1. Introduction

Coma aberration, in which an off-axis point light source produces multiple images of the source on the focal plane with various magnifications, stems from the imperfection of imaging or focusing devices [1]. Because material refraction indices are less than 1 in the X-ray regime, X-ray mirrors commonly depend on grazing-incidence configurations, which inherently produce coma aberration [2,3]. Because light sources for scanning X-ray microscopy are not infinitesimally small, coma aberration can broaden the image of the source at the focus. Such an expanded geometrical focus size limits the capability of X-ray nanofocusing mirrors to decrease the focus size to the diffraction limit. In addition, coma aberration arises when paraxial rays strike the reflective surface at an incidence angle different from the designed value. X-ray focusing mirrors thus require precise positioning in terms of rotation errors [4,5]. This requirement causes a narrow field of view when grazing incidence mirrors are used for imaging.

The wavelength in the soft-X-ray region (0.5 to 4 nm) is typically 10 to 20 times longer than that in the hard-X-ray region (<0.1 nm). Recently developed soft-X-ray nanofocusing mirrors are thus required to achieve a large numerical aperture (NA) [6–12]. Such mirrors often employ the Wolter geometry [9–12], which approximately satisfies the Abbe sine condition by combining an ellipsoidal and a hyperboloidal mirror [13]. The satisfaction of this condition allows the Wolter geometry to remove the primary coma aberration, which increases in proportion to the size of the light source [2]. However, the surfaces of Wolter mirrors have to be ultrasmooth in the tangential and sagittal directions. This geometry also produces singular points at the ellipsoid-hyperboloid boundary, which makes surface production difficult on monolithic substrates [10] or in tube-like axisymmetric shapes [9,14–16]. To facilitate mirror fabrication, the Kirkpatrick-Baez (KB) geometry is often employed to divide a doubly curved surface into vertically and horizontally curved surfaces in a sequentially crossed layout [17]. Advanced KB mirrors leverage this approach to divide Wolter mirrors into a pair of vertical and horizontal tandem mirrors [18–20]. Such mirrors increase the number of ultraprecision component mirrors to up to four. In addition, paired tandem mirrors generally require long focal lengths for the upstream mirrors due to the length of the downstream mirrors. Although advanced KB mirrors achieve hard-X-ray nanofocusing, such a design may not be suitable for large-NA designs [21], which are required for soft-X-ray nanofocusing.

Considering the fabrication challenges, KB mirrors can be advantageous for achieving a large NA with relatively simple designs. They consist of vertical and horizontal elliptic-cylindrical mirrors, which have been ultraprecisely fabricated with small-NA designs [22]. However, such mirrors do not satisfy the Abbe sine condition. The aberration caused by grazing incidence mirrors can be well formulated using geometrical optics theories [2,3]. However, few studies have investigated methods for simultaneously increasing the NA and suppressing aberration, which is necessary for large-NA focusing devices. The geometrical optics theories used in previous studies are insufficient to derive a strategy for achieving a large NA in grazing incidence setups; that is, they cannot be used to determine whether the mirror length should be extended, as done previously to avoid high thermal loads [23,24], or whether the grazing angle should be increased. If the severity of coma aberration is reduced and thus the field of view is expanded, large-NA KB mirrors will be advantageous for nanofocusing soft-X-rays. Such mirrors are robust to rotation errors as an off-axis point light source forms relatively well-combined focus spots.

In the present study, the suppression of coma aberration generated by elliptic-cylindrical mirrors is theoretically considered in order to derive an effective strategy for achieving large-NA X-ray mirrors. The effectiveness of the strategy is examined using ray-tracing and wave propagation simulations. A short-focal-length strategy is then investigated using an ultracompact Kirkpatrick-Baez (ucKB) mirror [25,26], which reduces the mirror lengths for ultrashort focal lengths of less than 10 mm. The focusing performance of the ucKB mirror is demonstrated in a synchrotron radiation experiment.

