions to this work. The authors declare no conflicts of interest.

References

1. A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2(4), 473–475 (1996). [CrossRef]  

2. C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81(17), 3287–3289 (2002). [CrossRef]  

3. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 2 42(Part 2, No. 7B), L866–L868 (2003). [CrossRef]  

4. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Zeigler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296 (2005). [CrossRef]  

5. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase Tomography by X-ray Talbot Interferometry for Biological Imaging,” Jpn. J. Appl. Phys., Part 1 45(6A), 5254–5262 (2006). [CrossRef]  

6. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2(4), 258–261 (2006). [CrossRef]  

7. M. Bech, T. Jensen, R. Feidenhans’l, O. Bunk, C. David, and F. Pfeiffer, “Soft-tissue phase-contrast tomography with an x-ray tube source,” Phys. Med. Biol. 54(9), 2747–2753 (2009). [CrossRef]  

8. T. H. Jensen, A. Böttiger, M. Bech, I. Zanette, T. Weitkamp, S. Rutishauser, C. David, E. Reznikova, J. Mohr, L. B. Christensen, E. V. Olsen, R. Feidenhans’l, and F. Pfeiffer, “X-ray phase-contrast tomography of porcine fat and rind,” Meat Sci. 88(3), 379–383 (2011). [CrossRef]  

9. K. Li, Y. Ge, J. Garrett, N. Bevins, J. Zambelli, and G.-H. Chen, “Grating-based phase contrast tomosynthesis imaging: Proof-of-concept experimental studies,” Med. Phys. 41(1), 011903 (2013). [CrossRef]  

10. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7(2), 134–137 (2008). [CrossRef]  

11. W. Yashiro, Y. Terui, K. Kawabata, and A. Momose, “On the origin of visibility contrast in x-ray Talbot interferometry,” Opt. Express 18(16), 16890–16901 (2010). [CrossRef]  

12. G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. A. Fasching, W. Haas, A. Hartmann, T. Michel, G. Pelzer, and M. Radicke, “Grating-based darkfield imaging of human breast tissue,” Z. Med. Phys. 23(3), 228–235 (2013). [CrossRef]  

13. T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Durst, P. A. Fasching, W. Haas, A. Hartmann, and G. Pelzer, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58(8), 2713–2732 (2013). [CrossRef]  

14. Z. Wang, N. Hauser, G. Singer, M. Trippel, R. A. Kubik-Huch, C. W. Schneider, and M. Stampanoni, “Non-invasive classification of microcalcifications with phase-contrast x-ray mammography,” Nat. Commun. 5(1), 3797 (2014). [CrossRef]  

15. S. Grandl, K. Scherer, A. Sztrókay-Gaul, L. Birnbacher, K. Willer, M. Chabior, J. Herzen, D. Mayr, S. D. Auweter, and F. Pfeiffer, “Improved visualization of breast cancer features in multifocal carcinoma using phase-contrast and dark-field mammography: an ex vivo study,” Eur. Radiol. 25(12), 3659–3668 (2015). [CrossRef]  

16. K. Scherer, L. Birnbacher, K. Willer, M. Chabior, J. Herzen, and F. Pfeiffer, “Correspondence: Quantitative evaluation of x-ray dark-field images for microcalcification analysis in mammography,” Nat. Commun. 7(1), 10863 (2016). [CrossRef]  

17. P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. 107(31), 13576–13581 (2010). [CrossRef]  

18. I. Zanette, M. Bech, A. Rack, G. L. Duc, P. Tafforeau, C. David, J. Mohr, F. Pfeiffer, and T. Weitkamp, “Trimodal low-dose x-ray tomography,” Proc. Natl. Acad. Sci. U. S. A. 109(26), 10199–10204 (2012). [CrossRef]  

19. H. Miao, L. Chen, E. E. Bennett, N. M. Adamo, A. A. Gomella, A. M. DeLuca, A. Patel, N. Y. Morgan, and H. Wen, “Motionless phase stepping in x-ray phase contrast imaging with a compact source,” Proc. Natl. Acad. Sci. 110(48), 19268–19272 (2013). [CrossRef]  

