Abstract

The correlation between the intensities of partially or fully developed far-field speckle patterns of two different wavelengths is investigated, assuming Gaussian field amplitudes. The degree of spectral speckle correlation depends mainly on the standard deviation of the surface height distribution, the wavelength difference, the angle of incidence, and the number of independent scattering cells in the illuminated surface spot. It provides a simple way of performing noncontact surface-roughness measurements with a variable measuring range. The theory, which has so far been restricted to fully developed speckle patterns with circular field statistics, is extended to partially developed speckles. Assumptions about the circularity or noncircularity of the speckle patterns are not necessary. An experiment has been carried out to support the theoretical results.

© 1985 Optical Society of America

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References

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  1. J. Ohtsubo, T. Asakura, “Statistical properties of laser speckles produced in the diffraction field,” Appl. Opt. 16, 1742–1753 (1977).
    [CrossRef] [PubMed]
  2. J. W. Goodman, “Statistical properties of laser sparkle patterns,” Tech. Rep. No. 2303-1 (Stanford Electronics Laboratories, Stanford, Calif., 1963).
  3. M. J. Elbaum, M. Greenebaum, M. King, “A wavelength diversity technique for reduction of speckle size,” Opt. Commun. 5, 171–174 (1972).
    [CrossRef]
  4. N. George, A. Jain, “Speckle reduction using multiple tones of illumination,” Appl. Opt. 12, 1202–1212 (1973).
    [CrossRef] [PubMed]
  5. G. Tribillon, “Correlation entre deux speckles obtenus avec deux longeurs d’onde. Application à la mesure de la rugosité moyenne,” Opt. Commun. 11, 172–174 (1974).
    [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. G. Bitz, “Verfahren zur Bestimmung von Rauheitskenngröβ en durch Specklekorrelation,” doctoral dissertation (Universität (TH) Karlsruhe, Karlsruhe, Federal Republic of Germany, 1982).
  8. H. M. Pedersen, “Second-order statistics of light diffracted from Gaussian, rough surfaces with application to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
    [CrossRef]
  9. D. Lèger, E. Mathieu, J. C. Perrin, “Optical surface roughness determination using speckle correlation techniques,” Appl. Opt. 14, 872–877 (1975).
    [CrossRef]
  10. D. Lèger, J. C. Perrin, “Real-time measurement of surface roughness by correlation of speckle patterns,” J. Opt. Soc. Am. 66, 1210–1217 (1976).
    [CrossRef]
  11. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).
  12. H. M. Pedersen, “Object-roughness dependence of partially developed speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
    [CrossRef]
  13. M. V. Berry, “The statistical properties of echoes diffracted from rough surfaces,” Phil. Trans. R. Soc. London 273, 611–654 (1973).
    [CrossRef]
  14. J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
    [CrossRef]
  15. J. S. Gradshteyn, J. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

1979 (1)

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

1977 (1)

1976 (2)

D. Lèger, J. C. Perrin, “Real-time measurement of surface roughness by correlation of speckle patterns,” J. Opt. Soc. Am. 66, 1210–1217 (1976).
[CrossRef]

H. M. Pedersen, “Object-roughness dependence of partially developed speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
[CrossRef]

1975 (3)

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

J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
[CrossRef]

D. Lèger, E. Mathieu, J. C. Perrin, “Optical surface roughness determination using speckle correlation techniques,” Appl. Opt. 14, 872–877 (1975).
[CrossRef]

1974 (1)

G. Tribillon, “Correlation entre deux speckles obtenus avec deux longeurs d’onde. Application à la mesure de la rugosité moyenne,” Opt. Commun. 11, 172–174 (1974).
[CrossRef]

1973 (2)

N. George, A. Jain, “Speckle reduction using multiple tones of illumination,” Appl. Opt. 12, 1202–1212 (1973).
[CrossRef] [PubMed]

M. V. Berry, “The statistical properties of echoes diffracted from rough surfaces,” Phil. Trans. R. Soc. London 273, 611–654 (1973).
[CrossRef]

1972 (1)

M. J. Elbaum, M. Greenebaum, M. King, “A wavelength diversity technique for reduction of speckle size,” Opt. Commun. 5, 171–174 (1972).
[CrossRef]

Asakura, T.

