Abstract

The accuracy of Young’s fringe method for reducing velocity field data is compromised by a spatially incoherent background field which originates in the random locations of the seeding particles. The probability density function for the spatial frequency cutoff of this background is derived as a function of the particle count, the distribution governing the power in each frequency interval is derived, and conditions are found under which the Van Cittert-Zernike theorem applies. The background field resembles the far field of a partially coherent source in the high particle count limit, but departs significantly at low and moderate counts.

© 1990 Optical Society of America

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References

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  1. R. J. Adrian, C. S. Yao, “Development of Pulsed Laser Velocimetry for Measurement of Turbulent Flow,” in Proceedings of the Eighth Biennial Symposium on Turbulence (University of Missouri, Rolla, 1983).
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).
  3. R. Dandliker, “Heterodyne Holographic Interferometry,” in Progress in Optics, E. Wolf, Ed., Vol. 17 (North-Holland, New York, 1980).
    [CrossRef]
  4. K. A. Stetson, “Effect of Scintillation Noise in Heterodyne Speckle Photogrammetry,” Appl. Opt. 23, 920–923 (1984).
    [CrossRef] [PubMed]
  5. R. D. Keane, R. J. Adrian, “Optimization of Particle Image Velocimeters,” Laser Institute of America 68, ICALEO (1989), p. 141.
  6. J. M. Coupland, C. J. D. Pickering, “Particle Image Velocimetry: Estimation of Measurement Confidence at Low Seeding Densities,” Opt. Lasers Eng. 9, 201 (1988).
    [CrossRef]
  7. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Topics in Applied Physics, J. C. Dainty, Ed., Vol. 9, (Springer, New York, 1975), Chap. 2.
    [CrossRef]
  8. E. Wolf, “Coherence and Radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  9. W. H. Carter, E. Wolf, “Coherence and Radiometry with Quasihomogeneous Planar Sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  10. J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688 (1965).
    [CrossRef]
  11. 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]
  12. R. Meynart, “Non-Gaussian Statistics of Speckle Noise of Young’s Fringes in Speckle Velocimetry,” Appl. Opt. 24, 1448–1453 (1985).
    [CrossRef] [PubMed]
  13. R. J. Adrian, “Statistical Properties of Particle Image Velocimetry Measurements in Turbulent Flow,” in Proceedings Fourth International Symposium on Applications of Laser Anemometry in Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1988).
  14. J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1968).
  15. C. J. D. Pickering, N. A. Halliwell, “Laser Speckle Photography and Particle Image Velocimetry: Photographic Film Noise,” Appl. Optics 23, 2961–2969 (1984).
    [CrossRef]
  16. W. Feller, Introduction to Probability Theory and Its Applications (Wiley, New York, 1950).
  17. S. H. Chen, P. Tartaglia, P. N. Pusey, “Light Scattering from Independent Particles—Nongaussian Correction to the Clipped Intensity Correlation Function,” J. Phys. A. 6, 490 (1973).
    [CrossRef]
  18. D. W. Schaefer, P. N. Pusey, “Statistics of Non-Gaussian Scattered Light,” Phys. Rev. Lett. 29, 8 (1972).
    [CrossRef]
  19. G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U.P., Cambridge, 1915).

1989

R. D. Keane, R. J. Adrian, “Optimization of Particle Image Velocimeters,” Laser Institute of America 68, ICALEO (1989), p. 141.

1988

J. M. Coupland, C. J. D. Pickering, “Particle Image Velocimetry: Estimation of Measurement Confidence at Low Seeding Densities,” Opt. Lasers Eng. 9, 201 (1988).
[CrossRef]

1985

1984

1978

1977

1973

S. H. Chen, P. Tartaglia, P. N. Pusey, “Light Scattering from Independent Particles—Nongaussian Correction to the Clipped Intensity Correlation Function,” J. Phys. A. 6, 490 (1973).
[CrossRef]

1972

D. W. Schaefer, P. N. Pusey, “Statistics of Non-Gaussian Scattered Light,” Phys. Rev. Lett. 29, 8 (1972).
[CrossRef]

1965

J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688 (1965).
[CrossRef]

Adrian, R. J.

R. D. Keane, R. J. Adrian, “Optimization of Particle Image Velocimeters,” Laser Institute of America 68, ICALEO (1989), p. 141.

R. J. Adrian, C. S. Yao, “Development of Pulsed Laser Velocimetry for Measurement of Turbulent Flow,” in Proceedings of the Eighth Biennial Symposium on Turbulence (University of Missouri, Rolla, 1983).

R. J. Adrian, “Statistical Properties of Particle Image Velocimetry Measurements in Turbulent Flow,” in Proceedings Fourth International Symposium on Applications of Laser Anemometry in Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1988).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Carter, W. H.

