Abstract

The autocorrelation function of the intensity scattered from cylindrical rough surfaces is analytically obtained with the Kirchhoff scalar diffraction theory. It is shown that, in contrast to the case in which planar rough surfaces scatter radiation, this function, related to the speckle size, depends on the statistical parameters that characterize the surface and on the scattering direction. This result suggests a new, to our knowledge, optical method that can be applied to the characterization of cylindrical rough surfaces, such as in on-line quality assessment, in manufacturing processes. The calculated theoretical expression was tested, showing good agreement with experiments.

© 2000 Optical Society of America

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References

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  1. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).
  2. J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).
    [CrossRef]
  3. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Adam Hilger, Bristol, UK, 1991).
  4. J. M. Bennett, Surface Finish and its Measurement, Vol. 2 of OSA Collected Works in Optics Series (Optical Society of America, Washington, D.C., 1992).
  5. G. Da Costa, J. Ferrari, “Anisotropic speckle patterns in the light scattered by rough cylindrical surfaces,” Appl. Opt. 36, 5231–5237 (1997).
    [CrossRef] [PubMed]
  6. F. Perez Quintián, M. A. Rebollo, N. G. Gaggioli, “Optical methods for on line surface wire testing,” in Proceedings of the 7th European Conference on Non Destructive Testing (7th ECNDT, Broendby, Denmark, 1998), pp. 2920–2926 (also available on line at http://www.ntd.net ).
  7. F. Perez Quintián, M. A. Rebollo, R. Berlasso, N. G. Gaggioli, “Roughness measurement on cylindrical surfaces by optical methods,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.48–5.57.
  8. F. Perez Quintián, M. A. Rebollo, M. T. Bernal, C. A. Raffo, “Measurements of high quality wire roughness,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.58–5.64.
  9. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics Series (Springer-Verlag, Berlin, 1975), pp. 9–75.

1997 (1)

Beckmann, P.

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

Bennett, J. M.

J. M. Bennett, Surface Finish and its Measurement, Vol. 2 of OSA Collected Works in Optics Series (Optical Society of America, Washington, D.C., 1992).

Berlasso, R.

F. Perez Quintián, M. A. Rebollo, R. Berlasso, N. G. Gaggioli, “Roughness measurement on cylindrical surfaces by optical methods,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.48–5.57.

Bernal, M. T.

F. Perez Quintián, M. A. Rebollo, M. T. Bernal, C. A. Raffo, “Measurements of high quality wire roughness,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.58–5.64.

Da Costa, G.

Ferrari, J.

Gaggioli, N. G.

F. Perez Quintián, M. A. Rebollo, R. Berlasso, N. G. Gaggioli, “Roughness measurement on cylindrical surfaces by optical methods,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.48–5.57.

F. Perez Quintián, M. A. Rebollo, N. G. Gaggioli, “Optical methods for on line surface wire testing,” in Proceedings of the 7th European Conference on Non Destructive Testing (7th ECNDT, Broendby, Denmark, 1998), pp. 2920–2926 (also available on line at http://www.ntd.net ).

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics Series (Springer-Verlag, Berlin, 1975), pp. 9–75.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Adam Hilger, Bristol, UK, 1991).

Perez Quintián, F.

F. Perez Quintián, M. A. Rebollo, R. Berlasso, N. G. Gaggioli, “Roughness measurement on cylindrical surfaces by optical methods,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.48–5.57.

F. Perez Quintián, M. A. Rebollo, N. G. Gaggioli, “Optical methods for on line surface wire testing,” in Proceedings of the 7th European Conference on Non Destructive Testing (7th ECNDT, Broendby, Denmark, 1998), pp. 2920–2926 (also available on line at http://www.ntd.net ).

F. Perez Quintián, M. A. Rebollo, M. T. Bernal, C. A. Raffo, “Measurements of high quality wire roughness,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.58–5.64.

Raffo, C. A.

F. Perez Quintián, M. A. Rebollo, M. T. Bernal, C. A. Raffo, “Measurements of high quality wire roughness,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.58–5.64.

Rebollo, M. A.

F. Perez Quintián, M. A. Rebollo, M. T. Bernal, C. A. Raffo, “Measurements of high quality wire roughness,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.58–5.64.

F. Perez Quintián, M. A. Rebollo, N. G. Gaggioli, “Optical methods for on line surface wire testing,” in Proceedings of the 7th European Conference on Non Destructive Testing (7th ECNDT, Broendby, Denmark, 1998), pp. 2920–2926 (also available on line at http://www.ntd.net ).

F. Perez Quintián, M. A. Rebollo, R. Berlasso, N. G. Gaggioli, “Roughness measurement on cylindrical surfaces by optical methods,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.48–5.57.

Spizzichino, A.

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

Appl. Opt. (1)

Other (8)

F. Perez Quintián, M. A. Rebollo, N. G. Gaggioli, “Optical methods for on line surface wire testing,” in Proceedings of the 7th European Conference on Non Destructive Testing (7th ECNDT, Broendby, Denmark, 1998), pp. 2920–2926 (also available on line at http://www.ntd.net ).

