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 logamplitude 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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  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).
  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).
  4. J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere, Topics in Applied Physics, Vol. 25 (Springer-Verlag, New York, 1978).
  5. V. A. Banakh et al., "Focused-laser-beam scintillations in the turbulent atmosphere," J. Opt. Soc. Am. 64, 516–518 (1974).
  6. M. Tur and M. J. Beran, "Propagation of a finite beam through a random medium," Opt. Lett. 5, 306–308 (1980).
  7. D. L. Fried and J. B. Seidman, "Laser-beam scintillation in the atmosphere," J. Opt. Soc. Am. 57, 181–185 (1967).
  8. A. Ishimaru, "Fluctuations of a focused beam wave for atmospheric turbulence probing," Proc. IEEE 57, 407–414 (1969).
  9. S. S. Khmelevtsov, "Propagation of laser radiation in a turbulent atmosphere," Appl. Opt. 12, 2421–2433 (1972).
  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).
  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).
  14. B. Steinle and H. P. Baltes, "Radiant intensity and spatial coherence for finite planar sources," J. Opt. Soc. Am. 67, 241–247 (1977).
  15. W. H. Carter and L. Wolf, "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785–796 (1977).
  16. S. M. Wandzura, "Meeting of quadratic structure functions," J. Opt. Soc. Am. 70, 745–747 (1980).
  17. R. L. Fante, "Some physical insights into beam propagation in strong turbulence," Radio Sci. 15, 757–762 (1980).
  18. J. C. Leader, "Beam properties of partially coherent curved beam waves in the turbulent atmosphere," J. Opt. Soc. Am. 70, 682–688 (1980).
  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).
  20. J. R. Kerr and J. R. Dunphy, "Experimental effects of finite transmitter apertures on scintillations," J. Opt. Soc. Am. 63, 1–8 (1973).
  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).
  22. Z. I. Feizulin and Yu. A. Krovtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 68–73 (1967).
  23. R. L. Fante, "Two-source spherical wave structure functions in atmospheric turbulence," J. Opt. Soc. Am. 66, 74 (1976).
  24. H. M. Pedersen, "Theory of speckle dependence on surface roughness," J. Opt. Soc. Am. 66, 1204–1210 (1976).
  25. Ref. 4, Chap. 2.
  26. V. Ya. Sedin et al.; "Intensity fluctuations of a focused laser beam in air," Radioteckh. Elektron. 15, 1290–1292 (1970).
  27. M. L. Gracheva 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).
  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).
  30. R. L. Fante, "Electric field spectrum and intensity covariance of a wave in a random medium," Radio Sci. 10, 77–85 (1975).
  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).
  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).
  35. R. A. Elliott, J. R. Kerr, and P. A. Pincus, "Optical propagation in laboratory-generated turbulence," Appl. Opt. 18, 3315–3323 (1979).
  36. J. C. Dainty, ed., Laser Speckle, Topics in Applied Physics, Vol. 9 (Springer-Verlag, New York, 1975).

1980 (6)

M. Tur and M. J. Beran, "Propagation of a finite beam through a random medium," Opt. Lett. 5, 306–308 (1980).

S. M. Wandzura, "Meeting of quadratic structure functions," J. Opt. Soc. Am. 70, 745–747 (1980).

R. L. Fante, "Some physical insights into beam propagation in strong turbulence," Radio Sci. 15, 757–762 (1980).

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

M. A. Plonus, C. F. Ouyang, and S. C. H. Wang, "Intensity properties of partially coherent beam waves," Appl. Opt. 19, 3082–3085 (1980).

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).

1979 (4)

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).

R. Dashen, "Path integral for waves in random media," J. Math. Phys. 20, 894–920 (1979).

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

R. A. Elliott, J. R. Kerr, and P. A. Pincus, "Optical propagation in laboratory-generated turbulence," Appl. Opt. 18, 3315–3323 (1979).

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).

1974 (3)

1973 (1)

1972 (2)

1970 (1)

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

1969 (1)

J. L. Hansen, "Radiative transfer by doubling very thin layers," Astrophys. J. 155, 565–573 (1969).

1967 (2)

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

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

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).

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).

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).

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

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

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).

Fried, D. L.

Gracheva, M. L.

M. L. Gracheva 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).

Ishimaru, A.

A. Ishimaru, "Fluctuations of a focused beam wave for atmospheric turbulence probing," Proc. IEEE 57, 407–414 (1969).

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).

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).

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

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).

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

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

Ochs, G. R.

Ouyang, C. F.

Pedersen, H. M.

Pincus, P. A.

Plonus, M. A.

Sedin, V. Ya.

V. Ya. Sedin 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).

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).

J. Math. Phys. (1)

R. Dashen, "Path integral for waves in random media," J. Math. Phys. 20, 894–920 (1979).

J. Opt. Soc. Am. (13)

V. A. Banakh et al., "Focused-laser-beam scintillations in the turbulent atmosphere," J. Opt. Soc. Am. 64, 516–518 (1974).

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

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

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

S. M. Wandzura, "Meeting of quadratic structure functions," J. Opt. Soc. Am. 70, 745–747 (1980).

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

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

J. R. Kerr, "Experiments on turbulence characteristics and multiwavelength scintillation phenomena," J. Opt. Soc. Am. 62, 1040–1049 (1972).

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).

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

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

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

H. M. Pedersen, "Theory of speckle dependence on surface roughness," J. Opt. Soc. Am. 66, 1204–1210 (1976).

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).

Opt. Lett. (1)

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).

R. L. Fante, "Some physical insights into beam propagation in strong turbulence," Radio Sci. 15, 757–762 (1980).

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

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).

Radioteckh. Elektron. (1)

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

Other (11)

M. L. Gracheva 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].

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

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).

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.

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).

A. Ishimaru, "Fluctuations of a focused beam wave for atmospheric turbulence probing," Proc. IEEE 57, 407–414 (1969).

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