Abstract

The consequences of partially developed speckle and the effects giving rise to bias errors in velocity determination are discussed with respect to robustness of a classical laser time-of-flight velocimetry (LTV) system. It is demonstrated that surface regimes exist that define the degree of partially developed speckle. These regimes are explored both theoretically and experimentally; surface models are developed to predict the resulting cross covariance from which velocity estimations can be obtained. The surface models describe the behavior of the cross covariance caused by reflection structures and with disparate lateral-roughness scales. In particular, it is shown that it is possible to obtain a twin-Gaussian cross covariance as a result of the presence of partially developed speckle. All models described are compared with experimental observations of the cross covariance for differing surface regimes. The objects are solid targets having lateral spatial correlations in reflection amplitude, height, or both, generally giving rise to partially developed speckle. In almost all cases good agreement with the corresponding theoretical predictions are found. Decorrelation caused by velocity misalignment is shown to shift the peak of the cross covariance significantly, giving a velocity bias. A corresponding theoretical model is developed and verified experimentally. Cross-talk measurements have been performed and compared with a theory developed herein. Both measurements and theory indicate that only spot sizes comparable with or larger than their corresponding separation will lead to a measurable peak shift of the time lag for the maximum of the cross covariance. We conclude that LTV systems will provide accurate velocity estimates under a wide variety of practical conditions.

© 1997 Optical Society of America

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References

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  1. D. H. Thompson, “A tracer-particle fluid velocity meter incorporating a laser,” J. Phys. E 1, 929–932 (1968).
    [Crossref]
  2. H. T. Yura, S. G. Hanson, “Laser time-of-flight velocimetry: analytical solution to the optical system based on ABCD matrices,” J. Opt. Soc. Am. A 10, 1918–1924 (1993).
    [Crossref]
  3. H. T. Yura, S. G. Hanson, L. Lading, “Comparison between the time-of-flight velocimeter and the laser Doppler velocimeter for measurements on solid surfaces,” in Optical Velocimetry, M. Pluta, J. K. Jabczynski, eds., Proc. SPIE2729, 91–102 (1995).
    [Crossref]
  4. H. T. Yura, S. G. Hanson, “Optical beam propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [Crossref]
  5. H. T. Yura, S. G. Hanson, L. Lading, “Laser Doppler velocimetry: analytical solution to the optical system including the effects of partial coherence of the target,” J. Opt. Soc. Am. A 12, 2040–2047 (1995).
    [Crossref]
  6. B. Saleh, Photoelectron Statistics, Vol. 6 of Springer Series in Optical Sciences (Springer-Verlag, Berlin1978), pp. 152–156.
  7. In the following we omit the contribution to the cross covariance that results from the cross product between the two terms in expression (22), as indicated in Eq. (10). This is because the width of the cross term is primarily determined by the smallest scale. In addition, the corresponding contribution to the magnitude of the cross covariance is small compared with the corresponding contribution of the direct component resulting from the smallest scale. Therefore we expect a negligible impact deriving from the cross term. This theoretical argument is substantiated by the experimental results presented here.
  8. H. T. Yura, S. G. Hanson, “Effects of receiver optics contamination on the performance of laser velocimeter systems,” J. Opt. Soc. Am. A 13, 1891–1902 (1996).
    [Crossref]

1996 (1)

1995 (1)

1993 (1)

1987 (1)

1968 (1)

D. H. Thompson, “A tracer-particle fluid velocity meter incorporating a laser,” J. Phys. E 1, 929–932 (1968).
[Crossref]

Hanson, S. G.

Lading, L.

H. T. Yura, S. G. Hanson, L. Lading, “Laser Doppler velocimetry: analytical solution to the optical system including the effects of partial coherence of the target,” J. Opt. Soc. Am. A 12, 2040–2047 (1995).
[Crossref]

H. T. Yura, S. G. Hanson, L. Lading, “Comparison between the time-of-flight velocimeter and the laser Doppler velocimeter for measurements on solid surfaces,” in Optical Velocimetry, M. Pluta, J. K. Jabczynski, eds., Proc. SPIE2729, 91–102 (1995).
[Crossref]

Saleh, B.

B. Saleh, Photoelectron Statistics, Vol. 6 of Springer Series in Optical Sciences (Springer-Verlag, Berlin1978), pp. 152–156.

Thompson, D. H.

D. H. Thompson, “A tracer-particle fluid velocity meter incorporating a laser,” J. Phys. E 1, 929–932 (1968).
[Crossref]

Yura, H. T.

