Abstract

Beam misalignments and bulk fluid velocities can influence the time history and intensity of laser-induced thermal acoustics (LITA) signals. A closed-form analytic expression for LITA signals incorporating these effects is derived, allowing the magnitude of beam misalignment and velocity to be inferred from the signal shape. It is demonstrated how instantaneous, nonintrusive, and remote measurement of sound speed and velocity (Mach number) can be inferred simultaneously from homodyne-detected LITA signals. The effects of different forms of beam misalignment are explored experimentally and compared with theory, with good agreement, allowing the amount of misalignment to be measured from the LITA signal. This capability could be used to correct experimental misalignments and account for the effects of misalignment in other LITA measurements. It is shown that small beam misalignments have no influence on the accuracy or repeatability of sound speed measurements with LITA.

© 1999 Optical Society of America

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References

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  1. T. J. Butenhoff, “Measurement of the thermal-diffusivity and speed of sound of hydrothermal solutions via the laser-induced grating technique,” Int. J. Thermophys. 16(1), 1–9 (1995).
    [CrossRef]
  2. B. Hemmerling, D. N. Kozlov, “Generation and temporally resolved detection of laser-induced gratings by a single, pulsed Nd:YAG laser,” Appl. Opt. 38, 1001–1007 (1999).
    [CrossRef]
  3. M. S. Brown, W. L. Roberts, “Single-point thermometry in high-pressure, sooting, premixed combustion environments,” J. Propul. Power 15, 119–127 (1999).
    [CrossRef]
  4. D. J. W. Walker, R. B. Williams, P. Ewart, “Thermal grating velocimetry,” Opt. Lett. 23, 1316–1318 (1998).
    [CrossRef]
  5. E. B. Cummings, I. A. Leyva, H. G. Hornung, “Laser-induced thermal acoustics (LITA) signals from finite beams,” Appl. Opt. 34, 3290–3302 (1995).
    [CrossRef] [PubMed]
  6. H. J. Eichler, P. Günther, D. W. Phol, Laser-Induced Dynamic Gratings (Springer-Verlag, New York, 1986).
    [CrossRef]
  7. D. B. Brayton, “Small particle signal characteristics of a dual-scatter laser velocimeter,” Appl. Opt. 13, 2346–2351 (1974).
    [CrossRef] [PubMed]
  8. A. E. Siegman, “Bragg diffraction of a Gaussian beam by a crossed-Gaussian volume grating,” J. Opt. Soc. Am. 67, 545–550 (1977).
    [CrossRef]
  9. B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).
  10. W. H. Press, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

1999

B. Hemmerling, D. N. Kozlov, “Generation and temporally resolved detection of laser-induced gratings by a single, pulsed Nd:YAG laser,” Appl. Opt. 38, 1001–1007 (1999).
[CrossRef]

M. S. Brown, W. L. Roberts, “Single-point thermometry in high-pressure, sooting, premixed combustion environments,” J. Propul. Power 15, 119–127 (1999).
[CrossRef]

1998

1995

E. B. Cummings, I. A. Leyva, H. G. Hornung, “Laser-induced thermal acoustics (LITA) signals from finite beams,” Appl. Opt. 34, 3290–3302 (1995).
[CrossRef] [PubMed]

T. J. Butenhoff, “Measurement of the thermal-diffusivity and speed of sound of hydrothermal solutions via the laser-induced grating technique,” Int. J. Thermophys. 16(1), 1–9 (1995).
[CrossRef]

1977

1974

Berne, B. J.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Brayton, D. B.

Brown, M. S.

M. S. Brown, W. L. Roberts, “Single-point thermometry in high-pressure, sooting, premixed combustion environments,” J. Propul. Power 15, 119–127 (1999).
[CrossRef]

Butenhoff, T. J.

T. J. Butenhoff, “Measurement of the thermal-diffusivity and speed of sound of hydrothermal solutions via the laser-induced grating technique,” Int. J. Thermophys. 16(1), 1–9 (1995).
[CrossRef]

Cummings, E. B.

Eichler, H. J.

H. J. Eichler, P. Günther, D. W. Phol, Laser-Induced Dynamic Gratings (Springer-Verlag, New York, 1986).
[CrossRef]

Ewart, P.

Günther, P.

H. J. Eichler, P. Günther, D. W. Phol, Laser-Induced Dynamic Gratings (Springer-Verlag, New York, 1986).
[CrossRef]

Hemmerling, B.

Hornung, H. G.

Kozlov, D. N.

Leyva, I. A.

Pecora, R.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Phol, D. W.

H. J. Eichler, P. Günther, D. W. Phol, Laser-Induced Dynamic Gratings (Springer-Verlag, New York, 1986).
[CrossRef]

Press, W. H.

W. H. Press, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

Roberts, W. L.

M. S. Brown, W. L. Roberts, “Single-point thermometry in high-pressure, sooting, premixed combustion environments,” J. Propul. Power 15, 119–127 (1999).
[CrossRef]

Siegman, A. E.

