Abstract

An expression is derived for the intensity correlation function of a partially coherent beam wave of arbitrary size and focus propagating through weak atmospheric turbulence. Because the derivation employs the quadratic structure function (QSF) approximation, it describes only long-term beam-wander effects. Calculated covariance results agree well with measured data and substantiate the importance of beam-wander effects on focused beam statistics. Illustrative results show the effects of the phase deviation, size, and phase curvature of the source as well as the strength of turbulence and range. The weak-turbulence formulas are used to calculate strong-turbulence statistics by iterative recalculation of the coherence parameters of the beam in a succession of weakly turbulent path intervals. Successively calculated values of the beam-phase deviation and correlation length effectively provide wave-tilt correlation information that is missing in the original QSF approximation. The iterative solution for the normalized beam-intensity variance saturates and asymptotically approaches unity in a manner predicted by other theories. Calculated covariance functions also exhibit the initial rapid falloff and subsequent long coherence tail typical of saturated covariance behavior. Some magnitude discrepancies between calculated results and reported measurements are apparent for strong turbulence conditions. The iteration analysis predicts that log-amplitude fluctuations are diminished for increasingly strong turbulence and that saturated conditions arise solely from phase effects, in agreement with Fante’s conclusions [ R, L. Fante, Radio Sci. 15, 757 ( 1980)].

© 1981 Optical Society of America

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References

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  1. D. A. deWolf, “Operator and diagram techniques for solving intensity moment equations in random media: applications to strong scattering,” Radio Sci. 14, 277–286 (1979).
    [CrossRef]
  2. V. Tatarskii, The Effects of Turbulent Atmospheres on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  3. R. Dashen, “Path integral for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
    [CrossRef]
  4. J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere, Topics in Applied Physics, Vol. 25 (Springer-Verlag, New York, 1978).
    [CrossRef]
  5. V. A. Banakh and et al., “Focused-laser-beam scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. 64, 516–518 (1974).
    [CrossRef]
  6. M. Tur and M. J. Beran, “Propagation of a finite beam through a random medium,” Opt. Lett. 5, 306–308 (1980).
    [CrossRef] [PubMed]
  7. D. L. Fried and J. B. Seidman, “Laser-beam scintillation in the atmosphere,” J. Opt. Soc. Am. 57, 181–185 (1967).
    [CrossRef]
  8. A. Ishimaru, “Fluctuations of a focused beam wave for atmospheric turbulence probing,” Proc. IEEE 57, 407–414 (1969).
    [CrossRef]
  9. S. S. Khmelevtsov, “Propagation of laser radiation in a turbulent atmosphere,” Appl. Opt. 12, 2421–2433 (1972).
    [CrossRef]
  10. Ref. 4, Chap. 5.
  11. J. C. Leader, “Intensity fluctuations resulting from partially coherent light propagating through atmospheric turbulence,” J. Opt. Soc. Am. 69, 73–84 (1979).
    [CrossRef]
  12. H. van de Hulst and K. Grossman, The Atmosphere of Venus and Mars, J. C. Brandt and M. B. McElroy, eds. (Gordon and Breach, New York, 1968).
  13. J. L. Hansen, “Radiative transfer by doubling very thin layers,” Astrophys. J. 155, 565–573 (1969).
    [CrossRef]
  14. B. Steinle and H. P. Baltes, “Radiant intensity and spatial coherence for finite planar sources,” J. Opt. Soc. Am. 67, 241–247 (1977).
    [CrossRef]
  15. W. H. Carter and L. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  16. S. M. Wandzura, “Meeting of quadratic structure functions,” J. Opt. Soc. Am. 70, 745–747 (1980).
    [CrossRef]
  17. R. L. Fante, “Some physical insights into beam propagation in strong turbulence,” Radio Sci. 15, 757–762 (1980).
    [CrossRef]
  18. J. C. Leader, “Beam properties of partially coherent curved beam waves in the turbulent atmosphere,” J. Opt. Soc. Am. 70, 682–688 (1980).
    [CrossRef]
  19. M. A. Plonus, C. F. Ouyang, and S. C. H. Wang, “Intensity properties of partially coherent beam waves,” Appl. Opt. 19, 3082–3085 (1980).
    [CrossRef] [PubMed]
  20. J. R. Kerr and J. R. Dunphy, “Experimental effects of finite transmitter apertures on scintillations,” J. Opt. Soc. Am. 63, 1–8 (1973).
    [CrossRef]
  21. The concept of a spatially partially coherent beam used in this paper is discussed in J. C. Leader, “Similarities and distinctions between coherence theory relations and laser scattering phenomena,” Opt. Eng. 19, 593–601 (1980).
    [CrossRef]
  22. Z. I. Feizulin and Yu. A. Krovtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 68–73 (1967).
    [CrossRef]
  23. R. L. Fante, “Two-source spherical wave structure functions in atmospheric turbulence,” J. Opt. Soc. Am. 66, 74 (1976).
    [CrossRef]
  24. H. M. Pedersen, “Theory of speckle dependence on surface roughness,” J. Opt. Soc. Am. 66, 1204–1210 (1976).
    [CrossRef]
  25. Ref. 4, Chap. 2.
  26. V. Ya. Sedin and et al., “Intensity fluctuations of a focused laser beam in air,” Radioteckh. Elektron. 15, 1290–1292 (1970).
  27. M. L. Gracheva and et al., “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation,” Akademiia Nauk SSSR. Otdelenie Okeanologii, Fizikii, Atmosfery i Geografii. [Preprint, Moscow, (1973). Aerospace Corp. translation no. LRG-73-T-28].
  28. R. S. Lawrence and J. W. Strohbehn, “A survey of clean-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
    [CrossRef]
  29. S. F. Clifford, G. R. Ochs, and R. S. Lawrence, “Saturation of optical scintillation by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
    [CrossRef]
  30. R. L. Fante, “Electric field spectrum and intensity covariance of a wave in a random medium,” Radio Sci. 10, 77–85 (1975).
    [CrossRef]
  31. V. I. Tatarskii, Wave Propagation in a Turbulent Medium, R. A. Silverman, trans. (Dover, New York, 1967).
  32. S. F. Clifford and H. T. Yura, “Equivalence of two theories of strong optical scintillation,” J. Opt. Soc. Am. 64, 1641–1644 (1974).
    [CrossRef]
  33. Ref. 4, Chap 3, p. 87.
  34. J. R. Kerr, “Experiments on turbulence characteristics and multiwavelength scintillation phenomena,” J. Opt. Soc. Am. 62, 1040–1049 (1972).
    [CrossRef]
  35. R. A. Elliott, J. R. Kerr, and P. A. Pincus, “Optical propagation in laboratory-generated turbulence,” Appl. Opt. 18, 3315–3323 (1979).
    [CrossRef] [PubMed]
  36. J. C. Dainty, ed., Laser Speckle, Topics in Applied Physics, Vol. 9 (Springer-Verlag, New York, 1975).

