Abstract

Combined computer simulations of the growth of multilayer mirrors and their exact differential reflection coefficients in the soft-x-ray–EUV range have been conducted. The proposed model describes the variation of the surface roughness of the multilayer Al/Zr mirror boundary profiles taking into account a random noise source. Theoretically calculated Al/Zr boundary profiles allow one to know real rough boundary statistics including rms roughnesses and correlation lengths and, to obtain rigorously EUV specular and diffuse reflection coefficients. The proposed integrated approach opens up a way to performing exact theoretical studies similar in accuracy to results obtained by quantitative microscopy investigations of nanoreliefs and synchrotron radiation measurements.

© 2015 Optical Society of America

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References

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  2. L. Goray and M. Lubov, “Nonlinear continuum growth model of multiscale reliefs as applied to rigorous analysis of multilayer short-wave scattering intensity. I. Gratings,” J. Appl. Cryst. 46(4), 926–932 (2013).
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    [Crossref] [PubMed]
  5. L. I. Goray, “Solution of the inverse problem of diffraction from low-dimensional periodically arranged nanocrystals,” Proc. SPIE 8083, 80830L (2011).
    [Crossref]
  6. I. Kozhevnikov, L. Peverini, and E. Ziegler, “Exact determination of the phase in time-resolved X-ray reflectometry,” Opt. Express 16(1), 144–149 (2008).
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  7. M.-A. Henn, H. Gross, F. Scholze, M. Wurm, C. Elster, and M. Bär, “A maximum likelihood approach to the inverse problem of scatterometry,” Opt. Express 20(12), 12771–12786 (2012).
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  8. L. I. Goray, “Application of the rigorous method to x-ray and neutron beam scattering on rough surfaces,” J. Appl. Phys. 108(3), 033516 (2010).
    [Crossref]
  9. L. I. Goray and G. Schmidt, “Boundary Integral Equation Methods for Conical Diffraction and Short Waves,” in Gratings: Theory and Numerical Applications, E. Popov, ed. (Presses universitaires de Provence, Sec. rev. ed., 2014).
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  13. D. G. Stearns, D. P. Gaines, D. W. Sweeney, and E. M. Gullikson, “Nonspecular x-ray scattering in a multilayer-coated imaging system,” J. Appl. Phys. 84(2), 1003–1028 (1998).
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  14. L. Peverini, E. Ziegler, and I. Kozhevnikov, “Dynamic scaling in ion etching of tungsten films,” Appl. Phys. Lett. 91(5), 053121 (2007).
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  16. Y. V. Yashchuk, V. V. Yashchuk, and Q. Zhong, “Reliable before-fabrication forecasting of expected surface slope distributions for x-ray optics,” Opt. Eng. 51(4), 046501 (2012).
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    [Crossref]
  20. E. L. Church, “Fractal surface finish,” Appl. Opt. 27(8), 1518–1526 (1988).
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  21. Q. Zhong, Z. Zhang, S. Ma, R. Qi, J. Li, Z. Wang, K. Le Guen, J.-M. André, and P. Jonnard, “The transition from amorphous to crystalline in Al/Zr multilayers,” J. Appl. Phys. 113(13), 133508 (2013).
    [Crossref]
  22. D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
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    [Crossref]
  28. K. Le Guen, M.-H. Hu, J.-M. André, P. Jonnard, S. K. Zhou, H. C. Li, J. T. Zhu, Z. S. Wang, N. Mahne, A. Giglia, and S. Nannarone, “Introduction of Zr in nanometric periodic Mg/Co multilayers,” Appl. Phys., A Mater. Sci. Process. 102(1), 69–77 (2011).
    [Crossref]
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2014 (2)

L. Goray and M. Lubov, “Nonlinear continual growth model of nonuniformly scaled reliefs as applied to the rigorous analysis of the X-ray scattering intensity of multilayer mirrors and gratings,” J. Surf. Investig. X-Ra. 8(3), 444–455 (2014).
[Crossref]

V. V. Yashchuk, Y. N. Tyurin, and A. Y. Tyurina, “Application of the time-invariant linear filter approximation to parametrization of surface metrology with high-quality x-ray optics,” Opt. Eng. 53(8), 084102 (2014).
[Crossref]

2013 (2)

L. Goray and M. Lubov, “Nonlinear continuum growth model of multiscale reliefs as applied to rigorous analysis of multilayer short-wave scattering intensity. I. Gratings,” J. Appl. Cryst. 46(4), 926–932 (2013).
[Crossref] [PubMed]

