Abstract

Peculiarities of X-ray diffraction from a rough surface at an extremely small grazing angle of an incident beam are theoretically studied. The interrelation of four diffraction channels (coherent reflectance, coherent transmittance, diffuse scattering in vacuum, and scattering into the matter depth) is analyzed for different limiting cases (large and small correlation length of roughness and large and extremely small grazing angle of incident radiation). Both the Debye-Waller and the Nevot-Croce factors are demonstrated to describe improperly the features of X-ray diffraction at extremely small grazing angles. More appropriate simple analytic expressions for the specular reflectivity and total integrated scattering in vacuum are obtained instead. Transformation of one limiting diffraction regime into another one with variation in the correlation length of roughness is discussed.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Effect of the surface roughness on X-ray absorption by mirrors operating at extremely small grazing angles

Mingwu Wen, Igor V. Kozhevnikov, Frank Siewert, Aleksey V. Buzmakov, Chun Xie, Qiushi Huang, Zhanshan Wang, Liubov Samoylova, and Harald Sinn
Opt. Express 26(16) 21003-21018 (2018)

Interference suppression of light backscattering through oblique deposition of a layered reflecting coating: bi-layer on a substrate

Jinlong Zhang, Han Wu, Igor V. Kozhevnikov, Shuaikai Shi, Xinbin Cheng, and Zhanshan Wang
Opt. Express 27(11) 15262-15282 (2019)

References

  • View by:
  • |
  • |
  • |

  1. M. Tolan, X-Ray Scattering from Soft-Matter Thin Films (Springer, 1999).
  2. U. Pietsch, V. Holy, and T. Baumbach, High-Resolution X-ray Scattering from Thin Films and Multilayers (Springer, 2004).
  3. V. E. Asadchikov, I. V. Kozhevnikov, Y. S. Krivonosov, R. Mercier, T. H. Metzger, C. Morawe, and E. Ziegler, “Application of X-ray scattering technique to the study of supersmooth surfaces,” Nucl. Instrum. Methods Phys. Res. A 530(3), 575–595 (2004).
    [Crossref]
  4. L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Roughness conformity during tungsten film growth: An in situ synchrotron x-ray scattering study,” Phys. Rev. B 72(4), 045445 (2005).
    [Crossref]
  5. H. Sinn, M. Dommach, X. Dong, D. La Civita, L. Samoylova, R. Villaneuva, and F. Yang, “X-ray optics and beam transport,” The European XFEL Technical Design Report, Hamburg, 2012.
  6. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).
  7. L. Névot and P. Croce, “Caracterisation des surfaces par reflexion rasante de rayons X. Application a l’etude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15(3), 761–779 (1980).
    [Crossref]
  8. R. Pynn, “Neutron scattering by rough surfaces at grazing incidence,” Phys. Rev. B Condens. Matter 45(2), 602–612 (1992).
    [Crossref] [PubMed]
  9. D. K. G. de Boer, “X-ray reflection and transmission by rough surfaces,” Phys. Rev. B Condens. Matter 51(8), 5297–5305 (1995).
    [Crossref] [PubMed]
  10. N. Alehyane, M. Arbaoui, R. Barchewitz, J.-M. André, F. E. Christensen, A. Hornstrup, J. Palmari, M. Rasigni, R. Rivoira, and G. Rasigni, “Extreme UV and x-ray scattering measurements from a rough LiF crystal surface characterized by electron micrography,” Appl. Opt. 28(10), 1763–1772 (1989).
    [Crossref] [PubMed]
  11. B. Aschenbach, H. Brauninger, G. Hasinger, and J. Trumper, “Measurements of X-ray scattering from Wolter type telescopes and various flat zerodur mirrors,” Proc. SPIE 257, 223–229 (1981).
    [Crossref]
  12. A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, and I. G. Yakushkin, “Phenomenon of total external reflection of X-rays,” Sov. Phys. JETP 62, 1225–1229 (1985).
  13. A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, and A. G. Turyansky, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).
  14. A. Andronow and M. Leontowiez, “Zur theorie der molekularen Lichtzerstreuung an Flüssigkeitsoberflächen,” Z. Phys. 38(6-7), 485–501 (1926).
    [Crossref]
  15. I. V. Kozhevnikov and M. V. Pyatakhin, “Use of DWBA and perturbation theory in x-ray control of the surface roughness,” J. XRay Sci. Technol. 8, 253–275 (1998).
  16. I. V. Kozhevnikov, “General laws of X-ray reflection from rough surfaces. I. Optical theorem. Energy conservation law,” Crystallogr. Rep. 55(4), 539–545 (2010).
    [Crossref]
  17. S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B Condens. Matter 38(4), 2297–2311 (1988).
    [Crossref] [PubMed]
  18. M. Bass, ed., Handbook of Optics (McGraw-Hill, 1995).
  19. A.-L. Barabǎsi, H.E. Stanley, Fractal Concepts in Surface Growth (Syndicate of the University of Cambridge, 1995).
  20. Y. Yoneda, “Anomalous surface reflection of X-rays,” Phys. Rev. 131(5), 2010–2013 (1963).
    [Crossref]
  21. 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]
  22. 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]
  23. D. H. Bilderback and S. Hubbard, “X-ray mirror reflectivities from 3.8 to 50 keV (3.3 Ǻ to 0.25 Ǻ). Part I -Float glass,” Nucl. Instrum. Methods 195(1-2), 85–89 (1982).
    [Crossref]

