Abstract

This study theoretically analyzes an increase in X-ray absorption by a grazing incidence mirror due to its surface roughness. We demonstrate that the increase in absorption can be several hundred times larger than predicted by the Nevot-Croce formula. As a result, absorption enhances by several times compared to a perfectly smooth mirror despite the extremely small grazing angle of an incident X-ray beam (a fraction of the critical angle of the total external reflection) and the high quality of the reflecting surface (the roughness height was 0.5 nm in modeling). The main contribution to the absorption increase was dictated by the mid-scale roughness (waviness) of the virgin substrate surface, whose quality thus defines an absorption enhancement. The approach was applied to the analysis of two real mirrors used in a synchrotron (BESSY-I) and a European X-ray free-electron laser (XFEL) beamline. The modern surface finishing technology of elastic emission machining provides extremely low substrate waviness, guaranteeing the negligible effect of the surface roughness on the absorption increase.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
    [Crossref]
  2. S. P. Hau-Riege, R. A. London, A. Graf, S. L. Baker, R. Soufli, R. Sobierajski, T. Burian, J. Chalupsky, L. Juha, J. Gaudin, J. Krzywinski, S. Moeller, M. Messerschmidt, J. Bozek, and C. Bostedt, “Interaction of short x-ray pulses with low-Z x-ray optics materials at the LCLS free-electron laser,” Opt. Express 18(23), 23933–23938 (2010).
    [Crossref] [PubMed]
  3. H. Yumoto, T. Koyama, S. Matsuyama, Y. Kohmura, K. Yamauchi, T. Ishikawa, and H. Ohashi, “Ellipsoidal mirror for two-dimensional 100-nm focusing in hard X-ray region,” Sci. Rep. 7(1), 16408 (2017).
    [Crossref] [PubMed]
  4. T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
    [Crossref] [PubMed]
  5. P. Beckmann and A. Spizzichino, The scattering of electromagnetic waves from rough surfaces (Pergamon, 1963).
  6. L. Névot and P. Croce, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates,” Rev. Phys. Appl. (Paris) 15(3), 761–779 (1980).
    [Crossref]
  7. M. Wen, I. V. Kozhevnikov, and Z. Wang, “Reflection of X-rays from a rough surface at extremely small grazing angles,” Opt. Express 23(19), 24220–24235 (2015).
    [Crossref] [PubMed]
  8. V. V. Yashchuk, L. Samoylova, and I. V. Kozhevnikov, “Specification of x-ray mirrors in terms of system performance: new twist to an old plot,” Opt. Eng. 54(2), 025108 (2015).
    [Crossref]
  9. A. -L. Barabǎsi and H. E. Stanley, Fractal concepts in surface growth (Syndicate of the University of Cambridge, 1995).
  10. The center for X-ray optics, http://henke.lbl.gov/optical_constants/getdb2/html .
  11. M. Störmer, F. Siewert, and H. Sinn, “Preparation and characterization of B4C coatings for advanced research light sources,” J. Synchrotron Radiat. 23(1), 50–58 (2016).
    [Crossref] [PubMed]
  12. L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Dynamic scaling of roughness at the early stage of tungsten film growth,” Phys. Rev. B 76(4), 045411 (2007).
    [Crossref]
  13. E. O. Filatova, L. Peverini, E. Ziegler, I. V. Kozhevnikov, P. Jonnard, and J.-M. André, “Evolution of surface morphology at the early stage of Al2O3 film growth on a rough substrate,” J. Phys. Condens. Matter 22(34), 345003 (2010).
    [Crossref] [PubMed]
  14. H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft X-ray monochromators,” Rev. Sci. Instrum. 66(1), 1–14 (1995).
    [Crossref]
  15. K. Boller, R.-P. Haelbich, H. Hogrefe, W. Jark, and C. Kunz, “Investigation of carbon contamination of mirror surfaces exposed to synchrotron radiation,” Nucl. Instrum. Methods A 208(1-3), 273–279 (1983).
    [Crossref]
  16. I. V. Kozhevnikov, A. V. Buzmakov, F. Siewert, K. Tiedtke, M. Störmer, L. Samoylova, and H. Sinn, “Growth of nano-dots on the grazing-incidence mirror surface under FEL irradiation,” J. Synchrotron Radiat. 23(1), 78–90 (2016).
    [Crossref] [PubMed]
  17. K. Yamauchi, H. Mimura, K. Inagaki, and Y. Mori, “Figuring with sub-nanometer-level accuracy by numerically controlled elastic emission machining,” Rev. Sci. Instrum. 73(11), 4028–4033 (2002).
    [Crossref]
  18. F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise X-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
    [Crossref] [PubMed]
  19. F. Siewert, T. Zeschke, T. Arnold, H. Paetzelt, and V. V. Yashchuk, “Linear chirped slope profile for spatial calibration in slope measuring deflectometry,” Rev. Sci. Instrum. 87(5), 051907 (2016).
    [Crossref] [PubMed]