Q. Zhong, Z. Zhang, S. Ma, R. Qi, J. Li, Z. Wang, K. Le Guen, J.-M. André, and P. Jonnard, “The transition from amorphous to crystalline in Al/Zr multilayers,” J. Appl. Phys. 113(13), 133508 (2013).
[Crossref]

2012 (2)

Y. V. Yashchuk, V. V. Yashchuk, and Q. Zhong, “Reliable before-fabrication forecasting of expected surface slope distributions for x-ray optics,” Opt. Eng. 51(4), 046501 (2012).
[Crossref]

M.-A. Henn, H. Gross, F. Scholze, M. Wurm, C. Elster, and M. Bär, “A maximum likelihood approach to the inverse problem of scatterometry,” Opt. Express 20(12), 12771–12786 (2012).
[Crossref] [PubMed]

2011 (4)

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

L. I. Goray, “Solution of the inverse problem of diffraction from low-dimensional periodically arranged nanocrystals,” Proc. SPIE 8083, 80830L (2011).
[Crossref]

K. Le Guen, M.-H. Hu, J.-M. André, P. Jonnard, S. K. Zhou, H. C. Li, J. T. Zhu, Z. S. Wang, N. Mahne, A. Giglia, and S. Nannarone, “Introduction of Zr in nanometric periodic Mg/Co multilayers,” Appl. Phys., A Mater. Sci. Process. 102(1), 69–77 (2011).
[Crossref]

D. L. Voronov, E. H. Anderson, R. Cambie, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Roughening and smoothing behavior of Al/Zr multilayers grown on flat and saw-tooth substrates,” Proc. SPIE 8139, 81390B (2011).
[Crossref]

2010 (1)

L. I. Goray, “Application of the rigorous method to x-ray and neutron beam scattering on rough surfaces,” J. Appl. Phys. 108(3), 033516 (2010).
[Crossref]

2009 (1)

L. I. Goray, “Specular and diffuse scattering from random asperities of any profile using the rigorous method for x-rays and neutrons,” Proc. SPIE 7390, 73900V (2009).
[Crossref]

2008 (1)

2007 (1)

L. Peverini, E. Ziegler, and I. Kozhevnikov, “Dynamic scaling in ion etching of tungsten films,” Appl. Phys. Lett. 91(5), 053121 (2007).
[Crossref]

2006 (1)

2004 (2)

J. F. Seely, L. Goray, D. L. Windt, B. Kjornrattanawanich, Y. Uspenskii, and A. V. Vinogradov, “Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings,” Proc. SPIE 5538, 43–52 (2004).
[Crossref]

V. E. Asadchikov, I. V. Kozhevnikov, Yu. 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. Inst. Meth. Phys. Res. A. 530, 575–595 (2004).

1998 (1)

D. G. Stearns, D. P. Gaines, D. W. Sweeney, and E. M. Gullikson, “Nonspecular x-ray scattering in a multilayer-coated imaging system,” J. Appl. Phys. 84(2), 1003–1028 (1998).
[Crossref]

1990 (1)

E. L. Church, P. Z. Takacs, and T. A. Leonard, “The prediction of BRDFs from surface profile measurements,” Proc. SPIE 1165, 136–150 (1990).
[Crossref]

1988 (1)

1983 (1)

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, and A. Llebaria, “Roughness spectrum and surface plasmons for surfaces of silver, copper, gold, and magnesium deposits,” Phys. Rev. B 27(2), 819–830 (1983).
[Crossref]

1957 (1)

W. W. Mullins, “Theory of thermal grooving,” J. Appl. Phys. 28(3), 333–339 (1957).
[Crossref]

Anderson, E. H.

D. L. Voronov, E. H. Anderson, R. Cambie, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Roughening and smoothing behavior of Al/Zr multilayers grown on flat and saw-tooth substrates,” Proc. SPIE 8139, 81390B (2011).
[Crossref]

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

André, J.-M.

Q. Zhong, Z. Zhang, S. Ma, R. Qi, J. Li, Z. Wang, K. Le Guen, J.-M. André, and P. Jonnard, “The transition from amorphous to crystalline in Al/Zr multilayers,” J. Appl. Phys. 113(13), 133508 (2013).
[Crossref]

K. Le Guen, M.-H. Hu, J.-M. André, P. Jonnard, S. K. Zhou, H. C. Li, J. T. Zhu, Z. S. Wang, N. Mahne, A. Giglia, and S. Nannarone, “Introduction of Zr in nanometric periodic Mg/Co multilayers,” Appl. Phys., A Mater. Sci. Process. 102(1), 69–77 (2011).
[Crossref]

Asadchikov, V. E.

