1. Introduction

Multi-projection X-ray microscopy methods provide important technological advancements for the visualization of fast stochastic processes. In the so-called single-shot multi-projection approach, a sample is held fixed in an X-ray probe beam, and volume information is obtained simultaneously in several defined directions without the need for scanning or rotating the sample. In the hard X-ray regime, multiple projections can be obtained simultaneously using crystal beam splitters. These optical elements split the primary beam into multiple directions as shown schematically in Fig. 1. When a sample that is smaller than the width of the beam is inserted just downstream of the splitter it is illuminated from multiple directions, providing access to 3D information in a single-shot exposure. Recently, various beam-splitting schemes for monochromatic or broadband X-ray beams have been proposed and experimentally demonstrated at synchrotron radiation facilities [1–4]. These use Laue or Bragg diffraction geometries in crystals to generate multiple X-ray probes. Such approaches are suitable for recording multiple projection images in a single-pulse exposure at an XFEL source [5]. With the megahertz pulse trains available at the European XFEL, it becomes possible to far exceed the current capabilities of standard tomographic methods where the sample is rapidly rotated in the beam. The highest speed in tomography reported so far is 208 tomographic frames per second [6]. Increasing these speeds is not only technologically challenging but may affect the dynamics of the samples due to the strong centrifugal forces induced by fast sample rotation.

 

Fig. 1. Schematic diagram of the stereographic X-ray phase imaging method, using a single phase grating made from diamond and a Si 001 crystal splitter. The −1−3 1 and 1 3 1 reflections of the crystal beam-splitter were used to produce two identical X-ray beams. The sample was placed in the overlap region of the beams just after the splitter and two X-ray interferograms were recorded simultaneously at the two detectors, at two different views. The angle between the two views was 70.2$^{\circ }$.

Download Full Size | PDF

X-ray multi-projection imaging to date has mainly relied upon contrast generated by the propagation of the spatially coherent beam passing through the sample. That is, a near-field hologram or Fresnel diffraction of the object is recorded. For a pure phase object, the intensity in such an image is proportional to the Laplacian of the phase [7] and is thus sensitive to sharp transitions in density, e.g. at object boundaries, at the expense of smooth density variations in the sample. However, the characterization of many fast processes, such as shock wave propagation or mixing of fluids, requires tracking and quantifying smooth changes in density. Phase-contrast imaging methodologies with the requisite sensitivity to these lower spatial frequencies exist that sense the first derivative of the phase. Many of these so-called differential phase-contrast methods utilize structures placed in the beam to modify the phase of the incident wavefront and are thus compatible with a single-shot acquisition. Among the many deterministic differential phase methods in the hard X-ray region, the most practical are based on periodic [8,9] or random phase modulators [10,11]. X-ray grating interferometry [12,13] has found broad application and is the method used here. The required phase gratings can be fabricated with high precision from low absorbing materials such as diamond [14] and which can withstand the high photon fluences of XFEL beams. Several single-shot schemes for phase-contrast imaging and wavefront sensing using a single phase grating have recently been demonstrated [8,9,15,16].

In this work, we combine multi-projection microscopy (in this case stereography) [3] with single-grating interferometry to show the feasibility of single-exposure stereographic X-ray phase-contrast imaging of a static phantom object. We placed the phase grating upstream of the beam-splitter so that it functions equivalently for all projections. We propose that this method can be applied at XFEL sources where it will enable time-resolved quantitative and highly sensitive phase imaging of fast stochastic processes. At the European XFEL, for example, imaging frame rates of up to several megahertz can be achieved [17].

