Abstract

Speckle phenomena result whenever spatially coherent radiation is reflected from a rough surface or propagated through a random medium such as atmospheric turbulence. Speckle characteristics are therefore a major concern in many laser-imaging or wave-propagation applications. We present the results of experimental measurements of target-induced speckle patterns produced in the laboratory from a variety of targets and illumination conditions. We then compare these experimental measurements with a theoretical model for the average speckle size as a function of intensity threshold level. Excellent agreement is obtained for intensity threshold levels greater than approximately twice the mean intensity level.

© 1994 Optical Society of America

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References

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  1. N. George, Speckle (The Institute of Optics, University of Rochester, Rochester, New York, 1979).
  2. J. C. Dainty, Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, New York, 1984), pp. 1–7.
  3. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 4, pp. 124–127; Chap. 7, 340–350.
  4. M. Francon, Laser Speckle and Applications in Optics (Academic, New York, 1979), pp. i–xi.
  5. D. Fink, S. N. Vodopia, “Coherent detection signal-to-noise ratio of an array of detectors,” Appl. Opt. 15, 453–454 (1976).
    [CrossRef] [PubMed]
  6. P. Gatt, W. P. Perez, D. A. Heimmermann, C. M. Stickley, “Hererodyne laser radar array receiver for the mitigation of target-induced speckle,” in Laser Radar Applications, G. W. Kamerman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1936, 157–164 (1993).
  7. P. Gatt, W. P. Perez, D. A. Heimmermann, C. M. Stickley, “Coherent laser radar array receivers: theory and experiment,” presented at the Optical Society of America Seventh Annual Conference on Coherent Laser Radar, Paris, France, 19–23 July 1993.
  8. J. Marron, G. M. Morris, “Correlation measurements using clipped laser speckle,” Appl. Opt. 25, 789–793 (1985).
    [CrossRef]
  9. J. Marron, “Correlation properties of clipped laser speckle,” J. Opt. Soc. Am. A 2, 1403–1410 (1986).
    [CrossRef]
  10. R. Barakat, “Clipped correlation functions of aperture integrated laser speckle,” Appl. Opt. 25, 3885–3888 (1986).
    [CrossRef] [PubMed]
  11. A. D. Ducharme, G. D. Boreman, D. R. Snyder, “Effects of intensity thresholding on the power spectrum of laser speckle,” Appl. Opt. 33, 2715–2720 (1994).
    [CrossRef] [PubMed]
  12. F. E. Kragh, “Excursion areas of intensity due to random optical waves,” M. S. thesis (University of Central Florida, Orlando, Fla., 1990).
  13. E. Jakeman, “Enhanced backscattering through a deep random phase screen,” J. Opt. Soc. Am. A 5, 1638–1648 (1988).
    [CrossRef]
  14. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic, New York, 1980), p. 940.
  15. D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960), Chap. 9, pp. 426–427.
  16. E. Jakeman, “Photon correlation,” in Photon Correlation and Light Beating SpectroscopyH. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1974), Chap. 2, pp. 75–149.
  17. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 9, p. 282.
  18. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), Chap. 6, p. 115.
  19. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 7, p. 213.
  20. R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), Chap. 2, p. 41.
  21. G. Cooper, C. McGillem, Probabilistic Methods of Signal and System Analysis (Harcourt Brace Jovanovich, San Diego, Calif., 1986), Chap. 5, pp. 126–129.

1994 (1)

1988 (1)

1986 (2)

1985 (1)

1976 (1)

Barakat, R.

Boreman, G. D.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), Chap. 6, p. 115.

Cooper, G.

G. Cooper, C. McGillem, Probabilistic Methods of Signal and System Analysis (Harcourt Brace Jovanovich, San Diego, Calif., 1986), Chap. 5, pp. 126–129.

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, New York, 1984), pp. 1–7.

Ducharme, A. D.

Fink, D.

Francon, M.