V. E. Asadchikov, I. V. Kozhevnikov, Yu. 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. Inst. Meth. Phys. Res. A. 530, 575–595 (2004).

Bär, M.

Cabrini, S.

Cambie, R.

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

D. L. Voronov, E. H. Anderson, R. Cambie, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Roughening and smoothing behavior of Al/Zr multilayers grown on flat and saw-tooth substrates,” Proc. SPIE 8139, 81390B (2011).
[Crossref]

Church, E. L.

E. L. Church, P. Z. Takacs, and T. A. Leonard, “The prediction of BRDFs from surface profile measurements,” Proc. SPIE 1165, 136–150 (1990).
[Crossref]

E. L. Church, “Fractal surface finish,” Appl. Opt. 27(8), 1518–1526 (1988).
[Crossref] [PubMed]

Content, D. A.

Dhuey, S. D.

Elster, C.

Gaines, D. P.

D. G. Stearns, D. P. Gaines, D. W. Sweeney, and E. M. Gullikson, “Nonspecular x-ray scattering in a multilayer-coated imaging system,” J. Appl. Phys. 84(2), 1003–1028 (1998).
[Crossref]

Giglia, A.

K. Le Guen, M.-H. Hu, J.-M. André, P. Jonnard, S. K. Zhou, H. C. Li, J. T. Zhu, Z. S. Wang, N. Mahne, A. Giglia, and S. Nannarone, “Introduction of Zr in nanometric periodic Mg/Co multilayers,” Appl. Phys., A Mater. Sci. Process. 102(1), 69–77 (2011).
[Crossref]

Goray, L.

L. Goray and M. Lubov, “Nonlinear continual growth model of nonuniformly scaled reliefs as applied to the rigorous analysis of the X-ray scattering intensity of multilayer mirrors and gratings,” J. Surf. Investig. X-Ra. 8(3), 444–455 (2014).
[Crossref]

L. Goray and M. Lubov, “Nonlinear continuum growth model of multiscale reliefs as applied to rigorous analysis of multilayer short-wave scattering intensity. I. Gratings,” J. Appl. Cryst. 46(4), 926–932 (2013).
[Crossref] [PubMed]

J. F. Seely, L. Goray, D. L. Windt, B. Kjornrattanawanich, Y. Uspenskii, and A. V. Vinogradov, “Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings,” Proc. SPIE 5538, 43–52 (2004).
[Crossref]

Goray, L. I.

L. I. Goray, “Solution of the inverse problem of diffraction from low-dimensional periodically arranged nanocrystals,” Proc. SPIE 8083, 80830L (2011).
[Crossref]

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

L. I. Goray, “Application of the rigorous method to x-ray and neutron beam scattering on rough surfaces,” J. Appl. Phys. 108(3), 033516 (2010).
[Crossref]

L. I. Goray, “Specular and diffuse scattering from random asperities of any profile using the rigorous method for x-rays and neutrons,” Proc. SPIE 7390, 73900V (2009).
[Crossref]

L. I. Goray, I. G. Kuznetsov, S. Yu. Sadov, and D. A. Content, “Multilayer resonant subwavelength gratings: effects of waveguide modes and real groove profiles,” J. Opt. Soc. Am. A 23(1), 155–165 (2006).
[Crossref] [PubMed]

Gross, H.

Gullikson, E. M.

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

D. L. Voronov, E. H. Anderson, R. Cambie, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Roughening and smoothing behavior of Al/Zr multilayers grown on flat and saw-tooth substrates,” Proc. SPIE 8139, 81390B (2011).
[Crossref]

D. G. Stearns, D. P. Gaines, D. W. Sweeney, and E. M. Gullikson, “Nonspecular x-ray scattering in a multilayer-coated imaging system,” J. Appl. Phys. 84(2), 1003–1028 (1998).
[Crossref]

Henn, M.-A.

Hu, M.-H.

K. Le Guen, M.-H. Hu, J.-M. André, P. Jonnard, S. K. Zhou, H. C. Li, J. T. Zhu, Z. S. Wang, N. Mahne, A. Giglia, and S. Nannarone, “Introduction of Zr in nanometric periodic Mg/Co multilayers,” Appl. Phys., A Mater. Sci. Process. 102(1), 69–77 (2011).
[Crossref]

Jonnard, P.

