Abstract

Double-exposure records in speckle photography or particle image velocimetry are often evaluated by analysis of the system of Young's diffraction fringes. Fringe spacing, necessary to calculate the displacement, is determined from the positions of fringe maxima or minima. These, however, are influenced by the diffraction halo function and by fringe visibility. A generalized theory of the effects is presented, including position dependent visibility and fringe phase. Evaluations are given for disk-shaped particle images in particle image velocimetry, and for coherent and incoherent speckle photography. Fringe shifts are determined numerically for commonly encountered values of fringe density and visibility thus presenting a basis for rapid assessment of accuracy in metrological experiments.

© 1989 Optical Society of America

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References

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  1. G. H. Kaufmann, “On the Numerical Processing of Speckle Photograph Fringes,” Opt. Laser Technol. 12, 207–209 (1980).
    [Crossref]
  2. C. S. Vikram, “Error in Speckle Photography of Lateral Sinusoidal Vibrations: a Simple Analytical Solution,” Appl. Opt. 21, 1710–1712 (1982).
    [Crossref] [PubMed]
  3. C. S. Vikram, “Simple Approach to Process Speckle-Photography Data,” Opt. Lett. 7, 374–375 (1982).
    [Crossref] [PubMed]
  4. C. S. Vikram, K. Vedam, “Processing Speckle Photography Data: Circular Imaging Aperture,” Appl. Opt. 22, 653–654 (1983).
    [Crossref] [PubMed]
  5. G. H. Kaufmann, “Numerical Processing of Speckle Photography Data by Fourier Transform,” Appl. Opt. 20, 4277–4280 (1981).
    [Crossref] [PubMed]
  6. J. M. Huntley, “Speckle Photography Fringe Analysis by the Walsh Transform,” Appl. Opt. 25, 382–386 (1986).
    [Crossref] [PubMed]
  7. C. S. Vikram, K. Vedam, “Speckle Photography of Lateral Sinusoidal Vibrations: Error due to Varying Halo Intensity,” Appl. Opt. 20, 3388–3391 (1981).
    [Crossref] [PubMed]
  8. S. A. Isacson, G. H. Kaufmann, “Two-Dimensional Digital Processing of Speckle Photography Fringes. 1: Diffraction Halo Influence for the Noise-Free Case,” Appl. Opt. 24, 189–193 (1985).
    [Crossref] [PubMed]
  9. S. A. Isacson, G. H. Kaufmann, “Two-Dimensional Digital Processing of Speckle Photography Fringes. 2: Diffraction Halo Influence for the Noisy Case,” Appl. Opt. 24, 1444–1447 (1985).
    [Crossref] [PubMed]
  10. H. D. Navone, G. H. Kaufmann, “Two-Dimensional Digital Processing of Speckle Photography Fringes. 3: Accuracy in Angular Determination,” Appl. Opt. 26, 154–156 (1987).
    [Crossref] [PubMed]
  11. C. S. Vikram, K. Vedam, “Selective Counting Path of Young's Fringes in Speckle Photography for Eliminating Diffraction Halo Effects,” Appl. Opt. 22, 2242–2243 (1983).
    [Crossref] [PubMed]
  12. C. Joenathan, R. S. Sirohi, “Elimination of Error in Speckle Photography,” Appl. Opt. 25, 1791–1794 (1986).
    [Crossref] [PubMed]
  13. R. Meynart, “Diffraction Halo in Speckle Photography,” Appl. Opt. 23, 2235–2236 (1984).
    [Crossref] [PubMed]
  14. J. Georgieva, “Diffraction Halo Effect in Speckle Photography,” Appl. Opt. 25, 3970–3971 (1986).
    [Crossref] [PubMed]
  15. K. Hinsch, W. Schipper, D. Mach, “Fringe Visibility in Speckle Velocimetry and the Analysis of Random Flow Components,” Appl. Opt. 23, 4460–4462 (1984).
    [Crossref] [PubMed]
  16. W. Arnold, K. Hinsch, D. Mach, “Turbulence Level Measurement by Speckle Velocimetry,” Appl. Opt. 25, 330–331 (1986).
    [Crossref] [PubMed]
  17. K. Hinsch, W. Arnold, W. Platen, “Flow Field Analysis by Large-Area Interrogation in Particle Image Velocimetry,” Opt. Laser Engin. 9, 229–243 (1988).
    [Crossref]
  18. W. Lauterborn, A. Vogel, “Modern Optical Techniques in Fluid Mechanics,” Ann. Rev. Fluid Mech. 16, 223–244 (1984).
    [Crossref]
  19. C. J. D. Pickering, N. A. Halliwell, “Speckle Photography in Fluid Flows: Signal Recovery with Two-Step Processing,” Appl. Opt. 23, 1128–1129 (1984).
    [Crossref] [PubMed]
  20. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), p. 361.
  21. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975), p. 41.
  22. F.-P. Chiang, D. W. Li, “Diffraction Halo Functions of Coherent and Incoherent Random Speckle Patterns,” Appl. Opt. 24, 2166–2171 (1985).
    [Crossref] [PubMed]
  23. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill Kogakusha, Tokyo, 1965), p. 153.