1980 (6)

1979 (4)

J. C. Leader, “Intensity fluctuations resulting from partially coherent light propagating through atmospheric turbulence,” J. Opt. Soc. Am. 69, 73–84 (1979).
[CrossRef]

R. A. Elliott, J. R. Kerr, and P. A. Pincus, “Optical propagation in laboratory-generated turbulence,” Appl. Opt. 18, 3315–3323 (1979).
[CrossRef] [PubMed]

D. A. deWolf, “Operator and diagram techniques for solving intensity moment equations in random media: applications to strong scattering,” Radio Sci. 14, 277–286 (1979).
[CrossRef]

R. Dashen, “Path integral for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

1977 (2)

1976 (2)

1975 (1)

R. L. Fante, “Electric field spectrum and intensity covariance of a wave in a random medium,” Radio Sci. 10, 77–85 (1975).
[CrossRef]

1974 (3)

1973 (1)

1972 (2)

1970 (2)

V. Ya. Sedin and et al., “Intensity fluctuations of a focused laser beam in air,” Radioteckh. Elektron. 15, 1290–1292 (1970).

R. S. Lawrence and J. W. Strohbehn, “A survey of clean-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

1969 (2)

J. L. Hansen, “Radiative transfer by doubling very thin layers,” Astrophys. J. 155, 565–573 (1969).
[CrossRef]

A. Ishimaru, “Fluctuations of a focused beam wave for atmospheric turbulence probing,” Proc. IEEE 57, 407–414 (1969).
[CrossRef]

1967 (2)

Z. I. Feizulin and Yu. A. Krovtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 68–73 (1967).
[CrossRef]

D. L. Fried and J. B. Seidman, “Laser-beam scintillation in the atmosphere,” J. Opt. Soc. Am. 57, 181–185 (1967).
[CrossRef]

Baltes, H. P.

Banakh, V. A.

Beran, M. J.

Carter, W. H.

Clifford, S. F.

Dashen, R.

R. Dashen, “Path integral for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

deWolf, D. A.

D. A. deWolf, “Operator and diagram techniques for solving intensity moment equations in random media: applications to strong scattering,” Radio Sci. 14, 277–286 (1979).
[CrossRef]

Dunphy, J. R.

Elliott, R. A.

Fante, R. L.