M. Francon, Laser Speckle and Applications in Optics (Academic, New York, 1979), pp. i–xi.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 7, p. 213.

Gatt, P.

P. Gatt, W. P. Perez, D. A. Heimmermann, C. M. Stickley, “Coherent laser radar array receivers: theory and experiment,” presented at the Optical Society of America Seventh Annual Conference on Coherent Laser Radar, Paris, France, 19–23 July 1993.

P. Gatt, W. P. Perez, D. A. Heimmermann, C. M. Stickley, “Hererodyne laser radar array receiver for the mitigation of target-induced speckle,” in Laser Radar Applications, G. W. Kamerman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1936, 157–164 (1993).

George, N.

N. George, Speckle (The Institute of Optics, University of Rochester, Rochester, New York, 1979).

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), Chap. 2, p. 41.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 4, pp. 124–127; Chap. 7, 340–350.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic, New York, 1980), p. 940.

Heimmermann, D. A.

P. Gatt, W. P. Perez, D. A. Heimmermann, C. M. Stickley, “Hererodyne laser radar array receiver for the mitigation of target-induced speckle,” in Laser Radar Applications, G. W. Kamerman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1936, 157–164 (1993).

P. Gatt, W. P. Perez, D. A. Heimmermann, C. M. Stickley, “Coherent laser radar array receivers: theory and experiment,” presented at the Optical Society of America Seventh Annual Conference on Coherent Laser Radar, Paris, France, 19–23 July 1993.

Jakeman, E.

E. Jakeman, “Enhanced backscattering through a deep random phase screen,” J. Opt. Soc. Am. A 5, 1638–1648 (1988).
[CrossRef]

E. Jakeman, “Photon correlation,” in Photon Correlation and Light Beating SpectroscopyH. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1974), Chap. 2, pp. 75–149.

Kragh, F. E.

F. E. Kragh, “Excursion areas of intensity due to random optical waves,” M. S. thesis (University of Central Florida, Orlando, Fla., 1990).

Marron, J.

McGillem, C.

G. Cooper, C. McGillem, Probabilistic Methods of Signal and System Analysis (Harcourt Brace Jovanovich, San Diego, Calif., 1986), Chap. 5, pp. 126–129.

Middleton, D.

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960), Chap. 9, pp. 426–427.

Morris, G. M.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 9, p. 282.

Perez, W. P.

P. Gatt, W. P. Perez, D. A. Heimmermann, C. M. Stickley, “Hererodyne laser radar array receiver for the mitigation of target-induced speckle,” in Laser Radar Applications, G. W. Kamerman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1936, 157–164 (1993).

P. Gatt, W. P. Perez, D. A. Heimmermann, C. M. Stickley, “Coherent laser radar array receivers: theory and experiment,” presented at the Optical Society of America Seventh Annual Conference on Coherent Laser Radar, Paris, France, 19–23 July 1993.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic, New York, 1980), p. 940.

Snyder, D. R.

Stickley, C. M.

P. Gatt, W. P. Perez, D. A. Heimmermann, C. M. Stickley, “Coherent laser radar array receivers: theory and experiment,” presented at the Optical Society of America Seventh Annual Conference on Coherent Laser Radar, Paris, France, 19–23 July 1993.

P. Gatt, W. P. Perez, D. A. Heimmermann, C. M. Stickley, “Hererodyne laser radar array receiver for the mitigation of target-induced speckle,” in Laser Radar Applications, G. W. Kamerman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1936, 157–164 (1993).

Vodopia, S. N.

Woods, R. E.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), Chap. 2, p. 41.

Appl. Opt. (4)

J. Opt. Soc. Am. A (2)

Other (15)

N. George, Speckle (The Institute of Optics, University of Rochester, Rochester, New York, 1979).

J. C. Dainty, Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, New York, 1984), pp. 1–7.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 4, pp. 124–127; Chap. 7, 340–350.

