Abstract

Simultaneous measurements of the spatial covariance of intensity and the angular spectrum of plane waves have been made for a diverging He–Ne laser beam propagating over a 1-km horizontal path. Most of the data are taken during strong scintillation. The measured angular spectrum is used to estimate the wave-structure function D(s). The propagation equation for the intensity covariance depends on the medium only through the longitudinal derivative of D(s). Thus, for a homogeneous medium, D(s) is sufficient to calculate the intensity covariance, although such a calculation has not been done in general. The estimated D(s) is used to calculate the intensity covariance by using weak scintillation theory and also by using an asymptotic theory for strong scintillation. The measured intensity covariances are then compared with these calculations. In both cases there is qualitative agreement, but the quantitative comparison is poor. It is shown that the inner scale of the turbulence is an important factor in weak scintillation, and it is argued that this inner scale is a dominant factor in strong scintillation.

© 1982 Optical Society of America

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References

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  1. V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Spring-field, Va., 1971).
  2. K. S. Gochelashvily and V. I. Shishov, “Laser beam scintillation beyond a turbulent layer,” Opt. Acta 18, 313–320 (1971).
    [CrossRef]
  3. K. S. Gochelashvily, V. G. Pevgov, and V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere (Fraunhofer zone of transmitter),” Kvantovaya Elektron. 1, 1156–1165 (1974) [Sov. J. Quantum Electron. 4, 632–637 (1974)].
  4. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
    [CrossRef]
  5. G. R. Ochs and R. S. Lawrence, “Saturation of laser beam scintillation under conditions of strong atmospheric turbulence,” J. Opt. Soc. Am. 59, 226–227 (1969).
    [CrossRef]
  6. G. R. Ochs, R. R. Bergman, and J. R. Snyder, “Laser beam scintillation over horizontal paths from 5.5 to 145 km,” J. Opt. Soc. Am. 59, 231–234 (1969).
    [CrossRef]
  7. J. Dunphy and J. Kerr, “Scintillation measurements for large integrated-path turbulence,” J. Opt. Soc. Am. 63, 981–986 (1973).
    [CrossRef]
  8. M. E. Gracheva, A. S. Gurvich, and A. S. Khrupin, “Correlation function of the light intensity in a turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 155–157 (1974) [Radiophys. Quantum Electron. 17, 120–122 (1974)].
  9. V. H. Rumsey, “Scintillations due to a concentrated layer with a power-law turbulence spectrum,” Radio Sci. 10, 107–114 (1975).
    [CrossRef]
  10. M. Marians, “Computed scintillation spectra for strong turbulence,” Radio Sci. 10, 115–119 (1975).
    [CrossRef]
  11. P. M. Livingston, D. H. Dietz, and A. Alcaraz, “Light propagation through a turbulent atmosphere: measurement of the optical filter function,” J. Opt. Soc. Am. 60, 925–935 (1970).
    [CrossRef]
  12. R. W. Lee and J. C. Harp, “Weak scattering in random media with applications to remote probing,” Proc. IEEE 57, 375–394 (1969).
    [CrossRef]
  13. R. J. Hill and S. F. Clifford, “Theory of saturation of optical scintillation by strong turbulence for arbitrary refractive index spectra,” J. Opt. Soc. Am. 71, 675–686 (1981).
    [CrossRef]
  14. R. J. Hill, “Theory of saturation of optical scintillation by strong turbulence: plane-wave variance and covariance and spherical-wave covariance,” J. Opt. Soc. Am. 72, 212–222 (1982).
    [CrossRef]
  15. V. V. Boronoev, N. Ts. Gomboev, V. L. Mironov, and E. A. Trubacheva, “Experimental determination of the fluctuation correlation scales of intensity and coherence of laser beams in turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 23, 322–325 (1980) [Radiophys. Quantum Electron. 23, 229–231 (1980)].

