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Abstract
In this work, quick imaging measurements at different X-ray penetration depths, including the total-reflection region and Bragg reflection angles, were used to characterize the X-ray multilayer inner structure. We then measured the interface replication factor, intrinsic layer growth, and layer inhomogeneity, as these medium-spatial-frequency structural parameters are worthy of attention for demanding X-ray imaging and focusing applications. We compared conventional analysis methods to the speckle scanning technique, as it provides important phase-contrast and dark-field information. The method proposed here remedies the limitations of conventional methods and affords the possibility of in-situ determination of complex buried layers, localized strain or defects, and other statistical anomalies.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
X-ray multilayers are capable of obtaining high photon flux for imaging experiments [1], large numeral apertures for diffraction-limited focusing [2], tailored spectral and/or polarization features [3], etc., in synchrotron light sources. In the past three decades, with the development of deposition and measurement techniques, the structural characterization methods of multilayers have been fully researched, and many multilayer pairs have approached their theoretical performance levels. It is worth noting that most of this previous research focused on the mutual relations between structural imperfections and reflectivity loss [4,5]. Limited by the effective spatial frequency, conventional measurements using grazing incidence X-ray reflectometry (GIXRR) or diffuse scattering [5], atomic force microscopy (AFM) [6], or transmission electron microscopy [7] were only able to determine high-spatial-frequency-localized structural deviations from the ideal multilayer structure. With the wide application of large-scale multilayers in the synchrotron radiation and space-observation fields, the lower spatial parts of imperfections are regarded as more important factors, especially for demanding imaging and diffraction-limited focusing applications. Normally, figure error and slope error are used to describe the deviation of a real surface relative to an ideal surface which removes the mirror shape in terms of height and slope. The appearance of apparent stripes resulting from mirror-figure errors was found in full-field imaging while using a multilayer monochromator [8], which degrades the imaging contrast and beam coherence. The medium-spatial-frequency figure errors in the multilayer Kirkpatrick-Baez (K-B) mirror were determined to be the main factor forming side lobes to broaden the diffraction-limited focusing spot [9]. In these situations, the medium spatial wavelength from 1 mm to 1 µm, in multilayers, is more worthy of study than high-spatial-frequency imperfections.
Recently, the cutting-edge polishing technology known as elastic emission machining (EEM) was reported to have realized figure correlation in spatial wavelengths longer than 0.3 mm [10]. Inserting an extra bimorph mirror upstream of the multilayers can partly compensate these wavefront errors in medium- and/or low-spatial-frequency ranges [2]. Accurate characterization is the basis of both polishing process optimization and phase compensation, but effective methods of determining the multilayer in this spatial frequency range are still lacking. Ex-situ metrologies, such as the long-trace profiler (LTP) [11], Fizeau interferometer [12], and nanometer optical component measuring machine [13] reach a high precision for surface measurement, but are still inaccessible in monitoring the real-time heat-load status and the deformation caused by gravity, mounting, or clamping. At-wavelength metrologies are still being developed, including pencil-beam scanning [14], Shack − Hartmann sensors [15], ptychography [16], grating-based techniques [17], speckle-based [18] techniques, and others. These methods are widely used to characterize surface figures to remedy some of the insufficiencies of ex-situ measurements. However, it is still challenging to clearly recognize multilayer inner structure features in medium-spatial-frequency regimes. A surface figure is partially replicated from the substrate and coatings [19], but normally the surface of a multilayer is easily influenced by relaxation, oxidation, and contamination [20], and is thereby very different from the underlying interfaces. Thus, it is not enough to simply deduce the inner structure based only on the surface-figure information. At-wavelength and some ex-situ metrologies can detect the information of inner structures, but cannot distinguish their interfacial wavy height profiles from complex interference signals through conventional single measurements.
As is well known, in the total-reflection region, the field strength drops off exponentially away from the surface [21], which is very different from the situation at larger grazing incident angles. This phenomenon (i.e., evanescence field) provides a simple method of distinguishing the structural information between inner films and surfaces. In this paper, a novel method based on the speckle scanning technique is demonstrated to extract the medium-spatial-frequency inner structural features of multilayers. By simple and rapid imaging measurements, roughness replication, intrinsic layer growth, and structure inhomogeneity can be evaluated quantitatively. Some preliminary attempts have been recorded in a conference article [22], but herein the samples, measurements, and calculations are significantly improved, corrected, and supplemented.
