Open Access
Add to BibSonomy
Abstract
Coherent diffraction imaging (CDI) is a powerful method for visualizing the structure of an object with a high spatial resolution that exceeds the performance limits of the lens. Single-frame CDI in the X-ray region has potential use for probing dynamic phenomena with a high spatiotemporal resolution. Here, we experimentally demonstrate a general method for single-frame X-ray CDI using a triangular aperture and a Fresnel zone plate. Using 5 keV synchrotron radiation X-rays, we reconstructed the object image of the locally illuminated area with a spatial resolution of higher than 50 nm and an exposure time of more than 0.1 s without prior information about the sample. After a 10 s exposure, a resolution of 17 nm was achieved. The present method opens new frontiers in the study of dynamics at the nanoscale by using next-generation synchrotron radiation X-rays/free-electron lasers as light sources.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Coherent diffraction imaging (CDI) is a powerful method for visualizing the structure of an object with a high spatial resolution that exceeds the performance limits of the lens [1], in which the object is irradiated with a coherent beam, and phase retrieval calculations are performed on the measured diffraction intensity pattern to obtain the object image. CDI is particularly useful in the extreme ultraviolet to X-ray region, in which it is difficult to fabricate lenses with high numerical apertures. Thus far, CDI has been applied to the observation of noncrystalline samples in physics, chemistry, materials science, nanoscience, geology, and biology, that are not accessible by traditional X-ray crystallography, electron, and probe microscopes [2]. There are several types of CDI according to their optical systems and reconstruction methods. Plane-wave CDI, in which a coherent planar beam is incident on the sample, can be used to observe isolated objects. The single-shot CDI schemes have been demonstrated using X-ray free electron lasers [3] and high-order harmonic lasers [4,5]. Scanning CDI, commonly referred to as ptychographic CDI [6], is superior in that it enables the observation of extended samples. Its disadvantage is that it is difficult to improve its temporal resolution since it is based on multiple-frame data collection. It is therefore desirable to establish a method of reconstructing the image of an extended object from a single-frame diffraction intensity pattern.
Some approaches to single-frame CDI have been proposed thus far, in which images are reconstructed by phase retrieval calculation with the real-space constraint that restricts the imaged object to a confined region called the “support”, that is, a support constraint. To apply the support constraint on the extended object, it is necessary to illuminate the sample with a top-hat beam. In keyhole CDI [7], the X-ray beam is focused and the object is placed downstream of the focus so that it is illuminated by a divergent wave, allowing us to illuminate small isolated areas within a large object. Also, the apodizing slits have been used to reduce the side-lobe intensity of the focused beam [8]. However, it is technically difficult to produce an ideal top-hat beam, and hence phase retrieval calculations remain ambiguous. In order to remove the inherent ambiguities, the use of randomized illumination, which is also referred to as a phase modulator, has been proposed, which allows us to reconstruct the image of an extended object with a relaxed support constraint [9].
Recently, we have proposed a practical method for single-frame CDI [10], in which a triangular aperture is used as a key element for the optical system. The image of a selected field of view of an extended object is reconstructed from the single-frame diffraction intensity pattern by phase retrieval calculation, and its spatial resolution is limited by both the sharpness of triangular aperture edges and the quality of the diffracted intensity pattern in the high-Q region. This method can be considered a fusion of in-line holography and CDI, and its concept is also analogous to holography with extended reference by autocorrelation linear differential operation [11]. In this study, we experimentally demonstrate single-frame CDI in the hard X-ray region using a high-precision triangular aperture fabricated by focused ion beam processing. Finally, the possibility of spatiotemporal resolution for next-generation synchrotron radiation X-rays/free-electron laser applications is discussed.
2. Fabrication and evaluation of triangular aperture
The high-precision triangular aperture was fabricated as follows. A platinum (Pt) foil was polished on both sides to a thickness of 20 $\mu$m (TDC Co., Ltd., Miyagi, Japan). The $R_a$ and $R_z$ values of the surface were about 1.0 nm and 6.0 nm, respectively. The Pt foil was fixed with an aluminum jig and processed by focused ion beam processing (Daiwa Techno Systems Co., Ltd., Ibaraki, Japan). Figure 1(a) shows a scanning ion microscopy (SIM) image of the aperture fabricated. One side of the triangle is approximately 10 $\mu$m and the corners have a radius of curvature of 700 nm. We know from computer simulations that this level of curvature does not adversely affect the image reconstruction [10].