M. Francon, Laser Speckle and Applications in Optics (Academic, New York, 1979), pp. i–xi.

P. Gatt, W. P. Perez, D. A. Heimmermann, C. M. Stickley, “Hererodyne laser radar array receiver for the mitigation of target-induced speckle,” in Laser Radar Applications, G. W. Kamerman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1936, 157–164 (1993).

P. Gatt, W. P. Perez, D. A. Heimmermann, C. M. Stickley, “Coherent laser radar array receivers: theory and experiment,” presented at the Optical Society of America Seventh Annual Conference on Coherent Laser Radar, Paris, France, 19–23 July 1993.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic, New York, 1980), p. 940.

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960), Chap. 9, pp. 426–427.

E. Jakeman, “Photon correlation,” in Photon Correlation and Light Beating SpectroscopyH. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1974), Chap. 2, pp. 75–149.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 9, p. 282.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), Chap. 6, p. 115.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 7, p. 213.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), Chap. 2, p. 41.

G. Cooper, C. McGillem, Probabilistic Methods of Signal and System Analysis (Harcourt Brace Jovanovich, San Diego, Calif., 1986), Chap. 5, pp. 126–129.

F. E. Kragh, “Excursion areas of intensity due to random optical waves,” M. S. thesis (University of Central Florida, Orlando, Fla., 1990).

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Figures (19)

Fig. 1
Fig. 1

Typical CLAR setup illustrating random phase and intensity variations at the receiver.

Fig. 2
Fig. 2

(a) Typical speckle pattern produced by illumination of a rough surface with laser light; (b) a schematic illustration of speckle size at a specific intensity threshold level.

Fig. 3
Fig. 3

Gamma probability-distribution function plotted for several values of M.

Fig. 4
Fig. 4

Contour plots of I = I thres. Points marked with an X satisfy Eq. (8), and points marked with an O satisfy Eq. (9).

Fig. 5
Fig. 5

Experimental configuration for laser speckle data collection.

Fig. 6
Fig. 6

Illustration of the relationship between the measurable speckle intensity data and the normalized PSD function of the field.

Fig. 7
Fig. 7

Calibration curve for curvature calculation; better than 2% accuracy is indicated for sampling densities exceeding 8 samples/speckle.

Fig. 8
Fig. 8

Horizontal and vertical profiles of the normalized autocovariance function of the intensity speckle pattern produced by illumination of a circular region on a sandblasted region of an aluminum target.

Fig. 9
Fig. 9

Truncated section of the recorded speckle pattern illustrated along with the measured PDF compared with an ideal negative exponential and the predicted PDF in the presence of error sources characterized by Eq. (29).

Fig. 10
Fig. 10

Comparison of theoretical predictions of average speckle size as a function of threshold level with experimental measurements for an isotropic speckle pattern.

Fig. 11
Fig. 11

Vertical and horizontal autocovariance function profiles for a speckle pattern produced by illumination of a rectangular target.

Fig. 12
Fig. 12

Truncated section of the recorded (nonisotropic) speckle pattern illustrated along with the normalized PDF compared with the PDF predicted in the presence of error sources characterized by Eq. (19).

Fig. 13
Fig. 13

Comparison of theoretical predictions of average speckle size as a function of threshold level with experimental measurements for a nonisotropic speckle pattern.

Fig. 14
Fig. 14

Normalized autocovariance function from which the second moment is determined.

Fig. 15
Fig. 15

Truncated section of the recorded (circularly polarized) speckle pattern illustrated along with the normalized PDF compared with a shaping constant of M = 2 and the prediction based on the known measurement noise PDF.

Fig. 16
Fig. 16

Comparison of the theoretical prediction of average speckle size as a function of the threshold level with experimental measurements for this speckle pattern.

Fig. 17
Fig. 17

Speckle pattern illustrated at four different threshold levels: (a) I th = Ī, (b) I th = 2Ī, (C) I th = 3Ī, (d) I th = 4Ī.

Fig. 18
Fig. 18

(a) Speckle-pattern normalized autocovariance function, (b) normalized PDF for computer-generated speckle.