1982 (1)

1981 (1)

1980 (1)

V. V. Boronoev, N. Ts. Gomboev, V. L. Mironov, and E. A. Trubacheva, “Experimental determination of the fluctuation correlation scales of intensity and coherence of laser beams in turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 23, 322–325 (1980) [Radiophys. Quantum Electron. 23, 229–231 (1980)].

1975 (3)

V. H. Rumsey, “Scintillations due to a concentrated layer with a power-law turbulence spectrum,” Radio Sci. 10, 107–114 (1975).
[CrossRef]

M. Marians, “Computed scintillation spectra for strong turbulence,” Radio Sci. 10, 115–119 (1975).
[CrossRef]

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

1974 (2)

M. E. Gracheva, A. S. Gurvich, and A. S. Khrupin, “Correlation function of the light intensity in a turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 155–157 (1974) [Radiophys. Quantum Electron. 17, 120–122 (1974)].

K. S. Gochelashvily, V. G. Pevgov, and V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere (Fraunhofer zone of transmitter),” Kvantovaya Elektron. 1, 1156–1165 (1974) [Sov. J. Quantum Electron. 4, 632–637 (1974)].

1973 (1)

1971 (1)

K. S. Gochelashvily and V. I. Shishov, “Laser beam scintillation beyond a turbulent layer,” Opt. Acta 18, 313–320 (1971).
[CrossRef]

1970 (1)

1969 (3)

Alcaraz, A.

Bergman, R. R.

Boronoev, V. V.

V. V. Boronoev, N. Ts. Gomboev, V. L. Mironov, and E. A. Trubacheva, “Experimental determination of the fluctuation correlation scales of intensity and coherence of laser beams in turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 23, 322–325 (1980) [Radiophys. Quantum Electron. 23, 229–231 (1980)].

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Clifford, S. F.

Dietz, D. H.

Dunphy, J.

Gochelashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

K. S. Gochelashvily, V. G. Pevgov, and V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere (Fraunhofer zone of transmitter),” Kvantovaya Elektron. 1, 1156–1165 (1974) [Sov. J. Quantum Electron. 4, 632–637 (1974)].

K. S. Gochelashvily and V. I. Shishov, “Laser beam scintillation beyond a turbulent layer,” Opt. Acta 18, 313–320 (1971).
[CrossRef]

Gomboev, N. Ts.

V. V. Boronoev, N. Ts. Gomboev, V. L. Mironov, and E. A. Trubacheva, “Experimental determination of the fluctuation correlation scales of intensity and coherence of laser beams in turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 23, 322–325 (1980) [Radiophys. Quantum Electron. 23, 229–231 (1980)].

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, and A. S. Khrupin, “Correlation function of the light intensity in a turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 155–157 (1974) [Radiophys. Quantum Electron. 17, 120–122 (1974)].

Gurvich, A. S.

M. E. Gracheva, A. S. Gurvich, and A. S. Khrupin, “Correlation function of the light intensity in a turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 155–157 (1974) [Radiophys. Quantum Electron. 17, 120–122 (1974)].

Harp, J. C.

R. W. Lee and J. C. Harp, “Weak scattering in random media with applications to remote probing,” Proc. IEEE 57, 375–394 (1969).
[CrossRef]

Hill, R. J.

Kerr, J.

Khrupin, A. S.

M. E. Gracheva, A. S. Gurvich, and A. S. Khrupin, “Correlation function of the light intensity in a turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 155–157 (1974) [Radiophys. Quantum Electron. 17, 120–122 (1974)].

Lawrence, R. S.

Lee, R. W.

R. W. Lee and J. C. Harp, “Weak scattering in random media with applications to remote probing,” Proc. IEEE 57, 375–394 (1969).
[CrossRef]

Livingston, P. M.

Marians, M.

M. Marians, “Computed scintillation spectra for strong turbulence,” Radio Sci. 10, 115–119 (1975).
[CrossRef]

Mironov, V. L.

V. V. Boronoev, N. Ts. Gomboev, V. L. Mironov, and E. A. Trubacheva, “Experimental determination of the fluctuation correlation scales of intensity and coherence of laser beams in turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 23, 322–325 (1980) [Radiophys. Quantum Electron. 23, 229–231 (1980)].