R. L. Fante, “Some physical insights into beam propagation in strong turbulence,” Radio Sci. 15, 757–762 (1980).
[CrossRef]

R. L. Fante, “Two-source spherical wave structure functions in atmospheric turbulence,” J. Opt. Soc. Am. 66, 74 (1976).
[CrossRef]

R. L. Fante, “Electric field spectrum and intensity covariance of a wave in a random medium,” Radio Sci. 10, 77–85 (1975).
[CrossRef]

Feizulin, Z. I.

Z. I. Feizulin and Yu. A. Krovtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 68–73 (1967).
[CrossRef]

Fried, D. L.

Gracheva, M. L.

M. L. Gracheva and et al., “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation,” Akademiia Nauk SSSR. Otdelenie Okeanologii, Fizikii, Atmosfery i Geografii. [Preprint, Moscow, (1973). Aerospace Corp. translation no. LRG-73-T-28].

Grossman, K.

H. van de Hulst and K. Grossman, The Atmosphere of Venus and Mars, J. C. Brandt and M. B. McElroy, eds. (Gordon and Breach, New York, 1968).

Hansen, J. L.

J. L. Hansen, “Radiative transfer by doubling very thin layers,” Astrophys. J. 155, 565–573 (1969).
[CrossRef]

Ishimaru, A.

A. Ishimaru, “Fluctuations of a focused beam wave for atmospheric turbulence probing,” Proc. IEEE 57, 407–414 (1969).
[CrossRef]

Kerr, J. R.

Khmelevtsov, S. S.

Krovtsov, Yu. A.

Z. I. Feizulin and Yu. A. Krovtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 68–73 (1967).
[CrossRef]

Lawrence, R. S.

S. F. Clifford, G. R. Ochs, and R. S. Lawrence, “Saturation of optical scintillation by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
[CrossRef]

R. S. Lawrence and J. W. Strohbehn, “A survey of clean-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Leader, J. C.

The concept of a spatially partially coherent beam used in this paper is discussed in J. C. Leader, “Similarities and distinctions between coherence theory relations and laser scattering phenomena,” Opt. Eng. 19, 593–601 (1980).
[CrossRef]

J. C. Leader, “Beam properties of partially coherent curved beam waves in the turbulent atmosphere,” J. Opt. Soc. Am. 70, 682–688 (1980).
[CrossRef]

J. C. Leader, “Intensity fluctuations resulting from partially coherent light propagating through atmospheric turbulence,” J. Opt. Soc. Am. 69, 73–84 (1979).
[CrossRef]

Ochs, G. R.

Ouyang, C. F.

Pedersen, H. M.

Pincus, P. A.

Plonus, M. A.

Sedin, V. Ya.

V. Ya. Sedin and et al., “Intensity fluctuations of a focused laser beam in air,” Radioteckh. Elektron. 15, 1290–1292 (1970).

Seidman, J. B.

Steinle, B.

Strohbehn, J. W.

R. S. Lawrence and J. W. Strohbehn, “A survey of clean-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Tatarskii, V.

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

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium, R. A. Silverman, trans. (Dover, New York, 1967).

Tur, M.

van de Hulst, H.

H. van de Hulst and K. Grossman, The Atmosphere of Venus and Mars, J. C. Brandt and M. B. McElroy, eds. (Gordon and Breach, New York, 1968).

Wandzura, S. M.

Wang, S. C. H.

Wolf, L.

Yura, H. T.

Appl. Opt. (3)

Astrophys. J. (1)

J. L. Hansen, “Radiative transfer by doubling very thin layers,” Astrophys. J. 155, 565–573 (1969).
[CrossRef]

J. Math. Phys. (1)

R. Dashen, “Path integral for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

J. Opt. Soc. Am. (13)

J. R. Kerr, “Experiments on turbulence characteristics and multiwavelength scintillation phenomena,” J. Opt. Soc. Am. 62, 1040–1049 (1972).
[CrossRef]

J. R. Kerr and J. R. Dunphy, “Experimental effects of finite transmitter apertures on scintillations,” J. Opt. Soc. Am. 63, 1–8 (1973).
[CrossRef]

S. F. Clifford, G. R. Ochs, and R. S. Lawrence, “Saturation of optical scintillation by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
[CrossRef]

S. F. Clifford and H. T. Yura, “Equivalence of two theories of strong optical scintillation,” J. Opt. Soc. Am. 64, 1641–1644 (1974).
[CrossRef]

H. M. Pedersen, “Theory of speckle dependence on surface roughness,” J. Opt. Soc. Am. 66, 1204–1210 (1976).
[CrossRef]

B. Steinle and H. P. Baltes, “Radiant intensity and spatial coherence for finite planar sources,” J. Opt. Soc. Am. 67, 241–247 (1977).
[CrossRef]

