Abstract

Within the framework of ABCD matrix theory, analytical expressions are derived for the time-lagged covariance of a classical laser Doppler velocimetry system as a function of the laser spot size, the limiting aperture, and the measurement aperture size. Both partial and fully developed speckle as well as planar and rotating targets, are considered. Further, error estimates are presented that indicate how well one can determine in practice the velocity of both planar and rotating targets, and a comparison with time-of-flight velocimetry is given.

© 1995 Optical Society of America

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References

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  1. L. E. Drain, The Laser Doppler Technique (Wiley, Chichester, 1980).
  2. L. Lading, “Principles of laser anemometry,” in Optical Diagnostics for Flow Processes, L. Lading, P. Buchhave, G. Wigley, eds. (Plenum, New York, 1994), pp. 85–125.
  3. H. T. Yura, S. G. Hanson, T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCDoptical systems,” J. Opt. Soc. Am. A 10, 316–323 (1993).
    [Crossref]
  4. R. V. Edwards, J. C. Angus, J. W. Dunning, “Spectral analysis of the signal from a laser Doppler flowmeter: time independent systems,” J. Appl. Phys. 42, 837–850 (1971); W. K. George, J. L. Lumley, “The laser Doppler velocimeter and its application to the measurement of turbulence,” J. Fluid Mech. 60, 312–362 (1973).
    [Crossref]
  5. L. Lading, R. V. Edwards, “Laser velocimeters: lower limits to uncertainty,” Appl. Opt. 32, 3855–3866 (1993).
    [PubMed]
  6. H. T. Yura, S. G. Hanson, “Laser-time-of-flight velocimetry: analytical solution to the optical system based on ABCDmatrices,” J. Opt. Soc. Am. A 10, 1918–1924 (1993).
    [Crossref]
  7. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.
  8. J. W. Goodman, Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.
  9. S. Wolfram, mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, Calif., 1991).
  10. For nonnormal angles of incidence, the factor of 2 in the exponent is replaced by (1 + cos β), where βis the angle between the direction of propagation and the normal to the surface.
  11. J. S. Bendat, A. G. Piersol, Random Data: Analyses and Measurement Procedures (Wiley, New York, 1986), Chap. 8.

1993 (3)

1971 (1)

R. V. Edwards, J. C. Angus, J. W. Dunning, “Spectral analysis of the signal from a laser Doppler flowmeter: time independent systems,” J. Appl. Phys. 42, 837–850 (1971); W. K. George, J. L. Lumley, “The laser Doppler velocimeter and its application to the measurement of turbulence,” J. Fluid Mech. 60, 312–362 (1973).
[Crossref]

Angus, J. C.

R. V. Edwards, J. C. Angus, J. W. Dunning, “Spectral analysis of the signal from a laser Doppler flowmeter: time independent systems,” J. Appl. Phys. 42, 837–850 (1971); W. K. George, J. L. Lumley, “The laser Doppler velocimeter and its application to the measurement of turbulence,” J. Fluid Mech. 60, 312–362 (1973).
[Crossref]

Bendat, J. S.

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

Drain, L. E.

L. E. Drain, The Laser Doppler Technique (Wiley, Chichester, 1980).

Dunning, J. W.

R. V. Edwards, J. C. Angus, J. W. Dunning, “Spectral analysis of the signal from a laser Doppler flowmeter: time independent systems,” J. Appl. Phys. 42, 837–850 (1971); W. K. George, J. L. Lumley, “The laser Doppler velocimeter and its application to the measurement of turbulence,” J. Fluid Mech. 60, 312–362 (1973).
[Crossref]

Edwards, R. V.

L. Lading, R. V. Edwards, “Laser velocimeters: lower limits to uncertainty,” Appl. Opt. 32, 3855–3866 (1993).
[PubMed]

R. V. Edwards, J. C. Angus, J. W. Dunning, “Spectral analysis of the signal from a laser Doppler flowmeter: time independent systems,” J. Appl. Phys. 42, 837–850 (1971); W. K. George, J. L. Lumley, “The laser Doppler velocimeter and its application to the measurement of turbulence,” J. Fluid Mech. 60, 312–362 (1973).
[Crossref]

Goodman, J. W.

J. W. Goodman, Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.

Grum, T. P.

Hanson, S. G.

Lading, L.

L. Lading, R. V. Edwards, “Laser velocimeters: lower limits to uncertainty,” Appl. Opt. 32, 3855–3866 (1993).
[PubMed]

L. Lading, “Principles of laser anemometry,” in Optical Diagnostics for Flow Processes, L. Lading, P. Buchhave, G. Wigley, eds. (Plenum, New York, 1994), pp. 85–125.

Piersol, A. G.

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

Siegman, A. E.

