Abstract

The statistical properties of speckle patterns generated from a rough surface under a fully developed static speckle-pattern illumination are examined. The roughness dependence of the intensity autocorrelation function is studied and utilized to characterize typical engineering surfaces with anisotropic roughness. The speckle patterns under investigation are recorded by use of a CCD technique and are then analyzed by digital image processing algorithms to obtain a parameter that describes the surface roughness. It is shown that an in-process surface inspection can be achieved by this method.

© 1999 Optical Society of America

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References

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  1. G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Springer Topics in Applied Physics Series (Springer-Verlag, Berlin, 1975), Chap. 3, pp. 78–122.
  2. H. M. Pedersen, “Second order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
    [CrossRef]
  3. G. R. C. Reddy, V. V. Rao, “Correlation of speckle patterns generated by a diffuser illuminated by partially coherent light,” Opt. Acta 30, 1213–1216 (1983).
    [CrossRef]
  4. K. A. O’Donnell, “Speckle statistics of doubly scattered light,” J. Opt. Soc. Am. 72, 1459–1463 (1982).
    [CrossRef]
  5. R. Barakat, “Second- and fourth-order statistics of doubly scattered speckle,” Opt. Acta 33, 79–89 (1986).
    [CrossRef]
  6. M. Giglio, S. Musazzi, U. Perini, “Surface roughness measurements by means of speckle wavelength decorrelation,” Opt. Commun. 28, 166–170 (1979).
    [CrossRef]
  7. P. Lehmann, S. Patzelt, A. Schöne, “Surface roughness measurement by means of polychromatic speckle elongation,” Appl. Opt. 36, 2188–2197 (1997).
    [CrossRef] [PubMed]
  8. T. V. Vorburger, E. C. Teague, “Optical techniques for on-line measurement of surface topography,” Precision Eng. 3, 61–83 (1981).
    [CrossRef]
  9. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford1963 Part 1, Chap. 5), pp. 80–98.
  10. K. Nakagawa, T. Yoshimura, T. Minemoto, “Surface roughness measurement using Fourier transformation of doubly scattered speckle pattern,” Appl. Opt. 32, 4898–4903 (1993).
    [CrossRef] [PubMed]
  11. T. Yoshimura, K. Kazuo, K. Nakagawa, “Surface roughness dependence of the intensity correlation function under speckle pattern illumination,” J. Opt. Soc. Am. A 7, 2254–2259 (1990).
    [CrossRef]
  12. L. Basano, S. Leporatti, P. Ottonello, V. Palestini, R. Rolandi, “Measurements of surface roughness: use of a CCD camera to correlate doubly scattered speckle patterns,” Appl. Opt. 34, 7286–7290 (1995).
    [CrossRef] [PubMed]
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York1996), Chap. 5, pp. 101–114; Chap. 8, pp. 232–237.
  14. A. T. Friberg, R. J. Sidol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  15. E. Menzel, B. Stoffregen, “Autocorrelation functions of a general scattering object and its averaged coherent image and diffraction patterns,” Optik 46, 203–210 (1976).
  16. J. Peklenik, “Neue statistische Verfahren zur topographischen Erfassung von Oberflächen 2.Teil,” WT Z. Ind. Fertigung 59, 633–637 (1969).

1997 (1)

1995 (1)

1993 (1)

1990 (1)

1986 (1)

R. Barakat, “Second- and fourth-order statistics of doubly scattered speckle,” Opt. Acta 33, 79–89 (1986).
[CrossRef]

1983 (1)

G. R. C. Reddy, V. V. Rao, “Correlation of speckle patterns generated by a diffuser illuminated by partially coherent light,” Opt. Acta 30, 1213–1216 (1983).
[CrossRef]

1982 (2)

A. T. Friberg, R. J. Sidol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

K. A. O’Donnell, “Speckle statistics of doubly scattered light,” J. Opt. Soc. Am. 72, 1459–1463 (1982).
[CrossRef]

1981 (1)

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

1979 (1)

M. Giglio, S. Musazzi, U. Perini, “Surface roughness measurements by means of speckle wavelength decorrelation,” Opt. Commun. 28, 166–170 (1979).
[CrossRef]

1976 (1)

E. Menzel, B. Stoffregen, “Autocorrelation functions of a general scattering object and its averaged coherent image and diffraction patterns,” Optik 46, 203–210 (1976).

1975 (1)

H. M. Pedersen, “Second order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[CrossRef]

1969 (1)

J. Peklenik, “Neue statistische Verfahren zur topographischen Erfassung von Oberflächen 2.Teil,” WT Z. Ind. Fertigung 59, 633–637 (1969).

Barakat, R.