Fig. 1. (a) Scanning ion microscopy (SIM) image of the triangular aperture fabricated by FIB processing on Pt foil. (b) Coherent X-ray diffraction pattern of the triangular aperture. (c) Cross section and line profiles of diffraction intensity patterns of the triangular aperture. Red line shows the cross section along the diagonal dotted line in (b). Black, green, gray, and blue lines show the numerically simulated line profiles with 0 nm, 10 nm, 13.5 nm, and 20 nm smoothed edges of triangular apertures, respectively.
Aperture steepness was evaluated and coherent diffraction intensity patterns from the samples were measured at BL29XUL [12] in SPring-8. Incident X-rays were monochromatized at an energy of 5 keV using a Si 111 double-crystal monochromator and Pt-coated mirrors. The monochromatized X-rays were irradiated into the aperture, and the forward diffraction intensity pattern was measured using a pixelated detector (EIGER 1M) placed 3.43 m downstream of the aperture. Figure 1(b) shows the coherent X-ray diffraction pattern of the triangular aperture. The streaks of the diffraction intensity pattern extend to 85.2 $\mu$m$^{-1}$ at spatial frequency. Figure 1(c) shows the cross section along the diagonal dotted line in Fig. 1(b) and the numerically simulated line profiles with 0 nm, 10 nm, 13.5 nm, and 20 nm smoothed edges of triangular apertures. The intensity profile of the experiment is in partial agreement with the simulated profile of the 13.5 nm smoothed edge.
3. CDI experiments
Figure 2 shows a schematic diagram of the single-frame CDI optical system. The $y$ direction is parallel to the optical axis. According to our previous simulation, the angular tolerance for the triangular aperture is less than 1 degree, which was fully satisfied in this experiment. Here, the distance between sample and detector (camera length) was limited by the length of the experimental hutch, and the triangular probe was reduced with a Fresnel zone plate (FZP) to ensure a sufficient oversampling ratio, although it is the most efficient way to place the triangular aperture just in front of the sample for using the incident photons. The FZP with a Ni thickness of 900 nm, a minimum zone width of 50 nm, and an outer diameter of 180 $\mu$m was placed 108 mm downstream of the aperture and 54 mm upstream of the sample. According to our previous simulation, the resolution of the lens does not compromise the achievable resolution in the single-frame CDI method. Although the probe sharpness is worse than the Rayleigh resolution of the zone plate $\sim$60 nm, the present single-frame CDI can achieve resolutions beyond 60 nm.
The aperture was scaled down to a factor of two on the sample surface. The FZP was located 50 $\mu$m off-axis laterally to the aperture beam so that the triangular illumination was incident outside the optics to separate the first-order focus from the direct beam. An order-sorting aperture (OSA) with a tungsten foil of 25 $\mu$m thickness and a pinhole of 10 $\mu$m diameter was placed about 1 mm upstream from the sample to block X-rays except for the first-order diffracted X-rays. A 200-nm-thick tantalum (Ta) X-ray test chart with a minimum structure of 50 nm was used as the sample. X-rays at an energy of 5 keV illuminated the sample, where the incident photon flux at the sample position was $\sim$ 1$\times 10^{7}$ photons/s. The diffraction intensity patterns were recorded using an in-vacuum pixelated detector (EIGER 1M, Dectris) with a pixel size of 75 $\mu$m that was placed 3.27 m downstream of the sample.
Fig. 2. Schematic diagram of single-frame CDI optical system with a triangular aperture and a Fresnel zone plate.
