Jun Yang(楊君), Jian-Heng Huang(黃建衡), Yao-Hu Lei(雷耀虎), Jing-Biao Zheng(鄭景標(biāo)),Yu-Zheng Shan(單雨征), Da-Yu Guo(郭大育), and Jin-Chuan Guo(郭金川)

Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province,College of Physics and Optoelectronic Engineering,Shenzhen University,Shenzhen 518060,China

Keywords: phase contrast,grating interferometer,fringe period,fringe visibility

1. Introduction

X-ray phase-contrast images and dark-field images obtained by a Talbot–Lau interferometer have potential applications in diagnosis of rheumatoid arthritis, microcalcifications in breast and lung diseases.[1–14]However,the Talbot–Lau interferometer is restricted by the small field of view in clinic applications. It is a great challenge to fabricate the analyzer grating over a large area and high aspect ratio. The inverse geometry Talbot–Lau interferometer,[15]which interchanges the position of the x-ray source and the detector, does not need any analyzer grating. However, its high system magnification will also reduce the field of view. Miaoet al.[16]used a dual phase grating interferometer to produce fringes with large periods. The dual phase grating interferometer does not use the analyzer grating,and it may provide a larger field of view and higher x-ray dose efficiency. Kagiaset al.[17]connected dual Talbot interferometers together for dark-field imaging. The idea of virtual structure source proposed in their paper is very instructive for understanding the imaging process.Leiet al.[18]realized a dual Talbot–Lau interferometer in which the addition of source grating overcame the limitation of flux of micro-focal spot source. Geet al.[7]proposed that under the optical reversibility principle,the phase gratings in the dual phase grating interferometer could be treated as thin lens.However,the aforementioned theoretical calculations were too complicated.[7,16,19]What’s more, the theoretical and experimental results given by different researchers were inconsistent.For example, the fringe period given by Miaoet al.[16]was twice as long as the fringe period given by Geet al.[7]In this paper, we will work to simplify the theoretical analysis and explain these different theoretical and experimental results.

2. Theoretical analysis

2.1. Dual phase grating interferometer

Considering Fig.1,let us assume that a point source with a wavelengthλis in the planex0oy0,the first phase grating G1 with periodp1is in the planex1oy1,the second phase grating G2 with periodp2is in the planex2oy2, and the detector is in the planex3oy3. The distance from the source to the first phase grating,the first phase grating to the second phase grating,and the second phase grating to the detector arez1,z2andz3,respectively.

Fig.1. Setup of the dual phase grating interferometer.

According to the Fourier series theory, the complex amplitude distributions of the first phase grating G1 and the second phase grating G2 may be expressed as

2.2. Relationship between the dual phase grating interferometer and the Talbot interferometer

2.3. Fringe periods of dual phase grating interferometer

Next,we will derive the fringe periods of the dual phase grating interferometer from Eq.(17). If we let

Next, we consider the maximum amplitude terms in Eq. (28) except for the constant terms. Since we use phase gratings with a duty cycle of 50%,the sum coefficientsn,m,s,tcannot be even. Therefore,there are six non-constant terms with the largest amplitude in Eq.(28),

Now we consider the case in which two phase gratings with very small period produce fringes with very large period. Under this condition, the difference frequency of two phase gratings imaging fringes will be much smaller than the other three frequencies. When the fringes of various frequencies are overlapped together,due to the focus size of the x-ray source and the integration effect of detector pixels,a low-pass filtering effect may be observed. When the system parameters satisfy certain conditions(discussed later),the difference frequency will be strengthened and other frequencies will be further suppressed. Hence, the final fringe frequency will be

2.4. Optimal visibility conditions of the dual phase grating interferometer

For the analysis of the optimal visibility conditions of the dual phase grating interferometer,two cases of dual phase grating ofπandπ/2 are discussed.

Case 1 Dualπ-phase grating interferometer.

Comparing Eqs.(34)and(43)–(46),it may be found that there is a negative sign before Eqs.(43)–(46),so it is necessary to add aπphase or a-πphase in Eqs.(43)–(46). It may also be observed that the initial phases of Eqs.(43)and(44)are opposite to each other,as is the case for Eqs.(45)and(46). The initial phase of formula(34)is zero. Therefore,if the fringes represented by Eqs.(43)–(46)and(34)are mutually reinforcing, they need to be in the same phase. Then, the following relations must be satisfied:

whereh1andh2are integers. Equation(47)may be simplified to

Here, Eqs. (48) are the optimal visibility conditions for dualπ-phase grating interferometer.Whenp1→∞orp2→∞,Eqs.(48)will transform into the optimal visibility condition of the Talbot interferometer.Next,we ascertain whether the third strongest fringe with the same frequency as Eq.(34)may satisfy the condition represented by Eq.(48).

