Abstract

There is still great interest in the determination of microtopographic properties of rough metallic surfaces from light scattering measurements. According to Beckmann–Kirchhoff theory a clear relationship is established between the in-plane angular scattered light intensity and the statistical properties of the surface. We discuss one way to invert this relationship, and we introduce a new iterative procedure to retrieve the height autocorrelation function even for a very rough metallic surface (rms surface roughness of the same order of the optical wavelength). The procedure is eventually applied to the experimental data of a known metallic surface for validation.

© 2009 Optical Society of America

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References

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  1. Lord Rayleigh, The Theory of Sound (Macmillan, 1896), Vol. II.
  2. Lord Rayleigh, “Polish,” Nature (London) 64, 385-388 (1901).
    [CrossRef]
  3. U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld's waves),” J. Opt. Soc. Am. 31, 213-222 (1941).
    [CrossRef]
  4. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, p. 351-378 (1951).
    [CrossRef]
  5. I. M. Fuks, “Theory of radio wave scattering at a rough sea surface,” Sov. Radiophys. 9, 513-519 (1966).
    [CrossRef]
  6. L. M. Brekhovskikh, “Difrakcya voln na nerovnoj poverhnosti: 1, obschaya teoriya,” Zh. Eksp. Teor. Fiz. 23, 275-288 (1952).
  7. M. A. Isakovich, “Wave scattering from a statistically rough surface,” Zh. Eksp. Teor. Fiz. 23, 305-314 (1952).
  8. C. Eckart, “The scattering of sound from the sea surface,” J. Acoust. Soc. Am. 25, 66-570 (1953).
    [CrossRef]
  9. H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. 101, 209-214 (1954).
  10. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).
  11. T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1-R40 (2004).
    [CrossRef]
  12. J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering theory for rough surfaces with large scattering and incident angles,” Opt. Eng. 46, 078002 (2007).
    [CrossRef]
  13. Y. Sun, “Statistical ray method for deriving reflection models of rough surfaces,” J. Opt. Soc. Am. A 24, 724-744 (2007).
    [CrossRef]
  14. J.-J. Greffet and M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmoltz's reciprocity principle and Kirchhoff's law,” J. Opt. Soc. Am. A 15, 2735-2744 (1998).
    [CrossRef]
  15. E. Marx, B. Leridon, T. R. Lettieri, J.-F. Song, and T. V. Vorburger, “Autocorrelation functions from optical scattering for one-dimensionally rough surfaces,” Appl. Opt. 32, 67-76 (1993).
    [CrossRef] [PubMed]
  16. E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125-136 (1979).
  17. J. M. Elson and J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116-124 (1979).
  18. J. C. Stover, S. A. Serati, and C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406-412 (1984).
  19. J. M. Elson and J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. 69, 31-47 (1979).
    [CrossRef]
  20. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965): ∫0∞Jo(βx)exp(−αx)xdx=1/[α2(1+(β/α)2)3/2].
  21. The intensity profiles are calculated numerically from Eq. , with particular care taken to check the convergence of the integral in the domain [0, ∞). In practice the domain of integration may be truncated to [0,L=1 mm] where L is the spot size. The integration of the autocorrelation function C(τ) is carefully evaluated only in the initial domain [0,τmax=10 μm] with fixed step of 10 nm, while in the final domain [τmax,L], to save computational time, we may reasonably assume C=0 during the integration of the χ function. Nevertheless, in the lucky case of the linear autocorrelation function Cb, the intensity profile may be expressed in the exact form in Eq. . For this reason we use this case as a reference case to check the validity of the inversion procedures.
  22. P. C. Hansen, “The truncated SVD as a method for regularization,” BIT 27, 534-553 (1987).
    [CrossRef]
  23. R. Li Voti, C. Sibilia, and M. Bertolotti, “Photothermal depth profiling by genetic algorithms and thermal wave backscattering,” Int. J. Thermophys. 26, 1833-1848 (2005).
    [CrossRef]
  24. M. Tomoda, R. Li Voti, O. Matsuda, and O. B. Wright, “Tomographic reconstruction of picosecond acoustic strain propagation,” Appl. Phys. Lett. 90, 041114 (2007).
    [CrossRef]
  25. K. SealA. K. Sarychev, H. Noh, D. A. Genov, A. Yamilov, V. M. Shalaev, Z. C. Ying, and H. Cao, “Near-field intensity correlations in semicontinuous metal-dielectric films,”Phys. Rev. Lett. 94, 226101 (2005).
    [CrossRef] [PubMed]

