Abstract

By use of path-integral methods, a general expression is obtained for the two-frequency, two-position mutual coherence function of an electromagnetic pulse propagating through turbulent atmosphere. This expression is valid for arbitrary models of refractive-index fluctuations, wide band pulses, and turbulence of arbitrary strength. The approach presented in this paper was examined in the cases of plane-wave, spherical wave, and Gaussian beam propagation in power-law turbulence and compared with existing numerical and exact results. A number of new results were obtained for the Gaussian beam pulse. Expressions derived here should be applicable to a wide range of practical pulse propagation problems.

© 2002 Optical Society of America

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References

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  1. C. Y. Young, A. Ishimaru, L. C. Andrews, “Two-frequency mutual coherence function of a Gaussian beam pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 35, 6522–6526 (1996).
    [CrossRef] [PubMed]
  2. C. Y. Young, L. C. Andrews, A. Ishimaru, “Time-of-arrival fluctuations of a space–time Gaussian pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 37, 7655–7660 (1998).
    [CrossRef]
  3. I. Sreenivasiah, “Two-frequency mutual coherence function and pulse propagation in continuous random media: forward and backscattering solutions,” Ph.D. dissertation (University of Washington, Seattle, Wash.1976).
  4. I. Sreenivasiah, A. Ishimaru, “Beam-wave two-frequency mutual-coherence function and pulse propaga-tion in random media: an analytic solution,” Appl. Opt. 18, 1613–1618 (1979).
    [CrossRef] [PubMed]
  5. I. Sreenivasiah, A. Ishimaru, S. T. Hong, “Two-frequency mutual coherence function and pulse propagation in a random medium: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
    [CrossRef]
  6. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U. S. Department of Commerce, Springfield, Va., 1971).
  7. C. H. Liu, “Pulse statistics in random media,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE, Bellingham, Wash., 1993), pp. 291–304.
  8. R. P. Feynman, A. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
  9. L. S. Brown, Quantum Field Theory (Cambridge U. Press, Cambridge, UK, 1995).
  10. R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
    [CrossRef]
  11. C. M. Rose, I. M. Besieris, “Nth-order multifrequency coherence functions: a functional path integral approach,” J. Math. Phys. 20, 1530–1538 (1979).
    [CrossRef]
  12. C. M. Rose, I. M. Besieris, “Nth-order multifrequency coherence functions: a functional path integral approach. II,” J. Math. Phys. 21, 2114–2120 (1980).
    [CrossRef]
  13. J. Gozani, “Pulsed beam propagation through random media,” Opt. Lett. 21, 1712–1714 (1996).
    [CrossRef] [PubMed]
  14. S. Wandzura, “Meaning of quadratic structure functions,” J. Opt. Soc. Am. 70, 745–747 (1980).
    [CrossRef]
  15. C. H. Liu, K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14, 925–931 (1979).
    [CrossRef]
  16. L. C. Lee, J. R. Jokipii, “Strong scintillations in atsrophysics. II. A theory of temporal broadening of pulses,” Astrophys. J. 201, 532–543 (1975).
    [CrossRef]
  17. K. Furutsu, “An analytical theory of pulse wave propagation in turbulent media,” J. Math. Phys. 20, 617–628 (1979).
    [CrossRef]
  18. J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution:I,” Waves Random Media 7, 79–93 (1997).
    [CrossRef]
  19. J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution: II,” Waves Random Media 7, 95–106 (1997).
    [CrossRef]
  20. J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: point source,” Waves Random Media 7, 107–117 (1997).
    [CrossRef]
  21. J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: beam waves,” Waves Random Media 8, 159–174 (1998).
    [CrossRef]
  22. J. Oz, E. Heyman, “Modal solution to the plane wave two-frequency mutual coherence function in random media,” Radio Sci. 31, 1907–1917 (1996).
    [CrossRef]
  23. R. L. Fante, “Two-position, two-frequency mutual-coherence function in turbulence,” J. Opt. Soc. Am. 71, 1446–1451 (1981).
    [CrossRef]
  24. J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), pp. 45–106.
  25. V. I. Tatarskii, M. I. Charnotskii, J. Gozani, V. U. Zavorotny, “Path integral approach to wave propagation in random media, Part I: derivations and various formulations,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 383–402.
  26. V. U. Zavorotny, M. I. Charnotskii, J. Gozani, V. I. Tatarskii, “Path integral approach to wave propagation in random media, Part II: exact formulations and heuristic approximations,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavor-otny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 403–421.
  27. J. Gozani, M. I. Charnotskii, V. I. Tatarskii, V. U. Zavorotny, “Path integral approach to wave propagation in random media, Part III: mixed representation; orthogonal expansion,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 422–441.
  28. A. Ishimaru, Wave Propagation and Scattering in Random Media (Institute of Electrical and Electronics Engineers, New York, 1997).
  29. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  30. R. J. Hill, S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. 68, 892–899 (1978).
    [CrossRef]