20. Y. Ge, K. Li, J. Garrett, and G.-H. Chen, “Grating based x-ray differential phase contrast imaging without mechanical phase stepping,” Opt. Express 22(12), 14246–14252 (2014). [CrossRef]  

21. T. Koehler, H. Daerr, G. Martens, N. Kuhn, S. Löscher, U. van Stevendaal, and E. Roessl, “Slit-scanning differential x-ray phase-contrast mammography: Proof-of-concept experimental studies,” Med. Phys. 42(4), 1959–1965 (2015). [CrossRef]  

22. M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. Noël, F. Pfeiffer, and J. Herzen, “Helical x-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6(1), 23953 (2016). [CrossRef]  

23. Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys., Part 2 46(No. 3), L89–L91 (2007). [CrossRef]  

24. S. Cartier, M. Kagias, A. Bergamaschi, Z. Wang, R. Dinapoli, A. Mozzanica, M. Ramilli, B. Schmitt, M. Brückner, and E. Fröjdh, “Micrometer-resolution imaging using mönch: towards G2-less grating interferometry,” J. Synchrotron Radiat. 23(6), 1462–1473 (2016). [CrossRef]  

25. A. Momose, H. Kuwabara, and W. Yashiro, “X-ray phase imaging using Lau effect,” Appl. Phys. Express 4(6), 066603 (2011). [CrossRef]  

26. Y. Du, X. Liu, Y. Lei, J. Guo, and H. Niu, “Non-absorption grating approach for x-ray phase contrast imaging,” Opt. Express 19(23), 22669–22674 (2011). [CrossRef]  

27. G. Zan, D. J. Vine, R. I. Spink, W. Yun, Q. Wang, and G. Wang, “Design optimization of a periodic microstructured array anode for hard x-ray grating interferometry,” Phys. Med. Biol. 64(14), 145011 (2019). [CrossRef]  

28. H. Miao, A. A. Gomella, K. J. Harmon, E. E. Bennett, N. Chedid, S. Znati, A. Panna, B. A. Foster, P. Bhandarkar, and H. Wen, “Enhancing tabletop x-ray phase contrast imaging with nano-fabrication,” Sci. Rep. 5(1), 13581 (2015). [CrossRef]  

29. H. Miao, A. Panna, A. A. Gomella, E. E. Bennett, S. Znati, L. Chen, and H. Wen, “A universal Moiré effect and application in X-ray phase-contrast imaging,” Nat. Phys. 12(9), 830–834 (2016). [CrossRef]  

30. M. Kagias, Z. Wang, K. Jefimovs, and M. Stampanoni, “Dual phase grating interferometer for tunable dark-field sensitivity,” Appl. Phys. Lett. 110(1), 014105 (2017). [CrossRef]  

31. F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31(1), 4–8 (1979). [CrossRef]  

32. A. Yan, X. Wu, and H. Liu, “Quantitative theory of x-ray interferometers based on dual phase grating: fringe period and visibility,” Opt. Express 26(18), 23142–23155 (2018). [CrossRef]  

33. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1993).

34. A. Yan, X. Wu, and H. Liu, “A general theory of interference fringes in x-ray phase grating imaging,” Med. Phys. 42(6Part1), 3036–3047 (2015). [CrossRef]  

35. A. Yan, X. Wu, and H. Liu, “Predicting visibility of interference fringes in x-ray grating interferometry,” Opt. Express 24(14), 15927–15939 (2016). [CrossRef]  

36. Y. Lei, X. Liu, J. Huang, Y. Du, J. Guo, Z. Zhao, and J. Li, “Cascade Talbot-Lau interferometers for x-ray differential phase-contrast imaging,” J. Phys. D: Appl. Phys. 51(38), 385302 (2018). [CrossRef]  

37. A. Yan, X. Wu, and H. Liu, “Clarification on generalized lau condition for x-ray interferometers based on dual phase gratings,” Opt. Express 27(16), 22727–22736 (2019). [CrossRef]  

38. V. Arrizon and J. Ojedacastaneda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A 9(10), 1801–1806 (1992). [CrossRef]  

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Figures (6)
Fig. 1.
Fig. 1. Illustration of the thin lens interpretation of a dual phase grating XPCI setups. The $G_1$ and $G_2$ are phase gratings. Herein, the solid arrows represent the forward imaging procedure, in which an arrayed source (in red color) positioned on the left hand side is assumed. Additionally, the backward imaging procedure is schemed by the dashed arrows, in which the arrayed source (in purple color) is assumed on the right hand side.

