1. Introduction

Since the discovery of x-rays by Röntgen in 1895, x-ray imaging has been widely used for nondestructive visualization of internal structures in various fields, such as industrial nondestructive testing, security screening, and medical diagnostics. However, as the contrast stems from differences in x-ray attenuation, weakly absorbing objects composed of low-Z atoms, such as soft biological tissues, do not (in principle) form clear images. To solve this problem, several recent studies have developed imaging techniques based on x-ray phase shift [1–4]. These so-called x-ray phase-contrast-imaging (XPCI) techniques are important because they are three orders of magnitude more sensitive to light elements than that of the absorption-contrast technique [5]. However, most of these recent techniques require synchrotron facilities because they need highly coherent x-rays to detect the x-rays slightly refracted by objects. An x-ray Talbot–Lau interferometer, which consists of three transmission gratings (source, phase, and absorption) may provide a technical solution to this problem [6]. From a series of moiré images, it provides three images: absorption, differential phase, and dark field [7]. Because it is compatible with conventional x-ray tubes, the Talbot–Lau interferometer is expected to be used in practical applications in various fields.

Fabricating the source and absorption gratings for a Talbot–Lau interferometer is a subject of current research because a high-aspect-ratio microstructure is needed to block hard x-rays and transmit them through narrow slits. Even when gold is used as a grating material, the grating thickness should be several tens of micrometers, and it is difficult to fabricate narrow slits having width less than 10 μm. Furthermore, the size of the absorption grating determines the field of view of XPCI, so fabricating large absorption gratings is a crucial development challenge [8].

To solve this problem, we proposed an alternative configuration of the Talbot–Lau interferometer that dispenses with the source and absorption gratings by using multiline metal targets embedded in a diamond substrate [9,10]. When irradiated by electrons, these metal targets create an array of x-ray line sources with sufficiently narrow line width and emit x-rays. The absorption grating is not required if the self-image is sufficiently magnified to be resolved directly by the detector, which may be done by reducing the distance between the target and the phase grating. By using multiline embedded targets with 1 μm linewidth as x-ray sources, we developed a compact interferometer that requires only a single transmission grating and that has a source–detector distance of less than 1 m [10].

In this work, we demonstrate a further advancement in the configuration of this interferometer, which consists of using a micrometal-array target and a two-dimensional (2D) phase grating (see Fig. 1). The checkerboard phase grating with π-shifting structures produces an interference distribution of periodic dots pattern at the Talbot distance. To magnify the self-image sufficiently to be resolved directly with a CCD detector, the phase grating is placed close to the embedded targets. By proper design of the pitch p₀ of the metal arrays, we obtain superimposed self-images from each target. For the one-dimensional (1D) Talbot–Lau interferometer with a line-patterned phase grating, the image contrast is only due to x-ray refraction and scattering in the direction perpendicular to the grating lines. This means that the sensitivity of the 1D interferometer depends on the orientation of the grating lines: in principle, refraction that is oriented parallel to the grating lines is not visible. In the 2D configuration, however, a 2D periodic intensity pattern (dot-shaped self-image) can be shifted and deformed in all directions on the CCD detector, which enables us to simultaneously retrieve the differential-phase and dark-field signals in all directions.

 

Fig. 1 Two-dimensional x-ray Talbot–Lau interferometer with single transmission grating and micro x-ray-source array.

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Momose et al. first demonstrated a 1D interferometer without an absorption grating by increasing the distance between the phase grating and the detector, and suggested an expansion to a 2D setup [11]. However, their interferometer was very long, and the 2D interferometer has yet to be demonstrated.

Recent publications have reported XPCI methods based on 2D gratings [12–16]. Sato et al. first demonstrated a 2D Talbot–Lau configuration, which consists of a conventional x-ray tube, two mesh-patterned amplitude gratings (source and absorption), and a checkerboard-patterned phase grating [12]. They obtained differential-phase-contrast and dark-field images by Fourier analysis of a moiré fringe pattern. Using this technique, they captured single-shot images without resorting to the multiple exposures required by the phase-stepping technique [13]. However, they pointed out the problem of long exposure time originated from the low aperture ratio of the source and the absorption gratings [14].

