Abstract

A simple, analytic, geometrical optics expression for the variance of the beam displacements caused by propagation through weak refractive turbulence described by the Kolmogorov spectrum is presented. The analytical formula includes the effect of the divergence or convergence of the initial beam. The formula is compared with numerical results obtained from a more complicated expression including effects of diffraction and strong path-integrated turbulence. The simple geometrical optics expression holds for apertures larger than the Fresnel zone size and larger than the ratio of the square of the Fresnel zone to the phase coherence length.

© 1990 Optical Society of America

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References

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  1. L. A. Chernov, Wave Propagation in a Random Medium (Dover, New York, 1967), p. 26.
  2. P. Beckmann, “Signal Degeneration in Laser Beams Propagated Through a Turbulent Atmosphere,” Radio Sci. 69D, 629–640 (1965).
  3. T. Chiba, “Spot Dancing of the Laser Beam Propagated Through the Atmosphere,” Appl. Opt. 10, 2456–2461 (1971).
    [CrossRef] [PubMed]
  4. G. A. Andreev, E. I. Gelfer, “Angular Random Walks of the Center of Gravity of the Cross Section of a Diverging Light Beam,” Radiophys. Quantum Electron. 14, 1145–1147 (1971).
    [CrossRef]
  5. M. A. Kallistratova, V. V. Pokasov, “Defocusing and Fluctuations of the Displacement of a Focused Laser Beam in the Atmosphere,” Radiophys. Quantum Electron. 14, 940–945 (1971).
    [CrossRef]
  6. J. A. Dowling, P. M. Livingston, “Behavior of Focused Beams in Atmospheric Turbulence: Measurements and Comments on the Theory,” J. Opt. Soc. Am. 63, 846–858 (1973).
    [CrossRef]
  7. J. R. Dunphy, J. R. Kerr, “Turbulence Effects on Target Illumination by Laser Sources: Phenomenological Analysis and Experimental Results,” Appl. Opt. 16, 1345–1358 (1977).
    [CrossRef] [PubMed]
  8. V. I. Klyatskin, A. I. Kon, “On the Displacement of Spatially-Bounded Light Beams in a Turbulent Medium in the Markovian-Random-Process Approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
    [CrossRef]
  9. A. I. Kon, V. L. Mironov, V. V. Nosov, “Dispersion of Light Beam Displacements in the Atmosphere with Strong Intensity Fluctuations,” Radiophys. Quantum Electron. 19, 722–725 (1976).
    [CrossRef]
  10. V. L. Mironov, V. V. Nosov, “On the Theory of Spatially Limited Light Beam Displacements in a Randomly Inhomogeneous Medium,” J. Opt. Soc. Am. 67, 1073–1080 (1977).
    [CrossRef]
  11. R. J. Cook, “Beam Wander in a Turbulent Medium: an Application of Ehrenfest’s Theorem,” J. Opt. Soc. Am. 65, 942–948 (1975).
    [CrossRef]
  12. R. L. Fante, “Electromagnetic Beam Propagation in Turbulent Media: An Update,” Proc. IEEE 68, 1424–1443 (1980).
    [CrossRef]
  13. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971), p. 76.
  14. R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, Degradation of Laser Systems by Atmospheric Turbulence (Defense Advanced Research Projects Agency, Report Number R-117-1-ARPA/RC, 1973) Chap. 7.
  15. S. M. Wandzura, “Meaning of Quadratic Structure Functions,” J. Opt. Soc. Am. 70, 745–747.
  16. R. G. Frehlich, “Intensity Covariance of a Point Source in a Random Medium with a Kolmogorov Spectrum and Inner Scale of Turbulence,” J. Opt. Soc. Am. 4, 360–366 (1987).
    [CrossRef]

1987 (1)

R. G. Frehlich, “Intensity Covariance of a Point Source in a Random Medium with a Kolmogorov Spectrum and Inner Scale of Turbulence,” J. Opt. Soc. Am. 4, 360–366 (1987).
[CrossRef]

1980 (1)