Walker, D. J. W.

Williams, R. B.

Appl. Opt.

Int. J. Thermophys.

T. J. Butenhoff, “Measurement of the thermal-diffusivity and speed of sound of hydrothermal solutions via the laser-induced grating technique,” Int. J. Thermophys. 16(1), 1–9 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Propul. Power

M. S. Brown, W. L. Roberts, “Single-point thermometry in high-pressure, sooting, premixed combustion environments,” J. Propul. Power 15, 119–127 (1999).
[CrossRef]

Opt. Lett.

Other

H. J. Eichler, P. Günther, D. W. Phol, Laser-Induced Dynamic Gratings (Springer-Verlag, New York, 1986).
[CrossRef]

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

W. H. Press, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

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Figures (10)

Fig. 1
Fig. 1

Frame of reference for LITA analysis. The beams’ diameters at their foci are ω and σ for the driver beams and interrogation beam, respectively.

Fig. 2
Fig. 2

Theoretical LITA signals for atmospheric air from Eq. (17) with θ = 1.23°, ω = 370 µm, σ = 700 µm, misalignment η̅ = -2ω, and with a fluid flow in +y direction with different Mach numbers: (a) M = 0, (b) M = 0.25, (c) M = 0.5, (d) M = 0.75, (e) M = 1.0, (f) M = 1.5, (g) M = 2.0, and (h) M = 3.0.

Fig. 3
Fig. 3

Schematic diagram of experimental LITA setup.

Fig. 4
Fig. 4

Experimental and theoretical LITA signals of atmospheric air: σ = 700 µm, ω = 370 µm, θ = 1.23°, and (a) η̅ = -1100 µm, (b) η̅ = -900 µm, (c) η̅ = -700 µm, (d) η̅ = -600 µm, (e) η̅ = -500 µm, and (f) η̅ = 0 µm. The top trace in each graph is the fitted theoretical signal, shifted for clarity.

Fig. 5
Fig. 5

Experimental signal (top) and fitted theoretical signal (bottom) for η̅ = -1000 µm with ω = 370 µm, σ = 700 µm, and θ = 1.23° in atmospheric air.

Fig. 6
Fig. 6

Measured versus true misalignment η̅. The dashed curves have a slope of ±1. Traces shown in Figs. 4(a)4(f) correspond to data points labeled A–F. The experimental conditions are the same as in Fig. 4.

Fig. 7
Fig. 7

Normalized peak signal intensity versus η̅. Traces shown in Figs. 4(a)4(f) correspond to data points labeled A–F. The experimental conditions are the same as in Fig. 4.

Fig. 8
Fig. 8

Uncertainty and error of sound speed versus η̅. The sound speed for η̅ = 0 is taken as reference. Three different fitting strategies were used. In the strategy denoted by float, η̅ is a floating fitting parameter; in fixed 0, η̅ is fixed at zero; and in fixed corr, η̅ was held constant at the correct preset value. The experimental conditions are the same as in Fig. 4.

Fig. 9
Fig. 9

Intensity of overlap of density grating with source beam from the inverse Fourier transform of Eqs. (14) for atmospheric air and at z = 0, i.e., along the grating center line. In the upper portion, the time history as a function of y position is plotted—on the left-hand side for perfect beam alignment and on the right-hand side with η̅ = ω. The experimental parameters are σ = ω = 500 µm, λ d = 589 nm, and θ = 0.8°. White corresponds to high intensities, black to low intensities. In the lower portion, the resulting LITA signals are shown if the receiver only detects the range y < 0 or only y > 0.

Fig. 10
Fig. 10

Two traces from a sweep through values of η̅. Left trace, η̅ = -650 µm; right trace, η̅ = +650 µm. A small pinhole with a diameter of only 40 µm in the receiver unit blocked portions of the signal beam. Compare with theoretical results in Fig. 9.

Equations (35)

