Abstract

Results of autocorrelation functions and power spectral densities of laser Doppler velocimeters in the reference-beam-heterodyning and the cross-beam-mixing mode are derived to handle turbulence measurements, Doppler ambiguity, spatial coherence, and effects of particle concentration and detection aperture. A useful definition of signal-to-noise ratio is given for both correlation and spectral analyses. The signal-to-noise calculation provides a direct comparison of the figures of merit between various arrangements of a laser Doppler velocimeter with methods for signal processing to measure optically the flow properties of a fluid under given experimental conditions.

© 1975 Optical Society of America

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References

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  1. Y. Yeh and H. Z. Commins, Appl. Phys. Lett. 4, 176 (1964).
    [Crossref]
  2. F. Durst, A. Melling, and J. H. Whitelaw, J. Fluid Mech. 56, 143 (1972).
    [Crossref]
  3. M. J. Rudd, Opt. Laser Tech. 3, 200 (1971).
    [Crossref]
  4. J. C. Owens, Proc. IEEE 57, 530 (1969).
    [Crossref]
  5. L. Lading, Appl. Opt. 10, 1943 (1971).
    [Crossref] [PubMed]
  6. W. M. Farmer and D. B. Brayton, Appl. Opt. 10, 2319 (1971).
    [Crossref] [PubMed]
  7. C. Y. She, Appl. Opt. 12, 2415 (1973).
    [Crossref] [PubMed]
  8. R. V. Edwards, J. C. Angus, and J. W. Dunning, Opto-Electronics 5, 119 (1973).
    [Crossref]
  9. R. J. Adrian and R. J. Goldstein, J. Phys. E 4, 505 (1971).
    [Crossref]
  10. L. E. Drain, J. Phys. D 5, 481 (1972).
    [Crossref]
  11. V. Degiorgio and J. B. Lastovka, Phys. Rev. A 4, 2033 (1971).
    [Crossref]
  12. J. C. Owens, Appl. Opt. 11, 2977 (1972).
    [Crossref] [PubMed]
  13. W. M. Farmer and D. B. Brayton, Appl. Opt. 11, 2978 (1972).
    [Crossref] [PubMed]
  14. A. J. Hughes and E. R. Pike, Appl. Opt. 12, 597 (1973).
    [Crossref] [PubMed]
  15. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 491.
  16. W. K. George and J. L. Lumley, J. Fluid Mech. 60, 321 (1973).
    [Crossref]
  17. A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962), p. 21.
  18. R. V. Edwards, J. C. Angus, M. J. French, and J. W. Dunning, J. Appl. Phys. 42, 837 (1971).
    [Crossref]
  19. C. Y. She and J. A. Lucero, Opt. Commun. 9, 300 (1973).
    [Crossref]
  20. G. B. Benedek, in Polarization Matière et Rayonnement Livre de Jubìlé en l’honneur du Professeur A. Kastler (Presses Universitaire de France, Paris, 1968), p. 149.
  21. C. P. Wang, J. Phys. E 5, 763 (1972).
    [Crossref]
  22. H. Z. Cummins and H. L. Swinney, Prog. Opt. 8, 133 (1970).
    [Crossref]

1973 (5)

C. Y. She, Appl. Opt. 12, 2415 (1973).
[Crossref] [PubMed]

R. V. Edwards, J. C. Angus, and J. W. Dunning, Opto-Electronics 5, 119 (1973).
[Crossref]

A. J. Hughes and E. R. Pike, Appl. Opt. 12, 597 (1973).
[Crossref] [PubMed]

W. K. George and J. L. Lumley, J. Fluid Mech. 60, 321 (1973).
[Crossref]

C. Y. She and J. A. Lucero, Opt. Commun. 9, 300 (1973).
[Crossref]

1972 (5)

