Abstract

The theory of optical propagation through atmospheric turbulence demonstrates the sensitivity of such quantities as log-amplitude variance and covariance to strength of refractive turbulence and transverse wind. We exploit this sensitivity by using a crossed-path technique to derive path profiles of these quantities. The results are insensitive to changes in the spatial spectrum of the refractive-index variations. The path resolution is easily varied by changing the receiver and transmitter separations and is ultimately limited by signal-to-noise considerations. The experimental results for horizontal paths, described here, will ultimately be used to indicate the feasibility of profiling on vertical paths with passive sources.

© 1974 Optical Society of America

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References

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  1. R. S. Lawrence, G. R. Ochs, S. F. Clifford, Appl. Opt. 11, 239 (1972).
    [CrossRef] [PubMed]
  2. L. Shen, IEEE Trans. Antennas Propag. AP-18, 493 (1970).
    [CrossRef]
  3. A. Ishimaru, IEEE Trans. Antennas Propag. AP-20, 10 (1972).
    [CrossRef]
  4. P. A. Mandics, R. W. Lee, A. T. Waterman, Radio Sci. 8, 185 (1973).
    [CrossRef]
  5. P. A. Mandics, Ph.D. Dissertation, Stanford University, Stanford, Calif. (1971).
  6. J. C. Harp, Ph.D. Thesis, AFCRL-71-0451, SU-SEL-71-042, Scientific Report 1 (1971).
  7. A. G. Kjelaas, G. R. Ochs, J. Appl. Meteor., 13, 242 (1974).
    [CrossRef]
  8. J. Vernin, F. Roddier, J. Opt. Soc. Am. 63, 270 (1973).
    [CrossRef]
  9. R. W. Lee, J. C. Harp, Proc. IEEE 57, 375 (1969).
    [CrossRef]
  10. A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 299 (1941).
  11. S. Corrsin, J. Appl. Phys. 22, 469 (1951).
    [CrossRef]
  12. S. F. Clifford, J. Opt. Soc. Am. 61, 1285 (1971).
    [CrossRef]
  13. R. S. Lawrence, G. R. Ochs, S. F. Clifford, J. Opt. Soc. Am. 60, 826 (1970).
    [CrossRef]
  14. S. F. Clifford, G. R. Ochs, R. S. Lawrence, J. Opt. Soc. Am., 64, 148 (1974).
    [CrossRef]

1974 (2)

1973 (2)

J. Vernin, F. Roddier, J. Opt. Soc. Am. 63, 270 (1973).
[CrossRef]

P. A. Mandics, R. W. Lee, A. T. Waterman, Radio Sci. 8, 185 (1973).
[CrossRef]

1972 (2)

1971 (1)

1970 (2)

1969 (1)

R. W. Lee, J. C. Harp, Proc. IEEE 57, 375 (1969).
[CrossRef]

1951 (1)

S. Corrsin, J. Appl. Phys. 22, 469 (1951).
[CrossRef]

1941 (1)

A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 299 (1941).

Clifford, S. F.

Corrsin, S.

S. Corrsin, J. Appl. Phys. 22, 469 (1951).
[CrossRef]

Harp, J. C.

R. W. Lee, J. C. Harp, Proc. IEEE 57, 375 (1969).
[CrossRef]

J. C. Harp, Ph.D. Thesis, AFCRL-71-0451, SU-SEL-71-042, Scientific Report 1 (1971).

Ishimaru, A.

A. Ishimaru, IEEE Trans. Antennas Propag. AP-20, 10 (1972).
[CrossRef]

Kjelaas, A. G.

A. G. Kjelaas, G. R. Ochs, J. Appl. Meteor., 13, 242 (1974).
[CrossRef]

Kolmogorov, A. N.

A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 299 (1941).

Lawrence, R. S.

Lee, R. W.

P. A. Mandics, R. W. Lee, A. T. Waterman, Radio Sci. 8, 185 (1973).
[CrossRef]

R. W. Lee, J. C. Harp, Proc. IEEE 57, 375 (1969).
[CrossRef]

Mandics, P. A.

