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The dynamic range of X-ray detectors is a key factor limiting both the spatial resolution and sensitivity of X-ray ptychography as well as the coherent flux of incident X-rays. Here, we propose a method for high-resolution and high-sensitivity X-ray ptychography named “dark-field X-ray ptychography”, which compresses the dynamic range of intensities of diffraction patterns. In this method, a small reference object is aligned upstream of the sample. The scattered X-rays from the object work as a reference beam for in-line holography. Ptychographic diffraction patterns including the in-line hologram are collected, and then the image of the sample is reconstructed by an iterative phasing method. This method allows us to obscure the low-Q region of the diffraction patterns using a beamstop since the in-line hologram complements structural information in the low-Q region, resulting in the compression of the dynamic range of intensities of diffraction patterns. A numerical study shows that the dynamic range of intensities of diffraction patterns is decreased by about three orders of magnitude.
© 2015 Optical Society of America
1. Introduction
Coherent X-ray diffraction imaging (CXDI) [1] is a lensless imaging technique based on coherent diffraction and phase retrieval calculation, which can achieve a high spatial resolution beyond that of conventional X-ray microscopes with lenses. CXDI enables highly sensitive observation using phase contrast, allowing us to observe weak-phase objects such as soft biological tissues. The original concept of CXDI is based on a plane-wave geometry [2], in which the sample is illuminated with an X-ray plane wave. Plane-wave CXDI has a significant limitation, that is, the sample must be an isolated object. Scanning CXDI, which is called X-ray ptychography [3], was a breakthrough that overcame this limitation. In X-ray ptychography, a probe is scanned across the sample and the diffraction pattern is observed at each beam position. The complex functions of both the sample and the probe are reconstructed by iterative phasing methods, e.g., an extended ptychographical iterative engine (ePIE) [4]. Since the initial demonstration of X-ray ptychography, it has been used to observe biological samples in two [5,6] and three dimensions [7]. The high sensitivity of X-ray ptychography has also enabled us to visualize the elemental distribution in materials at the nanoscale [8–10 ]. X-ray ptychography is now a promising tool for high-resolution and high-sensitivity X-ray imaging of various specimens in biology and materials science.
The high performance of X-ray ptychography relies on state-of-the-art technologies concerning X-ray source, optics, and detectors. In particular, the spatial resolution depends on the signal-to-noise (S/N) ratio of high-Q diffraction patterns. Therefore, highly intense incident coherent X-rays and highly efficient two-dimensional detectors are indispensable for improving the resolution. Advanced focusing optics such as refractive lenses [11], mirror optics [12], and Fresnel zone planes [13] have produced highly intense coherent X-ray beams from a low-emittance source at third-generation synchrotron facilities and have contributed to improving the resolution of X-ray ptychography down to 10 nm. The recent advances in X-ray detector techniques are also remarkable. The use of photon-counting pixel detectors such as Lambda [14], MM-PAD [15], and EIGER [16] has markedly improved the throughput of experiments.
For more advanced X-ray ptychography, a key factor is the “dynamic range” of the diffraction patterns. The dynamic range becomes wide with the improvement of the resolution since the diffraction intensities rapidly decay in the high-Q region. Recently, Wilke et al. [17] have reported the use of a semitransparent central stop for high-resolution X-ray ptychography. A semitransparent central stop in front of the detector was introduced, and then the intensities of transmitted X-rays were numerically estimated from the amount of X-ray absorption. Although this approach is very useful for strong-phase objects, the sensitivity of X-ray ptychography is limited when the signal in transmitted X-rays is buried within the photon-counting noises, which means that the dynamic range is correlated with the sensitivity as well as the resolution. Maiden et al. [18] have used a random hole array to reduce the dynamic range without sensitivity degradation. By randomizing the phase structure of the probe, the dynamic range of the diffraction patterns was reduced by an order of magnitude. In this approach, since the intensities of transmitted X-rays have to be measured, it is difficult to reduce the dynamic range by more than an order of magnitude. Thus, a novel approach for markedly reducing the dynamic range is required towards higher-resolution and higher-sensitivity X-ray ptychography.
In this paper, we propose a novel ptychographic approach named “dark-field X-ray ptychography”. This approach uses both a central stop and a small reference object, which can reduce the dynamic range of the diffraction patterns by three orders of magnitude without degradation of the sensitivity. To assess the effectiveness of the present approach, we perform a simulation-based study and then discuss its feasibility.
