Abstract

In the theory of atmospheric turbulence, the strength of the spatial variations of the index of refraction n is proportional to a parameter known as the atmospheric-structure constant. The atmospheric-structure constant is denoted Cn2(z) and is a function of position along the optical path z. The characteristics of the temporal variations of the index of refraction are related to both Cn2(z) and to the transverse wind velocity V(z). Current optical techniques for remotely sensing Cn2(z) and V(z) rely primarily on the spatial or temporal cross-correlation properties of the intensity of the optical field. In the remote-sensing technique proposed here, we exploit the correlation properties of the wave-front slope measured from two point sources to obtain profiles of Cn2(z) and V(z). The two sources are arranged to give crossed optical paths. The geometry of the crossed paths and the characteristics of the wave-front slope sensor determine the achievable resolution. The signal-to-noise ratio calculations indicate the need for multiple measurements to obtain useful estimates of the desired quantities.

© 1994 Optical Society of America

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References

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  1. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 11, p. 490.
  2. F. Merkle, “Adaptive optics,” Phys. World 66, 33–38 (1991).
  3. D. L. Fried, “Remote sensing of the optical strength of atmospheric turbulence and wind velocity,” Proc. IEEE 57, 415–420 (1969).
    [CrossRef]
  4. R. R. Beland, J. Krause-Polstorff, “Lidar measurement of optical turbulence: theory of crossed path technique,” tech rep. PL-TR-91-2139 (Phillips Laboratory, Directorate of Geophysics, Hanscom Air Force Base, Mass., July1991).
  5. R. S. Lawrence, G. R. Ochs, S. F. Clifford, “Use of scintillations to measure average wind across a light beam,” Appl. Opt. 11, 239–243 (1972).
    [CrossRef] [PubMed]
  6. T.-I. Wang, S. Clifford, G. Ochs, “Wind and refractive turbulence sensing using crossed laser beams,” Appl. Opt. 13, 2602–2608 (1974).
    [CrossRef] [PubMed]
  7. B. M. Welsh, “Sensing refractive-turbulence profiles using wave-front phase measurements from multiple reference sources,” Appl. Opt. 31, 7283–7291 (1992).
    [CrossRef] [PubMed]
  8. E. P. Wallner, “Optical wave-front correction using slope measurements,” J. Opt. Soc. Am. 12, 1771–1776 (1983).
    [CrossRef]
  9. R. Lutomirski, R. Buser, “Mutual coherence function of a finite optical beam and application to coherent detection,” Appl. Opt. 12, 2159–2160 (1973).
    [CrossRef]
  10. S. C. Koeffler, “Remote sensing of turbulence and transverse atmospheric wind profiles using optical reference sources,” M.S. thesis AFIT/GE/ENG/92-D-22 (School of Engineering, Air Force Institute of Technology Air University, Wright-Patterson Air Force Base, Ohio, December1992).
  11. G. Taylor, “Statistical theory of turbulence,” Proc. R. Soc. London 151, 421–478 (1935).
    [CrossRef]
  12. A. N. Kolmogoroff, “The local structure of turbulence in incompressible viscous fluids for very large reynolds numbers,” in Turbulence: Classic Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Interscience, New York, 1961), pp. 151–155.
  13. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8, p. 426.

1992 (1)

1991 (1)

F. Merkle, “Adaptive optics,” Phys. World 66, 33–38 (1991).

1983 (1)

E. P. Wallner, “Optical wave-front correction using slope measurements,” J. Opt. Soc. Am. 12, 1771–1776 (1983).
[CrossRef]

1974 (1)

1973 (1)

R. Lutomirski, R. Buser, “Mutual coherence function of a finite optical beam and application to coherent detection,” Appl. Opt. 12, 2159–2160 (1973).
[CrossRef]

1972 (1)

1969 (1)

D. L. Fried, “Remote sensing of the optical strength of atmospheric turbulence and wind velocity,” Proc. IEEE 57, 415–420 (1969).
[CrossRef]

1935 (1)

G. Taylor, “Statistical theory of turbulence,” Proc. R. Soc. London 151, 421–478 (1935).
[CrossRef]

Beland, R. R.

