Abstract

We have simulated optical propagation through atmospheric turbulence in which the spectrum near the inner scale follows that of Hill and Clifford [J. Opt. Soc. Am. 68, 892 (1978)] and the turbulence strength puts the propagation into the asymptotic strong-fluctuation regime. Analytic predictions for this regime have the form of power laws as a function of β02, the irradiance variance predicted by weak-fluctuation (Rytov) theory, and l0, the inner scale. The simulations indeed show power laws for both spherical-wave and plane-wave initial conditions, but the power-law indices are dramatically different from the analytic predictions. Let σI2-1=a(β02/βc2)-b(l0/Rf)c, where we take the reference value of β02 to be βc2=60.6, because this is the center of our simulation region. For zero inner scale (for which c=0), the analytic prediction is b=0.4 and a=0.17 (0.37) for a plane (spherical) wave. Our simulations for a plane wave give a=0.234±0.007 and b=0.50±0.07, and for a spherical wave they give a=0.58±0.01 and b=0.65±0.05. For finite inner scale the analytic prediction is b=1/6, c=7/18 and a=0.76 (2.07) for a plane (spherical) wave. We find that to a reasonable approximation the behavior with β02 and l0 indeed factorizes as predicted, and each part behaves like a power law. However, our simulations for a plane wave give a=0.57±0.03, b=0.33±0.03, and c=0.45±0.06. For spherical waves we find a=3.3±0.3, b=0.45±0.05, and c=0.8±0.1.

© 2000 Optical Society of America

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References

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  1. A. Consortini, F. Cochetti, J. H. Churnside, R. J. Hill, “Inner-scale effect on intensity variance measured for weak to strong atmospheric scintillation,” J. Opt. Soc. Am. A 10, 2354–2362 (1993).
    [CrossRef]
  2. S. M. Flatté, G. Y. Wang, J. Martin, “Irradiance variance of optical waves through atmospheric turbulence by numerical simulation and comparison with experiment,” J. Opt. Soc. Am. A 10, 2363–2370 (1993).
    [CrossRef]
  3. M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
    [CrossRef]
  4. R. Hill, S. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its applications to optical propagation,” J. Opt. Soc. Am. 68, 892–899 (1978).
    [CrossRef]
  5. J. M. Martin, S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
    [CrossRef] [PubMed]
  6. J. M. Martin, S. M. Flatté, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–847 (1990).
    [CrossRef]
  7. L. Andrews, R. Phillips, Laser Beam Propagation through Random Media (SPIE Publishing Services, Bellingham, Wash., 1999).
  8. S. M. Flatté, C. Bracher, G.-Y. Wang, “Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation,” J. Opt. Soc. Am. A 11, 2080–2092 (1994).
    [CrossRef]
  9. K. Gochelashvili, V. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).
  10. R. L. Fante, “Inner-scale size effect on the scintillations of light in the turbulent atmosphere,” J. Opt. Soc. Am. 73, 277–281 (1983).
    [CrossRef]
  11. R. Frehlich, “Intensity covariance of a point source in a random medium with a Kolmogorov spectrum and an inner scale of turbulence,” J. Opt. Soc. Am. A 4, 360–366 (1987).
    [CrossRef]
  12. L. Andrews, R. Phillips, C. Hopen, M. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
    [CrossRef]
  13. R. Dashen, G. Wang, S. M. Flatté, C. Bracher, “Moments of intensity and log intensity: new asymptotic results for waves in power-law media,” J. Opt. Soc. Am. A 10, 1233–1242 (1993).
    [CrossRef]

1999 (1)

1994 (1)

1993 (3)

1990 (1)

1988 (1)

1987 (1)

1983 (1)

1978 (1)

1974 (2)

M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[CrossRef]

K. Gochelashvili, V. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).

Al-Habash, M.

Andrews, L.

L. Andrews, R. Phillips, C. Hopen, M. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

L. Andrews, R. Phillips, Laser Beam Propagation through Random Media (SPIE Publishing Services, Bellingham, Wash., 1999).

Bracher, C.

Churnside, J. H.

Clifford, S.

Cochetti, F.

Consortini, A.

Dashen, R.

Fante, R. L.

Flatté, S. M.

Frehlich, R.

Gochelashvili, K.

K. Gochelashvili, V. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).

Gracheva, M.

M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[CrossRef]

Gurvich, A.

M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[CrossRef]

Hill, R.