2. Samples and measurements
2.1 Sample preparation
Multilayers were deposited by direct-current magnetron sputtering on polished, flat 30 mm × 20 mm × 3 mm silicon substrates at room temperature. Samples S1 – S3 are Cr/C multilayers, S4 – S6 Cr/B4C multilayers, S7 – S9 Ni/Ti multilayers, and S10 – S12 W/Si multilayers. These multilayers were all important material pairs in applications using synchrotron radiation sources [23], spallation neutron sources [24] or X-ray telescope for astronomical observation [25]. According to the order of the index, each material pair was deposited with designed thickness ratios (thickness of scattering layer to periodic thickness) of Γ = 0.4, 0.5, and 0.6. The periodic thickness and the bi-layer number of each multilayer were designed to be D = 5 nm and N = 30, respectively. After deposition, all of the multilayers were characterized by grazing-incidence X-ray reflection measurements to determine the periodic thicknesses, thickness ratios, and root-mean-square (RMS) interfacial widths, which are listed in Table 1.
2.2 Experimental setup
The speckle scanning measurements were carried out at beamline BL15U1 of the Shanghai Synchrotron Radiation Facility. An X-ray energy of 10 keV (wavelength λ = 0.124 nm) was selected using a double-crystal monochromator. The setup of the measurements is shown in Fig. 1. Abrasive paper with an average pore size of 0.7 µm was located at L1 = 5953 mm from the secondary source. The choice of the small pore size is to facilitate comparison with the transverse coherence length and to reduce the signal disturbance between the sample and abrasive paper at a similar spatial frequency. The selection of a suitable pore size satisfied a range between the Nyquist sampling criterion and the transverse coherence length. In this range, normally smaller pore sizes have better resolution and avoid aliasing. The multilayers were mounted with horizontal deflection on the sample manipulator at L2 = 53 mm downstream of the abrasive paper. The grazing-incidence angles for the experiment were at the total reflection θ = 0.2° and the first Bragg angle of θB ∼0.7°, satisfying the Bragg Eq. (2) DsinθB = λ. The choice of two angles was made to obtain speckle images with high signal-to-noise ratios. The detection system, placed at L3 = 1633 mm behind the multilayers, was a microscope objective lens (Optique Peter) of 5× magnification coupled to a charge-coupled-device (CCD) camera (PCO 4000). The effective pixel size was p = 1.48 µm and the exposure time 3 s. During the scanning measurements, taking into account motor return difference and repeatability, the abrasive paper was driven at a high precision by a KOHZU motor with a step size of μ = 0.4 µm along the transverse X direction. One hundred and one speckle patterns were recorded during each scan.
Fig. 1. (a) Schematic of the speckle scanning experiment and (b) geometric consideration in reflection plane.
3. Theory and analysis
3.1 Speckle scanning technique
The X-ray speckle-based technique, an important metrology technique, has recently been developed. Compared to the ptychography technique, it does not require high-coherence conditions and has been extended to lab-based microfocus X-ray sources with small transverse coherence length using sandpaper with small pore sizes [26]. Compared to a grating-based technique, its experimental setup is very simple [27] and has been proved to be very effective for in situ determination of the figure errors of mirrors [18,21]. A stationary abrasive paper, instead of a conventional phase grating, plays the role of a random phase object to modulate the wavefront of incoming partial-coherent beams. The superposition of the interference of the weak scattered beams from the abrasive paper and the strong transmitted beams produces a speckle intensity distribution on the detection plane. The figure errors and thin-film-thickness fluctuations produce additional phase differences for the reflected beam from the ideal mirror shape. These phase differences distort the speckle pattern, mainly through deviations, at the corresponding positions.
The first derivative of the wavefront phase, that is, wavefront slope φ, can be extracted by tracking the aforementioned speckle-pattern distortions. The speckle-pattern distortions are related not only to the upstream optics, but also to the measured multilayer. A single measurement suffers from low spatial resolution depending on the pixel size of the detector and the sample-to-detector distance. By scanning the abrasive paper along the transverse X direction to the incident beams, the spatial resolution can be improved by analyzing the self-correlation coefficients in a stack of speckle patterns at equidistant positions of the abrasive paper [18,28]. The intensity profiles of any specific column in a stack of speckle patterns can build up a new image; that is to say, a column of data in the original single measurement can be expanded into an image with the width of pattern number by the scanning measurement. As can be seen in Fig. 2, the ith and jth columns (Fig. 2(a)) are expanded into two reconstructed images in Fig. 2(c) and (d), respectively.