2. Theoretical considerations

Elliptical mirrors are designed with the positions of the light source and focus at the foci of an ellipse, as shown in Fig. 1(a). If an arbitrary point on the mirror surface is expressed as $(x,y)$, the semi-major axis $a$ and semi-minor axis $b$ for the designed ellipse are determined by $p(x)$, $q(x)$, and $\theta (x)$, which are the incident path length, exit path length, and grazing incidence angle (pitch), respectively. In-plane rotation and rotation around the X-ray path are later considered as yaw and roll, respectively (see Fig. 2(a)). $p$, $q$, and $\theta$ can be defined at the mirror center as $p_{\mathrm {c}}$, $q_{\mathrm {c}}$, and $\theta _{\mathrm {c}}$, respectively. These parameters satisfy

 

Fig. 1. Schematic diagram of coma aberration caused by rotated elliptical mirrors. (a) Ideal condition where elliptic-cylindrical mirror is placed at correct grazing incidence angle $\theta$. The light source and focus are at the foci of the designed ellipse. The radius of curvature $R_{\mathrm {t}}$ can be calculated using circles locally fitted to the elliptical mirror surface. (b) Focus shift $\Delta q$ caused by elliptic-cylindrical mirror that is rotated by $\Delta \theta$ in direction of grazing incidence angle. The rotation error produces discrepancies between the paraxial rays and the designed incidence direction. The value of $\Delta q$ depends on the position of the X-ray reflection on the focusing mirror, resulting in coma aberration.

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$R_{\mathrm {t}}$, which monotonically decreases with increasing $x$ in Eq. (2), can also be expressed using $p(x)$ and $q(x)$ [27].

Equation (3) is satisfied in the tangential direction over the entire surface of an ideally positioned elliptic mirror.

When a rotation error $\Delta \theta$ is added to $\theta$, the shift in the focus position $\Delta q$ can be derived from Eq. (3):

To focus incident X-rays from the light source to a single point, $\Delta q (\approx \Delta \theta \partial q/\partial \theta )$ is required to be constant regardless of $x$, as shown in Fig. 1(b). If $x_{1}$ and $x_{2}$ are arbitrary $x$ coordinates on the mirror surface, the relative change in the respective exit path lengths $\Delta q_{1}$ and $\Delta q_{2}$ is

As $x_1$ and $x_2$ are arbitrary, $R_{\mathrm {t1}} - R_{\mathrm {t2}}$ is not always zero according to Eq. (2). Equation (6) indicates that $\Delta q$ depends on $x$, namely the location of the X-ray reflection on the mirror. Rotated elliptical mirrors thus produce coma aberration because the incident rays stray from the designed incidence direction. The discrepancy between the designed and actual incidence directions is also caused by off-axis incident X-rays that are emitted from finite-size light sources.

From Eq. (6), the variation of $\Delta q$ can be suppressed by reducing the change in $R_{\mathrm {t}}$ over the mirror surface. This reduction makes elliptical mirrors more cylindrical. As discussed later, this approach results in mirrors designed with a configuration that is slightly close to normal-incidence configurations. $R_{\mathrm {t}}$ with a small variation can be designed by making $|\mathrm {d}R_{\mathrm {t}}/\mathrm {d}x|$ close to zero. Equation (2) yields

Here, $0<b/a<1$ in Fig. 1. $|\mathrm {d}R_{\mathrm {t}}/\mathrm {d}x|\to 0$ when $b/a\to 1$ or $b/a\to 0$. For $b/a\to 1$, the mirror shape becomes cylindrical as $\theta _{\mathrm {c}}$ increases according to Eq. (1). As the light source point corresponds to the focus point for $b/a\to 1$, the mirror accepts the incident X-rays from the point light source in a normal-incidence configuration.

