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Wavefront measurement for a hard-X-ray nanobeam using single-grating interferometry based on the Talbot effect and the Fourier transform method was demonstrated in the 1-km-long beamline of SPring-8. 10 keV X-rays were one-dimensionally focused down to 32 nm using a total-reflection elliptical mirror. An intentionally distorted wavefront was generated using a deformable mirror placed just upstream of the focusing mirror. The wavefront measured by interferometry was cross-checked with the phase retrieval method using intensity profiles around the beam waist. Comparison of the obtained wavefront errors revealed that they are in good agreement with each other and with the wavefront error estimated from the shape of the deformable mirror at a ~0.5 rad level.
©2012 Optical Society of America
1. Introduction
X-ray-based analysis methods such as X-ray absorption, X-ray fluorescence, X-ray diffraction and X-ray photoelectron spectroscopy offer powerful advantages for the investigation of the structure, elemental distribution and chemical bonding state of materials and biological samples. To enhance the spatial resolution and sensitivity, the use of focusing devices such as mirrors, zone planes, refractive lenses and multilayer Laue lenses is required. With regard to the focus size available using hard-X-ray focusing devices, a zone plate [1], a Laue lens [2] and a refractive lens [3] attained focus size of 20, 16 and 47 nm, respectively. Also, a total-reflection mirror [4] and a multilayer mirror [5] achieved focus size of 25 and 7 nm, respectively.
However, imperfections of the focusing device itself and its misalignment affect the minimum achievable spot size. The imperfections disturb the wavefront of the focused beam. Such imperfections must therefore be minimized by improving the fabrication method as well as the alignment procedure. Also, compensation of the wavefront error in a focusing optical system is carried out to achieve the minimum spot size [5, 6]. Such systems control the wavefront using a phase compensator. For such systems, precise information on the wavefront of the focused beam is required. There are several methods of characterizing a wavefront, such as the use of Hartmann-Shack wavefront sensors [7], phase retrieval methods [8], ptychography methods [9, 10] and X-ray interferometers [11].
In this study, we focused on single-grating interferometry [12] based on the Talbot effect [13] and the Fourier transform method [14, 15] to measure the wavefront error of a hard-X-ray nanobeam. Wavefront sensing [16–21] and phase imaging [11, 12, 22–26] using gratings in the hard-X-ray region have already been reported. The purpose of this study is to demonstrate the performance of single-grating interferometry for characterizing the wavefront of a hard-X-ray nanobeam. This method of interferometry, which involves a single grating placed downstream of the focus and an area detector placed downstream of the grating, can provide phase information on wavefront errors by analyzing the obtained interferogram with the Fourier transform method. This interferometric method has many advantages. The experimental setup is very simple. In general, a second absorption grating is often placed just in front of the detector to easily visualize phase information with a moiré fringe. Our interferometer is not equipped with a second grating because the fabrication of an absorption grating with a high aspect ratio is not straightforward and a fringe pattern magnified by a focusing optical system can be directly detected without the moiré effect. The method also has the potential for precise measurement owing to the simple setup and principle. Moreover, rapid measurement is possible because it has no moving components during the measurement and all the required information can be acquired during a single exposure. This advantage makes the method compatible with hard-X-ray free-electron lasers (XFELs), which have extremely brilliant and ultrashort pulses, and have recently started operation in SACLA [27] and LCLS [28]. In experiments using a focused XFEL, single-shot wavefront measurements will be required because XFEL pulses exhibit shot-to-shot variation.
We performed one-dimensional wavefront measurements for a nanobeam with a full width at half maximum (FWHM) of 32 nm, which was one-dimensionally focused by a total-reflection focusing mirror, with the interferometer located in the 1-km-long beamline (BL29XUL) of SPring-8 [29]. For the demonstration, a wavefront was modulated using a deformable mirror [6] placed just upstream of the focusing mirror. The wavefront was measured using a π/2-phase grating placed downstream of the focal point. Furthermore, it was cross-checked with the phase retrieval method using intensity profiles around the beam waist [8]. We discuss the performance of the interferometer by comparing the two sets of wavefront data and that estimated from the shape of the deformable mirror.