W. H. Carter and L. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
[CrossRef]

J. C. Leader, “Beam properties of partially coherent curved beam waves in the turbulent atmosphere,” J. Opt. Soc. Am. 70, 682–688 (1980).
[CrossRef]

J. C. Leader, “Intensity fluctuations resulting from partially coherent light propagating through atmospheric turbulence,” J. Opt. Soc. Am. 69, 73–84 (1979).
[CrossRef]

D. L. Fried and J. B. Seidman, “Laser-beam scintillation in the atmosphere,” J. Opt. Soc. Am. 57, 181–185 (1967).
[CrossRef]

S. M. Wandzura, “Meeting of quadratic structure functions,” J. Opt. Soc. Am. 70, 745–747 (1980).
[CrossRef]

V. A. Banakh and et al., “Focused-laser-beam scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. 64, 516–518 (1974).
[CrossRef]

R. L. Fante, “Two-source spherical wave structure functions in atmospheric turbulence,” J. Opt. Soc. Am. 66, 74 (1976).
[CrossRef]

Opt. Eng. (1)

The concept of a spatially partially coherent beam used in this paper is discussed in J. C. Leader, “Similarities and distinctions between coherence theory relations and laser scattering phenomena,” Opt. Eng. 19, 593–601 (1980).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (2)

R. S. Lawrence and J. W. Strohbehn, “A survey of clean-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

A. Ishimaru, “Fluctuations of a focused beam wave for atmospheric turbulence probing,” Proc. IEEE 57, 407–414 (1969).
[CrossRef]

Radio Sci. (3)

D. A. deWolf, “Operator and diagram techniques for solving intensity moment equations in random media: applications to strong scattering,” Radio Sci. 14, 277–286 (1979).
[CrossRef]

R. L. Fante, “Electric field spectrum and intensity covariance of a wave in a random medium,” Radio Sci. 10, 77–85 (1975).
[CrossRef]

R. L. Fante, “Some physical insights into beam propagation in strong turbulence,” Radio Sci. 15, 757–762 (1980).
[CrossRef]

Radiophys. Quantum Electron. (1)

Z. I. Feizulin and Yu. A. Krovtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 68–73 (1967).
[CrossRef]

Radioteckh. Elektron. (1)

V. Ya. Sedin and et al., “Intensity fluctuations of a focused laser beam in air,” Radioteckh. Elektron. 15, 1290–1292 (1970).

Other (9)

M. L. Gracheva and et al., “Similarity correlations and their experimental verification in the case of strong intensity fluctuations of the laser radiation,” Akademiia Nauk SSSR. Otdelenie Okeanologii, Fizikii, Atmosfery i Geografii. [Preprint, Moscow, (1973). Aerospace Corp. translation no. LRG-73-T-28].

Ref. 4, Chap. 2.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium, R. A. Silverman, trans. (Dover, New York, 1967).

Ref. 4, Chap 3, p. 87.

J. C. Dainty, ed., Laser Speckle, Topics in Applied Physics, Vol. 9 (Springer-Verlag, New York, 1975).

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

J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere, Topics in Applied Physics, Vol. 25 (Springer-Verlag, New York, 1978).
[CrossRef]

H. van de Hulst and K. Grossman, The Atmosphere of Venus and Mars, J. C. Brandt and M. B. McElroy, eds. (Gordon and Breach, New York, 1968).

Ref. 4, Chap. 5.

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Figures (17)

Fig. 1
Fig. 1

Propagation geometry.

Fig. 2
Fig. 2

Graphical depiction of the cross-phase log-amplitude structure function approximation.

Fig. 3
Fig. 3

Predictions of the calculated normalized intensity deviation as a function of the plane-wave log-intensity deviation using (a) the structure function approximations of this paper, (b) the small-aperture structure functions of Ref. 9, and (c) plane-wave and spherical-wave Rytov theory.

Fig. 4
Fig. 4

Predictions of the normalized intensity deviation as a function of phase curvature radius and beam radius for coherent turbulent-media propagation with σϕ = 0, λ = 1.0 μm, R = 3 km, and Cn2 = 10−15 m2/3.

Fig. 5
Fig. 5

(a) Predictions of the normalized intensity deviation as a function of source-phase variance and phase-curvature radius for a beam in vacuo with LA = 0.05 m, λ = 1.0 μm, ρu = 10−5 m, and R = 3 km. (b) Predictions of the normalized intensity deviation as a function of turbulence and phase-curvature radius for a coherent beam in turbulence with LA = 0.05 m, λ = 1.0 μm, σϕ = 0, and R = 3 km.