However, grazing incidence setups require $b/a\ll 1$ and $q/p\ll 1$, and thus $b/a\to 0$ is feasible. According to Eq. (1),

As grazing incidence mirrors require the focal length to be longer than half the mirror length ($q>0.5l_{\mathrm {m}}$), extremely short $q$ requires short mirrors. A short $q$ also increases $x$ over the mirror surface for a constant value of $a$. $R_{\mathrm {t}}$ monotonically decreases with increasing $x$ according to Eq. (2). Equation (3) becomes

Short-focal-length mirrors are robust to coma aberration. However, technical obstacles must be overcome to achieve highly curved surfaces for such mirrors. The value of $\theta$ for Ni-based total-refleciton coatings can reach up to 25 mrad for the soft-X-ray range of 300 eV to 1 keV [25]. The working distance (i.e., the distance between the downstream mirror end and the focal point) mostly determines the shortest focal length. A focal length of 2 mm, for example, results in a central tangential radius of curvature of 0.16 m.

3. Focusing simulation of ultracompact Kirkpatrick-Baez mirror

3.1 Simulation setup

The focusing performance of KB mirrors was numerically evaluated. The vertical and horizontal focus sizes were calculated using an in-house multicore simulation that combines ray tracing and wave propagation [28]. The simulation accepts rotation errors in the pitch, roll, and yaw directions for the vertically focusing mirror (VFM) and horizontally focusing mirror (HFM), as shown in Fig. 2(a). Strong diffraction in the soft-X-ray region can be evaluated by placing a grid at the downstream end of the VFM and then propagating the wavefield converted from the ray information at the grid. When a focusing mirror achieves a large NA, the reflected rays strongly converge. The conversion process removes the information about the directions of individual aberrated rays. A spherical grid was chosen to preserve the direction information about most of the reflected rays. The center of the spherical grid was set at the ray-tracing focus position shifted by the rotation error. The sphere radius was then optimized such that the spherical grid touched the downstream mirror end. Such optimization is not readily available for commonly used simulations. A simulation that combines ray tracing and wave propagation can delineate the focusing performance more precisely as the obtained results are defocus-free, unlike those reported in a previous study [29].

 

Fig. 2. Schematic diagram of beamline BL25SU-A with elliptic-cylindrical mirrors installed downstream in KB geometry. (a) Experimental setup. In the simulation described in Section 3, the light source (vertical and horizontal slits) was assumed to be located at a distance of 0 m. Ray tracing was performed from the light source to the focus to determine the minimum focus point. Wave propagation was then conducted using the ray intensity and phase on the spherical grid. The center of the spherical grid and the focal plane were set to the obtained minimum focus point. (b) Definition of focus size. The images show the intensity distributions at the focus with a VFM roll error of −10 mrad. The focus size in the ray-tracing simulation was evaluated using the maximum distance between rays in the vertical and horizontal directions. The focus size in the simulation that combined ray tracing and wave propagation was evaluated by fitting a two-dimensional Gaussian function to the calculated focus intensity distribution. The $X$- and $Y$-direction FWHM values are regarded as the horizontal and vertical focus sizes, respectively.

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Coma aberration can be evaluated using this simulation by expanding the size of the light sources with the mirrors correctly positioned or by rotating the mirrors with a single light source point. The ray-tracing simulation generated $8\times 10^6$ rays from the point light source. The number of rays was determined based on the time required for the conversion process from ray information to wavefield information. The spherical grid had around $3\times 10^3$ sampling points. More than 2000 rays were introduced at each sampling point to average the intensity and phase of rays that followed uncommon paths due to the rotation error. Rays were randomly generated so that the periodic ray arrival points on the grid surface would not produce diffraction. The distribution of the incident rays was fixed for all rotation errors. The rays were spread widely enough to illuminate all of the mirror surfaces regardless of the amount of rotation error. The optimal focus position was found by screening the focus size along the reflected X-ray path and then determining the minimum focus size, as routinely done in synchrotron radiation experiments. The updated focus position was then used in the wave propagation to determine the focal plane and generate the spherical grid. The wavefield converted from the ray information was propagated from the spherical grid to the updated focal plane based on the Fresnel-Kirchhoff diffraction formula [1].

Figure 2(b) shows examples of intensity distributions at the focus obtained using the ray-tracing and wave propagation simulations. The photon energy was set to 300 eV. For the focus sizes, the ray tracing and wave propagation employed the maximum distance between rays in the vertical and horizontal directions [29] and the full width at half maximum (FWHM) value of the Gaussian fitting function, respectively.