2. Experimental

Our experiments were carried out at the Diamond Light Source (UK) high-coherence beamline I13-1 [18]. The photon energy was set to 12.6 keV using a horizontal two-bounce Si 111 monochromator. A linear diamond transmission phase grating with a pitch of 4 µm and structure height of 3.8 µm was placed in front of the thin Si crystal beam-splitter. The diffraction vectors (+1 and −1 orders) of the phase grating were oriented in the vertical plane, perpendicular to the diffraction plane of the splitter to avoid Fourier filtering of the diffracted waves from the grating by narrow angular acceptance of the splitter in the diffraction plane. The splitter was produced in the laboratories of INFN-Ferrara (Italy) by anisotropic etching of an area of 5 mm $\times$ 5 mm into a 500 µm thick wafer to leave a 30 µm thick single-crystal membrane supported by a frame. The normal of the main face of the splitter coincides with the (001) crystallographic direction and the sides of the square membrane are parallel to the (0-10) and (001) directions. X-rays passing through a crystal generate a diffracted beam for every lattice plane that is oriented such that its Bragg diffraction condition is obeyed. The symmetries of families of planes can be exploited for selecting multiple planes that diffract at the same Bragg angle and specific photon energy to generate several beams with equal efficiency. The 113 family of planes was used in this experiment, following earlier practice [3]. The included angle between the (001) direction and the reciprocal lattice point of any of the eight reflections in this family is 17.55$^{\circ }$ and thus the diffraction condition is satisfied for a beam incident in the 100 direction with a wavelength commensurate with the 17.55$^{\circ }$ Bragg angle. We used the −1−3 and 1 3 1 reflections for the two projection imaging branches. These lie in a plane that is tilted by $\arctan 1/3 =$ 18.4$^{\circ }$ to the 001 plane. The splitter was therefore rotated around the beam axis by this angle to place these two diffracted beams in the horizontal diffraction plane (Fig. 1). The angle between the two chosen beams is 70.2$^{\circ }$. The splitter was slightly de-tuned to excite only two diffraction waves in a horizontal plane from −1−3 1 and 1 3 1 reflections rather than all 8 waves from the 113 family. The sample was placed approximately 0.6 mm downstream of the splitter in the beam overlap region so that it was illuminated simultaneously by the diffracted beams as well as the transmitted beam. The direct transmitted beam, which also illuminated the sample, was not used in this case due to the lack of a third X-ray detector. The sample consisted of soda lime glass spheres with a diameter of 100 µm glued onto a bundle of carbon fibres. Two simultaneous X-ray projection interferometric images were recorded using high-resolution indirect X-ray detectors. two high-resolution indirect-detection X-ray microscopes. Each microscope was composed of a scintillator coupled via diffraction-limited optics to a CCD detector (PCO4000, 4008 ${\times }$ 2672 pixels, 9 $\mathrm{\mu}$m pixel size). A total magnification of 20x was used in the experiment, providing an effective pixel size of 0.45 $\mathrm{\mu}$m. An exposure time of 5 s was used. A photograph of the experimental setup is shown in Fig. 2.

 

Fig. 2. Photograph of the set-up during the experiment at the Diamond Light Source I13 coherence branch beamline, showing the beam paths. The insert shows details of the grating, crystal beam-splitter, and sample.

Download Full Size | PDF

3. Results and discussion

The recorded interferograms were processed using the Fourier method [19,20] to extract the phase gradient and absorption images. Absorption images were obtained from zero order peak in the Fourier domain (DC component) and successive inverse Fourier transform. The phase gradient images were obtained by isolation of the first-order peak in the Fourier domain and its shift to zero frequency position. After the inverse Fourier transform, the differential phase image was obtained in the spatial domain followed by integration to obtain the phase map. The results are shown in Fig. 3. The phase profiles obtained from the two branches match well with the theoretical phase profiles of lime glass spheres.

 

Fig. 3. (aI, aII) Absorption images for branches 1 and 2. (bI, bII) Raw interferograms for the black rectangle in panels (aI) and (aII), respectively for branch 1 (bI) and branch 2 (bII). (cI, cII) Differential phase image reconstructed from the interferograms in the region of the black rectangle shown in panels (aI) and (aII), respectively for branch 1 (cI) and branch 2 (cII). (dI, dII) Comparison of the extracted differential phase (continuous red line), integrated phase (dashed blue line) and theoretically expected phase (blue shade) of the lime-glass spheres with a diameter of 100 $\mathrm{\mu}$m including thickness tolerance given by the manufacturer. The experimental plots represent an average over the columns in the red rectangles in panels (cI) and (cII).