Fig. 6
Fig. 6

Predictions of the normalized intensity deviation as a function of range and phase variance for a partially coherent focused beam with Lϕ = 3 km, ρu = 10−5 m, LA = 0.05 m, λ = 1.0 μm, and Cn2 = 5 × 10−6 m−2/3.

Fig. 7
Fig. 7

Comparison of the predictions of the normalized intensity-deviation variation with aperture size using the extended-aperture solution of this paper and the small-aperture analysis of Ref. 9. Values for the Rytov plane-wave and spherical-wave predictions are annotated. A coherent beam wave at a wavelength of 1 μm propagating through 5 km of homogeneous turbulence with Cn2 = 10−15 m−2/3 is assumed.

Fig. 8
Fig. 8

Comparison of the normalized intensity-deviation predictions of the extended theory (this paper) and Rytov theory for a focused beam in turbulence with the corresponding measurements of Ref. 26.

Fig. 9
Fig. 9

Predictions of the normalized collimated-beam covariance function at 5 km for three different turbulence strengths. A beam radius of LA = 0.5 m is assumed at a wavelength λ = 1.0 μm.

Fig. 10
Fig. 10

Comparison of the predictions for the collimated-beam normalized intensity covariance with experimental data of Gracheva et al.26 Experimental parameters assumed for the calculations are LA = 0.15 m, λ = 0.6328 μm, R = 0.65 km, and a nominal refractive-index structure constant, Cn2 = 5 × 10−14 m−2/3.

Fig. 11
Fig. 11

Comparison of the predictions for the normalized intensity-covariance function at a range of 1 km for collimated-, spherical-, and focused-beam conditions. A wavelength of 1 μm, a beam radius of LA = 15 cm, and a refractive-index structure constant of Cn2 = 10−15 m−2/3 are assumed for the calculations.

Fig. 12
Fig. 12

Focused-beam intensity-covariance predictions for various beam radii at a range of 1 km assuming λ = 1.0 μm and Cn2 = 10−15 m−2/3. Annotated arrows denote the predicted beam radius at the focus in turbulence. Predicted values for the normalized intensity deviation corresponding to the assumed beam radii are also shown.

Fig. 13
Fig. 13

Calculated values of the log-amplitude variance as a function of plane-wave log-intensity deviation σP.

Fig. 14
Fig. 14

Iteratively calculated values of the normalized intensity deviation and phase deviation for a succession of weakly turbulent path intervals of 0.5, 0.25, and 0.125 km assuming collimated-beam conditions (LA = 0.5 m) at a wavelength of 1.0 μm and a refractive-index structure constant of Cn2 = 10−15 m−2/3.

Fig. 15
Fig. 15

Iteratively calculated values of the normalized intensity deviation for a succession of weakly turbulent path intervals of 0.25 km assuming collimated-beam conditions (LA = 0.5 m) at a wavelength of 1.0 μm and a refractive-index structure constant of Cn2 = 10−14 m−2/3.

Fig. 16
Fig. 16

Calculated normalized intensity-covariance functions at a wavelength of 1.0 μm for two weakly turbulent path intervals of 0.5 km in strong turbulence. One path-interval calculation assumes Cn2 = 10−15 m−2/3 and a range of 25 km, corresponding to a plane-wave deviation of σP = 3, and the other assumes Cn2 = 10−14 m−2/3 and a range of 21 km, yielding σP = 11.

Fig. 17
Fig. 17

Calculated normalized intensity-covariance functions at a wavelength of 0.6328 μm and a range of 8.5 km for two weakly turbulent path intervals of 0.125 km in strong turbulence. One path-interval calculation assumes Cn2 = 10−14 m−2/3, corresponding to a plane-wave log-intensity deviation of σP = 5.4, and the other assumes Cn2 = 4.0 × 10−14 m−2/3, yielding σP = 10.7. Two abscissas are provided, indicating the value of the unitless parameter ρ(C2k3)3/11 for the two assumed turbulence strengths. The experimental results obtained by Gracheva et al.27 for the magnitude of the coherence tail in strong turbulence are also illustrated.

Tables (4)

Tables Icon

Table 1 QSF Weak-Turbulence Atmospheric Structure Functions

Tables Icon

Table 2 Transformed Propagation Coefficients

Tables Icon

Table 3 Transformed Source Coefficients

Tables Icon

Table 4 Discontinuous Integration Coefficients

Equations (65)

Equations on this page are rendered with MathJax. Learn more.