Ochs, G. R.

Pevgov, V. G.

K. S. Gochelashvily, V. G. Pevgov, and V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere (Fraunhofer zone of transmitter),” Kvantovaya Elektron. 1, 1156–1165 (1974) [Sov. J. Quantum Electron. 4, 632–637 (1974)].

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Rumsey, V. H.

V. H. Rumsey, “Scintillations due to a concentrated layer with a power-law turbulence spectrum,” Radio Sci. 10, 107–114 (1975).
[CrossRef]

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

K. S. Gochelashvily, V. G. Pevgov, and V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere (Fraunhofer zone of transmitter),” Kvantovaya Elektron. 1, 1156–1165 (1974) [Sov. J. Quantum Electron. 4, 632–637 (1974)].

K. S. Gochelashvily and V. I. Shishov, “Laser beam scintillation beyond a turbulent layer,” Opt. Acta 18, 313–320 (1971).
[CrossRef]

Snyder, J. R.

Tatarski, V. I.

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Spring-field, Va., 1971).

Trubacheva, E. A.

V. V. Boronoev, N. Ts. Gomboev, V. L. Mironov, and E. A. Trubacheva, “Experimental determination of the fluctuation correlation scales of intensity and coherence of laser beams in turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 23, 322–325 (1980) [Radiophys. Quantum Electron. 23, 229–231 (1980)].

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (2)

M. E. Gracheva, A. S. Gurvich, and A. S. Khrupin, “Correlation function of the light intensity in a turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 155–157 (1974) [Radiophys. Quantum Electron. 17, 120–122 (1974)].

V. V. Boronoev, N. Ts. Gomboev, V. L. Mironov, and E. A. Trubacheva, “Experimental determination of the fluctuation correlation scales of intensity and coherence of laser beams in turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 23, 322–325 (1980) [Radiophys. Quantum Electron. 23, 229–231 (1980)].

J. Opt. Soc. Am. (6)

Kvantovaya Elektron. (1)

K. S. Gochelashvily, V. G. Pevgov, and V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere (Fraunhofer zone of transmitter),” Kvantovaya Elektron. 1, 1156–1165 (1974) [Sov. J. Quantum Electron. 4, 632–637 (1974)].

Opt. Acta (1)

K. S. Gochelashvily and V. I. Shishov, “Laser beam scintillation beyond a turbulent layer,” Opt. Acta 18, 313–320 (1971).
[CrossRef]

Proc. IEEE (2)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

R. W. Lee and J. C. Harp, “Weak scattering in random media with applications to remote probing,” Proc. IEEE 57, 375–394 (1969).
[CrossRef]

Radio Sci. (2)

V. H. Rumsey, “Scintillations due to a concentrated layer with a power-law turbulence spectrum,” Radio Sci. 10, 107–114 (1975).
[CrossRef]

M. Marians, “Computed scintillation spectra for strong turbulence,” Radio Sci. 10, 115–119 (1975).
[CrossRef]

Other (1)

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Spring-field, Va., 1971).

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Figures (13)

Fig. 1
Fig. 1

The telescope system used to observe the angular spectrum.

Fig. 2
Fig. 2

The central cut through the composite angular spectrum P(κx, 0) for a, weak scintillation; b, strong scintillation. The regions of overlap between exposures are marked by horizontal ticks. The ordinate is 10 log10P(κ).

Fig. 3
Fig. 3

The mutual coherence functions Bs calculated from the angular spectra labeled a and b in Fig. 2. The dotted lines give the ±2σ confidence limits.

Fig. 4
Fig. 4

The estimated spherical-wave-structure functions Ds derived from the angular spectra labeled a and b in Fig. 2. The contribution to these estimates from the OTF is also shown. Errors in the estimation of the angular spectra give the ±2σ confidence limits (dashed lines). Truncation of the angular spectra biases these estimates below the points marked by arrows.

