Abstract

We describe a method of measuring spatiotemporal (ST) structure and covariance functions of the phase fluctuations in a collimated light beam propagated through a region of refractive index turbulence. The measurements are performed in a small wind tunnel, in which a turbulent temperature field is created using heated wires at the inlet of the test section. A collimated sheet of light is sent through the channel, and the phase fluctuations across the sheet are measured. The spatial phase structure function can be estimated from a series of images captured at an arbitrary frame rate by spatial phase unwrapping, whereas the ST structure function requires a time resolved measurement and a full three-dimensional unwrapping. The measured spatial phase structure function shows agreement with the Kolmogorov theory with a pronounced inertial subrange, which is taken as a validation of the method. Because of turbulent mixing in the boundary layers close to the walls of the channel, the flow will not obey the Taylor hypothesis of frozen turbulence. This can be clearly seen in the ST structure function calculated in a coordinate system that moves along with the bulk flow. At zero spatial separation, this function should always be zero according to the Taylor hypothesis, but due to the mixing effect there will be a growth in the structure function with increasing time difference depending on the rate of mixing.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  8. J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer-Verlag, 1978).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  19. H. Lycksam, M. Sjödahl, P. Gren, and J. Leblanc, “Wiener filtering of interferometry measurements through turbulent air using and exponential forgetting factor,” Appl. Opt. 47, 2971-2978 (2008).
    [CrossRef] [PubMed]

2008 (3)

2007 (1)

S. N. Vecherin, V. E. Ostashev, A. Ziemann, D. K. Wilson, K. Arnold, and M. Barth, “Tomographic reconstruction of atmospheric turbulence with the use of time-dependent stochastic inversion,” J. Acoust. Soc. Am. 122, 1416-1425(2007).
[CrossRef] [PubMed]

2006 (2)

S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119, 2579-2588 (2006).
[CrossRef]

S. Ozono, “Turbulence generated in active-grid mode using a multi-fan wind tunnel,” J. Wind Eng. Ind. Aerodyn. 94, 225-240 (2006).
[CrossRef]

2003 (1)

1998 (1)

1993 (1)

E. P. Magee and B. M. Welsh, “Characterization of laboratory generated turbulence by optical phase measurements,” Proc. SPIE 2005, 50-61(1993).
[CrossRef]

1991 (1)

H. Makita, “Realization of a large-scale turbulence field in a small wind tunnel,” Fluid Dyn. Res. 8, 53-64 (1991).
[CrossRef]

1982 (1)

1974 (1)

K. T. Knox and B. J. Thompson, “Recovery of images from astronomically degraded short exposure images,” Astrophys. J. 193L45-L48 (1974).
[CrossRef]

1970 (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85-87 (1970).

Arnold, K.

S. N. Vecherin, V. E. Ostashev, A. Ziemann, D. K. Wilson, K. Arnold, and M. Barth, “Tomographic reconstruction of atmospheric turbulence with the use of time-dependent stochastic inversion,” J. Acoust. Soc. Am. 122, 1416-1425(2007).
[CrossRef] [PubMed]

Barth, M.

S. N. Vecherin, V. E. Ostashev, A. Ziemann, D. K. Wilson, K. Arnold, and M. Barth, “Tomographic reconstruction of atmospheric turbulence with the use of time-dependent stochastic inversion,” J. Acoust. Soc. Am. 122, 1416-1425(2007).
[CrossRef] [PubMed]

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, 1960).

Corrsin, S.

S. Corrsin, “Local isotropy in turbulent shear flow,” NACA-RM-58B11, National Aeronautics and Space Administration, 1958.

Goedecke, G. H.

S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119, 2579-2588 (2006).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Gren, P.

Hibino, K.

Ina, H.

Knox, K. T.

K. T. Knox and B. J. Thompson, “Recovery of images from astronomically degraded short exposure images,” Astrophys. J. 193L45-L48 (1974).
[CrossRef]

Kobayashi, S.

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85-87 (1970).

Leblanc, J.

Li, X.

Lycksam, H.

Magee, E. P.

E. P. Magee and B. M. Welsh, “Characterization of laboratory generated turbulence by optical phase measurements,” Proc. SPIE 2005, 50-61(1993).
[CrossRef]

Makita, H.

