Abstract

To estimate the root mean square roughness (σ) of a surface from reflected power, it is necessary to know the diffuse reflectance (DR) and the reflectance (SSR) of a smooth surface made from the same material as the rough surface. In our study, σ is estimated from value of power reflected from one-dimensionally rough steel surfaces in the specular direction without considering SSR and DR. An expression describing dependence of an error of the estimation on SSR and DR is derived. Linear polarized light with λ=660nm and the azimuth of polarization of 49° was used in the experiment. The angle of incidence was varied from 30° to 74°. It was found that absolute relative errors caused by influence of SSR and DR are smaller than 0.03 in the angular ranges of 46-54° and 30-58° for σ=10.2nm and σ = 49.8nm, respectively. Out of these ranges, SSR is the main reason for the errors lying in the wide range of ~0.05-2.5.

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. P. Beckmann, and A. Spizzichino, The scattering of electromagnetic waves from rough surfaces (Pergamon Press, 1963), Chap. 5, Eq. (54) on p. 88 and Eq. (8) on p. 98.
  14. M. Born, and E. Wolf, Principles of Optics (Pergamon Press, 1968), Chap. 1.
  15. GOST 9378–75, Standards of a rough surface (in Russian).
  16. ISO 4288:1996(E), Geometrical product specifications (GPS) - Surface texture: Profile method - Rules and procedures for the assessment of surface texture.
  17. K. A. O’Donnell and E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4(7), 1194–1205 (1987).
    [CrossRef]

2010

A. Melninkaitis, T. Tolenis, L. Mažulė, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandrė, S. Kićas, and R. Drazdys, “Complex study of zirconia-silica and niobia-silica composite coatings produced by ion beam sputtering,” Proc. SPIE 7842, 784203, 784203-13 (2010).
[CrossRef]

2006

2002

S.-M. F. Nee and T.-W. Nee, “Principal Mueller matrix of reflection and scattering measured for a one-dimensional rough surface,” Opt. Eng. 41(5), 994–1001 (2002).
[CrossRef]

2001

1998

S.-M. F. Nee and T.-W. Nee, “Polarization of scattering by rough surfaces,” Proc. SPIE 3426, 169–180 (1998).
[CrossRef]

1997

M. Bjuggren, L. Krummrnacher, and L. Mattsson, “Noncontact surface roughness measurement of engineering surfaces by total integrating infrared scattering,” Precis. Eng. 20(1), 33–45 (1997).
[CrossRef]

1992

1990

E. Marx and T. V. Vorburger, “Direct and inverse problems for light scattered by rough surfaces,” Appl. Opt. 29(25), 3613–3626 (1990).
[CrossRef] [PubMed]

S. F. Nee and H. E. Bennett, “Characterization of optical blacks by infrared ellipsometry and reflectometry,” Proc. SPIE 1331, 249–260 (1990).
[CrossRef]

1987

1978

H. E. Bennett, “Scattering characteristics of optical materials,” Opt. Eng. 17, 480–488 (1978).

1972

1961

H. E. Bennett and J. O. Porteus “Relation between roughness and specular reflectance at normal incidence,” JOSA 51, 123–129 (1961), Eqs. (1) and (5).
[CrossRef]

Ahmad, A.

Bennett, H. E.

S. F. Nee and H. E. Bennett, “Characterization of optical blacks by infrared ellipsometry and reflectometry,” Proc. SPIE 1331, 249–260 (1990).
[CrossRef]

H. E. Bennett, “Scattering characteristics of optical materials,” Opt. Eng. 17, 480–488 (1978).

H. E. Bennett and J. O. Porteus “Relation between roughness and specular reflectance at normal incidence,” JOSA 51, 123–129 (1961), Eqs. (1) and (5).
[CrossRef]

Bennett, J. M.

Bjuggren, M.

M. Bjuggren, L. Krummrnacher, and L. Mattsson, “Noncontact surface roughness measurement of engineering surfaces by total integrating infrared scattering,” Precis. Eng. 20(1), 33–45 (1997).
[CrossRef]

Chipman, R. A.

Cho, H.-J.

Commandre, M.