3.2 Short focal length versus small grazing angle

Equation (8) indicates that a small $q$ or $\theta$ is advantageous for reducing coma aberration. These two strategies were examined using the ray-tracing simulation. Coma aberration was produced using a finite-size light source with an elliptic-cylindrical mirror positioned at the correct grazing incidence angle. The minimum focus size was obtained at the designed focus. For simplicity, only one-dimensional focusing was considered. The total flight path ($a=p_{\mathrm {c}}+q_{\mathrm {c}}$), NA, and light source size were fixed at 21900 mm, 0.0149, and 180 µm, respectively. The total flight path and the light source size were chosen to allow evaluation at a synchrotron radiation facility (see Section 4). An NA of 0.0149 was calculated under the assumptions that the mirror length of an X-ray mirror is the same as the focal length and that the grazing angle achievable in the soft-X-ray region is 25 mrad [25]. Figure 3(a) shows that the focus size is generally larger than the size of the source image, which was calculated as the light source size multiplied by the demagnification factor. However, the focus size shrinks approximately in proportion to $q$ regardless of $\theta _{\mathrm {c}}$. Notably, the spread of the focus size from the theoretical values becomes larger as $\theta _{\mathrm {c}}$ decreases. This result stems from $l_{\mathrm {m}}$ being extended to make the NA constant (NA=0.0149). When the NA was fixed at the value obtained after only $\theta _{\mathrm {c}}$ was changed ($l_{\mathrm {m}}:q=1:1$), the focus size was approximately unchanged regardless of $\theta _{\mathrm {c}}$. Figure 3(b) shows the focus size normalized by the source image size. The normalized focus size remained approximately constant for $q$ values larger than 1.5 mm. Light source size variation in the range of 1 to 1000 µm did not affect the normalized focus size, indicating that the primary coma aberration is dominant. These results demonstrate that a short-focal-length strategy is advantageous for reducing coma aberration and nanofocusing soft X-rays.

 

Fig. 3. Simulation results for focus size calculated using ray-tracing simulation. Only one elliptic-cylinder mirror was placed in this simulation for simplicity. The mirror was placed at the correct grazing angle to evaluate coma aberration produced by a finite-size light source. Grazing incidence angles of 5, 15, and 25 mrad were chosen. The total flight path ($a=p_{\mathrm {c}}+q_{\mathrm {c}}$) was fixed at 21900 mm. (a) Focus size versus $q$ for two cases. In one case, $l_{\mathrm {m}}$ was adjusted to maintain an NA of 0.0149. In the other case, the NA was not maintained at 0.0149; as a result, the ratio of the mirror length to the focal length ($l_{\mathrm {m}}:q$) became approximately constant. The light source size was set to 180 µm. The size of the source image was calculated from the light source size and the demagnification factor (180 µ$\mathrm {m} \times q_{\mathrm {c}}/p_{\mathrm {c}}$). (b) Focus size normalized by source image size. The annotated values show the converged normalized focus size. The NA was fixed at 0.0149. The normalized focus size is not affected by the light source size for a size range of 1 to 1000 µm, indicating that the primary coma aberration determines the focus size.

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3.3 Rotation error robustness

To verify the effectiveness of short-focal-length mirrors, the rotation error tolerance of a ucKB mirror, shown in Fig. 2(a), was examined. Table 1 shows the design parameters for the constituent mirrors of the ucKB mirror, namely the VFM and the HFM. Compared to a large KB mirror, the ucKB mirror lengths are reduced to achieve short focal lengths. Both constitutive mirrors of the ucKB mirror have sub-meter radii in the tangential direction. These design parameters were chosen to allow evaluation at a synchrotron radiation facility (see Section 4).

Table 1. Design parameters for ultracompact and large elliptical mirrors in simulation.

