Abstract

Single-integral equations are developed for the angle-of-arrival and beam-wander variances of propagated Gaussian and uniform-intensity beams through refractive turbulence with a finite outer scale. The equations developed for Gaussian beams include initial wave-front curvature, diffraction, and turbulent spreading effects. Form-fitting expressions are used in both cases to reduce second- and third-order integral equations into equivalent single-integral forms. Numerical calculations with these new approximations give results within a few percent of the results obtained by using the more rigorous multiple integral expressions. Spectral analysis indicates that the refractive-index turbule size with the greatest influence on beam wander and angle of arrival is proportional to the outer-scale size.

© 1992 Optical Society of America

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  1. F. K. Brunner, “Determination of line averages of sensible heat flux using an optical method,” Boundary Layer Meteorol. 22, 193–207 (1982).
    [CrossRef]
  2. J. H. Churnside, R. J. Lataitis, “Wander of an optical beam in the turbulent atmosphere,” Appl. Opt. 29, 926–930 (1990).
    [CrossRef] [PubMed]
  3. W. S. Lewellen, “Use of invariant modeling,” in Handbook of Turbulence, W. Frost, T. H. Moulden, eds. (Plenum, New York, 1977), Chap. 9.
    [CrossRef]
  4. C. A. Paulson, “The mathematical representation of wind speed and temperature profiles in the unstable atmospheric surface layer,” Boundary Layer Meteorol. 9, 857–861 (1970).
  5. A. S. Monin, A. M. Obukhov, “Dimensionless characteristics of turbulence in the surface layer,” Dokl. Akad. Nauk. SSSR 93, 223–226 (1953).
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    [CrossRef] [PubMed]
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    [CrossRef]
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  13. 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]
  14. A. Ishimaru, “The beam wave case and remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), Chap. 5.
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  15. R. J. Hill, S. F. Clifford, “Theory of saturation of optical scintillation by strong turbulence for arbitrary refractive-index spectra,” J. Opt. Soc. Am. 71, 675–686 (1981).
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  16. S. M. Wandzura, “Meaning of quadratic structure functions,” J. Opt. Soc. Am. 70, 745–747 (1980).
    [CrossRef]
  17. 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]
  18. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, Orlando, Fla., 1980).
  19. W. L. Wolfe, G. J. Zissis, The Infrared Handbook (Environmental Research Center of Michigan, Ann Arbor, Mich., 1989), Chap. 6.

1990

1982

F. K. Brunner, “Determination of line averages of sensible heat flux using an optical method,” Boundary Layer Meteorol. 22, 193–207 (1982).
[CrossRef]

1981

1980

1977

1976

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

1973

1972

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

G. A. Andreev, E. I. Gelfer, “Angular random walks on the center of gravity of the cross section of a diverging light beam,” Radiophys. Quantum Electron. 14, 1145–1147 (1971).
[CrossRef]

T. Chiba, “Spot dancing of the laser beam propagated through the atmosphere,” Appl. Opt. 10, 2456–2461 (1971).
[CrossRef] [PubMed]

1970

C. A. Paulson, “The mathematical representation of wind speed and temperature profiles in the unstable atmospheric surface layer,” Boundary Layer Meteorol. 9, 857–861 (1970).

1965

P. Beckmann, “Signal degeneration in laser beams propagated through the atmosphere,” Radio Sci. 69, 629–640 (1965).

1953

A. S. Monin, A. M. Obukhov, “Dimensionless characteristics of turbulence in the surface layer,” Dokl. Akad. Nauk. SSSR 93, 223–226 (1953).

Andreev, G. A.

G. A. Andreev, E. I. Gelfer, “Angular random walks on 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 the atmosphere,” Radio Sci. 69, 629–640 (1965).

Brunner, F. K.

F. K. Brunner, “Determination of line averages of sensible heat flux using an optical method,” Boundary Layer Meteorol. 22, 193–207 (1982).
[CrossRef]

Chernov, L. A.

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

Chiba, T.

Churnside, J. H.

Clifford, S. F.

Cook, R. J.

Dowling, J. A.

Dunphy, J. R.

Gelfer, E. I.

G. A. Andreev, E. I. Gelfer, “Angular random walks on the center of gravity of the cross section of a diverging light beam,” Radiophys. Quantum Electron. 14, 1145–1147 (1971).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, Orlando, Fla., 1980).