2. Single-grating interferometry
A single-grating interferometer applied to a quasi-spherical X-ray wave is considered. To simplify the demonstration, a one-dimensional system is discussed here and in the experimental part. Figure 1 shows a schematic of the interferometer. When a periodic grating is illuminated by a quasi-spherical wave, a clear interferogram, which is a magnified image of the grating and the so-called Talbot self-image, appears behind the grating. By placing the grating at distance of R1 downstream of the focal point, the interferogram I(x) is obtained at a distance of R2 ( = R1 + z1) downstream of the focal point, and is approximately given by [30]
where p’ is called the Talbot order, d is the pitch of the grating, λ is the wavelength and M is the magnification defined by R2/R1. an is the nth Fourier coefficient of the complex transmission function of the grating. Here, ϕ is the wavefront error of the spherical wave along the direction of x, which is assumed to be the imperfection in the optical system.This equation indicates that the modulated fringe pattern includes information on the differential wavefront error. In this study, the modulation of the pattern is extracted by the Fourier transform method. The wavefront error is calculated by integrating the result and then by subtracting a least-square parabola.
3. Phase retrieval method
To cross-check the wavefront error obtained by the interferometer, the phase retrieval method using intensity profiles around the beam waist was employed. The wavefront error can be obtained from the beam waist profile of a focused beam by using an advanced phase retrieval algorithm based on nonlinear optimization and angular spectrum methods [31]. In this study, the following procedure was performed. First, multiple intensity profiles along the optical axis around the focal point were measured by the dark-field knife-edge scan method [32]. The phase information of the wavefront was reconstructed from the obtained beam waist profile using the algorithm. Then, the wavefront error on the exit pupil, which is defined as the pupil just behind the focusing device, was calculated by Fraunhofer diffraction theory. In the phase retrieval algorithm, the reconstructed phase information is optimized by nonlinear optimization to minimize the square error between the measured and reconstructed intensity profiles around the focal point. Thus, owing to the use of multiple experimental data (10-20 intensity profiles around the focus are typically used), this method is robust with respect to measurement errors. In contrast, the acquisition of multiple data requires a long time, which makes the method unsuitable for rapid measurements.
4. Experimental setup
To evaluate the performance of a one-dimensional single-grating interferometer for a hard-X-ray nanobeam, the experimental setup shown in Fig. 2 was constructed in the 1-km-long beamline (BL29XUL) of SPring-8. X-rays were generated by a standard undulator at SPring-8 and monochromatized at 10 keV by a double-crystal [Si(111)] monochromator (DCM). An upstream slit with a size of 0.1 × 1 mm2 (H × V) was installed as a virtual light source 950 m upstream of the focusing mirror. An X-ray focusing mirror, whose parameters are listed in Table 1 , was used to produce an X-ray nanobeam with a numerical aperture (NA) of 2.9 × 10−3. The mirror was figured on a quartz glass substrate by numerically controlled elastic emission machining (NC-EEM) [33] and covered with a 100-nm-thick platinum layer by magnetron sputtering. Figure 3 shows the residual figure error of the mirror as measured by a microstitching interferometer (MSI) [34] and a relative angle determinable stitching interferometer (RADSI) [35]. The error is ~1 nm, corresponding to λ/9, except the high spatial frequency figure error. A monomorph piezoelectric deformable mirror consisting of a substrate (silicon block), 18 piezoelectric actuators (lead zirconate titanate) and electrodes, the design parameters of which are listed in Table 1, was placed just upstream of the focusing mirror to intentionally disturb the wavefront to demonstrate the sensing of the wavefront. The details are described elsewhere [36]. To measure the deformation of the deformable mirror directly, a Fizeau-type optical interferometer was employed. The deformable mirror is controlled by a feedback system that adjusts the applied voltages on the basis of the measured deformation in order to deform the deformable mirror to the target shape and keep it constant. A tantalum Ronchi grating (2.5 μm pitch; NTT Advanced Technology Corporation) fabricated on a thin SiC membrane (~1 μm thickness) was used for the experiments. The thickness of the grating was designed to be 1.4 μm, considering the density decrease of 5% occurring in the fabrication, so that it would function as a π/2-phase grating at 10 keV. The grating was installed 26.1 mm downstream of the focal point. This distance corresponds to p’ = 0.5. The grating was aligned carefully so that it was orthogonal to the focused beam to minimize systematic errors of the interferometer. A single line of the grating was also used as a phase object to perform the dark-field knife-edge scan method. When carrying out the dark-field knife-edge scan method, the grating is moved near the focal point. In this method, intensity profiles can be precisely measured by scanning the phase object with a piezo nanopositioner (P-363, PI Japan Co., Ltd.) and detecting only diffractions from the top of the phase object using a scintillation detector (SP-10, OHYO KOKEN KOGYO Co., Ltd). The formed self-image was recorded using an X-ray zooming tube and a CCD camera (C5333, Hamamatsu Photonics) placed 0.8 m downstream of the focal point. The effective pixel size and field of view (FOV) are 2.2 μm and 2.2 × 2.2 mm2, respectively. In this case, the magnification is 30.7. All interferograms in this study were obtained by stitching multiple images together due to the narrow FOV of the imaging system.