2. Principle of dark-field X-ray ptychography
Figure 1(a) shows a schematic view of focused X-ray ptychography in the forward geometry. The y direction is parallel to the optical axis. Incident X-rays are focused at the diffraction-limited focal spot size using focusing optics such as total-reflection mirrors. The spot size w along the z direction for a probe with wavelength λ is generally expressed as
for focusing optics with a rectangular aperture, where NA is the numerical aperture. A sample is placed on the focal plane and a two-dimensional detector is placed at the far field. In the detector plane, the one-dimensional size along the z direction of the bright-field image (Dbright) is where fb is the back focal length, a is the aperture size of the focusing optics, and L is the distance between the sample and the detector. The diffraction at the bright field predominantly includes structural information at a spatial scale larger than w. If the sample is a weak-scattering object, the diffraction intensities are several orders larger at the bright field than at the dark field. Therefore, X-ray detectors with a large dynamic range are required for high-resolution and high-sensitivity X-ray ptychography.
Fig. 1 (a–b) Schematic view of X-ray ptychography. (a) Focused X-ray ptychography. (b) Dark-field X-ray ptychography.
Figure 1(b) shows a schematic view of dark-field X-ray ptychography. In this approach, a cylindrical object with diameter d less than w is positioned upstream of the sample, which acts as a reference source for in-line holography. The position of the cylindrical object along the optical axis is adjusted so that the scattered X-rays from the cylindrical object (i.e., reference beam) illuminate an area on the sample larger than w. The in-line hologram patterns of the sample are formed at the detector plane in addition to the diffraction patterns. Here, the one-dimensional size along the z direction of the in-line hologram pattern is given by
Thus, this method allows us to obscure the diffraction patterns in the bright field using a beamstop. When B < Dhologram, where B is the one-dimensional size along the z direction of the beamstop, the in-line hologram can complement the structural information in the low-Q region. In particluar, when Dbright < B, a significant decrease in the dynamic range of the diffraction patterns is expected since the central spot with the highest intensity is completely blocked by the beamstop.3. Simulations
3.1. Conditions
The performance of dark-field X-ray ptychography was evaluated by numerical simulation. The optical parameters were selected to simulate our recent experiments at SPring-8. The X-ray energy was 6.5 keV. A focusing optics of NA 0.016 with a rectangular aperture was installed, producing a diffraction-limited focal spot size of 500 nm at 6.5 keV. The diameter, height, and material of the cylindrical object used as the reference source were assumed to be 100 nm, 500 nm, and gold, respectively. As the sample, we used a phase object of the well-known “Lena” image with 20-bit gray scale ranging from −0.01 to 0 rad. The detector had 656×656 pixels with a size of 40 μm across and was located 1374 mm downstream from the sample. The bright field occupied the central 11×11 pixels. Four kinds of beamstop were prepared to obscure the central 60×60 pixels, 180×180 pixels, 300×300 pixels, and 420×420 pixels. The diffraction patterns were calculated for the three arrangements shown in Table 1. Photon-counting noises were induced in the diffraction intensity, and the dynamic range of the diffraction pattern was fixed at (1 photon/pixel : 106 photons/pixel).
Table 1. Three arrangements for the calculation of diffraction patterns.
3.2. Illumination wave fields and diffraction patterns
Figure 2(a) shows the image consisting of 1056×1056 pixels. The sample was positioned at the focal plane. The reference source was located 500 μm upstream from the phase object. Figures 2(b) and 2(c) show the calculated intensity distributions of wave fields at the focal plane without and with the reference source, respectively. The intensity distribution represents Fraunhofer diffraction from a rectangular aperture, which is a general feature of a diffraction-limited focused beam with a rectangular aperture. The reference source generates a pattern of concentric circles, as shown in Fig. 2(c).
Fig. 2 (a) Original sample image (20-bit “Lena” image). The maximum value of the phase shift is 0.01 rad. (b–c) X-ray wave fields at the sample plane (b) without the reference source and (c) with the reference source. The scale bar is 1 μm.