Hill, R. J.

Ishimaru, A.

A. Ishimaru, “The beam wave case and remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), Chap. 5.
[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]

Lataitis, R. J.

Lewellen, W. S.

W. S. Lewellen, “Use of invariant modeling,” in Handbook of Turbulence, W. Frost, T. H. Moulden, eds. (Plenum, New York, 1977), Chap. 9.
[CrossRef]

Livingston, P. M.

Mironov, V. L.

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]

Monin, A. S.

A. S. Monin, A. M. Obukhov, “Dimensionless characteristics of turbulence in the surface layer,” Dokl. Akad. Nauk. SSSR 93, 223–226 (1953).

Nosov, V. V.

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]

Obukhov, A. M.

A. S. Monin, A. M. Obukhov, “Dimensionless characteristics of turbulence in the surface layer,” Dokl. Akad. Nauk. SSSR 93, 223–226 (1953).

Paulson, C. A.

C. A. Paulson, “The mathematical representation of wind speed and temperature profiles in the unstable atmospheric surface layer,” Boundary Layer Meteorol. 9, 857–861 (1970).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, Orlando, Fla., 1980).

Wandzura, S. M.

Wolfe, W. L.

W. L. Wolfe, G. J. Zissis, The Infrared Handbook (Environmental Research Center of Michigan, Ann Arbor, Mich., 1989), Chap. 6.

Zissis, G. J.

W. L. Wolfe, G. J. Zissis, The Infrared Handbook (Environmental Research Center of Michigan, Ann Arbor, Mich., 1989), Chap. 6.

Appl. Opt.

Boundary Layer Meteorol.

C. A. Paulson, “The mathematical representation of wind speed and temperature profiles in the unstable atmospheric surface layer,” Boundary Layer Meteorol. 9, 857–861 (1970).

F. K. Brunner, “Determination of line averages of sensible heat flux using an optical method,” Boundary Layer Meteorol. 22, 193–207 (1982).
[CrossRef]

Dokl. Akad. Nauk. SSSR

A. S. Monin, A. M. Obukhov, “Dimensionless characteristics of turbulence in the surface layer,” Dokl. Akad. Nauk. SSSR 93, 223–226 (1953).

J. Opt. Soc. Am.

Radio Sci.

P. Beckmann, “Signal degeneration in laser beams propagated through the atmosphere,” Radio Sci. 69, 629–640 (1965).

Radiophys. Quantum Electron.

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]

G. A. Andreev, E. I. Gelfer, “Angular random walks on the center of gravity of the cross section of a diverging light beam,” Radiophys. Quantum Electron. 14, 1145–1147 (1971).
[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

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, Orlando, Fla., 1980).

W. L. Wolfe, G. J. Zissis, The Infrared Handbook (Environmental Research Center of Michigan, Ann Arbor, Mich., 1989), Chap. 6.

W. S. Lewellen, “Use of invariant modeling,” in Handbook of Turbulence, W. Frost, T. H. Moulden, eds. (Plenum, New York, 1977), Chap. 9.
[CrossRef]

A. Ishimaru, “The beam wave case and remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), Chap. 5.
[CrossRef]

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

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Figures (2)

Fig. 1
Fig. 1

Comparison of the direct integration of Eq. (7) (test points denoted by + symbols) and approximation (9) (dashed curve).

Fig. 2
Fig. 2

Geometry of a portion of the optical path of the positions of the top and the bottom of a uniform-intensity cross-section beam.

Equations (26)