A. Melninkaitis, T. Tolenis, L. Mažulė, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandrė, S. Kićas, and R. Drazdys, “Complex study of zirconia-silica and niobia-silica composite coatings produced by ion beam sputtering,” Proc. SPIE 7842, 784203, 784203-13 (2010).
[CrossRef]

Crandall, D. G.

Drazdys, R.

A. Melninkaitis, T. Tolenis, L. Mažulė, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandrė, S. Kićas, and R. Drazdys, “Complex study of zirconia-silica and niobia-silica composite coatings produced by ion beam sputtering,” Proc. SPIE 7842, 784203, 784203-13 (2010).
[CrossRef]

Fu, X.

A. Melninkaitis, T. Tolenis, L. Mažulė, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandrė, S. Kićas, and R. Drazdys, “Complex study of zirconia-silica and niobia-silica composite coatings produced by ion beam sputtering,” Proc. SPIE 7842, 784203, 784203-13 (2010).
[CrossRef]

Gallais, L.

A. Melninkaitis, T. Tolenis, L. Mažulė, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandrė, S. Kićas, and R. Drazdys, “Complex study of zirconia-silica and niobia-silica composite coatings produced by ion beam sputtering,” Proc. SPIE 7842, 784203, 784203-13 (2010).
[CrossRef]

Hadaway, J. B.

Hensler, D. H.

Hummer, L. L.

Kicas, S.

A. Melninkaitis, T. Tolenis, L. Mažulė, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandrė, S. Kićas, and R. Drazdys, “Complex study of zirconia-silica and niobia-silica composite coatings produced by ion beam sputtering,” Proc. SPIE 7842, 784203, 784203-13 (2010).
[CrossRef]

Krummrnacher, L.

M. Bjuggren, L. Krummrnacher, and L. Mattsson, “Noncontact surface roughness measurement of engineering surfaces by total integrating infrared scattering,” Precis. Eng. 20(1), 33–45 (1997).
[CrossRef]

Lee, J.-C.

Mangote, B.

A. Melninkaitis, T. Tolenis, L. Mažulė, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandrė, S. Kićas, and R. Drazdys, “Complex study of zirconia-silica and niobia-silica composite coatings produced by ion beam sputtering,” Proc. SPIE 7842, 784203, 784203-13 (2010).
[CrossRef]

Marx, E.

Mattsson, L.

M. Bjuggren, L. Krummrnacher, and L. Mattsson, “Noncontact surface roughness measurement of engineering surfaces by total integrating infrared scattering,” Precis. Eng. 20(1), 33–45 (1997).
[CrossRef]

Mažule, L.

A. Melninkaitis, T. Tolenis, L. Mažulė, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandrė, S. Kićas, and R. Drazdys, “Complex study of zirconia-silica and niobia-silica composite coatings produced by ion beam sputtering,” Proc. SPIE 7842, 784203, 784203-13 (2010).
[CrossRef]

Melninkaitis, A.

A. Melninkaitis, T. Tolenis, L. Mažulė, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandrė, S. Kićas, and R. Drazdys, “Complex study of zirconia-silica and niobia-silica composite coatings produced by ion beam sputtering,” Proc. SPIE 7842, 784203, 784203-13 (2010).
[CrossRef]

Mendez, E. R.

Mirauskas, J.

A. Melninkaitis, T. Tolenis, L. Mažulė, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandrė, S. Kićas, and R. Drazdys, “Complex study of zirconia-silica and niobia-silica composite coatings produced by ion beam sputtering,” Proc. SPIE 7842, 784203, 784203-13 (2010).
[CrossRef]

Nee, S. F.

S. F. Nee and H. E. Bennett, “Characterization of optical blacks by infrared ellipsometry and reflectometry,” Proc. SPIE 1331, 249–260 (1990).
[CrossRef]

Nee, S.-M. F.

S.-M. F. Nee and T.-W. Nee, “Principal Mueller matrix of reflection and scattering measured for a one-dimensional rough surface,” Opt. Eng. 41(5), 994–1001 (2002).
[CrossRef]

S.-M. F. Nee and T.-W. Nee, “Polarization of scattering by rough surfaces,” Proc. SPIE 3426, 169–180 (1998).
[CrossRef]

S.-M. F. Nee, “Ellipsometric view on reflection and scattering from optical blacks,” Appl. Opt. 31(10), 1549–1556 (1992).
[CrossRef] [PubMed]

Nee, T.-W.