1998 (2)

C. Y. Young, L. C. Andrews, A. Ishimaru, “Time-of-arrival fluctuations of a space–time Gaussian pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 37, 7655–7660 (1998).
[CrossRef]

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: beam waves,” Waves Random Media 8, 159–174 (1998).
[CrossRef]

1997 (3)

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution:I,” Waves Random Media 7, 79–93 (1997).
[CrossRef]

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution: II,” Waves Random Media 7, 95–106 (1997).
[CrossRef]

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: point source,” Waves Random Media 7, 107–117 (1997).
[CrossRef]

1996 (3)

1981 (1)

1980 (2)

C. M. Rose, I. M. Besieris, “Nth-order multifrequency coherence functions: a functional path integral approach. II,” J. Math. Phys. 21, 2114–2120 (1980).
[CrossRef]

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

1979 (5)

C. H. Liu, K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14, 925–931 (1979).
[CrossRef]

I. Sreenivasiah, A. Ishimaru, “Beam-wave two-frequency mutual-coherence function and pulse propaga-tion in random media: an analytic solution,” Appl. Opt. 18, 1613–1618 (1979).
[CrossRef] [PubMed]

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

C. M. Rose, I. M. Besieris, “Nth-order multifrequency coherence functions: a functional path integral approach,” J. Math. Phys. 20, 1530–1538 (1979).
[CrossRef]

K. Furutsu, “An analytical theory of pulse wave propagation in turbulent media,” J. Math. Phys. 20, 617–628 (1979).
[CrossRef]

1978 (1)

1976 (1)

I. Sreenivasiah, A. Ishimaru, S. T. Hong, “Two-frequency mutual coherence function and pulse propagation in a random medium: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
[CrossRef]

1975 (1)

L. C. Lee, J. R. Jokipii, “Strong scintillations in atsrophysics. II. A theory of temporal broadening of pulses,” Astrophys. J. 201, 532–543 (1975).
[CrossRef]

Andrews, L. C.

Besieris, I. M.

C. M. Rose, I. M. Besieris, “Nth-order multifrequency coherence functions: a functional path integral approach. II,” J. Math. Phys. 21, 2114–2120 (1980).
[CrossRef]

C. M. Rose, I. M. Besieris, “Nth-order multifrequency coherence functions: a functional path integral approach,” J. Math. Phys. 20, 1530–1538 (1979).
[CrossRef]

Brown, L. S.

L. S. Brown, Quantum Field Theory (Cambridge U. Press, Cambridge, UK, 1995).

Charnotskii, M. I.

V. I. Tatarskii, M. I. Charnotskii, J. Gozani, V. U. Zavorotny, “Path integral approach to wave propagation in random media, Part I: derivations and various formulations,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 383–402.

V. U. Zavorotny, M. I. Charnotskii, J. Gozani, V. I. Tatarskii, “Path integral approach to wave propagation in random media, Part II: exact formulations and heuristic approximations,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavor-otny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 403–421.

J. Gozani, M. I. Charnotskii, V. I. Tatarskii, V. U. Zavorotny, “Path integral approach to wave propagation in random media, Part III: mixed representation; orthogonal expansion,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 422–441.

