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Abstract
We propose an extended X-ray adaptive zoom condenser that can form an intermediate virtual focus. The system comprises two deformable mirrors for focusing within a single dimension and can vary its numerical aperture (NA) without changing the positions of the light source, mirrors, or final focus. The desired system NA is achieved simply by controlling the mirror surfaces, which enables conversion between convex and concave forms, by varying the position of the intermediate virtual focus. A feasibility test at SPring-8 under a photon energy of 10 keV demonstrated that the beam size can be varied between 134 and 1010 nm.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since X-ray focusing optics can dramatically increase the sensitivity and spatial resolution of X-ray analysis, various X-ray focusing optics based on refraction, diffraction, and reflection have been developed all over the world [1–4]. For routine experiments, focused beams with diameters of 40–200 nm are commonly used in synchrotron radiation facilities and X-ray free-electron laser (XFEL) facilities [5–8]. In state-of-the-art research, nanobeams with a size of 10–30 nm are being employed [9–11].
Fig. 1. Schematics of possible combinations of elliptical and hyperbolic mirrors for adaptive zoom condenser system. Points S, Fi, and Ff are the light source, intermediate focus, and final focus (sample position), respectively. The ellipse and hyperbola are color-coded as blue and green, respectively. (a) Previous adaptive zoom condenser (Wolter IV-like) arrangement comprising two concave ellipses [21]. (b) Wolter II-like type arrangement comprising concave ellipse and convex hyperbola (Lh1 < Lh2). (c) Wolter III-like type arrangement comprising convex hyperbola and concave ellipse (Lh1 > Lh2). (d) Wolter II′-like type arrangement comprising concave ellipse and concave hyperbola (Lh1 < Lh2). Note that, similar to the relationship between the Wolter II and III systems, another system type (Wolter III′-like) exists with an arrangement of concave ellipses and hyperbolas in the opposite order to that of the Wolter II′-like system.
Current developments in X-ray focusing optics are aimed at (1) the realization of extreme focusing and (2) the realization of on-demand focusing. For the former, diffraction-limited focusing with focusing optics featuring substantial numerical apertures (NAs) have been widely adopted, and tiny focus sizes of 5–8 nm have also been reported [12–19]. For the latter, beam-shaping systems [20], variable-NA focusing systems [21], and more versatile adaptive systems that offer control over various optical parameters [22,23] have been developed.
Variable-NA focusing systems enable focused beams in experiments sensitive to beam divergence angles, such as coherent diffraction imaging experiments [24]. Additionally, an appropriate choice of the NA under diffraction-limited conditions can modulate the beam size, provided a sufficiently small light source is used [21]. The electromagnetic lenses function as variable-NA focusing systems for electron beams; however, there are few examples of the implementation of such a system for the X-ray region. Owing to the limited space in synchrotron radiation facilities and XFEL facilities, it is wasteful to install multiple focusing optics to obtain beams of different sizes. If a single focusing system can provide optimized beams with beam modulation according to the various samples’ requirements, we can expect improvements in measurement sensitivity and spatial resolution.
In our previous work [21], a variable-NA focusing system—termed an adaptive zoom condenser—comprising four concave deformable mirrors; i.e., two sets of adaptive Kirkpatrick–Baez (KB) mirrors, was proposed [Fig. 1(a)]. This optical system offers control of the NA, without the necessity of altering either the mirror or sample positions, simply via deforming the four mirrors. An intermediate focus of the real image occurs between the two KB mirrors. Shifting only the intermediate focus position can modify the downstream mirror's illumination area and change the NA without altering the final focus position. An X-ray focusing test using 10-keV X-rays at SPring-8 demonstrated the system's capacity to vary the beam size in the vertical and horizontal directions in the ranges of 165–1434 nm and 108–560 nm, respectively.