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Fig. 2.
Fig. 2. Visibility distribution maps with respect to: (a) the source width $\sigma$ and the source spacing $p_0$, (b) the duty cycle and the source spacing $p_0$. The phase grating periods are assumed as $p_1=4.364\;\mu m$ and $p_2=4.640\;\mu m$. In image (a), the upper-left dark region above the dashed line corresponds to $\sigma >p_0$ and should be avoided. In image (b), the three dashed curves correspond to three different $\sigma$ values. In addition, these visibility maps have been normalized by their maximum values.

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Fig. 3.
Fig. 3. Fringe period distribution maps obtained with respect to: (a) $p_1-p_2$ (when $p_2=4.640\;\mu m$) and $p_0$, (b) $p_2$ and $p_1$ (when $p_0=24\;\mu m$), and (c) $p_2$ and $p_1$ (when $p_0=60\;\mu m$). Plots in (b) and (c) correspond to the two circles marked on (a), respectively. As can be seen, the difference between $p_2$ and $p_1$ decreases as $p_0$ increases. Due to this reason, the bright band in plot (c) is less deviated from the diagonal direction than in plot (b).

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Fig. 4.
Fig. 4. Flowchart to estimate the key parameters of the dual phase grating interferometer system used in our experiments. In this example, the mean beam energy and grating parameters are predetermined. The system geometry and fringe period are parameters to be calculated.

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Fig. 5.
Fig. 5. Quantitative comparison results of the fringe period and fringe visibility between the experimental measurements (dots or error bars) and the theoretical calculations (solid lines). (a), (b) and (d) correspond to fringe period, fringe visibility and the fringe image measured from the forward imaging system, (c) and (e) are the measured fringe visibility and fringe image from the backward imaging system, The theoretical calculations were performed using polychromatic X-ray beams. The error bars in plots (b) and (c) were estimated from ten independent measurements on different region-of-interests (ROIs). The images in (d) and (e) have dimensions of 200 pixels by 300 pixels. The effective pixel dimensions are 7.40 $\mu m$ and 194.00 $\mu m$ in (d) and (e), correspondingly. On image (e), the four individual regions containing fringes are the active areas on our $G_0$ grating.

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Fig. 6.
Fig. 6. Imaging results of the PMMA rod obtained from the forward system setup with $G_0$ grating. Images in (a)-(c) correspond to the absorption, DPC, and DF contrast. Plots in (d)-(f) represent the vertically averaged line profiles of images (a)-(c), correspondingly. The effective image pixel dimension is 74.00 $\mu m$. The scale bar denotes 10.00 $mm$.

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Equations (36)