2007 (3)

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering theory for rough surfaces with large scattering and incident angles,” Opt. Eng. 46, 078002 (2007).
[CrossRef]

M. Tomoda, R. Li Voti, O. Matsuda, and O. B. Wright, “Tomographic reconstruction of picosecond acoustic strain propagation,” Appl. Phys. Lett. 90, 041114 (2007).
[CrossRef]

Y. Sun, “Statistical ray method for deriving reflection models of rough surfaces,” J. Opt. Soc. Am. A 24, 724-744 (2007).
[CrossRef]

2005 (2)

R. Li Voti, C. Sibilia, and M. Bertolotti, “Photothermal depth profiling by genetic algorithms and thermal wave backscattering,” Int. J. Thermophys. 26, 1833-1848 (2005).
[CrossRef]

K. SealA. K. Sarychev, H. Noh, D. A. Genov, A. Yamilov, V. M. Shalaev, Z. C. Ying, and H. Cao, “Near-field intensity correlations in semicontinuous metal-dielectric films,”Phys. Rev. Lett. 94, 226101 (2005).
[CrossRef] [PubMed]

2004 (1)

T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1-R40 (2004).
[CrossRef]

1998 (1)

1993 (1)

1987 (1)

P. C. Hansen, “The truncated SVD as a method for regularization,” BIT 27, 534-553 (1987).
[CrossRef]

1984 (1)

J. C. Stover, S. A. Serati, and C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406-412 (1984).

1979 (3)

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125-136 (1979).

J. M. Elson and J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116-124 (1979).

J. M. Elson and J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. 69, 31-47 (1979).
[CrossRef]

1966 (1)

I. M. Fuks, “Theory of radio wave scattering at a rough sea surface,” Sov. Radiophys. 9, 513-519 (1966).
[CrossRef]

1954 (1)

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. 101, 209-214 (1954).

1953 (1)

C. Eckart, “The scattering of sound from the sea surface,” J. Acoust. Soc. Am. 25, 66-570 (1953).
[CrossRef]

1952 (2)

L. M. Brekhovskikh, “Difrakcya voln na nerovnoj poverhnosti: 1, obschaya teoriya,” Zh. Eksp. Teor. Fiz. 23, 275-288 (1952).

M. A. Isakovich, “Wave scattering from a statistically rough surface,” Zh. Eksp. Teor. Fiz. 23, 305-314 (1952).

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, p. 351-378 (1951).
[CrossRef]

1941 (1)

1901 (1)

Lord Rayleigh, “Polish,” Nature (London) 64, 385-388 (1901).
[CrossRef]

Beckmann, P.

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

Bennett, J. M.

Bertolotti, M.

R. Li Voti, C. Sibilia, and M. Bertolotti, “Photothermal depth profiling by genetic algorithms and thermal wave backscattering,” Int. J. Thermophys. 26, 1833-1848 (2005).
[CrossRef]

Brekhovskikh, L. M.

L. M. Brekhovskikh, “Difrakcya voln na nerovnoj poverhnosti: 1, obschaya teoriya,” Zh. Eksp. Teor. Fiz. 23, 275-288 (1952).

Cao, H.

K. SealA. K. Sarychev, H. Noh, D. A. Genov, A. Yamilov, V. M. Shalaev, Z. C. Ying, and H. Cao, “Near-field intensity correlations in semicontinuous metal-dielectric films,”Phys. Rev. Lett. 94, 226101 (2005).
[CrossRef] [PubMed]

Church, E. L.

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125-136 (1979).

Davies, H.

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. 101, 209-214 (1954).

Eckart, C.