Clifford, S. F.

Dashen, R.

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

Fante, R. L.

Feynman, R. P.

R. P. Feynman, A. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

Furutsu, K.

K. Furutsu, “An analytical theory of pulse wave propagation in turbulent media,” J. Math. Phys. 20, 617–628 (1979).
[CrossRef]

Gozani, J.

J. Gozani, “Pulsed beam propagation through random media,” Opt. Lett. 21, 1712–1714 (1996).
[CrossRef] [PubMed]

V. U. Zavorotny, M. I. Charnotskii, J. Gozani, V. I. Tatarskii, “Path integral approach to wave propagation in random media, Part II: exact formulations and heuristic approximations,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavor-otny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 403–421.

J. Gozani, M. I. Charnotskii, V. I. Tatarskii, V. U. Zavorotny, “Path integral approach to wave propagation in random media, Part III: mixed representation; orthogonal expansion,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 422–441.

V. I. Tatarskii, M. I. Charnotskii, J. Gozani, V. U. Zavorotny, “Path integral approach to wave propagation in random media, Part I: derivations and various formulations,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 383–402.

Heyman, E.

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: beam waves,” Waves Random Media 8, 159–174 (1998).
[CrossRef]

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution:I,” Waves Random Media 7, 79–93 (1997).
[CrossRef]

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution: II,” Waves Random Media 7, 95–106 (1997).
[CrossRef]

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: point source,” Waves Random Media 7, 107–117 (1997).
[CrossRef]

J. Oz, E. Heyman, “Modal solution to the plane wave two-frequency mutual coherence function in random media,” Radio Sci. 31, 1907–1917 (1996).
[CrossRef]

Hibbs, A.

R. P. Feynman, A. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

Hill, R. J.

Hong, S. T.

I. Sreenivasiah, A. Ishimaru, S. T. Hong, “Two-frequency mutual coherence function and pulse propagation in a random medium: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
[CrossRef]

Ishimaru, A.

Jokipii, J. R.

L. C. Lee, J. R. Jokipii, “Strong scintillations in atsrophysics. II. A theory of temporal broadening of pulses,” Astrophys. J. 201, 532–543 (1975).
[CrossRef]

Lee, L. C.

L. C. Lee, J. R. Jokipii, “Strong scintillations in atsrophysics. II. A theory of temporal broadening of pulses,” Astrophys. J. 201, 532–543 (1975).
[CrossRef]

Liu, C. H.

C. H. Liu, K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14, 925–931 (1979).
[CrossRef]

C. H. Liu, “Pulse statistics in random media,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE, Bellingham, Wash., 1993), pp. 291–304.

Oz, J.

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: beam waves,” Waves Random Media 8, 159–174 (1998).
[CrossRef]

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: point source,” Waves Random Media 7, 107–117 (1997).
[CrossRef]

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution: II,” Waves Random Media 7, 95–106 (1997).
[CrossRef]

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution:I,” Waves Random Media 7, 79–93 (1997).
[CrossRef]

J. Oz, E. Heyman, “Modal solution to the plane wave two-frequency mutual coherence function in random media,” Radio Sci. 31, 1907–1917 (1996).
[CrossRef]

Rose, C. M.

C. M. Rose, I. M. Besieris, “Nth-order multifrequency coherence functions: a functional path integral approach. II,” J. Math. Phys. 21, 2114–2120 (1980).
[CrossRef]

C. M. Rose, I. M. Besieris, “Nth-order multifrequency coherence functions: a functional path integral approach,” J. Math. Phys. 20, 1530–1538 (1979).
[CrossRef]

Sreenivasiah, I.

