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

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References

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2018 (1)

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]

2016 (3)

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]

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]

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]

2015 (2)

S. Berujon and E. Ziegler, “Near-field speckle-scanning-based x-ray imaging,” Phys. Rev. A 92(1), 013837 (2015).
[Crossref]

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]

2014 (1)

2012 (1)

2011 (2)

2010 (4)

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]

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]

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]

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]

2009 (3)

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]

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]

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]

2007 (3)

H. Ju and B. J. Heuser, “Effect of hydrogen on Ni/Ti multilayer neutron monochromator performance,” Appl. Phys. Lett. 90(7), 073113 (2007).
[Crossref]

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]

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]

2006 (5)

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]

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).

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]

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]

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]

2004 (2)

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]

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]

2002 (1)

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]

1999 (1)

1992 (1)

D. G. Stearns, “X-ray scattering from interfacial roughness in multilayer structures,” J. Appl. Phys. 71(9), 4286–4298 (1992).
[Crossref]

1991 (1)

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]

1990 (1)

T. Jach and M. J. Bedzyk, “Grazing-angle X-ray standing waves,” Phys. Rev. B 42(8), 5399–5402 (1990).
[Crossref]

1988 (1)

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]

1980 (1)

1954 (1)

L. G. Parratt, “Surface studies of solids by total reflection of X-rays,” Phys. Rev. 95(2), 359–369 (1954).
[Crossref]

Alcock, S.

Asundi, A.

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]

Barshilia, H. C.

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]

Baumbach, T.

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]

Bedzyk, M. J.

T. Jach and M. J. Bedzyk, “Grazing-angle X-ray standing waves,” Phys. Rev. B 42(8), 5399–5402 (1990).
[Crossref]

Berujon, S.

Bijkerk, F.

Birch, J.

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]

Bucourt, S.

Bunk, O.

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]

Cauchon, G.

Chen, H.

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]

Chen, J.

Chinchio, E.

O. Hignette, A. Preund, and E. Chinchio, “Incoherent X-ray mirror surface metrology,” SPIE 3157 (1997)

Chitu, L.

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]

Christensen, F. E.

Cloetens, P.

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]

Couvet, L.

David, C.

Dierolf, M.

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]

Dietsch, R.

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]

Dovillaire, G.

Endo, K.

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]

Eriksson, F.

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]

Fu, C.

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]

Garoff, S.

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]

Ghafoor, N.

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]

Goodman, J.W.

J.W. Goodman, “Speckle Phenomena in Optics: Theory and Applications,” Roberts and Company, Greenwood Village USA (2007).

Goto, S.

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]

Grigoriev, D.

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]

Gullikson, E. M.

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]

Han, L.

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]

Handa, S.

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]

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]

Harada, T.

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Levecq, X.

Li, A.

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Liang, D.

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).

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Mashayekhi, A.

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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).
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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).
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Sano, Y.

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).
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Schneider, D.

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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).
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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).
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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]

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]

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]

Siffalovic, P.

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]

Silva, D. E.

Sinka, S. K.

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]

Sirota, E. B.

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]

Stanley, H. B.

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]

Stearns, D. G.

D. G. Stearns, “X-ray scattering from interfacial roughness in multilayer structures,” J. Appl. Phys. 71(9), 4286–4298 (1992).
[Crossref]

Sutter, J.

Tamasaku, K.

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]

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]

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]

Thibault, P.

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]

Tian, N.

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]

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).

Timmann, A.

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]

Tsuru, T.

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]

Ueno, K.

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]

van Schaik, W.

Vila-Comamala, J.

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]

Wang, H.

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]

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]

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]

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]

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]

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]

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]

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]

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).

Wang, J.

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]

Wang, J. Y.

Wang, Y.

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]

Wang, Z.

Weitkamp, T.

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]

Windt, D. L.

Wormeester, H.

Xie, H.

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]

Xue, C.

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]

Yabashi, M.

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]

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]

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]

Yamakawa, D.

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]

Yamamura, K.

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]

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]

Yamanoto, M.

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]

Yamauchi, K.

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]

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]

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]

Yan, S.

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).

Yang, C.

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]

Yokoyama, H.

