Abstract

Within various x-ray programs there exists a need for a detailed investigation of the surface roughness of mirrored surfaces over a wide spatial wavelength bandwidth, ranging from large scale figure error to microroughness. A number of methods exist to measure the surface roughness. Common to all methods is that they are bandwidth-limited. A crucial point in the analysis of data is, therefore, to specify accurately the wavelength bandwidth limitation and to determine the surface autocorrelation function within this bandwidth. We present a number of scattering measurements obtained using a triple-axis perfect-crystal x-ray diffractometer and the results of an autocorrelation function analysis. Furthermore, we present some measurements of integrated reflectivity, which we believe provide evidence for microroughness in the range from a few angstroms to tens of microns.

© 1988 Optical Society of America

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References

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  1. F. E. Christensen et al., “A Versatile Three/Four Crystal X-ray Diffractometer for X-ray Optical Elements. Performance and Applications,” Nucl. Intrum. Methods A256, 381 (1987).
    [CrossRef]
  2. A. H. Compton, S. K. Allison, X-rays in Theory and Experiment (Van Nostrand, New York, 1935).
  3. D. T. Cromer, D. Liberman, Los Alamos Scientific Laboratory Report, LA-4403 (1970).
  4. B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, ATP Conf. Proc. 75, Appendix (1981).
  5. H. Kiessig, Ann. Phys. 10, 769 (1931).
    [CrossRef]
  6. E. L. Church, “Role of Surface Topography in X-ray Scattering,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 96 (1984).
  7. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).
  8. R. Klos, “X-ray Scattering Theories,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 35 (1985).
  9. E. L. Church, P. Z. Takacs, “The Interpretation of Glancing Incidence Scattering Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, preprint, (1986).
  10. E. L. Church, P. Z. Takacs, “Statistical and Signal Processing Concepts in Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, preprint (1986).
  11. E. L. Church, “Comments on the Correlation Length,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 102 (1986).
  12. D. H. Bilderback, “Reflectance of X-ray Mirrors from 3.8 to 50 keV (3.3 to 0.25Å),” Proc. Soc. Photo-Opt. Instrum. Eng. 315, 90 (1981).
  13. L. A. Smirnov, T. D. Sotnikova, B. S. Anokhin, B. Z. Taibin, “Total External Reflection of X-rays from Rough Surfaces,” Opt. Spektrosk. 46, 593 (1979) [Opt. Spectrosc. (USSR) 46, 329 (1979)].
  14. J. Als-Nielsen, “Solid and Liquid Surfaces Studied by Synchrotron X-ray Diffraction,” in Structures and Dynamics of Surfaces, Vol. 2 (Springer-Verlag, New York, 1986).
  15. L. G. Parrat, “Surface Studies of Solids by Total Reflection of X-Rays,” Phys. Rev. 95, 359 (1954).
    [CrossRef]
  16. P. Croce, L. Nevot, B. Pardo, “Sur l’etude de la diffusion de rayons x sous angles resantes,” C. R. Acad. Sci. Paris Ser. B 274, 855 (1972).
  17. L. A. Smirnov, T. D. Sotnikova, Yu. I. Kogan, “Diffuse Scattering in Total External Reflection of X-rays from a Rough Surface,” Opt. Spectrosc. USSR 58, 239 (1985).
  18. R. Schiffer, “Reflectivity of a Slightly Rough Surface,” Appl. Opt. 26, 704 (1987).
    [CrossRef] [PubMed]
  19. A. N. Nigam, “Origin of Anomalous Surface Reflection of X-rays,” Phys. Rev. 138, A1189 (1965).
    [CrossRef]
  20. Y. Yoneda, “Anomalous Surface Reflection of X-rays,” Phys. Rev. 131, 2010 (1963).
    [CrossRef]

1987

F. E. Christensen et al., “A Versatile Three/Four Crystal X-ray Diffractometer for X-ray Optical Elements. Performance and Applications,” Nucl. Intrum. Methods A256, 381 (1987).
[CrossRef]

R. Schiffer, “Reflectivity of a Slightly Rough Surface,” Appl. Opt. 26, 704 (1987).
[CrossRef] [PubMed]

1986

E. L. Church, P. Z. Takacs, “The Interpretation of Glancing Incidence Scattering Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, preprint, (1986).

E. L. Church, P. Z. Takacs, “Statistical and Signal Processing Concepts in Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, preprint (1986).

E. L. Church, “Comments on the Correlation Length,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 102 (1986).

1985

R. Klos, “X-ray Scattering Theories,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 35 (1985).

L. A. Smirnov, T. D. Sotnikova, Yu. I. Kogan, “Diffuse Scattering in Total External Reflection of X-rays from a Rough Surface,” Opt. Spectrosc. USSR 58, 239 (1985).