Beckmann, P.

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

Berry, M. V.

M. V. Berry, “The statistical properties of echoes diffracted from rough surfaces,” Phil. Trans. R. Soc. London 273, 611–654 (1973).
[CrossRef]

Bitz, G.

G. Bitz, “Verfahren zur Bestimmung von Rauheitskenngröβ en durch Specklekorrelation,” doctoral dissertation (Universität (TH) Karlsruhe, Karlsruhe, Federal Republic of Germany, 1982).

Elbaum, M. J.

M. J. Elbaum, M. Greenebaum, M. King, “A wavelength diversity technique for reduction of speckle size,” Opt. Commun. 5, 171–174 (1972).
[CrossRef]

George, N.

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, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
[CrossRef]

J. W. Goodman, “Statistical properties of laser sparkle patterns,” Tech. Rep. No. 2303-1 (Stanford Electronics Laboratories, Stanford, Calif., 1963).

Gradshteyn, J. S.

J. S. Gradshteyn, J. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Greenebaum, M.

M. J. Elbaum, M. Greenebaum, M. King, “A wavelength diversity technique for reduction of speckle size,” Opt. Commun. 5, 171–174 (1972).
[CrossRef]

Jain, A.

King, M.

M. J. Elbaum, M. Greenebaum, M. King, “A wavelength diversity technique for reduction of speckle size,” Opt. Commun. 5, 171–174 (1972).
[CrossRef]

Lèger, D.

Mathieu, E.

Musazzi, S.

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

Ohtsubo, J.

Pedersen, H. M.

H. M. Pedersen, “Object-roughness dependence of partially developed speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
[CrossRef]

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

Perini, U.

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

Perrin, J. C.

Ryzhik, J. M.

J. S. Gradshteyn, J. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Spizzichino, A.

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

Tribillon, G.

G. Tribillon, “Correlation entre deux speckles obtenus avec deux longeurs d’onde. Application à la mesure de la rugosité moyenne,” Opt. Commun. 11, 172–174 (1974).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

Opt. Acta (1)

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

Opt. Commun. (5)

M. J. Elbaum, M. Greenebaum, M. King, “A wavelength diversity technique for reduction of speckle size,” Opt. Commun. 5, 171–174 (1972).
[CrossRef]

G. Tribillon, “Correlation entre deux speckles obtenus avec deux longeurs d’onde. Application à la mesure de la rugosité moyenne,” Opt. Commun. 11, 172–174 (1974).
[CrossRef]

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

H. M. Pedersen, “Object-roughness dependence of partially developed speckle patterns in coherent light,” Opt. Commun. 16, 63–67 (1976).
[CrossRef]

J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
[CrossRef]

Phil. Trans. R. Soc. London (1)

M. V. Berry, “The statistical properties of echoes diffracted from rough surfaces,” Phil. Trans. R. Soc. London 273, 611–654 (1973).
[CrossRef]

Other (4)

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

J. S. Gradshteyn, J. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

G. Bitz, “Verfahren zur Bestimmung von Rauheitskenngröβ en durch Specklekorrelation,” doctoral dissertation (Universität (TH) Karlsruhe, Karlsruhe, Federal Republic of Germany, 1982).

J. W. Goodman, “Statistical properties of laser sparkle patterns,” Tech. Rep. No. 2303-1 (Stanford Electronics Laboratories, Stanford, Calif., 1963).

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Figures (9)

Fig. 1
Fig. 1

Scattering geometry.

Fig. 2
Fig. 2

Normalized autocorrelation function of the surface: solid curves, Eq. (15) for various β; dashed curve, Eq. (16).