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T ( x ) = ∑ n = − ∞ n = ∞ a n e i 2 π n x p ,
U ( x ) = U 0 e i k z i λ z ∫ S ( ξ ) e i k [ ( x − ξ ) 2 2 z ] d ξ ,
S ( x ) = rect ( x σ ) ,
I 3 ( x 3 ) = U 0 2 ( d 1 + d 2 + d 3 ) 2 ∑ s = − ∞ s = ∞ ∑ r = − ∞ r = ∞ C s ( d 1 ( d 2 + d 3 ) s + d 1 d 3 p 1 p 2 r d 1 + d 2 + d 3 , λ , p 1 , ϕ 1 ) C r ( ( d 1 + d 2 ) d 3 r + d 1 d 3 p 2 p 1 s d 1 + d 2 + d 3 , λ , p 2 , ϕ 2 ) s i n c ( [ d 3 r p 2 + ( d 2 + d 3 ) s p 1 ] σ d 1 + d 2 + d 3 ) e i 2 π [ d 3 r p 2 + ( d 2 + d 3 ) s p 1 ] x s d 1 + d 2 + d 3 s i n c ( [ ( d 1 + d 2 ) r p 2 + d 1 s p 1 ] p d e l d 1 + d 2 + d 3 ) e i 2 π [ ( d 1 + d 2 ) r p 2 + d 1 s p 1 ] x 3 d 1 + d 2 + d 3 ,
C s ( d 1 ( d 2 + d 3 ) s + d 1 d 3 p 1 p 2 r d 1 + d 2 + d 3 , λ , p 1 , π ) = − 2 π s i n ( 2 π [ d 1 ( d 2 + d 3 ) s p 1 + d 1 d 3 r p 2 ] λ ( d 1 + d 2 + d 3 ) p 1 ) ,
C r ( ( d 1 + d 2 ) d 3 r + d 1 d 3 p 2 p 1 s d 1 + d 2 + d 3 , λ , p 2 , π ) = − 2 π s i n ( 2 π [ ( d 1 + d 2 ) d 3 r p 2 + d 1 d 3 s p 2 ] λ ( d 1 + d 2 + d 3 ) p 2 ) .
I 3 ( x 3 ) = U 0 2 ( d 1 + d 2 + d 3 ) 2 ( 1 + 8 π 2 s i n ( 4 π d 1 [ ( d 2 + d 3 ) − d 3 p 1 p 2 ] ( d 1 + d 2 + d 3 ) Z t , 1 ) s i n ( 4 π d 3 [ ( d 1 + d 2 ) − d 1 p 2 p 1 ] ( d 1 + d 2 + d 3 ) Z t , 2 ) s i n c ( 2 [ d 2 + d 3 − d 3 p 1 p 2 ] σ ( d 1 + d 2 + d 3 ) p 1 ) e i 4 π [ ( d 2 + d 3 ) − d 3 p 1 p 2 ] x s ( d 1 + d 2 + d 3 ) p 1 s i n c ( 2 [ ( d 1 + d 2 ) − d 1 p 2 p 1 ] p d e l ( d 1 + d 2 + d 3 ) p 2 ) e i 4 π [ ( d 1 + d 2 ) − d 1 p 2 p 1 ] x 3 ( d 1 + d 2 + d 3 ) p 2 ) ,
d 1 + r 1 d 1 p 1 = d 3 + r 2 d 3 p 2 .
r 1 = d 1 ( d 2 + d 3 − d 3 p 1 p 2 ) d 3 p 1 p 2 + d 1 ,
r 2 = d 3 ( d 1 + d 2 − d 1 p 2 p 1 ) d 1 p 2 p 1 + d 3 .
f 1 := d 1 ( d 2 + d 3 − d 3 p 1 p 2 ) ( d 1 + d 2 + d 3 ) = r 1 d 1 r 1 + d 1 = 1 1 d 1 + 1 r 1 ,
f 2 := d 3 ( d 1 + d 2 − d 1 p 2 p 1 ) ( d 1 + d 2 + d 3 ) = r 2 d 3 r 2 + d 3 = 1 1 d 3 + 1 r 2 .
I 3 ( x 3 ) = U 0 2 ( d 1 + d 2 + d 3 ) 2 ( 1 + 8 π 2 s i n ( 4 π f 1 Z t , 1 ) s i n ( 4 π f 2 Z t , 2 ) s i n c ( 2 f 1 σ d 1 p 1 ) e i 4 π f 1 x s d 1 p 1 s i n c ( 2 f 2 p d e l d 3 p 2 ) e i 4 π f 2 x 3 d 3 p 2 ) .