C. P. Wang, J. Phys. E 5, 763 (1972).
[Crossref]

J. C. Owens, Appl. Opt. 11, 2977 (1972).
[Crossref] [PubMed]

W. M. Farmer and D. B. Brayton, Appl. Opt. 11, 2978 (1972).
[Crossref] [PubMed]

L. E. Drain, J. Phys. D 5, 481 (1972).
[Crossref]

F. Durst, A. Melling, and J. H. Whitelaw, J. Fluid Mech. 56, 143 (1972).
[Crossref]

1971 (6)

M. J. Rudd, Opt. Laser Tech. 3, 200 (1971).
[Crossref]

L. Lading, Appl. Opt. 10, 1943 (1971).
[Crossref] [PubMed]

W. M. Farmer and D. B. Brayton, Appl. Opt. 10, 2319 (1971).
[Crossref] [PubMed]

V. Degiorgio and J. B. Lastovka, Phys. Rev. A 4, 2033 (1971).
[Crossref]

R. J. Adrian and R. J. Goldstein, J. Phys. E 4, 505 (1971).
[Crossref]

R. V. Edwards, J. C. Angus, M. J. French, and J. W. Dunning, J. Appl. Phys. 42, 837 (1971).
[Crossref]

1970 (1)

H. Z. Cummins and H. L. Swinney, Prog. Opt. 8, 133 (1970).
[Crossref]

1969 (1)

J. C. Owens, Proc. IEEE 57, 530 (1969).
[Crossref]

1964 (1)

Y. Yeh and H. Z. Commins, Appl. Phys. Lett. 4, 176 (1964).
[Crossref]

Adrian, R. J.

R. J. Adrian and R. J. Goldstein, J. Phys. E 4, 505 (1971).
[Crossref]

Angus, J. C.

R. V. Edwards, J. C. Angus, and J. W. Dunning, Opto-Electronics 5, 119 (1973).
[Crossref]

R. V. Edwards, J. C. Angus, M. J. French, and J. W. Dunning, J. Appl. Phys. 42, 837 (1971).
[Crossref]

Benedek, G. B.

G. B. Benedek, in Polarization Matière et Rayonnement Livre de Jubìlé en l’honneur du Professeur A. Kastler (Presses Universitaire de France, Paris, 1968), p. 149.

Brayton, D. B.

Commins, H. Z.

Y. Yeh and H. Z. Commins, Appl. Phys. Lett. 4, 176 (1964).
[Crossref]

Cummins, H. Z.

H. Z. Cummins and H. L. Swinney, Prog. Opt. 8, 133 (1970).
[Crossref]

Degiorgio, V.

V. Degiorgio and J. B. Lastovka, Phys. Rev. A 4, 2033 (1971).
[Crossref]

Drain, L. E.

L. E. Drain, J. Phys. D 5, 481 (1972).
[Crossref]

Dunning, J. W.

R. V. Edwards, J. C. Angus, and J. W. Dunning, Opto-Electronics 5, 119 (1973).
[Crossref]

R. V. Edwards, J. C. Angus, M. J. French, and J. W. Dunning, J. Appl. Phys. 42, 837 (1971).
[Crossref]

Durst, F.

F. Durst, A. Melling, and J. H. Whitelaw, J. Fluid Mech. 56, 143 (1972).
[Crossref]

Edwards, R. V.

R. V. Edwards, J. C. Angus, and J. W. Dunning, Opto-Electronics 5, 119 (1973).
[Crossref]

R. V. Edwards, J. C. Angus, M. J. French, and J. W. Dunning, J. Appl. Phys. 42, 837 (1971).
[Crossref]

Farmer, W. M.

French, M. J.

R. V. Edwards, J. C. Angus, M. J. French, and J. W. Dunning, J. Appl. Phys. 42, 837 (1971).
[Crossref]

George, W. K.

W. K. George and J. L. Lumley, J. Fluid Mech. 60, 321 (1973).
[Crossref]

Goldstein, R. J.

R. J. Adrian and R. J. Goldstein, J. Phys. E 4, 505 (1971).
[Crossref]

Hughes, A. J.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 491.