R. L. Fante, “Electromagnetic Beam Propagation in Turbulent Media: An Update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

1977 (2)

1976 (1)

A. I. Kon, V. L. Mironov, V. V. Nosov, “Dispersion of Light Beam Displacements in the Atmosphere with Strong Intensity Fluctuations,” Radiophys. Quantum Electron. 19, 722–725 (1976).
[CrossRef]

1975 (1)

1973 (1)

1972 (1)

V. I. Klyatskin, A. I. Kon, “On the Displacement of Spatially-Bounded Light Beams in a Turbulent Medium in the Markovian-Random-Process Approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[CrossRef]

1971 (3)

G. A. Andreev, E. I. Gelfer, “Angular Random Walks of the Center of Gravity of the Cross Section of a Diverging Light Beam,” Radiophys. Quantum Electron. 14, 1145–1147 (1971).
[CrossRef]

M. A. Kallistratova, V. V. Pokasov, “Defocusing and Fluctuations of the Displacement of a Focused Laser Beam in the Atmosphere,” Radiophys. Quantum Electron. 14, 940–945 (1971).
[CrossRef]

T. Chiba, “Spot Dancing of the Laser Beam Propagated Through the Atmosphere,” Appl. Opt. 10, 2456–2461 (1971).
[CrossRef] [PubMed]

1965 (1)

P. Beckmann, “Signal Degeneration in Laser Beams Propagated Through a Turbulent Atmosphere,” Radio Sci. 69D, 629–640 (1965).

Andreev, G. A.

G. A. Andreev, E. I. Gelfer, “Angular Random Walks of the Center of Gravity of the Cross Section of a Diverging Light Beam,” Radiophys. Quantum Electron. 14, 1145–1147 (1971).
[CrossRef]

Beckmann, P.

P. Beckmann, “Signal Degeneration in Laser Beams Propagated Through a Turbulent Atmosphere,” Radio Sci. 69D, 629–640 (1965).

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium (Dover, New York, 1967), p. 26.

Chiba, T.

Cook, R. J.

Dowling, J. A.

Dunphy, J. R.

Fante, R. L.

R. L. Fante, “Electromagnetic Beam Propagation in Turbulent Media: An Update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

Frehlich, R. G.

R. G. Frehlich, “Intensity Covariance of a Point Source in a Random Medium with a Kolmogorov Spectrum and Inner Scale of Turbulence,” J. Opt. Soc. Am. 4, 360–366 (1987).
[CrossRef]

Gelfer, E. I.

G. A. Andreev, E. I. Gelfer, “Angular Random Walks of the Center of Gravity of the Cross Section of a Diverging Light Beam,” Radiophys. Quantum Electron. 14, 1145–1147 (1971).
[CrossRef]

Huschke, R. E.

R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, Degradation of Laser Systems by Atmospheric Turbulence (Defense Advanced Research Projects Agency, Report Number R-117-1-ARPA/RC, 1973) Chap. 7.

Kallistratova, M. A.

M. A. Kallistratova, V. V. Pokasov, “Defocusing and Fluctuations of the Displacement of a Focused Laser Beam in the Atmosphere,” Radiophys. Quantum Electron. 14, 940–945 (1971).
[CrossRef]

Kerr, J. R.

Klyatskin, V. I.

V. I. Klyatskin, A. I. Kon, “On the Displacement of Spatially-Bounded Light Beams in a Turbulent Medium in the Markovian-Random-Process Approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[CrossRef]

Kon, A. I.

A. I. Kon, V. L. Mironov, V. V. Nosov, “Dispersion of Light Beam Displacements in the Atmosphere with Strong Intensity Fluctuations,” Radiophys. Quantum Electron. 19, 722–725 (1976).
[CrossRef]

V. I. Klyatskin, A. I. Kon, “On the Displacement of Spatially-Bounded Light Beams in a Turbulent Medium in the Markovian-Random-Process Approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[CrossRef]

Livingston, P. M.

Lutomirski, R. F.

