Abstract

The relationship between the height autocorrelation function of a one-dimensionally rough surface and the Fourier transform of the intensity distribution of the light scattered by that surface is tested experimentally. The theory is derived by using the Fraunhofer approximation, without recourse to the inconsistent Kirchhoff boundary conditions. In spite of the limitations imposed by the approximations used, the results obtained from optical data agree well with those obtained from stylus data, even for an autocorrelation length as small as the optical wavelength. However, this method should be limited to surfaces with rms roughness smaller than approximately 0.14 times the wavelength of light.

© 1993 Optical Society of America

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References

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  1. R. S. Sayles, “The profile as a random process,” in Rough Surfaces, T. R. Thomas, ed. (Longmans, New York, 1982), pp. 91–118.
  2. ANSI/ASME Committee B46.1, “Surface texture” (American Society of Mechanical Engineers, New York, 1985).
  3. E. Marx, T. V. Vorburger, “Direct and inverse problems for light scattered by a rough surface,” Appl. Opt. 29, 3613–3626 (1990); “Light scattered by random rough surfaces and roughness determination,” in Scatter from Optical Components, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1165, 72–86 (1989).
    [CrossRef] [PubMed]
  4. E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
  5. J. M. Elson, J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).
  6. J. C. Stover, S. A. Serati, C. H. Gillespie, “Calculation of surface statistics from light scatter,” Opt. Eng. 23, 406–412 (1984).
  7. J. M. Elson, 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]
  8. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon, London, 1963), Part I.
  9. H. E. Bennett, J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,” J. Opt. Soc. Am. 51, 123–129 (1961).
    [CrossRef]
  10. H. E. Bennett, “Measurement of smooth surface finishes,” Ind. Qual. Control 20(8), 1–6 (1964).
  11. A. Abdulkadir, R. C. Birkebak, “Optical surface roughness and slopes measurements with a double beam spectrometer,” Rev. Sci. Instrum. 45, 1356–1360 (1974).
    [CrossRef]
  12. E. L. Church, T. V. Vorburger, J. C. Wyatt, “Direct comparison of mechanical and optical measurements of the finish of precision machined optical surfaces,” Opt. Eng. 24, 388–395 (1985).
  13. E. L. Church, G. M. Sanger, P. Z. Takacs, “Comparison of Wyko and TIS measurements of surface finish,” in Metrology: Figure and Finish, B. E. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 65–73 (1987).
  14. E. L. Church, P. Z. Takacs, “Instrumental effects in surface finish measurements,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 46–55 (1988).
  15. P. J. Chandley, “Determination of the autocorrelation function of height on a rough surface from coherent light scattering,” Opt. Quantum Electron. 8, 329–333 (1976).
    [CrossRef]
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), especially pp. 42 and 61.
  17. A. Sommerfeld, Optics (Academic, New York, 1964), p. 198.
  18. J.-F. Song, “Random profile precision roughness calibration specimens,” Surf. Topogr. 1, 29–40 (1988).
  19. J. E. Harvey, “Light-scattering characteristics of optical surfaces,” in Stray Light Problems in Optical Systems, J. D. Lytle, H. E. Morrow, eds., Proc. Soc. Photo-Opt. Instrum. Eng.107, 41–47 (1977); “Surface scatter phenomena: a linear shift-invariant process,” in Scatter from Optical Components, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1165, 87–99 (1989).
  20. T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,” J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
    [CrossRef]

1990 (1)

1988 (1)

J.-F. Song, “Random profile precision roughness calibration specimens,” Surf. Topogr. 1, 29–40 (1988).

1985 (1)

E. L. Church, T. V. Vorburger, J. C. Wyatt, “Direct comparison of mechanical and optical measurements of the finish of precision machined optical surfaces,” Opt. Eng. 24, 388–395 (1985).

1984 (2)

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

T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,” J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
[CrossRef]

1979 (3)

J. M. Elson, 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]

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

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

1976 (1)

P. J. Chandley, “Determination of the autocorrelation function of height on a rough surface from coherent light scattering,” Opt. Quantum Electron. 8, 329–333 (1976).
[CrossRef]

1974 (1)

A. Abdulkadir, R. C. Birkebak, “Optical surface roughness and slopes measurements with a double beam spectrometer,” Rev. Sci. Instrum. 45, 1356–1360 (1974).
[CrossRef]

1964 (1)

H. E. Bennett, “Measurement of smooth surface finishes,” Ind. Qual. Control 20(8), 1–6 (1964).

1961 (1)

Abdulkadir, A.