Fig. 2. One speckle image recorded by the detector for a mirror at (a) the first Bragg angle and (b) total-reflection angle; the images are built up, respectively, from the (c) ith and (d) (j = i + 3)th columns of all 101 speckle patterns during the scan of abrasive paper.
Letting the position of the X-ray beams at the central mirror be 0, the mirror slope based on the geometrical relationship in Fig. 1(b) can be expressed to be the first derivative of the wavefront slope [18]:
3.2 Multilayer model
Owing to oxidation and contamination, a thin-film surface normally has figures that are different from the inner interfaces. Ex-situ metrologies are not able to obtain inner structural information. Previous studies based on in-situ metrologies did not distinguish between the inner interface profile information and the surface-figure information by the measurement at a single grazing-incidence angle. To detect the inner medium-spatial-frequency structural information accurately, more measurements are necessary to obtain sufficient information. Our ideas are based on the high sensitivity of the grazing-incidence angle on the penetration depth near the total-reflection region to extract the surface and inner interfacial profile information by angular change. Regarding multilayers working in the total-reflection region, X-rays as evanescent waves only penetrate at surface layers, and wavefront slope error mainly results from surface-figure error and upstream optics. When the grazing-incidence angle is several times the critical angle θc, X-rays can penetrate into all of the inner layers, so that the interface wavy-height fluctuations and the thickness inhomogeneity mainly modulate the wavefront slope. Based on these two different X-ray penetration models, the interference signal from the surface and inner structure can be distinguished by at least two measurements, respectively, in and beyond the total-reflection region. A schematic of the experiment is shown in Fig. 3. Since thickness uniformity for multilayer deposition technology satisfies a symmetrical distribution [31] and has a similar influence as the mirror shape on wavefront phase, it can be neglected for the medium-spatial-frequency range. Thus, in a kinematic approximation the wavefront phase error can be simplified to be:
Fig. 3. Through measurements of two different grazing-incidence angles in and beyond the total-reflection region, the surface and inner structure can be distinguished based on the multilayer model.
where, for the ith layer, hi = ηidi is the intrinsic roughness, di is the layer thickness, ηi is the growth rate of the intrinsic roughness, αi = exp(−ζidi) gives the average replication factor in a range of middle spatial frequencies and ζi is the diffusion parameter. For the medium-spatial-frequency regime, the replication factor, diffusion parameter, growth rate of intrinsic roughness, and layer thickness for each material are, respectively, defined as α1, α2, ζ1, ζ2, η1, η2, and d1 = ΓD, d2 = (1−Γ)D, respectively. Considering the exponential term (α1α2)N<<[2N(1−α1α2)]−1, the terms of the wavy-height functions in Eqs. (3) can be deduced to be:
3.3 Dark-field imaging
The speckle-based technique is able to retrieve multi-modal images, including absorption, phase-contrast, and dark-field information, at the same time. Phase-contrast images were used to determine figure errors and interfacial height profiles as described in Section 3.2. Dark-field images afford useful information on multilayers as well. The dark-field signal was proved to have relationships with the maximum cross-correlation coefficient C = max(Ii*Ij) by different expression forms [23,33], which is reduced by small-angle scattering from the sample.
For multilayers, the dark-field signal can be expressed as the product of the local fluctuations of interfacial scattering cross-section and inner electron density fluctuations [34]. The dark-field signal distorts the speckle patterns and thereby decreases the cross-correlation coefficient of speckles from adjacent positions. The interfacial scattering cross-section is approximately proportional to a structural factor [35] S(q)∝∬exp(q⊥2σ2(1−exp(-R/ξ)2H)) based on a fractal self-affine growth, where q⊥ denotes the wave vector along the normal direction of the multilayer surface, σ the local interfacial roughness, R the position vector, ξ the lateral coherence length, and H the Hurst exponent. Taking into account a lateral correlation length of tens to hundreds of nanometers, which is less than the medium spatial wavelength, the main contribution of scattering cross-section is the Debye-Waller factor exp(q⊥2σ2). The main influence of inner electron density is volume scattering, attributed to the random path lengths travelled by X-ray beams. The probability density function can be assumed to play the same role as the interfacial wavy-height function [36]. For the total-reflection region, the reduced maximum cross-correlation coefficient only relates to surface scattering, while at an angle much larger than the critical angle this loss results from not only all interfacial scattering but also from volume scattering. Compared to phase-contrast reconstruction, the dark-field signal provides complementary information on the layer density fluctuation.
