Abstract

A closed set of three simultaneous partial differential equations is derived for the solution of the average irradiance and the irradiance variance of focused laser beams in turbulence. The equations are uniformly valid for arbitrary scintillation levels. Examples of solutions calculated by finite difference techniques are shown to fit typical atmospheric and laboratory data well. Analytic solutions derived for asymptotic conditions are in good agreement with the related classic perturbation results. The model also provides interesting physical insight into the phenomena of saturation and supersaturation.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere, Volume 25 of Topics in Applied Physics (Springer-Verlag, New York, 1978).
    [CrossRef]
  2. J. W. Strohbehn, “Line-of-sight wave propagation through the turbulent atmosphere,” Proc. IEEE 56, 1301–1318 (1968).
    [CrossRef]
  3. R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
    [CrossRef]
  4. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
    [CrossRef]
  5. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
    [CrossRef]
  6. R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1442 (1980).
    [CrossRef]
  7. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1967).
  8. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  9. R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef] [PubMed]
  10. L. R. Bissonnette, “Angle of arrival and irradiance statistics of laser beams in turbulence,” DREV Rep. 4213/81, July1981Accession No. 81-03113 (Director of Scientific Information Services, Ottawa, Canada).
  11. L. R. Bissonnette, “Focused laser beams in turbulent media,” DREV Rep. 4178/80, December1980Accession No. 81-00157 (Director of Scientific Information Services, Ottawa, Canada).
  12. L. R. Bissonnette, “Modelling of laser beam propagation in atmospheric turbulence,” in Proceedings of the Second International Symposium on Gas Flow and Chemical Lasers, J. F. Wendt, ed. (Hemisphere, Washington, D.C., 1979), pp. 73–94.
  13. J. L. Lumley, “Computational modeling of turbulent flows,” in Advances in Applied Mechanics (Academic, New York, 1978), Vol. 18.
  14. P. Bradshaw, ed., Turbulence, Volume 12 of Topics in Applied Physics (Springer-Verlag, New York, 1976), Chaps. 1, 5, and 6.
    [CrossRef]
  15. J. O. Hinze, Turbulence, 2nd ed. (McGraw-Hill, New York, 1975), Chaps. 1 and 5.
  16. L. R. Bissonnette and P. L. Wizinowich, “Probability distribution of turbulent irradiance in a saturation regime,” Appl. Opt. 18, 1590–1599 (1979).
    [CrossRef] [PubMed]
  17. L. R. Bissonnette, “Atmospheric scintillation of optical and infrared waves: a laboratory simulation,” Appl. Opt. 16, 2242–2251 (1977).
    [CrossRef] [PubMed]
  18. W. P. Brown, “Second moment of a wave propagating in a random medium,” J. Opt. Soc. Am. 61, 1051–1059 (1971).
    [CrossRef]
  19. H. T. Yura, “Atmospheric turbulence induced beam spread,” Appl. Opt. 10, 2771–2773 (1971).
    [CrossRef] [PubMed]
  20. J. A. Dowling and 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]
  21. L. R. Bissonnette, “Log-normal probability distribution in strong irradiance fluctuations: an asymptotic analysis,” in AGARD Conference Proceedings No. 183 on Optical Propagation in the Atmosphere (National Technical Information Service, Springfield, Va., 1976).
  22. L. R. Bissonnette, “Propagation of adaptively corrected laser beams through a turbulent atmosphere,” J. Phys. (Paris) 41, C9/415–427 (1980).
    [CrossRef]
  23. L. R. Bissonnette, “Propagation model of adaptively corrected laser beams in turbulence,” DREV Rep. 4200/81, October1981Accession No. 82-00524 (Director of Scientific Information Services, Ottawa, Canada).
  24. 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]
  25. R. J. Hill and 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).
    [CrossRef]
  26. R. J. Hill, “Theory of saturation of optical scintillation by strong turbulence: plane-wave variance and covariance and spherical-wave covariance,” J. Opt. Soc. Am. 72, 212–222 (1982).
    [CrossRef]

1982 (1)

1981 (1)