View Table

Figure 4(a) shows that the focus size of the ucKB mirror is affected by rotation errors, especially that in the pitch direction. The spread of the geometrical focus size is proportional to the rotation error, which corresponds to the source size when magnified by the incident path length. When the geometrical focus size exceeds the diffraction limit, the focus size calculated from the simulation that combines ray tracing and wave propagation increases. For the VFM pitch, the spread of the vertical focus size stems from coma aberration caused by pitch error. The horizontal focus size also increases because rotation errors generate astigmatism (induced by discrepancies between the vertical and horizontal focus positions). The VFM has a shorter focal length than that of the HFM. Such a short focal length allows the VFM to robustly maintain the ideal focus size despite large rotation errors. The focal length of the ucKB mirror is 50 times shorter than that of the large KB mirror, even though these mirrors have the same NA. Figure 4(b) shows that the focus size of the ucKB mirror is nearly 50 times more tolerant against rotation errors compared to the large KB mirror, as expected from Fig. 3(a). A short-focal-length strategy is derived using Eqs. (6)–(8), which consider only the pitch errors. However, Fig. 4(b) shows that a short-focal-length strategy works well for rotation errors in all directions.

 

Fig. 4. Change in focus size caused by rotation errors added to pitch, roll, and yaw angles of VFM and HFM. The focus position was optimized based on the results of the ray-tracing simulation. RT+WP (ray tracing + wave propagation) shows the results of wave propagation using the ray information at the VFM downstream end. (a) Focus size spread caused by rotation errors. The $X$ and $Y$ coordinates are shown in Fig. 2. The vertical and horizontal minimum focus sizes are 119 and 336 nm, respectively. (b) Comparison of focus spread between ucKB and large KB mirrors. Only the vertical focus size is plotted.

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To precisely discuss the tolerance for rotation errors, the intensity distributions at the focus were examined using the simulation that combines ray tracing and wave propagation, as shown in Fig. 5. The intensity distributions indicate that the aberrated focus spot can be reasonably evaluated using the Gaussian function. The peak intensity decreases as the pitch and roll errors increase. The ucKB mirror is least affected by yaw errors. The Strehl ratio is defined as the ratio of the peak intensity of an aberrated point spread function to the maximum attainable intensity. According to Maréchal’s criterion, the Strehl ratio must be more than 80% to achieve nearly ideal performance for imaging optics [1,30]. Provided that this criterion also applies to focusing devices, the Strehl ratio determines the tolerance for rotation errors. The ucKB mirror can accept up to $\pm$0.8 and $\pm$7 mrad for the pitch and roll errors, respectively, of the VFM. It can also accept pitch and roll errors of approximately $-2$ to $+0.16$ and $-6.5$ to $+6.0$ mrad, respectively, for the HFM. According to Maréchal’s criterion, yaw errors do not significantly affect the focusing performance of the ucKB mirror. The same criterion was applied to determine the tolerance of the large KB mirror. The large KB mirror showed 50 times lower tolerance in all three rotation directions (for example, $\pm 0.016$, $\pm 0.14$, and $\pm 34$ mrad for the VFM pitch, roll, yaw errors, respectively).

 

Fig. 5. Simulation intensity distributions at focus with rotated component mirrors of ucKB mirrors. The focus position was optimized based on the results of the ray-tracing simulation. The aberrated intensity distributions are normalized by the ideal peak intensity, which was achieved with zero rotation errors (Strehl ratio). The annotated values in the intensity distribution maps show the rotation error (units: mrad).

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4. Synchrotron radiation experiments

The focusing performance of the ucKB mirror was experimentally evaluated at beamline BL25SU-A, SPring-8, Japan [31]. The beamline layout is shown in Fig. 2. The vertical slit opening size of 180 µm results in an energy resolution of $E/\Delta E=1300$ with the grating-mirror pair M21a-G3a. The photon energy was tuned to 300 eV.