Fig. 3
Fig. 3

Speckle contrast V as a function of Ωσh and N (λ = 0.6328 μm, α = 45°): solid curves, Ruffing/Fleischer; dashed curves, Goodman.

Fig. 4
Fig. 4

Degree of spectral speckle correlation γ12 versus σh for various N (α = 45°): dashed curves, Eq. (21); solid curves, Eq. (20).

Fig. 5
Fig. 5

γ12 and V as functions of σh for N = 100 (α = 45°).

Fig. 6
Fig. 6

Experimental arrangement used to measure the degree of spectral speckle correlation.

Fig. 7
Fig. 7

Surface height density of sample No. 3 (solid curve, measured; dashed curve, assumed in theory with σh from Table 1).

Fig. 8
Fig. 8

Theoretical and experimental results for γ12 as a function of σh (α = 45°).

Fig. 9
Fig. 9

gx) for various β versus Δx/r and the step function Ux − Δx0).

Tables (1)

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Table 1 Stylus Reference Measurements

Equations (38)

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u ( ξ , k ) = j k cos α 2 π R exp [ j k R ( 1 + ξ 2 2 R 2 ) ] × + A ( x ) exp [ j k ( 2 cos α ξ R sin α ) h ( x ) ] × exp ( j cos α k ξ R x ) d x ,
A ( x ) = exp ( x 2 / L 2 ) .
u ( k ) = C + A ( x ) exp [ j Ω h ( x ) ] d x ,
u ( k ) = Δ u ( k ) + u ( k ) ,
i ( k ) = u ( k ) u * ( k ) ,
γ 12 = i 1 i 2 i 1 i 2 [ ( i 1 2 i 1 2 ) ( i 2 2 i 2 2 ) ] 1 / 2 = C 12 ( C 11 C 22 ) 1 / 2 .
C 12 = Δ u 1 Δ u 1 * Δ u 2 Δ u 2 * + u 1 u 2 × ( Δ u 1 Δ u 2 + Δ u 1 Δ u 2 * + Δ u 1 * Δ u 2 + Δ u 1 * Δ u 2 * ) Δ u 1 Δ u 1 * Δ u 2 Δ u 2 * .
Δ u 1 Δ u 1 * Δ u 2 Δ u 2 * = Δ u 1 Δ u 1 * Δ u 2 Δ u 2 * + Δ u 1 Δ u 2 Δ u 1 * Δ u 2 * + Δ u 1 Δ u 2 * Δ u 1 * Δ u 2 .
C 12 = | u 1 u 2 * | 2 + | u 1 u 2 | 2 2 u 1 2 u 2 2 .
γ 12 = | u 1 u 2 * | 2 + | u 1 u 2 | 2 2 u 1 2 u 2 2 ( C 11 C 22 ) 1 / 2 ,
γ 12 = | u 1 u 2 * | 2 i 1 i 2 ,
V n = C n n 1 / 2 i n = ( | i n | 2 + | u n 2 | 2 2 u n 4 ) 1 / 2 i n ,
u n = A ( x ) exp [ j Ω n h ( x ) ] d x ,
| u n u m * | 2 = | + A ( x 1 ) A ( x 2 ) × exp { j [ Ω n h ( x 1 ) Ω m h ( x 2 ) ] } d x 1 d x 2 | 2 ,
| u n u m | 2 = | + A ( x 1 ) A ( x 2 ) × exp { j [ Ω n h ( x 1 ) + Ω m h ( x 2 ) ] } d x 1 d x 2 | 2 .