1980 (2)

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1442 (1980).
[CrossRef]

L. R. Bissonnette, “Propagation of adaptively corrected laser beams through a turbulent atmosphere,” J. Phys. (Paris) 41, C9/415–427 (1980).
[CrossRef]

1979 (1)

1977 (1)

1975 (2)

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

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

1974 (1)

1973 (1)

1971 (3)

1970 (1)

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

1968 (1)

J. W. Strohbehn, “Line-of-sight wave propagation through the turbulent atmosphere,” Proc. IEEE 56, 1301–1318 (1968).
[CrossRef]

Bissonnette, L. R.

L. R. Bissonnette, “Propagation of adaptively corrected laser beams through a turbulent atmosphere,” J. Phys. (Paris) 41, C9/415–427 (1980).
[CrossRef]

L. R. Bissonnette and P. L. Wizinowich, “Probability distribution of turbulent irradiance in a saturation regime,” Appl. Opt. 18, 1590–1599 (1979).
[CrossRef] [PubMed]

L. R. Bissonnette, “Atmospheric scintillation of optical and infrared waves: a laboratory simulation,” Appl. Opt. 16, 2242–2251 (1977).
[CrossRef] [PubMed]

L. R. Bissonnette, “Log-normal probability distribution in strong irradiance fluctuations: an asymptotic analysis,” in AGARD Conference Proceedings No. 183 on Optical Propagation in the Atmosphere (National Technical Information Service, Springfield, Va., 1976).

L. R. Bissonnette, “Propagation model of adaptively corrected laser beams in turbulence,” DREV Rep. 4200/81, October1981Accession No. 82-00524 (Director of Scientific Information Services, Ottawa, Canada).

L. R. Bissonnette, “Angle of arrival and irradiance statistics of laser beams in turbulence,” DREV Rep. 4213/81, July1981Accession No. 81-03113 (Director of Scientific Information Services, Ottawa, Canada).

L. R. Bissonnette, “Focused laser beams in turbulent media,” DREV Rep. 4178/80, December1980Accession No. 81-00157 (Director of Scientific Information Services, Ottawa, Canada).

L. R. Bissonnette, “Modelling of laser beam propagation in atmospheric turbulence,” in Proceedings of the Second International Symposium on Gas Flow and Chemical Lasers, J. F. Wendt, ed. (Hemisphere, Washington, D.C., 1979), pp. 73–94.

Brown, W. P.

Bunkin, F. V.

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

Clifford, S. F.

Dowling, J. A.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1442 (1980).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Gochelashvily, K. S.

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

Hill, R. J.

Hinze, J. O.

J. O. Hinze, Turbulence, 2nd ed. (McGraw-Hill, New York, 1975), Chaps. 1 and 5.

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 clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Livingston, P. M.

Lumley, J. L.

J. L. Lumley, “Computational modeling of turbulent flows,” in Advances in Applied Mechanics (Academic, New York, 1978), Vol. 18.

Lutomirski, R. F.

Ochs, G. R.

Prokhorov, A. M.

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

Shishov, V. I.

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

Strohbehn, J. W.

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

J. W. Strohbehn, “Line-of-sight wave propagation through the turbulent atmosphere,” Proc. IEEE 56, 1301–1318 (1968).
[CrossRef]

Tatarskii, V. I.

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

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

Wizinowich, P. L.

Yura, H. T.

Appl. Opt. (4)

J. Opt. Soc. Am. (5)

J. Phys. (Paris) (1)

L. R. Bissonnette, “Propagation of adaptively corrected laser beams through a turbulent atmosphere,” J. Phys. (Paris) 41, C9/415–427 (1980).
[CrossRef]

Proc. IEEE (5)

J. W. Strohbehn, “Line-of-sight wave propagation through the turbulent atmosphere,” Proc. IEEE 56, 1301–1318 (1968).
[CrossRef]

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

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

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1442 (1980).
[CrossRef]

Other (11)

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

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

L. R. Bissonnette, “Angle of arrival and irradiance statistics of laser beams in turbulence,” DREV Rep. 4213/81, July1981Accession No. 81-03113 (Director of Scientific Information Services, Ottawa, Canada).