2017 (1)

H. Yumoto, T. Koyama, S. Matsuyama, Y. Kohmura, K. Yamauchi, T. Ishikawa, and H. Ohashi, “Ellipsoidal mirror for two-dimensional 100-nm focusing in hard X-ray region,” Sci. Rep. 7(1), 16408 (2017).
[Crossref] [PubMed]

2016 (4)

T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
[Crossref] [PubMed]

F. Siewert, T. Zeschke, T. Arnold, H. Paetzelt, and V. V. Yashchuk, “Linear chirped slope profile for spatial calibration in slope measuring deflectometry,” Rev. Sci. Instrum. 87(5), 051907 (2016).
[Crossref] [PubMed]

M. Störmer, F. Siewert, and H. Sinn, “Preparation and characterization of B4C coatings for advanced research light sources,” J. Synchrotron Radiat. 23(1), 50–58 (2016).
[Crossref] [PubMed]

I. V. Kozhevnikov, A. V. Buzmakov, F. Siewert, K. Tiedtke, M. Störmer, L. Samoylova, and H. Sinn, “Growth of nano-dots on the grazing-incidence mirror surface under FEL irradiation,” J. Synchrotron Radiat. 23(1), 78–90 (2016).
[Crossref] [PubMed]

2015 (2)

V. V. Yashchuk, L. Samoylova, and I. V. Kozhevnikov, “Specification of x-ray mirrors in terms of system performance: new twist to an old plot,” Opt. Eng. 54(2), 025108 (2015).
[Crossref]

M. Wen, I. V. Kozhevnikov, and Z. Wang, “Reflection of X-rays from a rough surface at extremely small grazing angles,” Opt. Express 23(19), 24220–24235 (2015).
[Crossref] [PubMed]

2014 (1)

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise X-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

2010 (2)

2009 (1)

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

2007 (1)

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Dynamic scaling of roughness at the early stage of tungsten film growth,” Phys. Rev. B 76(4), 045411 (2007).
[Crossref]

2002 (1)

K. Yamauchi, H. Mimura, K. Inagaki, and Y. Mori, “Figuring with sub-nanometer-level accuracy by numerically controlled elastic emission machining,” Rev. Sci. Instrum. 73(11), 4028–4033 (2002).
[Crossref]

1995 (1)

H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft X-ray monochromators,” Rev. Sci. Instrum. 66(1), 1–14 (1995).
[Crossref]

1983 (1)

K. Boller, R.-P. Haelbich, H. Hogrefe, W. Jark, and C. Kunz, “Investigation of carbon contamination of mirror surfaces exposed to synchrotron radiation,” Nucl. Instrum. Methods A 208(1-3), 273–279 (1983).
[Crossref]

1980 (1)

L. Névot and P. Croce, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates,” Rev. Phys. Appl. (Paris) 15(3), 761–779 (1980).
[Crossref]

Altapova, V.

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

André, J.-M.

E. O. Filatova, L. Peverini, E. Ziegler, I. V. Kozhevnikov, P. Jonnard, and J.-M. André, “Evolution of surface morphology at the early stage of Al2O3 film growth on a rough substrate,” J. Phys. Condens. Matter 22(34), 345003 (2010).
[Crossref] [PubMed]

Arnold, T.

F. Siewert, T. Zeschke, T. Arnold, H. Paetzelt, and V. V. Yashchuk, “Linear chirped slope profile for spatial calibration in slope measuring deflectometry,” Rev. Sci. Instrum. 87(5), 051907 (2016).
[Crossref] [PubMed]

Baker, S. L.

Bigault, T.

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Dynamic scaling of roughness at the early stage of tungsten film growth,” Phys. Rev. B 76(4), 045411 (2007).
[Crossref]

Boller, K.

K. Boller, R.-P. Haelbich, H. Hogrefe, W. Jark, and C. Kunz, “Investigation of carbon contamination of mirror surfaces exposed to synchrotron radiation,” Nucl. Instrum. Methods A 208(1-3), 273–279 (1983).
[Crossref]

Bostedt, C.

Bozek, J.

Buchheim, J.

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise X-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

Burian, T.

S. P. Hau-Riege, R. A. London, A. Graf, S. L. Baker, R. Soufli, R. Sobierajski, T. Burian, J. Chalupsky, L. Juha, J. Gaudin, J. Krzywinski, S. Moeller, M. Messerschmidt, J. Bozek, and C. Bostedt, “Interaction of short x-ray pulses with low-Z x-ray optics materials at the LCLS free-electron laser,” Opt. Express 18(23), 23933–23938 (2010).
[Crossref] [PubMed]

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Buzmakov, A. V.