R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, Degradation of Laser Systems by Atmospheric Turbulence (Defense Advanced Research Projects Agency, Report Number R-117-1-ARPA/RC, 1973) Chap. 7.

Meecham, W. C.

R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, Degradation of Laser Systems by Atmospheric Turbulence (Defense Advanced Research Projects Agency, Report Number R-117-1-ARPA/RC, 1973) Chap. 7.

Mironov, V. L.

V. L. Mironov, V. V. Nosov, “On the Theory of Spatially Limited Light Beam Displacements in a Randomly Inhomogeneous Medium,” J. Opt. Soc. Am. 67, 1073–1080 (1977).
[CrossRef]

A. I. Kon, V. L. Mironov, V. V. Nosov, “Dispersion of Light Beam Displacements in the Atmosphere with Strong Intensity Fluctuations,” Radiophys. Quantum Electron. 19, 722–725 (1976).
[CrossRef]

Nosov, V. V.

V. L. Mironov, V. V. Nosov, “On the Theory of Spatially Limited Light Beam Displacements in a Randomly Inhomogeneous Medium,” J. Opt. Soc. Am. 67, 1073–1080 (1977).
[CrossRef]

A. I. Kon, V. L. Mironov, V. V. Nosov, “Dispersion of Light Beam Displacements in the Atmosphere with Strong Intensity Fluctuations,” Radiophys. Quantum Electron. 19, 722–725 (1976).
[CrossRef]

Pokasov, V. V.

M. A. Kallistratova, V. V. Pokasov, “Defocusing and Fluctuations of the Displacement of a Focused Laser Beam in the Atmosphere,” Radiophys. Quantum Electron. 14, 940–945 (1971).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971), p. 76.

Wandzura, S. M.

Yura, H. T.

R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, Degradation of Laser Systems by Atmospheric Turbulence (Defense Advanced Research Projects Agency, Report Number R-117-1-ARPA/RC, 1973) Chap. 7.

Appl. Opt. (2)

J. Opt. Soc. Am. (5)

Proc. IEEE (1)

R. L. Fante, “Electromagnetic Beam Propagation in Turbulent Media: An Update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

Radio Sci. (1)

P. Beckmann, “Signal Degeneration in Laser Beams Propagated Through a Turbulent Atmosphere,” Radio Sci. 69D, 629–640 (1965).

Radiophys. Quantum Electron. (4)

G. A. Andreev, E. I. Gelfer, “Angular Random Walks of the Center of Gravity of the Cross Section of a Diverging Light Beam,” Radiophys. Quantum Electron. 14, 1145–1147 (1971).
[CrossRef]

M. A. Kallistratova, V. V. Pokasov, “Defocusing and Fluctuations of the Displacement of a Focused Laser Beam in the Atmosphere,” Radiophys. Quantum Electron. 14, 940–945 (1971).
[CrossRef]

V. I. Klyatskin, A. I. Kon, “On the Displacement of Spatially-Bounded Light Beams in a Turbulent Medium in the Markovian-Random-Process Approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[CrossRef]

A. I. Kon, V. L. Mironov, V. V. Nosov, “Dispersion of Light Beam Displacements in the Atmosphere with Strong Intensity Fluctuations,” Radiophys. Quantum Electron. 19, 722–725 (1976).
[CrossRef]

Other (3)

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971), p. 76.

R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, Degradation of Laser Systems by Atmospheric Turbulence (Defense Advanced Research Projects Agency, Report Number R-117-1-ARPA/RC, 1973) Chap. 7.

L. A. Chernov, Wave Propagation in a Random Medium (Dover, New York, 1967), p. 26.

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Figures (4)

Fig. 1
Fig. 1

Ratio of the single-axis beam wander variance σ2 to the collimated beam value σ COL 2 as a function of the path length L divided by focal range F.

Fig. 2
Fig. 2

Ratio of the total beam wander variance s2 to the collimated beam value σ COL 2 for various values of a = λ L / D as a function of the path length L divided by the focal length F. Here D is the source diameter and λ is the beam wavelength. The variances were evaluated for an infinite outer scale and weak path-integrated turbulence.

