1. INTRODUCTION

X-ray microscopy not only enables nondestructive and high-spatial-resolution observation of the internal structure of a sample, but also enables elemental distribution and chemical state analysis. Therefore, X-ray microscopy is one of the most suitable techniques for operando observation of functional materials [1–3] and biological specimens [4–6]. However, although X-ray microscope development has been ongoing since the early 1900s, the spatial resolution of microscopes remains unsatisfactory [7–9]. The most critical issue causing resolution degradation is the fabrication error of the objective lens. The accuracy required for an X-ray objective lens is much higher than that for visible light because the wavelength of X-rays is 1000 times shorter than that of visible light, and the required accuracy is inversely proportional to the wavelength. For example, in the case of advanced Kirkpatrick-Baez (AKB) reflective objectives [10–12], which have attracted attention because of their high efficiency and relatively small chromatic aberration observation in the hard X-ray bandpass, the acceptable shape error on the mirror required to achieve a diffraction-limited resolution of 50 nm is at least 3 nm. An accuracy of 0.3 nm is required to achieve a 5 nm resolution. The ultimate accuracy is beyond the limits of the current fabrication technology [13–17], and it is almost impossible to achieve accuracy by considering the addition of deformation owing to gravity and retention.

To overcome this problem, we propose the introduction of X-ray adaptive mirrors in full-field X-ray microscopes. If the mirrors can be optimally deformed according to the detected wavefront aberration, fabrication errors and deformations owing to gravity and retention can be corrected in situ. Various deformable mirrors (DMs) have been proposed for X-rays [18–26]. Piezoelectric bimorph-type DMs, which can achieve high flexibility and precision, are promising candidates for adaptive mirrors. However, this system has several limitations. Lead zirconate titanate (PZT), commonly used as a deformation driver, is a ceramic material containing many domains and grains. This causes long-term deformation drift and significant hysteresis, making it difficult to control and maintain the deformation. Because PZT has a low Curie point of 600 K, it cannot be operated at high temperatures. In addition, PZT plates must be bonded to mirror substrates such as silicon and quartz glass to create a bimorph structure. The bonding of dissimilar materials causes thermal instability, and the adhesive also causes various problems, such as gas emission under ultrahigh vacuum, swelling owing to moisture, drift, and radiation damage. Another problem is that the bimorph structure cannot be deformed into high spatial frequency shapes. The bimorph structure deforms its shape using the bending moment generated by stretching/contracting the piezoelement in the surface-parallel direction. Because the applied voltages are proportional to the local curvature, forming high-spatial-frequency shapes with a large curvature requires a large voltage. This introduces new concerns such as high-voltage power supply cost and dielectric breakdown. Therefore, existing PZT-based piezoelectric bimorph mirrors are not suitable for high-resolution microscopy, and optimized high-performance DMs are required.

In this study, we propose a novel DM composed entirely of single-crystal lithium niobate (LN) that enables high-precision, high-stability, and high-spatial-frequency-controlled deformation, termed LNDM (Fig. 1). Unlike PZT, the surface of the single-crystal piezoelectric material can be smoothed by super polishing; thus, it can function as an actuator and a reflective surface for X-rays. This enables a simple structure consisting only of an LN substrate and electrodes, which solves various problems associated with conventional DMs. In addition, because single-crystal LN is a piezoelectric material comprising a single domain, it can be expected to deform with high precision without hysteresis or drift [27]. Furthermore, it exhibits a high Curie point (1500 K) and can operate at high temperatures, which will become important when brighter X-ray sources emerge. In addition, a surface-normal deformation system is employed for the proposed LNDM [Fig. 1(b)], where the expansion and contraction of the LN lead directly to the deformation of the surface shape, enabling good controllability of high-spatial-frequency shapes that are difficult to achieve with conventional bimorph mirrors based on the surface-parallel deformation system.

 

Fig. 1. (a) Schematic diagram of a LNDM. (b) Deformation mechanism of the mirror. Unlike bimorph mirrors, which expand and contract parallel to the surface, the mirror expands and contracts normal to the surface. (c) Photograph of the fabricated mirror. Scale bar is 20 mm.

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The paper is organized as follows. First, the design and fabrication of the proposed LNDM are described in detail. Subsequently, an adaptive reflection objective was constructed by replacing one mirror of the existing AKB mirror [12] with the LNDM. The linearity and stability of the adaptive mirror system were evaluated by using an X-ray grating interferometer constructed at SPring-8. In addition, the correction of wavefront aberrations due to mirror fabrication errors was demonstrated. Consequently, the wavefront aberration was successfully corrected with an accuracy of 0.4 rad, corresponding to a shape error of 0.67 nm. Subsequently, adaptive X-ray microscopy was performed by obtaining X-ray images of a test chart before and after wavefront correction. The quality of the X-ray image after correction was improved, demonstrating the effectiveness of adaptive X-ray microscopy with LNDM.