I. V. Kozhevnikov, A. V. Buzmakov, F. Siewert, K. Tiedtke, M. Störmer, L. Samoylova, and H. Sinn, “Growth of nano-dots on the grazing-incidence mirror surface under FEL irradiation,” J. Synchrotron Radiat. 23(1), 78–90 (2016).
[Crossref] [PubMed]

Chalupsky, J.

Chalupský, J.

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Croce, P.

L. Névot and P. Croce, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates,” Rev. Phys. Appl. (Paris) 15(3), 761–779 (1980).
[Crossref]

Falkenberg, G.

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise X-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

Filatova, E. O.

E. O. Filatova, L. Peverini, E. Ziegler, I. V. Kozhevnikov, P. Jonnard, and J.-M. André, “Evolution of surface morphology at the early stage of Al2O3 film growth on a rough substrate,” J. Phys. Condens. Matter 22(34), 345003 (2010).
[Crossref] [PubMed]

Gaudin, J.

S. P. Hau-Riege, R. A. London, A. Graf, S. L. Baker, R. Soufli, R. Sobierajski, T. Burian, J. Chalupsky, L. Juha, J. Gaudin, J. Krzywinski, S. Moeller, M. Messerschmidt, J. Bozek, and C. Bostedt, “Interaction of short x-ray pulses with low-Z x-ray optics materials at the LCLS free-electron laser,” Opt. Express 18(23), 23933–23938 (2010).
[Crossref] [PubMed]

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Gleeson, A. J.

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Graf, A.

Gudat, W.

H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft X-ray monochromators,” Rev. Sci. Instrum. 66(1), 1–14 (1995).
[Crossref]

Haelbich, R.-P.

K. Boller, R.-P. Haelbich, H. Hogrefe, W. Jark, and C. Kunz, “Investigation of carbon contamination of mirror surfaces exposed to synchrotron radiation,” Nucl. Instrum. Methods A 208(1-3), 273–279 (1983).
[Crossref]

Hájková, V.

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Hau-Riege, S. P.

Hellwig, C.

H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft X-ray monochromators,” Rev. Sci. Instrum. 66(1), 1–14 (1995).
[Crossref]

Hogrefe, H.

K. Boller, R.-P. Haelbich, H. Hogrefe, W. Jark, and C. Kunz, “Investigation of carbon contamination of mirror surfaces exposed to synchrotron radiation,” Nucl. Instrum. Methods A 208(1-3), 273–279 (1983).
[Crossref]

Inagaki, K.

K. Yamauchi, H. Mimura, K. Inagaki, and Y. Mori, “Figuring with sub-nanometer-level accuracy by numerically controlled elastic emission machining,” Rev. Sci. Instrum. 73(11), 4028–4033 (2002).
[Crossref]

Inubushi, Y.

T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
[Crossref] [PubMed]

Ishikawa, T.

H. Yumoto, T. Koyama, S. Matsuyama, Y. Kohmura, K. Yamauchi, T. Ishikawa, and H. Ohashi, “Ellipsoidal mirror for two-dimensional 100-nm focusing in hard X-ray region,” Sci. Rep. 7(1), 16408 (2017).
[Crossref] [PubMed]

Jark, W.

K. Boller, R.-P. Haelbich, H. Hogrefe, W. Jark, and C. Kunz, “Investigation of carbon contamination of mirror surfaces exposed to synchrotron radiation,” Nucl. Instrum. Methods A 208(1-3), 273–279 (1983).
[Crossref]

Jonnard, P.

E. O. Filatova, L. Peverini, E. Ziegler, I. V. Kozhevnikov, P. Jonnard, and J.-M. André, “Evolution of surface morphology at the early stage of Al2O3 film growth on a rough substrate,” J. Phys. Condens. Matter 22(34), 345003 (2010).
[Crossref] [PubMed]

Juha, L.

S. P. Hau-Riege, R. A. London, A. Graf, S. L. Baker, R. Soufli, R. Sobierajski, T. Burian, J. Chalupsky, L. Juha, J. Gaudin, J. Krzywinski, S. Moeller, M. Messerschmidt, J. Bozek, and C. Bostedt, “Interaction of short x-ray pulses with low-Z x-ray optics materials at the LCLS free-electron laser,” Opt. Express 18(23), 23933–23938 (2010).
[Crossref] [PubMed]

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Jung, C.

H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft X-ray monochromators,” Rev. Sci. Instrum. 66(1), 1–14 (1995).
[Crossref]

Jurek, M.

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Katayama, T.

T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
[Crossref] [PubMed]

Kim, J.

T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
[Crossref] [PubMed]

Kohmura, Y.