Lading, L.

Lastovka, J. B.

V. Degiorgio and J. B. Lastovka, Phys. Rev. A 4, 2033 (1971).
[Crossref]

Lucero, J. A.

C. Y. She and J. A. Lucero, Opt. Commun. 9, 300 (1973).
[Crossref]

Lumley, J. L.

W. K. George and J. L. Lumley, J. Fluid Mech. 60, 321 (1973).
[Crossref]

Melling, A.

F. Durst, A. Melling, and J. H. Whitelaw, J. Fluid Mech. 56, 143 (1972).
[Crossref]

Owens, J. C.

Papoulis, A.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962), p. 21.

Pike, E. R.

Rudd, M. J.

M. J. Rudd, Opt. Laser Tech. 3, 200 (1971).
[Crossref]

She, C. Y.

C. Y. She, Appl. Opt. 12, 2415 (1973).
[Crossref] [PubMed]

C. Y. She and J. A. Lucero, Opt. Commun. 9, 300 (1973).
[Crossref]

Swinney, H. L.

H. Z. Cummins and H. L. Swinney, Prog. Opt. 8, 133 (1970).
[Crossref]

Wang, C. P.

C. P. Wang, J. Phys. E 5, 763 (1972).
[Crossref]

Whitelaw, J. H.

F. Durst, A. Melling, and J. H. Whitelaw, J. Fluid Mech. 56, 143 (1972).
[Crossref]

Yeh, Y.

Y. Yeh and H. Z. Commins, Appl. Phys. Lett. 4, 176 (1964).
[Crossref]

Appl. Opt. (6)

Appl. Phys. Lett. (1)

Y. Yeh and H. Z. Commins, Appl. Phys. Lett. 4, 176 (1964).
[Crossref]

J. Appl. Phys. (1)

R. V. Edwards, J. C. Angus, M. J. French, and J. W. Dunning, J. Appl. Phys. 42, 837 (1971).
[Crossref]

J. Fluid Mech. (2)

F. Durst, A. Melling, and J. H. Whitelaw, J. Fluid Mech. 56, 143 (1972).
[Crossref]

W. K. George and J. L. Lumley, J. Fluid Mech. 60, 321 (1973).
[Crossref]

J. Phys. D (1)

L. E. Drain, J. Phys. D 5, 481 (1972).
[Crossref]

J. Phys. E (2)

C. P. Wang, J. Phys. E 5, 763 (1972).
[Crossref]

R. J. Adrian and R. J. Goldstein, J. Phys. E 4, 505 (1971).
[Crossref]

Opt. Commun. (1)

C. Y. She and J. A. Lucero, Opt. Commun. 9, 300 (1973).
[Crossref]

Opt. Laser Tech. (1)

M. J. Rudd, Opt. Laser Tech. 3, 200 (1971).
[Crossref]

Opto-Electronics (1)

R. V. Edwards, J. C. Angus, and J. W. Dunning, Opto-Electronics 5, 119 (1973).
[Crossref]

Phys. Rev. A (1)

V. Degiorgio and J. B. Lastovka, Phys. Rev. A 4, 2033 (1971).
[Crossref]

Proc. IEEE (1)

J. C. Owens, Proc. IEEE 57, 530 (1969).
[Crossref]

Prog. Opt. (1)

H. Z. Cummins and H. L. Swinney, Prog. Opt. 8, 133 (1970).
[Crossref]

Other (3)

G. B. Benedek, in Polarization Matière et Rayonnement Livre de Jubìlé en l’honneur du Professeur A. Kastler (Presses Universitaire de France, Paris, 1968), p. 149.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962), p. 21.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 491.

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Figures (5)

FIG. 1
FIG. 1

Kinematics and coordinate systems of a two-beam laser Dopplerjvelocimeter. z = z1 = z2 = zD into the paper; K = k 1 k 2 = q 1 q 2 ; K 1 = k 1 k D, and K 2 = k 2 k D.

