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X-ray Talbot interferometry, which uses two transmission gratings, has the advantage that broad energy bandwidth x-rays can be used. We demonstrate the use of white synchrotron radiation for high-speed X-ray phase imaging and tomography in combination with an X-ray Talbot interferometer. The moiré fringe visibility over 20% was attained, enabling quantitative phase measurement. X-ray phase images with a frame rate of 500 f/s and an X-ray phase tomogram with a scan time of 0.5 s were obtained successfully. This result suggests a breakthrough for time-resolved three-dimensional observation of objects that weakly absorb X-rays, such as soft material and biological objects.
©2009 Optical Society of America
1. Introduction
Since the early 1990s, techniques for X-ray phase imaging have been extensively studied to visualize weakly absorbing objects [1, 2, 3]. The image quality achieved by using X-ray phase information is innovative and a variety of applications is expected. The techniques of X-ray phase imaging enable us not only to take phase-contrast pictures simply but also to generate quantitative images of the shift of the X-ray phase. For example, based on the measurement of the X-ray phase shift in plural projection directions through a weakly absorbing object, three-dimensional reconstruction of the refractive index called X-ray phase tomography was attained [4, 5].
Most X-ray phase imaging activities were demonstrated by static observation. Particularly using crystal optics, such as a crystal interferometer and an angular analyzer, monochromatic plane-wave X-rays are required and sufficiently long exposures are needed for imaging [5, 6]. Although the edge-enhanced contrast generated through the propagation-based method [7] is observed in live videos [8] at synchrotron facilities, quantitative measurement of the phase shift is not performed with the videos. Thus, time-resolved (or high-speed) X-ray phase imaging/tomography is unexplored, and we aim at pioneering this field. Various applications of the technique are strongly expected to materials science (deformation, phase separation, phase transition, and so forth) especially of soft matter and also to in vivo biological imaging.
A breakthrough has been made by using the grating-based technique (Fig. 1), which uses an X-ray Talbot interferometer consisting of two transmission gratings [9, 10]. This technique has a prominent advantage that polychromatic X-rays are available because of the use of gratings. It has been theoretically shown that the image generated with X-rays of a bandwidth (ΔE/E) of 1/8 is almost comparable to the image obtained with monochromatic X-rays [11]. Even when white synchrotron radiation is used, X-ray phase imaging is actually possible, as reported in this paper, although the image quality decreases to some extent. The use of white synchrotron radiation implies that an extremely bright X-ray beam is available and high-speed X-ray phase imaging/tomography and furthermore time-resolved X-ray phase tomography are strongly expected. In this paper, we demonstrate X-ray phase imaging with an exposure of 1 ms and X-ray phase tomography with a scan time of 0.5 s.
2. Differential phase measurement with Talbot interferometer
The configuration of an X-ray Talbot interferometer is shown in Fig. 1. G2 must be an amplitude grating and G1 is normally a phase grating when X-rays are used. We used a phase grating that generates approximately a π/2 phase shift, and therefore the spacing between G1 and G2 was set to d 2/(2λ)≡z 0, where d and λ are the pitch of the gratings and X-ray wavelength, assuming the use of nearly a plane wave such as synchrotron radiation [9]. Setting the coordinates (x,y) so that the diffraction vector of the gratings is parallel to the x axis, a moiré fringe pattern given by
is observed behind G2, where the coefficient an(λ) is determined by the gratings and the spatial coherence of X-rays, and φx(x,y;λ) is the inclination of the wavefront in the x direction caused by the refraction at the object placed in front of G1. φx(x,y;λ) is related to the phase shift Φ(x,y;λ) caused by the object as
In the case of polychromatic illumination, the observed moiré pattern is written as
where P(λ) is the normalized spectrum function of the X-rays behind G2. Strictly speaking, the spectral efficiency of the image detector located behind G2 should be considered in P(λ). Here, we assume that Ĩ(x,y) can be expressed with
where φ̃x(x,y) is considered to be an average differential phase image, which is measured experimentally.