1988 (1)

K. Hinsch, W. Arnold, W. Platen, “Flow Field Analysis by Large-Area Interrogation in Particle Image Velocimetry,” Opt. Laser Engin. 9, 229–243 (1988).
[Crossref]

1987 (1)

1986 (4)

1985 (3)

1984 (4)

1983 (2)

1982 (2)

1981 (2)

1980 (1)

G. H. Kaufmann, “On the Numerical Processing of Speckle Photograph Fringes,” Opt. Laser Technol. 12, 207–209 (1980).
[Crossref]

Arnold, W.

K. Hinsch, W. Arnold, W. Platen, “Flow Field Analysis by Large-Area Interrogation in Particle Image Velocimetry,” Opt. Laser Engin. 9, 229–243 (1988).
[Crossref]

W. Arnold, K. Hinsch, D. Mach, “Turbulence Level Measurement by Speckle Velocimetry,” Appl. Opt. 25, 330–331 (1986).
[Crossref] [PubMed]

Chiang, F.-P.

Georgieva, J.

Goodman, J. W.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975), p. 41.

Halliwell, N. A.

Hinsch, K.

Huntley, J. M.

Isacson, S. A.

Joenathan, C.

Kaufmann, G. H.

Lauterborn, W.

W. Lauterborn, A. Vogel, “Modern Optical Techniques in Fluid Mechanics,” Ann. Rev. Fluid Mech. 16, 223–244 (1984).
[Crossref]

Li, D. W.

Mach, D.

Meynart, R.

Navone, H. D.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill Kogakusha, Tokyo, 1965), p. 153.

Pickering, C. J. D.

Platen, W.

K. Hinsch, W. Arnold, W. Platen, “Flow Field Analysis by Large-Area Interrogation in Particle Image Velocimetry,” Opt. Laser Engin. 9, 229–243 (1988).
[Crossref]

Schipper, W.

Sirohi, R. S.

Vedam, K.

Vikram, C. S.

Vogel, A.

W. Lauterborn, A. Vogel, “Modern Optical Techniques in Fluid Mechanics,” Ann. Rev. Fluid Mech. 16, 223–244 (1984).
[Crossref]

Ann. Rev. Fluid Mech. (1)

W. Lauterborn, A. Vogel, “Modern Optical Techniques in Fluid Mechanics,” Ann. Rev. Fluid Mech. 16, 223–244 (1984).
[Crossref]

Appl. Opt. (16)

C. J. D. Pickering, N. A. Halliwell, “Speckle Photography in Fluid Flows: Signal Recovery with Two-Step Processing,” Appl. Opt. 23, 1128–1129 (1984).
[Crossref] [PubMed]

C. S. Vikram, “Error in Speckle Photography of Lateral Sinusoidal Vibrations: a Simple Analytical Solution,” Appl. Opt. 21, 1710–1712 (1982).
[Crossref] [PubMed]

F.-P. Chiang, D. W. Li, “Diffraction Halo Functions of Coherent and Incoherent Random Speckle Patterns,” Appl. Opt. 24, 2166–2171 (1985).
[Crossref] [PubMed]

C. S. Vikram, K. Vedam, “Processing Speckle Photography Data: Circular Imaging Aperture,” Appl. Opt. 22, 653–654 (1983).
[Crossref] [PubMed]

G. H. Kaufmann, “Numerical Processing of Speckle Photography Data by Fourier Transform,” Appl. Opt. 20, 4277–4280 (1981).
[Crossref] [PubMed]

J. M. Huntley, “Speckle Photography Fringe Analysis by the Walsh Transform,” Appl. Opt. 25, 382–386 (1986).
[Crossref] [PubMed]

C. S. Vikram, K. Vedam, “Speckle Photography of Lateral Sinusoidal Vibrations: Error due to Varying Halo Intensity,” Appl. Opt. 20, 3388–3391 (1981).
[Crossref] [PubMed]