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

Wolfram, S.

S. Wolfram, mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, Calif., 1991).

Yura, H. T.

Appl. Opt. (1)

J. Appl. Phys. (1)

R. V. Edwards, J. C. Angus, J. W. Dunning, “Spectral analysis of the signal from a laser Doppler flowmeter: time independent systems,” J. Appl. Phys. 42, 837–850 (1971); W. K. George, J. L. Lumley, “The laser Doppler velocimeter and its application to the measurement of turbulence,” J. Fluid Mech. 60, 312–362 (1973).
[Crossref]

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

Other (7)

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

J. W. Goodman, Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.

S. Wolfram, mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Redwood City, Calif., 1991).

For nonnormal angles of incidence, the factor of 2 in the exponent is replaced by (1 + cos β), where βis the angle between the direction of propagation and the normal to the surface.

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

L. E. Drain, The Laser Doppler Technique (Wiley, Chichester, 1980).

L. Lading, “Principles of laser anemometry,” in Optical Diagnostics for Flow Processes, L. Lading, P. Buchhave, G. Wigley, eds. (Plenum, New York, 1994), pp. 85–125.

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Figures (4)

Fig. 1
Fig. 1

Schematic of a LDV system. Here the transmitter and the receiver are shown as separate units. Often they are combined and have a common optical axis.

Fig. 2
Fig. 2

Optical diagram of a LDV system. The limiting aperture of 1/e2 radius σ is positioned in the Fourier plane.

Fig. 3
Fig. 3

Reflected intensity distribution for Λ = σs/5.

Fig. 4
Fig. 4

Measurement geometry for a rotating cylindrical shaft.

Tables (1)

Tables Icon

Table 1 Minimum Radius of Curvature for Measurement on Cylindrical Surfaces in the Two Limiting Cases for LDV and Time-of-Flight Velocimetry

Equations (64)