In another report, Wen et al. proposed a 2D phase-imaging method based on Bucky grids [15]. They obtained phase-contrast and diffraction images by Fourier analyzing x-ray-attenuation images of the grids as opposed to self-images generated by x-ray interference. In their demonstration, they placed the grid exactly midway between the x-ray source and the detector. The grid could not be easily placed close to the x-ray source in their setup,　because the pitch of the grid had to be smaller in order to maintain the pitch of the grid image obtained at the detector position, resulting in extremely high aspect-ratio structures. This configuration had a disadvantage in terms of phase sensitivity when samples were placed close to the x-ray source so that the sample image would be magnified more than twice at the detector position. It is well known that, in Talbot interferometers, the phase sensitivity is high when samples are placed close to the phase grating, and decreases with increasing distance between the sample and phase grating [17,18]. The phase sensitivity similarly decreased with increasing distance in their setup.

Meanwhile in our setup, the phase grating was located very close to the x-ray source and the sample was placed downstream of the phase grating. Therefore, the phase sensitivity increased as the sample approached to the x-ray source, because the distance between the sample and the self-image increased [18].

2. Fabrication of embedded x-ray targets and phase grating

We fabricated a dot-patterned Cu target in a manner similar to that described in our previous papers [9,10]. The Cu targets were embedded in commercially available, electrically conductive (resistivity = 5 × 10⁻⁴ to 7 × 10⁻⁴ Ωm) polycrystalline diamond substrates of 10 mm in diameter and 0.5 mm thick. The patterned area was 5 × 5 mm². After wet cleaning the substrate, a 2-µm-thick SiO₂ layer was sputter deposited for use as a hard mask for reactive-ion etching (RIE) of the substrate. We then coated the SiO₂ surface with an 800-nm-thick photoresist layer and patterned by using optical lithography. We formed a patterned Cr layer, 300 nm thick, by using the lift-off process. The SiO₂ hard mask was patterned by RIE using CF₄ gas and with the Cr layer serving as the etching mask. Holes 1.8 μm in diameter and 3 µm deep were fabricated by RIE of the substrate with a mixed gas (O₂ + CF₄). After sputter deposition of the 1.5-µm-thick Cu layers, the surface Cu layers were removed by lifting off the SiO₂ layers. Figure 2(a) and 2(b) shows the fabricated Cu targets embedded into the polycrystalline diamond substrate. The Cu dot pattern is clearly visible over the entire diamond substrate. The period p₀ of the targets was 3.1 µm, and the dot size of each target was 1.8 µm.

 

Fig. 2 (a) External view and (b) scanning electron micrograph of micro-array targets embedded in diamond substrate. Scanning electron micrographs of checkerboard phase grating (c) surface and (d) cross section. The grating structure was etched into a (100) oriented silicon wafer. The wafer was 60 µm thick.

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The checkerboard phase grating was fabricated using photolithography and deep reactive-ion etching (DRIE). After wet cleaning the Si substrate, we created the checkerboard Cr layer (100 nm thick) on the Si substrate by photolithography, sputter deposition, and lift off. The Si substrate was etched by DRIE by using the Cr mask layer. The patterned area was 10 × 10 mm², although we used only 0.7 × 0.7 mm² of this area in the x-ray experiments of this study because of the short distance between the source and phase gratings. This short distance is of crucial importance for the practical use of the Talbot–Lau interferometer. Figure 2(c) and 2(d) shows a scanning electron micrograph (SEM) of the phase grating (after the Cr was removed). The grating period p₁ was 6.0 μm. For CuKα x-rays (8.0 keV photon energy), the required π-phase shift was attained with a structure height of 10 μm. However, the height of the Si pillars was slightly greater than expected because of the fluctuations in the DRIE conditions.