FIG. 2
FIG. 2

Square root of the coherence factor (κRΩ/κ) as a function of the normalized detection aperture μ = (μΩ/2Ωc)1/2 for uniform beam.

FIG. 3
FIG. 3

Square root of the coherence factor (κRΩ/k) as a function of the normalized detection aperture μ = (πΩ/2Ωc)1/2 for gaussian beam.

FIG. 4
FIG. 4

Square root of the coherence factor (κ(2)Ω/κΩc) vs μ = (πΩ/2Ωc)1/2 for both uniform and gaussian beams.

FIG. 5
FIG. 5

Illustration of signal and noise (a) spectral analysis and (b) correlation analysis.

Tables (6)

Tables Icon

TABLE I Electric-field amplitudes for Eq. (1).

Tables Icon

TABLE II The magnitude of photocurrents for Eq. (5).

Tables Icon

TABLE III One-particle and two-particle coherence factors.a

Tables Icon

TABLE IV Functions and parameters for Eqs. (7) and (8).

Tables Icon

TABLE V Functions and parameters for Eq. (8).

Tables Icon

TABLE VI Predetection signal-to-noise ratios.

Equations (38)

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E 1 ( R , t ) = E 1 ( R ) exp [ j ( ω t k 1 · r k | R r | ϕ ) ] z ˆ ,
E 2 ( R , t ) = E 2 ( R ) exp [ j ( ω t k 2 · r k | R r | ) ] z ˆ ,
E R ( R , t ) = E R ( R ) exp [ j ( ω t k R ) ] z ˆ ,
i ( t ) = κ | E R ( R , t ) + i N [ E 1 ( R , t ) + E 2 ( R , t ) ] i | 2 d A D ,
i ( t ) = i R ( t ) + i N [ i R 2 ( t ) + i R 1 ( t ) + i 1 ( t ) + i 2 ( t ) + i 12 ( t ) ] i + i N j i N [ 1 2 i 11 ( 2 ) ( t ) + 1 2 i 22 ( 2 ) ( t ) + i 12 ( 2 ) ( t ) ] i , j .
i R ( t ) = κ ( E R E R * ) d A D ,
i R 1 ( t ) = κ ( E R E 1 * + E R * E 1 ) d A D ,
i R 2 ( t ) = κ ( E R E 2 * + E R * E 2 ) d A D ,
i 1 ( t ) = κ ( E 1 E 1 * ) d A D ,
i 2 ( t ) = κ ( E 2 E 2 * ) d A D ,
i 12 ( t ) = κ ( E 2 E 1 * + E 2 * E 1 ) d A D ,
i 11 ( 2 ) ( t ) = κ [ E 1 ( 1 ) E 1 * ( 2 ) + E 1 * ( 1 ) E 1 ( 2 ) ] d A D ,
i 22 ( 2 ) ( t ) = κ [ E 2 ( 1 ) E 2 * ( 2 ) + E 2 * ( 1 ) E 2 ( 2 ) ] d A D ,
i 12 ( 2 ) ( t ) = κ [ E 1 ( 1 ) E 2 * ( 2 ) + E 1 * ( 1 ) E 2 ( 2 ) ] d A D ,
