Multibeam X-ray tomography optical system for narrow-energy-bandwidth synchrotron radiation

The design and evaluation experiments of a multibeam X-ray tomography optical system that can be used with synchrotron radiation from sources with a narrow energy bandwidth, i.e. undulator sources, are reported. It consists of silicon single crystals that diffract the incident X-rays to 27 beams, which are used to image a sample. The energy of the beams was aligned with an accuracy sufficient for use at typical undulator beamlines. Projection images of a test sample were collected and successfully reconstructed, showing the feasibility of a high-speed X-ray tomography instrument based on the optical system.


X
-ray tomography using synchrotron radiation has become a standard tool for investigating the threedimensional structure of objects with a spatial resolution on the micrometer order.Instruments are available at many beamlines at synchrotron radiation facilities.Many applications only involve static objects for which the measurement time is not critical, but recently interest in high-speed time-resolved observations is growing and efforts are being made to improve measurement methods in order to fulfill the need for temporal resolution. 1,2)[5][6][7][8][9][10] This is close to the limit regarding the temporal resolution, however, as rotation speeds on the order of 1 rotation/ms cause unacceptably large centrifugal forces for many systems. 9)he multibeam X-ray tomography method is an alternative approach, which avoids the rotation of the sample.Instead, it simultaneously obtains projection images in a wide angular range with multiple X-ray beams. 11,12)The number of projections in a tomography data set using the multibeam method is 20-30 projections, smaller than in typical highresolution measurements, but it is sufficient for reconstruction when state-of-the-art algorithms 13,14) are used.A temporal resolution of 0.5 ms was recently demonstrated with multibeam tomography, 15) which is the highest resolution for non-repeatable phenomena to date.In addition to the high temporal resolution, a further advantage is the potential to measure samples that cannot be rotated easily, like fluids.
A key element of the multibeam X-ray tomography method is the optical system that generates multiple beams.Two different designs have been proposed, both using diffraction in silicon single crystals.The first design employs arrays of micro-fabricated silicon blades in the so-called πpolarization diffraction geometry, 11,16) while in the second design individual silicon crystals are placed along the incident synchrotron radiation beam in a σ-polarization diffraction geometry. 17,18)Both designs need white synchrotron radiation, because each beam has a different energy due to the different Bragg angle of each crystal.Therefore, they cannot use undulator sources, whose energy spectrum consists of discrete peaks with a narrow width. 19)Undulator sources typically have a much higher brightness than sources that produce white synchrotron radiation, like bending magnets or wigglers.Such sources would therefore significantly increase the photon flux in the beams for imaging, if the energy bandwidth of the optical system is the same, thus improving the temporal and spatial resolutions.Quantification of the contrast is simplified as well, when all beams have the same energy.
Here, we report a multibeam optical system designed for narrow-energy-bandwidth synchrotron radiation sources that can provide 27 beams for tomographic measurements.Evaluation experiments show that it can be employed with typical undulator sources.
Figure 1(a) shows an illustration of the optical system and the setup for the evaluation experiments.The optical system consists of an array of thin silicon single crystals, which are fixed to aluminium holders, as shown in Fig. 1(b).At each crystal, X-rays in the incident beam are diffracted so that the diffracted beams pass through the sample position.They are used to image the sample from many directions simultaneously.
The setup is similar to that for white synchrotron radiation in Ref. 18, with the difference being that the Bragg reflections used for each beam were chosen so that the same energy is diffracted for all crystals.The positions and orientations of the crystals can be calculated using Eqs.(1-7)  in Ref. 18.The optical system uses σ-polarization diffraction geometry, so there is no decrease in diffracted intensity due to the polarization factor. 20)It was designed for an X-ray energy of 24.35 keV, which is the energy of the K absorption edge of Pd.
The crystals were placed along the downstream direction, without shifting them in the direction perpendicular to the incident beam as in Ref. 18, because typical undulator radiation beams have a small cross section.The intensity of the X-rays at an energy of 24.35 keV is estimated to be attenuated to about 30% in front of the most downstream crystal, due to the absorption by the crystals.