Fig. 3
Fig. 3

Ratio of the total beam wander variance s2 to the weak-scatter, geometrical optics value s 0 2 for various values of γ = L/F as a function of λ L / D. The variance s2 was evaluated for an infinite outer scale and weak path-integrated turbulence.

Fig. 4
Fig. 4

Ratio of the total beam wander variance s2 to the weak-scatter, geometrical optics value s 0 2 for various values of γ = L/F as a function of S/D, where S = λL/ρ0 and ρ0 is the spherical wave coherence length. The variance s2 was evaluated for D λ L and an infinite outer scale.

Equations (17)

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d θ = Δ n ( z ) w ( z ) d z .
d p = ( L - z ) d θ .
p = 0 L Δ n ( z ) w ( z ) ( L - z ) d z .
σ 2 = 0 L 0 L Δ n ( z 1 ) Δ n ( z 2 ) ( L - z 1 ) ( L - z 2 ) w ( z 1 ) w ( z 2 ) d z 1 d z 2 ,
Δ n = n [ z , ½ w ( z ) ] - n [ z , - ½ w ( z ) ]
Δ n ( z 1 ) Δ n ( z 2 ) = ½ { n [ z 1 , ½ w ( z 1 ) ] - n [ z 2 , - ½ w ( z 2 ) ] } 2 + { n [ z 1 , - ½ w ( z 1 ) ] - n [ z 2 , ½ w ( z 2 ) ] } 2 - { n [ z 1 , ½ w ( z 1 ) ] - n [ z 2 , ½ w ( z 2 ) ] } 2 - { n [ z 1 , ½ w ( z 1 ) ] - n [ z 2 , ½ w ( z 2 ) ] } 2 - { n [ z 1 , - ½ w ( z 1 ) ] - n [ z 2 , - ½ w ( z 2 ) ] } 2 .
Δ n ( z 1 ) Δ n ( z 2 ) = C n 2 ( { ( z 2 - z 1 ) 2 + ¼ [ w ( z 2 ) - w ( z 1 ) ] 2 } 1 / 3 - { ( z 2 - z 1 ) 2 + ¼ [ w ( z 2 ) - w ( z 1 ) ] 2 } 1 / 3 ) ,
w ( z ) = D 1 - z F ,
σ 2 = D - 1 / 3 0 L 0 L C n 2 ( z 1 ) ( L - z 1 ) 2 D - 2 × ( 1 - z F ) - 2 { [ ( z 2 - z 1 ) 2 + D 2 ( 1 - z F ) 2 ] 1 / 3 - ( z 2 - z 1 ) 2 / 3 } d z 1 d z 2 .
σ 2 = 2.92 D - 1 / 3 L 3 0 1 C n 2 ( x ) ( 1 - x ) 2 | 1 - L F x | 1 / 3 d x .
σ 2 = 0.97 C n 2 D - 1 / 3 L 3 F 2 1 ( 1 3 , 1 ; 4 ; L F ) ,
σ COL 2 = 0.97 C n 2 D - 1 / 3 L 3 ,
σ FOC 2 = 1.10 C n 2 D - 1 / 3 L 3 .
s 2 = 4 π 2 0 L d z ( L - z ) 2 0 d K K 3 Φ n ( K , z ) × exp { - K 2 D 2 8 [ ( 1 - z F ) 2 + 16 z 2 k 2 D 4 ] - π D ( K z / k ) } ,
D ( ρ ) = 8 π 2 k 2 0 z d z 0 d K K Φ n ( K , z ) [ 1 - J 0 ( K ρ z / z ) ] ,
s 2 = 5.1 L 3 D - 1 / 3 0 1 C n 2 ( x ) ( 1 - x ) 2 [ 1 f ( x ) ] 1 / 6 ,
f ( x ) = ( 1 - γ x ) 2 + 0.41 a 4 x 2 + 1.3 x 3 β 2 ,

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