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Ed1=E1+E1*, Ed2=E2+E2*,
E1=t22πω2expikdeˆ1·r-r1-ifdt×exp-eˆ1r-r1ω2,
E2=t22πω2expikdeˆ2·r-r2-ifdt×exp-eˆ2r-r2ω2,
eˆ1=cos θ cos ϕ1, sin θ, sin ϕ1,
eˆ2=cos θ cos ϕ2, - sin θ, sin ϕ2
I=|Ed1+Ed2|2=Ed1+Ed2Ed1+Ed2*2E1*E2+E1E2*=EdIdPdt,
Id=2πω2cos2kd-ξ cos θ+y-η¯sin θ×exp-2X2x-ξ¯-XY η2-2Y2y-η¯-YX ξ2-2Z2z-ζ¯2+ζ2,
X=ωsin θ, Y=ωcos θ, Z=ω,ξ¯=x1+x2/2, η¯=y1+y2/2, ζ¯=z1+z2/2, ξ=x1-x2/2, η=y1-y2/2, ζ=z1-z2/2.
Id=2πω2exp-2Y2y-η¯2+η2-2Z2z-ζ¯2+ζ2cosqψy-η¯,
Idq=YZ2ω2exp-2Y2 η2-2Z2 ζ2-Z28 qz2+iη¯qy+iζ¯qz×exp-Y28qy-qψ2+exp-Y28qy+qψ2.
Hθq, t=HθP1ΦP1+HθP2ΦP2+HθTΦT+HθDΦD,
Heq, t=HeP1ΦP1+HeP2ΦP2+HeTΦT,
ΦP1,2q, t=exp-Γq2t±icsqt+ivtqy+iwtqz, ΦTq, t=exp-DTq2t+ivtqy+iwtqz, ΦDq, t=exp-γθ+γnθt-Dsq2t+ivtqy+iwtqz,
HθP1=1+iΔ-G1-GΠ-iΠ21+Δ-G21-GΠ2+Π2=HθP2*, HθT=-11+Δ-G21-ΔΠ, HθD=Π21-GΠ2+Π21-ΔΠ, HeP1=i 1-iγΔ-G1+iΔ-G21+Δ-G2=HeP2*, HeT=γ-1Δ1+Δ-G2,
Π=csq/γθ+γnθ+Dsq2,
ρq, tρ=-ω2IdPdtHθUθ+HeUe,
ΦP1,P2,T,Dd=IdqΦP1,P2,T,Dq, t,
ΦP1,2dr, t=2πω2exp-2Y2 η2-2Z2 ζ2exp-Γqψ2t×cosy-v±cstqψ×exp-2Y2y-η¯+v±cst2-2Z2z-ζ¯+wt2,
ΦTdr, t=2πω2exp-2Y2 η2-2Z2 ζ2exp-DTqψ2t×cosy-vtqψ×exp-2Y2y-η¯+vt2-2Z2z-ζ¯+wt2,
ΦDdr, t=2πω2exp-2Y2 η2-2Z2 ζ2exp-DSqψ2t-γθ+γnθtcosy-vtqψ×exp-2Y2y-η¯+vt2-2Z2z-ζ¯+wt2.
EsR, t; q=-ks24πRcosks·R-f0tμq, t,
μr, t=χr, t; f0E0r, t.
ES=E0+E0*,
E0=12expik0eˆ0·r-if0texp-eˆ0rσ2,
eˆ0=cos ψ, sin ψ, 0.
I0=2πσ2exp-yσy2-zσz2,
σy=σsin ψ, σz=σ.
ΦP1,P2,T,Td,0q, t=NΨP1,P2,T,Dq,tΣP1,P2,T,Dt,
N=2/πσω2Y2σy2Y2+2σy21/2Z2σz2Z2+2σz21/2×exp-2Y2 η2-2Z2 ζ2,
ΨP1,2=exp-14Y2σy2Y2+2σy2qy-qψ2+i 2σy2Y2+2σy2η¯+v±cstqy-qψ×exp-14Z2σz2Z2+2σz2 qz2+i 2σz2Z2+2σz2ζ¯+wtqz,ΨT=exp-14Y2σy2Y2+2σy2qy-qψ2+i 2σy2Y2+2σy2η¯+vtqy-qψ×exp-14Z2σz2Z2+2σz2 qz2+i 2σz2Z2+2σz2ζ¯+wtqz=ΨD,
ΣP1,2=exp-Γqψ2t-2η¯+v±cst2Y2+2σy2-2Z2+2σz2ζ¯+wt2+iqψv±cst, ΣT=exp-DTqψ2t-2η¯+vt2Y2+2σy2-2ζ¯+wt2Z2+2σz2+iqψvt, ΣD=exp-Dsqψ2t-γθ+γnθt-2η¯+vt2Y2+2σy2-2ζ¯+wt2Z2+2σz2+iqψvt.
Esq, R, tP0t=-ks2ω24πR χf0expiks·R-f0tPdtAP1ΦP1d,0+AP2ΦP2d,0+ATΦTd,0+ADΦDd,0,
AP1,2=UθHθP1,2+UeHeP1,2, AT=UθHθT+UeHeT, AD=UθHθD,
homP02t=k0ω416π2 cos2 ψ |χf0|2×AP1ΦP1d,0+AP1ΦP2d,0+ATΦTd,0+ADΦDd,0×AP1*ΦP1d,0*+AP1*ΦP2d,0*+ATΦTd,0*+ADΦDd,0*.
homP02t=k024π2 cos2 ψ |χf0|2 exp-2Y2 η2-2Z2 ζ2×exp-8σy2Y2Y2+2σy2cst22×P1+P2T*+D*+P1*+P2*T+D+exp-8σy2Y2Y2+2σy2cst2P1P2*+P1*P2+P1P1*+P2P2*+TT*+TD*+T*D+DD*,

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