P. A. Mandics, R. W. Lee, A. T. Waterman, Radio Sci. 8, 185 (1973).
[CrossRef]

P. A. Mandics, Ph.D. Dissertation, Stanford University, Stanford, Calif. (1971).

Ochs, G. R.

Roddier, F.

Shen, L.

L. Shen, IEEE Trans. Antennas Propag. AP-18, 493 (1970).
[CrossRef]

Vernin, J.

Waterman, A. T.

P. A. Mandics, R. W. Lee, A. T. Waterman, Radio Sci. 8, 185 (1973).
[CrossRef]

Appl. Opt. (1)

Dokl. Akad. Nauk SSSR (1)

A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 299 (1941).

IEEE Trans. Antennas Propag. (2)

L. Shen, IEEE Trans. Antennas Propag. AP-18, 493 (1970).
[CrossRef]

A. Ishimaru, IEEE Trans. Antennas Propag. AP-20, 10 (1972).
[CrossRef]

J. Appl. Meteor. (1)

A. G. Kjelaas, G. R. Ochs, J. Appl. Meteor., 13, 242 (1974).
[CrossRef]

J. Appl. Phys. (1)

S. Corrsin, J. Appl. Phys. 22, 469 (1951).
[CrossRef]

J. Opt. Soc. Am. (4)

Proc. IEEE (1)

R. W. Lee, J. C. Harp, Proc. IEEE 57, 375 (1969).
[CrossRef]

Radio Sci. (1)

P. A. Mandics, R. W. Lee, A. T. Waterman, Radio Sci. 8, 185 (1973).
[CrossRef]

Other (2)

P. A. Mandics, Ph.D. Dissertation, Stanford University, Stanford, Calif. (1971).

J. C. Harp, Ph.D. Thesis, AFCRL-71-0451, SU-SEL-71-042, Scientific Report 1 (1971).

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Figures (11)

Fig. 1
Fig. 1

Schematic of the crossed-path geometry with two laser transmitters Ta and Tb illuminating detectors Ra and Rb, respectively.

Fig. 2
Fig. 2

Refractive-index structure parameter (Cn2) path-weighting functions for three different transmitter-receiver configurations. The transmitter and receiver separations are αt and α Fresnel zones, respectively, and β = 0.

Fig. 3
Fig. 3

Refractive-index structure parameter (Cn2) path-weighting functions showing the variability of the path resolution with increasing angle of intersection of the two optical paths. The transmitter and receiver separations are equal, α = αt and β = 0.

Fig. 4
Fig. 4

Refractive-index structure parameter (Cn2) path weighting functions for different assumed turbulence spectra. The receiver and transmitter separations α and αt are equal, and their displacements are in the same plane so that β = 0.

Fig. 5
Fig. 5

Wind weighting functions for various values of transmitter separation αt and receiver coordinates α and β.

Fig. 6
Fig. 6

Wind-weighting functions showing the variability of the path resolution with increasing angle of intersection of the two optical paths. The receiver coordinates are β = 0.3 Fresnel zones with variable α = αt.

Fig. 7
Fig. 7

Wind-weighting functions for different assumed turbulence spectra. The transmitter separation is αt = 1, and the receiver coordinates are α = 1 and β = 0.3. All quantities are in Fresnel zones.

Fig. 8
Fig. 8

Experimental arrangement to profile wind with crossed optical paths showing locations of in situ wind and Cn2 measurements.

Fig. 9
Fig. 9

Theoretical weighting function for Cn2, where α = αt = β0 = 1.8 and β = 0 Fresnel zones (solid line). The five experimental weights (crosses) were obtained from 4 h of data.

Fig. 10
Fig. 10

Theoretical weighting function for wind where αt = 1.8, α = 1.8, β = 0.3, and β0 = 1.8 Fresnel zones (solid-line). The crosses are experimental weights using single anemometers. Circles are weights obtained from averaging adjacent pairs of anemometers. All points are derived from 2 h of data.