2010 (2)

I. V. Kozhevnikov, “General laws of X-ray reflection from rough surfaces. I. Optical theorem. Energy conservation law,” Crystallogr. Rep. 55(4), 539–545 (2010).
[Crossref]

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]

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]

2005 (1)

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Roughness conformity during tungsten film growth: An in situ synchrotron x-ray scattering study,” Phys. Rev. B 72(4), 045445 (2005).
[Crossref]

2004 (1)

V. E. Asadchikov, I. V. Kozhevnikov, Y. S. Krivonosov, R. Mercier, T. H. Metzger, C. Morawe, and E. Ziegler, “Application of X-ray scattering technique to the study of supersmooth surfaces,” Nucl. Instrum. Methods Phys. Res. A 530(3), 575–595 (2004).
[Crossref]

1998 (1)

I. V. Kozhevnikov and M. V. Pyatakhin, “Use of DWBA and perturbation theory in x-ray control of the surface roughness,” J. XRay Sci. Technol. 8, 253–275 (1998).

1995 (1)

D. K. G. de Boer, “X-ray reflection and transmission by rough surfaces,” Phys. Rev. B Condens. Matter 51(8), 5297–5305 (1995).
[Crossref] [PubMed]

1992 (1)

R. Pynn, “Neutron scattering by rough surfaces at grazing incidence,” Phys. Rev. B Condens. Matter 45(2), 602–612 (1992).
[Crossref] [PubMed]

1989 (1)

1988 (2)

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, and A. G. Turyansky, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B Condens. Matter 38(4), 2297–2311 (1988).
[Crossref] [PubMed]

1985 (1)

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, and I. G. Yakushkin, “Phenomenon of total external reflection of X-rays,” Sov. Phys. JETP 62, 1225–1229 (1985).

1982 (1)

D. H. Bilderback and S. Hubbard, “X-ray mirror reflectivities from 3.8 to 50 keV (3.3 Ǻ to 0.25 Ǻ). Part I -Float glass,” Nucl. Instrum. Methods 195(1-2), 85–89 (1982).
[Crossref]

1981 (1)

B. Aschenbach, H. Brauninger, G. Hasinger, and J. Trumper, “Measurements of X-ray scattering from Wolter type telescopes and various flat zerodur mirrors,” Proc. SPIE 257, 223–229 (1981).
[Crossref]

1980 (1)

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

1963 (1)

Y. Yoneda, “Anomalous surface reflection of X-rays,” Phys. Rev. 131(5), 2010–2013 (1963).
[Crossref]

1926 (1)

A. Andronow and M. Leontowiez, “Zur theorie der molekularen Lichtzerstreuung an Flüssigkeitsoberflächen,” Z. Phys. 38(6-7), 485–501 (1926).
[Crossref]

Alehyane, N.