4. Results and discussions
In the experiment, considering both high angular sensitivity and the abundant search range for the maximum correlation coefficient, (j−i) = 3 was chosen for data analysis. In this case, the theoretical angular resolution was better than 8 nrad. The minimum spatial resolution to the mirror length was approximately 60 µm. Figure 4 (a) shows a comparison of the slope errors and maximum cross-correlation coefficients measured at the total-reflection region and first Bragg angle for the Ni/Ti multilayer (S7). The slope errors are influenced by the beam deflecting, whereas the correlation coefficients reveal the inhomogeneity of the structure. The results reveal that there is a certain degree of correlation between the surface and inner films for slope error, but in the inner layers the slope-error curve has larger fluctuation. The slope errors and maximum cross-correlation coefficients are from their different sensitive domains, but they also present obvious correlations, especially for the data measured in the Bragg-angle region. The difference of the slope errors between the interfaces and the surface mainly result from the inclination error and the differences in curvature of the interfaces and surface. As can be seen in Fig. 4(b), by removing the tilt and defocus modes in the Zernike function [37], the residual equivalent figure errors for the surface and inner interfaces have a similar profile. Figure 5 presents a comparisons of the slope errors and figure errors at the mirror center measured online at 0.2°, the first Bragg angle of 0.7°, and the ex-situ slope error measured by a Fizeau interferometer (Zygo Dynafiz) for the Cr/C multilayer (S1). The surface slope errors are in good agreement with the ex situ data. The mismatch between the online data at 0.7° and the surface data proves the high sensitivity of the speckle scanning technique to determine the difference between the surface and inner figure.
Fig. 4. (a) Comparison of the slope errors at the first Bragg angle (solid circle), the total-reflection region of 0.2° (solid square), the maximum correlation coefficients at the first Bragg angle (hollow circle) and the total-reflection region of 0.2° (hollow square) for the Ni/Ti multilayer (S7); (b) Comparison of the equivalent figure errors after removing the tilt and defocus modes in the Zernike function at the first Bragg angle and the total-reflection region of 0.2°.
Fig. 5. (a) Residual slope errors measured by the speckle scanning technique and (b) the figure errors at 0.2° and at 0.7° and with a Fizeau interferometer for Cr/C multilayer (S1).
The implicit medium-spatial-frequency information, such as intrinsic roughness, replication factor, and figure error of substrate, can be obtained by solving Eqs. (3). Figure 6 presents the results of Cr/C multilayers. The details of intrinsic roughness and replication factor can be found in Table 1. Cr/C, Cr/B4C, and W/Si multilayers have similar intrinsic roughnesses, but it becomes very large for Ni/Ti multilayers. Cr-based multilayers have good interface replication capability, which increases as the spacer-layer thickness increases, whereas the tendency of the replication rate is the opposite for Ni/Ti and W/Si multilayers. Peak-to-Valley (PV) figure error for all of the substrates is ∼ 6 nm.
Fig. 6. Mean intrinsic roughness is a period; upstream wavefront slope error and figure error of substrate vs corresponding position on Cr/C multilayers.
Table 1. Structural parameters including periodic thickness D, thickness ratio Γ, and mean RMS interfacial width $\overline \sigma $ obtained by fitting GIXRR curves; mean intrinsic roughness h and average replication factor α1α2 obtained by phase-contrast shift and maximum cross-correlation coefficients, respectively, at the total-reflection region and first Bragg angle.
