Abstract

It is well known that the three-dimensional dimension of laser speckle is a critical parameter in laser holographic interferometry, laser speckle photography, and holospeckle interferometry. In this paper a statistical analysis for computing the three-dimensional dimension of laser speckle is presented. Explicit formulas have been derived for laser speckle in a free space from a circular scattering area along with some supporting results from an experimental investigation.

© 1992 Optical Society of America

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References

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  1. L. T. Goldfischer, “Autocorrelation function and power spectral density of laser-produced speckle pattern,” J. Opt. Soc. Am. 55, 247–253 (1965).
    [CrossRef]
  2. L. H. Enloe, “Noise-like structure in the image of diffusely reflecting objects in coherent illumination,” Bell Syst. Tech. J. 46, 1479–1489 (1967).
  3. B. Eliason, F. M. Mottier, “Determination of the granular radiance distribution of diffuser and its use for vibration analysis,” J. Opt. Soc. Am. 61, 559–565 (1971).
    [CrossRef]
  4. G. P. Weigelt, B. Stoffregan, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik (Stuttgart) 48, 399–408 (1977).
  5. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin Heidelberg, 1975).
    [CrossRef]
  6. C. E. Halford, W. L. Gamble, N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).
  7. Q. B. Li, “A study of holo-speckle interferometry,” Ph.D. dissertation (State University of New York, Stony Brook, N.Y., 1986).
  8. F. P. Chiang, Q. B. Li, “Statistical analysis of holographic fringe formation and localization,” in Optical Testing and Metrology, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.661, 36–43 (1986).
  9. Q. B. Li, F. P. Chiang, “A new formula for fringe localization in holographic interferometry,” Opt. Lasers Eng. 8, 1–21 (1988).
  10. D. W. Li, F. P. Chiang, J. B. Chen, “Statistical analysis of one-beam subjective laser speckle interferometry,” J. Opt. Soc. Am. A 2, 657–666 (1985).
    [CrossRef]
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.
  12. F. P. Chiang, M. Pedretti, “Sensitivity increase of laser speckle interferometry by film nonlinearity,” Tech. Rep. 304 (State University of New York, Stony Brook, N.Y., 1977).
  13. F. P. Chiang, “A new family of two-dimensional and three-dimensional stress analysis techniques using laser speckles,” Solid Mech. Arch. 3, 1–32 (1978).

1988

Q. B. Li, F. P. Chiang, “A new formula for fringe localization in holographic interferometry,” Opt. Lasers Eng. 8, 1–21 (1988).

1987

C. E. Halford, W. L. Gamble, N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).

1985

1978

F. P. Chiang, “A new family of two-dimensional and three-dimensional stress analysis techniques using laser speckles,” Solid Mech. Arch. 3, 1–32 (1978).

1977

G. P. Weigelt, B. Stoffregan, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik (Stuttgart) 48, 399–408 (1977).

1971

1967

L. H. Enloe, “Noise-like structure in the image of diffusely reflecting objects in coherent illumination,” Bell Syst. Tech. J. 46, 1479–1489 (1967).

1965

Chen, J. B.

Chiang, F. P.

Q. B. Li, F. P. Chiang, “A new formula for fringe localization in holographic interferometry,” Opt. Lasers Eng. 8, 1–21 (1988).

D. W. Li, F. P. Chiang, J. B. Chen, “Statistical analysis of one-beam subjective laser speckle interferometry,” J. Opt. Soc. Am. A 2, 657–666 (1985).
[CrossRef]

F. P. Chiang, “A new family of two-dimensional and three-dimensional stress analysis techniques using laser speckles,” Solid Mech. Arch. 3, 1–32 (1978).

F. P. Chiang, M. Pedretti, “Sensitivity increase of laser speckle interferometry by film nonlinearity,” Tech. Rep. 304 (State University of New York, Stony Brook, N.Y., 1977).

F. P. Chiang, Q. B. Li, “Statistical analysis of holographic fringe formation and localization,” in Optical Testing and Metrology, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.661, 36–43 (1986).