R. R. Beland, J. Krause-Polstorff, “Lidar measurement of optical turbulence: theory of crossed path technique,” tech rep. PL-TR-91-2139 (Phillips Laboratory, Directorate of Geophysics, Hanscom Air Force Base, Mass., July1991).

Buser, R.

R. Lutomirski, R. Buser, “Mutual coherence function of a finite optical beam and application to coherent detection,” Appl. Opt. 12, 2159–2160 (1973).
[CrossRef]

Clifford, S.

Clifford, S. F.

Fried, D. L.

D. L. Fried, “Remote sensing of the optical strength of atmospheric turbulence and wind velocity,” Proc. IEEE 57, 415–420 (1969).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 11, p. 490.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8, p. 426.

Koeffler, S. C.

S. C. Koeffler, “Remote sensing of turbulence and transverse atmospheric wind profiles using optical reference sources,” M.S. thesis AFIT/GE/ENG/92-D-22 (School of Engineering, Air Force Institute of Technology Air University, Wright-Patterson Air Force Base, Ohio, December1992).

Kolmogoroff, A. N.

A. N. Kolmogoroff, “The local structure of turbulence in incompressible viscous fluids for very large reynolds numbers,” in Turbulence: Classic Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Interscience, New York, 1961), pp. 151–155.

Krause-Polstorff, J.

R. R. Beland, J. Krause-Polstorff, “Lidar measurement of optical turbulence: theory of crossed path technique,” tech rep. PL-TR-91-2139 (Phillips Laboratory, Directorate of Geophysics, Hanscom Air Force Base, Mass., July1991).

Lawrence, R. S.

Lutomirski, R.

R. Lutomirski, R. Buser, “Mutual coherence function of a finite optical beam and application to coherent detection,” Appl. Opt. 12, 2159–2160 (1973).
[CrossRef]

Merkle, F.

F. Merkle, “Adaptive optics,” Phys. World 66, 33–38 (1991).

Ochs, G.

Ochs, G. R.

Taylor, G.

G. Taylor, “Statistical theory of turbulence,” Proc. R. Soc. London 151, 421–478 (1935).
[CrossRef]

Wallner, E. P.

E. P. Wallner, “Optical wave-front correction using slope measurements,” J. Opt. Soc. Am. 12, 1771–1776 (1983).
[CrossRef]

Wang, T.-I.

Welsh, B. M.

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

E. P. Wallner, “Optical wave-front correction using slope measurements,” J. Opt. Soc. Am. 12, 1771–1776 (1983).
[CrossRef]

Phys. World (1)

F. Merkle, “Adaptive optics,” Phys. World 66, 33–38 (1991).

Proc. IEEE (1)

D. L. Fried, “Remote sensing of the optical strength of atmospheric turbulence and wind velocity,” Proc. IEEE 57, 415–420 (1969).
[CrossRef]

Proc. R. Soc. London (1)

G. Taylor, “Statistical theory of turbulence,” Proc. R. Soc. London 151, 421–478 (1935).
[CrossRef]

Other (5)

A. N. Kolmogoroff, “The local structure of turbulence in incompressible viscous fluids for very large reynolds numbers,” in Turbulence: Classic Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Interscience, New York, 1961), pp. 151–155.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8, p. 426.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 11, p. 490.

R. R. Beland, J. Krause-Polstorff, “Lidar measurement of optical turbulence: theory of crossed path technique,” tech rep. PL-TR-91-2139 (Phillips Laboratory, Directorate of Geophysics, Hanscom Air Force Base, Mass., July1991).

S. C. Koeffler, “Remote sensing of turbulence and transverse atmospheric wind profiles using optical reference sources,” M.S. thesis AFIT/GE/ENG/92-D-22 (School of Engineering, Air Force Institute of Technology Air University, Wright-Patterson Air Force Base, Ohio, December1992).