Figure 7 presents the maximum cross-correlation coefficients versus thickness ratio calculated at the first Bragg angle and total-reflection region. Table 1 presents the results and compares them with the structural parameters, including periodic thickness, thickness ratio, and mean RMS interfacial width obtained by fitting GIXRR curves based on Parratt’s recurrence formula [38], as shown in Fig. 8. The maximum cross-correlation coefficients in the total-reflection region reveal the local correlation of the surface wavy-height profile. Cr/C, Cr/B4C, and W/Si multilayers are shown to have a smoother or more homogeneous surface than Ni/Ti multilayers. The measurement at the first Bragg angle is slightly different and mirrors the interfacial wavy-height and density fluctuations. W/Si multilayers have large maximum cross-correlation coefficients that are always larger than 0.5. The results reveal that W/Si multilayers have very smooth or homogeneous interfaces and a dense structure. Cr/C and Cr/B4C multilayers have similar values that decrease with an increasing thickness ratio. In contrast, the maximum cross-correlation coefficients for Ni/Ti multilayers are worse. Ni/Ti multilayers clearly have larger interfacial roughness and more drastic oscillation (large standard deviation) of the dark-field signal may result from the inter-diffusion and alloy phases. It is clear that the variation change agrees with that of the replication factor shown in Table 1.
Fig. 7. Average maximum cross-correlation coefficients vs thickness ratio on multilayers calculated at first Bragg angle and total reflection.
Fig. 8. Grazing-incidence X-ray reflectivities and their fitting curves for multilayers with thickness ratio; Γ = 0.4.
GIXRR gives the results of mean interfacial widths, including the influences of roughness and interdiffusion. GIXRR results are more effective at a higher spatial frequency; nonetheless, the results agree on the tendency of dark-field signals. The ratios of the Debye-Waller factors based on GIXRR results for Cr/C, Cr/B4C, Ni/Ti, and W/Si multilayers are slightly different from those of the dark-field signal. Except for different spatial frequencies, the differences result from the surface oxidation and inter-diffusion features in four material pairs.
As a relatively new method, the speckle scanning technique has many advantages compared to conventional methods. GIXRR or diffuse scattering is the most typical and effective method of determining multilayer inner structures. However, the structure parameters determined by these methods cannot relate to the interfacial profiles along the lateral direction. The lateral information is statistical in the beam-footprint range and has to be expressed by an assumed model, such as a fractal model [35]. In strong contrast, the speckle scanning technique can obtain structural information in both the vertical and lateral directions. Other ex situ metrologies, such as AFM and LTP, can obtain accurate surface profiles. Stearn’s model [32] also builds a relationship between inner interfaces and surface profiles, but their known information is only limited to surfaces and is far less than that obtained by the speckle scanning technique. Frankly, the speckle scanning technique also has its limitations; that is, it is sensitive to figure and density fluctuations, but it cannot quantitatively obtain layer thickness and density information.
5. Summary
The speckle scanning imaging technique was developed to determine the inner structural features of Cr/C, Cr/B4C, Ni/Ti, and W/Si multilayers, which fill the gap in the medium-spatial-frequency regime. Compared to conventional methods, speckle scanning imaging is an intuitive method. Using a simplified multilayer model, the intrinsic roughness, replication factor, and figure error of a substrate are correlated with wavefront phase errors. Dark-field signals also afford useful information with which to estimate the fluctuation of inner layer density and interfacial roughness. Extracting these structural parameters can provide an evaluation method at medium-spatial frequencies to improve the application for multilayers in X-ray imaging and diffraction-limited focusing fields. In the future, more measurements for different angles are needed to solve more multilayer parameters and reduce the error of this method. Furthermore, a similar method is believed to have potential and be attractive for determining other complex lamellar and inclusion structures for various buried layers and localized strain or defects.
Funding
National Natural Science Foundation of China (11775295); Chinese Academy of Sciences (CAS) (Youth Innovation Promotion Funds).
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant No. 11775295) and the Funds from Youth Innovation Promotion Association, CAS. The authors would like to thank Hongchang Wang from the Diamond Synchrotron Radiation Facility for several illuminating discussions and Sebastien Berujon from the European Synchrotron Radiation Facility for detailed explanations. The authors are also grateful to Yu Zhu from Zygo China for the measurements with a Fizeau interferometer. We thank LetPub (www.letpub.com) for its linguistic assistance during the preparation of this manuscript.
