Abstract

The statistical description of atmospheric turbulence and laser propagation through the atmosphere is complicated by global intermittency, the slow variations in space and time of the basic turbulence parameters. The effects of global intermittency of atmospheric turbulence on the mutual coherence function and the irradiance spectrum are investigated with numerical simulation. The effects of global intermittency are negligible when the path length is of the order of 10 correlation scales of global intermittency. This condition is readily satisfied for typical atmospheric parameters. The higher moments of irradiance appear to be more sensitive to the effects of global intermittency.

© 1994 Optical Society of America

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References

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  1. A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2, Chap. 8, p. 385.
  2. R. G. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
    [CrossRef]
  3. L. Mahrt, “Intermittency of atmospheric turbulence,” J. Atmos. Sci. 46, 79–95 (1989).
    [CrossRef]
  4. V. I. Tatarskii, “Some new aspects in the problem of waves and turbulence,” Radio Sci. 22, 859–865 (1987).
    [CrossRef]
  5. V. I. Tatarskii, V. U. Zavorotnyi, “Wave propagation in random media with fluctuating turbulent parameters,” J. Opt. Soc. Am. A 2, 2069–2076 (1985).
    [CrossRef]
  6. J. Gozani, “Wave propagation in an intermittent quasi-homogeneous turbulent medium,” Opt. Lett. 17, 559–561 (1992).
    [CrossRef] [PubMed]
  7. D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
    [CrossRef]
  8. C. L. Rino, J. Owen, “Numerical simulations of irradiance scintillation using the power law phase screen model,” Radio Sci. 19, 891–908 (1984).
    [CrossRef]
  9. C. Macaskill, T. E. Ewart, “Computer simulation of two-dimensional random wave propagation,” IMA J. Appl. Math. 33, 1–15 (1984).
    [CrossRef]
  10. M. Spivack, B. J. Uscinski, “The split-step solution in random wave propagation,” J. Comp. Appl. Math. 27, 349–361 (1989).
    [CrossRef]
  11. M. Spivack, “Accuracy of the moments from simulation of waves in random media,” J. Opt. Soc. Am. A 7, 790–793 (1990).
    [CrossRef]
  12. 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]
  13. J. M. Martin, Stanley M. Flatté, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–847 (1990).
    [CrossRef]
  14. J. Martin, “Simulation of wave propagation in random: theory and applications,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds., Vol. 9 of SPIE Press Monograph Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 463–486.
  15. S. M. Flatté, G. 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]
  16. Wm. A. Coles, J. P. Filice, R. G. Frehlich, M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” submitted to Appl. Opt.
  17. S. M. Flatté, C. Bracher, G. Wang, “Probability density function of In irradiance for waves in three-dimensional random media by numerical simulation,” J. Opt. Soc. Am. A (to be published).
  18. V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Zh. Eksp. Teor. Fiz. 75, 56–65 (1978). [Sov. Phys. JETP 48, 27–31 (1978)].
  19. N. Boston, R. W. Burling, “An investigation of high-wavenumber temperature and velocity spectra in air,” J. Fluid Mech. 55, 473–492 (1972).
    [CrossRef]
  20. F. H. Champagne, C. A. Friehe, J. C. LaRue, J. C. Wyngaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34, 515–530 (1977).
    [CrossRef]
  21. R. M. Williams, C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. 83, 892–899 (1978).
  22. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
    [CrossRef]
  23. J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
    [CrossRef]
  24. J. L. Codona, R. G. Frehlich, “Scintillation from extended incoherent sources,” Radio Sci. 22, 469–480 (1987).
    [CrossRef]
  25. J. H. Churnside, R. J. Hill, “Probability density of irradiance for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
    [CrossRef]

1993

1992

R. G. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

J. Gozani, “Wave propagation in an intermittent quasi-homogeneous turbulent medium,” Opt. Lett. 17, 559–561 (1992).
[CrossRef] [PubMed]

1990

1989

M. Spivack, B. J. Uscinski, “The split-step solution in random wave propagation,” J. Comp. Appl. Math. 27, 349–361 (1989).
[CrossRef]

L. Mahrt, “Intermittency of atmospheric turbulence,” J. Atmos. Sci. 46, 79–95 (1989).
[CrossRef]

1988

1987

V. I. Tatarskii, “Some new aspects in the problem of waves and turbulence,” Radio Sci. 22, 859–865 (1987).
[CrossRef]

J. L. Codona, R. G. Frehlich, “Scintillation from extended incoherent sources,” Radio Sci. 22, 469–480 (1987).
[CrossRef]