From an optical viewpoint, this system is of interest as it has the potential for further improvements. A specific point of interest relates to the optical systems, which are achievable when the intermediate focus is located beyond the KB mirror pairs: when the intermediate focus passes beyond the mirror, the real image changes into an imaginary one, the passed mirror changes from a concave to a convex one, and its shape changes from elliptical to hyperbolic. Geometrically, the light forms a definite focus at a single point while maintaining a constant optical path length. Such an optical system closely resembles Wolter type II and III systems [25,26], and, depending on the details of the optical design, it can be considered equivalent to the same. This extended adaptive zoom condenser system design has the potential effect of reducing the overall length of the system, as compared to previous systems. When applied to XFEL focusing, the novel system should prove advantageous because a real intermediate image with ultrahigh power density will not appear, and the operation would be safer. By contrast, conventional systems form a real intermediate image, occasionally on the surface of a mirror or another optic, resulting in fatal damage to the optic. Furthermore, the combination of concave and convex surfaces should increase the degree of freedom in the optical design, facilitating the design of an optical system that can be adapted to allow optimal focusing for individual experiments.
In this paper, we propose an extended adaptive zoom condenser system and investigate its behavior via simulation. Additionally, herein, we report the demonstration of a one-dimensional adaptive zoom condenser at SPring-8 and the beam size controllability testing at a photon energy of 10 keV [27].
2. Adaptive zoom condenser
This optical system consists of two mirrors, which can be elliptical or hyperbolic, aligned facing each other in parallel planes to modulate the beam in a single dimension. When a real image is formed, as shown in Fig. 1(a), the system is equivalent to the previously proposed one [21] that is very similar to the so-called Wolter type IV system [28]. Here, we term this system a Wolter IV-like system because of an imperfection compared with the original type IV system, as described later. Here, we propose an extended system corresponding to the intermediate focus beyond the mirror and forming an imaginary image. To be precise, the imaginary image's position divides the modes of operation for the system into two groups. When the imaginary image is located near the mirror, we can design systems similar to Wolter type II and III [Figs. 1(b) and 1(c)]; here, we term them as Wolter II-like and Wolter III-like, respectively.
Moreover, when the imaginary image is located at a greater distance from the mirror, yet-unnamed systems, comprising a combination of concave-hyperbola and concave-ellipse surfaces, can be generated. For convenience, we term them as Wolter II′-like and III′-like systems. Thus, five types of systems can be realized depending on the position of the intermediate focus. In particular, when we use focusing optics, the four types—IV-like, II-like, III-like, and II′-like, shown in Figs. 1(a)–1(d), respectively—are achievable practically, and a high degree of beam-size variability can be gained by interchanging between them.
In this optical system, the coma aberration is almost entirely corrected only when the light source, intermediate focus, and final focus are precisely aligned. Under these special conditions, the concave-convex combinations are identical to those of the Wolter type II and Wolter type III systems, and the concave-concave combinations, forming a real image, are identical to those of the Wolter type IV system. The coma aberration appears gradually as the intermediate focal point deviates from the axis connecting the source and final focus. However, the coma aberration does not become more significant than that of KB mirrors if practical systems are designed. A coma-free condition does not exist, however, for the Wolter II′-like and III′-like systems. If the coma aberration in a system is almost entirely corrected, the system becomes insensitive to source position deviations from the optical axis, thereby alleviating the need for precise adjustments of the direction of the incident beam. However, the alignment of such a system can still be difficult because the relative position between the ellipse and hyperbola needs to be carefully aligned and fixed precisely [29].