H. Yumoto, T. Koyama, S. Matsuyama, Y. Kohmura, K. Yamauchi, T. Ishikawa, and H. Ohashi, “Ellipsoidal mirror for two-dimensional 100-nm focusing in hard X-ray region,” Sci. Rep. 7(1), 16408 (2017).
[Crossref] [PubMed]

Koyama, T.

H. Yumoto, T. Koyama, S. Matsuyama, Y. Kohmura, K. Yamauchi, T. Ishikawa, and H. Ohashi, “Ellipsoidal mirror for two-dimensional 100-nm focusing in hard X-ray region,” Sci. Rep. 7(1), 16408 (2017).
[Crossref] [PubMed]

T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
[Crossref] [PubMed]

Kozhevnikov, I.

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Dynamic scaling of roughness at the early stage of tungsten film growth,” Phys. Rev. B 76(4), 045411 (2007).
[Crossref]

Kozhevnikov, I. V.

I. V. Kozhevnikov, A. V. Buzmakov, F. Siewert, K. Tiedtke, M. Störmer, L. Samoylova, and H. Sinn, “Growth of nano-dots on the grazing-incidence mirror surface under FEL irradiation,” J. Synchrotron Radiat. 23(1), 78–90 (2016).
[Crossref] [PubMed]

V. V. Yashchuk, L. Samoylova, and I. V. Kozhevnikov, “Specification of x-ray mirrors in terms of system performance: new twist to an old plot,” Opt. Eng. 54(2), 025108 (2015).
[Crossref]

M. Wen, I. V. Kozhevnikov, and Z. Wang, “Reflection of X-rays from a rough surface at extremely small grazing angles,” Opt. Express 23(19), 24220–24235 (2015).
[Crossref] [PubMed]

E. O. Filatova, L. Peverini, E. Ziegler, I. V. Kozhevnikov, P. Jonnard, and J.-M. André, “Evolution of surface morphology at the early stage of Al2O3 film growth on a rough substrate,” J. Phys. Condens. Matter 22(34), 345003 (2010).
[Crossref] [PubMed]

Krzywinski, J.

Kunz, C.

K. Boller, R.-P. Haelbich, H. Hogrefe, W. Jark, and C. Kunz, “Investigation of carbon contamination of mirror surfaces exposed to synchrotron radiation,” Nucl. Instrum. Methods A 208(1-3), 273–279 (1983).
[Crossref]

London, R. A.

Matsuyama, S.

H. Yumoto, T. Koyama, S. Matsuyama, Y. Kohmura, K. Yamauchi, T. Ishikawa, and H. Ohashi, “Ellipsoidal mirror for two-dimensional 100-nm focusing in hard X-ray region,” Sci. Rep. 7(1), 16408 (2017).
[Crossref] [PubMed]

T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
[Crossref] [PubMed]

Messerschmidt, M.

Mimura, H.

K. Yamauchi, H. Mimura, K. Inagaki, and Y. Mori, “Figuring with sub-nanometer-level accuracy by numerically controlled elastic emission machining,” Rev. Sci. Instrum. 73(11), 4028–4033 (2002).
[Crossref]

Miura, T.

T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
[Crossref] [PubMed]

Moeller, S.

Mori, Y.

K. Yamauchi, H. Mimura, K. Inagaki, and Y. Mori, “Figuring with sub-nanometer-level accuracy by numerically controlled elastic emission machining,” Rev. Sci. Instrum. 73(11), 4028–4033 (2002).
[Crossref]

Névot, L.

L. Névot and P. Croce, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates,” Rev. Phys. Appl. (Paris) 15(3), 761–779 (1980).
[Crossref]

Ohashi, H.

H. Yumoto, T. Koyama, S. Matsuyama, Y. Kohmura, K. Yamauchi, T. Ishikawa, and H. Ohashi, “Ellipsoidal mirror for two-dimensional 100-nm focusing in hard X-ray region,” Sci. Rep. 7(1), 16408 (2017).
[Crossref] [PubMed]

T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
[Crossref] [PubMed]

Paetzelt, H.

F. Siewert, T. Zeschke, T. Arnold, H. Paetzelt, and V. V. Yashchuk, “Linear chirped slope profile for spatial calibration in slope measuring deflectometry,” Rev. Sci. Instrum. 87(5), 051907 (2016).
[Crossref] [PubMed]

Peatman, W. B.

H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft X-ray monochromators,” Rev. Sci. Instrum. 66(1), 1–14 (1995).
[Crossref]

Petersen, H.

H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft X-ray monochromators,” Rev. Sci. Instrum. 66(1), 1–14 (1995).
[Crossref]

Peverini, L.