J. Opt. Soc. Am. A (4)

J. Phys. E (1)

D. H. Thompson, “A tracer-particle fluid velocity meter incorporating a laser,” J. Phys. E 1, 929–932 (1968).
[Crossref]

Other (3)

H. T. Yura, S. G. Hanson, L. Lading, “Comparison between the time-of-flight velocimeter and the laser Doppler velocimeter for measurements on solid surfaces,” in Optical Velocimetry, M. Pluta, J. K. Jabczynski, eds., Proc. SPIE2729, 91–102 (1995).
[Crossref]

B. Saleh, Photoelectron Statistics, Vol. 6 of Springer Series in Optical Sciences (Springer-Verlag, Berlin1978), pp. 152–156.

In the following we omit the contribution to the cross covariance that results from the cross product between the two terms in expression (22), as indicated in Eq. (10). This is because the width of the cross term is primarily determined by the smallest scale. In addition, the corresponding contribution to the magnitude of the cross covariance is small compared with the corresponding contribution of the direct component resulting from the smallest scale. Therefore we expect a negligible impact deriving from the cross term. This theoretical argument is substantiated by the experimental results presented here.

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Figures (25)

Fig. 1
Fig. 1

Clean imaging system (receiver part of an LTV) with separate apertures in the object and Fourier planes.

Fig. 2
Fig. 2

Model for the one-scale surface having a height variation of scale ρ1 and a rms roughness σ1.

Fig. 3
Fig. 3

Model for the semirough reflection surface having reflective structures of size ρη, a height variation of scale ρξ, and a rms roughness σrms.

Fig. 4
Fig. 4

Model for the two-scale surface with elevation distributions σ1 and σ2 and their corresponding correlation lengths ρ1 and ρ2, respectively.

Fig. 5
Fig. 5

Schematic diagram of the optical setup for LTV measurements.

Fig. 6
Fig. 6

Effect of fully and partially developed speckle on the relative width of the cross covariance.

Fig. 7
Fig. 7

Normalized cross covariance obtained from a one-scale surface plotted versus the time delay.

Fig. 8
Fig. 8

Normalized cross covariances obtained from a reflection surface for (a) a single-mode and (b) a multimode system (N s ≈ 22). Parameters: σrms = 0.55 µm, ρξ = 20 µm, and ρη = 48 µm.

Fig. 9
Fig. 9

Normalized cross covariances obtained from a two-scale surface for (a) a single-mode and (b) a multimode system (N s ≈ 22). Parameters: σ1 = 0.056 µm, σ2 = 0.450 µm, ρ1 = 10 µm, and ρ2 = 219 µm.

Fig. 10
Fig. 10

Correlation geometry under velocity misalignment.

Fig. 11
Fig. 11

Effect of velocity misalignment on the decorrelation and bias (velocity-increase) errors for increasing angles of misalignment. The solid curves represent theoretical calculations, whereas the symbols represent measurements.

Fig. 12
Fig. 12

Cross-talk measurements for a relative spot size of 0.5.

Fig. 13
Fig. 13

Calculated bias errors generated by cross-talk effects for various relative spot sizes.

Fig. 14
Fig. 14

Comparison of the exact and approximate phase correlation functions for the one-scale model for s 1 = 0.5.

Fig. 15
Fig. 15

Comparison of the exact and approximate phase correlation functions for the one-scale model for s 1 = 1.

Fig. 16
Fig. 16

Comparison of the exact and approximate phase correlation functions for the one-scale model for s 1 = 5.

Fig. 17
Fig. 17

Comparison of the exact and approximate phase correlation functions for the reflection model for s 1 = 1 and ρξη = 0.2.

Fig. 18
Fig. 18

Comparison of the exact and approximate phase correlation functions for the one-scale model for s 1 = 0.5 and ρξη = 0.2.

Fig. 19
Fig. 19

Comparison of the exact and approximate phase correlation functions for the one-scale model for s 1 = 0.1 and ρξη = 0.2.

Fig. 20
Fig. 20

Comparison of the exact and approximate phase correlation functions for the two-scale model in the first regime (s 1 2 ≤ 1 and s 2 2 > 1) for s 1 = 1, s 2 = 2, and ρ12 = 0.2.

Fig. 21
Fig. 21

Comparison of the exact and approximate phase correlation functions for the two-scale model in the first regime (s 1 2 ≤ 1 and s 2 2 > 1) for s 1 = 0.5, s 2 = 4, and ρ12 = 0.5.

Fig. 22
Fig. 22

Comparison of the exact and approximate phase correlation functions for the two-scale model in the first regime (s 1 2 ≤ 1 and s 2 2 > 1) for s 1 = 0.1, s 2 = 4, and ρ12 = 0.5.

Fig. 23
Fig. 23

Comparison of the exact and approximate phase correlation functions for the two-scale model in the second regime (s 1 2 > 1 and s 2 2 > 1) for s 1 = 2, s 2 = 2, and ρ12 = 0.1.