Chen, S. H.

S. H. Chen, P. Tartaglia, P. N. Pusey, “Light Scattering from Independent Particles—Nongaussian Correction to the Clipped Intensity Correlation Function,” J. Phys. A. 6, 490 (1973).
[CrossRef]

Coupland, J. M.

J. M. Coupland, C. J. D. Pickering, “Particle Image Velocimetry: Estimation of Measurement Confidence at Low Seeding Densities,” Opt. Lasers Eng. 9, 201 (1988).
[CrossRef]

Dandliker, R.

R. Dandliker, “Heterodyne Holographic Interferometry,” in Progress in Optics, E. Wolf, Ed., Vol. 17 (North-Holland, New York, 1980).
[CrossRef]

Feller, W.

W. Feller, Introduction to Probability Theory and Its Applications (Wiley, New York, 1950).

Goodman, J. W.

J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688 (1965).
[CrossRef]

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Topics in Applied Physics, J. C. Dainty, Ed., Vol. 9, (Springer, New York, 1975), Chap. 2.
[CrossRef]

J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1968).

Halliwell, N. A.

C. J. D. Pickering, N. A. Halliwell, “Laser Speckle Photography and Particle Image Velocimetry: Photographic Film Noise,” Appl. Optics 23, 2961–2969 (1984).
[CrossRef]

Hinsch, K.

Keane, R. D.

R. D. Keane, R. J. Adrian, “Optimization of Particle Image Velocimeters,” Laser Institute of America 68, ICALEO (1989), p. 141.

Mach, D.

Meynart, R.

Pickering, C. J. D.

J. M. Coupland, C. J. D. Pickering, “Particle Image Velocimetry: Estimation of Measurement Confidence at Low Seeding Densities,” Opt. Lasers Eng. 9, 201 (1988).
[CrossRef]

C. J. D. Pickering, N. A. Halliwell, “Laser Speckle Photography and Particle Image Velocimetry: Photographic Film Noise,” Appl. Optics 23, 2961–2969 (1984).
[CrossRef]

Pusey, P. N.

S. H. Chen, P. Tartaglia, P. N. Pusey, “Light Scattering from Independent Particles—Nongaussian Correction to the Clipped Intensity Correlation Function,” J. Phys. A. 6, 490 (1973).
[CrossRef]

D. W. Schaefer, P. N. Pusey, “Statistics of Non-Gaussian Scattered Light,” Phys. Rev. Lett. 29, 8 (1972).
[CrossRef]

Schaefer, D. W.

D. W. Schaefer, P. N. Pusey, “Statistics of Non-Gaussian Scattered Light,” Phys. Rev. Lett. 29, 8 (1972).
[CrossRef]

Schipper, W.

Stetson, K. A.

Tartaglia, P.

S. H. Chen, P. Tartaglia, P. N. Pusey, “Light Scattering from Independent Particles—Nongaussian Correction to the Clipped Intensity Correlation Function,” J. Phys. A. 6, 490 (1973).
[CrossRef]

Watson, G. N.

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U.P., Cambridge, 1915).

Wolf, E.

Yao, C. S.

R. J. Adrian, C. S. Yao, “Development of Pulsed Laser Velocimetry for Measurement of Turbulent Flow,” in Proceedings of the Eighth Biennial Symposium on Turbulence (University of Missouri, Rolla, 1983).

Appl. Opt.

Appl. Optics

C. J. D. Pickering, N. A. Halliwell, “Laser Speckle Photography and Particle Image Velocimetry: Photographic Film Noise,” Appl. Optics 23, 2961–2969 (1984).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. A.

S. H. Chen, P. Tartaglia, P. N. Pusey, “Light Scattering from Independent Particles—Nongaussian Correction to the Clipped Intensity Correlation Function,” J. Phys. A. 6, 490 (1973).
[CrossRef]

Laser Institute of America

R. D. Keane, R. J. Adrian, “Optimization of Particle Image Velocimeters,” Laser Institute of America 68, ICALEO (1989), p. 141.

Opt. Lasers Eng.

J. M. Coupland, C. J. D. Pickering, “Particle Image Velocimetry: Estimation of Measurement Confidence at Low Seeding Densities,” Opt. Lasers Eng. 9, 201 (1988).
[CrossRef]

Phys. Rev. Lett.