S. A. Isacson, G. H. Kaufmann, “Two-Dimensional Digital Processing of Speckle Photography Fringes. 1: Diffraction Halo Influence for the Noise-Free Case,” Appl. Opt. 24, 189–193 (1985).
[Crossref] [PubMed]

S. A. Isacson, G. H. Kaufmann, “Two-Dimensional Digital Processing of Speckle Photography Fringes. 2: Diffraction Halo Influence for the Noisy Case,” Appl. Opt. 24, 1444–1447 (1985).
[Crossref] [PubMed]

H. D. Navone, G. H. Kaufmann, “Two-Dimensional Digital Processing of Speckle Photography Fringes. 3: Accuracy in Angular Determination,” Appl. Opt. 26, 154–156 (1987).
[Crossref] [PubMed]

C. S. Vikram, K. Vedam, “Selective Counting Path of Young's Fringes in Speckle Photography for Eliminating Diffraction Halo Effects,” Appl. Opt. 22, 2242–2243 (1983).
[Crossref] [PubMed]

C. Joenathan, R. S. Sirohi, “Elimination of Error in Speckle Photography,” Appl. Opt. 25, 1791–1794 (1986).
[Crossref] [PubMed]

R. Meynart, “Diffraction Halo in Speckle Photography,” Appl. Opt. 23, 2235–2236 (1984).
[Crossref] [PubMed]

J. Georgieva, “Diffraction Halo Effect in Speckle Photography,” Appl. Opt. 25, 3970–3971 (1986).
[Crossref] [PubMed]

K. Hinsch, W. Schipper, D. Mach, “Fringe Visibility in Speckle Velocimetry and the Analysis of Random Flow Components,” Appl. Opt. 23, 4460–4462 (1984).
[Crossref] [PubMed]

W. Arnold, K. Hinsch, D. Mach, “Turbulence Level Measurement by Speckle Velocimetry,” Appl. Opt. 25, 330–331 (1986).
[Crossref] [PubMed]

Opt. Laser Engin. (1)

K. Hinsch, W. Arnold, W. Platen, “Flow Field Analysis by Large-Area Interrogation in Particle Image Velocimetry,” Opt. Laser Engin. 9, 229–243 (1988).
[Crossref]

Opt. Laser Technol. (1)

G. H. Kaufmann, “On the Numerical Processing of Speckle Photograph Fringes,” Opt. Laser Technol. 12, 207–209 (1980).
[Crossref]

Opt. Lett. (1)

Other (3)

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill Kogakusha, Tokyo, 1965), p. 153.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), p. 361.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975), p. 41.

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Figures (5)

Fig. 1
Fig. 1

Halo influence on fringe position in particle image velocimetry (PIV) assuming circular disks as particle images. Plot of fringe shift vs position of correct fringe maximum relative to halo size. Parameters are fringe density, given by the number 2N of fringes within the halo, and fringe order n, visibility Vo = 1.

Fig. 2
Fig. 2

Comparison of fringe shifts in particle image velocimetry when evaluating fringe maxima (a) or fringe minima (b) of a pattern containing 2N = 20 fringes. Parameter is fringe visibility Vo; note that the shift in fringe minima vanishes for Vo = 1.

Fig. 3
Fig. 3

Halo influence on fringe position in laser speckle photography. Parameters are the same as in Fig. 1.

Fig. 4
Fig. 4

Comparison of fringe shifts in laser speckle photography for fringe maxima (a) or minima (b); all parameters are the same as in Fig. 2.

Fig. 5
Fig. 5

Relative error in fringe position vs total number of fringes 2N for coherent (broken line) and incoherent (solid lines) double exposure speckle metrology, n is the order of fringe used for evaluation, numerical data from8 are included.

Equations (44)