Bragg reflections from the Si(111) to the Si(21 3 1) reflection were used.Table S1 in the supplementary material lists all used reflections and the corresponding scattering angles, which determine the projection angles of the sample.The large number of crystals with different orientations that are needed to obtain these reflections were prepared from Si (100) and Si(110) wafers by microfabrication, similar to previous reports. 11,16)Figure 1(c) shows a part of the design for the mask used for preparing the crystals from the Si(100) wafer, illustrating the orientation of the crystals when diffracting planes parallel to the (011) and (001) planes are used.The number of the crystal was encoded in binary at the top of each crystal, in order to distinguish them after manufacturing.The thickness of the finished crystals were 40 μm to 120 μm.Thin crystals were used for most beams to reduce the absorption of the incident X-rays.However, the extinction length for Bragg reflection is large for large scattering angles, so thicker crystals were used to increase the diffracted intensity.For most beams using Laue diffraction geometry, symmetric reflections were used, but the Bragg geometry crystals mainly used asymmetric reflections (see Table S1 in the supplementary material for details).
The X-ray energy of 24.35 keV used here is a compromise between several factors, for example, the contrast in the targeted samples, the number of Bragg reflections in the scattering angle rangefrom 0°to 180°, the need to avoid too much absorption in the silicon crystals, and the energy spectrum of the synchrotron radiation source.In general, Xray energies in the range of 17 keV to 30 keV are most suitable.In order to use the optics with synchrotron radiation from an undulator source, the energy of all diffracted beams must be within the energy bandwidth of the undulator.Here we assume a bandwidth of ±100 eV.The accuracy of the Bragg angle of the crystals necessary to achieve this depends on the scattering angle.For example, it is about 0.6 mrad at a scattering angle of 9.3°, 8 mrad at a scattering angle of 91°, and 70 mrad at a scattering angle of 170°.The crystals must also be correctly oriented for the beams to pass through the sample position.For large scattering angles, this requirement is more stringent than the requirement on the energy, for example, at a scattering angle of 170°an accuracy in angle of about 1 mrad is necessary to achieve a position accuracy of 0.5 mm in our design.
Although the optical system is intended for undulator sources, it is easier to adjust and evaluate the individual crystals with white synchrotron radiation.Evaluation experiments of the optical system were conducted at beamline 14C of the Photon Factory at High Energy Accelerator Research Organization (KEK).The source of this beamline is a superconducting vertical wiggler, which produces comparatively  032002-2 © 2024 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd high energy X-rays with a vertical polarization direction. 21,22)he scattering plane for the σ-polarization diffraction geometry is therefore horizontal, simplifying the experimental setup.The white X-rays were attenuated with a 0.5 mm-thick aluminium plate placed upstream of the experimental hutch to reduce the heat load.The optical system was set up in the experimental hutch about 37 m from the source.A photograph of the optical system is shown in Figure 1(d).
The diffracted beams were observed with a detector consisting of a Gd 2 O 2 S:Tb scintillator (Mitsubishi Chemical Corporation, DRZ-HR, with a thickness of 50 μm), a lens-coupling system, and a high-speed CMOS camera (Photron FASTCAM Mini AX100).The effective pixel size was 20 × 20 μm 2 .The detector was positioned for each beam using a rotation stage centered at the sample position.
In preparation for the measurements, the crystals were aligned so that the diffracted beams have the correct X-ray energy and pass through the sample position.The X-ray energy was checked by placing a Pd foil (Nilaco, with a thickness of 8 μm) in the diffracted beam and rotating the optical system slightly while observing the beam.At the Pd edge, the beam intensity abruptly decreases due to the increased absorption of the Pd foil.
The aluminium holders for fixing the crystals were designed so that the crystals have the correct orientation, but the orientation of the crystals must be fine-tuned to achieve the required precision in energy and position of the diffracted beams.This was done by fixing thin spacers made of polyimide film or polyethylene naphthalate film between the crystal holders and their support, and by rotating the supports around a rotation shaft.
The beams from 27 crystals were measured, of which the Pd absorption edge was observed for 14 beams.The energy of 13 of these beams was within 100 eV of the absorption edge after alignment.The energy of one beam was 260 eV higher than the absorption edge, probably due to a problem with the holder.