I. Sreenivasiah, A. Ishimaru, “Beam-wave two-frequency mutual-coherence function and pulse propaga-tion in random media: an analytic solution,” Appl. Opt. 18, 1613–1618 (1979).
[CrossRef] [PubMed]

I. Sreenivasiah, A. Ishimaru, S. T. Hong, “Two-frequency mutual coherence function and pulse propagation in a random medium: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
[CrossRef]

I. Sreenivasiah, “Two-frequency mutual coherence function and pulse propagation in continuous random media: forward and backscattering solutions,” Ph.D. dissertation (University of Washington, Seattle, Wash.1976).

Strohbehn, J. W.

J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), pp. 45–106.

Tatarskii, V. I.

V. I. Tatarskii, M. I. Charnotskii, J. Gozani, V. U. Zavorotny, “Path integral approach to wave propagation in random media, Part I: derivations and various formulations,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 383–402.

V. U. Zavorotny, M. I. Charnotskii, J. Gozani, V. I. Tatarskii, “Path integral approach to wave propagation in random media, Part II: exact formulations and heuristic approximations,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavor-otny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 403–421.

J. Gozani, M. I. Charnotskii, V. I. Tatarskii, V. U. Zavorotny, “Path integral approach to wave propagation in random media, Part III: mixed representation; orthogonal expansion,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 422–441.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U. S. Department of Commerce, Springfield, Va., 1971).

Wandzura, S.

Yeh, K. C.

C. H. Liu, K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14, 925–931 (1979).
[CrossRef]

Young, C. Y.

Zavorotny, V. U.

J. Gozani, M. I. Charnotskii, V. I. Tatarskii, V. U. Zavorotny, “Path integral approach to wave propagation in random media, Part III: mixed representation; orthogonal expansion,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 422–441.

V. I. Tatarskii, M. I. Charnotskii, J. Gozani, V. U. Zavorotny, “Path integral approach to wave propagation in random media, Part I: derivations and various formulations,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 383–402.

V. U. Zavorotny, M. I. Charnotskii, J. Gozani, V. I. Tatarskii, “Path integral approach to wave propagation in random media, Part II: exact formulations and heuristic approximations,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavor-otny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 403–421.

Appl. Opt. (3)

Astrophys. J. (1)

L. C. Lee, J. R. Jokipii, “Strong scintillations in atsrophysics. II. A theory of temporal broadening of pulses,” Astrophys. J. 201, 532–543 (1975).
[CrossRef]

J. Math. Phys. (4)

K. Furutsu, “An analytical theory of pulse wave propagation in turbulent media,” J. Math. Phys. 20, 617–628 (1979).
[CrossRef]

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

C. M. Rose, I. M. Besieris, “Nth-order multifrequency coherence functions: a functional path integral approach,” J. Math. Phys. 20, 1530–1538 (1979).
[CrossRef]

C. M. Rose, I. M. Besieris, “Nth-order multifrequency coherence functions: a functional path integral approach. II,” J. Math. Phys. 21, 2114–2120 (1980).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Lett. (1)

Radio Sci. (3)

I. Sreenivasiah, A. Ishimaru, S. T. Hong, “Two-frequency mutual coherence function and pulse propagation in a random medium: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
[CrossRef]

C. H. Liu, K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14, 925–931 (1979).
[CrossRef]

J. Oz, E. Heyman, “Modal solution to the plane wave two-frequency mutual coherence function in random media,” Radio Sci. 31, 1907–1917 (1996).
[CrossRef]

Waves Random Media (4)

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution:I,” Waves Random Media 7, 79–93 (1997).
[CrossRef]

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution: II,” Waves Random Media 7, 95–106 (1997).
[CrossRef]

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: point source,” Waves Random Media 7, 107–117 (1997).
[CrossRef]

J. Oz, E. Heyman, “Modal theory for the two-frequency mutual coherence function in random media: beam waves,” Waves Random Media 8, 159–174 (1998).
[CrossRef]

Other (11)

I. Sreenivasiah, “Two-frequency mutual coherence function and pulse propagation in continuous random media: forward and backscattering solutions,” Ph.D. dissertation (University of Washington, Seattle, Wash.1976).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U. S. Department of Commerce, Springfield, Va., 1971).

C. H. Liu, “Pulse statistics in random media,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE, Bellingham, Wash., 1993), pp. 291–304.