Fig. 24
Fig. 24

Comparison of the exact and approximate phase correlation functions for the two-scale model in the second regime (s 1 2 > 1 and s 2 2 > 1) for s 1 = 5, s 2 = 4, and ρ12 = 0.5.

Fig. 25
Fig. 25

Comparison of the exact and approximate phase correlation functions for the two-scale model in the second regime (s 1 2 > 1 and s 2 2 > 1) for s 1 = 5, s 2 = 4, and ρ12 = 0.9.

Equations (47)

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Ciτ=i1ti2t+τ-i1ti2t+τ,
ijt=α  dpWjpIjp, t, j=1, 2,
Ijp,t= drUAjr, tGr, p2, j=1, 2,
Gr, p=-ik2πBexp-ik2BDp2-2r·p+Ar2,
Gr, p=-ik2πBexp-p/m-r2ω2,
UAjr, t=Ijr expiψr, t,
Ijr=2Psπrs2exp-2r±d2rs2, j=1, 2,
ψr, t+τ=ψr-vτ, t.
Ciτ= dp1  dp2Kp1, p2; τ,
Kp1, p2; τ=a  drUA1r, tUA2*rτ, t+τ×Gr, p1G*rτ, p22,
rτ=r+vτ
Ciτ=i2Nsexp-2d-vτ2rs2,
Ns=1+kσrs2f12.
Cψ1r, Δr, t, τ ψ r, tψ*r+Δr, t+τ=η2exp-2ikξr, t-ξr+Δr, t+τ,
Cψ1Δr, τ=exp-4k2σ121-exp-Δr-vτ2ρ12,
cs=s21-exp-s2,
Cψ1Δr, τĈψ1Δr, τ=1-exp-s2exp-Δr-vτ2ρ12cs+exp-s2.
C1τ=1-exp-s22×κ1+NsNs1+κ1exp-2d-ντ2rs21+1/κ1,
κ1=rs2cs2ρ12
CψRer, Δr, t, τ ψ r, tψ*r+Δr, t+τ=ηr, tηr+Δr, t+τ×exp-2ikξr, t-ξr+Δr, t+τ,
CψReΔr, τ=exp-Δr-vτ2/ρη2exp-s2×1-exp-Δr-vτ2/ρξ2,
CψReΔr, τĈψReΔr, τ=exp-s2exp-csΔr-vτ2ρη2+1-exp-s2exp-csΔr-vτ2ρξ2,
CReτ=exp-2s2κη+NsNs1+κηexp-Δr-vτ2rs21+1/κη+1-exp-s22κξ+NsNs1+κξ×exp-Δr-vτ2rs21+1/κξ,
κη=rs22ρη2,  κξ=rs2cs2ρξ2.
Cψ2r, Δr, t, τψ r, t ψ*r+Δr, t+τ=exp-2ikξr, t-ξr+Δr, t+τ,
Cψ2Δr, τ=exp-s21-σ12σrms2exp-Δr-vτ2ρ12-σ22σrms2exp-Δr-vτ2ρ22=exp-s121-exp-Δr-vτ2ρ12×exp-s221-exp-Δr-vτ2ρ22,
Cψ2Δr, τĈψ2Δr, τ=1-exp-s2-exp-s12×exp-Δr-vτ2ρ12 cs1+exp-s12×exp-Δr-vτ2ρ22 cs2+exp-s2,
Cψ2Δr, τĈψ2Δr, τ=exp-Δr-vτ2s1ρ12+s2ρ22,
C2scτ=1-exp-s2-exp-s122κ1+NsNs1+κ1×exp-2d-ντ2rs21+1/κ1+exp-2s12κ2+NsNs1+κ2×exp-2d-ντ2rs21+1/κ2,
κ1=rs2cs12ρ12,  κ2=rs2cs22ρ22.
C2scτ=κ+NsNs1+κexp-2d-ντ2rs21+1/κ,
κ=rs212s12ρ12+s22ρ22.
Cψ2Δr, τexp-s121-exp-Δr-vτ2ρ12×exp-s22Δr-vτ2ρ22,
Δττ0FWHM=ln2 rsd.
Δττ0FWHM=ln 2 rs21+1/κ1d,
Ciτ;νy=Df exp-νxτ-2drx2+νyτry2,
Df=exp-4d2νy2ry2νx2+rx2νy2,
tanθ=νyνx.
τest=2dνxνx2+νy2rx2ry2.
τest=2dνxνx2+νy2=2d cos2 θνx.
=νy2νx2.
Df=exp-2dsinθrs2.
i1t=α dp1-γI1p,t+γI2p,t,
i2t=α dp1-γI2p,t+γI1p,t,
Cixτ=1-γ2Ciτ+2γ1-γC1τ+γ2C2τ,
C1τ=i2Nsexp-ντ2rs2,
C2τ=i2Nsexp-2d+ντ2rs2.

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