H. Makita, “Realization of a large-scale turbulence field in a small wind tunnel,” Fluid Dyn. Res. 8, 53-64 (1991).
[CrossRef]

Ostashev, V. E.

S. N. Vecherin, V. E. Ostashev, and D. K. Wilson, “Three-dimensional acoustic travel-time tomography of the atmosphere,” Acta Acustica 94, 349-358 (2008).
[CrossRef]

S. N. Vecherin, V. E. Ostashev, A. Ziemann, D. K. Wilson, K. Arnold, and M. Barth, “Tomographic reconstruction of atmospheric turbulence with the use of time-dependent stochastic inversion,” J. Acoust. Soc. Am. 122, 1416-1425(2007).
[CrossRef] [PubMed]

S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119, 2579-2588 (2006).
[CrossRef]

Ozono, S.

S. Ozono, “Turbulence generated in active-grid mode using a multi-fan wind tunnel,” J. Wind Eng. Ind. Aerodyn. 94, 225-240 (2006).
[CrossRef]

Schedin, S.

Sjödahl, M.

Strohbehn, J. W.

J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer-Verlag, 1978).

Takeda, M.

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, 1961).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, 1971).

Thompson, B. J.

K. T. Knox and B. J. Thompson, “Recovery of images from astronomically degraded short exposure images,” Astrophys. J. 193L45-L48 (1974).
[CrossRef]

Vecherin, S. N.

S. N. Vecherin, V. E. Ostashev, and D. K. Wilson, “Three-dimensional acoustic travel-time tomography of the atmosphere,” Acta Acustica 94, 349-358 (2008).
[CrossRef]

S. N. Vecherin, V. E. Ostashev, A. Ziemann, D. K. Wilson, K. Arnold, and M. Barth, “Tomographic reconstruction of atmospheric turbulence with the use of time-dependent stochastic inversion,” J. Acoust. Soc. Am. 122, 1416-1425(2007).
[CrossRef] [PubMed]

S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119, 2579-2588 (2006).
[CrossRef]

Voronovich, A. G.

S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119, 2579-2588 (2006).
[CrossRef]

Welsh, B. M.

E. P. Magee and B. M. Welsh, “Characterization of laboratory generated turbulence by optical phase measurements,” Proc. SPIE 2005, 50-61(1993).
[CrossRef]

Wilson, D. K.

S. N. Vecherin, V. E. Ostashev, and D. K. Wilson, “Three-dimensional acoustic travel-time tomography of the atmosphere,” Acta Acustica 94, 349-358 (2008).
[CrossRef]

S. N. Vecherin, V. E. Ostashev, A. Ziemann, D. K. Wilson, K. Arnold, and M. Barth, “Tomographic reconstruction of atmospheric turbulence with the use of time-dependent stochastic inversion,” J. Acoust. Soc. Am. 122, 1416-1425(2007).
[CrossRef] [PubMed]

S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119, 2579-2588 (2006).
[CrossRef]

Yamauchi, M.

Ziemann, A.

S. N. Vecherin, V. E. Ostashev, A. Ziemann, D. K. Wilson, K. Arnold, and M. Barth, “Tomographic reconstruction of atmospheric turbulence with the use of time-dependent stochastic inversion,” J. Acoust. Soc. Am. 122, 1416-1425(2007).
[CrossRef] [PubMed]

Acta Acustica (1)

S. N. Vecherin, V. E. Ostashev, and D. K. Wilson, “Three-dimensional acoustic travel-time tomography of the atmosphere,” Acta Acustica 94, 349-358 (2008).
[CrossRef]

Appl. Opt. (4)

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85-87 (1970).

