Open Access
Add to BibSonomy
Abstract
We demonstrated that the combination of a hyperbolic convex and elliptical concave mirrors works as a compact reflective X-ray imaging system with a short optical focal length and large magnification factor. We performed an experiment to form a one-dimensional demagnified image with a demagnification factor of 321 within an approximately 2-m-long optical setup at an X-ray energy of 10 keV. The results showed that this imaging optics system is capable of providing a resolution of ~40 nm. From wavefront analysis, it was confirmed that the optics possessed a wide field-of-view with a significant reduction of comatic aberration.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
X-ray microscopy is a powerful method for investigating the structure, elemental distribution, and chemical composition of large volume materials with high spatial resolutions beyond those achieved by visible light microscope. This method has progressed with advanced X-ray optics and light sources of synchrotron radiation (SR) and X-ray free electron laser (XFEL) [1–4]. In a scanning hard X-ray microscope, high intensity and small focal spots are produced using X-ray focusing optics in combination with appropriate sources to yield enhanced spatial resolution and sensitivity. Among the various types of hard X-ray focusing optics [5–9], X-ray mirrors [10,11] facilitate high reflectivity, large spatial acceptance, and achromaticity (or less chromaticity in multilayer mirrors). Although precise fabrication of X-ray mirrors is challenging, recent progress in ultraprecise surface figuring [12–15] and measurement techniques [16–18] have led to the achievement of diffraction-limited X-ray focusing in the Kirkpatrick-Baez (KB) geometry [19], and to their widespread application in micro and nanofocusing of X-rays. A minimum spot size of 25 nm has been achieved using total-reflection mirrors [10] and 7 nm using multilayer mirrors [20].
As another type of an X-ray microscopic system, a full-field hard X-ray microscope which directly forms an image of a sample using imaging optics is a key technique for real-time, three-dimensional, and spectroscopic observation. In this case, mirrors are rarely utilized for the imaging optics because the KB geometry does not satisfy the Abbe’s sine condition and the resulting comatic aberration inhibits the formation of X-ray images over a wide field-of-view (FoV). To suppress comatic aberration, an advanced KB (AKB) geometry which consists of two pairs of elliptical and hyperbolic concave mirrors aligned perpendicularly to each other has been proposed [21,22]. In the mirror pair, a one-dimensional (1D) Wolter type I optics [23] is constructed, where the Abbe’s sine condition is satisfied. Consequently, the AKB geometry can attain a wide FoV due to the reduction of comatic aberration. A full-field X-ray microscope based on the AKB geometry was developed and a spatial resolution of less than 50 nm was achieved [24]. Furthermore, spectroscopic full-field X-ray XANES (X-ray absorption near edge structure) imaging and XRF (X-ray fluorescence) imaging have been successfully performed due in part to the achromaticity of the optical system [25,26].
However, it is difficult to attain a sufficiently large magnification factor using the KB and AKB geometries. In particular, in the conventional KB and AKB configuration where the mirrors are arranged sequentially, the mirror pair closer to the focal point limits the minimum focal length of the other mirror pair to be much larger. This results in a lower magnification factor for the latter. In fact, our previous full-field microscope based on the AKB geometry required a distance of several tens of meters between the mirrors and camera to achieve a magnification factor of ~200 with a long focal length of ~230 mm [24]. A more compact optical design is desirable for widespread application of the AKB geometry.
To address this problem, we recently proposed a novel AKB geometry based on a combination of concave and convex mirrors, which enable to achieve a short optical focal length and a large-magnification factor in a compact setup [27]. In this report, we present results based on this geometry using a 1D mirror pair for the first experimental demonstration. An alignment strategy was investigated based on the result of the wave-optical simulation. The line spread function (LSF) of the developed 1D mirror pair was experimentally evaluated using a demagnification geometry at an X-ray energy of 10 keV. Wavefront errors were also measured using single-grating interferometer. By comparing the experimental results with simulations, it was confirmed that proposed optics is able to achieve a nearly diffraction-limited resolution and wide FoV.
2. AKB geometry based on combination of concave and convex mirrors
Figure 1(a) shows a schematic diagram of the proposed geometry. This geometry is based on X-ray imaging optics and consist of two pairs of elliptical concave and hyperbolic convex mirrors, i.e., a 1D Wolter type III optics [23] arranged perpendicularly. The optical focal length, which determines the magnification factor, is given as the distance from the focal point to the principal plane. This is the intersection plane formed by extensions of the incident rays to the optical system and the emitted rays. The combination of the elliptical concave and hyperbolic convex mirrors enables the principal plane to be formed close to the focus (Fig. 1(b)), resulting in a short focal length and large magnification factor.