C. Eckart, “The scattering of sound from the sea surface,” J. Acoust. Soc. Am. 25, 66-570 (1953).
[CrossRef]

Elfouhaily, T. M.

T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1-R40 (2004).
[CrossRef]

Elson, J. M.

Fano, U.

Fuks, I. M.

I. M. Fuks, “Theory of radio wave scattering at a rough sea surface,” Sov. Radiophys. 9, 513-519 (1966).
[CrossRef]

Genov, D. A.

K. SealA. K. Sarychev, H. Noh, D. A. Genov, A. Yamilov, V. M. Shalaev, Z. C. Ying, and H. Cao, “Near-field intensity correlations in semicontinuous metal-dielectric films,”Phys. Rev. Lett. 94, 226101 (2005).
[CrossRef] [PubMed]

Gillespie, C. H.

J. C. Stover, S. A. Serati, and C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406-412 (1984).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965): ∫0∞Jo(βx)exp(−αx)xdx=1/[α2(1+(β/α)2)3/2].

Greffet, J.-J.

Guerin, C. A.

T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1-R40 (2004).
[CrossRef]

Hansen, P. C.

P. C. Hansen, “The truncated SVD as a method for regularization,” BIT 27, 534-553 (1987).
[CrossRef]

Harvey, J. E.

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering theory for rough surfaces with large scattering and incident angles,” Opt. Eng. 46, 078002 (2007).
[CrossRef]

Isakovich, M. A.

M. A. Isakovich, “Wave scattering from a statistically rough surface,” Zh. Eksp. Teor. Fiz. 23, 305-314 (1952).

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125-136 (1979).

Krywonos, A.

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering theory for rough surfaces with large scattering and incident angles,” Opt. Eng. 46, 078002 (2007).
[CrossRef]

Leridon, B.

Lettieri, T. R.

Li Voti, R.

M. Tomoda, R. Li Voti, O. Matsuda, and O. B. Wright, “Tomographic reconstruction of picosecond acoustic strain propagation,” Appl. Phys. Lett. 90, 041114 (2007).
[CrossRef]

R. Li Voti, C. Sibilia, and M. Bertolotti, “Photothermal depth profiling by genetic algorithms and thermal wave backscattering,” Int. J. Thermophys. 26, 1833-1848 (2005).
[CrossRef]

Marx, E.

Matsuda, O.

M. Tomoda, R. Li Voti, O. Matsuda, and O. B. Wright, “Tomographic reconstruction of picosecond acoustic strain propagation,” Appl. Phys. Lett. 90, 041114 (2007).
[CrossRef]

Nieto-Vesperinas, M.

Noh, H.

K. SealA. K. Sarychev, H. Noh, D. A. Genov, A. Yamilov, V. M. Shalaev, Z. C. Ying, and H. Cao, “Near-field intensity correlations in semicontinuous metal-dielectric films,”Phys. Rev. Lett. 94, 226101 (2005).
[CrossRef] [PubMed]

Rayleigh, Lord

Lord Rayleigh, “Polish,” Nature (London) 64, 385-388 (1901).
[CrossRef]

Lord Rayleigh, The Theory of Sound (Macmillan, 1896), Vol. II.

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, p. 351-378 (1951).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965): ∫0∞Jo(βx)exp(−αx)xdx=1/[α2(1+(β/α)2)3/2].

Sarychev, A. K.

K. SealA. K. Sarychev, H. Noh, D. A. Genov, A. Yamilov, V. M. Shalaev, Z. C. Ying, and H. Cao, “Near-field intensity correlations in semicontinuous metal-dielectric films,”Phys. Rev. Lett. 94, 226101 (2005).
[CrossRef] [PubMed]

Seal, K.

K. SealA. K. Sarychev, H. Noh, D. A. Genov, A. Yamilov, V. M. Shalaev, Z. C. Ying, and H. Cao, “Near-field intensity correlations in semicontinuous metal-dielectric films,”Phys. Rev. Lett. 94, 226101 (2005).
[CrossRef] [PubMed]

Serati, S. A.

J. C. Stover, S. A. Serati, and C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406-412 (1984).

Shalaev, V. M.

