Abstract

The roughness parameters of diamond-turned metal surfaces are obtained from the Fourier transform of the scattered light intensity. Theoretical consideration of the scattered light intensity and its Fourier transform is made. Under the condition of weak scattering, the Fourier transform of the scattered intensity is equivalent to the autocorrelation of the surface. From this autocorrelation function, we can derive the parameters of the surface roughness, namely, the amplitude and frequency of the periodic tool marks and the rms roughness and correlation length of the random surface component. The surface-roughness parameters of the diamond-turned surfaces of aluminum alloy are determined experimentally from the autocorrelation function. The data obtained by light scattering are compared with those of the conventional stylus method. They show qualitative agreement.

© 1986 Optical Society of America

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References

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  1. J. M. Bennett, “Measurement of the rms roughness, autocovariance function, and other statistical properties of optical surface using a FECO scanning interferometer,” Appl. Opt. 15, 2705–2721 (1976).
    [CrossRef] [PubMed]
  2. W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering,” Opt. Quantum Electron. 9, 269–287 (1977).
    [CrossRef]
  3. T. V. Vorburger, E. C. Teague, “Optical techniques for online measurement of surface topography,” Precis. Eng. 3, 61–83 (1981).
    [CrossRef]
  4. E. L. Church, J. M. Zavada, “Residual surface roughness of diamond-turned optics,” Appl. Opt. 14, 1788–1795 (1975).
    [CrossRef] [PubMed]
  5. E. L. Church, H. A. Jenkinson, J. M. Zavada, “Measurement of diamond-turned metal surfaces by differential light scattering,” Opt. Eng. 16, 360–374 (1977).
    [CrossRef]
  6. 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]
  7. K. H. Guenther, P. G. Wierer, J. M. Bennett, “Surface roughness measurements of low-scatter mirrors and roughness standards,” Appl. Opt. 23, 3820–3836 (1984).
    [CrossRef] [PubMed]
  8. P. Roche, E. Pelletier, “Characterizations of optical surfaces by measurement of scattering distribution,” Appl. Opt. 23, 3561–3566 (1984).
    [CrossRef] [PubMed]
  9. J. Ohtsubo, “Measurement of diamond-turned metal surface by light scattering,” Proc. Soc. Photo-Opt. Instrum. Eng. 558, 80–84 (1985).
  10. B. Ruffing, J. Fleischer, “Spectral correlation of partially or fully developed speckle patterns generated by rough surfaces,” J. Opt. Soc. Am. A 2, 1637–1643 (1985).
    [CrossRef]

1985 (2)

J. Ohtsubo, “Measurement of diamond-turned metal surface by light scattering,” Proc. Soc. Photo-Opt. Instrum. Eng. 558, 80–84 (1985).

B. Ruffing, J. Fleischer, “Spectral correlation of partially or fully developed speckle patterns generated by rough surfaces,” J. Opt. Soc. Am. A 2, 1637–1643 (1985).
[CrossRef]

1984 (2)

1981 (1)

T. V. Vorburger, E. C. Teague, “Optical techniques for online measurement of surface topography,” Precis. Eng. 3, 61–83 (1981).
[CrossRef]

1979 (1)

1977 (2)

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Measurement of diamond-turned metal surfaces by differential light scattering,” Opt. Eng. 16, 360–374 (1977).
[CrossRef]

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

1976 (1)

1975 (1)

Bennett, J. M.

Church, E. L.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Measurement of diamond-turned metal surfaces by differential light scattering,” Opt. Eng. 16, 360–374 (1977).
[CrossRef]

E. L. Church, J. M. Zavada, “Residual surface roughness of diamond-turned optics,” Appl. Opt. 14, 1788–1795 (1975).
[CrossRef] [PubMed]

Elson, J. M.

Fleischer, J.

Guenther, K. H.

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Measurement of diamond-turned metal surfaces by differential light scattering,” Opt. Eng. 16, 360–374 (1977).
[CrossRef]

Ohtsubo, J.

J. Ohtsubo, “Measurement of diamond-turned metal surface by light scattering,” Proc. Soc. Photo-Opt. Instrum. Eng. 558, 80–84 (1985).

Pelletier, E.

Roche, P.

Ruffing, B.

Teague, E. C.

T. V. Vorburger, E. C. Teague, “Optical techniques for online measurement of surface topography,” Precis. Eng. 3, 61–83 (1981).
[CrossRef]

Vorburger, T. V.

T. V. Vorburger, E. C. Teague, “Optical techniques for online measurement of surface topography,” Precis. Eng. 3, 61–83 (1981).
[CrossRef]

Welford, W. T.

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

Wierer, P. G.

