Abstract

A simple geometrical model was developed for calculation of the contrast of a polychromatic image-plane speckle pattern from a source of light with high spatial coherence. It is based on counting the number of independent speckle patterns that contribute to a given point in the image plane. This results in a simple equation for the contrast as a function of imaging geometry; relative orientation of the projection direction, observation direction, and specimen normal; bandwidth of the light source; and surface roughness. Its validity was established by comparison with an exact solution: rms errors in the calculated contrast were only 0.033 over a wide range of parameter values likely to be encountered in practice.

© 1999 Optical Society of America

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References

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  1. R. G. Dorsch, G. Häusler, J. M. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. 33, 1306–1314 (1994).
    [CrossRef] [PubMed]
  2. G. Häusler, “About the scaling behaviour of optical range sensors,” in Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 147–155.
  3. R. A. Sprague, “Surface roughness measurement using white light speckle,” Appl. Opt. 11, 2811–2816 (1972).
    [CrossRef] [PubMed]
  4. K. Nakagawa, T. Asakura, “Average contrast of white-light image speckle patterns,” Opt. Acta 26, 951–960 (1979).
    [CrossRef]
  5. K. Nakagawa, T. Asakura, “Contrast of white light speckle patterns at a defocused image plane,” Appl. Opt. 18, 3725–3728 (1979).
  6. H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” Opt. Acta 22, 15–24 (1975).
    [CrossRef]
  7. 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]
  8. G. Parry, “Speckle patterns in partially coherent light,” Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984), Chap. 3.
  9. T. S. McKechnie, “Image-plane speckle in partially coherent illumination,” Opt. Quantum Electron. 8, 61–67 (1976).
    [CrossRef]
  10. Y.-Q. Hu, “Dependence of polychromatic-speckle-pattern contrast on imaging and illumination directions,” Appl. Opt. 33, 2707–2714 (1994).
    [CrossRef] [PubMed]
  11. J. W. Goodman, “Statistical properties of laser speckle patterns,” Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984), Chap. 2.

1994 (2)

1979 (2)

K. Nakagawa, T. Asakura, “Contrast of white light speckle patterns at a defocused image plane,” Appl. Opt. 18, 3725–3728 (1979).

K. Nakagawa, T. Asakura, “Average contrast of white-light image speckle patterns,” Opt. Acta 26, 951–960 (1979).
[CrossRef]

1976 (1)

T. S. McKechnie, “Image-plane speckle in partially coherent illumination,” Opt. Quantum Electron. 8, 61–67 (1976).
[CrossRef]

1975 (2)

H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” Opt. Acta 22, 15–24 (1975).
[CrossRef]

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]

1972 (1)

Asakura, T.

K. Nakagawa, T. Asakura, “Average contrast of white-light image speckle patterns,” Opt. Acta 26, 951–960 (1979).
[CrossRef]

K. Nakagawa, T. Asakura, “Contrast of white light speckle patterns at a defocused image plane,” Appl. Opt. 18, 3725–3728 (1979).

Dorsch, R. G.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984), Chap. 2.

Häusler, G.

R. G. Dorsch, G. Häusler, J. M. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. 33, 1306–1314 (1994).
[CrossRef] [PubMed]

G. Häusler, “About the scaling behaviour of optical range sensors,” in Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 147–155.

Herrmann, J. M.

Hu, Y.-Q.

McKechnie, T. S.

T. S. McKechnie, “Image-plane speckle in partially coherent illumination,” Opt. Quantum Electron. 8, 61–67 (1976).
[CrossRef]

Nakagawa, K.

K. Nakagawa, T. Asakura, “Contrast of white light speckle patterns at a defocused image plane,” Appl. Opt. 18, 3725–3728 (1979).

K. Nakagawa, T. Asakura, “Average contrast of white-light image speckle patterns,” Opt. Acta 26, 951–960 (1979).
[CrossRef]

Parry, G.

G. Parry, “Speckle patterns in partially coherent light,” Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984), Chap. 3.

