Abstract

Within the framework of complex ABCD matrix theory, an analytical expression for the time-lagged covariance of a classical time-of-flight laser velocimetry system is obtained as a function of the laser spot size, the limiting aperture, and the measurement-aperture radii. The decorrelation effects of wave-front tilt and velocity misalignment are derived and discussed. In addition, error estimates are presented that indicate how well one can determine the precise location of the peak of the cross-covariance function.

© 1993 Optical Society of America

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References

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  1. L. H. Tanner, “A particle timing laser-velocimeter,” report (Department of Aeronautical Engineering, Queens University of Belfast, Belfast, 1972).
  2. R. Schodl, “The laser dual-focus flow velocimeter,” inNATO-AGARD Proceedings No. 193, Applications of Non-Intrusive Instrumentation in Fluid Flow Research, AGARD-CP-193 (NATO, Saint Louis, France, 1976), paper No. 21.
  3. L. Lading, “The time-of-flight laser anemometer,” inNATO-AGARD Proceedings No. 193, Applications of Non-Intrusive Instrumentation in Fluid Flow Research, AGARD-CP-193 (NATO, Saint Louis, France, 1976), paper No. 23.
  4. J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1986), Chap. 8.
  5. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 20.
  6. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  7. H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCDsystems,” J. Opt. Soc. Am. A 10, 316–323 (1993).
    [CrossRef]
  8. J. W. Goodman, Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), Chap. 2.
    [CrossRef]
  9. L. Lading, “Estimating time and time-lag in time-of-flight velocimetry,” Appl. Opt. 22, 3637–3643 (1983).
    [CrossRef] [PubMed]

1993 (1)

1987 (1)

1983 (1)

Bendat, J. S.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1986), Chap. 8.

Goodman, J. W.

J. W. Goodman, Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), Chap. 2.
[CrossRef]

Grum, T. P.

Hanson, S. G.

Lading, L.

L. Lading, “Estimating time and time-lag in time-of-flight velocimetry,” Appl. Opt. 22, 3637–3643 (1983).
[CrossRef] [PubMed]

L. Lading, “The time-of-flight laser anemometer,” inNATO-AGARD Proceedings No. 193, Applications of Non-Intrusive Instrumentation in Fluid Flow Research, AGARD-CP-193 (NATO, Saint Louis, France, 1976), paper No. 23.

Piersol, A. G.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1986), Chap. 8.

Schodl, R.

R. Schodl, “The laser dual-focus flow velocimeter,” inNATO-AGARD Proceedings No. 193, Applications of Non-Intrusive Instrumentation in Fluid Flow Research, AGARD-CP-193 (NATO, Saint Louis, France, 1976), paper No. 21.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 20.

Tanner, L. H.

L. H. Tanner, “A particle timing laser-velocimeter,” report (Department of Aeronautical Engineering, Queens University of Belfast, Belfast, 1972).

Yura, H. T.

Appl. Opt. (1)

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

Other (6)

J. W. Goodman, Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), Chap. 2.
[CrossRef]

L. H. Tanner, “A particle timing laser-velocimeter,” report (Department of Aeronautical Engineering, Queens University of Belfast, Belfast, 1972).

R. Schodl, “The laser dual-focus flow velocimeter,” inNATO-AGARD Proceedings No. 193, Applications of Non-Intrusive Instrumentation in Fluid Flow Research, AGARD-CP-193 (NATO, Saint Louis, France, 1976), paper No. 21.

L. Lading, “The time-of-flight laser anemometer,” inNATO-AGARD Proceedings No. 193, Applications of Non-Intrusive Instrumentation in Fluid Flow Research, AGARD-CP-193 (NATO, Saint Louis, France, 1976), paper No. 23.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1986), Chap. 8.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 20.

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Figures (3)

Fig. 1
Fig. 1

Basic setup for time-of-flight velocimetry system. Two laser spots, with spot 1 on the left and a mutual distance 2d, reflected from a diffusing target, are imaged onto two detectors.

Fig. 2
Fig. 2

Clean imaging system with separate apertures in object and Fourier planes. The system includes defocusing of the laser spots onto the detectors.

Fig. 3
Fig. 3

Measurement on a cylindrical target of radius R causes the scattering structure to be tilted as it passes from one spot to the next.

Equations (58)