R. Barakat, “Second- and fourth-order statistics of doubly scattered speckle,” Opt. Acta 33, 79–89 (1986).
[CrossRef]

Basano, L.

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford1963 Part 1, Chap. 5), pp. 80–98.

Friberg, A. T.

A. T. Friberg, R. J. Sidol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Giglio, M.

M. Giglio, S. Musazzi, U. Perini, “Surface roughness measurements by means of speckle wavelength decorrelation,” Opt. Commun. 28, 166–170 (1979).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York1996), Chap. 5, pp. 101–114; Chap. 8, pp. 232–237.

Kazuo, K.

Lehmann, P.

Leporatti, S.

Menzel, E.

E. Menzel, B. Stoffregen, “Autocorrelation functions of a general scattering object and its averaged coherent image and diffraction patterns,” Optik 46, 203–210 (1976).

Minemoto, T.

Musazzi, S.

M. Giglio, S. Musazzi, U. Perini, “Surface roughness measurements by means of speckle wavelength decorrelation,” Opt. Commun. 28, 166–170 (1979).
[CrossRef]

Nakagawa, K.

O’Donnell, K. A.

Ottonello, P.

Palestini, V.

Parry, G.

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Springer Topics in Applied Physics Series (Springer-Verlag, Berlin, 1975), Chap. 3, pp. 78–122.

Patzelt, S.

Pedersen, H. M.

H. M. Pedersen, “Second order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[CrossRef]

Peklenik, J.

J. Peklenik, “Neue statistische Verfahren zur topographischen Erfassung von Oberflächen 2.Teil,” WT Z. Ind. Fertigung 59, 633–637 (1969).

Perini, U.

M. Giglio, S. Musazzi, U. Perini, “Surface roughness measurements by means of speckle wavelength decorrelation,” Opt. Commun. 28, 166–170 (1979).
[CrossRef]

Rao, V. V.

G. R. C. Reddy, V. V. Rao, “Correlation of speckle patterns generated by a diffuser illuminated by partially coherent light,” Opt. Acta 30, 1213–1216 (1983).
[CrossRef]

Reddy, G. R. C.

G. R. C. Reddy, V. V. Rao, “Correlation of speckle patterns generated by a diffuser illuminated by partially coherent light,” Opt. Acta 30, 1213–1216 (1983).
[CrossRef]

Rolandi, R.

Schöne, A.

Sidol, R. J.

A. T. Friberg, R. J. Sidol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford1963 Part 1, Chap. 5), pp. 80–98.

Stoffregen, B.

E. Menzel, B. Stoffregen, “Autocorrelation functions of a general scattering object and its averaged coherent image and diffraction patterns,” Optik 46, 203–210 (1976).

Teague, E. C.

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

Vorburger, T. V.

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

Yoshimura, T.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

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

Opt. Acta (3)

H. M. Pedersen, “Second order statistics of light diffracted from Gaussian, rough surfaces with applications to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[CrossRef]

G. R. C. Reddy, V. V. Rao, “Correlation of speckle patterns generated by a diffuser illuminated by partially coherent light,” Opt. Acta 30, 1213–1216 (1983).
[CrossRef]

R. Barakat, “Second- and fourth-order statistics of doubly scattered speckle,” Opt. Acta 33, 79–89 (1986).
[CrossRef]

Opt. Commun. (2)

M. Giglio, S. Musazzi, U. Perini, “Surface roughness measurements by means of speckle wavelength decorrelation,” Opt. Commun. 28, 166–170 (1979).
[CrossRef]

A. T. Friberg, R. J. Sidol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Optik (1)

E. Menzel, B. Stoffregen, “Autocorrelation functions of a general scattering object and its averaged coherent image and diffraction patterns,” Optik 46, 203–210 (1976).

Precision Eng. (1)

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

WT Z. Ind. Fertigung (1)

J. Peklenik, “Neue statistische Verfahren zur topographischen Erfassung von Oberflächen 2.Teil,” WT Z. Ind. Fertigung 59, 633–637 (1969).

Other (3)

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Springer Topics in Applied Physics Series (Springer-Verlag, Berlin, 1975), Chap. 3, pp. 78–122.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford1963 Part 1, Chap. 5), pp. 80–98.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York1996), Chap. 5, pp. 101–114; Chap. 8, pp. 232–237.

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Figures (8)

Fig. 1
Fig. 1

Examples of (a) a ground surface with anisotropic roughness, (b) a nitrided surface with isotropic roughness.

Fig. 2
Fig. 2

Optical arrangements for the illumination of diffuse objects by spatially coherent light and the observation of the scattered light in the Fresnel diffraction plane: (a) transmitting rough object, (b) reflecting rough object (by use of a CCD array to record the intensity distribution in the observation plane).