Q. Zhong, Z. Zhang, S. Ma, R. Qi, J. Li, Z. Wang, K. Le Guen, J.-M. André, and P. Jonnard, “The transition from amorphous to crystalline in Al/Zr multilayers,” J. Appl. Phys. 113(13), 133508 (2013).
[Crossref]

K. Le Guen, M.-H. Hu, J.-M. André, P. Jonnard, S. K. Zhou, H. C. Li, J. T. Zhu, Z. S. Wang, N. Mahne, A. Giglia, and S. Nannarone, “Introduction of Zr in nanometric periodic Mg/Co multilayers,” Appl. Phys., A Mater. Sci. Process. 102(1), 69–77 (2011).
[Crossref]

Kjornrattanawanich, B.

J. F. Seely, L. Goray, D. L. Windt, B. Kjornrattanawanich, Y. Uspenskii, and A. V. Vinogradov, “Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings,” Proc. SPIE 5538, 43–52 (2004).
[Crossref]

Kozhevnikov, I.

I. Kozhevnikov, L. Peverini, and E. Ziegler, “Exact determination of the phase in time-resolved X-ray reflectometry,” Opt. Express 16(1), 144–149 (2008).
[Crossref] [PubMed]

L. Peverini, E. Ziegler, and I. Kozhevnikov, “Dynamic scaling in ion etching of tungsten films,” Appl. Phys. Lett. 91(5), 053121 (2007).
[Crossref]

Kozhevnikov, I. V.

V. E. Asadchikov, I. V. Kozhevnikov, Yu. 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. Inst. Meth. Phys. Res. A. 530, 575–595 (2004).

Krivonosov, Yu. S.

V. E. Asadchikov, I. V. Kozhevnikov, Yu. 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. Inst. Meth. Phys. Res. A. 530, 575–595 (2004).

Kuznetsov, I. G.

Le Guen, K.

Q. Zhong, Z. Zhang, S. Ma, R. Qi, J. Li, Z. Wang, K. Le Guen, J.-M. André, and P. Jonnard, “The transition from amorphous to crystalline in Al/Zr multilayers,” J. Appl. Phys. 113(13), 133508 (2013).
[Crossref]

K. Le Guen, M.-H. Hu, J.-M. André, P. Jonnard, S. K. Zhou, H. C. Li, J. T. Zhu, Z. S. Wang, N. Mahne, A. Giglia, and S. Nannarone, “Introduction of Zr in nanometric periodic Mg/Co multilayers,” Appl. Phys., A Mater. Sci. Process. 102(1), 69–77 (2011).
[Crossref]

Leonard, T. A.

E. L. Church, P. Z. Takacs, and T. A. Leonard, “The prediction of BRDFs from surface profile measurements,” Proc. SPIE 1165, 136–150 (1990).
[Crossref]

Li, H. C.

K. Le Guen, M.-H. Hu, J.-M. André, P. Jonnard, S. K. Zhou, H. C. Li, J. T. Zhu, Z. S. Wang, N. Mahne, A. Giglia, and S. Nannarone, “Introduction of Zr in nanometric periodic Mg/Co multilayers,” Appl. Phys., A Mater. Sci. Process. 102(1), 69–77 (2011).
[Crossref]

Li, J.

Q. Zhong, Z. Zhang, S. Ma, R. Qi, J. Li, Z. Wang, K. Le Guen, J.-M. André, and P. Jonnard, “The transition from amorphous to crystalline in Al/Zr multilayers,” J. Appl. Phys. 113(13), 133508 (2013).
[Crossref]

Llebaria, A.

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, and A. Llebaria, “Roughness spectrum and surface plasmons for surfaces of silver, copper, gold, and magnesium deposits,” Phys. Rev. B 27(2), 819–830 (1983).
[Crossref]

Lubov, M.

L. Goray and M. Lubov, “Nonlinear continual growth model of nonuniformly scaled reliefs as applied to the rigorous analysis of the X-ray scattering intensity of multilayer mirrors and gratings,” J. Surf. Investig. X-Ra. 8(3), 444–455 (2014).
[Crossref]

L. Goray and M. Lubov, “Nonlinear continuum growth model of multiscale reliefs as applied to rigorous analysis of multilayer short-wave scattering intensity. I. Gratings,” J. Appl. Cryst. 46(4), 926–932 (2013).
[Crossref] [PubMed]

Ma, S.

Q. Zhong, Z. Zhang, S. Ma, R. Qi, J. Li, Z. Wang, K. Le Guen, J.-M. André, and P. Jonnard, “The transition from amorphous to crystalline in Al/Zr multilayers,” J. Appl. Phys. 113(13), 133508 (2013).
[Crossref]

Mahne, N.

K. Le Guen, M.-H. Hu, J.-M. André, P. Jonnard, S. K. Zhou, H. C. Li, J. T. Zhu, Z. S. Wang, N. Mahne, A. Giglia, and S. Nannarone, “Introduction of Zr in nanometric periodic Mg/Co multilayers,” Appl. Phys., A Mater. Sci. Process. 102(1), 69–77 (2011).
[Crossref]

Mercier, R.