In order to measure the differential phase image, the fringe scanning method (or phase stepping technique) is normally adopted [9]. This method acquires several images by displacing one of the gratings against the other step by step. For high-speed imaging, however, this approach is obviously inadequate. Therefore, we applied the Fourier transform method [12] to X-ray Talbot interferometry to produce a differential phase image from a single moiré pattern.
Thus, the frame rate of differential phase images can be the same as that of moiré patterns. In order to apply the Fourier transform method, carrier fringes are introduced in the image by inclining G2 against G1 regarding the optical axis with an angle θ(≪1). Then, rotation moiré fringes are generated; such an image is given by
where
and fy≈θ/d is the spatial frequency of the moiré (carrier) fringes in the y direction.
The Fourier transform of eq. (5) with respect to y yields
where the subscript F indicates the Fourier transform of each term in eq. (5). When the spatial variation of φx(x,y) in the y direction is sufficiently slow, the terms on the right hand side of eq. (7) are separated on the f axis. Here, let us extract the term of n=1 and translate it by -fy toward the origin of the f axis. Subsequently, we obtain β 1F(x,f,z), the inverse Fourier transform of which yields β 1(x,y,z). Without the sample, we can obtain β 0 1 (x,y,z) in the same procedure. φ̃x(x,y) can be calculated with
The spatial resolution in the y direction is limited by double the carrier fringe spacing, according to the Fourier transform method. Finer carrier fringes are therefore preferable, and can be generated by increasing θ, although the generation of too fine moiré fringes may reduce fringe visibility. Because the differential phase shift is measured in the x direction, the spatial resolution limit does not affect the sensitivity to the X-ray refraction directly.
It is possible to generate carrier fringes by using gratings with slightly different pitches d 1 and d 2. In this case (θ=0), moiré fringes with a period of d 1 d 2/|d 1-d 2| are generated in the x direction. Therefore, this type of carrier fringes is not recommended because the differential phase contrast becomes blurry after the Fourier filtering.
For phase tomography, a moiré pattern video is recorded by rotating the object. Each frame is processed by the Fourier transform method, resulting a video of φ̃x(x,y). The reconstruction procedure for phase tomography from φ̃x(x,y) is the same as that described previously [11].
3. Experiment
Our X-ray Talbot interferometer imaging system consisted of a phase grating (G1) and an amplitude grating (G2), which were fabricated by X-ray lithography and gold electroplating on Si wafers 200 µm in thickness [13]. The pitch of G1 and G2 was 5.3 µm. The height of G1 was about 3.5 µm, which generates π/2 phase shift for about 36 keV X-rays. The height of G2 was about 30 µm.
We used a CMOS camera (pco.1200hs) combined with a Gd2O2S:Tb (P43) phosphor screen 10 µm in thickness through a reflecting mirror and coupling lenses. The CMOS camera had a 1280×1024 pixel array and the effective pixel size was 3.0 µm. The camera was operated with a frame rate of 500 f/s, and the exposure time per an image was set to 1 ms (1 ms blank between frames).
The X-ray Talbot interferometer was installed at BL28B2 of SPring-8, Japan, where white synchrotron radiation was available from a bending section of the storage ring. Except for beamline windows (Be and Kapton), no attenuators were used. The gratings were set so that their diffraction vector was in the vertical plane because the synchrotron radiation source size (or electron bunch size in the storage ring) is much smaller in the vertical direction than the horizontal direction, resulting in higher spatial coherence in the vertical direction. The X-ray image detector was placed about 10 cm behind G2, and a sample was placed about 10 cm in front of G1.
First, the visibility of the rotation moiré fringes was measured as a function of G1–G2 distance, z 0, as shown in Fig. 2. The visibility exceeded 20% in the wide range of G1–G2 distance. The distance that gave the peak in Fig. 2 is converted to the optimal wavelength 0.04 nm through d 2/(2λ)≡z 0 mentioned above. The refractive index is larger for X-rays of longer wavelength, and stronger phase contrast is generated. Therefore, we selected z 0=283 mm, which gives the longer optimal wavelength within the region exceeding 20% visibility, for the following experiments.