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]

N. Alehyane, M. Arbaoui, R. Barchewitz, J.-M. André, F. E. Christensen, A. Hornstrup, J. Palmari, M. Rasigni, R. Rivoira, and G. Rasigni, “Extreme UV and x-ray scattering measurements from a rough LiF crystal surface characterized by electron micrography,” Appl. Opt. 28(10), 1763–1772 (1989).
[Crossref] [PubMed]

Andronow, A.

A. Andronow and M. Leontowiez, “Zur theorie der molekularen Lichtzerstreuung an Flüssigkeitsoberflächen,” Z. Phys. 38(6-7), 485–501 (1926).
[Crossref]

Arbaoui, M.

Asadchikov, V. E.

V. E. Asadchikov, I. V. Kozhevnikov, Y. S. Krivonosov, R. Mercier, T. H. Metzger, C. Morawe, and E. Ziegler, “Application of X-ray scattering technique to the study of supersmooth surfaces,” Nucl. Instrum. Methods Phys. Res. A 530(3), 575–595 (2004).
[Crossref]

Aschenbach, B.

B. Aschenbach, H. Brauninger, G. Hasinger, and J. Trumper, “Measurements of X-ray scattering from Wolter type telescopes and various flat zerodur mirrors,” Proc. SPIE 257, 223–229 (1981).
[Crossref]

Barchewitz, R.

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]

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Roughness conformity during tungsten film growth: An in situ synchrotron x-ray scattering study,” Phys. Rev. B 72(4), 045445 (2005).
[Crossref]

Bilderback, D. H.

D. H. Bilderback and S. Hubbard, “X-ray mirror reflectivities from 3.8 to 50 keV (3.3 Ǻ to 0.25 Ǻ). Part I -Float glass,” Nucl. Instrum. Methods 195(1-2), 85–89 (1982).
[Crossref]

Brauninger, H.

B. Aschenbach, H. Brauninger, G. Hasinger, and J. Trumper, “Measurements of X-ray scattering from Wolter type telescopes and various flat zerodur mirrors,” Proc. SPIE 257, 223–229 (1981).
[Crossref]

Christensen, F. E.

Croce, P.

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

de Boer, D. K. G.

D. K. G. de Boer, “X-ray reflection and transmission by rough surfaces,” Phys. Rev. B Condens. Matter 51(8), 5297–5305 (1995).
[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]

Garoff, S.

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B Condens. Matter 38(4), 2297–2311 (1988).
[Crossref] [PubMed]

Hasinger, G.

B. Aschenbach, H. Brauninger, G. Hasinger, and J. Trumper, “Measurements of X-ray scattering from Wolter type telescopes and various flat zerodur mirrors,” Proc. SPIE 257, 223–229 (1981).
[Crossref]

Hornstrup, A.

Hubbard, S.

D. H. Bilderback and S. Hubbard, “X-ray mirror reflectivities from 3.8 to 50 keV (3.3 Ǻ to 0.25 Ǻ). Part I -Float glass,” Nucl. Instrum. Methods 195(1-2), 85–89 (1982).
[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]

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]

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Roughness conformity during tungsten film growth: An in situ synchrotron x-ray scattering study,” Phys. Rev. B 72(4), 045445 (2005).
[Crossref]

Kozhevnikov, I. V.

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]

I. V. Kozhevnikov, “General laws of X-ray reflection from rough surfaces. I. Optical theorem. Energy conservation law,” Crystallogr. Rep. 55(4), 539–545 (2010).
[Crossref]

V. E. Asadchikov, I. V. Kozhevnikov, Y. S. Krivonosov, R. Mercier, T. H. Metzger, C. Morawe, and E. Ziegler, “Application of X-ray scattering technique to the study of supersmooth surfaces,” Nucl. Instrum. Methods Phys. Res. A 530(3), 575–595 (2004).
[Crossref]

I. V. Kozhevnikov and M. V. Pyatakhin, “Use of DWBA and perturbation theory in x-ray control of the surface roughness,” J. XRay Sci. Technol. 8, 253–275 (1998).