Fig. 1 (a) Arrangement of an advanced KB optics based on concave and convex mirrors. (b) Cross section of a mirror pair. (c), (d) Schematic drawings of rays and angle between rays and optical axis of (c) an elliptical mirror and (d) an advanced KB optics based on Wolter type III optics. The relationships of φE > φ'E and θE < θ'E, where the subscript E indicated the ellipse, do not follow the Abbe’s sine condition. The relationships of φW > φ'W and θW > θ'W, where the subscript W indicated the Wolter type III, can follow the Abbe’s sine condition.
According to the Abbe’s sine condition, each ray in X-ray mirrors should obey the equation:
where M denotes a magnification factor of the optical system, φ and θ denote the angles between the rays and optical axis at object and image planes, respectively [28]. Although an elliptical mirror never satisfy the condition, the combination of concave and convex mirrors can follow the condition (see Fig. 1(c) and (d)). Consequently, the proposed AKB geometry is possible to reduce the comatic aberration.3. Design and mirror fabrication
For experimental demonstration, a 1D mirror pair which enables compact, high-resolution and large magnification imaging was designed using the following parameters: length of the axes of ellipse and hyperbola of ae = 1.72 × 10−2 m, be = 9.76 × 10−5 m, ah = 1.03 m and bh = 3.27 × 10−4 m in the equations x2 / ae2 + y2 / be2 = 1 and x2 / ah2 − y2 / bh2 = 1 where the subscripts e and h indicate the ellipse and hyperbola. These parameters result in an optical focal length of 6.5 mm, even with a long distance between the focus and the center of two mirrors of 20.15 mm. Consequently, this geometry can realize a magnification factor of 321 with a total length of approximately 2.1 m. The lengths of the mirrors are 8.2 mm (ellipse) and 8.6 mm (hyperbola), respectively. The glancing incident angle at the center of each mirror is 5.71 mrad (ellipse) and 2.34 mrad (hyperbola), which leads to a numerical aperture (NA) of 1.61 × 10−3 and a resolution of approximately 40 nm at an X-ray energy of 10 keV. Note that the conventional AKB system gives a magnification factor of only ~105 with this geometric configuration. More details of the design can be found in the paper regarding to a simulation study [27].
To confirm the comatic aberration of designed imaging optics, ray-tracing calculations were performed. The magnification factor was calculated using Eq. (1) with 2000 rays in the designed mirrors. For comparison, the same calculation with KB geometry which have a magnification factor of 321, NA of 1.61 × 10−3 and mirror length of 8.2 mm was also computed. Figure 2 shows the calculated results. The magnification factor hardly changed at each ray in the designed mirrors, although that in KB geometry drastically changed. It was confirmed that the designed mirrors almost satisfy the Abbe’s sine condition and eliminate the comatic aberration.
Fig. 2 Ray-tracing calculation results of magnification factor with designed mirror pair and an elliptical mirrors.
The mirror shape accuracy required to maintain a wavefront aberration less than λ/4 after the double reflection is 2 nm in peak-to-valley (PV) for both the concave and convex mirrors. The mirror shapes were fabricated on Cz -Si (001) substrates using an ion beam figuring system for surface figuring and a stitching interferometer for measurements [15–17]. The results of the mirror fabrication process are shown in Fig. 3. The residual errors from the ideal shapes were approximately 2 nm PV, which satisfied the required accuracy. After fabrication, the surfaces of the mirror were coated with a single layer of molybdenum to achieve a high reflectivity for hard X-rays [29].
Fig. 3 Measured shapes and residual errors from the ideal shape of the hyperbolic convex (left) and elliptical concave (right) mirror.
