Multi-beam X-ray optical system for high-speed tomography using a σ-polarization diffraction geometry

A multi-beam X-ray optical system using a σ-polarization diffraction geometry is proposed and its potential for high-speed tomography using synchrotron radiation is experimentally evaluated. Projection images of a sample are obtained simultaneously from different directions with X-ray beams generated by diffraction of a white synchrotron radiation beam at silicon single crystals. This makes it possible to record a tomographic dataset without rotation of the sample or X-ray source. Data sets of two samples obtained in a proof-of-principle experiment with an exposure time of 1 ms were successfully reconstructed using an advanced compressed-sensing algorithm.

A lthough X-ray computed tomography (CT) is a mature technique for observing the internal structure of objects three-dimensionally in life and material sciences, it is still challenging to achieve high temporal resolutions on the order of milliseconds in combination with micrometer spatial resolution. 1) When synchrotron radiation is used for X-ray CT measurements, usually the sample is rotated to obtain projection images in a range of 180°or more. For dynamic phenomena that are not easily repeatable, time resolution is achieved by rotating fast enough to avoid significant changes in the sample during the collection of a tomographic dataset. Large effort has been made to increase the measurement speed, and time resolutions of 50 ms or less were demonstrated about 10 years ago. 2,3) Recently, by combining experiments using intense synchrotron radiation with advanced analysis methods, even faster time resolutions down to one millisecond have become possible. [4][5][6][7][8][9][10][11] To realize this, the sample has to be rotated very fast (30 000 rpm for 1 ms time resolution), which causes high centrifugal forces that can disturb the structure of the sample. 9) The rotation also complicates control of the sample environment and the application of stimuli to the sample.
An approach that does not suffer from the problems caused by high-speed rotation is to use multiple X-ray beams to image the sample from different directions simultaneously. This is not easy, however, when a collimated, nearly parallel X-ray beam is used, like in synchrotron radiation CT. A first step in this direction was the demonstration of stereoscopic and triscopic imaging by Hoshino et al. achieved by using diffraction at silicon single crystals to split a synchrotron radiation beam into two or three beams. 12,13) Later, imaging with three beams generated by a single silicon crystal, with the potential for nine beams, was proposed and tested experimentally. 14) An analysis method advantageous for tomographic data with a small number of projections, but dense temporal information, was also recently proposed. 15) In the first demonstration of synchrotron radiation multibeam tomography without sample rotation, the time evolution of the three-dimensional structure of a test sample was reconstructed from projection images taken with X-ray beams simultaneously illuminating the sample in a wide angular range. 16) This was achieved by combining a multi-beam optical system consisting of arrays of crystalline blades attached to a bent silicon crystal 17,18) with a detection system capable of observing the projection images simultaneously. 19) A compressed-sensing algorithm 20,21) was used for the reconstruction. Recently, X-ray tomography with a time resolution of 0.5 ms was realized using this system. 22) A disadvantage of the previously reported optical system was, however, that it uses diffraction in a π-polarization geometry, i.e. with the polarization of the X-rays in the scattering plane. The diffracted intensity near a scattering angle of 90°decreases severely due to the polarization factor, 23) causing a missing angle of about 30°. This problem can be overcome with a σ-polarization geometry, in which the polarization of the X-rays is perpendicular to the scattering plane. We have recently reported preliminary results of a σ-polarization multi-beam optical system. 24) Here, we give further information about its design and evaluation, and show that this optical system can be used for multi-beam tomography without sample rotation with a millisecond time resolution by leveraging an advanced compressed-sensing reconstruction algorithm.
A schematic illustration of the optical system developed in this research is shown in Fig. 1. A white synchrotron radiation beam polarized in the z-direction is incident onto an array of silicon single crystals that are arranged along the propagation direction of the beam. At each crystal, an X-ray beam for imaging the sample is generated by diffraction. The crystals are oriented so that the diffracted beams cross at the sample position. Projection images of the sample in a wide angular range can be obtained without sample rotation by observing the diffracted beams. The crystals are shifted perpendicular to the incident beam to reduce the absorption.