2. LN-BASED MONOLITHIC DEFORMABLE MIRROR

We designed and fabricated the LNDM. The substrate size was 70 mm in length, 20 mm in width, and 5 mm thick. A 36°-rotated Y-cut single-crystal LN with a piezoelectric coefficient of approximately 37 pm/V (Yamaju Ceramics Co., Ltd.) was used to maximize the deformation. The crystal surface was processed into an elliptical shape with the same parameters as the horizontal elliptical mirror in [12] using the elastic emission machining (EEM) method [15]. The surface roughness after processing was below 0.2 nm in the root-mean-square (RMS), comparable to that of silicon or quartz glass mirrors, which are frequently used as X-ray mirrors. Gold was deposited on the bottom surface as an electrode to control its shape. Each of the 11 electrodes had an area of ${5}\;{\rm mm} \times {20}\;{\rm mm}$ and an interval of 1 mm. These were connected to a bipolar power supply that could apply a maximum (minimum) voltage of $\pm {500}\;{\rm V}$. Platinum was uniformly coated on the top surface with a thickness of approximately 100 nm. The layer serves as a reflective surface for X-rays and a ground for the bottom electrodes. The maximum deformation estimated by the finite element method was 9.3 nm in peak-to-valley (PV) value with a voltage of 500 V applied, and the minimum spatial wavelength was 10 mm. The only disadvantage compared to previous DMs is the small deformation; therefore, it is unsuitable for correcting large aberrations, which are often introduced by inaccurate mirror fabrication and mirror misalignment. However, correcting the slight shape errors (less than 10 nm) caused by ultraprecise mirror fabrication in X-ray microscopy is sufficient.

 

Fig. 2. Experimental setup for X-ray line-focusing.

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3. EXPERIMENTS AND RESULTS

A. Basic Performance Evaluation

An X-ray line-focusing experiment using LNDM was performed at the second experimental hatch (EH2) at BL29XU of SPring-8. Figure 2 shows the experimental setup. X-rays monochromatized to photon energy of 14.5 keV by a Si-111 double-crystal monochromator (DCM) were employed. A transport-channel slit placed immediately downstream of the DCM was used as a virtual light source. Wolter type-III advanced KB (AKB) mirror optics, which has the same parameters as those in [12], was constructed with a concave elliptical LNDM and a convex hyperbolic mirror to function as a horizontal adaptive one-dimensional lens for X-ray microscopy. The convex mirror is identical to that developed in a previous study and is not a DM. To control the shape of the DM, an X-ray grating interferometer [28], which can precisely measure the wavefront aberration with an accuracy of less than 0.045 rad (${\approx} \lambda /{140}$), was installed. A 1D grating with a period of 2.5 µm was positioned 112 mm downstream from the focal point, while a X-ray camera consisting of a CMOS camera with a pixel size of 6.5 µm, a scintillator (LuAG:Ce) with a thickness of 0.7 mm, and a lens ($\times {0.5}$) was placed 5.638 m downstream from the focal point. The deformation performance of the DM was evaluated by measuring the wavefront aberrations before and after applying the voltages.

 

Fig. 3. Evaluation results of deformation stability. (a) Measured wavefront changes from before the voltage application. The plots contain eight curves corresponding to 7 h of elapsed time. (b) Wavefront changes are obtained immediately after the voltage application. Legends indicate the elapsed hour.

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Fig. 4. Hysteresis evaluation results. (a) Wavefront changes when the voltage is stepped back and forth in 100 V increments from ${+}{500}\;{\rm V}$ to ${-}{500}\;{\rm V}$. (b) Plot of peak height in (a) at each voltage. (c) Plot of the nonlinear components in (b).

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First, the deformation stability was evaluated. A voltage of 500 V was applied to each electrode, and the wavefront was continuously measured every hour for 7 h. Figure 3(a) shows the wavefront aberration obtained by subtracting the wavefront measured before the voltage application. The single Gaussian peak-like deformation observed here is the deformation response introduced by the single electrode, which is consistent with the results measured by visible light interferometry (Verifire XPZ, Zygo Corporation) and calculated using the finite element method (both results are not shown here). All eight curves in the graph overlap completely and appear as a single line. To analyze the stability precisely, Fig. 3(b) was obtained by subtracting the wavefront measured immediately after the voltage application, showing a significantly slight drift of less than 0.1 rad in PV, corresponding to 0.17 nm as the shape height, throughout 7 h. The details are described in Supplement 1.

Next, hysteresis was evaluated by measuring the change in wavefront when the voltage was stepped forth and back in 100 V increments from ${+}{500}\;{\rm V}$ to ${-}{500}\;{\rm V}$ [Figs. 4(a) and 4(b)]. Eighteen curves are plotted in Fig. 4(a), and the two curves corresponding to each voltage were almost completely overlapped. For a precise analysis, the nonlinear components were obtained by approximating the curve in Fig. 4(b) as a straight line and extracting its deviation components. The nonlinear components, shown in Fig. 4(c), were 0.034 rad RMS, corresponding to a deformation height of 0.058 nm on the mirror. These results indicate that the hysteresis of the LNDM is negligibly small and that the deformation can be controlled with atomic accuracy.