4. Alignment strategy
Alignment of the relative position and angle between the two mirrors is important to obtain diffraction-limited performance. In this study, we performed wave-optical calculations with alignment errors in the θx and z-axes as shown in Fig. 4(a). In previous studies, these parameters were determined to significantly influence imaging performance [27,30]. Figure 4(b) presents the calculated results for the intensity distribution near the foci with/without the alignment errors. It was demonstrated that an error in either θx or the z-axis generates typical comatic aberration. Interestingly, we determined that an error in the θx axis of 100 μrad can be compensated with a translation of 1.35 μm along the z-axis. This simulated result suggests that the fine-tuning of only one parameter is sufficient to form a nearly diffraction-limited LSF profile while checking comatic aberration. To observe comatic aberration, X-ray wavefront sensing techniques [31–33] are useful. In this study, we employed X-ray single-grating interferometer [34] to perform fast and simple wavefront measurements.
Fig. 4 (a) Alignment errors in θx (left) and z (right) direction assumed in the wave-optical simulation. (b) Calculated intensity distributions near the foci with and without alignment errors for an X-ray energy of 10 keV. X-ray beams enter from the left side in each figure.
5. Experiment
Figure 5 shows the experimental setup which was installed at BL29XUL of SPring-8. We performed a 1D demagnification imaging of the virtual source which is generally employed for investigation of the optical characteristics [5,9,35,36]. The X-ray beam generated by an undulator was monochromatized at 10 keV using a double-crystal Si(111) monochromator. A cross-slit was used to form an X-ray source with a well-defined size. A demagnified image of the slit with a 10 μm width was generated at the focal point by the imaging optics which was arranged approximately 2 m downstream from the slit. As described in Sec. 3, the optical system had a magnification factor of 321 and NA of 1.61 × 10−3. These parameters give a demagnified slit width of 31.1 nm based on geometric calculations, and focal spot size of 35.2 nm at full width at half maximum (FWHM) based on the diffraction-limited calculation. A convolution of these two rectangular and sinc functions gives an expected focal spot size of approximately 40.6 nm. The intensity distribution on the focal plane is characterized by the knife-edge method with a 50-μm-diameter gold wire. A tantalum Ronchi grating with a 2.5 μm period (NTT Advanced Technology Corporation) and an X-ray camera (AA20MOD, Hamamatsu Photonics) were installed for wavefront measurement of the reflected X-rays. The grating and the camera were placed 25.7 mm and 1.3 m downstream from the focus, respectively, which results in the self-image of the grating with a Talbot order of 0.5.
Firstly, we carefully characterized the slit width along the vertical direction which is a critical parameter in this experiment, with sub-micron accuracy and set the width to 10.0 μm. The slit width along the horizontal direction was set to 50 μm to provide the sufficient X-ray intensity. After a rough alignment of the mirror pair, the wavefront aberration was measured using single-grating interferometer. A raw wavefront profile was obtained using a 25-step fringe scanning method. The wavefront error was calculated by subtracting a fitted quadratic function. The dashed blue line in Fig. 6(b) represented the wavefront error in the initial condition. The solid blue line represents the fitting result with a cubic function, which is equivalent to a comatic aberration. Middle-frequency wavefront errors originated from the slight shape errors of the mirrors. For the fine alignment, we adjusted the relative translation along the z-axis (Fig. 6(a)) with micrometer accuracy while measuring the wavefront error. The comatic aberration was gradually reduced using this procedure in good agreement with the wave-optical simulation, which is shown by the green and red lines in Fig. 6(b). Finally, we completed the fine alignment with a small wavefront error of approximately 1.6 rad in PV, which corresponds to λ/4 (the black lines in Fig. 6(b)). The obtained wavefront error had a uniformity along the horizontal direction with 0.1 rad accuracy in a range of +/− 1 mm, in which width of mirror area was 3 mm.
Fig. 6 (a) Optical geometry for the fine alignment. (b) Dependence of wavefront error on the translational deviation. Dashed and solid lines denote the raw wavefront errors and fitted cubic functions, respectively.
6. Results and discussions
We investigated a focal profile at the optimized glancing incident angle using the knife-edge scanning method. Figure 7 shows an intensity profile on the focal plane which was experimentally obtained together with that of based on the simulation. The experimental result was fitted with the Gaussian function, which gives a width of the 42.7 nm in FWHM. This result agrees well with the simulated value of 40.8 nm.
Fig. 7 Line profiles of demagnified images. Experimentally obtained values and the Gaussian fit are shown by red dots and line, respectively. The fitted width is 42.7 nm in FWHM. The black solid line indicates the profile calculated with the measured shape errors (shown in Fig. 2) and the experimental condition using our wave-optical simulator, with a width of 40.8 nm in FWHM.