E R E 1 * + E R E 1 = 2 a λ R 2 ( P 1 P 2 ) 1 / 2 exp [ x 1 2 + z 1 2 2 σ a ] × exp [ x D 2 + z D 2 2 S 2 ] cos [ k 1 · r + k | R r | k R + ϕ ] ,
cos [ k 1 · r + k | R r | k R + ϕ ] cos [ k 1 · r k y + ϕ + k r 2 2 R k R x x D k R z z D ] .
i R 1 = κ 2 a λ R 2 ( P 1 P 2 ) 1 / 2 exp [ x 1 2 + z 1 2 2 σ 2 ] cos [ ( k 1 k D ) · r + ϕ ] · x 0 x 0 z 0 z 0 exp [ x 0 2 + z 0 2 2 S 2 ] cos [ k R x x D + k R z z D ] d x D d z D = 2 κ R [ a λ λ σ 2 ( P 1 P 2 ) 1 / 2 ] exp [ x 1 2 + z 1 2 2 σ 2 ] cos [ ( k 1 k D ) · r + ϕ ] ,
i R ( t ) = κ I R Ω ,
i R 1 ( t ) = 2 κ R I R 1 Ω cos [ ( k 1 k D ) · r + ϕ ] ,
i R 2 ( t ) = 2 κ R I R 2 Ω cos [ ( k 2 k D ) · r ] ,
i 1 ( t ) = κ I 1 Ω ,
i 2 ( t ) = κ I 2 Ω ,
i 12 ( t ) = 2 κ I 12 Ω cos [ ( k 1 k 2 ) · r + ϕ ] ,
i 11 ( 2 ) ( t ) = 2 κ ( 2 ) I 11 ( 2 ) Ω cos [ ( k 1 k D ) · ( r 2 r 1 ) ] ,
i 22 ( 2 ) ( t ) = 2 κ ( 2 ) I 22 ( 2 ) Ω cos [ ( k 2 k D ) · ( r 2 r 1 ) ] ,
i 22 ( 2 ) ( t ) = 2 κ ( 2 ) I 12 ( 2 ) Ω cos [ ( k 1 · r 2 k 2 · r 1 ) k D · ( r 2 r 1 ) + ϕ ] .
i R B ( t ) = ( κ Ω ) [ I R + i N ( I 1 ) i ] + 2 ( κ R Ω ) i N ( I 1 ) i cos [ ( q 1 q 2 ) · r i + ϕ ] ,
R R B ( τ ) = ( κ Ω ) 2 { I R 2 + 2 I R I 1 N 1 + I 1 2 [ N 1 g 1 ( τ ) + N 1 ( N 1 1 ) ] } + 2 ( κ R Ω ) 2 I R 1 2 N 1 cos [ ( q 1 q 2 ) · X ( t ) ] g 1 ( τ ) cos [ ( q 1 q 2 ) · Δ r ( τ ) ] ,
p ( Δ r , τ / 0 , 0 ) = [ 2 π ξ 2 ( τ ) ] 3 / 2 exp [ | Δ r ( τ ) | 2 / 2 ξ 2 ( τ ) ] .
i C B ( t ) = ( κ Ω ) i N { I 1 + I 2 + 2 I 12 cos [ ( q 1 q 2 ) · r ( τ ) + ϕ ] } i + 2 ( κ ( 2 ) Ω ) i N j i N { 1 2 I 11 ( 2 ) cos [ ( k 1 k D ) · ( r ( 2 ) r ( 1 ) ) ] + 1 2 I 22 ( 2 ) · cos [ ( k 2 k D ) · ( r ( 2 ) r ( 1 ) ) ] + I 12 ( 2 ) cos [ ( k 1 · r ( 2 ) k 2 · r ( 1 ) ) · ( r ( 2 ) r ( 1 ) ) + ϕ ] } i , j ,