Figures 3(a)–3(c) show the diffraction patterns calculated at Arrangements 1, 2, and 3, respectively. The X-ray irradiation position is indicated by a yellow dot in Fig. 2(a). In Fig. 3(a), the diffraction intensities are buried in the photon-counting noises except in the central region since the diffraction intensities are five orders larger at the bright field than at the dark field. In Figs. 3(b) and 3(c), the beamstop of 60 60 pixels blocked the diffraction patterns in the region −4.57×10−3 nm−1 ≤qx,z≤4.57×10−3 nm−1, which resulted in the production of high-Q diffraction patterns with a high S/N ratio. In Fig. 3(c), the circular pattern due to the reference object appears to be predominant. Figure 3(d) shows the cross sections of the diffraction patterns with and without the sample in Arrangement 3 and their difference. The difference includes the significant signal originating from the sample.
Fig. 3 (a–c) Diffraction patterns with photon-counting noises calculated for Arrangements 1, 2, and 3. (a) Arrangement 1, (b) Arrangement 2, and (c) Arrangement 3. The incident beam illuminates the yellow point in Fig. 2. (d) Cross sections of the diffraction patterns with and without the sample in Arrangement 3 and their difference. The cross section is through P in (c).
3.3. Reconstructed images
Images were reconstructed from the ptychographic diffraction patterns using a ptychographic iterative engine [19]. The wave fields of Figs. 2(b) and 2(c) were used as the probe functions for the reconstruction without and with the reference source. A constant modulus and a phase map were used as an initial estiamte for the object. The wavefields of Fig. 2(a) were used for the illumination fucntions for Arrangements 1 and 2. On the other hand, the reconstruction in Arrangement 3 used the wavefield of Fig. 2(b) as the illumination function. The probe function was fixed in each iterative process. The iterative process for each reconstruction was continued for up to 5 × 103 iterations. Figures 4(a)–4(c) show the reconstructed images for Arrangements 1, 2, and 3, respectively. Figure 4(d) shows the original image for comparison. In Arrangement 1, the resolution is poor and many artifacts emerged owing to the poor S/N ratio of the high-Q diffraction intensities. In Arrangement 2, only the sharp-edge structures are reconstructed owing to the loss of low-Q diffraction. In Arrangement 3, a high-resolution and high-contrast image is reconstructed, which is in good agreement with the original image.
Fig. 4 (a–c) Reconstructed images for three arrangements. (a) Arrangement 1, (b) Arrangement 2, and (c) Arrangement 3. (d) Original sample image. The scale bar is 500 nm.
Next, we assessed the quality of the reconstructed images using the Fourier ring correlation (FRC) [20, 21]. The FRC is a cross-correlation coefficient between two independent images in the frequency space, which is also referred to as the Fourier shell correlation (FSC) in three dimensions. Recently, the FRC and FSC have been used to estimate the resolution in X-ray ptychography [13, 22]. Figure 5 shows FRC curves computed from each reconstructed image and the original sample image. In Arrangement 1, the correlation decays rapidly with increasing spatial frequency, and the FRC curve intersects the threshold of 0.5 at 35.7 nm. In Arrangement 2, the FRC curve is below the threshold in the low-Q region blocked by the beamstop. In Arrangement 3, the FRC curve was above 0.9 in the low-Q region. It is evident that the scattered X-rays from the cylindrical object work as a reference beam for in-line holography, and the inline hologram complements the structural information in the low-Q diffraction. The resolution is estimated to be 14.4 nm, which is comparable to that in Arrangement 2. A dynamic range of (1 photon/pixel : 109 photons/pixel) is required for the diffraction patterns to reconstruct the image with the same quality as that in Arrangement 3. The present simulation shows that the dynamic range of diffraction patterns is compressed by about three orders of magnitude.
Fig. 5 Fourier ring correlation (FRC) plot in the reconstructed images for Arrangements 1, 2, and 3. The blue dotted line shows the threshold value of 0.5.
Next, we evaluated the beamstop-size dependency of the reconstructed image in Arrangement 3. Figures 6(a)–6(c) show the reconstructed images for the beamstop of 180×180 pixels, 300×300 pixels, and 420×420 pixels, respectively. As increasing the size of the beamstop, the long-period artifacts emerge in the reconstructed images due to the loss of the low-Q information. Figure 6(d) shows the FRC curves computed from images of Fig. 4(c) and Figs. 6(a)–6(c). As increasing the size of the beamstop, the FRC value decreses in the low-Q region since the component of the in-line hologram decreases, while the FRC value in the high-Q region increases since the S/N ratio of the high-Q diffraciton intensities becomes high. Thus, in order to obtain the high FRC value in the low-Q and high-Q regions, the optimum sizes of both the beamstop and cylindrical object have to be determined by considering both the desired spatial resolution and the dynamic range of the detector.