To study the off-axis imaging characteristic, we investigated the dependence of the wavefront error on the glancing incident angle while maintaining the relative alignment of the two mirrors. Figure 8(a) represents the experimentally obtained wavefront errors with varying glancing incident angle errors from −0.7 to 0.7 mrad. It was determined that the wavefront errors in the low-spatial-frequency range are sufficiently suppressed in this range, which indicates the reduction of comatic aberration. A simulated result of the relationship between the angular deviation and the focal spot size in FWHM is shown in Fig. 8(b). The upper axis shows the FoV corresponding to the angular deviation. By comparing these results, we confirmed that the imaging optics has a wide FoV of ~11 μm.
Fig. 8 (a) Experimentally obtained wavefront errors with various glancing incident angles. (b) Calculated widths of LSFs in FWHM versus the angular deviation from the center of the FoV [27]. The upper axis indicates FoV for the magnifying imaging corresponding to the angular deviation. The colored circles and arrows with dashed lines indicate the corresponding wavefront errors in graphs (a).
In Fig. 8(b), the focal size appears to be broadened in a region outside of the FoV. However, the wavefront errors outside of the FoV did not contain the comatic aberration. According to our simulation for the focal surface, shown in in Fig. 9(a), the broadening in a region outside of the FoV is caused mainly by the aberration of the field curvature, rather than that due to comatic aberration [27]. To confirm this, we plotted the best focus position along the optical axis using the quadratic function extracted from the raw wavefront data, at each glancing incident angle, in Fig. 9(b). The results followed the simulated curve and clearly indicated the distinctive curve of the focal surface.
Fig. 9 (a) Distribution of demagnified beam size near the focus calculated using the ray-tracing simulator. The horizontal (vertical) direction shows the distance along (perpendicular to) the optical axis. The deep blue region indicates the focal surface [27]. (b) Displacement of the focal point versus angular deviation from the FoV center. The experimental results (red dots) are obtained from the radii of curvature of measured wavefronts. The simulated curve (blue line) corresponds to the calculated focal surface shown in (a).
7. Summary and outlook
We developed a 1D imaging system consisting of hyperbolic convex and elliptical concave mirrors. The mirrors were fabricated with a shape accuracy of 2 nm in PV, which is sufficiently small to facilitate diffraction-limited performance. Wave-optical simulations were performed for investigating the alignment procedure. The calculated results indicated that only one parameter should be tuned for the fine alignment. A demagnification experiment was performed to demonstrate the performance of our imaging system. The results showed that the system had a resolution of ~40 nm with a wide FoV of ~11 μm.
The imaging capability of the proposed optics was revealed with 1D experiment. For the 2-dimensional imaging, the assembling and immobilization of the mirror pair are important to solve the complexity of four independent mirrors. It has been confirmed that the assembling with sufficient accuracy is feasible in our previous study [27]. In fact, we have been developing the assembled type mirror pairs of concave and convex mirrors [37]. 2-dimensional imaging system will be available in the near future.
The large magnification factor achieved using this compact setup will drastically improve the versatility of full-field X-ray microscopy using mirror optics not only for SR and XFEL sources, but also laboratory sources.
Funding
Japan Society for the Promotion of Science (JSPS), Grants-in-Aid for Scientific Research (KAKENHI) (JP17H01073, JP26286077, JP16H06358). Grant-in-Aid for JSPS Research Fellow (JP16J00953). Japan Science and Technology Agency (JST), Adaptable and Seamless Technology Transfer Program through Target-driven R&D (A-STEP) (AS2915035S).
Acknowledgments
This research was partially supported by Shimadzu Science Foundation and JSPS Core-to-Core Program on International Alliance for Material Science in Extreme States with High Power Laser and XFEL. The use of BL29XUL at SPring-8 was supported by RIKEN. The authors acknowledge Mr. Shuhei Yasuda and Ms. Ikumi Tokuoka for their support. One of the authors (J. Y.) acknowledges the SACLA Research Support Program for Graduate Students.