There are unambiguous design guidelines for this optical system. First, we select and fix the reference positions on mirrors (e.g., the center of a mirror), mirror grazing-incidence angles, and the light source's positions and final focus. Next, we introduce the position of an intermediate focus as a variable. The intermediate focus must be located somewhere along the line between the two reference positions for the mirrors to ensure that the source's positions, mirrors, and final focus remain fixed. The appropriate quadratic curve for the shape of the mirror can be obtained using the geometric features of the ellipse and hyperbola, as follows:
The NA of the downstream mirror, i.e., the mirror closer to the final focus along the beam path, can be simply estimated from the overall demagnification, which is approximately given by the parameters L1 and L2 under conditions in which vignetting does not occur:
Table 1 shows the detailed design parameters of the zoom condenser used in the experiments described in section 4. The compatible requirements of both XFEL focusing [30] and high-energy X-ray focusing [31] led to this design. In both cases, we need a small grazing-incidence angle and a long footprint. We found that a photon energy of 66 keV was acceptable for this design utilizing platinum coatings. Figure 2 shows the relationship between Lu-i, defined as the distance between the upstream mirror and the intermediate focus (Fi), and the FWHM of the beam width at the final focus, as calculated via wave-optics simulations based on the Fresnel–Kirchhoff integral (λ = 0.124 nm). A negative value of Lu-i indicates the Wolter III-like condition. In the figure, the vertical dashed lines represent singular points and borders between the types mentioned above. The left and center vertical dashed lines correspond to the conditions under which the intermediate focus is located on the upstream and downstream mirror surfaces, respectively. The gray-shaded area around Lu-i = 0 represents the region in which vignetting occurs on the downstream mirror. Under these conditions, demagnification cannot determine the beam size; instead, the maximum NA generated by the downstream mirror determines the beam size. Furthermore, this condition should be avoided for practical uses due to flux loss. The horizontal dashed line represents the condition in which the first mirror's rays are almost parallel because the intermediate focus is in the far-field. The points indicated by the arrows in the graph represent the system modes used in the following experiment (Wolter III-like and II-like types). Figure 3 and Table 2 show the corresponding mirror shapes, and their radius of curvature and slope, respectively. The Wolter type III-like mode (large NA mode) corresponds to the largest achievable NA, and under this condition, the downstream mirror is illuminated entirely. By contrast, in the Wolter type II-like mode (small NA mode), the smaller NA implies that approximately 1/9 of the downstream mirror's surface is illuminated.
Fig. 2. Relationship between Lu-i, the distance between the upstream mirror and the intermediate focus (Fi), and the FWHM beam width at the final focus.
Fig. 3. Target shapes for the upstream and downstream mirrors for the (a) large- and (b) small-NA modes.
Table 1. Design parameters
Table 2. Radius of curvature and slope of the mirrors
3. Ultra-precise deformable mirrors based on a mechanical bender and piezoelectric bimorph mirrors
The deformable mirrors used in the experiment are hybrid systems developed by our group; each consists of a piezoelectric bimorph mirror and a mechanical bender [32]. The mechanical bender is used to deform the mirror into low-spatial-frequency shapes, described by polynomial functions with a degree of up to three. The piezoelectric bimorph mirror is used to obtain shapes with higher-spatial-frequency components. The most significant advantage of this system is that the deformation of the piezoelectric elements can be minimized; thus, deformation drift, which occurs in proportion to the amount of deformation, can be significantly reduced. In addition, this system has the additional advantage that long deformable mirrors can be used. To fabricate long piezoelectric bimorph mirrors, piezoelectric plates with lengths greater than ∼400 mm are necessary for conventional piezoelectric bimorph mirrors. However, these cannot be fabricated as long monolithic plates, and hence, we must instead utilize multiple-segmented plates. Because there is no bending moment in the gap area between the segmented plates after bonding to the mirror substrate, non-negligible deformation errors occur; this is termed as the junction effect [33]. By contrast, in the hybrid system, this effect is greatly reduced because the bending moment is provided by the mechanical bender. As constraints on the manufacturing of piezoelectric plates are removed, it will become possible to fabricate very long deformable mirrors (>1 m in length).
Detailed information related to the hybrid deformable system was reported in our previous study [32]. The parameters are summarized as follows, and a 3D model of this system is presented in Fig. 4. The effective surface of the deformable mirror has a length of 400 mm, a width of 30 mm, and a thickness of 30 mm. Lead zirconate titanate (PZT) piezoelectric elements (length = 200 or 100 mm, width = 10 mm, thickness = 1 mm, piezoelectric coefficient = −1.35 × 10–10 m/V) were bonded to the sides of the substrate. The substrate material is ClearCeram-Z (Ohara Inc.), and the coating material on the reflective surface is molybdenum. The substrate surface, which was prepared via conventional polishing, has a root mean square (rms) micro-roughness of approximately 0.3 nm. The expected minimum deformable spatial wavelength is 40 mm, which is equivalent to 10 sinusoidal waves at the entrance pupil because 20 electrodes are placed over a 400-mm-area on the PZT plate. The two actuators (PI-Japan, P-841) that drive the mechanical bender are mounted on both sides of the substrate, and these provide different bending moments at both ends to create a mirror shape with a cubic function. According to finite element analysis, the hybrid system has the potential to be deformed into the target shape with an accuracy of at least 2 nm in peak-to-valley (PV).