E. O. Filatova, L. Peverini, E. Ziegler, I. V. Kozhevnikov, P. Jonnard, and J.-M. André, “Evolution of surface morphology at the early stage of Al2O3 film growth on a rough substrate,” J. Phys. Condens. Matter 22(34), 345003 (2010).
[Crossref] [PubMed]

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Dynamic scaling of roughness at the early stage of tungsten film growth,” Phys. Rev. B 76(4), 045411 (2007).
[Crossref]

Samoylova, L.

I. V. Kozhevnikov, A. V. Buzmakov, F. Siewert, K. Tiedtke, M. Störmer, L. Samoylova, and H. Sinn, “Growth of nano-dots on the grazing-incidence mirror surface under FEL irradiation,” J. Synchrotron Radiat. 23(1), 78–90 (2016).
[Crossref] [PubMed]

V. V. Yashchuk, L. Samoylova, and I. V. Kozhevnikov, “Specification of x-ray mirrors in terms of system performance: new twist to an old plot,” Opt. Eng. 54(2), 025108 (2015).
[Crossref]

Sankari, R.

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise X-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

Siewert, F.

F. Siewert, T. Zeschke, T. Arnold, H. Paetzelt, and V. V. Yashchuk, “Linear chirped slope profile for spatial calibration in slope measuring deflectometry,” Rev. Sci. Instrum. 87(5), 051907 (2016).
[Crossref] [PubMed]

I. V. Kozhevnikov, A. V. Buzmakov, F. Siewert, K. Tiedtke, M. Störmer, L. Samoylova, and H. Sinn, “Growth of nano-dots on the grazing-incidence mirror surface under FEL irradiation,” J. Synchrotron Radiat. 23(1), 78–90 (2016).
[Crossref] [PubMed]

M. Störmer, F. Siewert, and H. Sinn, “Preparation and characterization of B4C coatings for advanced research light sources,” J. Synchrotron Radiat. 23(1), 50–58 (2016).
[Crossref] [PubMed]

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise X-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

Sinn, H.

I. V. Kozhevnikov, A. V. Buzmakov, F. Siewert, K. Tiedtke, M. Störmer, L. Samoylova, and H. Sinn, “Growth of nano-dots on the grazing-incidence mirror surface under FEL irradiation,” J. Synchrotron Radiat. 23(1), 78–90 (2016).
[Crossref] [PubMed]

M. Störmer, F. Siewert, and H. Sinn, “Preparation and characterization of B4C coatings for advanced research light sources,” J. Synchrotron Radiat. 23(1), 50–58 (2016).
[Crossref] [PubMed]

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Sobierajski, R.

S. P. Hau-Riege, R. A. London, A. Graf, S. L. Baker, R. Soufli, R. Sobierajski, T. Burian, J. Chalupsky, L. Juha, J. Gaudin, J. Krzywinski, S. Moeller, M. Messerschmidt, J. Bozek, and C. Bostedt, “Interaction of short x-ray pulses with low-Z x-ray optics materials at the LCLS free-electron laser,” Opt. Express 18(23), 23933–23938 (2010).
[Crossref] [PubMed]

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Soufli, R.

Störmer, M.

M. Störmer, F. Siewert, and H. Sinn, “Preparation and characterization of B4C coatings for advanced research light sources,” J. Synchrotron Radiat. 23(1), 50–58 (2016).
[Crossref] [PubMed]

I. V. Kozhevnikov, A. V. Buzmakov, F. Siewert, K. Tiedtke, M. Störmer, L. Samoylova, and H. Sinn, “Growth of nano-dots on the grazing-incidence mirror surface under FEL irradiation,” J. Synchrotron Radiat. 23(1), 78–90 (2016).
[Crossref] [PubMed]

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise X-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Tiedtke, K.

I. V. Kozhevnikov, A. V. Buzmakov, F. Siewert, K. Tiedtke, M. Störmer, L. Samoylova, and H. Sinn, “Growth of nano-dots on the grazing-incidence mirror surface under FEL irradiation,” J. Synchrotron Radiat. 23(1), 78–90 (2016).
[Crossref] [PubMed]

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Togashi, T.

T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
[Crossref] [PubMed]

Toleikis, S.

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Tono, K.

T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
[Crossref] [PubMed]

Tschentscher, Th.

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Vyšín, L.

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Wabnitz, H.

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

Wang, Z.

Wen, M.

Yabashi, M.

T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
[Crossref] [PubMed]

Yamauchi, K.

H. Yumoto, T. Koyama, S. Matsuyama, Y. Kohmura, K. Yamauchi, T. Ishikawa, and H. Ohashi, “Ellipsoidal mirror for two-dimensional 100-nm focusing in hard X-ray region,” Sci. Rep. 7(1), 16408 (2017).
[Crossref] [PubMed]

T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
[Crossref] [PubMed]

K. Yamauchi, H. Mimura, K. Inagaki, and Y. Mori, “Figuring with sub-nanometer-level accuracy by numerically controlled elastic emission machining,” Rev. Sci. Instrum. 73(11), 4028–4033 (2002).
[Crossref]

Yashchuk, V. V.