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I ( k ) = H ( k ) { 1 + V ( k ) cos [ k d o + φ ( k ) ] } ,
H ( k n ) H ( k n ) = V o d o sin k n d o 1 + V o cos k n d o ,
sin k d o k d o k n o d o n = 0 , ± 1 , k d o 2 π n , cos k d o 1.
H ( k ) = H ( k n o ) + H ( k n o ) ( k k n o ) , H ( k ) = H ( k n o ) .
h ( k ) = H ( k ) H ( k )
h ( k ) = H ( k n o ) H ( k n o ) + ( k k n o )
h ( k n o ) + k n k n o = V o + 1 V o d o 2 ( k n k n o ) .
Δ k n 2 + h ( k n o ) Δ k n 1 + V o V o 1 d o 2 = 0 ,
Δ k n = h ( k n o ) 2 ± { h 2 ( k n o ) 4 + V o + 1 V o 1 d o 2 } 1 / 2 .
n = 1 2 h ( k n o ) k o ± { 1 4 h 2 ( k n o ) k o 2 + V o + 1 V o 1 4 π 2 } 1 / 2 .
sin k d o k d o + k n 1 d o n = 0 , ± 1 , k d o + ( 2 n + 1 ) π cos k d o 1
h ( k n 1 ) + k n k n 1 = V o 1 V o d o 2 ( k n k n 1 )
Δ k n 2 + h ( k n 1 ) Δ k n V o 1 V o 1 d o 2 = 0 , and
n = 1 2 h ( k n 1 ) k o ± { 1 4 h 2 ( k n 1 ) k o 2 + V o 1 V o 1 4 π 2 } 1 / 2 .
1 4 π 2 V o + 1 V o 1 4 h 2 ( k n o ) k o 2
k o 2 π 2 h 2 ( k n o ) V o V o + 1 ,
n ( V o ) = V o + 1 2 V o n ( 1 ) .
n ( V o ) = V o 1 V o n ( 0.5 ) ,
V o 1 V o 1 4 π 2 1 4 h 2 ( k n o ) k o 2 .
tan [ k n d o + φ ( k n ) ] = V ( k n ) V ( k n ) [ d o + φ ( k n ) ] .
[ k n d o + φ ( k n ) n 2 π ] [ d o + φ ( k n ) ] = V ( k n ) V ( k n ) ,
[ k n d o + φ ( k n ) + ( 2 n + 1 ) π ] [ d o + φ ( k n ) ] = V ( k n ) V ( k n ) ,
t ( r ) = circ ( r / R ) = { 1 r / R 1 0 elsewhere . }
H ( k ) = 4 π 2 R 2 J 1 2 ( R ρ k ) R 2 ρ k 2 ,
H ( k x ) = 4 π 2 R 2 [ J 1 ( ζ ) / ζ ] 2 .
J l 1 ( z ) + J l + 1 ( z ) = 2 l z J l ( z ) J l ( z ) = J l 1 ( z ) 1 z J l ( z ) ,
H ( k x ) = 8 π R 3 J 1 ( ζ ) ζ J 2 ( ζ ) ζ ,
h ( k x ) = 1 2 R J 1 ( ζ ) J 2 ( ζ ) ,
n = ± 0.0652 N J 1 ( ζ n o ) J 2 ( ζ n o ) { [ 0.0652 N J 1 ( ζ n o ) J 2 ( ζ n o ) ] 2 + 0.0253 V o + 1 V o } 1 / 2 ,
n = ± 0.0652 N J 1 ( ζ n 1 ) J 2 ( ζ n 1 ) { [ 0.0652 N J 1 ( ζ n 1 ) J 2 ( ζ n 1 ) ] 2 + 0.0253 V o 1 V o } 1 / 2 .
H ( k ) = cos 1 k k R k k R ( 1 k 2 k R 2 ) 1 / 2 ,
H ( k ) = 2 k R ( 1 k 2 ) 1 / 2 ,
h ( k ) = k R 2 [ cos 1 ( k ) ( 1 k 2 ) 1 / 2 k ] .
1 2 h ( k n o ) k o = N 4 { cos 1 n / N [ 1 ( n / N ) 2 ] 1 / 2 n N } ,
H i ( k ) = H o 2 ( k ) .
H i ( k ) = 2 H o ( k ) H o ( k ) ,
h i ( k ) = H i ( k ) H i ( k ) = H o 2 ( k ) 2 H o ( k ) H o ( k ) = 1 2 H o ( k ) H o ( k ) h i ( k ) = 1 2 h o ( k ) .
Δ k = 6 N ( N 2 1 ) [ ( N 1 ) ( k N k 1 ) + ( N 3 ) ( k n 1 k 2 ) + ] .
V ( k ) = a exp ( k 2 / 2 σ 2 ) .
V ( k ) V ( k ) = k / σ 2 .
k n ( d o 2 + 1 σ 2 ) = n 2 π d o ,
k n = n 2 π d o d o 2 + 1 / σ 2 .
n = n { 1 1 + 1 / σ 2 d o 2 1 } .
n = ( n + 1 2 ) { 1 1 1 / σ 2 d o 2 1 } .

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