S.-M. F. Nee and T.-W. Nee, “Principal Mueller matrix of reflection and scattering measured for a one-dimensional rough surface,” Opt. Eng. 41(5), 994–1001 (2002).
[CrossRef]

S.-M. F. Nee and T.-W. Nee, “Polarization of scattering by rough surfaces,” Proc. SPIE 3426, 169–180 (1998).
[CrossRef]

O’Donnell, K. A.

Pezzaniti, J. L.

Porteus, J. O.

H. E. Bennett and J. O. Porteus “Relation between roughness and specular reflectance at normal incidence,” JOSA 51, 123–129 (1961), Eqs. (1) and (5).
[CrossRef]

Shin, M.-J.

Sirutkaitis, V.

A. Melninkaitis, T. Tolenis, L. Mažulė, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandrė, S. Kićas, and R. Drazdys, “Complex study of zirconia-silica and niobia-silica composite coatings produced by ion beam sputtering,” Proc. SPIE 7842, 784203, 784203-13 (2010).
[CrossRef]

Tolenis, T.

A. Melninkaitis, T. Tolenis, L. Mažulė, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandrė, S. Kićas, and R. Drazdys, “Complex study of zirconia-silica and niobia-silica composite coatings produced by ion beam sputtering,” Proc. SPIE 7842, 784203, 784203-13 (2010).
[CrossRef]

Vorburger, T. V.

Wilkes, D. R.

Zerrad, M.

A. Melninkaitis, T. Tolenis, L. Mažulė, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandrė, S. Kićas, and R. Drazdys, “Complex study of zirconia-silica and niobia-silica composite coatings produced by ion beam sputtering,” Proc. SPIE 7842, 784203, 784203-13 (2010).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

JOSA

H. E. Bennett and J. O. Porteus “Relation between roughness and specular reflectance at normal incidence,” JOSA 51, 123–129 (1961), Eqs. (1) and (5).
[CrossRef]

Opt. Eng.

H. E. Bennett, “Scattering characteristics of optical materials,” Opt. Eng. 17, 480–488 (1978).

S.-M. F. Nee and T.-W. Nee, “Principal Mueller matrix of reflection and scattering measured for a one-dimensional rough surface,” Opt. Eng. 41(5), 994–1001 (2002).
[CrossRef]

Precis. Eng.

M. Bjuggren, L. Krummrnacher, and L. Mattsson, “Noncontact surface roughness measurement of engineering surfaces by total integrating infrared scattering,” Precis. Eng. 20(1), 33–45 (1997).
[CrossRef]

Proc. SPIE

A. Melninkaitis, T. Tolenis, L. Mažulė, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandrė, S. Kićas, and R. Drazdys, “Complex study of zirconia-silica and niobia-silica composite coatings produced by ion beam sputtering,” Proc. SPIE 7842, 784203, 784203-13 (2010).
[CrossRef]

S. F. Nee and H. E. Bennett, “Characterization of optical blacks by infrared ellipsometry and reflectometry,” Proc. SPIE 1331, 249–260 (1990).
[CrossRef]

S.-M. F. Nee and T.-W. Nee, “Polarization of scattering by rough surfaces,” Proc. SPIE 3426, 169–180 (1998).
[CrossRef]

Other

P. Beckmann, and A. Spizzichino, The scattering of electromagnetic waves from rough surfaces (Pergamon Press, 1963), Chap. 5, Eq. (54) on p. 88 and Eq. (8) on p. 98.

M. Born, and E. Wolf, Principles of Optics (Pergamon Press, 1968), Chap. 1.

GOST 9378–75, Standards of a rough surface (in Russian).

ISO 4288:1996(E), Geometrical product specifications (GPS) - Surface texture: Profile method - Rules and procedures for the assessment of surface texture.

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the goniometric instrument: (1) laser, (2) and (6) diaphragms, (3) Glan prism, (4) sample, (5) neutral density filter, (7) silicon diode, (8) photosensor, and (9) computer.

Fig. 2
Fig. 2

Theoretical values of Δ σ ( θ 1 , θ 2 , α ) obtained from Eq. (12) (solid line) and the experimental values of Δ σ ( θ 1 , θ 2 , α ) corresponding to the experimental values of A 0 ( θ 1 , θ 2 , α ) + A D ( θ 1 , θ 2 ) . The square and triangle symbols correspond to the experimental values for σ = 10.2nm and σ = 49.8nm, respectively.