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C i ( τ ) = i ( t ) i ( t + τ ) - i ( t ) i ( t + τ ) ,
i ( t ) = α d p W ( p ) I ( p , t ) ,
α = q η / h ν ,
W ( p ) = exp ( - 2 p 2 / σ a 2 ) ,
I ( p , t ) = | d r U 0 ( r , t ) G ( r , p ) | 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 ,
D = - f 1 / f 2 = - 1 / m ,
G ( r , p ) = ( - i k / 2 π B ) exp [ - ( x + r ) 2 / ω 2 ] ,
ω = 2 f 1 / k σ ,
x = p / m .
U 0 ( r , t ) = U i ( r ) ψ ( r , t ) ,
I 0 ( r ) = U i ( r ) 2 = 4 P 0 π σ s 2 ( κ x r x / 2 ) exp ( - 2 r 2 σ s 2 ) ,
κ x = 2 π / Λ ,
B ψ ( r 1 , r 2 ) = ( 4 π k 2 ) { 2 π r c 2 exp [ - 2 ( r 1 - r 2 ) 2 r c 2 ] } ,
ψ ( r , t + τ ) = ψ ( r - v τ , t ) .
U 0 ( r 1 , t ) U 0 * ( r 2 , t + τ ) U 0 * ( r 2 , t ) U 0 ( r 1 , t + τ ) ,
U 0 ( r 1 , t ) U 0 * ( r 2 , t + τ ) U 0 * ( r 2 , t ) U 0 ( r 1 , t + τ ) = U 0 ( r 1 , t ) U 0 * ( r 2 , t ) U 0 * ( r 2 , t + τ ) U 0 ( r 1 , t + τ ) + U 0 ( r 1 , t ) U 0 * ( r 2 , t + τ ) U 0 * ( r 2 , t ) U 0 ( r 1 , t + τ ) .
C i ( τ ) = d p 1 W ( p 1 ) d p 2 W ( 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 0 ( r 1 , t ) U 0 * ( r 2 , t + τ ) × U 0 * ( r 2 , t ) U 0 ( r 1 , t + τ ) .
K ( p 1 , p 2 ; τ ) = | d r 1 U i ( r 1 ) G ( r 1 , p 1 ) × d r 2 U i * ( r 2 ) G * ( r 2 , p 2 ) B ψ ( r 1 , r 2 τ ) | 2 ,
r 2 τ = r 2 + v τ .
C i ( τ ) = i 2 N [ 1 + cos ( κ x v x τ ) 2 ] exp [ - ( v t ) 2 σ s 2 + r c 2 ] ,
i = d p W ( p ) d r 1 d r 2 B ψ ( r 1 , r 2 ) U i ( r 1 ) U i * ( r 2 ) × G ( r 1 , p ) G * ( r 2 , p ) = α P 0 ( σ 2 f 1 ) 2 1 1 + r c 2 / ω 2 + r c 2 / σ s 2 ,
N = ( 1 + σ s 2 / ω 2 ) ( 1 + r c 2 / σ s 2 ) 1 + r c 2 / ω 2 + r c 2 / σ s 2 .
b = 1 / 2 R .
U 0 ( r , t ) = U i ( r ) exp ( 2 i k b r x 2 ) ( ψ r , t ) .
C i ( τ ) = i 2 N [ 1 + cos ( κ x ω 0 R τ ) 2 ] exp [ - ( ω 0 R τ ) 2 Δ 2 ] ,
1 Δ 2 = 1 σ s 2 + 1 ω 2 + σ 0 2 + ( k σ s ω / R ) 2 σ s 2 + ω 2 ,
i = α P 0 ( σ 2 f 1 ) 2 1 1 + ( ω s 2 / σ a 2 ) ,
N = N c ( ω 2 + σ 0 2 ( ω s / m ) 2 + σ 0 2 ) 1 / 2 ,
N c = 1 + ( σ s ω ) 2 = 1 + ( k σ σ s 2 f 1 ) 2 ,
ω s 2 = ( m σ s ) 2 + ( 2 f 2 k σ ) 2 ,
σ 0 = σ a m .
C i ( τ ) = i 2 N c [ 1 + cos ( κ x ω 0 R τ ) 2 ] exp { - ( ω 0 R τ σ s ) 2 × [ 1 + ( k σ s ω / R ) 2 1 + ω 2 / σ 2 2 ] } ( σ a ) ,
( k ω σ s / R min ) 2 1 + ω 2 / σ s 2 > 1.
R min = k ω σ s ,
R min = k σ s 2 .
S ( f ) = - C i ( τ ) exp ( - 2 π i f τ ) d τ ,
S ( f ) = i 2 2 π Δ f N exp [ - ( f - f 0 ) 2 Δ f 2 ] ,
f 0 = { v x / Λ ( planner E ) ω 0 R / Λ ( rotating targets ) ,
Δ f = { v / π σ s 2 + r c 2 ( planar targets ) ω 0 R / π Δ ( rotating targets ) .
var [ S ^ ( f ) ] = [ S ^ ( f ) - S ^ ( f ) ] 2 = S 2 ( f ) / B e T ,
S ^ ( f ) = [ 1 - ( f - f 0 ) 2 Δ f 2 ] S ( f 0 ) .
var [ S ^ ( f ) ] = ( f - f 0 ) 4 Δ f 4 S ( f 0 ) 2 .
σ f 0 2 = 1 / 3 Δ f 2 var [ S ^ ( f 0 ) ] 1 / 2 / S ( f 0 ) .
σ f 0 2 = Δ f 2 3 B e T .
σ v 2 v 2 1 N f 2 B e T ,
B e = v ( 1 / σ s 2 + 1 / ω 2 ) 1 / 2 ,
σ v 2 v 2 1 N f 2 B e T ( v x v ) 2 ,
ψ ( r ) = ψ ( r ) exp [ i ϕ ( r ) ] ,
ϕ ( r ) = k ( 1 + cos β ) h ( r ) ,
σ ϕ 2 = [ k ( 1 + cos β ) σ h ] 2 .
B ψ ( r 1 , r 2 ) = exp { i [ ϕ ( r 1 ) - ϕ ( r 2 ) ] } = exp { - σ ϕ 2 [ 1 - b h ( r 1 , r 2 ) ] } ,
b h ( r ) = exp [ - 2 ( r / r h ) 2 ] ,
B ψ ( r ) = exp ( - σ ϕ 2 { 1 - exp [ - 2 ( r / r h ) 2 ] } ) .
B ψ ( r ) = exp [ - 2 ( r / r c ) 2 ] ,
r c = r h σ ϕ = r h k ( 1 + cos β ) σ h
B ψ ( r 1 , r 2 ) = ( 4 π k 2 ) { 2 π r c 2 exp [ - 2 ( r 1 - r 2 ) 2 r c 2 ] } .
ψ ( r , t ) = ρ ( r , t ) ,
K ( p 1 , p 2 ; τ ) = | d r 1 ρ ( r 1 ) 1 / 2 U i ( r 1 ) G ( r 1 , p 1 ) × d r 2 ρ ( r 2 ) 1 / 2 U i * ( r 2 ) G * ( r 2 , p 2 ) × B ψ ( r 1 , r 2 τ ) | 2 .
K ( p 1 , p 2 ; τ ) = | 4 π k 2 d r ρ ( r ) ρ ( r τ ) U i ( r ) U i * ( r τ ) × G ( r , p 1 ) G * ( r τ , p 2 ) | 2 ,
r τ = r + v τ .

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