Fig. 6 (a–c) Reconstructed images for Arrangements 3. Sizes of the beamstop are 180×180 pixels for (a), 300×300 pixels for (b), and 420×420 pixels for (c). The scale bar is 500 nm. (d) The beamstop-size dependency of FRC in the reconstructed images in Arrangements 3
4. Summary and Conclusions
We have proposed dark-field X-ray ptychography for high-resolution and high-sensitivity imaging. In this method, a small reference object is aligned upstream of the sample and the scattered X-rays from the object work as a reference beam for in-line holography. This method allows us to obscure the low-Q region of the diffraction patterns using a beamstop since the in-line hologram complements the loss, resulting in the compression of the dynamic range of intensities of diffraction patterns. The numerical study showed that the dynamic range of diffraction patterns was decreased by about three orders of magnitude. A weak-phase object ranging from −0.01 to 0 rad was reconstructed at ∼15 nm resolution from ptychographic diffraction patterns with a dynamic range of (1 photon/pixel : 106 photons/pixel). In the present simulation, in order to simply show the performance of dark-field X-ray ptychography, known probe functions were used for the image reconstruction. In experiments, the exact probe function has to beforehand be determined in the measurement of known objects, and then the probe function is updated in parallel with the reconstruction of the unknown objects using ePIE. Our supplemental simulation confirms that this procedure works well.
The present method can be applied to other forward geometries of CXDI such as plane-wave CXDI [2] and Fresnel CXDI [23]. In particular, for single-shot plane-wave CXDI using X-ray free-electron lasers [24, 25], the present method is useful since extremely high performance detectors are required for the measurement. An experimental difficulty in dark-field X-ray ptychography is producing a cylindrical object with high accuracy since the surface roughness causes the parasitic scattering of X-rays. It is necessary to produce the surface of a cylindrical object with accuracy better than the desired resolution. A modern e-beam lithography technology will enable us to fabricate the cylindrical object. In addition, vibrations and drifts of the cylindrical object have to be concerned in the experiment since the vibration reduces the contrast of hologram patterns and the drift limits the achievable spatial resolution. For precise measurements of both the in-line hologram and diffraction patterns in dark-field X-ray ptychography, a high-stability optical system under a constant temperature is required. We believe that dark-field X-ray ptychography allows us the observation of soft biological tissues and the spectroscopic imaging of materials at the nanoscale.
Acknowledgments
This work was supported by KAKENHI (Grant Nos. 25709057, 26600143, and 26106515) and JSPS Fellows (Grant No. 25·2959), the X-ray Free Electron Laser Priority Strategy Program of MEXT, and the SENTAN of JST.
References and links
1. H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nat. Photonics 4, 833–839 (2010). [CrossRef]
2. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (2010). [CrossRef]
3. J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007). [CrossRef] [PubMed]
4. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009). [CrossRef] [PubMed]
5. K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U. S. A. 107, 529–534 (2010). [CrossRef]
6. E. Lima, A. Diaz, M. Guizar-Sicairos, S. Gorelick, P. Pernot, T. Schleier, and A. Menzel, “Cryo-scanning x-ray diffraction microscopy of frozen-hydrated yeast,” J. Microsc. 249, 1–7 (2013). [CrossRef]
7. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010). [CrossRef] [PubMed]
8. Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, K. Yamauchi, and T. Ishikawa, “Multiscale element mapping of buried structures by ptychographic x-ray diffraction microscopy using anomalous scattering,” Appl. Phys. Lett. 99, 131905 (2011). [CrossRef]
9. M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S.-C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208202 (2011). [CrossRef]