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]

Yumoto, H.

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]

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]

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]

Zeitoun, P.

Zeschke, T.

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]

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]

Zhang, J.

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]

Zhang, Z.

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]

Zhao, X.

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).

Zhu, J.

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]

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]

Ziegler, E.

S. Berujon and E. Ziegler, “Near-field speckle-scanning-based x-ray imaging,” Phys. Rev. A 92(1), 013837 (2015).
[Crossref]

S. Berujon and E. Ziegler, “Grating-based at-wavelength metrology of hard x-ray reflective optics,” Opt. Lett. 37(21), 4464–4466 (2012).
[Crossref]

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]

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]

AIP Conf. Proc. (1)

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]

Appl. Opt. (2)

Appl. Phys. Lett. (3)

H. Ju and B. J. Heuser, “Effect of hydrogen on Ni/Ti multilayer neutron monochromator performance,” Appl. Phys. Lett. 90(7), 073113 (2007).
[Crossref]

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]

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]

J. Appl. Phys. (2)

D. G. Stearns, “X-ray scattering from interfacial roughness in multilayer structures,” J. Appl. Phys. 71(9), 4286–4298 (1992).
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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).
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J. Synchrotron Radiat. (4)

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]

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]

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]

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]

Meas. Sci. Technol. (1)

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]

Nat. Phys. (1)

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]

Nucl. Instrum. Methods Phys. Res., Sect. A (1)

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]

Opt. Express (6)

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L. G. Parratt, “Surface studies of solids by total reflection of X-rays,” Phys. Rev. 95(2), 359–369 (1954).
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Phys. Rev. A (1)

S. Berujon and E. Ziegler, “Near-field speckle-scanning-based x-ray imaging,” Phys. Rev. A 92(1), 013837 (2015).
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Phys. Rev. B (2)

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).
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Research Report of the Faculty of Engineering Niigata University (1)

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).

Rev. Sci. Instrum. (1)

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]

Sci. Rep. (1)

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]

Surf. Coat. Technol. (2)

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).
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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).
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Thin Solid Films (1)

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).
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Ultramicroscopy (1)

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]

Vacuum (1)

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]

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O. Hignette, A. Preund, and E. Chinchio, “Incoherent X-ray mirror surface metrology,” SPIE 3157 (1997)

J.W. Goodman, “Speckle Phenomena in Optics: Theory and Applications,” Roberts and Company, Greenwood Village USA (2007).

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).

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Figures (8)

Fig. 1.
Fig. 1. (a) Schematic of the speckle scanning experiment and (b) geometric consideration in reflection plane.
Fig. 2.
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.
Fig. 3.
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.
Fig. 4.
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.
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).
Fig. 6.
Fig. 6. Mean intrinsic roughness is a period; upstream wavefront slope error and figure error of substrate vs corresponding position on Cr/C multilayers.
Fig. 7.
Fig. 7. Average maximum cross-correlation coefficients vs thickness ratio on multilayers calculated at first Bragg angle and total reflection.
Fig. 8.
Fig. 8. Grazing-incidence X-ray reflectivities and their fitting curves for multilayers with thickness ratio; Γ = 0.4.

Tables (1)

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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.

Equations (5)

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d Ψ i d x = k φ i k 2 X i det x i L 3 y i ,
x i = m = 0 i ε i , j μ / ( j i ) .
{ Δ Ψ = Δ Ψ s , 2 N + Δ Ψ u = 2 k sin θ Δ f 2 N + Δ Ψ u  if  θ < θ c , Δ Ψ = 1 2 N i = 1 2 N Δ Ψ s , i + Δ Ψ u = 2 k sin θ 1 2 N i = 1 2 N Δ f i + Δ Ψ u  if  θ > θ c ,
Δ f i ( x ) = h i ( x ) + α i 1 Δ f i 1 ( x ) ,
{ Δ f 2 N ( 1 1 α 1 α 2 ) ( h 1 + h 2 ) , 1 2 N i = 1 2 N Δ f i 1 2 ( 1 α 1 α 2 ) ( h 1 + h 2 ) + 1 2 N ( 1 α 1 α 2 ) f 0 .

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