Fig. 11
Fig. 11

Theoretical curve α is the wind weighting function for αt = 1.8, α = 3.6, β = 0.6, and β0 = 3.6 Fresnel zones. Curve b is for αt = 1.8, α = 0.9, β = 0.15, and β0 = 0.9. The experimental weights from 2 h of data are shown as crosses and circles.

Equations (20)

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d E a = 1 ikads exp [ i K 2 s ( L s ) 2 k L ] cos { K · [ ρ 0 s L + 1 2 ρ t ( 1 s L ) + b ] } .
d E b = ikads exp [ i K 2 s ( L S ) 2 k L ] cos { K · [ y 0 ( s / 2 L ) 1 2 ρ t ( 1 s / L ) + b ] } .
d P = | d E | 1 = [ Re 2 ( d E ) + Im 2 ( d E ) ] 1 / 2 1 ,
d P a = kads sin [ K 2 s ( L s ) 2 k L ] cos { K · [ ρ 0 ( s / L ) + 1 2 ρ t ( 1 s / L ) + b ] }
d P a = kads sin [ K 2 s ( L s ) 2 k L ] cos { K · [ y 0 ( s / 2 L ) 1 2 ρ t ( 1 s / L ) + b ] } .
d C a b = k 2 0 L d s 1 sin [ K 2 s 1 ( L s 1 ) 2 k L ] 0 L d s 2 sin [ k 2 s 2 ( L s 2 ) 2 k L ] < a ( K , s 1 ) a * ( K , s 2 ) cos { K · [ ρ 0 s 1 / L + 1 2 ρ t ( 1 s 1 / L ) + b 1 ] } cos { K · [ y 0 ( s 2 / 2 L ) 1 2 ρ t ( 1 s 2 / L ) + b 2 ] } > .
d C a b = 2 π k 2 0 L d s Φ n ( K , s ) d 2 K sin 2 [ K 2 s 1 ( L s ) 2 k L ] cos { K · [ ( x 0 y 0 ) s / L + ρ t ( 1 s / L ) ] } .
Φ n ( K ) = 0.033 C n 2 ( s ) K 11 / 3 ; L 0 1 K l 0 1 ,
C a b = 0.132 π 2 k 2 0 L d s C n 2 ( s ) 0 d K K 8 / 3 sin 2 [ K 2 s ( L s ) 2 k L ] J 0 [ K | ( x 0 y 0 ) s / L + ρ t ( 1 s / L ) | ] .
C a b ( x 0 y 0 , ρ t ) = 0 L d s C n 2 ( s ) W a b ( s , x 0 , y 0 , ρ t ) ,
W a b = 0.132 π 2 k 2 0 d K K 8 / 3 sin 2 [ K 2 s ( L s ) 2 k L ] J 0 [ K | ( x 0 y 0 ) s / L + ρ t ( 1 s / L ) | ] .
W a b = 0 d y y 8 / 3 sin 2 [ y 2 ( s / L ) ( 1 s / L ) 4 π ] J 0 [ y | ( β α ) s / L + α t ( 1 s / L ) | ] .
( s / L ) peak = α t / ( α t + α ) .
M a b = 0 L d s C n 2 ( s ) v ( s ) · W a b ( s ) ,
M a b = C a b ( τ ) / τ | τ = 0 ,
W a b = 0.132 π 2 k 2 0 d K K 5 / 3 sin 2 [ K 2 s ( L s ) 2 k L ] J 1 ( K r ) r ˆ .
r = ( x 0 y 0 ) s / L + ρ t ( 1 s / L ) .
W a b = { 1 + β 2 [ α α t ( L / s 1 ) ] 2 } 1 / 2 0 d y y 5 / 3 sin 2 [ y 2 s / L ( 1 s / L ) 4 π ] J 1 [ y | ( β α ) s / L + α t ( 1 s / L ) | ] .
A 0 = i = 1 N K i A i ,
C n 2 = B σ χ 2 ρ ,

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