1984

E. L. Church, “Role of Surface Topography in X-ray Scattering,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 96 (1984).

1981

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, ATP Conf. Proc. 75, Appendix (1981).

D. H. Bilderback, “Reflectance of X-ray Mirrors from 3.8 to 50 keV (3.3 to 0.25Å),” Proc. Soc. Photo-Opt. Instrum. Eng. 315, 90 (1981).

1979

L. A. Smirnov, T. D. Sotnikova, B. S. Anokhin, B. Z. Taibin, “Total External Reflection of X-rays from Rough Surfaces,” Opt. Spektrosk. 46, 593 (1979) [Opt. Spectrosc. (USSR) 46, 329 (1979)].

1972

P. Croce, L. Nevot, B. Pardo, “Sur l’etude de la diffusion de rayons x sous angles resantes,” C. R. Acad. Sci. Paris Ser. B 274, 855 (1972).

1965

A. N. Nigam, “Origin of Anomalous Surface Reflection of X-rays,” Phys. Rev. 138, A1189 (1965).
[CrossRef]

1963

Y. Yoneda, “Anomalous Surface Reflection of X-rays,” Phys. Rev. 131, 2010 (1963).
[CrossRef]

1954

L. G. Parrat, “Surface Studies of Solids by Total Reflection of X-Rays,” Phys. Rev. 95, 359 (1954).
[CrossRef]

1931

H. Kiessig, Ann. Phys. 10, 769 (1931).
[CrossRef]

Allison, S. K.

A. H. Compton, S. K. Allison, X-rays in Theory and Experiment (Van Nostrand, New York, 1935).

Als-Nielsen, J.

J. Als-Nielsen, “Solid and Liquid Surfaces Studied by Synchrotron X-ray Diffraction,” in Structures and Dynamics of Surfaces, Vol. 2 (Springer-Verlag, New York, 1986).

Anokhin, B. S.

L. A. Smirnov, T. D. Sotnikova, B. S. Anokhin, B. Z. Taibin, “Total External Reflection of X-rays from Rough Surfaces,” Opt. Spektrosk. 46, 593 (1979) [Opt. Spectrosc. (USSR) 46, 329 (1979)].

Beckmann, P.

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

Bilderback, D. H.

D. H. Bilderback, “Reflectance of X-ray Mirrors from 3.8 to 50 keV (3.3 to 0.25Å),” Proc. Soc. Photo-Opt. Instrum. Eng. 315, 90 (1981).

Christensen, F. E.

F. E. Christensen et al., “A Versatile Three/Four Crystal X-ray Diffractometer for X-ray Optical Elements. Performance and Applications,” Nucl. Intrum. Methods A256, 381 (1987).
[CrossRef]

Church, E. L.

E. L. Church, P. Z. Takacs, “The Interpretation of Glancing Incidence Scattering Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, preprint, (1986).

E. L. Church, P. Z. Takacs, “Statistical and Signal Processing Concepts in Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, preprint (1986).

E. L. Church, “Comments on the Correlation Length,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 102 (1986).

E. L. Church, “Role of Surface Topography in X-ray Scattering,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 96 (1984).

Compton, A. H.

A. H. Compton, S. K. Allison, X-rays in Theory and Experiment (Van Nostrand, New York, 1935).

Croce, P.

P. Croce, L. Nevot, B. Pardo, “Sur l’etude de la diffusion de rayons x sous angles resantes,” C. R. Acad. Sci. Paris Ser. B 274, 855 (1972).

Cromer, D. T.

D. T. Cromer, D. Liberman, Los Alamos Scientific Laboratory Report, LA-4403 (1970).

Fujikawa, B. K.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, ATP Conf. Proc. 75, Appendix (1981).

Henke, B. L.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, ATP Conf. Proc. 75, Appendix (1981).

Kiessig, H.

H. Kiessig, Ann. Phys. 10, 769 (1931).
[CrossRef]

Klos, R.

R. Klos, “X-ray Scattering Theories,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 35 (1985).

Kogan, Yu. I.

L. A. Smirnov, T. D. Sotnikova, Yu. I. Kogan, “Diffuse Scattering in Total External Reflection of X-rays from a Rough Surface,” Opt. Spectrosc. USSR 58, 239 (1985).

Lee, P.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, ATP Conf. Proc. 75, Appendix (1981).

Liberman, D.

D. T. Cromer, D. Liberman, Los Alamos Scientific Laboratory Report, LA-4403 (1970).

Nevot, L.

P. Croce, L. Nevot, B. Pardo, “Sur l’etude de la diffusion de rayons x sous angles resantes,” C. R. Acad. Sci. Paris Ser. B 274, 855 (1972).

Nigam, A. N.