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L 0 = 1.68 k vK z ( 1 - 16 ζ ) 1 / 4 ,
σ w 2 = 4 π 2 0 L d s ( L - s ) 2 0 d κ κ 3 Φ n ( κ , s ) × exp [ - V ( κ , , s ) ] ,
Φ n ( κ , s ) = 0.033 C n 2 ( s ) [ κ 2 + 1 L 0 2 ( s ) ] - 11 / 6 × exp ( - κ 2 l 0 2 / 35 ) ,
V ( κ , , r 0 ) = κ 2 D e 2 8 [ ( 1 - s F ) 2 + ( 4 s k D e 2 ) 2 ] + 2 π s L ( 2.1 κ s k r 0 ) 5 / 3 ,
D 2 = D e 2 8 ( 1 - s F ) 2 + 1 8 ( 4 s k ) 2 ( 1 D e 2 + 13.85 r 0 2 s L ) + l 0 2 35 .
0 d κ κ 3 exp { - κ 2 D 2 } ( κ 2 + L 0 - 2 ) 11 / 6 = D - 1 / 3 2 0 u exp ( - u ) d u ( u + A ) 11 / 6 .
0 u exp ( - u ) d u ( u + A ) 11 / 6 = Γ ( 1 6 ) F 1 1 ( 11 6 , 5 6 ; A ) + [ Γ ( - 1 / 6 ) Γ ( 11 / 6 ) ] A 1 / 6 F 1 1 ( 2 , 7 6 ; A ) ,
σ w 2 = 0 L D - 1 / 3 C n 2 B ( D 2 L 0 2 ) ( L - s ) 2 d s .
B ( A ) { 3.626 - 4.69 A 1 / 6 + 1.9 A 0.82 exp ( - 0.6 A ) 0 A < 0.4 11.43 exp ( - 4.402 A 0.2268 ) - 0.107 exp ( - 6 A ) 0.4 A < 19 0.6514 A - 11 / 6 19 A .
σ α 2 = 0 L D - 1 / 3 C n 2 B ( D 2 L 0 2 ) d s .
d α x = Δ n ( s ) D ( s ) d s ,
σ α x 2 = 0 L 0 L Δ n ( s 1 ) Δ n ( s 2 ) D ( s 1 ) D ( s 2 ) d s 1 d s 2 ,
Δ n ( s 1 ) Δ n ( s 2 ) = ½ [ n ( s 1 + ) - n ( s 2 - ) ] 2 + ½ [ n ( s 1 - ) - n ( s 2 + ) ] 2 - ½ [ n ( s 1 + ) - n ( s 2 + ) ] 2 - ½ [ n ( s 1 - ) - n ( s 2 - ) ] 2 .
Δ n ( s 1 ) Δ n ( s 2 ) = S n ( r 2 ) - S n ( r 1 ) .
S n ( r ) = 8 π 0 d κ κ 2 Φ n ( κ ) [ 1 - sin ( κ r ) κ r ] .
σ α x 2 = 8 π 0 L d s 1 D ( s 1 ) 0 d κ κ 2 Φ n ( κ , s 1 ) × 0 L d s 2 D ( s 2 ) [ sin ( κ r 1 ) κ r 1 - sin ( κ r 2 ) κ r 2 ] .
1 D 1 - d x { sin ( κ x ) κ x - sin [ κ ( D 1 2 + x 2 ) 1 / 2 ] κ ( D 1 2 + x 2 ) 1 / 2 } = π κ D 1 Ψ ( κ D 1 ) ,
0.033 π C n 2 D 1 2 / 3 0 2 π d v v ( v 2 + D 1 2 L 0 2 ) - 11 / 6 × exp [ - v 2 ( l 0 2 / 35 D 1 2 ) ] Ψ ( v ) ,
σ α x 2 = 0 L D - 1 / 3 C n 2 G ( D 2 L 0 2 ) d s ,
G ( A ) { 2.843 - 2.337 A 1 / 6 + 0.15 A 0.84 × exp ( - A / 55.6 ) , 0 A < 1 0.659 - 0.257 [ ln ( A ) ] 0.875 , 1 A < 3 3.083 exp ( - 1.505 A 0.3005 ) + 0.00026 exp ( A / 70.9 ) , 3 A < 100 0 , A > 100 .
σ α x 2 = 0 L D - 1 / 3 C n 2 H ( D 2 l 0 2 ) d s ,
H ( A ) { 3.279 A 1 / 6 A 0.01 3.84 A 0.105 - 0.85 , 0.01 < A 0.1 2.843 / ( 0.125 A - 0.49 + 0.93 ) , 0.1 < A 1 2.843 - 0.156 / A , 1 < A
σ α x 2 = 0 L D - 1 / 3 C n 2 [ G ( D 2 L 0 2 ) + H ( D 2 l 0 2 ) - 2.843 ] d s .
σ w x 2 = 0 L C n 2 D 1 / 3 [ G ( D 2 L 0 2 ) + H ( D 2 l 0 2 ) - 2.843 ] ( L - s ) 2 d s .
B chiba ( A ) = 3.591 - 5.699 A 1 / 6 .
B cook ( A ) = 3.629 - 4.690 A 1 / 6 + 5.644 A 5 / 6 - 5.675 A .

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