References
1. R. Hettel, “DLSR design and plans: an international overview,” J. Synchrotron Radiat. 21(5), 843–855 (2014). [CrossRef] [PubMed]
2. M. Eriksson, J. F. van der Veen, and C. Quitmann, “Diffraction-limited storage rings - A window to the science of tomorrow,” J. Synchrotron Radiat. 21(5), 837–842 (2014). [CrossRef] [PubMed]
3. M. Yabashi, K. Tono, H. Mimura, S. Matsuyama, K. Yamauchi, T. Tanaka, H. Tanaka, K. Tamasaku, H. Ohashi, S. Goto, and T. Ishikawa, “Optics for coherent X-ray applications,” J. Synchrotron Radiat. 21(5), 976–985 (2014). [CrossRef] [PubMed]
4. M. Yabashi and H. Tanaka, “The next ten years of X-ray science,” Nat. Photonics 11(1), 12–14 (2017). [CrossRef]
5. A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature 384(6604), 49–51 (1996). [CrossRef]
6. C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Küchler, “Hard x-ray nanoprobe based on refractive x-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005). [CrossRef]
7. C. David, S. Gorelick, S. Rutishauser, J. Krzywinski, J. Vila-Comamala, V. A. Guzenko, O. Bunk, E. Färm, M. Ritala, M. Cammarata, D. M. Fritz, R. Barrett, L. Samoylova, J. Grünert, and H. Sinn, “Nanofocusing of hard X-ray free electron laser pulses using diamond based Fresnel zone plates,” Sci. Rep. 1(1), 57 (2011). [CrossRef] [PubMed]
8. W. Chao, P. Fischer, T. Tyliszczak, S. Rekawa, E. Anderson, and P. Naulleau, “Real space soft x-ray imaging at 10 nm spatial resolution,” Opt. Express 20(9), 9777–9783 (2012). [CrossRef] [PubMed]
9. H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, “Focusing of hard x-rays to 16 nanometers with a multilayer Laue lens,” Appl. Phys. Lett. 92(22), 221114 (2008). [CrossRef]
10. H. Mimura, H. Yumoto, S. Matsuyama, Y. Sano, K. Yamamura, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Efficient focusing of hard x rays to 25 nm by a total reflection mirror,” Appl. Phys. Lett. 90(5), 051903 (2007). [CrossRef]
11. H. Yumoto, H. Mimura, T. Koyama, S. Matsuyama, K. Tono, T. Togashi, Y. Inubushi, T. Sato, T. Tanaka, T. Kimura, H. Yokoyama, J. Kim, Y. Sano, Y. Hachisu, M. Yabashi, H. Ohashi, H. Ohmori, T. Ishikawa, and K. Yamauchi, “Focusing of X-ray free-electron laser pulses with reflective optics,” Nat. Photonics 7(1), 43–47 (2013). [CrossRef]
12. K. Yamauchi, H. Mimura, K. Inagaki, and Y. Mori, “Figuring with subnanometer-level accuracy by numerically controlled elastic emission machining,” Rev. Sci. Instrum. 73(11), 4028–4033 (2002). [CrossRef]
13. K. Yamamura, K. Yamauchi, H. Mimura, Y. Sano, A. Saito, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Fabrication of elliptical mirror at nanometer-level accuracy for hard X-ray focusing by numerically controlled plasma chemical vaporization machining,” Rev. Sci. Instrum. 74(10), 4549–4553 (2003). [CrossRef]
14. A. Shorey, W. Kordonski, and M. Tricard, “Magnetorheological finishing of large and lightweight optics,” Proc. SPIE 5533, 99–107 (2004). [CrossRef]
15. J. Yamada, S. Matsuyama, Y. Sano, and K. Yamauchi, “Development of ion beam figuring system with electrostatic deflection for ultraprecise X-ray reflective optics,” Rev. Sci. Instrum. 86(9), 093103 (2015). [CrossRef] [PubMed]
16. K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for X-ray reflective optics,” Rev. Sci. Instrum. 74(5), 2894–2898 (2003). [CrossRef]
17. H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard X-ray reflective optics,” Rev. Sci. Instrum. 76(4), 045102 (2005). [CrossRef]
18. F. Siewert, T. Noll, T. Schlegel, T. Zeschke, and H. Lammert, “The nanometer optical component measuring machine: a new sub-nm topography measuring device for X-ray optics at BESSY,” AIP Conf. Proc. 705, 847–850 (2004). [CrossRef]
19. P. Kirkpatrick and A. V. Baez, “Formation of optical images by X-rays,” J. Opt. Soc. Am. 38(9), 766–774 (1948). [CrossRef] [PubMed]
20. H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-X-ray focusing,” Nat. Phys. 6(2), 122–125 (2010). [CrossRef]
21. R. Kodama, Y. Katori, T. Iwai, N. Ikeda, Y. Kato, and K. Takeshi, “Development of an advanced Kirkpatrick-Baez microscope,” Opt. Lett. 21(17), 1321–1323 (1996). [CrossRef] [PubMed]
22. R. Sauneuf, J. M. Dalmasso, T. Jalinaud, J. P. Le Breton, D. Schirmann, J. P. Marioge, F. Bridou, G. Tissot, and J. Y. Clotaire, “Large-field high-resolution X-ray microscope for studying laser plasmas,” Rev. Sci. Instrum. 68(9), 3412–3420 (1997). [CrossRef]
23. H. Wolter, “Spiegelsysteme streifenden Einfalls als abbildende Optiken für Röntgenstrahlen,” Ann. Phys. 445(1-2), 94–114 (1952). [CrossRef]
24. 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] [PubMed]
25. S. Matsuyama, J. Yamada, S. Yasuda, H. Okada, Y. Kohmura, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Development of full-field x-ray fluorescence microscope using total-reflection mirrors,” presented at SPIE Optics+Photonics 2017, Optical Engineering+Applications, X-Ray Nanoimaging: Instruments and Methods III, 10389–5, San Diego, California, USA, 6–10 August (2017).
