Abstract

The power spectral density (PSD), in its two-dimensional form, has been designated as the preferred quantity for specifying surface roughness on a draft international drawing standard for surface texture. The correct calculation of the one-dimensional PSD from discrete surface profile data is given, and problems in using fast Fourier-transform routines that are given in some of the standard reference books are flagged. The method given here contains the correct normalizing factors. Two ways to reduce the variance of the PSD estimate are suggested. Examples are shown of the variance reduction possible in the PSD’s.

© 1995 Optical Society of America

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References

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  1. “Optics and optical instruments—indications in optical drawings,” in Draft International Standard ISO 10110 Part 8: Surface Texture (International Organization for Standardization, ISO/TC 172/SC 1/WG 2, Geneva, Switzerland).
  2. J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989), pp. 28–29, 44–50.
  3. E. L. Church, P. Z. Takacs, “The optimal estimation of finish parameters,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1530, 71–78 (1991).
  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, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. 69, 31–47 (1979).
    [CrossRef]
  6. J. M. Elson, J. M. Bennett, J. C. Stover, “Wavelength and angular dependence of light scattering from beryllium: comparison of theory and experiment,” Appl. Opt. 32, 3362–3376 (1993); see also references by Church et al. in this reference.
    [CrossRef] [PubMed]
  7. S. L. Marple, Digital Spectral Analysis with Applications (Prentice-Hall, Englewood Cliffs, N.J., 1987), pp. 152–158.
  8. W. H. Press, S. A. Teukolsky, W. T. Vettering, B. P. Flannery, Numerical Recipes in Fortran, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 543–551.
  9. J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley-Interscience, New York, 1971), pp. 322–330.
  10. For example, a commercial algorithm in Ref. 8 called SPCTRM.FOR yields smoothed PSD estimates that are one sided.

1993

1979

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).

Bendat, J. S.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley-Interscience, New York, 1971), pp. 322–330.

Bennett, J. M.

Church, E. L.

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, P. Z. Takacs, “The optimal estimation of finish parameters,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1530, 71–78 (1991).

Elson, J. M.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vettering, B. P. Flannery, Numerical Recipes in Fortran, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 543–551.

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).

Marple, S. L.

S. L. Marple, Digital Spectral Analysis with Applications (Prentice-Hall, Englewood Cliffs, N.J., 1987), pp. 152–158.

Mattsson, L.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989), pp. 28–29, 44–50.

Piersol, A. G.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley-Interscience, New York, 1971), pp. 322–330.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vettering, B. P. Flannery, Numerical Recipes in Fortran, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 543–551.

Stover, J. C.

Takacs, P. Z.

E. L. Church, P. Z. Takacs, “The optimal estimation of finish parameters,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1530, 71–78 (1991).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vettering, B. P. Flannery, Numerical Recipes in Fortran, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 543–551.

Vettering, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vettering, B. P. Flannery, Numerical Recipes in Fortran, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 543–551.

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.

J. Opt. Soc. Am.

Opt. Eng.

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

Other

S. L. Marple, Digital Spectral Analysis with Applications (Prentice-Hall, Englewood Cliffs, N.J., 1987), pp. 152–158.

W. H. Press, S. A. Teukolsky, W. T. Vettering, B. P. Flannery, Numerical Recipes in Fortran, The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992), pp. 543–551.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley-Interscience, New York, 1971), pp. 322–330.

For example, a commercial algorithm in Ref. 8 called SPCTRM.FOR yields smoothed PSD estimates that are one sided.

“Optics and optical instruments—indications in optical drawings,” in Draft International Standard ISO 10110 Part 8: Surface Texture (International Organization for Standardization, ISO/TC 172/SC 1/WG 2, Geneva, Switzerland).

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989), pp. 28–29, 44–50.

E. L. Church, P. Z. Takacs, “The optimal estimation of finish parameters,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1530, 71–78 (1991).

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of a normally incident beam on a rough surface described by the random variable z(x) and scattered light at positive and negative scattering angles θ.

Fig. 2
Fig. 2

Graph of log(PSD) versus spatial frequency for sample 18 calculated from surface profile data. The digitization interval Δx = 0.37 μm, and the profile length is 1000 μm. The averaged and the unaveraged PSD’s are shown where the averaged curve is intentionally displaced 1 order of magnitude above the unaveraged curve.

Fig. 3
Fig. 3

Same as Fig. 2, except Δx = 0.037 μm, and the profile length is 100 μm.

Fig. 4
Fig. 4

Graph of log(PSD) versus spatial frequency for sample 18, with the averaged curves in Figs. 2 and 3 combined. The calculated values marked by ○ and △ are for Δx = 0.37 and 0.037 μm, respectively.

