1 INTRODUCTION

Recently, various methods of hard X-ray phase-contrast imaging have been developed (see [1, 2] and references therein). The effectiveness of such methods is based on the fact that the sensitivity of phase contrast when investigating the internal structure of objects made of light elements (in particular, soft biological tissues) using hard X-ray radiation exceeds the absorption contrast sensitivity by three orders of magnitude [3]. One of the first and most common methods of phase-contrast imaging is the interferometric method based on a triple Laue-case interferometer (the so-called LLL-interferometer) [4, 5]. With this scheme, the interference pattern formed behind the third block of the interferometer between the beams is recorded: (a) the wave passing through the first block and reflected from the second and third blocks of the interferometer, and (b) the wave reflected from the first and second, and passing through the third block. The phase object under study is placed between the second and third blocks of the interferometer, in the path of one of the mentioned beams—the object wave (see Fig. 1). The phase shift of the object wave introduced by the test object leads to a redistribution of the registered interference fringes, which, in turn, serves as the basis for reconstructing the spatial distribution of the phase shift (see the algorithms presented in [6, 7]).

Fig. 1.
figure 1

Phase-contrast imaging device based on triple Laue-case (LLL) interferometer.

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Among the main advantages of such a device are its high stability and simplicity of adjustment, which is achieved due to the monolithic nature of the entire interferometer. The interferometer operates in the amplitude-division mode, which significantly reduces the requirements for the coherency of the initial radiation. As a result, such devices operate with laboratory sources of hard X-ray radiation and were implemented long before the advent of synchrotron sources of hard X-ray radiation [4].

As a disadvantage of the method, we note the limited space for placing the test object (about or less than one cm). To mitigate this limitation, an interferometer with a modified geometry has been proposed [8], as well as a non-monolithic interferometer made of two monocrystals [9]. The stability of the non-monolithic interferometer is ensured by a complex feedback control system. Another disadvantage of the scheme is the low resolution (~30 µm), which is caused by the “blurring” of the object wave during diffraction on the third block of the interferometer (analyzer). The reason for this is the peculiarity of Bragg diffraction, according to which a small angular displacement of radiation incident on a crystal results in a displacement of the crystal beam by an angle exceeding the initial displacement by 4–5 orders of magnitude. As a result, a small angular divergence of the object wave, caused by refraction on the test object, leads to a large angular displacement of the rays in the analyzer, consequently, blurring the interference pattern to the size 2t tanθB, where θB is the Bragg angle of the diffraction, and t is the thickness of crystal blocks, particularly the analyzer. One way to prevent this is to use a three-block Bragg interferometer, in which the object wave is reflected from the surface of the analyzer without penetrating deep into the plate [10]. Another method is to reduce the thickness of the analyzer to values of 40–100 µm to improve the resolution to ~10 µm [11, 12]. The disadvantage of this method is the difficulty of manufacturing such a device and its low accuracy.

Back in the 70s of the last century, it was shown that a two-block crystal system with parallel blocks of equal thickness, oriented according to Laue diffraction geometry (LL-system), acts as a kind of diffraction lens [13]. The δ-shaped wave incident on it, undergoing sequential diffraction on both blocks of the interferometer, is again focused on the output surface of the second block, which was experimentally proven in [14]. As a result, the LL-system is capable of transmitting an X-ray image from the input surface of the system to the output surface. This was also demonstrated experimentally [15]. Based on the diffraction focusing of X-rays in the LL-system, a compact X-ray spectrometer was proposed and experimentally implemented [16]. In [17], the phenomenon of diffraction focusing was studied in more detail. In particular, the dependence of the focusing characteristics on various deviations of the real experimental setup from the ideal one was studied: the difference in the thickness of the system blocks, the displacement of the incident beam from the exact Bragg direction, the width of the incident beam, etc. In [18], it is proposed to use the LL-system to suppress diffraction blurring of the object wave in the analyzer of the LLL-interferometer, and thereby increase the resolution of the interferometer in phase contrast imaging.

In [19] the feasibility of using an LL-system for the X-ray image transmission was considered by means of numerical simulation. In particular, it has been shown that the low coherence of the initial radiation results in the suppression of interference distortions, and, thereby, an increase in the quality and unambiguity of the resulting image.

The presented work aims to study the possibility of increasing the resolution of the above-presented device for phase-contrast imaging based on an LLL-interferometer by using an LL-system.