10. D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. L. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nat. Photonics 8, 765–769 (2014). [CrossRef]
11. A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100, 253112 (2012). [CrossRef]
12. Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, Y. Senba, H. Ohashi, K. Yamauchi, and T. Ishikawa, “Towards high-resolution ptychographic x-ray diffraction microscopy,” Phys. Rev. B 83, 214109 (2011). [CrossRef]
13. J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive X-ray optics by ptychographic coherent diffractive imaging,” Opt. Express 19, 21333–21344 (2011). [CrossRef] [PubMed]
14. R. N. Wilke, J. Wallentin, M. Osterhoff, D. Pennicard, A. Zozulya, M. Sprung, and T. Salditt, “High-flux ptychographic imaging using the new 55 μm-pixel detector ‘Lambda’ based on the Medipix3 readout chip,” Acta Crystallogr. A 70, 552–562 (2014). [CrossRef]
15. K. Giewekemeyer, H. T. Philipp, R. N. Wilke, A. Aquila, M. Osterhoff, M. W. Tate, K. S. Shanks, A. V. Zozulya, T. Salditt, S. M. Gruner, and A. P. Mancuso, “High-dynamic-range coherent diffractive imaging: ptychography using the mixed-mode pixel array detector,” J. Synchrotron Rad. 21, 1167–1174 (2014). [CrossRef]
16. M. Guizar-Sicairos, I. Johnson, A. Diaz, M. Holler, P. Karvinen, H. Stadler, R. Dinapoli, O. Bunk, and A. Menzel, “High-throughput ptychography using Eiger: scanning X-ray nano-imaging of extended regions,” Opt. Express 22, 14859–14870 (2014). [CrossRef] [PubMed]
17. R. N. Wilke, M. Vassholz, and T. Salditt, “Semi-transparent central stop in high-resolution X-ray ptychography using Kirkpatrick–Baez focusing,” Acta Crystallogr. A 69, 490–497 (2013). [CrossRef]
18. A. M. Maiden, G. R. Morrison, B. Kaulich, A. Gianoncelli, and J. M. Rodenburg, “Soft X-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013). [CrossRef] [PubMed]
19. H. M. L. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy 103, 153–164 (2005). [CrossRef] [PubMed]
20. W. O. Saxton and W. Baumeister, “The correlation averaging of a regularly arranged bacterial cell envelope protein,” J. Microsc. 127, 127–138 (2013). [CrossRef]
21. G. Cardone, K. Grünewald, and A. C. Steven, “A resolution criterion for electron tomography based on cross-validation,” J. Struct. Biol. 151, 117–129 (2005). [CrossRef] [PubMed]
22. M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4, 3857 (2014). [CrossRef] [PubMed]
23. B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Pelle, M. A. Pfeifer, M. De Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Photonics 4, 394–398 (2008).
24. M. M. Seibert, T. Ekeberg, F. R. N. C. Maia, M. Svenda, J. Andreasson, O. Jonsson, D. Odić, B. Iwan, A. Rocker, D. Westphal, M. Hantke, D. P. DePonte, A. Barty, J. Schulz, L. Gumprecht, N. Coppola, A. Aquila, M. Liang, T. A. White, A. Martin, C. Caleman, S. Stern, C. Abergel, V. Seltzer, J.-M. Claverie, C. Bostedt, J. D. Bozek, S. Boutet, A. A. Miahnahri, M. Messerschmidt, J. Krzywinski, G. Williams, K. O. Hodgson, M. J. Bogan, C. Y. Hampton, R. G. Sierra, D. Starodub, I. Andersson, S. Bajt, M. Barthelmess, J. C. H. Spence, P. Fromme, U. Weierstall, R. Kirian, M. Hunter, R. B. Doak, S. Marchesini, S. P. Hau-Riege, M. Frank, R. L. Shoeman, L. Lomb, S. W. Epp, R. Hartmann, D. Rolles, A. Rudenko, C. Schmidt, L. Foucar, N. Kimmel, P. Holl, B. Rudek, B. Erk, A. Homke, C. Reich, D. Pietschner, G. Weidenspointner, L. Struder, G. Hauser, H. Gorke, J. Ullrich, I. Schlichting, S. Herrmann, G. Schaller, F. Schopper, H. Soltau, K.-U. Kuhnel, R. Andritschke, C.-D. Schroter, F. Krasniqi, M. Bott, S. Schorb, D. Rupp, M. Adolph, T. Gorkhover, H. Hirsemann, G. Potdevin, H. Graafsma, B. Nilsson, H. N. Chapman, and J. Hajdu, “Single mimivirus particles intercepted and imaged with an x-ray laser,” Nature (London) 470, 78–82 (2011). [CrossRef]
25. Y. Takahashi, A. Suzuki, N. Zettsu, T. Oroguchi, Y. Takayama, Y. Sekiguchi, A. Kobayashi, M. Yamamoto, and M. Nakasako, “Coherent diffraction imaging analysis of shape-controlled nanoparticles with focused hard x-ray free-electron laser pulses,” Nano Lett. 13, 6028–6032 (2013). [CrossRef] [PubMed]