Fig. 5
Fig. 5

Graph of log(PSD) versus spatial frequency for sample 8 calculated from surface profile data. The digitization interval Δx = 0.37 μm, and the profile length is 1000 μm. The averaged and unaveraged PSD’s are shown where the averaged curve is intentionally displaced 1 order of magnitude above the unaveraged curve.

Fig. 6
Fig. 6

Same as Fig. 5, except Δx = 0.037 μm, and the profile length is 100 μm.

Fig. 7
Fig. 7

Graph of log(PSD) versus spatial frequency for sample 8, combining the averaged curves in Figs. 5 and 6. The calculated values marked by ○ and △ are for Δx = 0.37 and 0.037 μm, respectively.

Fig. 8
Fig. 8

Graph of log(PSD) versus spatial frequency on a linear scale for sample 18 calculated from surface profile data. The digitization interval Δx = 0.037 μm, and the profile length is 160 μm. The unaveraged PSD was calculated from a single profile. PSD’s calculated from 20 profiles measured at different places on the same surface were averaged to give the upper curve, which is intentionally displaced 1 order of magnitude above the unaveraged curve.

Equations (23)

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Z ( k ) = 0 L d x z ( x ) exp ( - i k x ) ,
δ N 2 = 1 N n = 0 N - 1 z 2 ( n ) .
Z ( m ) = Δ x Z ^ ( m ) ,             - N / 2 m N / 2 ,
Z ^ ( m ) = n = 0 N - 1 z ( n ) exp ( - 2 π i m n / N ) ,             - N / 2 m N / 2.
BRDF = ( 2 π / λ ) 3 cos θ F ( θ , λ ) Z ( k ) 2 L ,
PSD = Z ( k ) 2 L ,
Z ( 2 π f ) 2 L Z ( m ) 2 N Δ x = Δ x N | n = 0 N - 1 z ( n ) exp ( - 2 π i m n / N ) | 2 .
P N ( m ) = Δ x N | n = 0 N - 1 z ( n ) exp ( - 2 π i m n / N ) | 2 = N Δ x P ^ N ( m ) ,             - N 2 m N 2 ,
P ^ N ( m ) = [ Z ^ ( m ) 2 N 2 ] ,             - N 2 m N 2 .
- f c f c d f Z ( 2 π f ) 2 L = 1 L - f c f c d f | 0 L d x z ( x ) exp ( - 2 π i f x ) | 2 .
( 1 N Δ x ) 2 [ ½ Z ( - N / 2 ) 2 + m = - ( N / 2 ) + 1 ( N / 2 ) - 1 Z ( m ) 2 + ½ Z ( N / 2 ) 2 ] = 1 N 2 n = 0 N - 1 n ' = 0 N - 1 z ( n ) z ( n ) { ½ exp [ - π i ( n - n ) ] + m = - ( N / 2 ) + 1 ( N / 2 ) - 1 exp [ 2 π i m ( n - n ) / N ] + 1 / 2 exp [ π i ( n - n ) ] } .
( 1 N Δ x ) 2 [ ½ Z ( - N / 2 ) 2 + m = - ( N / 2 ) + 1 ( N / 2 ) - 1 Z ( m ) 2 + ½ Z ( N / 2 ) 2 ] = δ N 2 .
1 N Δ x [ ½ P N ( - N / 2 ) + m = - ( N / 2 ) + 1 ( N / 2 ) - 1 P N ( m ) + ½ P N ( N / 2 ) ] = δ N 2 .
P ^ N ( m ) = ( 1 / N 2 ) Z ^ ( m ) 2 ,             m = 0 or N / 2 ,
P ^ N ( m ) = ( 2 / N 2 ) Z ^ ( m ) 2 ,             m = 1 , 2 , , ( N / 2 ) - 1 ,
P ^ N ( m ) = ( 1 / N 2 ) Z ^ ( m ) 2 ,             m = 0 or N / 2 ,
P ^ N ( m ) = ( 1 / N 2 ) [ Z ^ ( m ) 2 + Z ^ ( N - m ) 2 ] ,             m = 1 , 2 , , ( N / 2 ) - 1.
m = 0 N / 2 P ^ N ( m ) = δ N 2 .
P M ( k ) ( m ) = Δ x M | n = 0 M - 1 z ( n ) w ( n ) exp ( 2 π i m n / M ) | 2 ,             - M 2 m M 2 ,
P A ( m ) = S K k = 1 K P M ( k ) ( m ) ,
1 M Δ x [ ½ P A ( - M / 2 ) + m = ( - M / 2 ) + 1 ( M / 2 ) - 1 P A ( m ) + ½ P A ( M / 2 ) ] = S K k = 1 K δ k 2 = S δ ¯ 2 ,
S δ ¯ 2 = δ N 2 .
w ( n ) = 1 - 2 M - 1 | n - M - 1 2 | .

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