Fig. 4. 3D model of a hybrid system consisting of piezoelectric bimorph mirror and mechanical bender. The entire system (left) and its interior (right). The upper-right schematic presents the side view of the bimorph mirror, indicating the electrodes, PZTs, and PZT plate gaps.
4. Experiment and results
A proof-of-concept experiment was performed with the optical setup in the third experimental hutch (EH3), placed ∼100 m downstream from an undulator of the SPring-8 BL29XUL beamline [34] (Fig. 5). X-rays with an energy of 10 keV, produced by a SPring-8 standard undulator, were monochromatized (ΔE/E ≈ 0.01%) with a Si 111 double crystal monochromator (DCM). A transport channel (TC) slit placed just downstream of the DCM (∼50 m downstream of the undulator) was used as a virtual source for the focusing system, in which the horizontal slit width was less than 10 µm. Consequently, the incident beam just before the mirrors was uniform across the target mirror area. The mirrors were arranged according to the design parameters listed in Table 1. Bending moments for the bender and voltages for the piezo-bimorph mirrors, determined in advance via off-line experiments using a Fizeau interferometer (VeriFire XPZ, Zygo Corp.), were applied, and the mirrors were deformed into the designed shapes.
Fig. 5. Schematic of the experimental setup. The image of the TC slit was demagnified using the deformable mirrors (DM1 and DM2) on the downstream plane coinciding with both the beam monitor (BM) and the wire. The ion chamber and photodiode (PD) were used for the wire scanning method. The values are in m.
Deformable mirrors must be used in combination with a wavefront measurement method. In this study, wavefront aberrations were measured by the X-ray pencil beam method [35,36], utilizing an incident slit mounted on a translation stage positioned just upstream of the mirrors and a beam monitor placed in the focal plane. This method can be understood as an experimental ray-tracing method that directly provides the local direction of the wavefront. The final deformation adjustment was made based on the inverse matrix method and the piezo response function with respect to the voltage [33]. Due to the hysteresis and deformation drift caused by the piezoelectric elements, this optimization process was repeated several times until the required accuracy was achieved. The deformation responses of the bender and the piezoelectric bimorph mirrors thus obtained agreed reasonably with the expected responses. In this experiment, only the upstream mirror was finely adjusted until the wavefront aberration was reduced to λ/4, based on the concept of wavefront compensation. Figure 6 shows the shape error for the upstream mirror, as determined via the X-ray pencil beam method before and after fine adjustment. Note that these are not real shape errors for the upstream mirror but represent the approximate sum of the shape errors for the two mirrors because both of them contribute to the measured wave-front aberration. Therefore, this procedure cannot uniquely identify the shape errors for each mirror. Finally, shape errors of 10.7 nm in PV (2.8 nm in rms) for the large-NA mode and 6.1 nm in PV (1.3 nm in rms) for the small-NA mode were successfully achieved; the shape error height (PV) tolerance corresponding to a wavefront accuracy of λ/4 for the upstream mirror was 21 nm. The achieved wavefront aberration was 0.81 rad in PV (0.21 rad in rms) for the large-NA mode and 0.46 rad in PV (0.099 rad in rms) for the small-NA mode, respectively. This can lead to a Strehl ratio exceeding 0.8.
Fig. 6. Shape errors for the upstream mirror, measured using the X-ray pencil beam method before and after fine adjustment for the (a) large NA and (b) small NA modes.