A. Abdulkadir, R. C. Birkebak, “Optical surface roughness and slopes measurements with a double beam spectrometer,” Rev. Sci. Instrum. 45, 1356–1360 (1974).
[CrossRef]

Beckmann, P.

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

Bennett, H. E.

Bennett, J. M.

Birkebak, R. C.

A. Abdulkadir, R. C. Birkebak, “Optical surface roughness and slopes measurements with a double beam spectrometer,” Rev. Sci. Instrum. 45, 1356–1360 (1974).
[CrossRef]

Chandley, P. J.

P. J. Chandley, “Determination of the autocorrelation function of height on a rough surface from coherent light scattering,” Opt. Quantum Electron. 8, 329–333 (1976).
[CrossRef]

Church, E. L.

E. L. Church, T. V. Vorburger, J. C. Wyatt, “Direct comparison of mechanical and optical measurements of the finish of precision machined optical surfaces,” Opt. Eng. 24, 388–395 (1985).

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

E. L. Church, G. M. Sanger, P. Z. Takacs, “Comparison of Wyko and TIS measurements of surface finish,” in Metrology: Figure and Finish, B. E. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 65–73 (1987).

E. L. Church, P. Z. Takacs, “Instrumental effects in surface finish measurements,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 46–55 (1988).

Elson, J. M.

Gillespie, C. H.

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

Gilsinn, D. E.

T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,” J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), especially pp. 42 and 61.

Harvey, J. E.

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” in Stray Light Problems in Optical Systems, J. D. Lytle, H. E. Morrow, eds., Proc. Soc. Photo-Opt. Instrum. Eng.107, 41–47 (1977); “Surface scatter phenomena: a linear shift-invariant process,” in Scatter from Optical Components, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1165, 87–99 (1989).

Jenkinson, H. A.

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

Marx, E.

McLay, M. J.

T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,” J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
[CrossRef]

Porteus, J. O.

Sanger, G. M.

E. L. Church, G. M. Sanger, P. Z. Takacs, “Comparison of Wyko and TIS measurements of surface finish,” in Metrology: Figure and Finish, B. E. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 65–73 (1987).

Sayles, R. S.

R. S. Sayles, “The profile as a random process,” in Rough Surfaces, T. R. Thomas, ed. (Longmans, New York, 1982), pp. 91–118.

Scire, F. E.

T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,” J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
[CrossRef]

Serati, S. A.

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

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1964), p. 198.

Song, J.-F.

J.-F. Song, “Random profile precision roughness calibration specimens,” Surf. Topogr. 1, 29–40 (1988).

Spizzichino, A.

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

Stover, J. C.

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

Takacs, P. Z.

E. L. Church, G. M. Sanger, P. Z. Takacs, “Comparison of Wyko and TIS measurements of surface finish,” in Metrology: Figure and Finish, B. E. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 65–73 (1987).

E. L. Church, P. Z. Takacs, “Instrumental effects in surface finish measurements,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 46–55 (1988).

Teague, E. C.

T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,” J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
[CrossRef]

Vorburger, T. V.

E. Marx, T. V. Vorburger, “Direct and inverse problems for light scattered by a rough surface,” Appl. Opt. 29, 3613–3626 (1990); “Light scattered by random rough surfaces and roughness determination,” in Scatter from Optical Components, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1165, 72–86 (1989).
[CrossRef] [PubMed]

E. L. Church, T. V. Vorburger, J. C. Wyatt, “Direct comparison of mechanical and optical measurements of the finish of precision machined optical surfaces,” Opt. Eng. 24, 388–395 (1985).

T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,” J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
[CrossRef]

Wyatt, J. C.

E. L. Church, T. V. Vorburger, J. C. Wyatt, “Direct comparison of mechanical and optical measurements of the finish of precision machined optical surfaces,” Opt. Eng. 24, 388–395 (1985).

Zavada, J. M.

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

Appl. Opt. (1)

Ind. Qual. Control (1)

H. E. Bennett, “Measurement of smooth surface finishes,” Ind. Qual. Control 20(8), 1–6 (1964).

J. Opt. Soc. Am. (2)

J. Res. Natl. Bur. Stand. (1)

T. V. Vorburger, E. C. Teague, F. E. Scire, M. J. McLay, D. E. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,” J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
[CrossRef]

Opt. Eng. (4)

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

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

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

E. L. Church, T. V. Vorburger, J. C. Wyatt, “Direct comparison of mechanical and optical measurements of the finish of precision machined optical surfaces,” Opt. Eng. 24, 388–395 (1985).