A. N. Nigam, “Origin of Anomalous Surface Reflection of X-rays,” Phys. Rev. 138, A1189 (1965).
[CrossRef]

Pardo, B.

P. Croce, L. Nevot, B. Pardo, “Sur l’etude de la diffusion de rayons x sous angles resantes,” C. R. Acad. Sci. Paris Ser. B 274, 855 (1972).

Parrat, L. G.

L. G. Parrat, “Surface Studies of Solids by Total Reflection of X-Rays,” Phys. Rev. 95, 359 (1954).
[CrossRef]

Schiffer, R.

Shimabukuro, R. L.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, ATP Conf. Proc. 75, Appendix (1981).

Smirnov, L. A.

L. A. Smirnov, T. D. Sotnikova, Yu. I. Kogan, “Diffuse Scattering in Total External Reflection of X-rays from a Rough Surface,” Opt. Spectrosc. USSR 58, 239 (1985).

L. A. Smirnov, T. D. Sotnikova, B. S. Anokhin, B. Z. Taibin, “Total External Reflection of X-rays from Rough Surfaces,” Opt. Spektrosk. 46, 593 (1979) [Opt. Spectrosc. (USSR) 46, 329 (1979)].

Sotnikova, T. D.

L. A. Smirnov, T. D. Sotnikova, Yu. I. Kogan, “Diffuse Scattering in Total External Reflection of X-rays from a Rough Surface,” Opt. Spectrosc. USSR 58, 239 (1985).

L. A. Smirnov, T. D. Sotnikova, B. S. Anokhin, B. Z. Taibin, “Total External Reflection of X-rays from Rough Surfaces,” Opt. Spektrosk. 46, 593 (1979) [Opt. Spectrosc. (USSR) 46, 329 (1979)].

Spizzichino, A.

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

Taibin, B. Z.

L. A. Smirnov, T. D. Sotnikova, B. S. Anokhin, B. Z. Taibin, “Total External Reflection of X-rays from Rough Surfaces,” Opt. Spektrosk. 46, 593 (1979) [Opt. Spectrosc. (USSR) 46, 329 (1979)].

Takacs, P. Z.

E. L. Church, P. Z. Takacs, “Statistical and Signal Processing Concepts in Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, preprint (1986).

E. L. Church, P. Z. Takacs, “The Interpretation of Glancing Incidence Scattering Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, preprint, (1986).

Tanaka, T. J.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, ATP Conf. Proc. 75, Appendix (1981).

Yoneda, Y.

Y. Yoneda, “Anomalous Surface Reflection of X-rays,” Phys. Rev. 131, 2010 (1963).
[CrossRef]

Ann. Phys.

H. Kiessig, Ann. Phys. 10, 769 (1931).
[CrossRef]

Appl. Opt.

ATP Conf. Proc.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, ATP Conf. Proc. 75, Appendix (1981).

C. R. Acad. Sci. Paris Ser. B

P. Croce, L. Nevot, B. Pardo, “Sur l’etude de la diffusion de rayons x sous angles resantes,” C. R. Acad. Sci. Paris Ser. B 274, 855 (1972).

Nucl. Intrum. Methods

F. E. Christensen et al., “A Versatile Three/Four Crystal X-ray Diffractometer for X-ray Optical Elements. Performance and Applications,” Nucl. Intrum. Methods A256, 381 (1987).
[CrossRef]

Opt. Spectrosc. USSR

L. A. Smirnov, T. D. Sotnikova, Yu. I. Kogan, “Diffuse Scattering in Total External Reflection of X-rays from a Rough Surface,” Opt. Spectrosc. USSR 58, 239 (1985).

Opt. Spektrosk.

L. A. Smirnov, T. D. Sotnikova, B. S. Anokhin, B. Z. Taibin, “Total External Reflection of X-rays from Rough Surfaces,” Opt. Spektrosk. 46, 593 (1979) [Opt. Spectrosc. (USSR) 46, 329 (1979)].

Phys. Rev.

L. G. Parrat, “Surface Studies of Solids by Total Reflection of X-Rays,” Phys. Rev. 95, 359 (1954).
[CrossRef]

A. N. Nigam, “Origin of Anomalous Surface Reflection of X-rays,” Phys. Rev. 138, A1189 (1965).
[CrossRef]

Y. Yoneda, “Anomalous Surface Reflection of X-rays,” Phys. Rev. 131, 2010 (1963).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

E. L. Church, “Role of Surface Topography in X-ray Scattering,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 96 (1984).

R. Klos, “X-ray Scattering Theories,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 35 (1985).

E. L. Church, P. Z. Takacs, “The Interpretation of Glancing Incidence Scattering Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, preprint, (1986).