The position and orientation of crystal i was calculated from the desired projection angle, which is equal to the scattering angle 2θ i of the diffracted beam, the distance a from the sample to the plane of the crystals and the incident beam (the working distance), and the width b that the crystals are shifted perpendicular to the incident white synchrotron beam. We define a coordinate system with the x-axis pointing from the sample to the incident white synchrotron radiation beam, the y-axis in the propagation direction of the incident beam, the z-axis in the polarization direction of the incident beam, and the origin at the sample position. The distance d i from crystal i to the point (a, 0, 0) and its position vector (x i , y i , z i ) are given by the following equations: Each crystal must be oriented so that the diffracted beams pass through the sample position. The orientation was obtained from the scattering vector q i that satisfies this condition for the diffracted beam originating from crystal i. q i points in the direction of the normaln i of the diffracting crystal planes. Unit vectors in the direction of the incoming beamk i in, and diffracted beamk i out, can be expressed asˆ( The diffracting crystal plane is perpendicular to the crystal surface in the case of the Laue diffraction geometry and parallel to the surface in the case of the Bragg diffraction geometry. The optical system used for the proof-of-principle experiment consisted of aluminum holders, in which depressions for holding the crystals were cut. The position and shape of the depressions were calculated so that the crystals satisfy Eqs. (2)-(4) and (7). 80 μm thick silicon single crystals ((Sharan Instr., floating zone, (001) surface) were fixed with silver paste (Dotite D-550, Fujikura Kasei) to the holders. Each crystal was oriented so that X-rays diffracted to a reflection of the 00n (n = 4, 8, 12...) family pass through the sample position. The Laue geometry was used for scattering angles up to 130°, and the Bragg geometry for larger angles.
Proof-of-principle experiments were conducted at beamline 14C of the Photon Factory at KEK. The synchrotron beam has a vertical polarization direction because the source is a vertical wiggler. 25,26) An aluminum attenuator (thickness 0.5 mm) was inserted in the X-ray beam upstream of the experimental hutch to reduce the heat load. The optical system was placed about 36 m from the source, with the xy plane in Fig. 1 horizontal and the tomographic plane inclined 4.1°from the horizontal. Further details and a photograph of the optics are given in our previous report. 24) Two test samples, a tungsten wire (Nilaco, diameter 50 μm) and a glass capillary (PB Probeta, inner diameter 0.4 mm, outer diameter 0.8 mm), were employed to evaluate the optical system. To observe the projection images of the samples in the diffracted beams, the X-rays were converted to visible light with a Gd 2 O 2 S:Tb scintillator (Mitsubishi Chemical Corporation, DRZ-HR, thickness 50 μm), which was transmitted to a high-speed CMOS camera (Photron FASTCAM Mini AX100) using a lens-coupling system (effective pixel size 20 × 20 μm 2 ).
Projection images of the samples were recorded sequentially with an exposure time of 1 ms by moving the detector to the appropriate position for each beam. Transmittance images were calculated by normalizing the projection images with images without the sample, after subtracting the background intensity (dark images for the wire, dark images and the average background intensity close to the beam for the capillary). Images from 28 beams were used for the analysis of the tungsten wire, and from 27 beams for the capillary. Representative transmittance images for the two samples are shown in Fig. 2; transmittance images for the other beams are shown in the supplementary material.
Each X-ray beam has a different energy because the Bragg angle differs. The effective X-ray energy of each beam was estimated by measuring the absorption by thin aluminum films. The dependence of the effective energy of each beam on the scattering angle is shown in the supplementary material. The effective energy is high for small scattering angles, so the contrast of the samples is small. For larger angles, the effective energy decreases, increasing the contrast, but the incident intensity decreases. The samples are visible up to the highest scattering angles, but when approaching 180°the intensity becomes weak compared to the background, resulting in a small signal-noise ratio.