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Figures (9)

Fig. 1
Fig. 1

Measurement geometry.

Fig. 2
Fig. 2

Path weighting function wC(z) for L1/L2 = 0.2.

Fig. 3
Fig. 3

Path weighting function wC(z) for L1/L2 = 0.5.

Fig. 4
Fig. 4

Path weighting function wC(z) for L1/L2 = 0.8.

Fig. 5
Fig. 5

RMS width of wC(z).

Fig. 6
Fig. 6

Measurement geometry showing aperture and source offset in the, y direction.

Fig. 7
Fig. 7

y-Directed component of the wind-path weighting function wV(z) for L1/L2 = 0.5 and Δpx = Δx = 10L2.

Fig. 8
Fig. 8

y-Directed component of the wind-path weighting function wV(z) for L1/L2 = 0.5 and Δpy, = Δy = 0.25L2.

Fig. 9
Fig. 9

Single measurement SNR versus zs/zrms for L1/L2 = 0.5.

Equations (26)

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C s = d z C n 2 ( z ) w C ( z ) ,
C s = d z C n 2 ( z ) V ( z ) w V ( z ) .
C s = ( s 1 s 1 ) ( s 2 s 2 ) .
C s = d z C n 2 ( z ) w C ( z ) ,
s n = d 2 x ( W n s ( x ) d ̂ ) ϕ n ( x ) + α n .
ϕ 1 ( x ) ϕ 2 ( x ) = 1 2 D 12 ( x , x ) + 1 2 ϕ 1 2 ( x ) + 1 2 ϕ 2 2 ( x ) ,
D 12 ( x , x ) = [ ϕ 1 ( x ) ϕ 2 ( x ) ] 2 .
D 12 ( x x ) = 8 π 2 k 2 d z d K Φ n ( K , 0 , z ) × { 1 J 0 [ K | Δ p ( z z s ) + ( x x ) ( 1 z z s ) | ] } ,
w C ( z ) = 1.46 k 2 d 2 x d 2 x × | Δ p ( z z s ) + ( x x ) ( 1 z z s ) | 5 / 3 × [ W L 1 s ( x ) W L 1 s ( x Δ x ) + W L 2 s ( x ) W L 2 s ( x Δ x ) 2 W L 2 s ( x ) W L 1 s ( x Δ x ) ] ,
C s ( 0 ) = [ s 1 ( t ) s 1 ( t ) ] [ s 2 ( t + τ ) s 2 ( t + τ ) ] τ | τ = 0 = d z C n 2 ( z ) V ( z ) w V ( z ) ,
w V ( z ) = 2.43 k 2 d 2 x d 2 x × | Δ p z z s + ( x x ) ( 1 z z s ) | 2 / 3 × [ W L 1 s ( x ) W L 1 s ( x Δ x ) + W L 2 s ( x ) W L 2 s ( x Δ x ) 2 W L 2 s ( x ) W L 1 s ( x Δ x ) ] r ̂ ,
SNR = | C s | ( C s 2 C s 2 ) 1 / 2 ,
C s = ( s 1 s 1 ) ( s 2 s 2 ) ,
C s = d 2 x d 2 x ϕ 1 ( x ) ϕ 2 ( x ) × [ W L 1 s ( x ) W L 1 s ( x Δ x ) + W L 2 s ( x ) W L 2 s ( x Δ x ) W L 2 s ( x ) W L 1 s ( x Δ x ) W L 1 s ( x ) W L 2 s ( x Δ x ) ] + α 1 α 2 + α 1 α 2 α 1 α 2 α 1 α 2 ,