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, and A. G. Turyansky, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, and I. G. Yakushkin, “Phenomenon of total external reflection of X-rays,” Sov. Phys. JETP 62, 1225–1229 (1985).

Krivonosov, Y. S.

V. E. Asadchikov, I. V. Kozhevnikov, Y. S. Krivonosov, R. Mercier, T. H. Metzger, C. Morawe, and E. Ziegler, “Application of X-ray scattering technique to the study of supersmooth surfaces,” Nucl. Instrum. Methods Phys. Res. A 530(3), 575–595 (2004).
[Crossref]

Leontowiez, M.

A. Andronow and M. Leontowiez, “Zur theorie der molekularen Lichtzerstreuung an Flüssigkeitsoberflächen,” Z. Phys. 38(6-7), 485–501 (1926).
[Crossref]

Mercier, R.

V. E. Asadchikov, I. V. Kozhevnikov, Y. S. Krivonosov, R. Mercier, T. H. Metzger, C. Morawe, and E. Ziegler, “Application of X-ray scattering technique to the study of supersmooth surfaces,” Nucl. Instrum. Methods Phys. Res. A 530(3), 575–595 (2004).
[Crossref]

Metzger, T. H.

V. E. Asadchikov, I. V. Kozhevnikov, Y. S. Krivonosov, R. Mercier, T. H. Metzger, C. Morawe, and E. Ziegler, “Application of X-ray scattering technique to the study of supersmooth surfaces,” Nucl. Instrum. Methods Phys. Res. A 530(3), 575–595 (2004).
[Crossref]

Morawe, C.

V. E. Asadchikov, I. V. Kozhevnikov, Y. S. Krivonosov, R. Mercier, T. H. Metzger, C. Morawe, and E. Ziegler, “Application of X-ray scattering technique to the study of supersmooth surfaces,” Nucl. Instrum. Methods Phys. Res. A 530(3), 575–595 (2004).
[Crossref]

Névot, L.

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

Palmari, J.

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]

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Roughness conformity during tungsten film growth: An in situ synchrotron x-ray scattering study,” Phys. Rev. B 72(4), 045445 (2005).
[Crossref]

Pyatakhin, M. V.

I. V. Kozhevnikov and M. V. Pyatakhin, “Use of DWBA and perturbation theory in x-ray control of the surface roughness,” J. XRay Sci. Technol. 8, 253–275 (1998).

Pynn, R.

R. Pynn, “Neutron scattering by rough surfaces at grazing incidence,” Phys. Rev. B Condens. Matter 45(2), 602–612 (1992).
[Crossref] [PubMed]

Rasigni, G.

Rasigni, M.

Rivoira, R.

Sagitov, S. I.

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, and A. G. Turyansky, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).

Sinha, S. K.

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B Condens. Matter 38(4), 2297–2311 (1988).
[Crossref] [PubMed]

Sirota, E. B.

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B Condens. Matter 38(4), 2297–2311 (1988).
[Crossref] [PubMed]

Stanley, H. B.

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B Condens. Matter 38(4), 2297–2311 (1988).
[Crossref] [PubMed]

Trumper, J.

B. Aschenbach, H. Brauninger, G. Hasinger, and J. Trumper, “Measurements of X-ray scattering from Wolter type telescopes and various flat zerodur mirrors,” Proc. SPIE 257, 223–229 (1981).
[Crossref]

Turyansky, A. G.

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, and A. G. Turyansky, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).

Vinogradov, A. V.

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, and A. G. Turyansky, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, and I. G. Yakushkin, “Phenomenon of total external reflection of X-rays,” Sov. Phys. JETP 62, 1225–1229 (1985).

Yakushkin, I. G.

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, and I. G. Yakushkin, “Phenomenon of total external reflection of X-rays,” Sov. Phys. JETP 62, 1225–1229 (1985).