Opt. Quantum Electron. (1)

P. J. Chandley, “Determination of the autocorrelation function of height on a rough surface from coherent light scattering,” Opt. Quantum Electron. 8, 329–333 (1976).
[CrossRef]

Rev. Sci. Instrum. (1)

A. Abdulkadir, R. C. Birkebak, “Optical surface roughness and slopes measurements with a double beam spectrometer,” Rev. Sci. Instrum. 45, 1356–1360 (1974).
[CrossRef]

Surf. Topogr. (1)

J.-F. Song, “Random profile precision roughness calibration specimens,” Surf. Topogr. 1, 29–40 (1988).

Other (8)

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” in Stray Light Problems in Optical Systems, J. D. Lytle, H. E. Morrow, eds., Proc. Soc. Photo-Opt. Instrum. Eng.107, 41–47 (1977); “Surface scatter phenomena: a linear shift-invariant process,” in Scatter from Optical Components, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1165, 87–99 (1989).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), especially pp. 42 and 61.

A. Sommerfeld, Optics (Academic, New York, 1964), p. 198.

E. L. Church, G. M. Sanger, P. Z. Takacs, “Comparison of Wyko and TIS measurements of surface finish,” in Metrology: Figure and Finish, B. E. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 65–73 (1987).

E. L. Church, P. Z. Takacs, “Instrumental effects in surface finish measurements,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 46–55 (1988).

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

R. S. Sayles, “The profile as a random process,” in Rough Surfaces, T. R. Thomas, ed. (Longmans, New York, 1982), pp. 91–118.

ANSI/ASME Committee B46.1, “Surface texture” (American Society of Mechanical Engineers, New York, 1985).

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Figures (5)

Fig. 1
Fig. 1

Comparison between the ACF obtained by using (a) Eq. (3) or (b) Eq. (6) from optical data and the ACF obtained from stylus data for a relatively smooth surface.

Fig. 2
Fig. 2

Comparison between the ACF obtained from optical data by using (a) Eq. (3), (b) Eq. (3), (c) Eq. (6), or (d) Eq. (3) and the ACF obtained from stylus data for a moderately rough surface.

Fig. 3
Fig. 3

Comparison between the ACF obtained from optical data by using Eq. (3) and the ACF obtained from stylus data for the rougher surfaces for (a) normal incidence and (b) optical data only.

Fig. 4
Fig. 4

Geometrical configuration for the diffraction problem. The illuminated region ∑ is produced by a light with a wave vector k normal to the mean surface Π.

Fig. 5
Fig. 5

Measured and computed angular distributions of scattered light intensities for NASA, with and without the Fresnel term, for (a) a relatively rough surface and (b) a relatively smooth surface.

Tables (1)

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Table 1 Comparison of Stylus and Optical ACL’s

Equations (43)