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C i ( τ ) = i 1 ( t ) i 2 ( t + τ ) - i 1 ( t ) i 2 ( t + τ ) ,
i j ( t ) = α d p W j ( p ) I j ( p , t ) ,             j = 1 , 2 ,
α = q η h ν ,
W j ( p ) = exp [ - 2 ( p x m d ) 2 σ a 2 - 2 p y 2 σ a 2 ] ,             j = 1 , 2 ,
I j ( p , t ) = | d r U A j ( r , t ) G ( r , p ) | 2 ,             j = 1 , 2 ,
G ( r , p ) = - i k 2 π B exp [ - i k 2 B ( D p 2 - 2 r · p + A r 2 ) ] ,
A = - f 2 / f 1 = - m ,
B = - 2 i f 1 f 2 k σ 2 ( 1 - i δ ) ,
D = - f 1 / f 2 = - 1 / m ,
δ = k σ 2 2 f 2 2 δ f ,
G ( r , p ) = ( - i k / 2 π B ) exp [ - ( x + r ) 2 ω ˜ 2 ] ,
ω ˜ 2 = ω 1 2 ( 1 - i δ ) ,
ω 1 = 2 f 1 k σ ,
x - p m .
U A j ( r , t ) = I j ( r ) ψ ( r , t ) ,             j = 1 , 2 ,
I 1 = 2 P o π σ s 2 exp [ - 2 ( r + d ) 2 σ s 2 ] ,
I 2 = 2 P o π σ s 2 exp [ - 2 ( r - d ) 2 σ s 2 ] ,
ψ ( r , t + τ ) = ψ ( r - v τ , t ) .
U A 1 ( r 1 , t ) U A 2 * ( r 2 , t + τ ) U A 1 * ( r 2 , t ) U A 2 ( r 1 , t + τ ) ,
U A 1 ( r 1 , t ) U A 2 * ( r 1 , t + τ ) U A 1 * ( r 2 , t ) U A 2 ( r 2 , t + τ ) = U A 1 ( r 1 , t ) U A 1 * ( r 2 , t ) U A 2 * ( r 2 , t + τ ) U A 2 ( r 1 , t + τ ) + U A 1 ( r 1 , t ) U A 2 * ( r 2 , t + τ ) U A 1 * ( r 2 , t ) U A 2 ( r 1 , t + τ ) .
C i ( τ ) = d p 1 d p 2 W 1 ( p 1 ) W 2 ( p 2 ) K ( p 1 , p 2 ; τ ) ,
K ( p 1 , p 2 ; τ ) = d r 1 d r 2 d r 1 d r 2 × G ( r 1 , p 1 ) G * ( r 2 , p 1 ) G ( r 1 , p 2 ) G * ( r 2 , p 2 ) × U A 1 ( r 1 , t ) U A 2 * ( r 2 , t + τ ) × U A 1 * ( r 2 , t ) U A 2 ( r 1 , t + τ ) .
K ( p 1 , p 2 ; τ ) = | a d r [ I 1 ( r ) I 2 ( r τ ) ] 1 / 2 G ( r , p 1 ) G * ( r τ , p 2 ) | 2 ,
r τ = r + v τ
C i ( τ ) = const . d r d r [ I 1 ( r ) I 2 ( r τ ) I 1 ( r ) I 2 ( r τ ) ] 1 / 2 H ( r , r ) ,
H ( r , r ) = exp { - [ σ o 2 ( r - r ) 2 ω 1 2 ( σ o 2 + ω ˜ 2 ) + ( r + d ) 2 ω 1 2 + σ o 2 + ( r + d ) 2 ω 1 2 + σ o 2 + ( r τ - d ) 2 ω 1 2 + σ o 2 + ( r - d ) 2 ω 1 2 + σ o 2 ] } ,
σ o = σ a m
C i ( τ ) = i 2 N exp [ - ( v τ - 2 d ) 2 Δ 2 ] ,
1 Δ 2 = 1 σ s 2 + 1 ω 1 2 + σ o 2 ,
i = i 1 ( t ) = i 2 ( t + τ ) = α P o ( σ 2 4 f 1 2 ) 1 1 + ( ω s 2 / σ a 2 ) ,
ω s 2 = m 2 σ s 2 + ( 2 f 2 k σ ) 2
ω s 2 = m 2 σ s 2 ( N c N c - 1 ) ,
N c = 1 + k 2 σ 2 σ s 2 4 f 1 2 .
1 Δ d f 2 = Re [ 1 ω 1 2 ( 1 - i δ ) + σ o 2 + 1 σ s 2 ] = ω 1 2 + σ o 2 ( ω 1 2 + σ o 2 ) 2 + δ 2 ω 1 4 + 1 σ s 2 .
P ( ω ) = - i ( t ) i ( t + τ ) exp ( i ω τ ) d τ .
p ( f ) = i 2 [ 1 π N f o exp ( - f 2 f o 2 ) + δ ( 2 π f ) ] ,
f o = ( 1 2 π ) 2 v Δ d f = v π Δ d f ,
C i ( τ ; θ ) = exp ( - θ 2 / θ o 2 ) C i ( τ ) ,
θ o = 2 k β Δ d f .
R min = d k β Δ d f .
r τ = ( r x + v τ ) e ^ x + v τ e ^ y ,
C i ( τ ; v ) = exp [ - ( δ y / Δ d f ) 2 ] C i ( τ ) ,
δ y = v τ ,
F d = exp [ - ( θ v θ v d ) 2 ] ,
θ v = v v ,
θ v d = Δ d f 2 d
Δ d f = σ s / N c σ s , N c 1 ω 1 , N c 1 } ,
θ v d = σ s / ( 2 d N c ) σ s / 2 d ,             N c 1 ω 1 / 2 d ,             N c 1.
θ v d = σ s σ o / 2 d ( σ o 2 + σ s 2 ) 1 / 2 σ s / 2 d ,             σ o σ s σ o / 2 d ,             σ o σ s .
i 1 , 2 ( t ) = i 1 , 2 ( t ) - 1 T 0 T i 1 , 2 ( s ) d s .
C ^ 12 ( τ ) = 1 T 0 T i 1 ( t ) i 2 ( t + τ ) d t .
σ o 2 ( τ ) Var [ C ^ 12 ( τ ) ] 1 T - [ C 11 ( t ) C 22 ( t ) + C 12 ( τ + t ) C 12 ( τ - t ) ] d t ,
σ o 2 = o 2 ( τ ) C 12 2 ( τ ) ,
o 2 ( τ ) = π 2 ( Δ τ T ) [ 1 + C 11 ( 0 ) C 22 ( 0 ) C 12 2 ( τ ) ] = π 2 ( Δ τ T ) { 1 + F - 2 exp [ ( τ - τ o ) 2 Δ τ 2 ] } ,
o 2 ( τ o ) = π 2 ( Δ τ T ) ( 1 + F - 2 ) .
C ^ 12 ( τ ) = [ 1 - ( τ - τ o ) 2 Δ τ 2 ] C 12 ( τ o ) .
o 2 ( τ o ) = ( τ ^ - τ o ) 4 Δ τ 4 .
σ τ o 2 = 1 / 3 Δ τ 2 o ( τ o ) ,

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