4. Ptychographic image reconstruction
Before the single-frame CDI measurements, we performed X-ray ptychography measurement to obtain probe functions. The sample was illuminated in 15$\times$15 overlapping fields of view, separated by 500 nm in the horizontal and vertical directions. The exposure time at each position was 10 s. The sample and probe images were reconstructed by the mixed-state reconstruction method [13]. Figure 3(a) shows the reconstructed phase image of the Ta test chart. Numeric characters and radial structures, including minimum structures, are clearly visualized. Figure 3(b) shows the intensity distributions of reconstructed probe functions. Each probe function is divided into four modes that are orthogonal to each other. All probes exhibit a half-sized triangular aperture imaged using the FZP, and the first mode populates 90.9% of all photons. Higher-order probe functions result from imperfections in the spatial coherence of the X-rays illuminating the aperture and/or from parasitic scattering from the inner walls of the aperture.
Fig. 3. (a) Reconstructed phase image of the sample with a phase distribution of -0.27 to 0.27 rad. (b) Four probe modes reconstructed. The percentage values indicate the population of each mode.
5. Image reconstruction in single-frame CDI
The single-frame CDI measurements were performed at three scanning points in the ptychographic scan and 28 exposure times ranging from 10 ms to 10 s to evaluate the spatiotemporal resolution. In the image reconstruction of single-frame CDI, a single-object function $T_{\textbf {r}}$ is reconstructed from a single diffraction pattern $I_{\textbf {q}}$ using the probe functions $P^{(k)}_{\textbf {r}}$ with mode $k=$1, 2, 3, and 4 shown in Fig. 3(b). Here, we employ a phase retrieval algorithm based on the reciprocal-space constraint expressed as
Figure 4(a) shows three typical reconstructed images of a Ta test chart obtained by single-frame CDI using a 10 s exposure diffraction pattern. The upper images in Fig. 4(a) have the same scale as those in Figs. 3(a) and 3(b), and the bottom images in Fig. 4(a) show threefold enlargement of the area enclosed by the dashed line in the upper images in Fig. 4(a). The numeric and radial structures in the illuminated area are distinctly visualized, and the 50 nm pitched minimum structures are vividly resolved. The contrasts are similar to those in the corresponding area of the reconstructed image of ptychography in Fig. 3(a). The expected phase shift is 0.54 rad at 5 keV X-ray beam, which is calculated from a 200-nm-thick Ta plate. There are differences between the average bright area and the average dark area of the phase image in Fig. 4(a) : 0.45 rad for the left, 0.47 rad for the center, and 0.48 rad for the right, which are similar to the expected phase shift of the test chart. The amounts of phase shifts in Fig. 3(a) and Fig. 4(a) showed good agreement in the range of $\pm$ 0.04 rad, indicating that the present single-frame CDI is highly quantitative. We also evaluated the oversampling ratio ($\sigma$), which is defined as
Fig. 4. (a) Reconstructed phase images obtained by single-frame CDI with a phase distribution of -0.27 to 0.27 rad. The bottom images are magnified threefold. (b) Line profiles along green, blue, and red lines in the left bottom reconstructed image of (a).
6. Evaluation of spatiotemporal resolution
Next, we evaluated the spatiotemporal resolution of the object image, where the FWHM was calculated for each exposure time along the lines in Fig. 4(a). Figure 5(a) shows the dependence of the spatial resolution on the exposure time in the range from 10 ms to 10 s. The point shows the average FWHM and the error bars correspond to the difference between the maximum and minimum FWHMs for each exposure time. The spatial resolution is lowest at an exposure time of 10 s and increases with decreasing exposure time. When the exposure time is less than 1 s, the resolution is less than 50 nm, and even at 10 ms, the resolution remains higher than 100 nm. Figure 5(b) shows the reconstructed phase images for exposure times of 10 ms, 100 ms, 1 s, and 10 s. The object image for an exposure time of 1 s shows a similar contrast to that for 10 s and remains vivid at 100 ms. Even for 10 ms, the numerical and radial structures can be recognized. Thus far, a sub-50 nm resolution has been achieved at an exposure time of 10 ms using a transmission X-ray microscopy (TXM) system equipped with an FZP [14]. In the present single-frame CDI, the resolution is limited by the signal-to-noise ratio of the diffraction intensity in the high-Q region. Further improvement in resolution can be expected by increasing the fluence of coherent X-rays irradiated onto the sample. In addition, the present single-frame CDI is characterized by a higher quantitative phase image quality than Zernike phase contrast imaging with TXM [15].
