Abstract

Irradiance variance for optical propagation through atmospheric turbulence is calculated by numerical simulation. Let l0 be the inner scale, Rf be the Fresnel scale, and β02 be the weak-fluctuation irradiance variance at zero inner scale. Then results in the strong-focusing regime just past the peak can be summarized by σI2 = 1.74 − 0.092β0 + 0.60(l0/Rf) for a plane wave and σI2 = 3.02 − 0.35β0 + 5.56(l0/Rf) for a point source. These numerical results are in excellent agreement with experimental results.

© 1993 Optical Society of America

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  1. M. E. Gracheva, A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Radiophys. Quantum Electron. 8, 511–525 (1965).
  2. V. I. Tatarskii, The Effects of Thrbulent Atmospheres on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  3. I. G. Yakushkin, “Asymptotic calculations of intensity fluctuations in a turbulent medium for long paths,” Radiophys. Quantum Electron. 18, 1224–1229 (1975).
    [CrossRef]
  4. 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]
  5. M. E. Gracheva, A. S. Gurvich, S. O. Lomadze, VI. V. Pokasov, A. S. Khrupin, “Probability distribution of strong fluctuation of light intensity in the atmosphere,” Radiophys. Quantum Electron. 17, 83–87 (1974).
    [CrossRef]
  6. G. Parry, “Measurement of atmospheric turbulence-induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
    [CrossRef]
  7. R. L. Philips, L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,”J. Opt. Soc. Am. 71, 1440–1445 (1981).
    [CrossRef]
  8. W. R. Coles, R. G. Frehlich, “Simultaneous measurements of angular scattering and intensity scintillation in the atmosphere,”J. Opt. Soc. Am. 72, 1042–1048 (1982).
    [CrossRef]
  9. K. S. Gochelashvili, V. I. Shishov, “Saturated fluctuations in the laser radiation intensity in a turbulent media,” Sov. Phys. JETP 39, 605–609 (1974).
  10. V. I. Tatarskii, A. S. Gurvich, B. S. Elepov, VI. V. Pokasov, K. K. Sabelfeld, “Space structure of strong intensity fluctuations of light in a turbulent medium,” Opt. Acta 26, 531–542 (1979).
    [CrossRef]
  11. A. M. Whitman, M. J. Beran, “Two-scale solution for atmospheric scintillation,” J. Opt. Soc. Am. A 2, 2133–2143 (1985).
    [CrossRef]
  12. A. M. Whitman, M. J. Beran, “First-order correction for the scintillation index and correlation of intensity function,” J. Opt. Soc. Am. A 9, 974–977 (1992).
    [CrossRef]
  13. A. M. Whitman, M. J. Beran, “Two-scale solution for atmospheric scintillation from a point source,” J. Opt. Soc. Am. A 5, 735–737 (1988).
    [CrossRef]
  14. M. J. Beran, A. M. Whitman, “Effect of the turbulence inner scale on scintillation in the atmosphere,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1115, 2–10 (1989).
    [CrossRef]
  15. R. J. Hill, S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,”J. Opt. Soc. Am. 68, 892–899 (1978).
    [CrossRef]
  16. R. J. Hill, “Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes,” Waves Random Media 2, 179–201 (1992).
    [CrossRef]
  17. 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]
  18. 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]
  19. R. Dashen, G. Y. 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]
  20. A. Consortini, F. Cochetti, J. H. Churnside, R. J. Hill, “Inner-scale effect on irradiance variance measured for weak-to-strong atmospheric scintillation,” J. Opt. Soc. Am. A 10, 2354–2362 (1993).
    [CrossRef]
  21. R. J. Hill, “Models of the scalar spectrum for turbulent advection,”J. Fluid Mech. 68, 541–562 (1978).
    [CrossRef]
  22. A. S. Gurvich, Institute for Atmospheric Physics, Moscow (personal communication 1992).