F. Siewert, T. Zeschke, T. Arnold, H. Paetzelt, and V. V. Yashchuk, “Linear chirped slope profile for spatial calibration in slope measuring deflectometry,” Rev. Sci. Instrum. 87(5), 051907 (2016).
[Crossref] [PubMed]

V. V. Yashchuk, L. Samoylova, and I. V. Kozhevnikov, “Specification of x-ray mirrors in terms of system performance: new twist to an old plot,” Opt. Eng. 54(2), 025108 (2015).
[Crossref]

Yumoto, H.

H. Yumoto, T. Koyama, S. Matsuyama, Y. Kohmura, K. Yamauchi, T. Ishikawa, and H. Ohashi, “Ellipsoidal mirror for two-dimensional 100-nm focusing in hard X-ray region,” Sci. Rep. 7(1), 16408 (2017).
[Crossref] [PubMed]

T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
[Crossref] [PubMed]

Zeschke, T.

F. Siewert, T. Zeschke, T. Arnold, H. Paetzelt, and V. V. Yashchuk, “Linear chirped slope profile for spatial calibration in slope measuring deflectometry,” Rev. Sci. Instrum. 87(5), 051907 (2016).
[Crossref] [PubMed]

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise X-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

Ziegler, E.

E. O. Filatova, L. Peverini, E. Ziegler, I. V. Kozhevnikov, P. Jonnard, and J.-M. André, “Evolution of surface morphology at the early stage of Al2O3 film growth on a rough substrate,” J. Phys. Condens. Matter 22(34), 345003 (2010).
[Crossref] [PubMed]

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Dynamic scaling of roughness at the early stage of tungsten film growth,” Phys. Rev. B 76(4), 045411 (2007).
[Crossref]

Appl. Phys. Lett. (1)

J. Chalupský, V. Hájková, V. Altapova, T. Burian, A. J. Gleeson, L. Juha, M. Jurek, H. Sinn, M. Störmer, R. Sobierajski, K. Tiedtke, S. Toleikis, Th. Tschentscher, L. Vyšín, H. Wabnitz, and J. Gaudin, “Damage of amorphous carbon induced by soft x-ray femtosecond pulses above and below the critical angle,” Appl. Phys. Lett. 95(3), 031111 (2009).
[Crossref]

J. Phys. Condens. Matter (1)

E. O. Filatova, L. Peverini, E. Ziegler, I. V. Kozhevnikov, P. Jonnard, and J.-M. André, “Evolution of surface morphology at the early stage of Al2O3 film growth on a rough substrate,” J. Phys. Condens. Matter 22(34), 345003 (2010).
[Crossref] [PubMed]

J. Synchrotron Radiat. (3)

M. Störmer, F. Siewert, and H. Sinn, “Preparation and characterization of B4C coatings for advanced research light sources,” J. Synchrotron Radiat. 23(1), 50–58 (2016).
[Crossref] [PubMed]

I. V. Kozhevnikov, A. V. Buzmakov, F. Siewert, K. Tiedtke, M. Störmer, L. Samoylova, and H. Sinn, “Growth of nano-dots on the grazing-incidence mirror surface under FEL irradiation,” J. Synchrotron Radiat. 23(1), 78–90 (2016).
[Crossref] [PubMed]

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise X-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

Nucl. Instrum. Methods A (1)

K. Boller, R.-P. Haelbich, H. Hogrefe, W. Jark, and C. Kunz, “Investigation of carbon contamination of mirror surfaces exposed to synchrotron radiation,” Nucl. Instrum. Methods A 208(1-3), 273–279 (1983).
[Crossref]

Opt. Eng. (1)

V. V. Yashchuk, L. Samoylova, and I. V. Kozhevnikov, “Specification of x-ray mirrors in terms of system performance: new twist to an old plot,” Opt. Eng. 54(2), 025108 (2015).
[Crossref]

Opt. Express (2)

Phys. Rev. B (1)

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Dynamic scaling of roughness at the early stage of tungsten film growth,” Phys. Rev. B 76(4), 045411 (2007).
[Crossref]

Rev. Phys. Appl. (Paris) (1)

L. Névot and P. Croce, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates,” Rev. Phys. Appl. (Paris) 15(3), 761–779 (1980).
[Crossref]

Rev. Sci. Instrum. (4)