Fig. 5
Fig. 5

A representative set of plane-wave-structure functions Dp estimated from spherical-wave-structure functions Ds as shown in Fig. 4. The dashed line has a slope of 2, and the dotted line has a slope of 5/3.

Fig. 6
Fig. 6

The intensity detector positions and covariance weights for the main array. The spacings marked by dashed lines are available but unused.

Fig. 7
Fig. 7

Intensity covariances in strong scintillation corresponding to angular spectrum b in Fig. 2. Here the secondary array is necessary to measure structure near the origin.

Fig. 8
Fig. 8

Measured intensity covariances. The lower curves are consecutive 150-sec averages. Time increases upward. The average of the lower traces is plotted at the top. All covariances are normalized for plotting. These data were taken in weak scintillation simultaneously with the angular spectrum labeled a in Fig. 2.

Fig. 9
Fig. 9

A sample of the intensity time series for one detector in strong scintillation corresponding to angular spectrum b of Fig. 2. The traces are continuous, and time increases to the bottom.

Fig. 10
Fig. 10

The spatial correlation of intensity and its Fourier transform, the spatial spectrum of intensity in weak scintillation, corresponding to angular spectrum a of Fig. 2. Here the ordinate is 10 log10 (spectral density). The crosses on the correlation give the 3σ confidence limits.

Fig. 11
Fig. 11

The spatial correlation and its Fourier transform in strong scintillation, corresponding to angular spectrum b of Fig. 2. The scale is the same as that of Fig. 10.

Fig. 12
Fig. 12

The observed scintillation indices versus Ds(rf). The solid lines plotted for Ds(rf) > 10 are from strong asymptotic theories for power-law-structure functions of exponent α. The weak-scintillation index is plotted as a solid line for the actual Ds(s) and as a dashed line for the Kolmogorov spectrum with no inner scale. The data are plotted as crossed error bars.

Fig. 13
Fig. 13

The observed spatial scale of intensity normalized by the Fresnel radius rf (1.08 cm for these data) versus Ds(rf). The data are plotted as crossed error bars. The two scales from the strong asymptotic theory are plotted as small circles. The weak-scintillation scale is plotted as a solid line for the actual Ds(s) and as a dashed line for the Kolmogorov spectrum with no inner scale.

Equations (12)

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D ( s , z ) = 4 π k 2 - + d 2 q [ 1 - cos ( q · s ) ] Φ n ( q ; q z = 0 , z ) .
B p ( s ) = E ( r ) E * ( r + s ) / I 0 = exp [ - ½ D p ( s ) ] ,
D p ( s ) = 0 L d z D ( s , z ) .
B s ( s ) = E ( r - ½ s ) E * ( r + ½ s ) / I 0 = exp [ - ½ D s ( s ) ] ,
D s ( s ) = 0 L d z D ( s z L , z ) .
D p ( s ) = L D ( s ) ,             D s ( s ) = L 0 l d α D ( α s ) .
D ˜ s ( s ) = D s ( s ) - 2 log [ OTF ( s ) ] .
Φ n ( κ ) = 0.033 C n 2 κ - 11 / 3 , D n ( r ) = C n 2 r 2 / 3 .
C 12 = d 2 s C I ( s ) w 12 ( s ) .
w 12 ( s ) = d 2 r A 1 ( r ) A 2 ( r + s ) / d 2 r 1 A 1 ( r 1 ) d 2 r 2 A 2 ( r 2 ) .
m s 2 = [ 2 α cos ( α π / 4 ) Γ 3 ( 1 + α / 2 ) / Γ ( α + 1 ) ] D s ( r f ) .
m s 2 = 1 + N ( α ) D s ( r f ) - ( 2 / α ) ( 2 - α ) , N ( α ) = ( 2 α + 1 / π α ) ( α + 1 ) sin ( α π / 2 ) Γ 2 ( 1 + α / 2 ) × Γ ( 4 / α - 1 ) Γ 2 ( α - 1 ) Γ - 1 ( 2 α - 2 ) .