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I sp / I tot = exp [ - ( 4 π σ cos θ i / λ ) 2 ] ,
C ( τ ) = 1 + log [ C v ( τ ) + exp ( - 4 σ 2 k 2 ) ] / 4 σ 2 k 2 ,             k = 2 π / λ ,
C v ( τ ) = - I ( R 0 ξ / k ) exp ( i ξ τ ) d ξ ,
exp [ - 4 σ 2 k 2 ( 1 - 1 / e ) ] - exp ( - 4 σ 2 k 2 ) = 0.1 [ 1 - exp ( - 4 σ 2 k 2 ) ] .
C ( τ ) C v ( τ ) / 4 σ 2 k 2 .
C v ( τ ) = - I 0 [ θ ( u ) ] exp ( i u τ ) d u ,             u = k sin θ .
σ ( 1 N k = 1 N z k 2 ) 1 / 2 ,
C j = 1 σ 2 ( N - j ) k = 1 N - j z k z j + k ,             j = 0 , 1 , ,
I ( ξ ) = I 0 ( θ ) cos θ ,             ξ = R 0 tan θ ,
( 2 + k 2 ) U ( x ) = 0.
U ( x ) = S [ U ( x ) n G ( x , x ) - U ( x ) G ( x , x ) n ] d S ,
u ( x ) ( 2 + k 2 ) v ( x ) - v ( x ) ( 2 + k 2 ) u ( x ) = · [ u ( x ) v ( x ) - v ( x ) u ( x ) ]
( 2 + k 2 ) G ( x , x ) = - δ ( x - x ) .
G ( x , x ) = exp ( i k R ) / 4 π R - exp ( i k R I ) / 4 π R I ,
G ( x , x ) = exp ( i k R ) / 4 π R - exp ( i k R I ) / 4 π R I ,
U ( x ) = - Σ U ( x ) G ( x , x ) n d S U ( x ) G ( x , x ) n d S .
G ( x , x ) n = 2 n ^ · R ^ ( i k - 1 R ) exp ( i k R ) 4 π R ,
R = [ R 0 2 + ( ξ - x ) 2 + ( η - y ) 2 ] 1 / 2 R 0 [ 1 + 1 2 ( ξ - x R 0 ) 2 + 1 2 ( η - y R 0 ) 2 ]
U ( ξ , η , R 0 ) Σ U 0 ( x , y ) exp ( i k R 0 ) i λ R 0 × exp { i ( k / 2 R 0 ) [ ( ξ - x ) 2 + ( η - y ) 2 ] } d x d y .
U 0 ( x , y ) = A exp [ - 2 i k ζ ( x , y ) ] ,
U ( ξ , η , R 0 ) A i π η sin ( k Y η 2 R 0 ) exp [ i k ( R 0 + ξ 2 + η 2 2 R 0 ) ] × - X X exp { - i k [ 2 z ( x ) + ξ x R 0 ] } d x = F ( ξ , η ) V ( ξ ) ,
I ( ξ ) = B [ V ( ξ ) 2 - V ( ξ ) 2 ] = B - X X - X X C v ( τ ) exp [ - i k ξ ( x 1 - x 2 ) / R 0 ] d x 1 d x 2 ,
B = - F ( ξ , η ) 2 d η = ( A π ) 2 - [ 1 η sin ( k Y η 2 R 0 ) ] 2 d η = k Y A 2 2 π R 0 ,
C v ( τ ) = exp { - 4 σ 2 k 2 [ 1 - C ( τ ) ] } - exp ( - 4 σ 2 k 2 ) ,             τ = x 1 - x 2 ,
C ( τ ) = 1 + log [ C v ( τ ) + exp ( - 4 σ 2 k 2 ) ] / 4 σ 2 k 2 .
C v ( 0 ) = 1 - exp ( - 4 σ 2 k 2 ) .
I ( ξ ) + B - 2 X 2 X C v ( τ ) exp ( - i k ξ τ / R 0 ) ( 2 X - τ ) d τ .
I ( ξ ) 2 X B - C v ( τ ) exp ( - i k ξ τ / R 0 ) d τ .
C v ( τ ) = 1 2 X Y A 2 - I ( ξ ) exp ( i k ξ τ / R 0 ) d ξ .
C v ( τ ) = - I ( R 0 ξ / k ) exp ( i ξ τ ) d ξ = - I ( R 0 ξ / k ) cos ( ξ τ ) d ξ ,
C v ( τ ) = [ 1 - exp ( - 4 σ 2 k 2 ) ] C v ( τ ) / C v ( 0 ) .
F ( β - β 0 ) = I ( θ ) / cos θ ,
sin θ = sin θ - sin θ s ,
k = 2 π / λ ,             R = R 0 - x ,             R = R .
R = R 0 ( 1 - 2 k · x / k R 0 + r 2 / R 0 2 ) 1 / 2 ,
R 0 = R 0 ,             r = x ,             k = k k ^ ,             k ^ = R ^ 0 = R 0 / R 0 .
k R k R 0 - k · x + ½ [ k 2 r 2 - ( k · x ) 2 ] / k R 0 .
I 0 ( θ ) = F ( θ ) 2 - C v ( τ , θ ) exp ( i v x τ ) d τ .
C v ( τ , θ ) = B exp ( σ 2 v z 2 ) { exp [ σ 2 v z 2 C ( τ ) ] - 1 } ,
F ( θ ) = [ 1 + cos ( θ i - θ ) ] / [ 2 π cos θ i ( cos θ i + cos θ ) ] ,
v x ( θ ) = - k ( sin θ i + sin θ ) , v z ( θ ) = - k ( cos θ i + cos θ ) .
I 0 [ θ ( u ) ] = 1 2 π - C v ( τ ) exp ( - i u τ ) d τ .
C v ( τ ) = - I 0 [ θ ( u ) ] exp ( i u τ ) d u .

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