T. Koyama, H. Yumoto, T. Miura, K. Tono, T. Togashi, Y. Inubushi, T. Katayama, J. Kim, S. Matsuyama, M. Yabashi, K. Yamauchi, and H. Ohashi, “Damage threshold of coating materials on x-ray mirror for x-ray free electron laser,” Rev. Sci. Instrum. 87(5), 051801 (2016).
[Crossref] [PubMed]

H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft X-ray monochromators,” Rev. Sci. Instrum. 66(1), 1–14 (1995).
[Crossref]

F. Siewert, T. Zeschke, T. Arnold, H. Paetzelt, and V. V. Yashchuk, “Linear chirped slope profile for spatial calibration in slope measuring deflectometry,” Rev. Sci. Instrum. 87(5), 051907 (2016).
[Crossref] [PubMed]

K. Yamauchi, H. Mimura, K. Inagaki, and Y. Mori, “Figuring with sub-nanometer-level accuracy by numerically controlled elastic emission machining,” Rev. Sci. Instrum. 73(11), 4028–4033 (2002).
[Crossref]

Sci. Rep. (1)

H. Yumoto, T. Koyama, S. Matsuyama, Y. Kohmura, K. Yamauchi, T. Ishikawa, and H. Ohashi, “Ellipsoidal mirror for two-dimensional 100-nm focusing in hard X-ray region,” Sci. Rep. 7(1), 16408 (2017).
[Crossref] [PubMed]

Other (3)

P. Beckmann and A. Spizzichino, The scattering of electromagnetic waves from rough surfaces (Pergamon, 1963).

A. -L. Barabǎsi and H. E. Stanley, Fractal concepts in surface growth (Syndicate of the University of Cambridge, 1995).

The center for X-ray optics, http://henke.lbl.gov/optical_constants/getdb2/html .

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Figures (9)

Fig. 1
Fig. 1 (a) Correction to the specular reflectance and (b) the TIS in the vacuum of a B4C-coated mirror at the photon energy E = 3 keV and different grazing angles of an incident beam (from 1 to 15 mrad) versus the correlation length of roughness. The TIS and δR values were normalized to the TIS calculated using the DW factor (Eq. (2)). The model (6) of the PSD-function was used, assuming the fractal parameter to be equal to α = 0.8. The horizontal black dashed line (DW) corresponds to the TIS value calculated for the limiting cases of very large correlation lengths, while blue and red dashed lines (NC) in the left graph indicate correction to the reflectivity calculated using the Nevot-Croce formula for extremely small correlation length at θ0 = 15 or 1 mrad. The vertical dashed-dotted lines show the values of the correlation length, when the parameter μ0 = 1 for 15 mrad grazing incidence angles. Here and below, the dielectric constants of materials were taken from [10].
Fig. 2
Fig. 2 The relative increase in the absorptivity of B4C-coated mirrors versus the correlation length of roughness. Calculations were performed at the photon energy E = 3 keV and different grazing angles of the incident beam, varying from 1 to 15 mrad. The model of the PSD-function in Eq. (6) was used, assuming the rms roughness and the fractal parameter to be equal to σ = 0.5 nm and α = 0.8. The solid curves were calculated with the use of exact formulas from [7], while the dashed curves of the same colors were calculated using simple Eq. (7).
Fig. 3
Fig. 3 The dependence of the relative increase in the absorptivity versus the roughness correlation length of (a) C-coated and (b) B4C-coated mirrors operating at the 1 mrad grazing incidence angle and different photon energies. The model (6) of the PSD-function was used, assuming the rms roughness and the fractal parameter to be equal to σ = 0.5 nm and α = 0.8. The solid colored curves were calculated with the use of exact formulas from [7], while the black dotted curves – using simple Eq. (7).
Fig. 4
Fig. 4 The dependence of the relative increase in the absorptivity versus the roughness correlation length of C-coated mirror operated at the 1 mrad grazing incidence angle and the 10 keV photon energy, calculations were performed using different fractal parameter α. The model (6) of the PSD-function was used, assuming the rms roughness and the fractal parameter to be equal to σ = 0.5 nm and α = 0.8. The solid colored curves were calculated using exact formulas from [7], while the dashed and dotted black curves were calculated using simple Eq. (7).
Fig. 5
Fig. 5 The dependence of the relative absorption increases of a B4C-coated mirror vs the maximum spatial frequency of the calculated roughness spectrum. Calculations were performed for different photon energies as indicated in the graph. Model (6) of the PSD function was used, assuming the RMS roughness and the fractal parameter were equal to σ = 0.5 nm and α = 0.8, respectively. The grazing angle of the incident radiation was θ0 = 1 mrad and the correlation length of the roughness was ξ = 30 μm (a) or 5 μm (b). The vertical dotted line indicates the frequency ν = 1/ξ.
Fig. 6
Fig. 6 The measured 1D PSD functions of the two samples. S1 was fabricated using conventional superpolishing technology, while S2 was produced using elastic emission machining. The measurements were obtained using a long trace profilometer (LPT), a slope measuring profiler (SMP), a white light interferometer (WLI) at various magnifications, and an atomic force microscope (AFM). The black dashed curves are approximations of the experimental PSD functions by superposition of the models determined in Eq. (6).
Fig. 7
Fig. 7 The dependence of the relative increase in the absorption of S1 (a) and S2 (b) vs the maximum spatial frequency of the roughness spectrum. For S1, calculations were performed using an approximation of the experimental PSD function by a sum of the models (curve 1). The contribution of summands with correlation lengths of ξ2 = 500 μm, ξ3 = 20 μm, and ξ4 = 0.11 μm are indicated by dashed curves 2, 3, and 4. For S2, calculations were performed for experimental PSD function directly (curve 1) and for its approximation by a sum of the model functions (curve 2). The contribution of summands with correlation lengths of ξ3 = 25 μm and ξ4 = 0.35 μm are indicated by dashed curves 3 and 4. The samples were supposed to be covered by B4C coating.
Fig. 8
Fig. 8 Relative increase in X-ray absorption of S1 (a) and S2 (b) vs the photon energy at the grazing angle θ0 = 0.7θC. Calculations were performed for the experimental PSD function (curve 1) and for its approximation by a sum of the model functions (curve 2). Individual contributions of different summands are also shown by dashed curves 3 and 4. When calculating, the silicon substrates were supposed to be covered by B4C coatings.
Fig. 9
Fig. 9 Relative increase in X-ray absorption of S1 (a) and S2 (b) vs the photon energy at different grazing angles. The stars indicate the maximum value of the relative absorption increase. When calculating, the sample surface was supposed to be covered by B4C reflecting coating.