K. SealA. K. Sarychev, H. Noh, D. A. Genov, A. Yamilov, V. M. Shalaev, Z. C. Ying, and H. Cao, “Near-field intensity correlations in semicontinuous metal-dielectric films,”Phys. Rev. Lett. 94, 226101 (2005).
[CrossRef] [PubMed]

Sibilia, C.

R. Li Voti, C. Sibilia, and M. Bertolotti, “Photothermal depth profiling by genetic algorithms and thermal wave backscattering,” Int. J. Thermophys. 26, 1833-1848 (2005).
[CrossRef]

Song, J.-F.

Spizzichino, A.

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

Stover, J. C.

J. C. Stover, S. A. Serati, and C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406-412 (1984).

Sun, Y.

Tomoda, M.

M. Tomoda, R. Li Voti, O. Matsuda, and O. B. Wright, “Tomographic reconstruction of picosecond acoustic strain propagation,” Appl. Phys. Lett. 90, 041114 (2007).
[CrossRef]

Vernold, C. L.

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering theory for rough surfaces with large scattering and incident angles,” Opt. Eng. 46, 078002 (2007).
[CrossRef]

Vorburger, T. V.

Wright, O. B.

M. Tomoda, R. Li Voti, O. Matsuda, and O. B. Wright, “Tomographic reconstruction of picosecond acoustic strain propagation,” Appl. Phys. Lett. 90, 041114 (2007).
[CrossRef]

Yamilov, A.

K. SealA. K. Sarychev, H. Noh, D. A. Genov, A. Yamilov, V. M. Shalaev, Z. C. Ying, and H. Cao, “Near-field intensity correlations in semicontinuous metal-dielectric films,”Phys. Rev. Lett. 94, 226101 (2005).
[CrossRef] [PubMed]

Ying, Z. C.

K. SealA. K. Sarychev, H. Noh, D. A. Genov, A. Yamilov, V. M. Shalaev, Z. C. Ying, and H. Cao, “Near-field intensity correlations in semicontinuous metal-dielectric films,”Phys. Rev. Lett. 94, 226101 (2005).
[CrossRef] [PubMed]

Zavada, J. M.

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125-136 (1979).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

M. Tomoda, R. Li Voti, O. Matsuda, and O. B. Wright, “Tomographic reconstruction of picosecond acoustic strain propagation,” Appl. Phys. Lett. 90, 041114 (2007).
[CrossRef]

BIT (1)

P. C. Hansen, “The truncated SVD as a method for regularization,” BIT 27, 534-553 (1987).
[CrossRef]

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, p. 351-378 (1951).
[CrossRef]

Int. J. Thermophys. (1)

R. Li Voti, C. Sibilia, and M. Bertolotti, “Photothermal depth profiling by genetic algorithms and thermal wave backscattering,” Int. J. Thermophys. 26, 1833-1848 (2005).
[CrossRef]

J. Acoust. Soc. Am. (1)

C. Eckart, “The scattering of sound from the sea surface,” J. Acoust. Soc. Am. 25, 66-570 (1953).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Nature (London) (1)

Lord Rayleigh, “Polish,” Nature (London) 64, 385-388 (1901).
[CrossRef]

Opt. Eng. (4)

E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125-136 (1979).

J. M. Elson and J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116-124 (1979).

J. C. Stover, S. A. Serati, and C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406-412 (1984).

J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering theory for rough surfaces with large scattering and incident angles,” Opt. Eng. 46, 078002 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

K. SealA. K. Sarychev, H. Noh, D. A. Genov, A. Yamilov, V. M. Shalaev, Z. C. Ying, and H. Cao, “Near-field intensity correlations in semicontinuous metal-dielectric films,”Phys. Rev. Lett. 94, 226101 (2005).
[CrossRef] [PubMed]

Proc. Inst. Electr. Eng. (1)

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. 101, 209-214 (1954).

Sov. Radiophys. (1)

I. M. Fuks, “Theory of radio wave scattering at a rough sea surface,” Sov. Radiophys. 9, 513-519 (1966).
[CrossRef]

Waves Random Media (1)

T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1-R40 (2004).
[CrossRef]

Zh. Eksp. Teor. Fiz. (2)

L. M. Brekhovskikh, “Difrakcya voln na nerovnoj poverhnosti: 1, obschaya teoriya,” Zh. Eksp. Teor. Fiz. 23, 275-288 (1952).