26. S. Matsuyama, S. Yasuda, J. Yamada, Y. Kohmura, M. Yabashi, T. Ishikawa, and K. Yamauchi are preparing a manuscript to be called “Novel full-field X-ray fluorescence microscopy using total reflection imaging mirrors.”
27. J. Yamada, S. Matsuyama, Y. Sano, and K. Yamauchi, “Simulation of concave-convex imaging mirror system for development of a compact and achromatic full-field x-ray microscope,” Appl. Opt. 56(4), 967–974 (2017). [CrossRef] [PubMed]
28. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
29. J. Yamada, S. Matsuyama, Y. Sano, and K. Yamauchi, “Simulation of concave-convex imaging mirror system for development of a compact and achromatic full-field x-ray microscope,” Appl. Opt. 56(4), 967–974 (2017). [CrossRef] [PubMed]
30. S. Matsuyama, M. Fujii, and K. Yamauchi, “Simulation study of four-mirror alignment of advanced Kirkpatrick-Baez optics,” Nucl. Instrum. Methods Phys. Res. A 616(2-3), 241–245 (2010). [CrossRef]
31. M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, and P. Sauvageot, “X-ray active mirror coupled with a Hartmann wavefront sensor,” Nucl. Instrum. Methods Phys. Res. A 616(2–3), 162–171 (2010). [CrossRef]
32. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009). [CrossRef] [PubMed]
33. S. Berujon, H. Wang, S. Alcock, and K. Sawhney, “At-wavelength metrology of hard X-ray mirror using near field speckle,” Opt. Express 22(6), 6438–6446 (2014). [CrossRef] [PubMed]
34. S. Matsuyama, H. Yokoyama, R. Fukui, Y. Kohmura, K. Tamasaku, M. Yabashi, W. Yashiro, A. Momose, T. Ishikawa, and K. Yamauchi, “Wavefront measurement for a hard-X-ray nanobeam using single-grating interferometry,” Opt. Express 20(22), 24977–24986 (2012). [CrossRef] [PubMed]
35. S. Matsuyama, N. Kidani, H. Mimura, Y. Sano, Y. Kohmura, K. Tamasaku, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Hard-X-ray imaging optics based on four aspherical mirrors with 50 nm resolution,” Opt. Express 20(9), 10310–10319 (2012). [CrossRef] [PubMed]
36. S. Matsuyama, T. Wakioka, N. Kidani, T. Kimura, H. Mimura, Y. Sano, Y. Nishino, M. Yabashi, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “One-dimensional Wolter optics with a sub-50 nm spatial resolution,” Opt. Lett. 35(21), 3583–3585 (2010). [CrossRef] [PubMed]
37. J. Yamada, S. Matsuyama, K. Hata, R. Hirose, Y. Takeda, Y. Kohmura, M. Yabashi, K. Omote, T. Ishikawa, and K. Yamauchi, “Reflective Imaging Optics Using Concave and Convex Mirrors for a Compact and Achromatic Full-field X-ray Microscope,” Microsc. Microanal. 24(S2Suppl 2), 276 (2018). [CrossRef]