E. L. Church, P. Z. Takacs, “Statistical and Signal Processing Concepts in Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, preprint (1986).

E. L. Church, “Comments on the Correlation Length,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 102 (1986).

D. H. Bilderback, “Reflectance of X-ray Mirrors from 3.8 to 50 keV (3.3 to 0.25Å),” Proc. Soc. Photo-Opt. Instrum. Eng. 315, 90 (1981).

Other

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

A. H. Compton, S. K. Allison, X-rays in Theory and Experiment (Van Nostrand, New York, 1935).

D. T. Cromer, D. Liberman, Los Alamos Scientific Laboratory Report, LA-4403 (1970).

J. Als-Nielsen, “Solid and Liquid Surfaces Studied by Synchrotron X-ray Diffraction,” in Structures and Dynamics of Surfaces, Vol. 2 (Springer-Verlag, New York, 1986).

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Figures (9)

Fig. 1
Fig. 1

Experimental arrangements: (a) high resolution channelcut perfect crystal arrangement; (b) integrated reflectivity arrangement.

Fig. 2
Fig. 2

High resolution scattering data for the Pt mirror: θi = 0.05°; θi = 0.1°; θi = 0.2°; and θi = 0.3°. Filled points are scattering away from mirror. Diamonds are scattering toward mirror. Full and dash–dot lines are fit, the full line being away from mirror. The dashed line is the resolution curve.

Fig. 3
Fig. 3

Same as Fig. 2 for θi = 0.4°, θi = 0.5°, θi = 0.6°, θi = 0.7°.

Fig. 4
Fig. 4

High resolution scattering data for the EXOSAT Au test mirror: θi = 0.4, 0.5, and 0.65°. Symbols as in Fig. 2.

Fig. 5
Fig. 5

High resolution scattering data for the ROSAT Au test mirror: θi = 0.05°; θi = 0.2°; θi = 0.65°. Symbols as in Fig. 2.

Fig. 6
Fig. 6

(a) Integrated reflectivity of the Italian Au standard mirror. The full line is the fit with the following parameters: σ = 15.5 Å; thickness = 507 Å. (b) Specular (lower points) and integrated (upper points) reflectivity of the ROSAT mirror. The dashed line is in both cases the classical Fresnel curve assuming a thick Au layer.

Fig. 7
Fig. 7

Low resolution scattering data from the ROSAT mirror. Incident angles are given in the figure.

Fig. 8
Fig. 8

Surface scattering geometry.

Fig. 9
Fig. 9

Three fits to the Pt 0.3° data. It is clear that the modified Bessel correlation function provides the best fit to the data.

Tables (4)

Tables Icon

Table I Test Flats Discussed in the Text

Tables Icon

Table II Parameters of Various Correlation Function Fits

Tables Icon

Table III Overview of Pt Data Analysis

Tables Icon

Table IV Overview of EXOSAT Au and ROSAT Au Analysis

Equations (13)

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

I M ( θ , ϕ ) = W ( θ S , ϕ S ) R i ( θ θ S ) R 0 ( ϕ ϕ S ) d w , d w = cos θ S d θ S d ϕ S .
I M ( θ ) = S ( θ S ) R i ( θ θ S ) d θ S ,
S ( θ S ) G 2 exp ( γ ) [ sinc 2 ( p L ) + γ L cos ( p τ ) C ( τ ) d τ ] ,
σ 2 C ( τ ) = 1 2 L L L Z ( x ) Z ( x + τ ) d x ; L ,
p = k · ( cos θ S cos θ i ) ; k = 2 π / λ ,
Δ 2 γ = ( 2 k σ sin θ i ) 2 , G = F R f ( θ i ) , F = 1 sin θ i 1 cos ( θ i + θ S ) sin θ i + sin θ S ,
K B . S . = [ 1 2 ( 1 + θ S θ i ) ] 4 .
K R . R . = ( θ S θ i ) R ( θ ) S R ( θ ) i .
I . exponential c ( τ ) exp ( | τ | ξ ) ; II . Gaussian c ( τ ) exp ( 1 2 ) ( τ ξ ) 2 ; III . modified Bessel 10 c ( τ ) | τ | ν K ν ( τ ξ ) .
exponential : S ( θ S ) [ sinc 2 ( p 1 ) + γ L · ξ ( 1 + p ξ ) 2 ] ,
Gaussian : S ( θ S ) { sinc 2 ( p 1 ) + γ L ξ π 2 exp [ 1 2 ( p ξ ) 2 ] } ,
modified Bessel : S ( θ S ) { sinc 2 ( p 1 ) + γ L π Γ ( ν + 1 2 ) Γ ( ν ) ξ [ 1 + ( p ξ ) 2 ] ν + 1 2 } ,
Δ θ = λ d θ i ,

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