Yoneda, Y.

Y. Yoneda, “Anomalous surface reflection of X-rays,” Phys. Rev. 131(5), 2010–2013 (1963).
[Crossref]

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]

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Roughness conformity during tungsten film growth: An in situ synchrotron x-ray scattering study,” Phys. Rev. B 72(4), 045445 (2005).
[Crossref]

V. E. Asadchikov, I. V. Kozhevnikov, Y. S. Krivonosov, R. Mercier, T. H. Metzger, C. Morawe, and E. Ziegler, “Application of X-ray scattering technique to the study of supersmooth surfaces,” Nucl. Instrum. Methods Phys. Res. A 530(3), 575–595 (2004).
[Crossref]

Zorev, N. N.

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, and A. G. Turyansky, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, and I. G. Yakushkin, “Phenomenon of total external reflection of X-rays,” Sov. Phys. JETP 62, 1225–1229 (1985).

Appl. Opt. (1)

Crystallogr. Rep. (1)

I. V. Kozhevnikov, “General laws of X-ray reflection from rough surfaces. I. Optical theorem. Energy conservation law,” Crystallogr. Rep. 55(4), 539–545 (2010).
[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. XRay Sci. Technol. (1)

I. V. Kozhevnikov and M. V. Pyatakhin, “Use of DWBA and perturbation theory in x-ray control of the surface roughness,” J. XRay Sci. Technol. 8, 253–275 (1998).

Nucl. Instrum. Methods (1)

D. H. Bilderback and S. Hubbard, “X-ray mirror reflectivities from 3.8 to 50 keV (3.3 Ǻ to 0.25 Ǻ). Part I -Float glass,” Nucl. Instrum. Methods 195(1-2), 85–89 (1982).
[Crossref]

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

V. E. Asadchikov, I. V. Kozhevnikov, Y. S. Krivonosov, R. Mercier, T. H. Metzger, C. Morawe, and E. Ziegler, “Application of X-ray scattering technique to the study of supersmooth surfaces,” Nucl. Instrum. Methods Phys. Res. A 530(3), 575–595 (2004).
[Crossref]

Phys. Rev. (1)

Y. Yoneda, “Anomalous surface reflection of X-rays,” Phys. Rev. 131(5), 2010–2013 (1963).
[Crossref]

Phys. Rev. B (2)

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]

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Roughness conformity during tungsten film growth: An in situ synchrotron x-ray scattering study,” Phys. Rev. B 72(4), 045445 (2005).
[Crossref]

Phys. Rev. B Condens. Matter (3)

R. Pynn, “Neutron scattering by rough surfaces at grazing incidence,” Phys. Rev. B Condens. Matter 45(2), 602–612 (1992).
[Crossref] [PubMed]

D. K. G. de Boer, “X-ray reflection and transmission by rough surfaces,” Phys. Rev. B Condens. Matter 51(8), 5297–5305 (1995).
[Crossref] [PubMed]

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B Condens. Matter 38(4), 2297–2311 (1988).
[Crossref] [PubMed]

Proc. SPIE (1)

B. Aschenbach, H. Brauninger, G. Hasinger, and J. Trumper, “Measurements of X-ray scattering from Wolter type telescopes and various flat zerodur mirrors,” Proc. SPIE 257, 223–229 (1981).
[Crossref]

Rev. Phys. Appl. (1)

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

Sov. Phys. JETP (2)

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, and I. G. Yakushkin, “Phenomenon of total external reflection of X-rays,” Sov. Phys. JETP 62, 1225–1229 (1985).

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, and A. G. Turyansky, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).

Z. Phys. (1)

A. Andronow and M. Leontowiez, “Zur theorie der molekularen Lichtzerstreuung an Flüssigkeitsoberflächen,” Z. Phys. 38(6-7), 485–501 (1926).
[Crossref]

Other (6)

M. Bass, ed., Handbook of Optics (McGraw-Hill, 1995).

A.-L. Barabǎsi, H.E. Stanley, Fractal Concepts in Surface Growth (Syndicate of the University of Cambridge, 1995).