Finally, wavefront compensation was demonstrated. Figure 5(a) shows the changes in the wavefront when 500 V was applied to one electrode, representing the piezo response function of the DM [19]. The deformation adjustment was made based on the inverse matrix method and the piezo response function, which allowed accurate wavefront modification. Consequently, the wavefront aberration was successfully corrected from 3 rad to 0.4 rad in PV, corresponding to a shape accuracy of 0.67 nm.

 

Fig. 5. Results of shape correction. (a) Measured deformation when 500 V was applied to each of the 11 electrodes (piezo response function). (b) Wavefront aberration values measured before and after shape correction.

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B. Adaptive X-Ray Microscope

A proof-of-principle experiment using an adaptive X-ray microscope was performed at the EH2 of BL29XU at SPring-8. To the best of our knowledge, this is the first attempt at adaptive wavefront compensation using a full-field hard X-ray microscope with a high spatial resolution. The experimental setup is shown in Fig. 6. The same AKB mirrors described in published papers [12] were used, except that the horizontal elliptical mirror was replaced with an LNDM. Therefore, the horizontal mirror pair is the same as discussed in Section 3.A, and the vertical mirror pair is the same as in published papers. The source was also monochromatized to 14.5 keV. A Siemens star test pattern (XRESO-50HC, tantalum thickness 500 nm, NTT Advanced Technology Corporation) with a minimum line width of 50 nm was used as the sample. A rotating diffuser was placed in front of the sample to reduce the unwanted speckles in the images. The section between the sample and the camera was placed under helium. Furthermore, the bright-field images were captured with a scintillator-based X-ray camera consisting of LuAG (5 µm thick), a CMOS camera (${2048} \times {2048}\;{\rm pixels}$, ${6.5} \times {6.5}\;\unicode{x00B5}{\rm m}^2/{\rm pixel}$), and a lens ($\times {10}$) [29]. In this experiment, a wave-propagation-based phase imaging method was employed to obtain a phase image of the sample, resulting in clear images without unwanted interference caused by highly coherent X-rays. A total of 31 images were acquired while moving the sample at 50 µm increments along the optical axis. The phase images were reconstructed using a previously described method [30]. Phase images were acquired before and after wavefront compensation by applying the voltage pattern obtained in the experiment described in Section 3.A.

 

Fig. 6. Experimental setup for the adaptive X-ray microscope.

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Figures 7(a) and 7(b) show the reconstructed phase images before and after deformation, respectively. In addition, Figs. 7(c) and 7(d) present enlarged images, specifically showing improved image quality in the horizontal direction owing to the installation of the LNDM for horizontal imaging. Before the correction, aberration-derived blurring occurred, particularly noticeable in the distortion of the dots (see the character “.2” in the X-ray images of (c) and line profiles of (e)). However, these distortions disappeared after correction, resulting in a clearer image. These results provide clear evidence for the effectiveness of the adaptive X-ray microscope using the new DM.

 

Fig. 7. Results of the adaptive X-ray microscope. The reconstructed phase images (a) before and (b) after the deformation. Scale bar is 2 µm. (c) and (d) were enlarged images of (a) and (b), respectively. Scale bar is 0.5 µm. The gray scales represent phase shift. (e) Line profiles of the horizontal direction of the allowed dots in (c) and (d). (f) Line profiles of the vertical direction of the allowed dots in (c) and (d).

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4. DISCUSSION AND OUTLOOK

The developed LNDM provided sufficient performance for adaptive X-ray microscopy with high resolution; however, there is still room for improvement. Figure 5 demonstrates that the adaptive mirror system can provide wavefronts with an accuracy better than $\lambda /{4}$ after the wavefront compensation, which can satisfy the accuracy to realize the diffraction-limited performance for this X-ray microscope. However, the accuracy was not sufficiently high to achieve the ultimate resolution, such as sub-5-nm. The reason for this limitation lies in the spatial frequency that can be effectively corrected, which is determined by the electrode width and pitch. Finite element simulations have shown that a more accurate deformation can be easily achieved if the electrode width is reduced from 5 mm to 2 mm, maintaining an interval of 1 mm. With this improvement, the achievable shape accuracy was estimated to be 0.27 nm, assuming the same shape error (“before deformation”) as in Fig. 5(b) (see Fig. S2 in Supplement 1). This accuracy would enable us to reach a 5 nm spatial resolution, corresponding to the diffraction limit. Such high-spatial-frequency shape control is difficult for conventional bimorph mirrors and can be further improved by refining the LNDM design.

In parallel with the development of DMs, a wavefront-sensing technology dedicated to full-field X-ray microscopy has been developed. Based on the DM fabrication technology established here and these wavefront measurement techniques, we plan to develop an ultrahigh-resolution X-ray microscope equipped with an adaptive mirror system to achieve a sub-5-nm resolution in the future. We anticipate that ultrahigh-resolution X-ray microscopes, comparable to electron microscopes, will advance novel applications for future synchrotron X-rays and X-ray free-electron lasers.