1993 (2)

1992 (2)

R. J. Hill, “Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes,” Waves Random Media 2, 179–201 (1992).
[CrossRef]

A. M. Whitman, M. J. Beran, “First-order correction for the scintillation index and correlation of intensity function,” J. Opt. Soc. Am. A 9, 974–977 (1992).
[CrossRef]

1990 (1)

1988 (2)

1985 (1)

1983 (1)

1982 (1)

1981 (2)

G. Parry, “Measurement of atmospheric turbulence-induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[CrossRef]

R. L. Philips, L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,”J. Opt. Soc. Am. 71, 1440–1445 (1981).
[CrossRef]

1979 (1)

V. I. Tatarskii, A. S. Gurvich, B. S. Elepov, VI. V. Pokasov, K. K. Sabelfeld, “Space structure of strong intensity fluctuations of light in a turbulent medium,” Opt. Acta 26, 531–542 (1979).
[CrossRef]

1978 (2)

1975 (1)

I. G. Yakushkin, “Asymptotic calculations of intensity fluctuations in a turbulent medium for long paths,” Radiophys. Quantum Electron. 18, 1224–1229 (1975).
[CrossRef]

1974 (2)

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

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

1965 (1)

M. E. Gracheva, A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Radiophys. Quantum Electron. 8, 511–525 (1965).

Andrews, L. C.

Beran, M. J.

Bracher, C.

Churnside, J. H.

Clifford, S. F.

Cochetti, F.

Coles, W. R.

Consortini, A.

Dashen, R.

Elepov, B. S.

V. I. Tatarskii, A. S. Gurvich, B. S. Elepov, VI. V. Pokasov, K. K. Sabelfeld, “Space structure of strong intensity fluctuations of light in a turbulent medium,” Opt. Acta 26, 531–542 (1979).
[CrossRef]

Fante, R. L.

Flatté, S. M.

Frehlich, R. G.

Gochelashvili, K. S.

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

Gracheva, M. E.

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

M. E. Gracheva, A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Radiophys. Quantum Electron. 8, 511–525 (1965).

Gurvich, A. S.

V. I. Tatarskii, A. S. Gurvich, B. S. Elepov, VI. V. Pokasov, K. K. Sabelfeld, “Space structure of strong intensity fluctuations of light in a turbulent medium,” Opt. Acta 26, 531–542 (1979).
[CrossRef]

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

M. E. Gracheva, A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Radiophys. Quantum Electron. 8, 511–525 (1965).

A. S. Gurvich, Institute for Atmospheric Physics, Moscow (personal communication 1992).

Hill, R. J.

A. Consortini, F. Cochetti, J. H. Churnside, R. J. Hill, “Inner-scale effect on irradiance variance measured for weak-to-strong atmospheric scintillation,” J. Opt. Soc. Am. A 10, 2354–2362 (1993).
[CrossRef]

R. J. Hill, “Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes,” Waves Random Media 2, 179–201 (1992).
[CrossRef]

R. J. Hill, S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,”J. Opt. Soc. Am. 68, 892–899 (1978).
[CrossRef]

R. J. Hill, “Models of the scalar spectrum for turbulent advection,”J. Fluid Mech. 68, 541–562 (1978).
[CrossRef]

Khrupin, A. S.

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

Lomadze, S. O.

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

Martin, J. M.

Parry, G.

G. Parry, “Measurement of atmospheric turbulence-induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[CrossRef]

Philips, R. L.

Pokasov, VI. V.

V. I. Tatarskii, A. S. Gurvich, B. S. Elepov, VI. V. Pokasov, K. K. Sabelfeld, “Space structure of strong intensity fluctuations of light in a turbulent medium,” Opt. Acta 26, 531–542 (1979).
[CrossRef]

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

Sabelfeld, K. K.

V. I. Tatarskii, A. S. Gurvich, B. S. Elepov, VI. V. Pokasov, K. K. Sabelfeld, “Space structure of strong intensity fluctuations of light in a turbulent medium,” Opt. Acta 26, 531–542 (1979).
[CrossRef]

Shishov, V. I.

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

Tatarskii, V. I.

V. I. Tatarskii, A. S. Gurvich, B. S. Elepov, VI. V. Pokasov, K. K. Sabelfeld, “Space structure of strong intensity fluctuations of light in a turbulent medium,” Opt. Acta 26, 531–542 (1979).
[CrossRef]

V. I. Tatarskii, The Effects of Thrbulent Atmospheres on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Wang, G. Y.

Whitman, A. M.

Yakushkin, I. G.