V. E. Asadchikov, I. V. Kozhevnikov, Yu. 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. Inst. Meth. Phys. Res. A. 530, 575–595 (2004).

Metzger, T. H.

V. E. Asadchikov, I. V. Kozhevnikov, Yu. 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. Inst. Meth. Phys. Res. A. 530, 575–595 (2004).

Morawe, C.

V. E. Asadchikov, I. V. Kozhevnikov, Yu. 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. Inst. Meth. Phys. Res. A. 530, 575–595 (2004).

Mullins, W. W.

W. W. Mullins, “Theory of thermal grooving,” J. Appl. Phys. 28(3), 333–339 (1957).
[Crossref]

Nannarone, S.

K. Le Guen, M.-H. Hu, J.-M. André, P. Jonnard, S. K. Zhou, H. C. Li, J. T. Zhu, Z. S. Wang, N. Mahne, A. Giglia, and S. Nannarone, “Introduction of Zr in nanometric periodic Mg/Co multilayers,” Appl. Phys., A Mater. Sci. Process. 102(1), 69–77 (2011).
[Crossref]

Padmore, H. A.

D. L. Voronov, E. H. Anderson, R. Cambie, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Roughening and smoothing behavior of Al/Zr multilayers grown on flat and saw-tooth substrates,” Proc. SPIE 8139, 81390B (2011).
[Crossref]

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

Palmari, J. P.

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, and A. Llebaria, “Roughness spectrum and surface plasmons for surfaces of silver, copper, gold, and magnesium deposits,” Phys. Rev. B 27(2), 819–830 (1983).
[Crossref]

Peverini, L.

I. Kozhevnikov, L. Peverini, and E. Ziegler, “Exact determination of the phase in time-resolved X-ray reflectometry,” Opt. Express 16(1), 144–149 (2008).
[Crossref] [PubMed]

L. Peverini, E. Ziegler, and I. Kozhevnikov, “Dynamic scaling in ion etching of tungsten films,” Appl. Phys. Lett. 91(5), 053121 (2007).
[Crossref]

Qi, R.

Q. Zhong, Z. Zhang, S. Ma, R. Qi, J. Li, Z. Wang, K. Le Guen, J.-M. André, and P. Jonnard, “The transition from amorphous to crystalline in Al/Zr multilayers,” J. Appl. Phys. 113(13), 133508 (2013).
[Crossref]

Rasigni, G.

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, and A. Llebaria, “Roughness spectrum and surface plasmons for surfaces of silver, copper, gold, and magnesium deposits,” Phys. Rev. B 27(2), 819–830 (1983).
[Crossref]

Rasigni, M.

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, and A. Llebaria, “Roughness spectrum and surface plasmons for surfaces of silver, copper, gold, and magnesium deposits,” Phys. Rev. B 27(2), 819–830 (1983).
[Crossref]

Sadov, S. Yu.

Salmassi, F.

D. L. Voronov, E. H. Anderson, R. Cambie, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Roughening and smoothing behavior of Al/Zr multilayers grown on flat and saw-tooth substrates,” Proc. SPIE 8139, 81390B (2011).
[Crossref]

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

Scholze, F.

Seely, J. F.

J. F. Seely, L. Goray, D. L. Windt, B. Kjornrattanawanich, Y. Uspenskii, and A. V. Vinogradov, “Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings,” Proc. SPIE 5538, 43–52 (2004).
[Crossref]

Stearns, D. G.

D. G. Stearns, D. P. Gaines, D. W. Sweeney, and E. M. Gullikson, “Nonspecular x-ray scattering in a multilayer-coated imaging system,” J. Appl. Phys. 84(2), 1003–1028 (1998).
[Crossref]

Sweeney, D. W.

D. G. Stearns, D. P. Gaines, D. W. Sweeney, and E. M. Gullikson, “Nonspecular x-ray scattering in a multilayer-coated imaging system,” J. Appl. Phys. 84(2), 1003–1028 (1998).
[Crossref]

Takacs, P. Z.

E. L. Church, P. Z. Takacs, and T. A. Leonard, “The prediction of BRDFs from surface profile measurements,” Proc. SPIE 1165, 136–150 (1990).
[Crossref]

Tyurin, Y. N.

V. V. Yashchuk, Y. N. Tyurin, and A. Y. Tyurina, “Application of the time-invariant linear filter approximation to parametrization of surface metrology with high-quality x-ray optics,” Opt. Eng. 53(8), 084102 (2014).
[Crossref]

Tyurina, A. Y.