I ( P ) I ( P ) = k 4 cos 2 θ cos 2 θ ( 2 π ) 4 ( R R ) 2 - - × d 2 r 1 d 2 r 2 d 2 r 3 d 2 r 4 Γ 4 S ( r 1 , r 2 , r 3 , r 4 ) H ( r 1 , r 2 , r 3 , r 4 , P , P ) × exp [ i k ( r 1 - P - r 2 - P + r 3 - P - r 4 - P ) ] ,
H ( r 1 , r 2 , r 3 , r 4 , P , P ) = exp { χ 2 - 1 2 [ D 1 ( r 12 , 0 ) + D 1 ( r 13 , P - ) + D 1 ( r 14 , P - ) + D 1 ( r 23 , P - ) + D 1 ( r 24 , P - ) + D 1 ( r 34 , 0 ) ] + D S ( r 24 , P - ) + D S ( r 13 , P - ) + i [ D χ S ( r 24 , P - ) - D χ S ( r 13 , P - ) ] } ,
U S ( r ) = U 0 exp [ i ϕ ( r ) - k α 2 r 2 ] ,
α = 2 k L A 2 + i L ϕ ,
Γ 4 S ( r 1 , r 2 , r 3 , r 4 ) = U 0 4 exp ( - k 2 ( α r 1 2 + α * r 2 2 + α r 3 2 + α * r 4 2 ) - 2 σ ϕ 2 ) ( ( χ + ) 2 { exp [ - ( r 12 2 + r 34 2 ) ρ u - 2 ] + exp [ - ( r 14 2 + r 23 2 ) ρ u - 2 } + ( χ - ) 2 exp [ - ( r 13 2 + r 24 2 ) ρ u - 2 ] + χ + [ exp ( - r 12 2 ρ u - 2 ) + exp ( - r 34 2 ρ u - 2 ) + exp ( - r 14 2 ρ u - 2 ) + exp ( - r 23 2 ρ u - 2 ) ] + χ - [ exp ( - r 13 2 ρ u - 2 ) + exp ( - r 24 2 ρ u - 2 ) ] + 1 ) ,
χ + = ( e σ ϕ 2 - 1 ) ,             χ - = ( e - σ ϕ 2 - 1 ) ,             σ ϕ 2 = ϕ 2 ,
x 2 = P ˆ - 2 + P ˆ - · r ˆ m n + r ˆ m n 2 ,
ρ 1 = ( 3 / 8 1.455 k 2 R C n 2 ) - 3 / 5 , ρ S = ( 0.875 k 13 / 6 R 5 / 6 C n 2 ) - 1 / 2 , ρ χ S = ( 0.234 k 13 / 6 R 5 / 6 C n 2 ) - 1 / 2 , A = 0.485 18 ,
D S ( x ) 2 ρ 1 - 2 x 2
D 1 ( x ) 2 ρ 1 - 2 x 2
D χ S ( x ) { ρ χ S - 2 x 2 for x L m 7.46 χ 2 for x > L M ,
ρ 1 = ( r 1 + r 2 + r 3 + r 4 ) / 2 , ρ 3 = ( r 1 - r 3 - r 2 + r 4 ) / 2 , ρ 2 = ( r 1 - r 3 + r 2 - r 4 ) / 2 , ρ 4 = ( r 1 + r 3 - r 2 - r 4 ) / 2.
J = - - d 2 r 1 d 2 r 2 d 2 r 3 d 2 r 4 × exp [ m = 0 4 n = 0 4 ( a m n x m x n + b m n y m y n ) ]
J = - - d 2 ρ 1 d 2 ρ 2 d 2 ρ 3 d 2 ρ 4 × exp [ m = 0 4 n = 0 4 ( α m n ρ x m ρ x n + β m n ρ x m ρ y n ) ] ,
α i j = ρ i k a k l ρ l j , β i j = ρ i k β k l ρ l j ,
x 0 = y 0 = 1 , a m n a n m , b m n b n m for m n , a m m < 0 , b m m < 0 for m > 0 ,
α m n ( l ) = α m n propagation + α ( l ) source , β m m ( l ) = β m n propagation + β ( l ) source ,
ρ 2 = 1 2 ( r 13 + r 24 )
ρ 3 = 1 2 ( r 13 - r 24 ) ,
J = - d 2 ρ 2 d 2 ρ 3 exp [ - α 22 ( ± ) ρ 2 2 - α 33 ( ± ) ρ 3 2 + 2 α 23 ( ± ) ρ 2 · ρ 3 + 2 β 20 ( ± ) · ρ 2 + 2 β 30 ( ± ) · ρ 3 ] Θ [ ± ( L m 2 - ρ 2 + ρ 3 2 ) ] × Θ [ ± ( L m 2 - ρ 2 - ρ 3 2 ) ] ,