F. Perez Quintián, M. A. Rebollo, R. Berlasso, N. G. Gaggioli, “Roughness measurement on cylindrical surfaces by optical methods,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.48–5.57.

F. Perez Quintián, M. A. Rebollo, M. T. Bernal, C. A. Raffo, “Measurements of high quality wire roughness,” in Proceedings of the International Symposium on Laser Metrology for Precision Measurement and Inspection in Industry (International Measurement Confederation, Earlangen, Germany, 1999), pp. 5.58–5.64.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics Series (Springer-Verlag, Berlin, 1975), pp. 9–75.

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

J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).
[CrossRef]

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Adam Hilger, Bristol, UK, 1991).

J. M. Bennett, Surface Finish and its Measurement, Vol. 2 of OSA Collected Works in Optics Series (Optical Society of America, Washington, D.C., 1992).

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Figures (9)

Fig. 1
Fig. 1

Scheme of the light scattering from a cylinder. A plane wave represented by wave vector i is incident over a rough cylinder. The scattered waves are represented by vectors and also by directions (θ, ϕ) (the usual spherical coordinates), each of which determines a point on the observation plane at the far field. The dotted line corresponds to the circumference θ = α.

Fig. 2
Fig. 2

Light scattered from a rough cylinder. The screen was placed on the observation plane of Fig. 1, and the photograph was taken from behind. The small rectangles [(a) and (b)] are magnified in Fig. 6, below.

Fig. 3
Fig. 3

(a) Illustration of the vector = [h(φ, z), φ, z] that ends in a point on the cylindrical rough surface, where the function h(φ, z) is assumed to be a random process. (b) Representation of the variables τ and ζ over the cylinder surface, which are used instead of the usual cylindrical coordinates (φ, z) to solve the integrals in Eq. (3).

Fig. 4
Fig. 4

Geometrical representation of the parameters involved in Eq. (14), used to express the difference between the directions (θ + δθ, ϕ + δϕ) and (θ, ϕ) as a difference between points on the observation plane.

Fig. 5
Fig. 5

Experimental setup for registering the speckle pattern. The lens behind the pupil is placed to validate the Fraunhoffer approximation at the plane where the camera is placed.

Fig. 6
Fig. 6

Images of the speckles produced by sample B at (a) ϕ = 0° and (b) ϕ = 90°. The images are magnifications of the rectangles shown in Fig. 2. (c) and (d) are further magnifications of (a) and (b), respectively.

Fig. 7
Fig. 7

Autocorrelation functions of images (a) and (b) of Fig. 6. (a) Radial (horizontal) experimental autocorrelation functions. (b) Azimuthal (vertical) experimental autocorrelation functions. Squares correspond to image on left-hand side of Fig. 6 (ϕ = 0°); circles correspond to image on right-hand side of Fig. 6 (ϕ = 90°).

Fig. 8
Fig. 8

Experimental autocorrelation functions for the four samples at different values of ϕ, with their respective theoretical fitting curves.

Fig. 9
Fig. 9

Linear fit corresponding to Eq. (24) for the four samples.

Tables (1)

Tables Icon

Table 1 Comparison between the Characteristics of the Samples Used in the Experiment, when Measured with a Mechanical Profilometer and when Measured with Our Method

Equations (31)