L. R. Bissonnette, “Focused laser beams in turbulent media,” DREV Rep. 4178/80, December1980Accession No. 81-00157 (Director of Scientific Information Services, Ottawa, Canada).

L. R. Bissonnette, “Modelling of laser beam propagation in atmospheric turbulence,” in Proceedings of the Second International Symposium on Gas Flow and Chemical Lasers, J. F. Wendt, ed. (Hemisphere, Washington, D.C., 1979), pp. 73–94.

J. L. Lumley, “Computational modeling of turbulent flows,” in Advances in Applied Mechanics (Academic, New York, 1978), Vol. 18.

P. Bradshaw, ed., Turbulence, Volume 12 of Topics in Applied Physics (Springer-Verlag, New York, 1976), Chaps. 1, 5, and 6.
[CrossRef]

J. O. Hinze, Turbulence, 2nd ed. (McGraw-Hill, New York, 1975), Chaps. 1 and 5.

L. R. Bissonnette, “Propagation model of adaptively corrected laser beams in turbulence,” DREV Rep. 4200/81, October1981Accession No. 82-00524 (Director of Scientific Information Services, Ottawa, Canada).

L. R. Bissonnette, “Log-normal probability distribution in strong irradiance fluctuations: an asymptotic analysis,” in AGARD Conference Proceedings No. 183 on Optical Propagation in the Atmosphere (National Technical Information Service, Springfield, Va., 1976).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Normalized beam radius plotted versus normalized propagation distance; b = 0.0016, f = ∞, η0 = 7.0. ■, Data measured in laboratory turbulence. —, Finite-difference solution calculated from the present model.

Fig. 2
Fig. 2

Same as Fig. 1 except for the conditions b = 0.0064, f = ∞, and η0 = 7.0.

Fig. 3
Fig. 3

Same as Fig. 1 except for the conditions b = 0.0023, f = 3.13, and η0 = 8.5.

Fig. 4
Fig. 4

Same as Fig. 1 except for the conditions b = 0.0021, f = 4.41, and η0 = 8.5.

Fig. 5
Fig. 5

Focal-plane beam radius normalized by the diffraction-limited radius wd (=F/kw0) plotted versus the index structure parameter Cn; k = 1.32 × 107 m−1. ▼, F = 1.57 m and w0 = 1.90 mm. ●, F = 3.00 m and w0 = 2.44 mm. —, Finite-difference solution calculated from the present model.

Fig. 6
Fig. 6

Normalized-irradiance standard deviation plotted versus normalized propagation distance. Data measured in laboratory turbulence on the axis of a 25-mm-diameter collimated beam. ○, η0 = 4.0 and b = 0.0020. Δ, η0 = 5.3 and b = 0.0015.∇, η0 = 7.0 and b = 0.0011. —, Finite-difference solution calculated from the present model: curve No. 1 for η0 = 4.0 and b = 0.0020 and curve No. 2 for η0 = 7.0 and b = 0.0011.

Fig. 7
Fig. 7

Normalized-irradiance standard deviation plotted versus normalized propagation distance. Data measured in the atmosphere and reprinted from Fig. 19 of Ref. 4. —, Finite-difference solution calculated from the present model for plane waves (f = ∞ and b = 0). – – –, Asymptotic solutions given in Ref. 4.

Fig. 8
Fig. 8

Comparison of the theoretical beam-spread formula of this paper (θB) and of Ref. 19 (θY) with the empirical regression formula (θDL) proposed in Ref. 20. The propagation distance is 1750 m. (a) λ = 0.63 μm, (b) λ = 10.6 μm.