V. V. Yashchuk, Y. N. Tyurin, and A. Y. Tyurina, “Application of the time-invariant linear filter approximation to parametrization of surface metrology with high-quality x-ray optics,” Opt. Eng. 53(8), 084102 (2014).
[Crossref]

Uspenskii, Y.

J. F. Seely, L. Goray, D. L. Windt, B. Kjornrattanawanich, Y. Uspenskii, and A. V. Vinogradov, “Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings,” Proc. SPIE 5538, 43–52 (2004).
[Crossref]

Varnier, F.

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, and A. Llebaria, “Roughness spectrum and surface plasmons for surfaces of silver, copper, gold, and magnesium deposits,” Phys. Rev. B 27(2), 819–830 (1983).
[Crossref]

Vinogradov, A. V.

J. F. Seely, L. Goray, D. L. Windt, B. Kjornrattanawanich, Y. Uspenskii, and A. V. Vinogradov, “Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings,” Proc. SPIE 5538, 43–52 (2004).
[Crossref]

Voronov, D. L.

D. L. Voronov, E. H. Anderson, R. Cambie, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Roughening and smoothing behavior of Al/Zr multilayers grown on flat and saw-tooth substrates,” Proc. SPIE 8139, 81390B (2011).
[Crossref]

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

Wang, Z.

Q. Zhong, Z. Zhang, S. Ma, R. Qi, J. Li, Z. Wang, K. Le Guen, J.-M. André, and P. Jonnard, “The transition from amorphous to crystalline in Al/Zr multilayers,” J. Appl. Phys. 113(13), 133508 (2013).
[Crossref]

Wang, Z. S.

K. Le Guen, M.-H. Hu, J.-M. André, P. Jonnard, S. K. Zhou, H. C. Li, J. T. Zhu, Z. S. Wang, N. Mahne, A. Giglia, and S. Nannarone, “Introduction of Zr in nanometric periodic Mg/Co multilayers,” Appl. Phys., A Mater. Sci. Process. 102(1), 69–77 (2011).
[Crossref]

Warwick, T.

D. L. Voronov, E. H. Anderson, R. Cambie, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Roughening and smoothing behavior of Al/Zr multilayers grown on flat and saw-tooth substrates,” Proc. SPIE 8139, 81390B (2011).
[Crossref]

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

Windt, D. L.

J. F. Seely, L. Goray, D. L. Windt, B. Kjornrattanawanich, Y. Uspenskii, and A. V. Vinogradov, “Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings,” Proc. SPIE 5538, 43–52 (2004).
[Crossref]

Wurm, M.

Yashchuk, V. V.

V. V. Yashchuk, Y. N. Tyurin, and A. Y. Tyurina, “Application of the time-invariant linear filter approximation to parametrization of surface metrology with high-quality x-ray optics,” Opt. Eng. 53(8), 084102 (2014).
[Crossref]

Y. V. Yashchuk, V. V. Yashchuk, and Q. Zhong, “Reliable before-fabrication forecasting of expected surface slope distributions for x-ray optics,” Opt. Eng. 51(4), 046501 (2012).
[Crossref]

D. L. Voronov, E. H. Anderson, R. Cambie, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Roughening and smoothing behavior of Al/Zr multilayers grown on flat and saw-tooth substrates,” Proc. SPIE 8139, 81390B (2011).
[Crossref]

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

Yashchuk, Y. V.

Y. V. Yashchuk, V. V. Yashchuk, and Q. Zhong, “Reliable before-fabrication forecasting of expected surface slope distributions for x-ray optics,” Opt. Eng. 51(4), 046501 (2012).
[Crossref]

Zhang, Z.

Q. Zhong, Z. Zhang, S. Ma, R. Qi, J. Li, Z. Wang, K. Le Guen, J.-M. André, and P. Jonnard, “The transition from amorphous to crystalline in Al/Zr multilayers,” J. Appl. Phys. 113(13), 133508 (2013).
[Crossref]

Zhong, Q.

Q. Zhong, Z. Zhang, S. Ma, R. Qi, J. Li, Z. Wang, K. Le Guen, J.-M. André, and P. Jonnard, “The transition from amorphous to crystalline in Al/Zr multilayers,” J. Appl. Phys. 113(13), 133508 (2013).
[Crossref]

Y. V. Yashchuk, V. V. Yashchuk, and Q. Zhong, “Reliable before-fabrication forecasting of expected surface slope distributions for x-ray optics,” Opt. Eng. 51(4), 046501 (2012).
[Crossref]

Zhou, S. K.