The focused intensity profile was measured at the focal plane using the dark-field wire scanning method [37] immediately after fine adjustment (Fig. 7). The solid line in the figure represents the best-fit diffraction-limited profile. The full widths at half maximum (FWHMs) of the beams were found to be 134 and 1010 nm; these values are almost the same as the expected diffraction-limited values (114 nm and 1055 nm, respectively). Thus, we confirmed that the proposed system could successfully function as expected. The slight mismatch between the observed and ideal values may have been a result of the difference between the actual NA produced and the designed value, and the measurement error of the wire scanning method, which may have resulted from scratches on the wire surface; vibrations of the wire and the mirrors could also have been a contributing factor. If it is assumed that there are no measurement errors in the wire scan, the possible NA error is 18% for the large-NA mode and 4% for the small-NA mode. In addition, there are relatively large side wings, as depicted in Fig. 7(a), for which the ratio of the peak area and integrated side wings was approximately 1:1. This could be attributed to the remaining middle spatial frequency shape errors, especially those smaller than the electrode period. Measuring and correcting the remaining shape errors at these middle spatial frequencies is difficult, because the spatial frequency exceeds one that can be corrected by the deformable mirrors. Moreover, pencil beam scans perform poorly in terms of sensing high spatial frequency shape errors.
Fig. 7. Beam profiles at the focus, measured using the dark-field wire scanning method for the (a) large NA and (b) small NA modes. These plots comprise all the results obtained via multiple wire scans.
5. Discussion and summary
5.1 Stability of deformation
In order to use an adaptive zoom condenser in practical experiments, determining the deformation stability is very important. Because our system included hybrid deformable mirror systems in which the piezoelectric plates produce deformation drift, significant suppression of this drift must be achieved. The deformation drift was estimated from the wavefront aberration acquired after 24 h of fine-tuning. Deformation drift up to a third-order polynomial was concluded to originate from the mechanical bender. This is because only the bender introduced shape deformation up to a third-order polynomial, whereas the shape deformation afforded by the piezoelectric bimorph mirrors did not include polynomials up to the third order. These facts strongly support that these deformation drifts were caused by the bender, especially the temperature change of the bender. This was excluded from the estimation because the problem can be overcome by improving the mechanical bender and temperature stabilization. To achieve this, we plan to change the materials used to construct the body of the bender. As shown in Fig. 8, the substantial average deformation drift speed, defined as the change in shape at each point divided by the elapsed time, is 0.2 nm/h (PV) and 0.07 nm/h (rms), and thus, very high deformation stability was achieved.
Fig. 8. Deformation drift speed estimated from wavefront aberration acquired after 24 h of fine-tuning. Deformation drift with polynomials up to the third order was excluded from this estimation.
5.2 Time required to change the optical system
In this experiment, each of the wavefront measurements and deformations was performed manually by a human under all the cases. Therefore, approximately half a day was required to modify the optical system, including both the coarse and fine adjustments. If the deformation adjustments were to be implemented automatically, without human intervention, it is expected that the required changes would be completed in less than 1 h. In addition, the X-ray pencil-beam method is significantly time-consuming because of the need to mechanically scan the slit. The combination of grating interferometry and an FFT method [38] would allow the determination of the shape error from a single fringe image. This would make it possible to loop through deformation-wavefront measurements very quickly, and thus, the time required for adjustment could be significantly reduced in the near future.
5.3 Summary and outlook
In this paper we proposed a novel adaptive zoom condenser system that can vary the NA and beam size by controlling the deformable mirrors via the position control of the intermediate virtual focus. A proof-of-concept experiment was performed using hybrid deformable mirrors developed in-house. The variable-NA (or variable-beam-size) focusing systems can provide versatile X-ray nano-beams without any change in the setup, which is compatible with the next-generation low-emittance synchrotron radiation sources [39] and XFELs. The techniques of adaptive control of optical systems and wavefronts are now being developed worldwide [40–42]. Continuous improvements will make them more user-friendly while providing high-performance in the near future. We believe that these new optics will introduce novel avenues for research in X-ray science.
Funding
Japan Society for the Promotion of Science (JP16H06358, JP17H01073, JP19J200094, JP20K21146); Adaptable and Seamless Technology Transfer Program through Target-Driven R and D (A-STEP) (AS2915035S).
Acknowledgments
We are grateful to Dr. Takumi Goto and Mr. Koki Matsumoto for their considerable support. We would like to acknowledge the TOYAMA Corporation for helping with the development of the mechanical bender system. The use of the BL29XUL beamline at SPring-8 was supported by RIKEN.
Disclosures
The authors declare no conflicts of interest.
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