M. Tolan, X-Ray Scattering from Soft-Matter Thin Films (Springer, 1999).

U. Pietsch, V. Holy, and T. Baumbach, High-Resolution X-ray Scattering from Thin Films and Multilayers (Springer, 2004).

H. Sinn, M. Dommach, X. Dong, D. La Civita, L. Samoylova, R. Villaneuva, and F. Yang, “X-ray optics and beam transport,” The European XFEL Technical Design Report, Hamburg, 2012.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1 Scheme illustrating reflection of X-ray wave from a rough surface.
Fig. 2
Fig. 2 Scheme illustrating different limiting cases of X-ray diffraction from a rough surface.
Fig. 3
Fig. 3 The total integral scattering (TIS) and specular reflectance ( δ R ) of an Au-coated mirror at the photon energy E = 3 keV and with the grazing angle of an incident beam of θ 0 = 50 mrad versus the correlation length of roughness. The TIS and δ R values were normalized to the DW factor (Eq. (13)). The model (14) of the PSD-function was used, assuming the fractal parameter to be equal to α = 0.8 . Curves 1 and 2 were calculated using simplified analytic expressions (25) and (27). The horizontal dashed lines show the value of δ R calculated for the limiting cases of very large and extremely small correlation lengths (Eqs. (12) and (13)). The vertical dashed-dotted line shows the value of the correlation length when the parameter μ 0 = 1 .
Fig. 4
Fig. 4 The total integral scattering (TIS) and specular reflectance ( δ R ) of an Au-coated mirror at the photon energy E = 3 keV and the extremely small grazing angle of an incident beam θ 0 = 1 mrad versus the correlation length of roughness. The TIS and δ R values were normalized to the DW factor (Eq. (13)). The model (14) of the PSD-function was used, assuming the fractal parameter to be equal to α = 0.8 . Curves 1–4 were calculated using simplified analytic expressions (23), (25), (27). The horizontal dashed lines show the value of δ R calculated for the limiting cases of very large and extremely small correlation lengths (Eqs. (12) and (13)). The vertical dashed-dotted line shows the value of the correlation length when the parameter μ с = 1 .

Tables (1)

Tables Icon

Table 1 The total integrated scattering in vacuum ( TIS ) and into the matter depth ( TIS + ), as well as corrections to the coherent reflectance ( δ R ) and transmittance ( δ T ), calculated for a number of limiting cases.

Equations (31)