J = 1 4 - d 2 r 13 d 2 r 24 exp [ - 1 2 A 1 r 13 2 - 1 2 A 2 r 24 2 - B 1 · r 13 - B 2 · r 24 + C 12 r 13 · r 24 ] × Θ [ ± ( L m 2 - r 13 2 ) ] Θ [ ± ( L m 2 - r 24 2 ) ] ,
A 1 = 1 2 ( α 22 + α 33 ) - α 23 ,             A 2 = 1 2 ( α 22 + α 33 ) + α 23 , B 1 = - ( β 20 + β 30 ) ,             B 2 = β 30 - β 20 ,
C 12 = α 33 - α 22 .
U ( A n , B n , L m ) - d 2 r exp ( - A n r 2 / 2 - i B n i · r ) × Θ ( L m 2 - r 2 ) ,
B n i = Im B n .
U ( A n , B n , L m ) = 2 π L m exp [ - ( B n 2 / 2 A n ) ] A n × 0 d B J 1 ( B L m ) I 0 ( B B n i / A n ) exp [ - ( B 2 / 2 A n ) ]
U ( A n , 0 , L m ) = 2 π A n [ 1 - exp ( - L m 2 2 A n ) ] .
V ( A n , B n , L m ) - d 2 r exp ( - A n r 2 / 2 - i B n i · r ) × Θ [ - ( L m 2 - r 2 ) ] = - d 2 r exp ( - A n r 2 / 2 - i B n i · r ) × [ 1 - Θ ( L m 2 - r 2 ) ] = U ( A n , B n , ) - U ( A n , B n , L m ) = 2 π A n exp [ - ( B n 2 / 2 A n ) ] - U ( A n , B n , L m ) .
J = 1 4 [ U ( A 1 + + , B 1 + + , L m ) U ( A 2 + + , B 2 + + , L m ) + U ( A 1 + - , B 1 + - , L m ) V ( A 2 + - , B 2 + - , B 2 + - , L m ) × exp ( i 7.46 χ 2 ) + V ( A 1 - + , B 1 - + , L m ) × U ( A 2 - + , B 2 - + , L m ) exp ( - i 7.46 χ 2 ) + V ( A 1 - - , B 1 - - , L m ) V ( A 2 - - , B 2 - - , L m ) ] ,
U ( A 2 + + , B 2 + + , L m ) = U * ( A 1 + + , B 1 + + , L m ) , U ( A 1 + - , B 1 + 1 , L m ) = U ( A 1 + + , B 1 + + , L m ) , U ( A 2 - + , B 2 - + , L m ) = U ( A 2 + + , B 2 + + , L m ) , U ( A 1 - + , B 1 - + , L m ) = U * ( A 2 + - , B 2 + - , L m ) = U ( A 1 - - , B 1 - - , L m ) = U * ( A 2 - - , B 2 - - , L m ) .
J = - d 2 ρ 1 d 2 ρ 4 exp ( - α 11 ρ 4 2 - α 44 ρ 4 2 + 2 α 23 ρ 1 · ρ 4 ) = π 2 ( α 11 α 44 - α 14 2 ) - 1 = π 2 [ L A - 4 ( 4 L A 2 / ρ 1 2 + 1 ) + k 2 4 ( R - 1 + L ϕ - 1 ) 2 ] - 1 ,
I ( P ) I ( P ) coherent = U 0 4 k 4 e 4 χ 2 ( 2 π R ) - 4 J J .
J = π 2 [ L A - 4 + k 2 4 ( L ϕ - 1 - R - 1 + 4 A ρ χ S - 2 k - 1 ) 2 ] - 1
I 2 coherent = U 0 4 k 4 e 4 χ 2 ( 2 R ) - 4 × [ L A - 4 ( 4 L A 2 / ρ 1 2 + 1 ) + k 2 4 ( R - 1 - L ϕ - 1 ) 2 ] - 1 × [ L A - 4 + k 2 4 ( L ϕ - 1 - R - 1 + 4 A ρ χ S - 2 k - 1 ) 2 ] - 1 for L A ( λ R ) 1 / 2 .