J. H. Churnside, R. J. Hill, “Probability density of irradiance for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
[CrossRef]

1986

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

1985

1984

C. L. Rino, J. Owen, “Numerical simulations of irradiance scintillation using the power law phase screen model,” Radio Sci. 19, 891–908 (1984).
[CrossRef]

C. Macaskill, T. E. Ewart, “Computer simulation of two-dimensional random wave propagation,” IMA J. Appl. Math. 33, 1–15 (1984).
[CrossRef]

1983

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

1978

V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Zh. Eksp. Teor. Fiz. 75, 56–65 (1978). [Sov. Phys. JETP 48, 27–31 (1978)].

R. M. Williams, C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. 83, 892–899 (1978).

1977

F. H. Champagne, C. A. Friehe, J. C. LaRue, J. C. Wyngaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34, 515–530 (1977).
[CrossRef]

1975

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

1972

N. Boston, R. W. Burling, “An investigation of high-wavenumber temperature and velocity spectra in air,” J. Fluid Mech. 55, 473–492 (1972).
[CrossRef]

Boston, N.

N. Boston, R. W. Burling, “An investigation of high-wavenumber temperature and velocity spectra in air,” J. Fluid Mech. 55, 473–492 (1972).
[CrossRef]

Bracher, C.

S. M. Flatté, C. Bracher, G. Wang, “Probability density function of In irradiance for waves in three-dimensional random media by numerical simulation,” J. Opt. Soc. Am. A (to be published).

Bunkin, F. V.

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

Burling, R. W.

N. Boston, R. W. Burling, “An investigation of high-wavenumber temperature and velocity spectra in air,” J. Fluid Mech. 55, 473–492 (1972).
[CrossRef]

Champagne, F. H.

F. H. Champagne, C. A. Friehe, J. C. LaRue, J. C. Wyngaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34, 515–530 (1977).
[CrossRef]

Churnside, J. H.

Codona, J. L.

J. L. Codona, R. G. Frehlich, “Scintillation from extended incoherent sources,” Radio Sci. 22, 469–480 (1987).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

Coles, Wm. A.

Wm. A. Coles, J. P. Filice, R. G. Frehlich, M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” submitted to Appl. Opt.

Creamer, D. B.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

Ewart, T. E.

C. Macaskill, T. E. Ewart, “Computer simulation of two-dimensional random wave propagation,” IMA J. Appl. Math. 33, 1–15 (1984).
[CrossRef]

Filice, J. P.

Wm. A. Coles, J. P. Filice, R. G. Frehlich, M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” submitted to Appl. Opt.

Flatté, S. M.

S. M. Flatté, G. 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]

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]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

S. M. Flatté, C. Bracher, G. Wang, “Probability density function of In irradiance for waves in three-dimensional random media by numerical simulation,” J. Opt. Soc. Am. A (to be published).

Flatté, Stanley M.

Frehlich, R. G.

R. G. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

J. L. Codona, R. G. Frehlich, “Scintillation from extended incoherent sources,” Radio Sci. 22, 469–480 (1987).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

Wm. A. Coles, J. P. Filice, R. G. Frehlich, M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” submitted to Appl. Opt.

Friehe, C. A.

F. H. Champagne, C. A. Friehe, J. C. LaRue, J. C. Wyngaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34, 515–530 (1977).
[CrossRef]

Gochelashvily, K. S.

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

Gozani, J.

Henyey, F. S.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

Hill, R. J.

Knepp, D. L.

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

LaRue, J. C.

F. H. Champagne, C. A. Friehe, J. C. LaRue, J. C. Wyngaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34, 515–530 (1977).
[CrossRef]

Macaskill, C.

C. Macaskill, T. E. Ewart, “Computer simulation of two-dimensional random wave propagation,” IMA J. Appl. Math. 33, 1–15 (1984).
[CrossRef]

Mahrt, L.

L. Mahrt, “Intermittency of atmospheric turbulence,” J. Atmos. Sci. 46, 79–95 (1989).
[CrossRef]

Martin, J.

S. M. Flatté, G. 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]

J. Martin, “Simulation of wave propagation in random: theory and applications,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds., Vol. 9 of SPIE Press Monograph Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 463–486.

Martin, J. M.

Monin, A. S.

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2, Chap. 8, p. 385.

Owen, J.