Eliason, B.

Enloe, L. H.

L. H. Enloe, “Noise-like structure in the image of diffusely reflecting objects in coherent illumination,” Bell Syst. Tech. J. 46, 1479–1489 (1967).

Gamble, W. L.

C. E. Halford, W. L. Gamble, N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).

George, N.

C. E. Halford, W. L. Gamble, N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).

Goldfischer, L. T.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin Heidelberg, 1975).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

Halford, C. E.

C. E. Halford, W. L. Gamble, N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).

Li, D. W.

Li, Q. B.

Q. B. Li, F. P. Chiang, “A new formula for fringe localization in holographic interferometry,” Opt. Lasers Eng. 8, 1–21 (1988).

F. P. Chiang, Q. B. Li, “Statistical analysis of holographic fringe formation and localization,” in Optical Testing and Metrology, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.661, 36–43 (1986).

Q. B. Li, “A study of holo-speckle interferometry,” Ph.D. dissertation (State University of New York, Stony Brook, N.Y., 1986).

Mottier, F. M.

Pedretti, M.

F. P. Chiang, M. Pedretti, “Sensitivity increase of laser speckle interferometry by film nonlinearity,” Tech. Rep. 304 (State University of New York, Stony Brook, N.Y., 1977).

Stoffregan, B.

G. P. Weigelt, B. Stoffregan, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik (Stuttgart) 48, 399–408 (1977).

Weigelt, G. P.

G. P. Weigelt, B. Stoffregan, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik (Stuttgart) 48, 399–408 (1977).

Bell Syst. Tech. J.

L. H. Enloe, “Noise-like structure in the image of diffusely reflecting objects in coherent illumination,” Bell Syst. Tech. J. 46, 1479–1489 (1967).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

C. E. Halford, W. L. Gamble, N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).

Opt. Lasers Eng.

Q. B. Li, F. P. Chiang, “A new formula for fringe localization in holographic interferometry,” Opt. Lasers Eng. 8, 1–21 (1988).

Optik (Stuttgart)

G. P. Weigelt, B. Stoffregan, “The longitudinal correlation of a three-dimensional speckle intensity distribution,” Optik (Stuttgart) 48, 399–408 (1977).

Solid Mech. Arch.

F. P. Chiang, “A new family of two-dimensional and three-dimensional stress analysis techniques using laser speckles,” Solid Mech. Arch. 3, 1–32 (1978).

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

F. P. Chiang, M. Pedretti, “Sensitivity increase of laser speckle interferometry by film nonlinearity,” Tech. Rep. 304 (State University of New York, Stony Brook, N.Y., 1977).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin Heidelberg, 1975).
[CrossRef]

Q. B. Li, “A study of holo-speckle interferometry,” Ph.D. dissertation (State University of New York, Stony Brook, N.Y., 1986).

F. P. Chiang, Q. B. Li, “Statistical analysis of holographic fringe formation and localization,” in Optical Testing and Metrology, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.661, 36–43 (1986).

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Figures (5)

Fig. 1
Fig. 1

Illustration for the analysis of the three-dimensional dimension of laser speckle.

Fig. 2
Fig. 2

Coefficient C (for calculating speckle length) versus observation angle θr.

Fig. 3
Fig. 3

Diffraction halos from two specklegrams recorded in (a) normal and (b) 45° directions.

Fig. 4
Fig. 4

Optical setup for the measurement of speckle length.

Fig. 5
Fig. 5

Young’s fringe patterns from three double-exposure specklegrams given different longitudinal translations w: (a) 1.016 mm, (b) 1.524 mm, and (c) 2.032 mm.