ϵ = 8 π 2 s i n ( 4 π f 1 Z t , 1 ) s i n ( 4 π f 2 Z t , 2 ) s i n c ( 2 f 1 σ d 1 p 1 ) s i n c ( 2 f 2 p d e l d 3 p 2 ) .
f 1 o p = 2 l 1 + 1 8 Z t , 1 = 2 l 1 + 1 8 p 1 2 λ , l 1 ∈ Z
f 2 o p = 2 l 2 + 1 8 Z t , 2 = 2 l 2 + 1 8 p 2 2 λ , l 2 ∈ Z
r 1 o p = f 1 o p d 1 d 1 − f 1 o p ,
r 2 o p = f 2 o p d 3 d 3 − f 2 o p .
p f = d 3 p 2 2 f 2 = d 1 + d 2 + d 3 2 ( d 1 + d 2 p 2 − d 1 p 1 ) .
p f = ( d 1 + d 2 + d 3 ) p 2 d 2 .
p 0 = d 1 p 1 2 f 1 = d 1 + d 2 + d 3 2 ( d 2 + d 3 p 1 − d 3 p 2 ) .
ϵ = 8 π 2 s i n ( 4 π f 1 Z t , 1 ) s i n ( 4 π f 2 Z t , 2 ) s i n c ( σ p 0 ) s i n c ( p d e l p f ) .
p f = p 0 p 1 p 2 2 p 0 ( p 1 − p 2 ) + p 1 p 2 ,
p 0 = p f p 1 p 2 2 p f ( p 2 − p 1 ) + p 1 p 2 .
U 1 ( x 1 , y 1 ) = U 0 e i k d 1 d 1 e i k [ ( x 1 − x s ) 2 2 d 1 + ( y 1 − y s ) 2 2 d 1 ] ,
U 1 ′ ( x 1 , y 1 ) = U 1 ( x 1 , y 1 ) T 1 ( x 1 ; p 1 ) .
U 2 ( x 2 , y 2 ) = U 0 e i k ( d 1 + d 2 ) d 1 + d 2 ∑ n = − ∞ n = ∞ a n e i 2 π n ( − d 1 n π + k p 1 x s ) k p 1 2 e i k [ [ x 2 − ( x s − 2 π d 1 n k p 1 ) ] 2 2 ( d 1 + d 2 ) + ( y 2 − y s ) 2 2 ( d 1 + d 2 ) ] .
U 2 ′ ( x 2 , y 2 ) = U 2 ( x 2 , y 2 ) T 2 ( x 2 ; p 2 ) .
U 3 ( x 3 , y 3 ) = U 0 e i k ( d 1 + d 2 + d 3 ) d 1 + d 2 + d 3 ∑ n = − ∞ n = ∞ ∑ m = − ∞ m = ∞ a n b m e i 2 π n ( − d 1 n π + k p 1 x s ) k p 1 2 e i 2 π m [ − ( d 1 + d 2 ) m π + k p 2 ( x s − 2 π d 1 n k p 1 ) ] k p 2 2 e i k [ [ x 3 − ( x s − 2 π d 1 n k p 1 − 2 π ( d 1 + d 2 ) m k p 2 ) ] 2 2 ( d 1 + d 2 + d 3 ) + ( y 3 − y s ) 2 2 ( d 1 + d 2 + d 3 ) ] .
I 3 ( x 3 , y 3 ) = U 0 2 ( d 1 + d 2 + d 3 ) 2 ∑ n = − ∞ n = ∞ ∑ n ′ = − ∞ n ′ = ∞ ∑ m = − ∞ m = ∞ ∑ m ′ = − ∞ m ′ = ∞ a n a n ′ ∗ b m b m ′ ∗ e − i π [ ( d 1 + d 2 ) d 3 ( m 2 − m ′ 2 ) p 1 2 + 2 d 1 d 3 ( m n − m ′ n ′ ) p 1 p 2 + d 1 ( d 2 + d 3 ) ( n 2 − n ′ 2 ) p 2 2 ] λ ( d 1 + d 2 + d 3 ) p 1 2 p 2 2 e i 2 π [ d 3 ( m − m ′ ) p 1 + ( d 2 + d 3 ) ( n − n ′ ) p 2 ] x s ( d 1 + d 2 + d 3 ) p 1 p 2 e i 2 π [ ( d 1 + d 2 ) ( m − m ′ ) p 1 + d 1 ( n − n ′ ) p 2 ] x 3 ( d 1 + d 2 + d 3 ) p 1 p 2 .