M. A. Isakovich, “Wave scattering from a statistically rough surface,” Zh. Eksp. Teor. Fiz. 23, 305-314 (1952).

Other (4)

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

Lord Rayleigh, The Theory of Sound (Macmillan, 1896), Vol. II.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965): ∫0∞Jo(βx)exp(−αx)xdx=1/[α2(1+(β/α)2)3/2].

The intensity profiles are calculated numerically from Eq. , with particular care taken to check the convergence of the integral in the domain [0, ∞). In practice the domain of integration may be truncated to [0,L=1 mm] where L is the spot size. The integration of the autocorrelation function C(τ) is carefully evaluated only in the initial domain [0,τmax=10 μm] with fixed step of 10 nm, while in the final domain [τmax,L], to save computational time, we may reasonably assume C=0 during the integration of the χ function. Nevertheless, in the lucky case of the linear autocorrelation function Cb, the intensity profile may be expressed in the exact form in Eq. . For this reason we use this case as a reference case to check the validity of the inversion procedures.

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Figures (13)

Fig. 1
Fig. 1

Surface reflection is closely related to the microscopic surface profile and to the direction of the incident wave.

Fig. 2
Fig. 2

Normalized angular scattered intensity versus scattering angle (degrees) for λ = 633 nm and autocorrelation length Λ = 10 μ m . Curve (a) σ = 200 nm and C a ( τ ) = exp ( τ Λ ) ; curve (b) σ = 200 nm and C b ( τ ) = 1 τ Λ ; dashed curve σ = 300 nm (the linear and the exponential case merge).

Fig. 3
Fig. 3

(a) Autocorrelation function versus distance: solid curve, C = 1 τ Λ with Λ = 10 μ m ; filled circles, reconstructed C IF for σ = 200 nm ; inverted triangles, reconstructed C IF for σ = 300 nm . Inset, magnification of the initial behavior. (b) χ function versus distance for the same case as (a): lines represent the calculated χ from Eq. (7) for σ = 200 nm (solid line) and σ = 300 nm (dashed line); symbols represent the reconstructed χ IF for σ = 200 nm (filled circles) and σ = 300 nm (inverted triangles).

Fig. 4
Fig. 4

Normalized scattering intensity versus scattering angle for the same case as in Fig. 3; curves represent the normalized intensity calculated by Eq. (6) with σ = 200 nm (solid), and σ = 300 nm (dashed); symbols represent the reconstructed I IF for σ = 200 nm (filled circles) and σ = 300 nm (inverted triangles).

Fig. 5
Fig. 5

(a) Singular functions versus distance (micrometers) for the same case as in Fig. 3: curves (1) k = 1 ; (2) k = 2 , (3) k = 20 . (b) Weight of the kth singular function versus index k calculated by Eq. (12).

Fig. 6
Fig. 6

(a) Reconstructed autocorrelation function versus distance for the case of Fig. 3: (1) linear profile; circles, C IF ; (2) C reconstructed by SVD with k opt = 17 ; (3) C reconstructed by SVD with k = 20 . (b) Normalized scattering intensity versus scattering angle: (1) calculated by Eq. (6); circles, I IF ; (2) reconstructed intensity with k opt = 17 ; (3) reconstructed intensity with k = 20 .

Fig. 7
Fig. 7

(a) Reconstructed autocorrelation function versus distance for the case of Fig. 3: solid line, linear profile; circles, C IF ; (1) reconstructed C 1 after the application of one cycle; (2) reconstructed C 2 after the application of two cycles; (3) reconstructed C 3 after the application of four cycles. (b) Normalized scattering intensity versus scattering angle: symbols and curves are as in (a).

Fig. 8
Fig. 8

(a) AFM xy surface image of the Pd sample. The scanned area is 15 μ m × 15 μ m . (b) Same AFM image in 3D. (c) Stylus scan over a 3 mm line. The surface roughness (micrometers) is plotted versus the offset (micrometers). Insets, magnifications of five different zones A, D, E, G, I.