D. W. Schaefer, P. N. Pusey, “Statistics of Non-Gaussian Scattered Light,” Phys. Rev. Lett. 29, 8 (1972).
[CrossRef]

Proc. IEEE

J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688 (1965).
[CrossRef]

Other

R. J. Adrian, C. S. Yao, “Development of Pulsed Laser Velocimetry for Measurement of Turbulent Flow,” in Proceedings of the Eighth Biennial Symposium on Turbulence (University of Missouri, Rolla, 1983).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

R. Dandliker, “Heterodyne Holographic Interferometry,” in Progress in Optics, E. Wolf, Ed., Vol. 17 (North-Holland, New York, 1980).
[CrossRef]

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Topics in Applied Physics, J. C. Dainty, Ed., Vol. 9, (Springer, New York, 1975), Chap. 2.
[CrossRef]

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U.P., Cambridge, 1915).

W. Feller, Introduction to Probability Theory and Its Applications (Wiley, New York, 1950).

R. J. Adrian, “Statistical Properties of Particle Image Velocimetry Measurements in Turbulent Flow,” in Proceedings Fourth International Symposium on Applications of Laser Anemometry in Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1988).

J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (4)

Fig. 1
Fig. 1

Young’s fringe geometry. Scattering from randomly distributed particles in the image plane gives rise to a spatially incoherent background field in the transform plane.

Fig. 2
Fig. 2

Probability that the grain size exceeds the laser speckle size for various mean particle populations.

Fig. 3
Fig. 3

Tenth percentile SNR (coherent power/estimated noise power) as a function of mean particle population.

Fig. 4
Fig. 4

Tenth percentile of long wavelength modulation scale as a function of particle population.

Tables (1)

Tables Icon

Table I Fifth Percentile SNR for Eighteen Particle Image Pairs as a Function of Resolution Parameter

Equations (51)