I 3 ( x 3 , y 3 ) = U 0 2 ( d 1 + d 2 + d 3 ) 2 ∑ n ′ + s = − ∞ n ′ + s = ∞ ∑ n ′ = − ∞ n ′ = ∞ ∑ m ′ + r = − ∞ m ′ + r = ∞ ∑ m ′ = − ∞ m ′ = ∞ a n ′ + s a n ′ ∗ b m ′ + r b m ′ ∗ e − i 2 π s ( 2 n + s ) λ 2 ( d 1 + d 2 + d 3 ) d 1 ( d 2 + d 3 ) + d 1 d 3 p 1 p 2 r s p 1 2 e − i 2 π r ( 2 m + r ) λ 2 ( d 1 + d 2 + d 3 ) ( d 1 + d 2 ) d 3 + d 1 d 3 p 2 p 1 s r p 2 2 e − i 2 π [ ( d 2 + d 3 ) s p 1 + d 3 r p 2 ] x s ( d 1 + d 2 + d 3 ) e − i 2 π [ ( d 1 + d 2 ) r p 2 + d 1 s p 1 ] x 3 ( d 1 + d 2 + d 3 ) .
∑ l = − ∞ l = ∞ C l ( d , λ , p g , Δ ϕ g ) = ∑ l = − ∞ l = ∞ ∑ n = − ∞ n = ∞ g l + n g n ∗ e − i 2 π l ( l + 2 n ) λ d 2 p g 2 ,
C l ( d , λ , p g , Δ ϕ g ) = { 1 l = 0 − ( 1 − c o s Δ ϕ g ) ⋅ ( − 1 ) ⌊ 4 k λ d / p g 2 ⌋ ⋅ s i n ( 4 k 2 π λ d / p g 2 ) k π l = 2 k ≠ 0 − i 2 s i n Δ ϕ g ⋅ s i n ( 4 π λ d / p g 2 ⋅ ( k + 1 / 2 ) 2 ) π ( 2 k + 1 ) l = 2 k + 1 } ,
I 3 ( x 3 , y 3 ) = U 0 2 ( d 1 + d 2 + d 3 ) 2 ∑ s = − ∞ s = ∞ ∑ r = − ∞ r = ∞ C s ( d 1 ( d 2 + d 3 ) + d 1 d 3 p 1 p 2 r s d 1 + d 2 + d 3 , λ , p 1 , ϕ 1 ) C r ( ( d 1 + d 2 ) d 3 + d 1 d 3 p 2 p 1 s r d 1 + d 2 + d 3 , λ , p 2 , ϕ 2 ) e i 2 π [ d 3 r p 2 + ( d 2 + d 3 ) s p 1 ] x s d 1 + d 2 + d 3 e i 2 π [ ( d 1 + d 2 ) r p 2 + d 1 s p 1 ] x 3 d 1 + d 2 + d 3 .
I 3 ( x 3 , y 3 ) = U 0 2 ( d 1 + d 2 + d 3 ) 2 ∑ s = − ∞ s = ∞ ∑ r = − ∞ r = ∞ C s ( d 1 ( d 2 + d 3 ) s + d 1 d 3 p 1 p 2 r d 1 + d 2 + d 3 , λ , p 1 , ϕ 1 ) C r ( ( d 1 + d 2 ) d 3 r + d 1 d 3 p 2 p 1 s d 1 + d 2 + d 3 , λ , p 2 , ϕ 2 ) 1 σ ∫ x s − σ 2 x s + σ 2 e − i 2 π [ ( d 2 + d 3 ) s p 1 + d 3 r p 2 ] t ( d 1 + d 2 + d 3 ) d t 1 p d e l ∫ x 3 − p d e l 2 x 3 + p d e l 2 e − i 2 π [ ( d 1 + d 2 ) r p 2 + d 1 s p 1 ] v ( d 1 + d 2 + d 3 ) d v ,
I 3 ( x 3 , y 3 ) = U 0 2 ( d 1 + d 2 + d 3 ) 2 ∑ s = − ∞ s = ∞ ∑ r = − ∞ r = ∞ C s ( d 1 ( d 2 + d 3 ) + d 1 d 3 p 1 p 2 r s d 1 + d 2 + d 3 , λ , p 1 , ϕ 1 ) C r ( ( d 1 + d 2 ) d 3 + d 1 d 3 p 2 p 1 s r d 1 + d 2 + d 3 , λ , p 2 , ϕ 2 ) s i n c ( [ d 3 r p 2 + ( d 2 + d 3 ) s p 1 ] σ d 1 + d 2 + d 3 ) e i 2 π [ d 3 r p 2 + ( d 2 + d 3 ) s p 1 ] x s d 1 + d 2 + d 3 s i n c ( [ ( d 1 + d 2 ) r p 2 + d 1 s p 1 ] p d e l d 1 + d 2 + d 3 ) e i 2 π [ ( d 1 + d 2 ) r p 2 + d 1 s p 1 ] x 3 d 1 + d 2 + d 3 .
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