3. X-ray phase imaging

The x-ray imaging experiments were done with the embedded Cu targets and a phase grating that we fabricated. Electrons irradiated the multidot Cu targets, creating x-rays that radiated out the other side of the diamond substrate; these rays were used for imaging. The irradiated area size was 0.7 mm (horizontal) by 0.8 mm (vertical). X-rays were recorded by using a cooled silicon CCD (BITRAN BQ-52E) equipped with a Kodak chip having 1,024 × 1,024 pixels (pixel size: 24 µm × 24 µm), in which no scintillator was used (i.e., direct-conversion x-ray CCD).

Figures 3(a) and 3(b) show the self-images of G₁ with a 100 µm pitch, which is seen through and outside of a polymer sphere 3 mm in diameter. The source–G₁ and G₁–detector distances were R = 3.0 cm and Z_T = 97 cm, respectively, so the self-image was magnified 32 times. We clearly see the self-image of the dots pattern formed by the checkerboard phase grating with π-shifting structures [16]. Figure 3(c) shows the intensity profiles of the self-image in the horizontal and vertical directions, and the pitches correspond to approximately four pixels of the detector. The self-image covered the entire field of view of the image detector (2.5 cm × 2.5 cm) with a high visibility of 38% in each direction. The self-image has a clear visibility of about 30% even at 10 cm from the optical axis, which indicates a very wide field of view of 20 × 20 cm² in this setup. Figure 3(d) shows how the x-ray-intensity peaks in the self-image displace due to refraction from the sample. The peaks are displaced radially inside the spherical sample. At the marginal region of the sphere, the self-image moved more than near the center of the sphere, which means x-rays are refracted mainly in the marginal region. The magnitude of the displacement of the peaks is 3.5–4.0 μm in the marginal region.

 

Fig. 3 (a) Self-image of polymer sphere (polymethylmethacrylate) obtained from 2D configuration. The distance between the objects and detector was 50 cm. The scale indicates the length at the position of the sample. (b) Magnified image of the rectangular area shown in Fig. 3 (a). The scale indicates the length at the detector plane. (c) Intensity profile along horizontal (red) and vertical (blue) directions shown in Fig. 3 (b). (d) Vector image: arrows show the magnitude and direction of the displacement of self-image caused by x-ray refraction in the sample. The exposure time was 800 s.

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To derive absorption, differential-phase contrast (DPC), and dark-field images from the single observed image, we did a least-squares fit of a 4 pixel × 4 pixel region of data around each pixel. If we define the fitting function as F(x, y) = A_xsin(x − φ_x) + A_ysin(y − φ_y) + C, then the transmission coefficient T, the refraction angle in each direction (Φ_x, Φ_y), and the visibility reduction (V_x, V_y) are given by T = C^s/C^r, Φ_i = p₂(φ_i^s − φ_i^r)/(2πd), V_i = A_i^sC^r/(A_i^rC^s), where d is the distance between sample and detector, p₂ is the pitch of the self-image, the superscripts s and r denote the values measured with the specimen in place and with no sample (i.e., reference), and the subscript i denotes the x or y direction. This means that we can simultaneously obtain five different contrast images.

Figure 4 shows the respective images obtained by using this procedure. These images were obtained with an x-ray tube input power of 60 W (20 kV, 3 mA). The types of images obtained are basically the same as for a conventional Talbot–Lau interferometer [7]. Typical shaded DPC images are obtained, and a void in the polyethylene sphere is visible. Additionally, a crack in the rice grain is clearly visible in the y direction in the dark-field image, but not in the x direction [cf. Figures 4(e) and 4(f)]. This result indicates the superiority of the 2D configuration for imaging samples with anisotropic microstructure.

 

Fig. 4 Images of rice grain and polyethylene sphere: (a) self-image, (b) absorption image, DPC images along (c) x direction and (d) y direction, dark-field images in (e) x direction and (f) y direction. The distance between the objects and detector was 50 cm. The scale indicates the length at the sample position. The ranges are 20%–120% in transmission, −15–15 µrad in refraction angle, and 20%–120% in visibility. The exposure time was 800 s.