Fig. 19
Fig. 19

Comparison of the theoretical prediction with actual data on computer-generated speckle.

Equations (31)

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d λ L / D .
A ¯ = Speckle area above threshold Total number of speckles .
A ¯ = Speckle area above threshold Total number of speckle = A tot N ¯ ,
A tot = S I thres p I ( I ) d I ,
p I I = ( M / I ¯ ) M I M - 1 exp [ - M ( I / I ¯ ) ] Γ ( M )             for I 0 ,
A tot = S Γ ( M , M I thres / I ¯ ) Γ ( M ) ,
Γ ( M , M I thres / I ¯ ) = M I thres / I ¯ exp ( - t ) t M - 1 d t
I = I thres ,             I x = 0 ,             I y > 0 ,             I x x < 0 ,
I = I thres ,             I x = 0 ,             I y > 0 ,             I x x > 0.
N ¯ = N a ¯ - N b ¯ .
I = I thres ,             I x = 0 ,             I y > 0 ,             I x x = 0
N a = S δ [ I ( x , y ) - I thres , I x ( x , y ) ] step [ I y ( x , y ) ] × step [ - I x x ( x , y ) ] J ( x , y ) d x d y ,
N b = S δ [ I ( x , y ) - I thres , I x ( x , y ) ] step [ I y ( x , y ) ] × step [ I x x ( x , y ) ] J ( x , y ) d x d y .
N ¯ = - S I y > 0 all I x x I y I x x p I I x I y I x x ( I thres , 0 , I y , I x x ) d I y d I x x ,
N ¯ = - S E { I y + I x x I = I thres and I x = 0 } p I I x ( I thres , 0 ) ,
N ¯ = S ( λ x x λ y y ) 1 / 2 I thres M - 1 ( M / I ¯ ) M π Γ ( M ) × ( 2 M I thres / I ¯ - 2 M + 1 ) exp [ - M I thres / I ¯ ] .
λ x x = x x ( I ¯ / 2 M ) ,             λ y y = y y ( I ¯ / 2 M ) ,
x x = - + - + ω x 2 s ( ω x , ω y ) d ω x d ω y , y y = - + - + ω y 2 s ( ω x , ω y ) d ω x d ω y .
A ¯ = π exp [ M ( I thres / I ¯ ) ] Γ [ M , M ( I thres / I ¯ ) ] ( x x y y ) 1 / 2 [ M ( I thres / I ¯ ] M - 1 [ 2 M ( I thres / I ¯ ) - 2 M + 1 ] .
M = ( I ¯ ) 2 / σ I 2 ,             σ I 2 = ( I 2 ) ¯ - ( I ¯ ) 2 .
R E ( x , y ) = [ R I ( x , y ) - ( I ¯ ) 2 ] 1 / 2 .
R E ( x , y ) = I ¯ [ c I ( x , y ) / σ I 2 ] 1 / 2 / ( M ) 1 / 2 .
σ E 2 = ( E 2 ) ¯ - ( E ¯ ) 2 = I ¯ ,             E ¯ = 0 ,
R E ( x , y ) / σ E 2 = [ c I ( x , y ) / σ I 2 ] 1 / 2 / ( M ) 1 / 2 .
R E ( x , y ) S ( ω x , ω y ) .
R E ( x , y ) / σ E 2 s ( ω x , ω y ) .
- 1 4 π 2 f x x ( 0 , 0 ) = - ω x 2 F ( ω x , ω y ) d ω x d ω y .
I measured ( x , y ) = I speckle ( x , y ) + I noise ( x , y ) ,
P I measured ( I ) = P I speckle ( I ) * P I noise ( I ) .
P I noise ( I ) = ( 1 / b ) exp [ - π ( x - x 0 ) / b ] 2 , b = 0.14 I ¯ ,             x 0 = b .
x x = 68.12 + 1.91 % = 69.42 , y y = 203.36 + 4.24 % = 211.98.

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