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

DW : R ( θ 0 ) = R F ( θ 0 ) e ( 4 π σ sin θ 0 / λ ) 2 ; TIS ( θ 0 ) = R F ( θ 0 ) [ 1 e ( 4 π σ sin θ 0 / λ ) 2 ] ; ξ
NC : R ( θ 0 ) = R F ( θ 0 ) e ( 4 π σ / λ ) 2 sin θ 0 Re ε cos 2 θ 0 ; TIS ( θ 0 ) = 0 ; ξ 0
Φ ( θ , φ ) = 1 W i n c d W s c a t d Ω = k 5 | 1 ε | 2 ( 4 π ) 2 κ ( q 0 ) | t ( q 0 ) t ( q ) | 2 PSD 2 D ( ν )
Φ + ( θ , φ ) = 1 W i n c d W s c a t d Ω + = k 5 ε | 1 ε | 2 ( 4 π ) 2 κ ( q 0 ) | t ( q 0 ) t ˜ ( q ) | 2 PSD 2 D ( ν ) Re κ + ( q ) κ + ( q ) t ( q ) = 2 κ ( q ) κ ( q ) + κ + ( q ) ; t ˜ ( q ) = 2 κ + ( q ) κ ( q ) + κ + ( q ) q = k cos θ ; κ ( q ) = k 2 q 2 ; κ + ( q ) = k 2 ε q 2 ; k = 2 π / λ
PSD 2 D ( ν ) = C ( ρ ) exp ( 2 π i ν ρ ) d 2 ρ = 2 π 0 C ( ρ ) J 0 ( 2 π ν ρ ) ρ d ρ
2 π ν = q q 0 ; q 0 = k ( cos θ 0 , 0 ) ; q = k ( cos θ cos φ , cos θ sin φ )
TIS ( q 0 ) W s c a t W i n c = Ω Φ ( θ , φ ) d Ω = Ω Φ ( θ , φ ) d 2 q k κ ( q ) = = κ ( q 0 ) π 2 | κ ( q 0 ) + κ + ( q 0 ) | 2 Ω κ ( q ) | κ ( q ) κ + ( q ) | 2 PSD 2 D ( ν ) d 2 q
TIS + ( q 0 ) W s c a t W i n c = Ω + Φ + ( θ , φ ) d Ω = Ω + Φ + ( θ , φ ) d 2 q k κ + ( q ) ε = = κ ( q 0 ) π 2 | κ ( q 0 ) + κ + ( q 0 ) | 2 Ω + Re κ + ( q ) | κ ( q ) κ + ( q ) | 2 PSD 2 D ( ν ) d 2 q
R ( q 0 ) = R F ( q 0 ) δ R ( q 0 ) , δ R ( q 0 ) = R F ( q 0 ) 4 κ ( q 0 ) Re κ + ( q 0 ) σ 2 + + R F ( q 0 ) κ ( q 0 ) π 2 Ω [ κ ( q ) Re κ + ( q ) ] PSD 2 D ( ν ) d 2 q
T ( q 0 ) = T F ( q 0 ) δ T ( q 0 ) , δ T ( q 0 ) = T F ( q 0 ) [ κ ( q 0 ) κ + ( q 0 ) ] 2 σ 2 + + T F ( q 0 ) κ ( q 0 ) κ + ( q 0 ) 2 π 2 Ω + [ κ ( q ) Re κ + ( q ) ] PSD 2 D ( ν ) d 2 q
R F ( q 0 ) = | κ ( q 0 ) κ + ( q 0 ) κ ( q 0 ) + κ + ( q 0 ) | 2 , T F ( q 0 ) = | 2 κ ( q 0 ) κ ( q 0 ) + κ + ( q 0 ) | 2 Re κ + ( q 0 ) κ ( q 0 )
δ R ( q 0 ) = 4 κ ( q 0 ) Re κ + ( q 0 ) σ 2 R F ( q 0 ) , TIS ( q 0 ) = 0 δ T ( q 0 ) = 4 κ ( q 0 ) Re κ + ( q 0 ) σ 2 R F ( q 0 ) , TIS + ( q 0 ) = 0
δ R ( q 0 ) = TIS ( q 0 ) = [ 2 κ ( q 0 ) σ ] 2 R F ( q 0 ) δ T ( q 0 ) = TIS + ( q 0 ) = [ 2 κ ( q 0 ) σ ] 2 R F ( q 0 ) Re κ + ( q 0 ) κ ( q 0 )
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