Tables (2)

Tables Icon

Table 1 The effect of the surface roughness on the absorptivity of a C-coated mirror operating at the 1 mrad grazing incidence angle

Tables Icon

Table 2 The effect of the surface roughness on the absorptivity of a B4C-coated mirror operating at the 1 mrad grazing incidence angle

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

R F ( θ 0 < < θ c ) 1 4 θ 0 Im ( 1 ε ) 1 / 2 1 θ 0 2 γ / δ 3 / 2
DW = ( 4 π σ sin θ 0 λ ) 2 ; R ( θ 0 ) = R F ( θ 0 ) ( 1 DW ) ; TIS ( θ 0 ) = R F ( θ 0 ) DW ; ξ
NC = ( 4 π σ λ ) 2 sin θ 0 Re ε cos 2 θ 0 ; R ( θ 0 ) = R F ( θ 0 ) ( 1 NC ) ; TIS ( θ 0 ) = 0 ; ξ 0
A = 1 R ( θ 0 ) = θ 0 2 γ δ 3 / 2 [ 1 + ( 2 π σ θ c λ ) 2 ] ; θ 0 < < θ c ; γ / δ < < 1 ; ξ 0
μ 0 = ξ sin 2 θ 0 2 λ and μ c = ξ ( 1 ε ) 2 λ
PSD 2 D ( ν ) = σ 2 ξ 2 α π ( 1 + ν 2 ξ 2 ) α + 1 ; PSD 1 D ( ν ) = 2 π Γ ( α + 1 / 2 ) Γ ( α ) σ 2 ξ ( 1 + ν 2 ξ 2 ) α + 1 / 2
δ R TIS R F = ( k σ θ c ) 2 sin θ 0 ( 2 γ δ 3 / 2 + k ξ π μ c F ( τ ) τ μ c ( τ + τ μ c ) 2 d τ ) ; Im ( μ c ) = 0
F ( τ ) = 2 π Γ ( α + 1 / 2 ) Γ ( α ) ( 1 + τ 2 ) α 1 / 2
δ R TIS R F = ( k σ θ c ) 2 sin θ 0 ( 2 γ δ 3 / 2 + k ξ G 1 ) ; G 1 = 1 4 π 0 F ( τ ) τ d τ ; μ 0 < < μ c < < 1
δ R TIS R F = ( k σ θ с ) 2 sin θ 0 [ 2 γ δ 3 / 2 + ( 4 π k ξ δ ) 2 α 1 δ G 2 ] ; μ 0 < < 1 , μ c > > 1 G 2 = 2 π Γ ( α + 1 / 2 ) Γ ( α ) 1 η 1 ( η + η 1 ) 2 d η η 1 + 2 α
ξ ~ λ δ ~ E ; and max ( A A F ) ~ ξ λ ~ E
θ 0 (in mrad) ( 18 24 ) / E (in keV), i .e ., θ 0 ( 0.6 0.8 ) θ С ( E )

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