References
1. Y. Wang, S. Narayanan, J. Liu, D. Shu, A. Mashayekhi, J. Qian, and J. Wang, “A sagittally focusing double-multilayer monochromator for ultrafast X-ray imaging applications,” J. Synchrotron Radiat. 14(1), 138–143 (2007). [CrossRef]
2. H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-X-ray focusing,” Nat. Phys. 6(2), 146 (2010). [CrossRef]
3. Z. Wang, H. Wang, J. Zhu, and Z. Zhang, “Broadband Mo/Si multilayer transmission phase retarders for the extreme ultraviolet,” Appl. Phys. Lett. 90(3), 031901 (2007). [CrossRef]
4. F. Eriksson, N. Ghafoor, F. Schäders, E. M. Gullikson, and J. Birch, “Interface engineering of short-period Ni/V multilayer X-ray mirrors,” Thin Solid Films 500(1-2), 84–95 (2006). [CrossRef]
5. P. Siffalovic, E. Majkova, L. Chitu, M. Jergel, S. Luby, J. Keckes, G. Maier, A. Timmann, S. V. Roth, T. Tsuru, T. Harada, M. Yamanoto, and U. Heinzmann, “Characterization of Mo/Si soft X-ray multilayer mirrors by grazing-incidence small-angle X-ray scattering,” Vacuum 84(1), 19–25 (2009). [CrossRef]
6. H. C. Barshilia and K. S. Rajam, “‘Characterization of Cu/Ni multilayer coatings by nanoindentation and atomic force microscopy,” Surf. Coat. Technol. 155(2-3), 195–202 (2002). [CrossRef]
7. H. Jiang, A. Michette, S. Pfauntsch, Z. Wang, and D. Li, “Determination of the evolution of layer thickness errors and interfacial imperfections in ultrathin sputtered Cr/C multilayers using high-resolution transmission electron microscopy,” Opt. Express 19(12), 11815–11824 (2011). [CrossRef]
8. A. Rack, T. Weitkamp, M. Riotte, D. Grigoriev, T. Rack, L. Helfen, T. Baumbach, R. Dietsch, T. Holz, M. Krämer, F. Siewert, M. Meduňa, P. Cloetens, and E. Ziegler, “Comparative study of multilayers used in monochromators for synchrotron-based coherent hard X-ray imaging,” J. Synchrotron Radiat. 17(4), 496–510 (2010). [CrossRef]
9. H. Yumoto, H. Mimura, S. Matsuyama, S. Handa, Y. Sano, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Development of scanning x-ray fluorescence microscope with spatial resolution of 30 nm using Kirkpatrick-Baez mirror optics,” Rev. Sci. Instrum. 77(6), 063712 (2006). [CrossRef]
10. H. Mimura, K. Yamauchi, K. Yamamura, A. Kubota, S. Matsuyama, Y. Sano, K. Ueno, K. Endo, Y. Nishino, K. Tamasaku, M. Yabashi, T. Ishikawa, and Y. Mori, “Image quality improvement in a hard X-ray projection microscope using total reflection mirror optics,” J. Synchrotron Radiat. 11(4), 343–346 (2004). [CrossRef]
11. Y. Senba, H. Kishimoto, H. Ohashi, H. Yumoto, T. Zeschke, F. Siewert, S. Goto, and T. Ishikawa, “Upgrade of long trace profiler for characterization of high-precision X-ray mirrors at SPring-8,” Nucl. Instrum. Methods Phys. Res., Sect. A 616(2-3), 237–240 (2010). [CrossRef]
12. X. Zhao, “Study on high accurate laser diode interferometer for on-machine surface profile measurement,” Research Report of the Faculty of Engineering Niigata University 55, 17–18 (2006).