The absorption edge was not found for most of the large scattering angle beams.For these beams, the energy changes only slowly with the Bragg angle, so the change in intensity at the edge is smeared out and difficult to observe.In addition, the signal/noise ratio of these beams is small, and for some beams other harmonics contribute significantly to the beam intensity.However, as discussed above, the fact that the beam passes near the sample position is sufficient to ensure that the energy is within the required bandwidth, if the correct Bragg reflection is used.We therefore concluded that all crystals except for one were aligned with a sufficient accuracy for use with an undulator source.
A molybdenum wire (Nilaco, with a diameter of 30 μm) was used as a test sample.The main purpose of this sample was to check whether reconstruction of the projection images obtained with the diffracted beams is possible.
Projection images of the Mo wire were obtained with exposure times from 1/3000 s to 1/10 s. Figure 2 shows examples for transmittance images of the Mo wire obtained with different beams.Transmittance images from the other beams are shown in Fig. S1 in the supplementary materials.At large scattering angles, the diffracted beams become weak and there is a significant background, resulting in a small signal/noise ratio.Images from two beams were excluded from the analysis, because exposure times longer than 1/10 s were necessary to see the wire.The contrast of the wire is not the same in the transmittance images, because some beams contain a significant amount of other harmonics besides the desired energy, and because of differences in the background.
A few X-ray beams were visibly distorted, probably due to the strain introduced by fixing the crystals.The crystals were fixed to the holders with silver paste (Dotite D-550, Fujikura Kasei), and the strain seemed to mainly occur when there was silver paste in more than one spot or in a larger area on the crystal.
In preparation for the tomographic reconstruction, the projection images were first processed similarly to previous reports, 11,18) by aligning the tip of the wire to the same position and correcting for the difference in the absorption of the sample for different beams.
Next, we explain the image reconstruction method used in this paper.The proposed optical system suffers from two major drawbacks.The first drawback is that only a small number of projection data can be measured.The second drawback is described as follows.The intensity of the diffracted beams is strongly dependent on the projection angle (i.e.scattering angle).The projection data corresponding to the weak beams are contaminated by many abnormal data values due to statistical noise as can be seen in the large projection angles in the sinogram of Fig. 3(a).The image reconstructed from such projection data using the Filtered Back Projection (FBP) method is severely degraded by streak artifacts as shown in the example tomogram of Fig. 3(b).
In our previous work, 18) we employed an advanced reconstruction method called ABD-CT (ABnormal data Detected CT) 23,24) which is able to overcome this drawback.In this work, we used a similar reconstruction method, which can be considered an extension of the previous method.Basically, this method is based on Compressed Sensing (CS).be a projection data vector.Let A denote a system matrix relating x and b.In CS, image reconstruction is performed by minimizing the cost function expressed as where the first term is the Total Variation (TV) regularization term and the second term is the least-squares (i.e.L 2 -norm) data fidelity term. 25)However, this method does not work well for the data measured by the proposed optical system, because the L 2 -norm data fidelity is very weak to the existence of abnormal data in the measurement.
A key to obtaining a reasonable reconstruction lies in how to exclude the abnormal data from being used during the reconstruction.In ABD-CT, to identify locations of abnormal data in the sinogram space and exclude them from the data fitting, the cost function is modified as where the data fidelity term was changed from the L 2 -norm to the so-called L 0 -norm.In statistics, machine learning, and CS fields, it is well-known that the data fidelity term constructed by the L p -norm with p = 0 or 1 has a power to exclude the abnormal data in the data fitting.In our previous work, 18) we have used the L 1 -norm data fidelity.From a theoretical point of view, however, the power to identify and neglect the abnormal data is stronger with the L 0 -norm than with the L 1 -norm. 24)Therefore, we have used the L 0 -norm in this work.
In image reconstruction, the minimization of Eq. ( 2) is performed by using an iterative method.There exist a variety of iterative methods to solve reconstruction problems formulated as TV minimization.Among them, for its fast convergence, we have used the FBP-preconditioned iterative method 13,14) [originally developed to minimize Eq. ( 1)], modified in such a way that it can handle the case of L 0 -norm data fidelity.The details will be published elsewhere.