Hill, R. J.

Hopen, C.

Khrupin, A.

M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[CrossRef]

Lomadze, S.

M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[CrossRef]

Martin, J.

Martin, J. M.

Phillips, R.

L. Andrews, R. Phillips, C. Hopen, M. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

L. Andrews, R. Phillips, Laser Beam Propagation through Random Media (SPIE Publishing Services, Bellingham, Wash., 1999).

Pokasov, V.

M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[CrossRef]

Shishov, V.

K. Gochelashvili, V. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).

Wang, G.

Wang, G. Y.

Wang, G.-Y.

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (7)

Radiophys. Quantum Electron. (1)

M. Gracheva, A. Gurvich, S. Lomadze, V. Pokasov, A. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
[CrossRef]

Sov. Phys. JETP (1)

K. Gochelashvili, V. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent medium,” Sov. Phys. JETP 39, 605–609 (1974).

Other (1)

L. Andrews, R. Phillips, Laser Beam Propagation through Random Media (SPIE Publishing Services, Bellingham, Wash., 1999).

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Figures (6)

Fig. 1
Fig. 1

Variance-preserving spectra of irradiance for plane-wave and spherical-wave simulations. The integral under each curve is the value of the second moment of irradiance, I2, for the corresponding turbulence parameters. All the curves shown here have β02=100 and, from top to bottom at κRf=10, have inner scale 8, 4, and 0 mm. The irradiance variance is given by σI2=I2-1. The dotted–dashed sections of the curves are extrapolations obtained by using an analytic low-wave-number asymptote (described further in the text). Asymptotic theory describes the difference between irradiance variance and unity; therefore in this paper we study the difference between the integral of the irradiance spectrum (shown here) and 2.

Fig. 2
Fig. 2

Spectra of irradiance from plane-wave simulations for different values of turbulence strength and inner scale. The values of β02 are 0.01, 30, 60, and 90. The order is from top to bottom at κRf=1. Each spectrum has been divided by its respective β02. For zero inner scale, it is expected that the high-wave-number asymptote of all the curves will be the same. The dotted–dashed sections of the curves represent extensions at low wave numbers by analytic means explained in the text.

Fig. 3
Fig. 3

Spectra of irradiance from spherical-wave simulations for different values of turbulence strength and inner scale. All else as in Fig. 2.

Fig. 4
Fig. 4

Irradiance variance minus unity from plane-wave simulations for different values of turbulence strength and inner scale. The dashed lines are from asymptotic theory (as β02 becomes large).12 It is seen that the simulations yield power-law behavior but with power-law indices that are quite different from expectations.

Fig. 5
Fig. 5

Irradiance variance from spherical-wave simulations for different values of turbulence strength and inner scale. The dashed lines are from asymptotic theory (as β02 becomes large).12 The solid symbols are from previously published simulations.2 The open symbols are from simulations done for this publication. It is seen that the new simulations yield power-law behavior but with power-law indices that are quite different from expectations.

Fig. 6
Fig. 6

Irradiance variance as a function of inner scale compared with analytic asymptotic theory. The variances come from the entries to Tables 2 and 3, that is, the values of the fitted lines at β02=60.6. It is seen that the simulation results fall roughly, but not exactly, on power-law behavior as a function of inner scale and that the power-law indices disagree with analytic predictions.

Tables (4)

Tables Icon

Table 1 Reference Numbers Used in Empirical Formulas for Screen Size Ns and Fresnel Length Rf for Cases with Nonzero Inner Scale

Tables Icon

Table 2 Results of Fits to Irradiance Variances from Plane-Wave Simulations to Eq. (9) a

Tables Icon

Table 3 Results of Fits to Irradiance Variances from Spherical-Wave Simulations to Eq. (9)a

Tables Icon

Table 4 Results of Common-Slope Fits to All Nonzero Inner Scales for a Given Initial Condition, by Use of Eq. (10)a

Equations (10)

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β02=1.23Cn2k7/6L11/6,
β02=0.50Cn2k7/6L11/6.
σI2=β02ns-11/6.
ns>(10β02)6/11.
Rf=Rfrβ02βr2g.
Ns=Nsrβ02βr2g+0.6,
Rf=Rfrβ02βr2gl0lrd
Ns=Nsrβ02βr20.6RfRfr.
σI2-1=aβc2β02b,
σI2-1=aβc2β02bl0Rfc.

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