π π PSD 2 D ( ν ) d φ = ( 2 π ) 2 0 C ( ρ ) J 0 ( q 0 ρ ) J 0 ( q ρ ) ρ d ρ ; q 0 = k cos θ 0 ; q = k cos θ
π π PSD 2 D ( ν ) d φ π q 0 q PSD 1 D ( p ) ; PSD 1 D ( p ) = 4 0 C ( ρ ) cos ( 2 π p ρ ) d ρ ; 2 π p = | q q 0 |
μ 0 = ξ sin 2 θ 0 2 λ and μ c = ξ ( 1 ε ) 2 λ
F ( τ ) = 2 π Γ ( α + 1 / 2 ) Γ ( α ) 1 ( 1 + τ 2 ) α + 1 / 2 ; α > 0
TIS ( q 0 ) R F ( q 0 ) = 4 π k κ ( q 0 ) σ 2 k ξ [ 0 ξ q 0 / 2 π μ 0 + τ | μ 0 + τ μ 0 μ c + τ μ 0 μ 0 μ c | 2 F ( τ ) d τ + 0 μ 0 μ 0 τ | μ 0 τ μ 0 μ c τ μ 0 μ 0 μ c | 2 F ( τ ) d τ ]
δ R ( q 0 ) R F ( q 0 ) = 4 π k κ ( q 0 ) σ 2 k ξ Re [ 2 μ 0 μ c + 0 ξ q 0 / 2 π ( μ 0 + τ μ 0 μ c + τ ) F ( τ ) d τ + 0 μ 0 ( μ 0 τ μ 0 μ c τ ) F ( τ ) d τ ]
TIS + ( q 0 ) R F ( q 0 ) = 4 π k κ ( q 0 ) σ 2 k ξ Re [ 0 ξ q 0 / 2 π μ 0 μ c + τ | μ 0 + τ μ 0 μ c + τ μ 0 μ 0 μ c | 2 F ( τ ) d τ + 0 μ 0 μ 0 μ c τ | μ 0 τ μ 0 μ c τ μ 0 μ 0 μ c | 2 F ( τ ) d τ ]
δ T ( q 0 ) R F ( q 0 ) = 4 π k κ ( q 0 ) σ 2 k ξ Re μ 0 μ c R e { 2 + 2 μ 0 μ 0 μ c [ 0 ξ q 0 / 2 π ( μ 0 + τ μ 0 μ c + τ ) F ( τ ) d τ + 0 μ 0 ( μ 0 τ μ 0 μ c τ ) F ( τ ) d τ ] }
TIS ( θ 0 ) R F ( θ 0 ) f 1 ( k σ ) 2 sin θ 0 k ξ ; δ R ( θ 0 ) R F ( θ 0 ) ( k σ ) 2 ( f 1 k ξ + 2 Re ε 1 ) sin θ 0
TIS + ( θ 0 ) R F ( θ 0 ) 2 ( k σ ) 2 sin θ 0 Re ε 1 ; δ T ( θ 0 ) R F ( θ 0 ) 0 ; f 1 = 4 π 0 τ F ( τ ) d τ
TIS ( θ 0 ) R F ( θ 0 ) TIS + ( θ 0 ) R F ( θ 0 ) f 2 ( k σ ) 2 k ξ sin θ 0 | sin θ 0 + ε cos 2 θ 0 | 2
δ T ( θ 0 ) R F ( θ 0 ) ( 2 k σ ) 2 sin θ 0 [ 1 + f 2 k ξ ( sin θ 0 + Re ε cos 2 θ 0 ) ] Re ε cos 2 θ 0
δ R ( θ 0 ) R F ( θ 0 ) ( 2 k σ ) 2 sin θ 0 [ Re ε cos 2 θ 0 + f 2 2 k ξ Re ( 1 ε ) ] ; f 2 = 1 4 π 0 F ( τ ) τ d τ
TIS ( θ 0 ) R F ( θ 0 ) TIS + ( θ 0 ) R F ( θ 0 ) f 2 ( k σ ) 2 δ k ξ sin θ 0 δ R ( θ 0 ) R F ( θ 0 ) 2 ( k σ ) 2 ( δ f 2 k ξ + γ δ 1 / 2 ) sin θ 0 ; δ T ( θ 0 ) R F ( θ 0 ) 2 ( k σ ) 2 γ δ 1 / 2 sin θ 0
TIS ( θ 0 ) R F ( θ 0 ) TIS + ( θ 0 ) R F ( θ 0 ) 2 ( k σ ) 2 f 2 k ξ ( 2 sin 2 θ 0 δ ) sin θ 0
δ R ( θ 0 ) R F ( θ 0 ) ( 2 k σ ) 2 sin θ 0 Re ε cos 2 θ 0 + 2 ( k σ ) 2 f 2 δ k ξ sin θ 0
δ T ( θ 0 ) R F ( θ 0 ) ( 2 k σ ) 2 sin θ 0 Re ε cos 2 θ 0 + 2 ( k σ ) 2 f 2 k ξ ( 4 sin 2 θ 0 3 δ ) sin θ 0

Metrics