Fig. 3
Fig. 3

Experimental values of A 0 ( θ 1 , θ 2 , α ) , A D ( θ 1 , θ 2 ) , A 0 ( θ 1 , θ 2 , α ) + A D ( θ 1 , θ 2 ) , and Δ σ ( θ 1 , θ 2 , α ) for σ=10.2nm (a) and σ=49.8nm (b). The, , “, 8, and x symbols correspond to Δ σ ( θ 1 , θ 2 , α ) , A 0 ( θ 1 , θ 2 , α ) , A D ( θ 1 , θ 2 ) , and A 0 ( θ 1 , θ 2 , α ) + A D ( θ 1 , θ 2 ) , respectively.

Equations (17)

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R ( θ ) = R 0 ( θ ) [ R S ( θ ) + R D ( θ ) ]
R ( θ ) = R 0 ( θ ) R S ( θ ) ,
P ( θ , α ) = P S ( θ , α ) + P D ( θ , α ) = P I R 0 ( θ , α ) [ R S ( θ ) + R D ( θ ) ] ,
ln [ P ( θ 1 , α ) P ( θ 2 , α ) ] = ln [ R 0 ( θ 1 , α ) R 0 ( θ 2 , α ) ] + ( 4 π σ λ ) 2 ( cos 2 θ 2 cos 2 θ 1 ) + ln [ 1 + R D ( θ 1 ) / R S ( θ 1 ) 1 + R D ( θ 2 ) / R S ( θ 2 ) ] = ( 4 π σ λ ) 2 ( cos 2 θ 2 cos 2 θ 1 ) [ 1 + A 0 ( θ 1 , θ 2 , α ) + A D ( θ 1 , θ 2 ) ] ,
A 0 ( θ 1 , θ 2 , α ) = ln [ R 0 ( θ 1 , α ) / R 0 ( θ 2 , α ) ] ( 4 π σ / λ ) 2 ( cos 2 θ 2 cos 2 θ 1 )
A D ( θ 1 , θ 2 ) = ln { [ 1 + R D ( θ 1 ) / R S ( θ 1 ) ] / [ 1 + R D ( θ 2 ) / R S ( θ 2 ) ] } ( 4 π σ / λ ) 2 ( cos 2 θ 2 cos 2 θ 1 ) .
| A 0 ( θ 1 , θ 2 , α ) + A D ( θ 1 , θ 2 ) | < < 1 ,
ln P ( θ 1 , α ) P ( θ 2 , α ) ( 4 π σ λ ) 2 ( cos 2 θ 2 cos 2 θ 1 ) .
σ ( θ 1 , θ 2 , α ) = λ 4 π ln [ P ( θ 1 , α ) / P ( θ 2 , α ) ] cos 2 θ 2 cos 2 θ 1 .
σ = λ 4 π ln [ P ( θ 1 , α ) / P ( θ 2 , α ) ] ( cos 2 θ 2 cos 2 θ 1 ) [ 1 + A 0 ( θ 1 , θ 2 , α ) + A D ( θ 1 , θ 2 ) ] .
Δ σ ( θ 1 , θ 2 , α ) = [ σ σ ( θ 1 , θ 2 , α ) ] / σ
Δ σ ( θ 1 , θ 2 , α ) = 1 1 + A 0 ( θ 1 , θ 2 , α ) + A D ( θ 1 , θ 2 ) .
ln R 0 ( θ 1 , α ) R 0 ( θ 2 , α ) = ln P S ( θ 1 , α ) P S ( θ 2 , α ) ( 4 π σ λ ) 2 ( cos 2 θ 2 cos 2 θ 1 ) ,
R D ( θ 1 ) R S ( θ 1 ) = P D ( θ 1 , α ) P S ( θ 1 , α ) ,
R D ( θ 2 ) R S ( θ 2 ) = P D ( θ 2 , α ) P S ( θ 2 , α ) ,
P S ( θ 1 , α ) = P ( θ 1 , α ) P D ( θ 1 4 , α )
P S ( θ 2 , α ) = P ( θ 2 , α ) P D ( θ 2 4 , α ) .

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