Equations (26)

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R I ( P 0 , P 1 ) = I ( P 0 ) I ( P 1 ) + J A ( P 0 , P 1 ) 2 ,
A ( P 0 ) = Ω a ( Q ) exp ( j k L 0 ) L 0 d σ
A ( P 1 ) = Ω a ( Q ) exp ( j k L 1 ) L 1 d σ ,
a ( Q ) = D ( Q ) j λ · exp ( j k L s ) L s · ( cos θ r + cos θ i ) 2 = K 1 D ( Q ) exp ( j k L s ) L s ,
J A ( P 0 , P 1 ) = Ω Ω a ( Q ) a * ( Q ) × exp [ - j k ( L i - L 0 ) ] L 0 L 1 d σ d σ = K 1 2 Ω Ω D ( Q ) D * ( Q ) × exp [ - j k ( L s - L s ) ] L s L s × exp [ - j k ( L 1 - L 0 ) ] L 0 L 1 d σ d σ .
D ( Q ) D * ( Q ) = C δ ( Q - Q ) ,
J A ( P 0 , P 1 ) = K 3 2 Ω exp [ - j k ( L 1 - L 0 ) ] d σ ,
L 0 = [ x 2 + ( y - y 0 ) 2 + z 0 2 ] 1 / 2 = [ L o p 2 + x 2 + y 2 - 2 y 0 y ] 1 / 2 = L o p ( 1 + x 2 + y 2 cos 2 θ r 2 L o p 2 - sin θ r L o p y ) .
L 1 = [ ( x - Δ ξ ) 2 + ( y - y 0 ) 2 + z 0 2 ] 1 / 2 = L o p ( 1 + x 2 + y 2 cos 2 θ r 2 L o p 2 - Δ ξ x L o p 2 + Δ ξ 2 2 L o p 2 - sin θ r L o p y ) .
L 1 - L 0 = - Δ ξ x L o p + Δ ξ 2 2 L o p .
J A ( P 0 , P 1 ) = K 3 2 Ω exp [ - j k ( - Δ ξ x L o p + Δ ξ 2 2 L o p ) d σ = K 3 2 exp ( - j k Δ ξ 2 2 L o p ) × 0 2 π 0 D 0 / 2 exp ( j 2 π Δ ξ r cos θ λ L o p ) r d r d θ .
0 2 π 0 D 0 / 2 exp ( j 2 π Δ ξ r cos θ λ L o p ) × r d r d θ = π 8 D 0 2 ( 2 J 1 ( π ρ ) π ρ ) ,
J A ( P 0 , P 1 ) = K 2 exp ( j k Δ ξ 2 2 L o p ) [ 2 J 1 ( π ρ ) π ρ ] ,
K 2 = π K 3 2 ( D 0 2 / 8 ) and ρ = Δ ξ D 0 / λ L o p .
R I ( P 0 , P 1 ) = I ( P 0 ) I ( P 1 ) + K 4 [ 2 J 1 ( π ρ ) π ρ ] 2 .
Δ ξ = 1.22 λ ( L o p / D 0 ) .
L 1 - L 0 = - Δ η cos θ r L o p y + Δ η 2 2 L o p ,
Δ η = 1.22 λ ( L o p D 0 cos θ r ) .
Δ ξ = 1.22 λ ( L o p D 0 ) , Δ η = 1.22 λ ( L o p D 0 cos θ r ) .
Δ ξ = Δ η = 1.22 λ ( L o p / D 0 )
L 1 - L 0 = Δ ζ - Δ ζ L o p 2 · x 2 + y 2 cos 2 θ r 2 .
J A ( P 0 , P 1 ) = K 3 2 exp ( - j k Δ ζ ) × Ω exp ( j k Δ ζ L o p 2 · x 2 + y 2 cos 2 θ r 2 ) d σ = K 3 2 exp ( - j k Δ ζ ) × 0 2 π 0 D 0 / 2 exp ( j k Δ ζ r 2 L o p 2 · cos 2 θ + cos 2 θ r sin 2 θ 2 ) × r d r d θ .
Δ ζ = 8 λ C ( L o p / D 0 ) 2 .
Δ ζ = 8 λ ( L o p / D 0 ) 2 .
Δ ξ = 2 λ L / D x ,
Δ η = 2 λ L / D y ,

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