I coherent = U 0 2 [ ρ 1 - 2 ( 2 R / k L A ) 2 + 1 + i α R 2 ] - 1 .
σ I 2 = I 2 I 2 - 1 ,
σ P = ( 1.23 k 7 / 6 C n 2 R 11 / 6 ) 1 / 2 .
D S = ( × 2.91 C n 2 R k 2 L A 5 / 3 ) 1 / 2 .
b I ( δ R ) = I ( - δ / 2 ) I ( δ / 2 ) - I ( - δ / 2 ) I ( δ / 2 ) I 2 ( 0 ) - I ( 0 ) 2 ,
L A ( R ) = L A ( Z 2 ρ 0 - 2 + 1 + i α R 2 ) 1 / 2 ,
L ϕ ( R ) = R ( Z 2 ρ 0 - 2 + 1 + i α R 2 ) ( Im α R - α 2 R 2 - ½ Z 2 ρ 0 - 2 ) - 1 ,
ρ u ( R ) = Z ( Z 2 ρ 0 - 2 + 1 + i α R 2 ) 1 / 2 × [ ρ 0 - 2 Z 2 1 + i α R 2 + ρ 0 - 2 Z 2 ( R Im α + ¾ Z 2 ρ 0 - 2 ) ] - 1 / 2 ,
Z 2 = 2 ( 2 R / k L A ) 2 .
Γ 4 S ( R ; r 1 , r 2 , r 3 , r 4 ) = exp [ 4 χ 2 ( R ) ] × Γ 4 S [ L A ( R ) , L ϕ ( R ) , ρ u ( R ) , U 0 4 ( R ) , r 1 , r 2 , r 3 , r 4 ] ,
U 0 4 ( R ) = I ( R ) 2 ,
σ I 2 ( σ ϕ ) = ( 1 - e - 2 σ ϕ 2 ) 2 .
σ I 2 ( σ ϕ , χ 2 ) = e 4 χ 2 ( e - 4 σ ϕ 2 - 2 e - 2 σ ϕ 2 + 2 ) - 1.
σ I 2 ( σ ϕ , χ 2 ) σ ϕ 0 e 4 χ 2 - 1 ,
σ I 2 ( σ ϕ ) = [ σ I 2 ( σ ϕ , χ 2 ) + 1 ] e 4 χ 2 - 1 ,
σ ϕ 2 = - 1 2 ln [ 1 - σ I ( σ ϕ ) ] .
χ 2 ( R ) = 0.132 π 2 k 2 R C n 2 0 1 d u 0 d κ κ - 8 / 3 × M [ κ R u ( 1 - u ) / k , R ] sin 2 [ κ 2 u ( 1 - u ) R 2 k ]
M 2 ( x , R ) = exp [ - ( x / ρ 1 ) 2 ] .
χ 2 ( R ) = 2.95 σ χ 2 0 1 d u 0 d y sin 2 y y 11 / 6 × [ u ( 1 - u ) ] 5 / 6 exp [ - α y u ( 1 - u ) ] ,
α = 2 R k ρ 1 2 = 11.72 ( σ χ 2 ) 6 / 5
σ χ 2 = 0.124 k 7 / 6 R 11 / 6 C n 2
χ 2 2.95 σ χ 2 0 1 d u 0 d y sin 2 y y 11 / 6 [ u ( 1 - u ) ] 5 / 6             for α 1 ,
χ 2 = 2.95 σ χ 2 [ - Γ ( - 5 / 6 ) cos - 5 π / 12 2 1 / 6 ] × [ Γ ( 11 / 6 ) Γ ( 11 / 6 ) Γ ( 22 / 6 ) ] = σ χ 2 ,
0 d y sin 2 y y 11 / 6 exp [ - α y u ( 1 - u ) ] α 0 d y y 2 y 11 / 6 exp [ - α y u ( 1 - u ) ]
χ 2 α 2.95 σ χ 2 Γ ( 7 / 6 ) α 7 / 6 Γ ( 2 / 3 ) Γ ( 2 / 3 ) Γ ( 4 / 3 ) = 0.32 ( σ χ 2 ) - 2 / 5 .
M 5 / 3 ( x , R ) = exp [ - ( x / ρ 1 ) 5 / 3 ]
0 d y sin 2 y y 11 / 6 exp { - [ α y u ( 1 - u ) ] 5 / 6 } α 0 d y y 2 y 11 / 6 exp { [ α y u ( 1 - u ) ] 5 / 6 } = 6 5 [ α u ( 1 - u ) ] - 7 / 6 Γ ( 7 / 5 ) .
χ 2 α 0.36 ( σ χ 2 ) - 2 / 5             for M ( x , R ) = M 5 / 3 ( x , R )
Δ χ 2 i + 1 = χ 2 ( R i + Δ R ) - χ 2 ( R i ) ,
σ I 2 ( σ ϕ , χ 2 ) 2 e 4 χ 2 - 1 2 ( 4 χ 2 + 1 ) - 1 = 1 + 8 χ 2             for σ P 1.
σ I 2 = 1 + 2.565 ( σ χ 2 ) - 2 / 5 .