Pedersen, H. M.

H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” Opt. Acta 22, 15–24 (1975).
[CrossRef]

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]

Sprague, R. A.

Appl. Opt. (4)

Opt. Acta (3)

K. Nakagawa, T. Asakura, “Average contrast of white-light image speckle patterns,” Opt. Acta 26, 951–960 (1979).
[CrossRef]

H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” Opt. Acta 22, 15–24 (1975).
[CrossRef]

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]

Opt. Quantum Electron. (1)

T. S. McKechnie, “Image-plane speckle in partially coherent illumination,” Opt. Quantum Electron. 8, 61–67 (1976).
[CrossRef]

Other (3)

J. W. Goodman, “Statistical properties of laser speckle patterns,” Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984), Chap. 2.

G. Häusler, “About the scaling behaviour of optical range sensors,” in Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 147–155.

G. Parry, “Speckle patterns in partially coherent light,” Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984), Chap. 3.

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Figures (2)

Fig. 1
Fig. 1

Simple model for speckle contrast. The rectangle is the cross section through the cylinder that contains scatterers contributing to a speckle pattern in the image plane. The horizontal lines are cross sections through planes of constant optical path length, which separate the measurement region into strata of depth δ. Light scattered from any two points within a single stratum will add coherently in the image plane.

Fig. 2
Fig. 2

Speckle contrast as a function of θ2, ψ, σ, W, Q, and R. (a) ψ = 0, Q = 32 mm, R = 1.2 m. (σ, W) = (0.2 µm, 0.1 µm-1), ●; (3 µm, 0.1 µm-1), ■; (0.2 µm, 2.5 µm-1), ▼; (3 µm, 2.5 µm-1), ▲. (b) ψ = θ, σ = 0.3 µm, Q = 32 mm, R = 1.2 m. W = 0.1 µm-1, ●; 0.5 µm-1, ■; 1.5 µm-1, ▼; 2.5 µm-1, ▲. (c) ψ = θ, σ = 0.3 µm, W = 2.5 µm-1, Q = 32 mm. R = 0.6 m, ●; 1.2 m, ■; 2 m, ▼; 3 m, ▲. (d) ψ = θ, W = 2.5 µm-1, Q = 32 mm, R = 1.2 m. σ = 0.2 µm, ●; 0.5 µm, ■; 1.0 µm, ▼; 2.0 µm, ▲. Open symbols show the corresponding results with the simple model [relations (11) and (12)].

Equations (13)

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= C λ 2 π   sin   u 0   sin   2 θ ,
s 0 k = 1 2 π 1 / 2 W exp - k - k 0 2 / 2 W 2 ,
C 2 θ 1 ,   θ 2 = - S 2 2 π W 2 exp - k 1 - k 0 2 + k 2 - k 0 2 2 W 2 × exp - σ 2 cos   θ 1 + cos   θ 2 2 × k 2 - k 1 2 / 2 d k 1 d k 2 ,
S = a / b = λ 2 / λ 1 ,     0 < d b - a = b / a = λ 1 / λ 2 ,     b - a d < 0 = a 2   cos - 1 a 2 + d 2 - b 2 / 2 a | d | + b 2   cos - 1 × b 2 + d 2 - a 2 / 2 b | d | - | d | a 2 - a 2 + d 2 - b 2 2 / 2 d 2 1 / 2 / π ab ,     | b - a | < | d | < b + a = 0 ,     b + a | d |
d = 1 2 π sin   θ 2 cos   θ 1 - tan   θ 1 k 2 - k 1 ,
δ = l c / 2   cos   θ
s = 1.22 λ R / 2 Q
z = r   sin ψ + tan - 1 h / ρ ,
N = 1 + 2 r   sin ψ + tan - 1 h / ρ δ .
N 1 + 4 ρ ψ + h / ρ l c .
N 1 + α s ψ W + β σ W ,
C = 1 / N ,
ψ = θ 2 - θ 1 / 2 , θ = θ 2 + θ 1 / 2 .

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