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Figure 5 compares the phase images calculated by the difference method. In the conventional analysis, the phase image is calculated by 1D integration [Fig. 5(a)]. However, this integration gives a phase image that suffers unavoidably from artifacts because of the noise in the DPC image. Recently, a 2D method for phase retrieval that combines images of local phase gradients in two orthogonal directions was proposed by Kottler et al. [19]. Figure 5(b) shows the clear phase image calculated by this method. The experimentally observed maximum of the integrated phase shift is φ = 180π. The real part of the refraction index is typically expressed as n = 1 − δ, and the total phase shift through the center of the sphere is given by φ = 2πdδ/λ for a homogenous sphere with diameter d (3 mm), where λ is the wavelength of the x-rays (0.154 nm). From this expression, we obtain δ = 4.7 × 10⁻⁶, which is consistent with published values. Therefore, we simultaneously obtain not only five different images but also images of good quality and an artifact-free phase image.

 

Fig. 5 (a) Integrated phase image of the polymer sphere (polymethylmethacrylate) [φ (π)] in the x direction. (b) Phase image obtained using 2D algorithm [19]. The scale indicates the length at the position of the sample. The images are represented with a linear grayscale.

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In the demonstration by Kottler et al. [19], 2D DPC images were obtained with the phase-stepping technique in two orthogonal directions using 1D gratings. They turned the sample by 90° in their experiments. In contrast, the technique we proposed herein could create 2D DPC images from a single raw image, similar to the demonstration of 1D single-grating interferometer shown in our previous paper [10]. The visibility of the self-image decreased, but still remained greater than 30% in the 2D experiments. This indicates the potential to the practical use of 2D x-ray phase contrast imaging.

In the measurements demonstrated in this work the exposure time of the individual raw images was 8.0 s, and 100 images were averaged to reduce the statistical noise of the raw image. The long exposure time was mainly originated from high signal-to-noise ratio (SNR) (V_SNR) of our CCD detector, which was expressed by V_SNR = σ/μ, where σ is standard deviation of the noise and μ is the average intensity. Observed images were suffered from high statistical noise, which was not related with the x-ray source and grating, but probably due to the radiation damage on the CCD chip of our detector. The SNR was typically 0.18 (σ = 5200 and μ = 29000 counts) in the image observed for 8.0-sec exposure time by the CCD detector placed 1 m from the x-ray source under the input power of 60 W (20 kV, 3 mA). In order to reduce the SNR of the image, 100 images were integrated, resulting in the total exposure time of 800 seconds. We believe that the use of low SNR detector drastically reduces the exposure time.

When we discuss the phase sensitivity of Talbot–Lau interferometer, we often use the ratio of Δ/P as the indicator of the phase sensitivity, where Δ is the shift of the self-image of the phase grating caused by x-ray refraction in the sample and P is the pitch of the self-image. In order to enhance the phase sensitivity, the phase grating with small pitch is placed near the absorption grating. In our setup the phase grating with 6 μm pitch was placed close to the x-ray source. The large pitch of phase grating and the small distance between the source and phase gratings lead to the large P, resulting in low Δ/P value. However, when we place the sample close to the phase grating in our setup, we obtain large Δ due to the large distance between the sample and the self-image. Therefore, we can compensate the disadvantage originating from large P.

In addition, we observe the self-image directly by 2D detector. The phase measurement is to measure Δ of the spots (2D self-image) in this case. This is similar to the measurement of a position of a light spot by four-element quadrant photodiodes. We can estimate the slight shift of a light spot by comparison of the signals from the four quadrants. Based on the analogy with this optical position sensing, we consider that Δ of the self-image are more important than the Δ/P value to discuss the phase sensitivity. The sensitivity depends on the visibility of the self-image and the detector performance (noise level and spatial resolution). If the detector has high performance, our setup would show high phase sensitivity, because our setup allows large Δ keeping the large distance between the sample and the self-image.

4. Conclusion

We demonstrated a 2D Talbot–Lau interferometer by using a micro-array target and 2D phase grating. This configuration dispenses with the need for source and absorption gratings, which were heretofore required high-aspect-ratio microstructures. We directly resolve the self-image with a dots pattern of the phase grating by using the x-ray image detector and simultaneously obtain six contrast images (including the artifact-free phase image). These results indicate that the proposed configuration should make 2D XPCI practical for a wide range of applications.