The preparation of the projection images for tomographic reconstruction was conducted in a similar way to a previous report. 17) The tip of the wire or the top and axis of the capillary was used as reference positions to establish a common coordinate system. To account for the difference in absorption due to the different X-ray energies, the peak absorbance of the wire in each transmittance image was adjusted to the same value before calculating sinograms. For the capillary, the sinograms were normalized so that the sum of the logarithmic transmittance in each row is the same value. 22) The data set consists of a small number of projections compared to standard X-ray CT measurements and has a low signal-noise ratio in some projections. While it was possible to reconstruct the wire with the widely-used Filtered BackProjection (FBP) algorithm when the signal-noise ratio was improved by averaging 100 projection images (total exposure time 100 ms), 24) the wire was almost indiscernible when reconstructing the 1 ms projection images, due to severe artifacts. It is therefore necessary to use advanced reconstruction methods to achieve the highest possible time resolution. We employed the Abnormal Data Detected CT (ABD-CT) 27,28) image reconstruction method, which is an improved variant of the total variation (TV) reconstruction method. We describe the principle of ABD-CT below. In the standard TV reconstruction method, image reconstruction is performed by minimizing the least-squares data fidelity term penalized with TV, which can be expressed as where the J-dimensional vector x contains the image to be reconstructed, the I-dimensional vector b contains the projection data, and A denotes a system matrix relating x and b. In multi-beam CT using the proposed σ-polarization optics, the standard TV method as well as the FBP method produces a degraded image with severe streak artifacts due to the following reason. The intensity of the diffracted beams is strongly dependent on the projection angle so beams for some angles become very weak. Consequently, projection data corresponding to the angles of weak beams are contaminated with high noise as in the top part of the example sinogram shown in Fig. 3. With the standard TV reconstruction, the effect of these erroneous data is spread over the whole image. To achieve successful reconstruction from such data, ABD-CT modifies the cost function so that the minimization problem becomes where we note that the L 2 -norm was replaced by the L 1 -norm in the data fidelity term. The use of the L 1 -norm identifies the location of erroneous data (outlier) bins in the sinogram and excludes them from the data fitting. In machine learning and statistics, this property of the L 1 -norm is well-known and is called "robust data fitting". We refer to Ref. 29 for unfamiliar readers. In Refs. 27 and 28, Kudo et al. used this idea for tomography reconstruction for the first time and called it "Fault-Tolerant reconstruction" or "ABD-CT". The power of neglecting the erroneous data (outlier) with high noise is surprisingly large. Thanks to this property, we succeeded in reconstructing a reasonable image with ABD-CT even for short exposure times. A three-dimensional rendering and a tomogram of the tungsten wire sample reconstructed with the ABD-CT method are shown in Figs. 4(a) and 4(b), and a tomogram of the glass capillary in (c). They were reconstructed from a set of projection images recorded with an exposure time of 1 ms. The wire is clearly visible and the reconstruction has a similar quality to that reconstructed with the FBP method using 100 ms exposure time projection images. 24) The structure of the glass capillary is overall correctly reconstructed. The artifacts outside of the sample and the density variation in the capillary walls are probably caused by the low signal-noise ratio of some beams. This proves the feasibility of millisecond time resolution tomography using  the present optical system in combination with an advanced reconstruction algorithm. Advantages of the present optical system are a larger working distance and a larger field of view than realizable with the π-polarization optics. 17) The crystals were shifted perpendicular to the beam to reduce absorption, but the amount of shift can be adjusted to the width of the incident white X-ray beam. If the crystals are not shifted, the size of the field of view can be on the order of the cross-section of the incident beam, which is advantageous if a small synchrotron radiation beam is used.
In the present optical system, separate crystals are used to generate each beam, so the properties of each beam can be optimized independently. 24) This is in contrast to the π-polarization optics, 17) where a number of beams were generated with the same crystal. For example, the X-ray energy of the low scattering angle beams is high, resulting in low contrast. Using lower index scattering planes, (220) or (111), for these beams would reduce the X-ray energy and thereby improve the contrast. Crystal planes and scattering angles could also be chosen so that the X-ray energy is the same for each beam, as mentioned in the previous report. 24) Compared to the π-polarization optics, 17) the intensity at scattering angles around 90°is much stronger. The absence of an intensity decrease due to the polarization factor is to some extent negated by the need to use larger scattering angles up to 180°, however, for which the beam intensities are weak too. The intensities of these beams could be increased by using slightly strained and/or thicker crystals. A way to avoid the back-scattering angles would be to split the incident beam, for example by reflection of part of the beam at a mirror, or by diffraction at a single crystal beam splitter. Then, two optical systems with scattering angles up to 90°could be used to image the sample in a 180°range.
In conclusion, the evaluation of a multi-beam X-ray optical system based on a σ-polarization diffraction geometry showed its potential for high-speed tomography with millisecond time resolution. A key element to achieve this was the application of an advanced compressed-sensing reconstruction algorithm. The optical system has advantages over a previously proposed optical system based on π-polarization.