C s = d 2 x d 2 x ϕ 1 ( x ) ϕ 2 ( x ) × [ W L 1 s ( x ) W L 1 s ( x Δ x ) + W L 2 s ( x ) W L 2 s ( x Δ x ) W L 2 s ( x ) W L 1 s ( x Δ x ) W L 1 s ( x ) W L 2 s ( x Δ x ) ] .
C s = d 2 x d 2 x ϕ 1 ( x ) ϕ 2 ( x ) × [ W L 1 s ( x ) W L 1 s ( x Δ x ) + W L 2 s ( x ) W L 2 s ( x Δ x ) 2 W L 2 s ( x ) W L 1 s ( x Δ x ) ] .
C s = 1 2 d 2 x d 2 x D 12 ( x x ) × [ W L 1 s ( x ) W L 1 s ( x Δ x ) + W L 2 s ( x ) W L 2 s ( x Δ x ) 2 W L 2 s ( x ) W L 1 s ( x Δ x ) ] .
C s = 4 π 2 k 2 d 2 x d 2 x d z d K Φ n ( K , 0 , z ) × { 1 J 0 [ K | Δ p ( z z s ) + ( x x ) ( 1 z z s ) | ] } × [ W L 1 s ( x ) W L 1 s ( x Δ x ) + W L 2 s ( x ) W L 2 s ( x Δ x ) 2 W L 2 s ( x ) W L 1 s ( x Δ x ) ] ,
Φ n ( K , K z , z ) = Φ n ( K , K z ) C n 2 ( z ) .
Φ n ( K , K z ) = 0.033 K 11 / 3 ,
C s = 1.46 k 2 d 2 x d 2 x d z C n 2 ( z ) × | Δ p ( z z s ) + ( x x ) ( 1 z z s ) | 5 / 3 × [ W L 1 s ( x ) W L 1 s ( x Δ x ) + W L 2 s ( x ) W L 2 s ( x Δ x ) 2 W L 2 s ( x ) W L 1 s ( x Δ x ) ] ,
w C ( z ) = 1.46 k 2 d 2 x d 2 x × | Δ p ( z z s ) + ( x x ) ( 1 z z s ) | 5 / 3 × [ W L 1 s ( x ) W L 1 s ( x Δ x ) + W L 2 s ( x ) W L 2 s ( x Δ x ) 2 W L 2 s ( x ) W L 1 s ( x Δ x ) ] .
C s ( τ ) = d 2 x d 2 x ϕ 1 ( x , t ) ϕ 2 ( x , t + τ ) × [ W L 1 s ( x ) W L 1 s ( x Δ x ) + W L 2 s ( x ) W L 2 s ( x Δ x ) 2 W L 2 s ( x ) W L 1 s ( x Δ x ) ] .
C s ( τ ) = 1.46 k 2 d 2 x d 2 x d z C n 2 ( z ) × | Δ p ( z z s ) + ( x x ) ( 1 z z s ) V ( z ) τ | 5 / 3 × [ W L 1 s ( x ) W L 1 s ( x Δ x ) + W L 2 s ( x ) W L 2 s ( x Δ x ) 2 W L 2 s ( x ) W L 1 s ( x Δ x ) ] .
C s ( τ ) τ | τ = 0 = 2.43 k 2 d 2 x d 2 x d z C n 2 ( z ) × | Δ p ( z z s ) + ( x x ) ( 1 z z s ) | 1 / 3 × V ( z ) [ Δ p ( z z s ) + ( x x ) ( 1 z z s ) ] × [ W L 1 s ( x ) W L 1 s ( x Δ x ) + W L 2 s ( x ) W L 2 s ( x Δ x ) 2 W L 2 s ( x ) W L 1 s ( x Δ x ) ] .
w v ( z ) = 2.43 k 2 d 2 x d 2 x × | Δ p ( z z s ) + ( x x ) ( 1 z z s ) | 2 / 3 × [ W L 1 s ( x ) W L 1 s ( x Δ x ) + W L 2 s ( x ) W L 2 s ( x Δ x ) 2 W L 2 s ( x ) W L 1 s ( x Δ x ) ] r ̂ ,

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