Fig. 3. Image processing for obtaining a differential phase image from a raw image with carrier fringes. The original moiré image (a), whose close up is shown in the inset, is Fourier-transformed as (b). Only the first order is extracted as in (c), and the calculation of the inverse Fourier-transform of (c) yields a differential phase image (d). Note that the x axis is parallel to the vertical direction.
Fig. 4. Demonstration with a polypropylene sphere. (a)–(d): Differential phase images of a moving polypropylene sphere across the field of view with the speeds of (a) 85 mm/s (Media 1), (b) 170 mm/s (Media 2), (c) 340 mm/s (Media 3), and (d) 850 mm/s (Media 4), respectively. (e) Phase tomogram of the sphere acquired with a 0.5 s exposure (Media 5). Media 1–4 are given with ×3/100 slow-motion videos.
A typical moiré pattern is shown in Fig. 3(a) with a polypropylene sphere 2.4 mm in diameter. The period of the rotation moiré fringes was 50 µm (θ≈0.11 rad). In order to obtain a differential phase image of this sphere, Fig. 3(a) was Fourier-transformed, resulting in Fig. 3(b). Only the first order was extracted and translated to the origin, as shown in Fig. 3(c). Through the calculation of the argument of the inverse transform of Fig. 3(c), a differential phase image Fig. 3(d) was obtained.
A demonstration experiment of high-speed X-ray phase imaging/tomography was performed using the polypropylene sphere, as shown in Fig. 4. Figures 4(a)–(d) show the sphere moving across the field of view with the speeds of 85, 170, 340, and 850 mm/s, respectively (Media 1–4). The sphere had a void structure inside, which could be revealed in the images except for Fig. 4(d). Thus, high-speed X-ray phase imaging was successful with 1 ms exposure. This result implies that the scan of X-ray phase tomography can be shortened considerably. We rotate the polypropylene sphere on its own axis with a speed of 1 rps, and 250 frames, which corresponds to 180° rotation, were recorded with 0.5 s. Figure 4(e) (Media 5) is a rendering view of the reconstructed phase tomogram, where the voids in the sphere are depicted.
4. Discussion
The time resolution of the imaging is currently determined by the X-ray image detector. In the experiment, we used a normal phosphor screen (P43), whose decay time was about 1 ms. In the presented experiment, the camera frame rate was set to match with the decay time. In order to improve the time resolution, we will use a screen of a shorter decay time, such as P46.
One attractive future of the presented technique is time-resolved X-ray phase tomography. We believe that our scan time (0.5 s) of phase tomography has reached the level where we can have time-resolved phase tomography in prospect. The experimental procedure for time-resolved phase tomography is the same as that described in this paper, except that a video is recorded during several turns of the sample within the capacitance of data storage. When a phase tomogram is reconstructed using the data at angular positions from Θ0 to Θ0+π, the next phase tomogram is reconstructed from Θ0+α to Θ0+π+α, where α is for example 10°. Thus, in the reconstruction of a series of tomograms, the projection image data are used repeatedly. We have started preliminary study of time-resolved X-ray phase tomography using live worm and polymers suffering radiation damage by the irradiation by white synchrotron radiation, the results of which will be reported in the near future.
5. Conclusion
White synchrotron radiation was available with the X-ray Talbot interferometer, and high-speed X-ray phase imaging/tomography was successful. This result opens up real-time observation with highly-sensitive X-ray phase imaging and quick scan X-ray phase tomography. This achievement advances X-ray phase imaging/tomography done for static cases so far to an imaging method for dynamic observation of weakly absorbing object.
Acknowledgments
We are grateful for the technical support for the experiment at SPring-8 by Drs. K. Kajiwara, K. Umetani, and Y. Imai. The experiment at SPring-8 was performed under the approval of SPring-8 committee 2007B1240 and 2008A1317. This study was financially supported by Japan Science and Technology Agency (JST).
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