C. L. Rino, J. Owen, “Numerical simulations of irradiance scintillation using the power law phase screen model,” Radio Sci. 19, 891–908 (1984).
[CrossRef]

Paulson, C. A.

R. M. Williams, C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. 83, 892–899 (1978).

Prokhorov, A. M.

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

Rino, C. L.

C. L. Rino, J. Owen, “Numerical simulations of irradiance scintillation using the power law phase screen model,” Radio Sci. 19, 891–908 (1984).
[CrossRef]

Shishov, V. I.

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

Spivack, M.

M. Spivack, “Accuracy of the moments from simulation of waves in random media,” J. Opt. Soc. Am. A 7, 790–793 (1990).
[CrossRef]

M. Spivack, B. J. Uscinski, “The split-step solution in random wave propagation,” J. Comp. Appl. Math. 27, 349–361 (1989).
[CrossRef]

Tatarskii, V. I.

Uscinski, B. J.

M. Spivack, B. J. Uscinski, “The split-step solution in random wave propagation,” J. Comp. Appl. Math. 27, 349–361 (1989).
[CrossRef]

Wang, G.

S. M. Flatté, G. 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]

S. M. Flatté, C. Bracher, G. Wang, “Probability density function of In irradiance for waves in three-dimensional random media by numerical simulation,” J. Opt. Soc. Am. A (to be published).

Williams, R. M.

R. M. Williams, C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. 83, 892–899 (1978).

Wyngaard, J. C.

F. H. Champagne, C. A. Friehe, J. C. LaRue, J. C. Wyngaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34, 515–530 (1977).
[CrossRef]

Yadlowsky, M.

Wm. A. Coles, J. P. Filice, R. G. Frehlich, M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” submitted to Appl. Opt.

Yaglom, A. M.

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2, Chap. 8, p. 385.

Zavorotnyi, V. U.

V. I. Tatarskii, V. U. Zavorotnyi, “Wave propagation in random media with fluctuating turbulent parameters,” J. Opt. Soc. Am. A 2, 2069–2076 (1985).
[CrossRef]

V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Zh. Eksp. Teor. Fiz. 75, 56–65 (1978). [Sov. Phys. JETP 48, 27–31 (1978)].

Appl. Opt.

IMA J. Appl. Math.

C. Macaskill, T. E. Ewart, “Computer simulation of two-dimensional random wave propagation,” IMA J. Appl. Math. 33, 1–15 (1984).
[CrossRef]

J. Atmos. Sci.

R. G. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

L. Mahrt, “Intermittency of atmospheric turbulence,” J. Atmos. Sci. 46, 79–95 (1989).
[CrossRef]

F. H. Champagne, C. A. Friehe, J. C. LaRue, J. C. Wyngaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34, 515–530 (1977).
[CrossRef]

J. Comp. Appl. Math.

M. Spivack, B. J. Uscinski, “The split-step solution in random wave propagation,” J. Comp. Appl. Math. 27, 349–361 (1989).
[CrossRef]

J. Fluid Mech.

R. M. Williams, C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. 83, 892–899 (1978).

N. Boston, R. W. Burling, “An investigation of high-wavenumber temperature and velocity spectra in air,” J. Fluid Mech. 55, 473–492 (1972).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Proc. IEEE

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

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

Radio Sci.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

J. L. Codona, R. G. Frehlich, “Scintillation from extended incoherent sources,” Radio Sci. 22, 469–480 (1987).
[CrossRef]

C. L. Rino, J. Owen, “Numerical simulations of irradiance scintillation using the power law phase screen model,” Radio Sci. 19, 891–908 (1984).
[CrossRef]

V. I. Tatarskii, “Some new aspects in the problem of waves and turbulence,” Radio Sci. 22, 859–865 (1987).
[CrossRef]

Zh. Eksp. Teor. Fiz.

V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Zh. Eksp. Teor. Fiz. 75, 56–65 (1978). [Sov. Phys. JETP 48, 27–31 (1978)].

Other

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2, Chap. 8, p. 385.

Wm. A. Coles, J. P. Filice, R. G. Frehlich, M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” submitted to Appl. Opt.

S. M. Flatté, C. Bracher, G. Wang, “Probability density function of In irradiance for waves in three-dimensional random media by numerical simulation,” J. Opt. Soc. Am. A (to be published).

J. Martin, “Simulation of wave propagation in random: theory and applications,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds., Vol. 9 of SPIE Press Monograph Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 463–486.