R. P. Feynman, A. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

L. S. Brown, Quantum Field Theory (Cambridge U. Press, Cambridge, UK, 1995).

J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), pp. 45–106.

V. I. Tatarskii, M. I. Charnotskii, J. Gozani, V. U. Zavorotny, “Path integral approach to wave propagation in random media, Part I: derivations and various formulations,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 383–402.

V. U. Zavorotny, M. I. Charnotskii, J. Gozani, V. I. Tatarskii, “Path integral approach to wave propagation in random media, Part II: exact formulations and heuristic approximations,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavor-otny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 403–421.

J. Gozani, M. I. Charnotskii, V. I. Tatarskii, V. U. Zavorotny, “Path integral approach to wave propagation in random media, Part III: mixed representation; orthogonal expansion,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 422–441.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Institute of Electrical and Electronics Engineers, New York, 1997).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

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Figures (3)

Fig. 1
Fig. 1

λ0=0.6943 μm, Z=10 km, and C2=4.856×10-14 m-2/3. Absolute magnitude and phase of Γ5/3sph(R=R0=p=ξ=0; k0, Δ)/Γ5/3sph(R=R0=p=ξ=0; k0, Δ=0) versus Δn. Solid curves, C˜=C˜max; dashed curves, C˜=0; dashed–dotted curves, HF approximation.

Fig. 2
Fig. 2

Absolute magnitude and phase of Γ5/3pl(p; k0, Δ) versus Δn. Solid curves, ξn=0; dashed curves, ξn=1; dashed–dotted curves, ξn=2; long-dashed–dotted curves, a ξn=0 numerical solution from Ref. 28.

Fig. 3
Fig. 3

λ0=0.6943 μm, Z=10 km, and C2=4.856×10-14 m-2/3. (a) Absolute magnitude and phase of Γ5/3Gaus(R=p=0; k0, Δ)/Γ5/3Gaus(R=p=0; k0, Δ=0) versus Δn for a collimated beam. Solid curves, W0=4 cm; dashed curves, W0=10 cm; dashed–dotted curves, W0=25 cm; long-dashed curves, W0=1 m; long-dashed–dotted curves, W010 m. (b) Absolute magnitude and phase of Γ5/3Gaus(R, p=0; k0, Δ)/Γ5/3Gaus(R, p=0; k0, Δ=0) versus Δn for a collimated beam. W0=10 cm: solid curves, R=0 cm, dashed curves, R=15 cm. W0=25 cm: dashed–dotted curves, R=0 cm; long-dashed–dotted curves, R=15 cm. (c) Absolute magnitude and phase of Γ5/3Gaus(R, p=0; k0, Δ)/Γ5/3Gaus(R, p=0; k0, Δ=0) versus Δn for a beam focused at Z. Solid curves, W0=10 and 25 cm and R=0 cm. Dashed phase curve, W0=10 and 25 cm and R=15 cm. Dashed absolute magnitude curve, W0=10 cm and R=15 cm. Long-dashed–dotted absolute magnitude curve, W0=25 cm and R=15 cm.

Equations (80)