Download Full Size | PDF

Further, we reconstructed a 3D model of the sample using a stereographic point-based technique [21–23]. This relies on finding low-level primitives or features of the object that can be unambiguously identified in several of the multi-view images. As an example, it is possible to fit circles to the borders of the projections of the glass spheres to locate their centers by triangulation. We consider a coordinate system for the model where $Z$ is in the direction of the incident beam, $Y$ is vertical, and $X$ horizontal. The two projections are labeled $1$ and $2$, with direction cosines in the $Z$ direction of $\cos \alpha$ and opposite $X$ components (here, $\alpha = 35.1^{\circ }$, see Fig. 1). Under the assumption that the projections are formed with collimated beams, the $Y$ coordinate of a sphere is the same in the two projections: $Y=Y_1=Y_2$. A sphere is identified in the two projections at positions $x_1$ and $x_2$ (in the rotated coordinate systems of those projections), giving rays at positions $X_{1P} = x_1/\cos \alpha + X_{10}$ and $X_{2P} = x_2/\cos \alpha + X_{20}$ in the global coordinate system, where $X_{10}$ and $X_{20}$ give the origins of the two projection images in global coordinates. The global $X$ and $Z$ coordinates of the sphere can then be determined by finding intersections of rays described as

Equating these gives a solution for the intersection at

This solution can be applied to any pair of points in the two projections that are identified with the same object, to locate that object in three dimensions. This was carried out for the centers of the spheres and some identifiable features of the carbon fibres. The two branches differ in the position of the sample illumination, but this does not affect the 3D reconstructed image. This difference produces just a horizontal translation of the 2D images over the field-of-view of the camera. This effect is indistinguishable from a translation of the cameras, so it is accounted for within the treatment of the positioning errors of the cameras. This translation does not affect the 3D reconstructed image since its only effect is a translation of the entire 3D image in the 3D space, which is eliminated by shifting the axis origin. The result of the reconstructed model is shown in Fig. 4, which is plausible since none of the spheres are overlapping with each other or the fibre bundle. This approach relies upon unambiguously identifying pairs of points in the two projections, which is not always possible. Nevertheless, it may provide a model that is a good enough starting point for further refinement through statistical methods [24], which was not performed in this work.

 

Fig. 4. Stereographic images of the sample with projection angle +35.1$^{\circ }$ (a) and −35.1$^{\circ }$ (b). 3D reconstruction of the sample as seen from these observation angles (c,d). The carbon fiber filament is approximated to a cylinder to increase its transparency so to ease the identification of the spheres The soda-lime glass spheres are non-intersecting and separated from the structure of the carbon fibre thread, composed of a multitude of micrometer fibres (The 3D volume can be seen under various views rendered in the video in Visualization 1).

Download Full Size | PDF

4. Conclusions

In summary, we have successfully demonstrated combining phase imaging using grating interferometry with stereographic image acquisition, to provide a way to obtain three-dimensional structural information in a single shot. From the simultaneously recorded interferograms, the phase information was recovered and the 3D information was accurately obtained. We propose to use this method for stereographic or multi-projection microscopy at megahertz XFEL sources for the visualization of complex dynamic phenomena in three dimensions, such as liquid mixing or shock-wave propagation. It should be noted that the proposed method has limitations that need to be addressed and investigated in order to achieve a practical implementation at XFEL sources. The most significant is the photon energy acceptance of the splitter crystal, which limits the spectral bandwidth of the diffracted beams and fixed photon energy in the case of multi-wave excitation. The loss of flux outside this bandwidth could be reduced by generating pulses of narrower bandwidth to begin with, as are obtained by seeded SASE beams [25]. However, these issues could be mitigated by employing other splitter schemes that are tunable in photon energy and projection angle.