Astrophys. J. (1)

K. T. Knox and B. J. Thompson, “Recovery of images from astronomically degraded short exposure images,” Astrophys. J. 193L45-L48 (1974).
[CrossRef]

Fluid Dyn. Res. (1)

H. Makita, “Realization of a large-scale turbulence field in a small wind tunnel,” Fluid Dyn. Res. 8, 53-64 (1991).
[CrossRef]

J. Acoust. Soc. Am. (2)

S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119, 2579-2588 (2006).
[CrossRef]

S. N. Vecherin, V. E. Ostashev, A. Ziemann, D. K. Wilson, K. Arnold, and M. Barth, “Tomographic reconstruction of atmospheric turbulence with the use of time-dependent stochastic inversion,” J. Acoust. Soc. Am. 122, 1416-1425(2007).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Wind Eng. Ind. Aerodyn. (1)

S. Ozono, “Turbulence generated in active-grid mode using a multi-fan wind tunnel,” J. Wind Eng. Ind. Aerodyn. 94, 225-240 (2006).
[CrossRef]

Proc. SPIE (1)

E. P. Magee and B. M. Welsh, “Characterization of laboratory generated turbulence by optical phase measurements,” Proc. SPIE 2005, 50-61(1993).
[CrossRef]

Other (6)

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, 1961).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, 1971).

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, 1960).

J. W. Goodman, Statistical Optics (Wiley, 1985).

J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer-Verlag, 1978).

S. Corrsin, “Local isotropy in turbulent shear flow,” NACA-RM-58B11, National Aeronautics and Space Administration, 1958.

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Figures (8)

Fig. 1
Fig. 1

Sketch of the experimental setup: λ / 2 , half-wave plate; P, thin film polarizer; CL1 and CL2,  cylindrical lenses; L1, L2, and L3,  ordinary single lenses; D, diffuser plate; BS, beam splitter cube; and CMOS, high-speed camera.

Fig. 2
Fig. 2

Sketch of the wind tunnel used to produce the turbulent air: S, settling chamber; C, contraction cone; HW, heated canthal wires inside the test section—T; RS, rotatable slits; DI, air diffuser; and F, fan.

Fig. 3
Fig. 3

Light propagation geometry. The refractive index fluctuations in the turbulent region of size L are n ( x , y , z , t ) , and ϕ ( x , y , t ) are the phase fluctuations of the wave in the detector plane. ρ is the vector between two general points P 1 and P 2 in the detection plane with coordinates ( x 1 , y 1 ) and ( x 2 , y 2 ) .

Fig. 4
Fig. 4

Part of a measured phase volume where A, B, C, and D are phase elements at different points in the image.

Fig. 5
Fig. 5

Spatial phase structure function with a flow velocity of 8.2 m / s . The power generated in the heated wires was 600 W . Also shown are theoretical curves fitted from Eq. (2) with C n 2 = 3.8 × 10 11 m 2 / 3 and l 0 0.5 mm .

Fig. 6
Fig. 6

Spatial phase structure function with a flow velocity of 2.7 m / s . The power generated in the heated wires was 400 W . Also shown are theoretical curves fitted from Eq. (2) with C n 2 = 3.4 × 10 11 m 2 / 3 and l 0 0.5 mm .

Fig. 7
Fig. 7

(a) ST phase structure and (b) covariance function for a flow velocity of 4.6 m / s . The amount of power generated in the heated wires was 0.45 kW .

Fig. 8
Fig. 8

ST phase structure function at zero separation in a coordinate system moving along with the flow.

Equations (9)

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D ϕ S ( x 1 , y 1 , x 2 , y 2 ) = [ ϕ ( x 1 , y 1 , t ) ϕ ( x 2 , y 2 , t ) ] 2 ,
D ϕ S ( ρ ) = 2.91 · k 2 · C n 2 · L · ρ 5 / 3 l 0 ρ L 0 D ϕ S ( ρ ) = 3.44 · k 2 · C n 2 · L · l 0 1 / 3 · ρ 2 ρ l 0 .
D ϕ ST ( x 1 , y 1 , x 2 , y 2 , τ ) = [ ϕ ( x 1 , y 1 , t ) ϕ ( x 2 , y 2 , t + τ ) ] 2 .
D ϕ ST ( ρ , τ ) = D ϕ S ( | ρ V 0 · τ | ) ,
V 0 = ρ min τ .
C ϕ ( x 1 , y 1 , x 2 , y 2 , τ ) = ϕ ( x 1 , y 1 , t ) · ϕ ( x 2 , y 2 , t + τ ) .
C ϕ ( ρ , τ ) = 1 2 · [ D ϕ ( , ) D ϕ ( ρ , τ ) ] ,
W ϕ ( f ) = 1 v 0 · V ϕ ( f v 0 ) ,
f max v 0 l 0 .

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