Zavada, J. M.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Measurement of diamond-turned metal surfaces by differential light scattering,” Opt. Eng. 16, 360–374 (1977).
[CrossRef]

E. L. Church, J. M. Zavada, “Residual surface roughness of diamond-turned optics,” Appl. Opt. 14, 1788–1795 (1975).
[CrossRef] [PubMed]

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

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

Opt. Eng. (1)

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Measurement of diamond-turned metal surfaces by differential light scattering,” Opt. Eng. 16, 360–374 (1977).
[CrossRef]

Opt. Quantum Electron. (1)

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

Precis. Eng. (1)

T. V. Vorburger, E. C. Teague, “Optical techniques for online measurement of surface topography,” Precis. Eng. 3, 61–83 (1981).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

J. Ohtsubo, “Measurement of diamond-turned metal surface by light scattering,” Proc. Soc. Photo-Opt. Instrum. Eng. 558, 80–84 (1985).

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Figures (8)

Fig. 1
Fig. 1

A schematic of a diamond-turned surface.

Fig. 2
Fig. 2

Schematic illustration of the experimental setup.

Fig. 3
Fig. 3

The scattered light intensity from a diamond-turned aluminum surface. A direct mapping of the power spectral density of the surface roughness on a logarithmic scale against a spatial frequency.

Fig. 4
Fig. 4

The Fourier transform of the power spectrum shown in Fig. 3, which is equivalent to the autocorrelation function of the surface.

Fig. 5
Fig. 5

The correlation function due to the random roughness component only. It is also plotted as a logarithmic scale.

Fig. 6
Fig. 6

The output signal of the diamond-turned metal surface obtained from the stylus instrument.

Fig. 7
Fig. 7

The power spectrum of the signal shown in Fig. 5.

Fig. 8
Fig. 8

The correlation function of the random roughness component only. The periodic components of the power spectrum are suppressed.

Tables (2)

Tables Icon

Table 1 Roughness Parameters of the Diamond-Turned Metal Surfaces Determined by the Optical Method

Tables Icon

Table 2 Comparison of the Roughness Parameters between the Optical and Stylus Methods

Equations (19)

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A ( u ) = A 0 ( x ) exp ( x 2 / w 0 2 i k u x / R ) d x ,
A 0 ( x ) = exp [ i ϕ ( x ) i θ ( x ) ] .
ϕ ( 0 ) ϕ ( x ) = σ ϕ 2 exp ( x 2 / r 0 2 ) .
θ ( x ) = n = 0 a n cos ( 2 π n x / p ) .
I ( u ) = d x 1 d x 2 [ 1 + a 2 cos ( 2 π x 1 / p ) cos ( 2 π x 2 / p ) ] × exp [ i ϕ ( x 1 ) i ϕ ( x 2 ) ] × exp [ ( x 1 2 + x 2 2 ) / w 0 2 i k u ( x 1 x 2 ) / R ] .
exp [ i ϕ ( x 1 ) i ϕ ( x 2 ) ] exp ( σ ϕ 2 ) { 1 + [ exp ( σ ϕ 2 ) 1 ] × exp [ g ( σ ϕ 2 ) ( x 1 x 2 ) 2 / r 0 2 ] } ,
g ( x ) = 1 + 0.313 x + 0.0887 x 2 .
I ( u ) = π w 0 2 exp ( σ ϕ 2 ) { exp [ ( k w 0 u / R ) 2 / 2 ] + a 2 exp [ ( k w 0 u / R ) 2 / 2 ] / × 2 * [ δ ( u 2 π R / k p ) + δ ( u + 2 π R / k p ) ] + [ exp ( σ ϕ 2 ) 1 ] r 0 / 2 w 0 exp [ ( k r 0 u / R ) 2 / 4 ] } ,
r 0 = r 0 / [ g ( σ ϕ 2 ) ] 1 / 2 .
C ( X ) = F T [ I ( u ) ] = exp ( Z 2 / 2 w 0 2 ) + a 2 cos ( 2 π Z / p ) exp ( Z 2 / 2 w 0 2 ) + [ exp ( σ ϕ 2 ) 1 ] exp ( Z 2 / r 0 2 ) ,
Z = λ R X
C ( Z ) = 1 + a 2 cos ( 2 π Z / p ) + σ ϕ 2 exp ( Z 2 / r 0 2 ) .
a p = a / 2 k ,
σ h = σ ϕ / 2 k .
R ( x 1 , x 2 ) = exp [ i ϕ ( x 1 ) i ϕ ( x 2 ) ] = exp { [ ϕ ( 0 ) 2 ϕ ( x 1 ) ϕ ( x 2 ) 2 ] } ,
ϕ ( x 1 ) ϕ ( x 2 ) = σ ϕ 2 exp [ ( x 1 x 2 ) 2 / r 0 2 ] ,
R ( x 1 , x 2 ) = R ( x 1 x 2 ) exp ( σ ϕ 2 ) { 1 + [ exp ( σ ϕ 2 ) 1 ] × exp [ g ( σ ϕ 2 ) ( x 1 x 2 ) 2 / r 0 2 ] } ,
g ( x ) = x / [ 1 exp ( x ) ] .
g ( x ) = 1 + 0.313 x + 0.0887 x 2

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