Equations (45)

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

3 2 E + k 2 N 2 n 0 2 E = 0 ,
E = A exp [ j k ( z + ϕ ) ] ,
( z + V ) V = ( N n 0 ) / n 0 ,
( z + V ) A + 1 2 A · V j 2 k 2 A = 0 ,
N = N + n , n = 0 ,
V = V + v , v = 0 ,
A = A + a , a = 0 ,
( z + V · ) V = 1 2 v · v + ( N n 0 ) / n 0 ,
( z + V · ) A + 1 2 A · V j 2 k 2 A = · v a + 1 2 a · v ,
( z + V ) a a * + a a * V j 2 k ( a * 2 a a 2 a * ) = v a a * a v A * a * v A 1 2 A * a v 1 2 A a * v ,
( z + V · ) a a + a a · V j 2 k 2 a a + j k a · a = · v a a 2 a v · A A a · v .
v a ( v · v I ) 1 / 2 ,
a ( z 1 , r 1 ) a * ( z 2 , r 2 ) = F [ z 1 , z 2 ; ( r 1 + r 2 ) / 2 ] f ( z 1 , z 2 ; | r 1 r 2 | ) ,
a ( z 1 , r 1 ) a ( z 2 , r 2 ) = H [ z 1 , z 2 ; ( r 1 + r 2 ) / 2 ] h ( z 1 , z 2 ; | r 1 r 2 | ) ,
z V + V · V = ( N n 0 ) / n 0 ,
V = r / ( z F ) .
a · v = K ( z ) A ,
a v = R ( z ) · A ,
a a v = 1 2 R ( z ) · a a ,
a a * v = 1 2 Re { R ( z ) } · a a * .
K ( z ) = 3.24 C n 2 n 0 2 l 1 7 / 3 0 z d u 0 u d ζ ( F ζ ) 2 G ( γ ) ( F z ) ( F + z 2 u ) ,
R ( z ) = 2.43 C n 2 n 0 2 l 1 1 / 3 δ 0 z d u 0 u d ζ ( F ζ ) 2 H ( γ ) ( F z ) ( F + z 2 u ) ,
G ( γ ) = ( j γ ) 2 { 1 2 3 i γ + [ 5 / 12 + j γ + 3 γ 2 ] e j γ ( j γ ) 5 / 6 Γ ( 5 / 6 , j γ ) } ,
H ( γ ) = ( j γ ) { 1 + [ 5 / 6 + j γ ] e j γ ( j γ ) 5 / 6 Γ ( 5 / 6 , j γ ) } ,
γ = k l 1 2 2 ( z u ) ( F + z 2 u ) ( F u ) ( F ζ ) 2 ,
lim ρ 0 ρ 2 a ( r + 1 2 ρ ) a ( r 1 2 ρ ) = T ( z ) a a ,
T ( z ) = { C F z / 2 F z k z , z z A C F z / 2 F z k z z A 2 , z > z A ,
a · a 1 4 2 a a = T ( z ) a a .
j 2 k [ a * 2 a a 2 a * ] θ 2 z 2 F F z 2 a a * ,
η = z / z A ,
ρ = r / w 0 ,
f = F / z A ,
b = z A / k w 0 2 ,
η 0 = 0.123 k l 0 2 / z A .
I = A A * + a a * ,
σ I 2 = a a * 2 + a a a a * + 2 A A * a a * + A A a a * + A * A * a a ,
C = 0.15 0.01 j .
I 1 σ 2 ( η ) exp [ ρ 2 / σ 2 ( η ) ] ,
σ I 2 F ( η ) exp [ 2 ρ 2 / σ 2 ( η ) ] ,
σ 2 ( η ) = ( f η ) 2 f 2 + b 2 η 2 + ( f η ) 2 0 η δ : Re { z A R ( z A ζ ) / w 0 2 } ( f ζ ) 2 d ζ ,
θ B 2 = 7.27 k 1 / 6 z 5 / 6 ( C n / n 0 ) 2 .
θ Y 2 = 15.4 k 2 / 5 z 6 / 5 ( C n / n 0 ) 12 / 5 .
θ DL 2 = 2.9 × 10 6 k 1 / 3 z ( C n / n 0 ) 6 / 5 .
σ I / I = 1.66 k 7 / 12 z 11 / 12 C n / n 0 .
[ σ I / I ] p = 1.11 k 7 / 12 z 11 / 12 C n / n 0 .