Fig. 3
Fig. 3

Lateral autocorrelation functions of the speckle intensities assuming that r ss/r si = 0.1, N 0 = 1000 (solid curves), N 0 = 100 (dashed curves), and N 0 = 10 (dotted curves): (a) 〈ϕ21/2 = 1 rad with the γ definition for an isotropic surface [Eq. (12)], (b) 〈ϕ21/2 = 1 rad with the γ definition according to Eq. (14), (c) 〈ϕ21/2 = 2π rad with the γ definition according to Eq. (14), (d) 〈ϕ21/2 = 1/2π rad with the γ definition according to Eq. (14).

Fig. 4
Fig. 4

CCD images (768 × 576 pixels) of speckle patterns produced by: (a) a silicon wafer (Ra < 10 nm), (b) a ground metallic surface of roughness Ra ≈ 25 nm, (c) a ground metallic surface of roughness Ra ≈ 50 nm, (d) a ground metallic surface of roughness Ra ≈ 100 nm.

Fig. 5
Fig. 5

Two-dimensional ACF’s of 256 × 256 pixel arrays of the speckle patterns according to Figs. 3(a)3(d) in the vicinity of the origin located at pixel position (32, 32).

Fig. 6
Fig. 6

Comparison of theoretical results [graphs of Figs. 3(b) and 3(d) (N 0 = 1000)] and experimental results: 〈ϕ21/2 ≈ 1.5 (asterisks), and 〈ϕ21/2 < 0.3 (filled circles).

Fig. 7
Fig. 7

Gray-level data of CCD line number 260 of the speckle patterns shown in Figs. 4(a)4(d).

Fig. 8
Fig. 8

One-dimensional ACF’s of 512 pixels of the CCD line according to Figs. 7(a)7(d) in the vicinity of the origin located at pixel position 128.

Tables (1)

Tables Icon

Table 1 Dependence of the Mean Speckle Widths X0 and Y0 on the Surface Roughness

Equations (22)

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EX= EIxexpiϕxK0x, Xdx.
K0x, X=k2πiL expikLexpik2L |x-X|2,
EX1E*X2= EIx1EI*x2expiϕx2-ϕx1K0x1, X1K0*x2, X2dx1dx2,
EIx1EI*x2=exp-|x1+x2|2/2d2×exp-|x2-x1|2/2rsi2,
ϕx=kn-1hx
ϕx=-2khxcos Θi
expiϕx2-ϕx1=exp-ϕ2×1-exp-|x2-x1|2/Rcx2×exp-|y2-y1|2/Rcy2.
ϕ2x1=ϕ2x2=ϕ2.
expiϕx2-ϕx1exp-ϕ2+1-exp-ϕ2×exp-ε2|x2-x1|2/Rcx2×exp-ε2|y2-y1|2/Rcy2,
ε=1 for ϕ21ϕ21/2 for ϕ21.
ERE*0=12rss2 expik2L |R|2exp-|R|22rss2exp-ϕ212rsi2+12rss2 exp|R|212rsi2+12rss214rss4-k216L2-ik4Lrss2+1-exp-ϕ212rsi2+12rss2+ε2Rcx212rsi2+12rss2+ε2Rcy2 expRx212rsi2+12rss2+ε2Rcx2+Ry212rsi2+12rss2+ε2Rcy2×14rss4-k216L2-ik4Lrss2,
g2R-1=IRI0I02-1=|ERE*0|2|E0E*0|2=γ exp-R2/2rss2+rsi2+1-γ×exp-R2/2rss2+rsr22,
γ=rsi2/rsi2+rss2exp-ϕ2rsi2/rsi2+rss2exp-ϕ2+rsr2/rsr2+rss21-exp-ϕ2,
rsr=rsi/1+2ε2N01/2,  N0=rsi/Rc2.
γ=rsi/rsi2+rss2exp-ϕ2rsi/rsi2+rss2exp-ϕ2+rsr/rsr2+rss21-exp-ϕ2,
rsr=rsi/1+2ε2N01/2,  N0=rsi/Rcx2.
Ra=1l0l |hx|dx.
ρm, n=k=1Ml=1N ΔIk, lΔIk+m, l+n,
ΔIm, n=Im, n-I¯
g2mΔX, nΔY-1ρm, nρ0, 0,
g2X, 0-11-axX2-bxX,  g20, Y-11-ayY2-byY.
ax=-121+ρ2, 0ρ0, 0-2 ρ1, 0ρ0, 0,  bx=123+ρ2, 0ρ0, 0-4 ρ1, 0ρ0, 0,  ay=-121+ρ0, 2ρ0, 0-2 ρ0, 1ρ0, 0,  by=123+ρ0, 2ρ0, 0-4 ρ0, 1ρ0, 0.

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