13. F. Siewert, T. Noll, T. Schlegel, T. Zeschke, and H. Lammert, “The Nanometer Optical Component Measuring Machine: a new Sub-nm Topography Measuring Device for X-ray Optics at BESSY,” AIP Conf. Proc. 705, 847–850 (2004). [CrossRef]
14. O. Hignette, A. Preund, and E. Chinchio, “Incoherent X-ray mirror surface metrology,” SPIE 3157 (1997)
15. P. Mercère, M. Idir, T. Moreno, G. Cauchon, G. Dovillaire, X. Levecq, L. Couvet, S. Bucourt, and P. Zeitoun, “Automatic alignment of a Kirkpatrick-Baez active optic by use of a soft-x-ray Hartmann wavefront sensor,” Opt. Lett. 31(2), 199–201 (2006). [CrossRef]
16. C. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, and F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard X-ray optics,” Ultramicroscopy 110(4), 325–329 (2010). [CrossRef]
17. H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, “X-ray wavefront characterization using a rotating shearing interferometer technique,” Opt. Express 19(17), 16550–16559 (2011). [CrossRef]
18. S. Berujon, H. Wang, S. Alcock, and K. Sawhney, “At-wavelength metrology of hard X-ray mirror using near field speckle,” Opt. Express 22(6), 6438–6446 (2014). [CrossRef]
19. D. E. Savage, J. Kleiner, N. Schimke, Y.-H. Phang, T. Jankowski, J. Jacobs, R. Kariotis, and M. G. Lagally, “Determination of roughness correlations in multilayer films for x-ray mirrors,” J. Appl. Phys. 69(3), 1411–1424 (1991). [CrossRef]
20. J. Chen, E. Louis, C. J. Lee, H. Wormeester, R. Kunze, H. Schmidt, D. Schneider, R. Moors, W. van Schaik, M. Lubomska, and F. Bijkerk, “Detection and characterization of carbon contamination on EUV multilayer mirrors,” Opt. Express 17(19), 16969–16979 (2009). [CrossRef]
21. H. Wang, J. Sutter, and K. Sawhney, “Advanced in situ metrology for x-ray beam shaping with super precision,” Opt. Express 23(2), 1605–1614 (2015). [CrossRef]
22. H. Jiang, S. Yan, D. Liang, N. Tian, H. Wang, and A. Li, “X-ray multilayer mid-frequency characterizations using speckle scanning techniques,” Proc. SPIE 10385, Advances in Metrology for X-Ray and EUV Optics VII, 103850Q (2017).
23. H. Jiang, H. Wang, J. Zhu, C. Xue, J. Zhang, N. Tian, and A. Li, “Thickness-dependent structural characteristics for a sputtering-deposited chromium monolayer and Cr/C and Cr/Sc multilayers,” J. Synchrotron Radiat. 25(3), 785–792 (2018). [CrossRef]
24. H. Ju and B. J. Heuser, “Effect of hydrogen on Ni/Ti multilayer neutron monochromator performance,” Appl. Phys. Lett. 90(7), 073113 (2007). [CrossRef]
25. P. H. Mao, F. A. Harrison, D. L. Windt, and F. E. Christensen, “Optimization of graded multilayer designs for astronomical x-ray telescopes,” Appl. Opt. 38(22), 4766–4775 (1999). [CrossRef]
26. H. Wang, Y. kashyap, and K. Sawhney, “From synchrotron radiation to lab source: advanced speckle-based X-ray imaging using abrasive paper,” Sci. Rep. 6(1), 20476 (2016). [CrossRef]
27. Y. kashyap, H. Wang, and K. Sawhney, “Experimental comparison between speckle and grating-based imaging technique using synchrotron radiation X-rays,” Opt. Express 24(16), 18664–18673 (2016). [CrossRef]
28. T. Jach and M. J. Bedzyk, “Grazing-angle X-ray standing waves,” Phys. Rev. B 42(8), 5399–5402 (1990). [CrossRef]
29. B. Pan, K. Qian, H. Xie, and A. Asundi, “Topical review: two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009). [CrossRef]
30. S. Berujon and E. Ziegler, “Grating-based at-wavelength metrology of hard x-ray reflective optics,” Opt. Lett. 37(21), 4464–4466 (2012). [CrossRef]
31. C. Fu, C. Yang, L. Han, and H. Chen, “The thickness uniformity of films deposited by magnetron sputtering with rotation and revolution,” Surf. Coat. Technol. 200(12-13), 3687–3689 (2006). [CrossRef]
32. D. G. Stearns, “X-ray scattering from interfacial roughness in multilayer structures,” J. Appl. Phys. 71(9), 4286–4298 (1992). [CrossRef]
33. S. Berujon and E. Ziegler, “Near-field speckle-scanning-based x-ray imaging,” Phys. Rev. A 92(1), 013837 (2015). [CrossRef]
34. H. Wang, Y. kashyap, and K. Sawhney, “Quantitative X-ray dark-field and phase tomography using single directional speckle scanning technique,” Appl. Phys. Lett. 108(12), 124102 (2016). [CrossRef]
35. S. K. Sinka, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38(4), 2297–2311 (1988). [CrossRef]
36. J.W. Goodman, “Speckle Phenomena in Optics: Theory and Applications,” Roberts and Company, Greenwood Village USA (2007).
37. J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19(9), 1510–1518 (1980). [CrossRef]
38. L. G. Parratt, “Surface studies of solids by total reflection of X-rays,” Phys. Rev. 95(2), 359–369 (1954). [CrossRef]