Fig. 5. Evaluation of the spatiotemporal resolution of reconstructed images in single-frame CDI. (a) Dependence of the spatial resolution on the exposure time in the range from 10 ms to 10 s. The point is the average FWHM of three line profiles, and the length of the error bars corresponds to the difference between the maximum and minimum FWHMs for each exposure time. The dashed line indicates 50 nm as the minimum zone width of the FZP. (b) The reconstructed phase images obtained by single-frame CDI for exposure times of 10 ms, 100 ms, 1 s, and 10 s. The bottom images are magnified threefold.
7. Conclusion
In conclusion, we have experimentally demonstrated the single-frame CDI of an extended object in the hard X-ray region using a high-precision triangular aperture fabricated by FIB processing and using an FZP. We successfully reconstructed object images with resolutions on the order of ten nanometers from single-frame diffracted intensity patterns of 10 s exposures, and resolutions higher than 50 nm, which is the minimum zone width of the FZP, were maintained for exposures less than 1 s. Thus far, X-ray photon correlation spectroscopy (XPCS) [16], which measures the temporal changes in speckle patterns produced when coherent light is scattered by a disordered system, has so far been applied to relatively slow glassy relaxation processes, with characteristic time scales ranging from tens of milliseconds to thousands of seconds. The present spatiotemporal resolution of single-frame CDI will cover a portion of the spatiotemporal scale of XPCS so far. For example, single-frame CDI with the present spatiotemporal resolution will allows for the slow dynamics research of soft materials such as colloids [17] and polymers [18] in a real-space movie. Although, currently, the spatiotemporal resolution is limited by the coherent flux of incident X-rays, the use of diffraction-limited storage rings or free-electron lasers in combination with advanced imaging mirror optics [19] can markedly improve the spatiotemporal resolution. In addition, reconstruction algorithms which exploit redundancy in successive images in order to improve convergence would be useful [20–22]. In the near future, single-frame CDI will allow us to visualize dynamics on a time scale of 1-10 ms with a spatial resolution of 10 nm. We believe that this method will open the frontier of research on dynamics in biological and materials sciences.
Funding
Japan Society for the Promotion of Science (JP18H05253, JP19H05814, JP20K15375, JP20K20523); Dynamic Alliance for Open Innovation Bridging Human, Environment and Materials from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
Acknowledgments
We thank Professor Tetsuya Ishikawa for stimulating discussions.
Disclosures
The authors declare no conflicts of interest.
References
1. H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nat. Photonics 4(12), 833–839 (2010). [CrossRef]
2. J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348(6234), 530–535 (2015). [CrossRef]
3. A. Pryor, A. Rana, R. Xu, J. Rodriguez, Y. Yang, M. Gallagher-Jones, H. Jiang, K. Kanhaiya, M. Nathanson, J. Park, S. Kim, S. Kim, D. Nam, Y. Yue, J. Fan, Z. Sun, B. Zhang, D. Gardner, C. Dias, Y. Joti, T. Hatsui, T. Kameshima, Y. Inubushi, K. Tono, J. Lee, M. Yabashi, C. Song, T. Ishikawa, H. Kapteyn, M. Murnane, H. Heinz, and J. Miao, “Single-shot 3d coherent diffractive imaging of core-shell nanoparticles with elemental specificity,” Sci. Rep. 8(1), 8284 (2018). [CrossRef]