Table 1. Design parameters of the focusing and deformable mirrors
5. Experimental results and discussion
Just before the main experiments, the alignments of the focusing mirror, particularly the incident glancing angle and focal length, were finely adjusted to minimize the beam size at the focal point. Figure 4 shows the obtained intensity profile at the focal point together with the calculated profile. Focusing with an FWHM of 32 nm was achieved. In contrast, the obtained focus size is not in good agreement with the ideal one. The discrepancy is due to the measurement error caused by the vibration and waviness of the side wall of the grating used as a knife edge. Waviness on the side wall with an amplitude of several nanometers seems to occur when the grating was fabricated by reactive ion etching.
Fig. 4 Measured and calculated intensity profiles at the focal point. The scanning pitch is 5 nm. The ideal profile with a 17 nm FWHM was calculated using the Fresnel-Kirchhoff integral.
First, an experiment to confirm the intrinsic wavefront error of the focusing system was performed with the deformable mirror set to a flat shape. Figure 5(a) shows the obtained interferogram. The gradient intensity distribution along the horizontal direction is derived from the distribution of reflectivity on the focusing mirror, which was caused by the distribution of the incident angle. The obtained fringe pattern has a visibility of ~80%. This demonstrates that the grating was successfully fabricated as expected and that the positions of the grating and detector are correct. Additionally, the wavefront error was cross-checked by the phase retrieval method. A beam waist profile was acquired by the dark-field knife-edge scan method (Fig. 6(a) , left). After the phase information was reconstructed, the wavefront error on the exit pupil of the focusing mirror was calculated. The beam waist profile calculated with the reconstructed phase information is compared with the experimental one to confirm the accuracy of the method (Fig. 6(a), right). They are in good agreement, meaning that the phase information was successfully reconstructed by the phase retrieval method. Figure 7 shows a comparison of the wavefront errors obtained by these ways. It was found that they are in good agreement and that the difference between the red and black lines is less than ~0.5 rad. Also, the intrinsic wavefront error of the focusing system appears to have only a comatic aberration caused by minor misalignment of the incident angle of the focusing mirror. This is reasonable considering the figure errors of the focusing mirror measured in advance. In addition, the small wavefront error of less than 1.6 rad ( = λ/4) strongly indicates that a diffraction-limited FWHM of 17 nm was achieved.
Fig. 5 Interferograms obtained (a) when the deformable mirror was set to a flat shape and (b) when the deformable mirror was deformed. The line profiles represent the intensity distribution at the center of the map. The horizontal axis represents the pixel position on the detector, corresponding to 2.2 μm/pixel. Vertical scale bar = 50 μm. Horizontal scale bar = 0.5 mm.