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I ( ξ , η ) = I 1 i m δ ( ξ - ξ i , η - η i )
p ( μ = m ) = exp ( - s ) s m m ! ,
I ( x , y ) = C ( m + i j exp { 2 π i [ x ( ξ i - ξ j ) + y ( η i - η j ) ] / λ z } ) ,
r ( Δ x , Δ y ) = I ( x , y ) I ( x + Δ x , y + Δ y ) .
k x = max λ z / 2 π ξ i - ξ j , k y = max λ z / 2 π η i - η j .
( N - k - m + 1 ) ( k + m - 2 k ) / ( N m ) .
M < l < N ( N - l + 1 ) ( l - 2 m - 2 ) = ( N m ) - N ( M - 2 ) ( M - m ) / ( m - 1 ) ! + ( M - 1 ) ( M - m ) / m ( m - 2 ) !
m l N ( N - l + 1 ) ( L - 2 m - 2 ) = ( N m ) ,
1 - m ( M - 2 ) ( M - m ) / ( N - 1 ) ( N - m + 1 ) + ( m - 1 ) ( M - 1 ) ( M - m ) / N ( N - m + 1 ) .
p ( r < α ) = α m .
α = 1 - 1 / ( m + 1 ) ,
p ( r < α ) = m 2 p ( r < α μ = m ) p ( μ = m ) = m 2 α m exp ( - s ) s m / m ! = exp ( α - 1 ) s - ( 1 + α s ) exp ( - s ) ,
p ( r x ) < α and r y < β ) = p ( r y < α and r y < β μ = m ) p ( μ = m ) = p ( r x < α μ = m ) p ( r y < β μ = m ) p ( μ = m ) = exp [ α β - 1 ) s ] - ( 1 + α β s ) exp ( - s ) .
s = - ln p 1 - α .
s = 10 ln 10 ~ 23.
E = x k exp ( 2 π i x k Δ ξ / λ z )
I = j , k x j x k exp ( 2 π i x ( k - j ) Δ ξ / λ z ) = l j - k = l x j x k exp ( 2 π i x l Δ ξ / λ z )
S l = x 1 x l + 1 + x 2 x l + 2 + + x N - l x N .
S l - Δ l + S l - Δ l + 1 + S l + Δ l .
N Δ ξ = L , l Δ ξ = κ L , 2 Δ l Δ ξ = Δ κ L .
p ( x i = 0 ) = 1 - σ Δ ξ + O ( Δ ξ 2 ) p ( x i = 1 ) = σ Δ ξ + O ( Δ ξ 2 ) .
p ( x i = 0 ) = ( 1 - ½ σ Δ ξ ) 2 + O ( Δ ξ 2 ) , p ( x i = 1 ) = σ Δ ξ ( 1 - ½ σ Δ ξ ) + O ( Δ ξ 3 ) , p ( x i 2 ) = ¼ σ 2 Δ ξ 2 + O ( Δ ξ 3 ) .
p ( x i x j = 0 ) = 2 p ( x i = 0 & x j 0 ) + p ( x i = x j = 0 ) = 1 - σ 2 Δ ξ 2 , p ( x i x j = 1 ) = σ 2 Δ ξ 2 .
p ( x i x j = 0 & x j x k = 0 ) = p ( x j = 0 ) + p ( x j 0 & x j = 0 & x k = 0 ) = 1 - 2 σ 2 Δ ξ 2 , p ( x i x j = 0 ) p ( x j x k = 0 ) = ( 1 - σ 2 Δ ξ 2 ) 2 = 1 - 2 σ 2 Δ ξ 2 , p ( x i x j = 0 & x j x k = 1 ) = p ( x j 0 & x i = 0 & x k = 1 ) = σ 2 Δ ξ 2 , p ( x i x j = 0 ) p ( x j x k = 1 ) = ( 1 - σ 2 Δ ξ 2 ) σ 2 Δ ξ 2 = σ 2 Δ ξ 2 , p ( x i x j = 1 & x j x k = 1 ) = p ( x i = x j = x k = 1 ) = O ( Δ ξ 3 ) , p ( x i x j = 1 ) p ( x j x k = 1 ) = O ( Δ ξ 4 ) ,
p ( x i x j = n & x j x k = m ) = p ( x i x j = n ) p ( x j x k = m ) + O ( Δ ξ 3 )
M ¯ = ( N - l - Δ l ) + + ( N - l + Δ l ) = L 2 Δ ξ 2 [ ( 1 - κ ) Δ κ - 1 2 Δ κ 2 ] .
p ( S l - Δ l + + S l + Δ l = n ) = ( M ¯ n ) ( σ 2 Δ ξ 2 ) n ( 1 - σ 2 Δ ξ 2 ) M ¯ - n + O ( M ¯ Δ ξ 3 ) .
( M ¯ n ) ( σ 2 Δ ξ 2 ) n ( 1 - σ 2 Δ ξ 2 ) M ¯ - n 1 n ! [ σ 2 L 2 ( 1 - κ ) Δ κ ] n × exp [ σ 2 L 2 ( 1 - κ ) Δ κ ] O ( M ¯ Δ ξ 3 ) 0 ,
( 1 / Δ κ ( 1 - a ) m ) / ( 1 / Δ κ m ) ~ ( Δ κ ) a m ,
SNR ~ m / m 2 Δ κ = 1 / m Δ κ .
E ( ξ , η ) E ( ξ , η ) = I δ ( ξ - ξ , η - η ) .
ξ j = ξ ¯ J + Δ ξ j .
I ( x ) = I j , k exp [ 2 π i x ( ξ j - ξ k ) / λ z = I J , K j J , K K exp [ 2 π i x ( ξ ¯ J - ξ ¯ K ) / λ z ] × exp [ 2 π i x ( Δ ξ j - Δ ξ k ) / λ z ] ,
I J K ( x ) = I exp [ 2 π i x ( ξ ¯ J - ξ K ) / λ z ] Z J K ( x ) = j J , k K exp [ 2 π i x ( Δ ξ j - Δ ξ k ) / λ z ] ,
I ( x ) = J , K I J K ( x ) Z J K ( x ) .
I J K ( x ) = I exp [ 2 π i x L ( J - K ) / N λ z ] .
E J = j J exp ( 2 π i x Δ ξ j / λ / z ) .
( E J + E K ) ( E J * + E K * ) = E J E J * + E K E K * + 2 Z J K .
S 2 = ( N / M 2 ) 1 i N ( X i - m / N ) 2
χ 2 = m S 2
m = m 1 + m 2 + + m N , m 1 m 2 m N
m k + 1 = = m N = 0 , m k 0.
I ( m 1 , , m k ) ( N k ) m ! / m 1 ! m k ! ,
1 k C - 1 exp ( - μ ) μ k / k ! ~ Z ( μ - c c ) ,
μ = m 2 Δ κ , c - 1 = m / 2 ,
m 2 Δ κ - ( m + 1 ) / 2 ~ 1.28 ( m + 1 ) / 2 ,
I ( x , y ) = | 1 k m exp [ i ( x ξ k + y n k ) ] | 2 ,
p ( x < r m steps ) = 0 [ J 0 ( ρ ) ] m J 1 ( ρ r ) r d ρ .
p ( x < r ) = r 0 J 1 ( u ) exp { s [ J 0 ( u / r ) - 1 ] } d u
p ( I ) = 0 J 1 ( u ) exp { s [ J 0 ( u / λ I ) - 1 ] } d u .
P ( I ) = [ 1 - exp ( - I / s ) ] + I exp ( - I / s ) ( 1 - I / 2 s ) / 4 s 2 - I exp ( - I / s ) ( 1 - I / s + I 2 / 6 s 2 ) / 6 s 3 + O ( 1 / s 3 ) .

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