4. E. Malm, H. Wikmark, B. Pfau, P. Villanueva-Perez, P. Rudawski, J. Peschel, S. Maclot, M. Schneider, S. Eisebitt, A. Mikkelsen, A. L’Huillier, and P. Johnsson, “Singleshot polychromatic coherent diffractive imaging with a high-order harmonic source,” Opt. Express 28(1), 394–404 (2020). [CrossRef]
5. A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J.-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J.-F. Hergott, A.-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-Shot Diffractive Imaging with a Table-Top Femtosecond Soft X-Ray Laser-Harmonics Source,” Phys. Rev. Lett. 103(2), 028104 (2009). [CrossRef]
6. F. Pfeiffer, “X-ray ptychography,” Nat. Photonics 12(1), 9–17 (2018). [CrossRef]
7. B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4(5), 394–398 (2008). [CrossRef]
8. K. P. Khakurel, T. Kimura, H. Nakamori, T. Goto, S. Matsuyama, T. Sasaki, M. Takei, Y. Kohmura, T. Ishikawa, K. Yamauchi, and Y. Nishino, “Generation of apodized X-ray illumination and its application to scanning and diffraction microscopy,” J. Synchrotron Radiat. 24(1), 142–149 (2017). [CrossRef]
9. F. Zhang, B. Chen, G. R. Morrison, J. Vila-Comamala, M. Guizar-Sicairos, and I. K. Robinson, “Phase retrieval by coherent modulation imaging,” Nat. Commun. 7(1), 13367 (2016). [CrossRef]
10. J. Kang, S. Takazawa, N. Ishiguro, and Y. Takahashi, “Single-frame coherent diffraction imaging of extended objects using triangular aperture,” Opt. Express 29(2), 1441–1453 (2021). [CrossRef]
11. M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express 15(26), 17592–17612 (2007). [CrossRef]
12. K. Tamasaku, Y. Tanaka, M. Yabashi, H. Yamazaki, N. Kawamura, M. Suzuki, and T. Ishikawa, “SPring-8 RIKEN beamline III for coherent X-ray optics,” Nucl. Instrum. Methods Phys. Res., Sect. A 467-468, 686–689 (2001). [CrossRef]
13. P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494(7435), 68–71 (2013). [CrossRef]
14. M. Ge, D. S. Coburn, E. Nazaretski, W. Xu, K. Gofron, H. Xu, Z. Yin, and W.-K. Lee, “One-minute nano-tomography using hard X-ray full-field transmission microscope,” Appl. Phys. Lett. 113(8), 083109 (2018). [CrossRef]
15. S. Flenner, M. Storm, A. Kubec, E. Longo, F. Döring, D. M. Pelt, C. David, M. Müller, and I. Greving, “Pushing the temporal resolution in absorption and Zernike phase contrast nanotomography: enabling fast in situ experiments,” J. Synchrotron Radiat. 27(5), 1339–1346 (2020). [CrossRef]
16. O. G. Shpyrko, “X-ray photon correlation spectroscopy,” J. Synchrotron Radiat. 21(5), 1057–1064 (2014). [CrossRef]
17. S. B. Dierker, R. M. Pindak, R. adn Fleming, I. K. Robinson, and L. Berman, “X-Ray Photon Correlation Spectroscopy Study of Brownian Motion of Gold Colloids in Glycerol,” Phys. Rev. Lett. 75(3), 449–452 (1995). [CrossRef]
18. P. Falus, M. A. Borthwick, and S. G. J. Mochrie, “Fluctuation Dynamics of Block Copolymer Vesicles,” Phys. Rev. Lett. 94(1), 016105 (2005). [CrossRef]
19. S. Matsuyama, S. Yasuda, J. Yamada, H. Okada, Y. Kohmura, M. Yabashi, T. Ishikawa, and K. Yamauchi, “50-nm-resolution full-field X-ray microscope without chromatic aberration using total-reflection imaging mirrors,” Sci. Rep. 7(1), 46358 (2017). [CrossRef]
20. Y. H. Lo, L. Zhao, M. Gallagher-Jones, A. Rana, J. J. Lodico, W. Xiao, B. C. Regan, and J. Miao, “In situ coherent diffractive imaging,” Nat. Commun. 9(1), 1826 (2018). [CrossRef]
21. X. Tao, Z. Xu, H. Liu, C. Wang, Z. Xing, Y. Wang, and R. Tai, “Spatially correlated coherent diffractive imaging method,” Appl. Opt. 57(22), 6527–6533 (2018). [CrossRef]
22. G. N. Hinsley, C. M. Kewish, and G. A. van Riessen, “Dynamic coherent diffractive imaging using unsupervised identification of spatiotemporal constraints,” Opt. Express 28(24), 36862–36872 (2020). [CrossRef]