Fig. 6 Measured and recovered beam waist profiles (a) when the deformable mirror was set to a flat shape and (b) when the deformable mirror was deformed. The “measured” beam waist profiles were acquired by the dark-field knife-edge scan method. The “recovered” profiles were calculated when the phase information was reconstructed from the measured beam waist profile with the phase retrieval method. X-rays come from the left and travel to the right. The data were acquired with scanning pitches of 12 μm and 5 nm along the optical axis and vertical direction, respectively. Vertical scale bar = 100 nm. Horizontal scale bar = 50 μm.
Fig. 7 Comparison of the wavefront errors on the exit pupil obtained by single-grating interferometry and the phase retrieval method. The phase error was calculated by integrating the data extracted by the Fourier transform method and then by multiplying by the coefficient of ∆x/(p’∙d∙M), where ∆x is the pixel size of the used detector. Then, it was corrected by subtracting a least-square parabola. The horizontal axis indicates the position normalized by the pupil width. The major waviness seems to mean a comatic aberration. The minor waviness except the comatic aberration does not seem to be a phase error caused by the figure error of the focusing mirror because they are not in good agreement with the residual figure errors of the focusing mirror measured in advance.
Next, an intentionally distorted wavefront was generated by deforming the deformable mirror from a flat shape to a shape having two bumps with ~18 nm height over a 100 mm length at the center. The interferogram modulated by the wavefront error was acquired and analyzed (Fig. 5(b)). The beam waist profile was also measured, as described above for cross-checking (Fig. 6(b)). Furthermore, the wavefront error was simply estimated from the shape of the deformable mirror. The three wavefront errors are compared in Fig. 8 . They can be seen to be in good agreement at the ~0.5 rad level. This consistency demonstrates that all of the methods used to obtain the wavefront error were performed accurately. The accuracy of the interferometry method easily satisfies with the accuracy of ~1.6 rad ( = λ/4) required to achieve a nearly ideal focus size. The discrepancy with the other two wavefront errors seems to be introduced by minor measurement errors such as distortion of the grating, blurring of the area detector and stitching errors. The other two results also seem to include the measurement errors in acquiring the beam waist profile and vibrations due to no use of a vibration isolation table for the Fizeau interferometer. We concluded that the single-grating interferometer can be applied for precise wavefront sensing for a nanobeam with a large NA.
Fig. 8 Comparison of the wavefront errors on the exit pupil obtained by the three methods. The horizontal axis indicates the position normalized by the pupil width.
6. Summary and outlook
We constructed a single-grating interferometer with a π/2-phase grating to measure the wavefront of an X-ray nanobeam. The results obtained by analyzing the interferogram using the Fourier transform method were cross-checked with the wavefront error obtained by the phase retrieval method. It was found that the results were consistent with each other and also with the wavefront error intentionally introduced by the use of a deformable mirror. This demonstrates that the interferometer can measure the wavefront of a nanobeam very precisely. In addition, this method is less sensitive to the vibration and imperfection of the phase object than the knife-edge scan method. This is an important advantage for characterizing a nanobeam with a large NA.
For practical applications, a detector with a wider FOV should be used. In our preliminary tests, a performance equivalent to the above result was achieved using an X-ray imaging system (AA20MOD and ORCA-R2, Hamamatsu Photonics, pixel size: 6.5 μm/pixel, FOV: 8.7 × 6.7 mm2). Also, calibration of the interferometer will be required to improve the accuracy. In the development of extreme-ultraviolet (EUV) lithography, much research on EUV grating interferometers, including calibration techniques, has been carried out [35]. These calibration techniques are applicable to the wavefront measurement of a hard X-ray nanobeam.
We plan to apply the single-grating interferometer to phase compensation for a nanobeam and XFEL focusing. In the latter case, the method is highly advantageous, not only in that wavefront sensing of a single pulse is possible but also that the interferometer can withstand exposure to an extremely intense beam because the grating is placed at the off-focus position.
Acknowledgments
This research was supported by CREST from the Japan Science and Technology Agency, a Grant-in-Aid for Scientific Research (S) (23226004) and the Global COE Program “Center of Excellence for Atomically Controlled Fabrication Technology” from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The use of the 1-km-long beamline (BL29XUL SPring-8) was supported by RIKEN. The authors would like to acknowledge T. Kimura for providing experimental advice.
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