The technical challenges to achieve a small tangential radius of curvature were overcome and paired sub-meter-radius mirrors were fabricated. The fabrication of the VFM is described in a previous study [32]. The same fabrication techniques were applied to the HFM. The figure errors for both mirrors are sufficiently small for diffraction-limited focus at 300-eV photon energy according to Rayleigh’s quarter wavelength rule (see Supplement 1 for the fabrication results). To precisely handle the paired small mirrors, a compact manipulator is necessary. However, manipulator designs become complicated if three rotational stages are assigned to each mirror with their rotational centers fixed at the mirror center. The roll errors are relative angle changes between the VFM and the HFM. Therefore, the rotational stage for the VFM roll can be skipped. The mirrors can be easily placed in a yaw direction within the tolerance described in Section 3. The manipulator holds five piezo-driven rotational stages, but only those for the VFM pitch, HFM pitch, and HFM roll were designed to have their rotational centers at their respective mirror centers [25].

The ucKB mirror was precisely mounted on the manipulator and adjusted using the Foucault knife-edge test. The focus size was then obtained using a knife-edge scanning method while the rotation errors were varied. The knife-edge position was fixed along the X-ray path direction (no focus optimization was done). Figure 6 shows the focus size and normalized peak intensity obtained using the ucKB mirror, whose components were rotated using the developed manipulator. The peak intensity is normalized by the maximum intensity obtained without any rotation errors. The minimum vertical and horizontal focus sizes were 141$\pm$5 and 403$\pm$6 nm, respectively; the vertical focus size was diffraction-limited (vertical and horizontal diffraction-limited FWHM sizes of 139 and 346 nm, respectively; see Supplement 1 for the focusing profiles). The effect of rotation errors on the ucKB mirror was larger than that shown in Fig. 4. This sensitivity to error resulted from the knife-edge position not being optimized to the minimum focus size; the expanded focus size stems from the defocus and coma aberration caused by a single rotated mirror. Nevertheless, the rotation error tolerance of the ucKB mirror was higher than that of the KB mirror in Fig. 4(b). According to previous studies [4,5,21,33,34], KB and advanced KB mirrors typically show rotation pitch error tolerances of 2 µrad and 0.05 to 0.7 mrad, respectively. Although the results of these previous studies were obtained for hard X-rays and cannot be directly compared with the present results, the ucKB mirror in the present study shows remarkable tolerance to rotation errors.

 

Fig. 6. Simulation and experiment results of focus size and intensity at focus point obtained using rotated ucKB mirrors. The manipulator used in the synchrotron radiation experiment had precise piezo-driven rotational stages for the VFM pitch, HFM pitch, and HFM roll. The focus size was determined using a knife-edge scanning method. The focal length was not optimized and the knife-edge position was constant in the X-ray path direction. The focus size and normalized peak intensity (Strehl ratio) were obtained through (a) vertical scanning with VFM pitch errors, (b) horizontal scanning with HFM pitch errors, (c) vertical scanning with HFM roll errors, and (d) horizontal scanning with HFM roll errors. The angle differences in (a) and (b) indicate that the VFM and HFM were positioned with respective pitch errors of 30 and −100 µrad.

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A simulation that combines ray tracing and wave propagation was performed to increase the reliability of the results. In this simulation, the focused wavefields were also propagated to the constant focal plane. The simulation did not consider the figure errors as they were negligible at this photon energy. The simulation results are in good agreement with the experimental results. A comparison between the simulation and experimental results indicates that the minimum focus size (maximum peak intensity) was achieved at slightly different angles. The differences were estimated to be 30 and −100 µrad for the VFM and HFM pitch directions, respectively. These differences originate from the limitation of the Foucault knife-edge test, which observes the response of the focus size. They are negligible considering the achieved focus size and the Strehl ratio shown in Fig. 6. The ucKB mirror was precisely aligned using the compact manipulator.

The ucKB mirror can be advantageous for soft-X-ray nanofocusing, which requires large-NA designs. After the photon energy was increased from 300 eV to 500 eV, the focusing profiles were obtained using a knife-edge scanning method. In Fig. 7, the focus size is slightly larger than the diffraction limit, but the vertical and horizontal FWHM focus sizes are 104$\pm$6 and 340$\pm$11 nm, respectively. The differences between the achieved focus size and the diffraction-limited focus size stem from the mirror figure errors. According to Rayleigh’s quarter wavelength rule, the acceptable maximum figure error for 500 eV is within 12.4 nm. The figure errors of the VFM and HFM were larger than this threshold. Such figure errors cause diffraction outside the main focus peak. The fabrication technique thus has to be improved to achieve diffraction-limited focusing of high-energy soft X-rays.