Fig. 9
Fig. 9

Normalized autocorrelation function versus autocorrelation distance τ. The autocorrelation ζ ( x ) ζ ( x + τ ) is calculated from the data shown in Fig. 8c in the five zones labeled A, D, E, G, I; av represents the averaged autocorrelation curve.

Fig. 10
Fig. 10

(a) Schematic setup and (b) experimental setup for the in-plane light scattering measurements.

Fig. 11
Fig. 11

In-plane scattered light intensity versus scattering angle for Pd sample. The symbols refer to different incident angles: circles, θ i = 0 ° ; squares, θ i = 34 ° ; inverted triangles, θ i = 50 ° .

Fig. 12
Fig. 12

(a) Normalized autocorrelation function versus autocorrelation distance τ for Pd sample: solid curve, averaged autocorrelation function calculated by stylus as shown in Fig. 9; circles, C IF autocorrelation calculated by Eq. (9); curve (1) smooting of C IF . (b) In-plane normalized scattered light intensity versus scattering angle (degrees) for Pd sample and for θ i = 0 ° : open circles,k experimental results; filled circles, I IF calculated from C IF in (a); (1) I IF calculated by smooting C IF ; solid curve, I stylus calculated from C stylus in (a).

Fig. 13
Fig. 13

(a) Reconstructed autocorrelation function versus autocorrelation distance τ, for Pd sample: solid curve, averaged autocorrelation function calculated by stylus as shown in Fig. 9; circles, C IF autocorrelation calculated by Eq. (9); dashed curve, reconstructed C 1 profile after the application of one SVD cycle; solid curve, reconstructed C 2 profile after the application of two SVD cycles. (b) In-plane normalized scattered light intensity versus scattering angle (degrees) for Pd sample and for θ i = 0 ° : open circles, experimental results; filled circles, I IF calculated from C IF ; dashed curve, intensity I C 1 calculated from C 1 ; solid curve, intensity I C 2 calculated from C 2 .

Tables (1)

Tables Icon

Table 1 Results from Stylus Measurements a

Equations (13)

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

V B = A exp i k i ρ ,
ψ = exp [ i k r ρ ] r ρ exp i k r 2 + ρ 2 2 r ρ r exp [ i k r i k o ρ ] r ,
V P = 1 4 π S ( V B , tot ψ n ψ V B , tot n ) d S = F D S exp [ i ( k i k o ) ρ ] d S ,
I P = V P V P * = F 2 S S exp [ i ( k i k o ) ( ρ ρ ) ] d S d S ,
I ( θ o ) = 2 π F 2 0 J o [ 2 π λ ( sin θ i sin θ o ) τ ] χ ( θ i , θ o , σ λ , C ( τ ) ) τ d τ ,
I b ( θ o ) = 2 π F 2 μ 2 ( cos θ i + cos θ o ) 4 ( λ 2 π ) 2 1 [ 1 + ( sin θ i sin θ o ) 2 ( cos θ i + cos θ o ) 4 μ 2 ] 3 2 ,
χ exp [ ( 4 π σ cos θ i λ ) 2 ( 1 C ( τ ) ) ] .
χ IF ( τ ) = 2 π 0 J o ( 2 π p τ ) I ( p ) F 2 p d p ,
C IF ( τ ) = 1 ( λ 4 π σ cos θ i ) 2 ln ( χ IF ( 0 ) χ IF ( τ ) ) .
I = 2 π F 2 0 J o ( 2 π p τ ) exp [ g ( 1 C IF Δ C ) ] τ d τ ,
Δ I = 2 π F 2 0 J o ( 2 π p τ ) exp [ g ( 1 C IF ) ] g Δ C τ d τ .
Δ C ( τ ) = k a k V k ( τ ) , where weights a k = I ( θ o ) , U k ( θ o ) μ k
Δ I n = I I n 1 = 2 π F 2 0 J o ( 2 π p τ ) exp [ g ( 1 C n 1 ) ] g Δ C n τ d τ .

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