I. G. Yakushkin, “Asymptotic calculations of intensity fluctuations in a turbulent medium for long paths,” Radiophys. Quantum Electron. 18, 1224–1229 (1975).
[CrossRef]

Appl. Opt. (1)

J. Fluid Mech. (1)

R. J. Hill, “Models of the scalar spectrum for turbulent advection,”J. Fluid Mech. 68, 541–562 (1978).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (6)

Opt. Acta (2)

G. Parry, “Measurement of atmospheric turbulence-induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[CrossRef]

V. I. Tatarskii, A. S. Gurvich, B. S. Elepov, VI. V. Pokasov, K. K. Sabelfeld, “Space structure of strong intensity fluctuations of light in a turbulent medium,” Opt. Acta 26, 531–542 (1979).
[CrossRef]

Radiophys. Quantum Electron. (3)

I. G. Yakushkin, “Asymptotic calculations of intensity fluctuations in a turbulent medium for long paths,” Radiophys. Quantum Electron. 18, 1224–1229 (1975).
[CrossRef]

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

M. E. Gracheva, A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Radiophys. Quantum Electron. 8, 511–525 (1965).

Sov. Phys. JETP (1)

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

Waves Random Media (1)

R. J. Hill, “Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes,” Waves Random Media 2, 179–201 (1992).
[CrossRef]

Other (3)

A. S. Gurvich, Institute for Atmospheric Physics, Moscow (personal communication 1992).

V. I. Tatarskii, The Effects of Thrbulent Atmospheres on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

M. J. Beran, A. M. Whitman, “Effect of the turbulence inner scale on scintillation in the atmosphere,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1115, 2–10 (1989).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Spectra of refractive index for different turbiilence models from Eq. (1.1). The dashed curves are from the traditional spectrum, and the solid curves are from the atmospheric spectrum.

Fig. 2
Fig. 2

Example profiles for a point-source beam simulation. (a) Irradiance profile: the thick solid curve is the result for no fluctuations; and the thinner curve is for a case with β0 = 4.47 and l0 = 4 mm. (b) Irradiance-variance profile: normalized so that the irradiance averaged over all the realizations and over the azimuth is unity at every r. Points with r below the cutoff are used to calculate the average irradiance variance.

Fig. 3
Fig. 3

Irradiance variances from numerical simulation with the atmospheric spectrum at different inner scales and as a function of turbulence strength. The solid curves connect the points to guide the eye; the dashed lines are best linear fits to the data with 3 < β0 < 8, constrained to a common slope. (a) Plane wave; the Fresnel scale Rf = 13 mm, the same as that in the experiment of Gracheva et al.5 (b) Point source; the Fresnel scale is 10 mm, the same as that in the experiment of Consortini et al.20

Fig. 4
Fig. 4

Irradiance variance as a function of inner scale. The values are obtained from the intercept of the linear fits in Fig. 3 with β0 = 5.48. Circles: point source; squares: plane wave. The lines are from the best-fit empirical formulas.

Fig. 5
Fig. 5

Near-plane-wave irradiance variances as a function of turbulence strength for different inner scales. The parameters of the simulations are given in Table 1. Open circles are data from the experiment of Gracheva et al.5 The long-dashed curve represents the plane-wave asymptotic theory with zero inner scale. In each plot the three curves starting from lower σ2 have inner scales of 2,5 (short-dashed curve), and 8 mm. (a) Plane wave, (b) extended beam.

Fig. 6
Fig. 6

Near-point-source irradiance variances as a function of turbulence strength for different inner scales. The parameters of the simulations are given in Table 2. Symbols other than closed circles are data from the experiment of Consortini et al.20 The dashed curve represents the asymptotic theory with zero inner scale.

Fig. 7
Fig. 7

Irradiance variance as a function of inner scale. All the data have been translated to β0 = 5.48. The simulation results are as in Fig. 4. Closed circles are data from Consortini et al.20 The outer error bars represent the rms deviation of the data points from their mean. The inner error bars represent the error on the mean itself.

Fig. 8
Fig. 8

Example of variability of irradiance variance in a simulation with β0 = 4.47 and l0 = 5 mm. (a) Variance along a horizontal line as a function of vertical position. (b) Histogram of frequency of occurrence as a function of variance. The mean value of the variance for this realization is 4.6, and the rms fluctuation in the variance is 7.2.

Tables (3)

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Table 1 Simulation Parameters

Tables Icon

Table 2 Simulation Parameters

Tables Icon

Table 3 Empirical Constants in the Formula for Irradiance Variance

Equations (6)

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Ψ ( K ) = 0.033 C n 2 K - 11 / 3 F ( K l 0 ) ,
K f R f - 1 ( k / L ) 1 / 2 ,
Φ = exp [ - ½ ( r / σ ) 8 ] ,
Φ = exp ( - r 2 / 2 σ 2 ) cos ( r 2 / 2 ρ 2 ) ,
β 0 2 α C n 2 k 7 / 6 L 11 / 6 ,
σ I 2 = s 0 - s b β 0 + s l ( l 0 / R f ) ,

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