K. Le Guen, M.-H. Hu, J.-M. André, P. Jonnard, S. K. Zhou, H. C. Li, J. T. Zhu, Z. S. Wang, N. Mahne, A. Giglia, and S. Nannarone, “Introduction of Zr in nanometric periodic Mg/Co multilayers,” Appl. Phys., A Mater. Sci. Process. 102(1), 69–77 (2011).
[Crossref]

Zhu, J. T.

K. Le Guen, M.-H. Hu, J.-M. André, P. Jonnard, S. K. Zhou, H. C. Li, J. T. Zhu, Z. S. Wang, N. Mahne, A. Giglia, and S. Nannarone, “Introduction of Zr in nanometric periodic Mg/Co multilayers,” Appl. Phys., A Mater. Sci. Process. 102(1), 69–77 (2011).
[Crossref]

Ziegler, E.

I. Kozhevnikov, L. Peverini, and E. Ziegler, “Exact determination of the phase in time-resolved X-ray reflectometry,” Opt. Express 16(1), 144–149 (2008).
[Crossref] [PubMed]

L. Peverini, E. Ziegler, and I. Kozhevnikov, “Dynamic scaling in ion etching of tungsten films,” Appl. Phys. Lett. 91(5), 053121 (2007).
[Crossref]

V. E. Asadchikov, I. V. Kozhevnikov, Yu. 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. Inst. Meth. Phys. Res. A. 530, 575–595 (2004).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

L. Peverini, E. Ziegler, and I. Kozhevnikov, “Dynamic scaling in ion etching of tungsten films,” Appl. Phys. Lett. 91(5), 053121 (2007).
[Crossref]

Appl. Phys., A Mater. Sci. Process. (1)

K. Le Guen, M.-H. Hu, J.-M. André, P. Jonnard, S. K. Zhou, H. C. Li, J. T. Zhu, Z. S. Wang, N. Mahne, A. Giglia, and S. Nannarone, “Introduction of Zr in nanometric periodic Mg/Co multilayers,” Appl. Phys., A Mater. Sci. Process. 102(1), 69–77 (2011).
[Crossref]

J. Appl. Cryst. (1)

L. Goray and M. Lubov, “Nonlinear continuum growth model of multiscale reliefs as applied to rigorous analysis of multilayer short-wave scattering intensity. I. Gratings,” J. Appl. Cryst. 46(4), 926–932 (2013).
[Crossref] [PubMed]

J. Appl. Phys. (4)

L. I. Goray, “Application of the rigorous method to x-ray and neutron beam scattering on rough surfaces,” J. Appl. Phys. 108(3), 033516 (2010).
[Crossref]

W. W. Mullins, “Theory of thermal grooving,” J. Appl. Phys. 28(3), 333–339 (1957).
[Crossref]

D. G. Stearns, D. P. Gaines, D. W. Sweeney, and E. M. Gullikson, “Nonspecular x-ray scattering in a multilayer-coated imaging system,” J. Appl. Phys. 84(2), 1003–1028 (1998).
[Crossref]

Q. Zhong, Z. Zhang, S. Ma, R. Qi, J. Li, Z. Wang, K. Le Guen, J.-M. André, and P. Jonnard, “The transition from amorphous to crystalline in Al/Zr multilayers,” J. Appl. Phys. 113(13), 133508 (2013).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Surf. Investig. X-Ra. (1)

L. Goray and M. Lubov, “Nonlinear continual growth model of nonuniformly scaled reliefs as applied to the rigorous analysis of the X-ray scattering intensity of multilayer mirrors and gratings,” J. Surf. Investig. X-Ra. 8(3), 444–455 (2014).
[Crossref]

Nucl. Inst. Meth. Phys. Res. A. (1)

V. E. Asadchikov, I. V. Kozhevnikov, Yu. 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. Inst. Meth. Phys. Res. A. 530, 575–595 (2004).