2 EXPERIMENTAL SETUP AND NUMERICAL SIMULATION

In the proposed device of phase contrast imaging, the third block of the LLL-interferometer (analyzer) is replaced by the above-mentioned two-block (LL) system (the so-called LLL+L interferometer). The purpose of this modification is to suppress diffraction blurring of part of the object wave caused by diffraction on the third block, through its subsequent diffraction on the fourth block. A schematic diagram and the path of the rays in such a device are shown in Fig. 2. Numerical modeling of such a device was carried out. The following testing phase objects are considered: a wire with a rectangular cross-section, directed perpendicular to the scattering plane, and a one-dimensional mesh of such wires. It is assumed that the phase shift of the object wave rays crossing the wire is –π rad. The phase object is placed on a substrate, causing a phase shift of –π rad. Numerical simulation of the phase contrast imaging was carried out for both the conventional scheme based on the LLL-interferometer and a modified one based on the LLL+L interferometer. As the initial radiation, we used both a plane monochromatic wave incident under the exact Bragg condition (coherent radiation) and radiation from a laboratory X-ray tube with a source size of 400 μm (here, we refer to the size of the transverse projection), taking into account the natural width of the spectral line of characteristic radiation. The distance of the laboratory source from the interferometer is chosen to be 1 m, so, the incident radiation can be considered incoherent. The Si(220) reflection of the characteristic radiation \({\text{Mo}}{{K}_{{{{\alpha }_{1}}}}}\) is considered. The thickness of the crystal plates is selected \(t = 12.25\,\Lambda = \) \({\text{446}}{\text{.8}}\;\mu {\text{m}}\), where \(\Lambda = {\text{36}}{\text{.48}}\;\mu {\text{m}}\) is the extinction length of Bragg diffraction. The computation results are presented in Fig. 3 (for phase objects in the form of a single wire) and Fig. 4 (for phase objects in the form of a one-dimensional mesh of such wires). Figures (a) and (c) show computations of phase contrast imaging using a three-block interferometer, while Figs. (b) and (d) show calculations using a four-block interferometer. Thick solid lines represent computations for incoherent radiation, and thin lines—coherent radiation.

Fig. 2.
figure 2

Modified device of phase-contrast imaging based on adding a fourth block to the LLL-interferometer (LLL+L interferometer).

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Fig. 3.
figure 3

Computed intensity distribution on the detector, depending on the x-coordinate directed along the line of intersection of the detector with the scattering plane, for a phase object in the form of a single wire directed perpendicular to the scattering plane. The width of the wires is (a, b) 7 and (c, d) 70 µm for (a, c) three-block and (b, d) four-block interferometers. Thick solid lines correspond to incoherent radiation, thin lines correspond to coherent radiation, and the dotted line shows the phase shifts caused by the test objects (without taking into account the –π rad phase shift of the substrate).

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Fig. 4.
figure 4

Computed intensity distribution on the detector, depending on the x-coordinate directed along the line of intersection of the detector with the scattering plane, for a phase object in the form of a one-dimensional mesh of wires with widths of (a, b) 10 and (c, d) 70 μm for (a, c) three-block and (b, d) four-block interferometers. The distances between the wires in the mesh are (a, b) 30 and (c, d) 50 μm. Thick solid lines correspond to incoherent radiation, thin lines correspond to coherent radiation, and the dotted line shows the phase shifts caused by the test objects (without taking into account the –π rad phase shift of the substrate).

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As can be seen from Fig. 3a, in the case of a thin single wire, the image obtained by a three-block interferometer is highly blurred and has a width \(\Delta \simeq 175\;\mu {\text{m}}\), that is close to the sum of the width of the wire (d), and the length of the base of the Bormann triangle \(2t\tan {{\theta }_{{\text{B}}}}\). This is explained by the fact that the small width of the wire results in a large angular spread of the object wave diffracted from it, which, when propagating in the analyzer, fills the Bormann triangle. In the case of a four-block interferometer, after diffraction on the fourth block, the image narrows to the width of the wire. In this case, the image is equally sharp for both incoherent and coherent radiation. This is caused by the low background level when imaging a δ-shaped wave with a two-block LL-system [13].

As the thickness of the wire increases, the angular broadening of the object wave diffracted by the wire decreases. As a result, in the case of a three-block interferometer and coherent radiation, the image narrows to the width of the wire (see Fig. 3c). However, interference effects distort the image, and in the mentioned case leads to the splitting of the wire image into two parts.

Interference distortions also arise with a test object in the form of a one-dimensional mesh when using coherent radiation, both in the case of three-block and four-block interferometers (see Fig. 4). In particular, in the considered case with the three-block interferometer, these distortions are perceived as artifacts resembling thin wires between real wires. However, when using a four-block interferometer with incoherent radiation, the distortions are minimized, and the artifacts disappear.

3 CONCLUSION

By means of numerical simulation, the feasibility of using a modified triple Laue-case interferometer for high-resolution X-ray phase contrast imaging has been demonstrated. The proposed scheme uses an incoherent X-ray source, which makes the device compact, with the possibility of using laboratory radiation sources.