Tomograms of the same slice reconstructed with the different compressed sensing methods are compared in Figs.3(c), 3(d) and 3(e).The Mo wire is clearly visible in all of them, but streak artifacts and a distortion of the wire caused by the abnormal data are visible when using the L 2 -norm data fidelity [Fig.3(c)].The streak artifacts are completely suppressed with the ABD-CT method using the L 1 -norm [Fig.3(d)] or L 0 -norm data fidelity [Fig.3(e)].They both tend to reconstruct the shape of the wire closer to the circular cross section, but the L 0 -norm suffers from less artifacts in the background.Similar observations could be found in the other slices.The same is true for the threedimensional reconstruction in Fig. 3(f), showing the power of the reconstruction method.The successful reconstruction indicates that the principle of the optical system reported here is sound.
To realize a high temporal resolution multibeam X-ray tomography instrument using undulator radiation, the optical system reported here must be combined with a multibeam detection system. 26,27)This could be done in a similar way to a multibeam instrument using white synchrotron radiation in the π-polarization diffraction geometry. 15)n the present experiment, comparatively long exposure times of 1/10 s were necessary for the beams with a large scattering angle.For these beams, the intensity is weak, but there is a large background due to the white synchrotron radiation beam, which results in a low signal/noise ratio.The intensity of the diffracted beams will be much higher, if radiation from a high-brightness undulator source is used.In addition, the background will be reduced significantly, because the energy bandwidth of the incident beam is much smaller.Exposure times shorter by several orders of magnitude can be expected to be sufficient in this case.
An instrument based on the optical system described here would be useful for samples containing mainly light elements up to aluminum or calcium, using absorption contrast or The optical system was designed so that the energy of all diffracted beams is exactly the same.In practice, however, some amount of energy difference is necessary, because upstream crystals diffract the X-rays in the energy width of their Bragg peak out of the incident beam; therefore, downstream crystals should be aligned to diffract a slightly different energy to avoid a decrease in intensity.In the present setup, the intensity decrease caused by the diffraction loss would be avoided due to the unintentional misalignment of the crystals.
After submission of the present paper, X-ray multi-projection imaging at an undulator source using three beams was reported. 28)The 3D reconstruction was obtained from only two beams and had to be reconstructed using a deep-learning algorithm 29) due to the very low number of projections in a comparatively small angular range.The present optical system provides about 10 times more projections in an angular range close to 180°, which makes it suitable for general tomography.
In conclusion, a multibeam X-ray optical system for highspeed tomography that is suitable for synchrotron radiation with a narrow energy bandwidth was reported.Evaluation experiments using white synchrotron radiation showed that it can be aligned with a sufficient precision to be used with typical undulator sources.The structure of a test sample was successfully reconstructed from projection images using an advanced compressed sensing algorithm.

Fig. 1 .
Fig. 1.(a) Illustration of the optical system and experimental setup (bottom view).Silicon crystals are drawn in blue and the incident and diffracted X-ray beams in red.(b) A detailed view of the crystals that generate the beams with scattering angles of 55.6°, 62.6°and 68.5°, and their holders.(c) A part of the design for the mask used for preparing the crystals from the Si(100) wafer, illustrating the orientation of the crystals when diffracting planes parallel to the (011) and (001) planes are used.(d) A photograph of the optical system during the evaluation experiment using white synchrotron radiation with a vertical polarization direction.The scattering plane is therefore horizontal.(e) A closeup of the crystals close to the sample position.The crystals that generate the beams from 55.6°to 91.1°are visible, with the sample in the foreground.

Fig. 2 . 3 ©
Fig. 2. Transmittance images of the Mo wire observed with different beams.The projection angles (design values) and the Miller indices of the reflections are indicated on the left, the exposure times are on the right.(grayscale 0-1.2).

Fig. 3 . 4 ©
Fig. 3. (a) A sinogram of the Mo wire.(b-e) Tomograms of a slice obtained with different reconstruction methods: (b) the FBP method, (c) Compressed Sensing using TV with the L 2 -norm data fidelity, (d) the ABD-CT method using the L 1 -norm data fidelity, and (e) L 0 -norm data fidelity.(f) Three-dimensional reconstruction obtained with the ABD-CT method using the L 0 -norm data fidelity.