3. Simulations and experiments

The simulations and experiments were performed with an x-ray tube (L9421-02, Hamamatsu Photonics K. K., Japan)with 5 μm focal spot(40 kVp,200 μA).The pixel size of the flat panel detector (Shadobox 6K HS, Teledyne Dalsa, USA)was 50 μm with field of view of 14.7×11.52 cm2. The two phase gratings with period of 3 μm wereπ-shift phase gratings at 28 keV, and they all had a duty cycle of 50%. The setup was arranged as follows:z1=1.2698 m,z2=0.026 m andz3=1.2698 m. By the illumination of the polychromatic x-ray,the simulation fringe with period of 300 μm is shown in Fig.2(a). Figure 2(c)presents the experimental result attained by the abovementioned simulation parameters. The period of the fringe perpendicular to the red line is 281.57 μm,which is 95%of the fringe period in Fig.2(a). The error mainly comes from the measurement of the distance between different components.

In Figs. 2(a) and 2(c), the positions of the dualπ-phase gratings are not nearby the positions where the fringe visibility is optimal. Therefore, the imaging period is twice as long as the theoretical result,such as the results given by Miaoet al.[16]and Wanget al.[17]When the positions of the dualπ-phase gratings are nearby the positions where the fringe visibility is optimal, e.g.,z1= 2.5395 m,z2= 0.0513 m,z3=2.5395 m, the fringe period in Fig. 2(b) is the same as that under monochromatic x-ray illumination.The experimental results given by Leiet al.[18]and Geet al.[7]could well prove the above theory. Note that only the effect of the spectrum could not double the fringe period, and the positions of the phase gratings must also be considered.A similar situation is whether monochromatic or polychromatic x-ray is used,the period of self-imaging of aπ-phase grating in the Talbot–Lau interferometer at the fractional Talbot order is 0.5Mp,notMp,whereMis the system magnification andpis the period of theπ-phase grating.

Surprisingly, the fringe shown in Fig. 2(c) has a twodimensional structure,although the phase gratings used in the experiment are one-dimensional. In Fig. 2(c), the period of lattice structure perpendicular to the blue line is 722.34 μm,which is 2.57 times longer than the period of the fringe perpendicular to the red line. The lattice structure along the blue line is produced by the supporting structure of the phase gratings,denoted by the arrow in Fig. 2(d), and its period is 7.6 μm.The ratio of the period of the supporting structure to the period of the grating bar is 2.53:1, which is very close to the ratio of the fringe periods imaged by the two structures. Note that the dark area of the fringe is wider than the bright area of the fringe in Fig.2(c). The reason is that the fringe visibility in Fig. 2(c) is very low (about 4% on average). In this situation, the low-intensity peaks in the fringe appear in black in the image,while the high-intensity peaks in the fringe appear in white.

Fig.2. (a)Simulation fringe obtained at z1 =1.2698 m,z2 =0.026 m and z3 =1.2698 m. (b)Simulation fringe obtained at z1 =2.5395 m,z2 =0.0513 m and z3 =2.5395 m. (c)Experimental result. The numbers beside the lines denote the number of pixels contained in the lines.(d)The SEM of the phase grating.

Fig. 3. (a) Differential phase image and (b) absorption image of the duck claws.

Two duck claws are imaged in the dual-phase grating interferometer. A phase-contrast image and an absorption image are retrieved by Fourier transform,as shown in Fig.3.The distance between the duck claws and the detector is 100 cm. Ten images are acquired for noise reduction, and the acquisition time of each image is 15 s. The cartilage in the joint of the duck claws is clearly visible in the differential phase image,and the effective field of view is 4.8×7 cm2. By increasing the area of the phase grating, the field of view is further improved.

4. Conclusion

We have successfully obtained the fringe period and the optimal visibility conditions of the dual phase grating interferometer. The dualπ-phase grating interferometer and the dualπ/2-phase grating interferometer have different fringe periods and optimal visibility conditions. Under the illumination of polychromatic x-ray, the fringe period is twice as long as the theoretical result, when the positions ofπ-phase gratings are far away from the positions where the fringe visibility is optimal. The phase gratings used in this study are the two halves of a phase grating with a diameter of 5 inch.[20]If complete 8-inch phase gratings are used,the field of view can be further expanded to meet imaging requirement of human fingers.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11674232, 62075141, and 12075156)and the Foundation of Shenzhen Science and Technology Bureau,China(Grant No.20200812122925001).