Opt. Eng. (2)

Y. V. Yashchuk, V. V. Yashchuk, and Q. Zhong, “Reliable before-fabrication forecasting of expected surface slope distributions for x-ray optics,” Opt. Eng. 51(4), 046501 (2012).
[Crossref]

V. V. Yashchuk, Y. N. Tyurin, and A. Y. Tyurina, “Application of the time-invariant linear filter approximation to parametrization of surface metrology with high-quality x-ray optics,” Opt. Eng. 53(8), 084102 (2014).
[Crossref]

Opt. Express (3)

Phys. Rev. B (1)

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, and A. Llebaria, “Roughness spectrum and surface plasmons for surfaces of silver, copper, gold, and magnesium deposits,” Phys. Rev. B 27(2), 819–830 (1983).
[Crossref]

Proc. SPIE (5)

J. F. Seely, L. Goray, D. L. Windt, B. Kjornrattanawanich, Y. Uspenskii, and A. V. Vinogradov, “Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings,” Proc. SPIE 5538, 43–52 (2004).
[Crossref]

L. I. Goray, “Specular and diffuse scattering from random asperities of any profile using the rigorous method for x-rays and neutrons,” Proc. SPIE 7390, 73900V (2009).
[Crossref]

L. I. Goray, “Solution of the inverse problem of diffraction from low-dimensional periodically arranged nanocrystals,” Proc. SPIE 8083, 80830L (2011).
[Crossref]

E. L. Church, P. Z. Takacs, and T. A. Leonard, “The prediction of BRDFs from surface profile measurements,” Proc. SPIE 1165, 136–150 (1990).
[Crossref]

D. L. Voronov, E. H. Anderson, R. Cambie, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Roughening and smoothing behavior of Al/Zr multilayers grown on flat and saw-tooth substrates,” Proc. SPIE 8139, 81390B (2011).
[Crossref]

Other (7)

U. Pietsch, V. Holy, and T. Baumbach, High-Resolution X-Ray Scattering: From Thin Films to Lateral Nanostructure, (Springer-Verlag, 2004).

L. I. Goray and G. Schmidt, “Boundary Integral Equation Methods for Conical Diffraction and Short Waves,” in Gratings: Theory and Numerical Applications, E. Popov, ed. (Presses universitaires de Provence, Sec. rev. ed., 2014).

M. Pelliccione and L. Toh-Ming, Evolution of Thin Film Morphology: Modeling and Simulations (Springer, 2008).

The Center for X-Ray Optics, X-ray Interactions With Matter, available from: http://henke.lbl.gov/optical_constants/ .

E. D. Palik, ed., Handbook of Optical Constant of Solids (Academic, 1985).

A. A. Maradudin, ed., Light Scattering and Nanoscale Surface Roughness (Springer, 2007).

Web site, http://www.pcgrate.com/ .

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Figures (7)

Fig. 1
Fig. 1 Rms roughness σ of the Al/Zr mirror obtained for different source noise parameters q and lg vs. total multilayer thickness H.
Fig. 2
Fig. 2 σ vs. H variations of the Al/Zr mirror obtained for different source noise parameters lg.
Fig. 3
Fig. 3 Calculated PSD functions on boundaries of the Al/Zr multilayer structure of different thickness H.
Fig. 4
Fig. 4 Calculated and analytical PSD functions of an Al/Zr mirror. Combined PSD function is obtained as a sum of three analytical PSD functions: ABC, Shifted Gaussian and Fractal.
Fig. 5
Fig. 5 Spectral reflectances of Al/Zr multilayer mirrors, which were calculated for an incident angle θ = 5° using: perfect boundaries (green line); the rigorous approach averaged over seven grown model realizations (points); Debye-Waller amplitude factors with rms roughness of 0.45 nm (blue line).
Fig. 6
Fig. 6 The same as in Fig. 5, but for θ = 30°.
Fig. 7
Fig. 7 Scattering intensity of the Al/Zr mirror which was calculated rigorously by averaging over seven grown model realizations for an incident angle θ = 5° and λ = 20 nm vs. angle of scattered radiation θs.

Equations (10)

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

h / t =g(x,t)+ ν 2 1+ [ h(x,t) ] 2 K(x) ν 4 1+ [ h(x,t) ] 2 [ 2 K(x) / x 2 ]
g(x,t)= g 0 +Δg(x,t), g(x,t) = g 0
K(x)= 2 h(x,t) ( 1+h(x,t) ) 3/2 2 h(x,t).
h / t =g(x,t)+ ν 2 2 h(x,t) ν 4 4 h(x,t).
S ABC ( f x )= A [ 1+ ( B f x ) 2 ] ( C+1 ) /2 , ξ ABC = ( C1 ) 2 B 2 2 π 2 C ,
S SG ( f x )= π 3/2 σ 2 ξ SG { exp[ ξ 2 ( f x f SG ) 2 /4 ]+exp[ ξ 2 ( f x + f SG ) 2 /4 ] }
S Fr ( f x )= P f x n .
ξ Fr = ( n1 ) 2 L 2(2n1) ,
S An ( f x )= S ABC ( f x )+ S SG ( f x )+ S Fr ( f x ).
C(x)= σ 2 exp{ x 2D ξ 2D },

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