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γABθ, ϕ; θ, ϕ=IAθ, ϕIBθ, ϕ-IAθ, ϕIBθ, ϕIA2θ, ϕ-IAθ, ϕ2IB2θ, ϕ-IBθ, ϕ21/2=|EAθ, ϕEB*θ, ϕ|2IAθ, ϕIBθ, ϕ,
Eθ, ϕ=-2i expikR0E0k4πR0aθ, ϕ, αsin α-cθ, αcos α×φ,zexpikcθ, αz+aθ, ϕ, αh cos φ+bθ, ϕhφ, zsin φ)aθ, ϕ, αcos φ+bθ, ϕsin φ hφ, zdzdφ,
aθ, ϕ, α=-sin θ cos ϕ-sin α, bθ, ϕ=-sin θ sin ϕ, cθ, α=cos α-cos θ.
Eθ, ϕE*θ, ϕ=Kθ, ϕK*θ, ϕφ1,z1dz1dφ1×φ2,z2dz2dφ2 expikcz2-cz1×expikh2Cφ2-h1Cφ1h1h2Cφ1Cφ2,
Kθ, ϕ, α=-2i expikR0E0k4πR0 ×aθ, ϕ, αsin α-cθ, αcos α, h1,2=hφ1,2, z1,2, Cθ, ϕ, φ1,2, α=aθ, ϕ, αcos φ1,2+bθ, ϕsin φ1,2,
c=cos α-cos θ=cos α-cosθ+δθcos α-cos θ+sin θδθ, Cφ1=aθ+δθ, ϕ+δϕcos φ1+bθ+δθ, ϕ+δϕsin φ1Cφ1+sin θ sinϕ-φ1δϕ-cos θ×cosϕ-φ1δθ, Cφ2=aθ, ϕcos φ2+bθ, ϕsin φ2Cφ1+τ/hsin ζb cos φ1-a sin φ1,
τ cos ζ=z2-z1, τ sin ζ=hφ2-φ1.
Eθ, ϕE*θ+δθ, ϕ+δϕ  h ζ,τ τdτdζ φ1,z1dz1dφ1 expikcτ cos ζexp-ikcz1 sin θδθ×exp(ikh2Cφ1+τ/hsin ζdφ1-h1Cφ1+Δφ1)Cφ1+Δφ1Cφ1+τ/hsin ζdφ1.
Δφ1sin θ sinϕ-φ1δϕ-cos θ cosϕ-φ1δθ, dφ1b cos φ1-a sin φ1,
-L/2L/2exp-ikcz1 sin θδθdz1=L sinkc sin θδθL/2kc sin θδθL/2.
expikh2Cφ2-h1Cφ1=expikhCφ2-Cφ1exp-½ σ2k2C2φ2-2γz2, φ2, z1, φ1Cφ2Cφ1+C2φ1,
γz2, φ2, z1, φ1=exp-τ2T21-τ2T2.
exp(ikh2Cφ1+τ/asin ζdφ1-h1Cφ1 +Δφ1)=expikhCφ2-Cφ1 ×exp-½ σ2k2Cφ2-Cφ12 ×exp-σ2k2τ2/T2Cφ2Cφ1.
expikh2Cφ2-h1Cφ1=expikτ sin ζdφ1×expikτ sin ζdφ1exp-ikhΔφ1×expσ2k2τhsin ζdφ1Δφ1×exp-12 σ2k2τhsin ζdφ12×exp-σ2k2τ2T2C2φ1+Cφ1τh×sin ζdφ1+Cφ1Δφ1+τhsin ζdφ1Δ(φ1.
expikh2Cφ2-h1Cφ1=expikτ sin ζdφ1×exp-ikhΔφ1exp-σ2k2τ2/T2×C2φ1+Cφ1Δφ1.
Eθ, ϕE*θ+δθ, ϕ+δϕ  hL ×sinkc sin θδθL/2kc sin θδθL/2 φ1,ζ,τ ×expikτc cos ζ+dφ1sin ζexp-ikhΔφ1C2φ1+Cφ1Δφ1 ×exp-σ2k2τ2/T2C2φ1+Cφ1Δφ1τdτdζdφ1.
Eθ, ϕE*θ+δθ, ϕ+δϕ  L sinkc sin θδθL/2kc sin θδθL/2×φ1,τ2πJ0kτc2+d2φ11/2C2φ1+Cφ1Δφ1exp-ikhΔφ1×exp-σ2k2τ2/T2C2φ1+Cφ1Δφ1τdτdφ1,
Eθ, ϕE*θ+δθ, ϕ+δϕ  hL ×sinkc sin θδθL/2kc sin θδθL/2T2σ2k2 ×φ1exp-T2c2+d2φ14σ2C2φ1+Cφ1Δφ1 × exp-ikhΔφ1C2φ1+Cφ1Δφ12dφ1.
Δφ1sin θ sinϕ-φ1δϕ-cos θ cosϕ-φ1δθ, N sinϕ-φ1-M cosϕ-φ1=-M2+N21/2 cosδ+ϕ-φ1-Δ0 cosδ+ϕ-φ1,
Eθ, ϕE*θ, ϕ+δϕ  φ1=ϕ-π/2π/2 exp-T2d2φ14σ2C2φ1 ×exp-ikhΔ0 sinϕ-φ1C4φ1dφ1.
φ1=ϕ/2+.
Cφ1=Cϕ/2cos , d2φ1=C2ϕ/2sin2 , sinϕ-φ1=cos  sinϕ/2-cosϕ/2sin .
Eθ, ϕE*θ, ϕ+δϕ  =ϕ/2-π/2π/2-ϕ/2exp-T24σ2tan2 ×exp-ikhΔ0cos  sinϕ2-cosϕ2sin C4ϕ/2cos4 d.
cos  sinϕ/2-cosϕ/2sin sinϕ/2-cosϕ/2tan .
Eθ, ϕE*θ, ϕ+δϕ  -exp-T2 tan2 4σ2×tan2 +12 exp-ikhΔ0 cosϕ/2tan d,
Eθ, ϕE*θ, ϕ+δϕ  -exp-T2ξ24σ2×ξ2+1exp-ikhΔ0 cosϕ2 ξdξ,
Eθ, ϕE*θ, ϕ+δϕ  exp-σkhΔ0 cosϕ/2T2.
γ=exp-2σkhΔ0 cosϕ/2T2.
Δ0=T2σ1kh1cosϕ/2.
Wp/D=Δ0,
W=T2σ1khDp1cosϕ/2.

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