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Figures (7)

Fig. 1
Fig. 1

Estimates of the MCF (•) from simulations in a weak-scattering regime for homogeneous and intermittent turbulence compared with the theoretical prediction equation (9) when homogeneous statistics are assumed.

Fig. 2
Fig. 2

Estimates of the spatial irradiance spectrum (•) from simulations in a weak-scattering regime for homogeneous turbulence and intermittent turbulence compared with the theoretical predictions of Born theory equation (16) when homogeneous statistics are assumed.

Fig. 3
Fig. 3

Estimates of the PDF of irradiance fluctuations (•) from simulations in a weak-scattering regime for homogeneous turbulence and intermittent turbulence compared with the theoretical predictions of the log-normal distribution.

Fig. 4
Fig. 4

Estimates of the MCF (•) from simulations in a strong-scattering regime for homogeneous turbulence and intermittent turbulence compared with the theoretical prediction equation (9) when homogeneous statistics are assumed.

Fig. 5
Fig. 5

Estimates of the spatial irradiance spectrum (•) from simulations in a strong-scattering regime for homogeneous turbulence and intermittent turbulence compared with the theoretical predictions for the low-frequency (lf) region equation (17) and the high-frequency (hf) region equation (19) when homogeneous statistics are assumed.

Fig. 6
Fig. 6

Estimates of the PDF of irradiance fluctuations (•) from simulations in a strong-scattering regime for homogeneous turbulence and intermittent turbulence compared with the theoretical predictions of the log normally modulated exponential distribution.

Fig. 7
Fig. 7

Estimates of the PDF of In irradiance fluctuations (•) from simulations in a strong-scattering regime for homogeneous turbulence and intermittent turbulence compared with the theoretical predictions of the log normally modulated exponential distribution.

Equations (19)

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MCF ( s , R ) = E ( x , R ) E * ( x + s , R ) ,
MCF ( s , R ) S S = M C ^ F ( s , R ) = I exp [ - 1 2 0 R d ^ ( s , z ) d z ] ,
d ^ ( s , z ) = 4 π k 2 - [ 1 - cos ( s . q ) ] Φ ^ n [ q , q z = 0 , z ] d q
Φ ^ n ( q , q z , z ) = 1 ( 2 π ) 3 - B ^ n ( r , s , z ) × exp ( - i q · r - i q z s ) d r d s .
B ^ n ( s , r , z ) = n ( x , z ) n ( x + s , z + r ) S S
Φ ^ n [ q , q z , z ] = 0.033 C n 2 ( z ) [ q 2 + ( 2 π / L 0 ) 2 ] - 11 / 6 f [ q l 0 ( z ) ] ,
f ( x ) = ( 1 + a 1 x + a 2 x 2 + a 3 x 3 ) exp ( - x ) ,
a 1 = 1.4284 ,             a 2 = 1.1987 ,             a 3 = 0.1414.
MCF ( s , R ) = I exp [ - R d ( s ) / 2 ] ,
d ( s ) = 4 π k 2 - [ 1 - cos ( s · q ) ] Φ n [ q , q z = 0 ] d q .
Φ n ( q , q z ) = 0.033 C n 2 [ q 2 + ( 2 π / L 0 ) 2 ] - 11 / 6 h ( q l ¯ 0 ) ,
ln ( l ¯ 0 ) = ln ( l 0 )
B I ( x 1 , x 2 , R ) = [ I ( x 1 , R ) - I ( x 1 , R ) ] [ I ( x 2 , R ) - I ( x 2 , R ) ] I 2 .
B I ( s , R ) = - Φ I ( q , R ) exp ( i q · s ) d q .
Φ ^ I ( q , R ) = 8 π k 2 0 R Φ ^ n ( q , z ) sin 2 [ q 2 2 k ( R - z ) ] d z .
Φ I ( q , R ) = 4 π k 2 R Φ n ( q ) [ 1 - k q 2 R sin ( q 2 R k ) ] ,
Φ I If ( q , R ) = 8 π k 2 Φ n ( q ) 0 R sin 2 [ q 2 2 k g ( z 1 , z 1 ) ] × exp [ - 0 R d [ q k g ( z 1 , z 1 ) d z ] d z 1 ,
g ( z , z 1 ) = z 1 - R z < z 1 , = z - R z > z 1 .
Φ I hf ( q , R ) = I 2 2 π 0 exp [ - R d ( s ) ] J 0 ( sq ) sds ,

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