 

Fig. 7. Vertical and horizontal focusing profiles acquired using knife-edge scanning method. The photon energy was set to 500 eV. The FWHM focus sizes of the Gaussian-fit profiles were 104$\pm$6 and 340$\pm$11 nm for the VFM and the HFM, respectively.

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5. Discussion and conclusions

Although the Wolter geometry is often employed for large-NA soft-X-ray focusing devices, the KB geometry can be advantageous in terms of ease of fabrication and accessibility to large-NA designs. However, the KB geometry is based on grazing-incidence configurations, which inherently produce severe coma aberration for large sources and off-axis points. Previous studies formulated the behavior of rays reflected on such grazing incidence mirrors. However, such geometrical optics theories are insufficient to derive a strategy for simultaneously increasing NA and suppressing aberration. In the present study, coma aberration was described using the relationship between the incident and exit path lengths, the radius of curvature, and the grazing angle on the tangentially focusing mirror. The results show that a short focal length or small grazing angle is effective for large-NA elliptic-cylindrical mirrors. The results of ray tracing show that large-NA designs require a short focal length at a large grazing angle rather than a long mirror length at a small grazing angle. The ucKB mirror was examined using simulations that combined ray tracing and wave propagation. It was found that it has high tolerance for rotation errors ($\pm 0.8$, $\pm 7$, $\pm 0.16$, and $\pm 6$ mrad are acceptable for the VFM pitch, VFM roll, HFM pitch, and HFM roll, respectively). The yaw errors do not practically affect the focus size. The results of the simulations were confirmed in a synchrotron radiation experiment at beamline BL25SU-A, SPring-8, Japan. The ucKB mirror reduced the amount of aberration and robustly achieved diffraction-limited focusing at 300-eV photon energy. Knife-edge scanning was performed at a constant position in the present study; no focus optimization was done. To optimize the focus size, the beam waist can be determined using ptychography-based techniques [35]. The vertical and horizontal FWHM focus sizes were 104$\pm$6 and 340$\pm$11 nm, respectively, at 500-eV photon energy.

The ucKB mirror can achieve a high demagnification ratio ($p/q$), which can be a decisive factor for beamline design. A high magnification ratio allows for the reduction of the geometrical focus size and the total beamline length ($a=p+q$). Besides suppressing the primary coma aberration, the short-focal-length design can reduce the drift of the focus spot caused by the oscillation of the stages as such drift is often amplified in proportion to the focal length. The short mirror length required for a short-focal-length strategy limits the spatial acceptance of the ucKB mirror. However, with the large grazing angle allowed in the soft-X-ray region, millimeter-scale mirrors moderately accept X-rays and achieve focusing throughput superior or comparable to that for diffractive nanofocusing devices (i.e., zone plates) [36,37]. ucKB mirrors are an alternative to tube-like X-ray mirrors [38] or capillary optics [6] for simultaneously achieving high focus intensity and small focus size. The large-grazing-angle design can expose the mirror surface to a relatively high power density. We did not observe any significant thermal drift in the focus position in 48-hour experiments.

Unlike diffractive nanofocusing devices, X-ray mirrors are achromatic, allowing X-ray absorption spectroscopy to be performed with samples at a constant position regardless of photon energy. The combination of ucKB mirrors and X-ray techniques that can use polychromatic X-rays would allow the production of multicolor nanoprobes, which can enhance the total probe intensity or simultaneously scan a sample at multiple photon energies. ucKB mirrors are robust to rotation errors and can thus accept a slight mismatch in the position of several light sources and form a nanoscale focus spot at an almost constant position. Multicolor nanoprobes could be used for the soft-X-ray fluorescence technique, which is a photon-hungry method compatible with polychromatic X-rays but is mostly performed with monochromatic X-rays. ucKB mirrors, which are based on a short-focal-length strategy, can be advantageous for achromatic soft-X-ray nanofocusing.