R C B ( τ ) = ( κ Ω ) 2 { I 1 2 [ N 1 g 1 ( τ ) + N 1 ( N 1 1 ) ] + I 2 2 [ N 2 g 2 ( τ ) + N 2 ( N 2 1 ) ] + 2 I 1 I 2 [ N 12 g 2 ( τ ) + ( N 1 N 2 N 12 ) ] + 2 I 12 2 N 12 · cos [ ( q 1 q 2 ) · X ( τ ) ] g ( τ ) cos ( q 1 q 2 ) · Δ r ( τ ) } + ( κ ( 2 ) Ω ) 2 { ( I 11 ( 2 ) ) 2 N 1 ( N 1 1 ) g 11 ( τ ) cos [ ( k 1 k D ) · ( Δ r ( τ 2 ) Δ r ( τ 1 ) ) ] + ( I 22 ( 2 ) ) 2 N 2 ( N 2 1 ) g 22 ( τ ) cos [ ( k 2 k D ) · ( Δ r ( τ 2 ) Δ r ( τ 1 ) ) ] + 4 ( I 12 ( 2 ) ) 2 ( N 1 N 2 N 12 ) g 12 ( τ ) cos [ ( k 1 k D ) · Δ r ( τ 2 ) ] + ( k 2 k D ) · Δ r ( τ 1 ) · cos [ ( q 1 q 2 ) · X ( τ ) ] } .
R C B ( τ ) = i C D C 2 + I 2 exp [ β 2 τ 2 ] [ ( κ Ω ) 2 2 ( N + N 12 ) + ( κ ( 2 ) Ω ) 2 N ( N 1 ) { exp ( ξ 2 ( τ ) K 1 2 ) + exp ( ξ 2 ( τ ) K 2 2 ) } ] + I 2 exp [ β 2 τ 2 ] cos ( K V τ ) · [ ( κ Ω ) 2 2 N 12 exp { ξ 2 ( τ ) K 2 / 2 } + ( κ ( 2 ) Ω ) 2 4 ( N 2 N 12 ) · exp { ξ 2 ( τ ) ( K 1 2 + K 2 2 ) / 2 } ] + G e i C D C δ ( τ ) ,
β 2 = V 2 / 2 ( σ / cos θ ) 2 , i C D C 2 = I 2 ( κ Ω ) 2 [ 2 N ( N 1 ) + 2 ( N 2 + N 12 ) ] .
R R B ( τ ) = i R D C 2 + I 2 ( κ Ω ) 2 N 1 exp [ β 2 τ 2 ] + I R 1 2 ( κ R Ω ) 2 2 N 1 cos ( K V τ ) exp [ β 2 τ 2 / 2 ] · exp [ 1 2 ξ 2 ( τ ) K 2 ] + G e i E D C δ ( τ ) ,
S C B ( ω ) = 2 π i C D C 2 δ ( ω ) + I 2 { ( κ Ω ) 2 2 ( N + N 12 ) ( π / β ) exp ( ω 2 / 4 β 2 ) + ( κ ( 2 ) Ω ) 2 N ( N 1 ) [ ( π / α 1 ( K 1 ) ) exp ( ω 2 / 4 α 2 ( K 1 ) ) + ( π / α 1 ( K 2 ) ) exp ( ω 2 / 4 α 1 2 ( K 2 ) ) ] + ( κ Ω ) 2 N 12 ( π / α 1 ( K / 2 ) ) [ exp ( ( ω K V ) 2 / 4 α 1 2 ( K / 2 ) ) + exp ( ( ω + K V ) 2 / 4 α 1 2 ( K / 2 ) ) ] + ( κ ( 2 ) Ω ) 2 2 ( N 2 N 12 ) · ( π / α 1 ( [ K 1 2 + K 2 2 ] 1 / 2 / 2 ) ) × [ exp ( ( ω + K V ) 2 / 4 α 1 2 ( [ K 1 2 + K 2 2 ] 1 / 2 / 2 ) ) + exp ( ( ω + K V ) 2 / 4 α 1 2 ( [ K 1 2 + K 2 2 ] 1 / 2 / 2 ) ) ] } + G e i C D C ,
S R B ( ω ) = 2 π i R D C 2 δ ( ω ) + I 2 ( κ Ω ) 2 N 1 ( π / β ) exp ( ω 2 / 4 β 2 ) + I R 1 2 ( κ R Ω ) 2 N 1 ( π / α 2 ( K ) ) × exp [ ( ω K V ) 2 / 4 α 2 2 ( K ) ] + exp [ ( ω + K V ) 2 / 4 α 2 2 ( K ) ] + G e i R D C ,
( S / N ) po = ( S / N ) pr 1 + ( S / N ) pr ( T B T A ) ,
( S / N ) po = ( S / N ) pr 1 + ( S / N ) pr ( T B τ c ) ,