M ( j ω n ) = exp [ j ω n h ( x ) ] = exp ( 1 2 ω n 2 σ h 2 )
M ( j ω n , j ω m ) = exp { j [ ω n h ( x 1 ) + ω m h ( x 2 ) ] } = exp { 1 2 σ h 2 [ ω n 2 + 2 ω n ω m ρ ( x 1 x 2 ) + ω m 2 ] } ,
u n = exp ( ½ β n n ) + exp ( x 2 / L 2 ) d x = x L exp ( ½ β n n ) ,
| u n u m * | 2 = F | + A ( x 1 ) A ( x 1 Δ x ) × exp [ β n m ρ ( Δ x ) ] d x 1 d Δ x | 2 ,
| u n u m | 2 = F | + A ( x 1 ) A ( x 1 Δ x ) × exp [ β n m ρ ( Δ x ) ] d x 1 d Δ x | 2 ,
ρ ( Δ x ) = 1 β n m ln { 1 + exp ( a n m Δ x 2 / r 2 ) [ exp ( β n m ) 1 ] } ,
ρ ( Δ x ) = exp ( Δ x 2 / r 2 ) ,
| u n u m * | 2 = F | + A ( x 1 ) A ( x 1 Δ x ) × { 1 + exp ( a n m Δ x 2 / r 2 ) [ exp ( β n m ) 1 ] } d x 1 d Δ x | 2 ,
| u n u m | 2 = F | + A ( x 1 ) A ( x 1 Δ x ) × { 1 + exp ( a n m Δ x 2 / r 2 ) × [ exp ( β n m ) 1 ] } 1 d x 1 d Δ x | 2 .
| u n u m * | 2 = F π 2 L 4 { 1 + [ exp ( β n m ) 1 ] ( 1 + 2 a n m N 2 ) 1 / 2 } 2 ,
| u n u m | 2 = F π 2 L 4 { 1 + [ exp ( β n m ) 1 ] erf ( ψ n m / N ) } 2 ,
γ 12 = π 2 L 4 F ( { 1 + [ exp ( β 12 ) 1 ] ( 1 + 2 a 12 N 2 ) 1 / 2 } 2 + { 1 + [ exp ( β 12 ) 1 ] erf ( ψ 12 / N ) } 2 2 ) ( C 11 C 22 ) 1 / 2 .
γ 12 = exp ( 2 β 12 ) exp ( β 11 ) exp ( β 22 ) = exp [ 4 cos 2 α σ h 2 ( k 1 k 2 ) 2 ] .
V = ( { 1 + [ exp ( β ) 1 ] ( 1 + 2 a N 2 ) 1 / 2 } 2 + { 1 + [ exp ( β ) 1 ] erf ( ψ / N ) } 2 2 ) 1 / 2 { 1 + [ exp ( β ) 1 ] ( 1 + 2 a N 2 ) 1 / 2 } ,
| u n u m | 2 = F π L 2 2 | 2 0 exp ( Δ x 2 / 2 L 2 ) × { 1 + exp ( a n m Δ x 2 / r 2 ) × [ exp ( β n m ) 1 ] } 1 d Δ x | 2 .
g ( Δ x ) = { 1 + exp ( a n m Δ x 2 / r 2 ) [ exp ( β n m ) 1 ] } 1
g ( Δ x = 0 ) = exp ( β n m ) , g ( Δ x ) = 1 ,
U ( Δ x Δ x 0 ) = { 1 Δ x Δ x 0 exp ( β n m ) Δ x < Δ x 0 .
Δ x 0 = r { ln [ exp ( 2 β n m ) 1 exp ( β n m ) 1 ] a n m } 1 / 2 .
| u n u m | 2 = F π L 2 2 | 2 [ exp ( β n m ) × 0 Δ x 0 exp ( Δ x 2 / 2 L 2 ) d Δ x + Δ x 0 0 exp ( Δ x 2 / 2 L 2 ) d Δ x ] | 2 .
0 u exp ( q 2 x 2 ) d x = π 2 q erf ( q u ) , q > 0 ,
| u n u m | 2 = F π 2 L 4 { 1 + [ exp ( β n m ) 1 ] erf ( ψ n m / N ) } 2 .
ψ n m = Δ x 0 2 r .

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