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

2iku(r, z, k)z+Δu(r, z, k)+k2˜u(r, z, k)=0,
u(r, z=0, k)=u0(r).
u(R, Z, k)
=d2R0u0(R0)D2sδ[s(0)-R0]δ[s(Z)-R]expik20Zdzdsdz2+˜(s, z).
ΓpN(R1, R2, R10, R20; k1, k2)
u(R1, R10, k1)u*(R2, R20, k2)uf(R1, R10, k1)uf*(R2, R20, k2),
uf(R, R0, k)=-ik2πZexpik2Z|R-R0|2
˜(r, z)˜(r, z)=δ(z-z)A(|r-r|, z).
ΓpN=exp-(k1-k2)2ZA(0)8I{Ap}I{0},
I{Ap}=fD2s1fD2s2 exp0Zdzik12s˙12(z)-ik22s˙22(z)+k1k24Ap|s1(z)-s2(z)|p,
A(s)A(0)+Ap|s|p,with1<p2.
v=s1-s2,
u=(k1s1-k2s2)/(k1-k2),
I{Ap}Iu{0}Iv{Ap}=fD2u expi2(k1-k2)0Zdzu˙2fD2v×exp0Zdz-i2k1k2k1-k2v˙2+k1k24Ap|v|p.
ΓpN(R1, R2, R10, R20; k0, Δ)=exp-Δ2ZA(0)8Iv{Ap}Iv{0},
Iv{Ap}Iv{0}
=fD2v exp0Zdz-i2k1k2k1-k2v˙2+k1k24Ap|v|pfD2v exp0Zdz-i2k1k2k1-k2v˙2.
k0=(k1+k2)/2,
Δ=k1-k2.
v(z)=zZ(R1-R2)+1-zZ(R10-R20)+n=1an sinnπzZ.
ΓpN(R1-R2, R10-R20; k0, Δ)=exp-Δ2ZA(0)8expk1k24Ap×0ZdzzZ(R1-R2)+1-zZ(R10-R20)p.
Iv{A}=ik1k22πΔZηsin ηexp-ik1k22ΔZηsin η×[(ξ2+p2)cos η-2ξp].
Γ2N(p, ξ; k0, Δ)=exp-Δ2ZA(0)8ηsin ηexpik1k22ΔZ(ξ-p)2×exp-ik1k22ΔZηsin η[(ξ2+p2)cos η-2ξp].
R=(R1+R2)/2,p=R1-R2,
R0=(R10+R20)/2,ξ=R10-R20,
Γ2sph(R, R0, p, ξ; k0, Δ)=uf(R1, R10, k1)uf*(R2, R20, k2)×Γ2N(R, R0, p, ξ; k0, Δ)=k02-Δ2/4(2πZ)2ηsin ηexp-Δ2ZA(0)8×expiΔ2Z(R-R0)2expik0Z(R-R0)(p-ξ)×expik022ΔZ(p-ξ)2exp-i2k02-Δ2/4ΔZηsin η×[(ξ2+p2)cos η-2ξp].
Γ2pl(p; k0, Δ)=d2R0d2ξΓ2sph(R, R0, p, ξ; k0, Δ)=exp-Δ2ZA(0)81cos η×expi2(k02-Δ2/4)ηΔZtan η p2.
Γ2Gaus(R, p; k0, Δ)
=d2R0d2ξ exp-1W02+ik12F0|R10|2
×exp-1W02-ik22F0|R20|2×Γ2sph(R, R0, p, ξ; k0, Δ),
Γ2Gaus(R, p; k0, Δ)=k1k24Z2ηsin ηexp-Δ2ZA(0)8ik1k22ΔZαη cot η+αβ4+k024F˜2-1×expiΔ2ZR2expik0ZR·pexpik022ΔZp2exp(ΔR+k0p)24αZ2exp-ik1k22ΔZη cot η p2×exp-k1k2ΔZηsin ηp-k02ΔZp-k0ZR+ik02αF˜Z(ΔR+k0p)22ik1k2ΔZη cot η+β+k02αF˜2,
1/F˜=1/F0-1/Z,
α=2/W02+iΔ/2F˜,
β=2/W02+i(Δ/2F0-2k02/ΔZ).
z=iz¯,
Ap=-iA¯p.
η=iZ¯-A¯2Δ21/2iη¯.
Iv{Ap}=fD2v exp-0ZdzLp,
Lp=k1k22(k1-k2)v˙2-k1k24Ap|v|2.
ΓpNΓ2N=fD2v exp-0ZdzL2exp-k1k240Zdz(A2|v|2-Ap|v|p)fD2v exp-0ZdzL2exp-k1k240Zdz(A2|v|2-Ap|v|p),
exp(J)exp-k1k240Zdz(A2|v|2-Ap|v|p)exp-k1k240Zdz(A2|v|2-Ap|v|p).
ΓpNΓ2N exp(J).
Apρp(2π)d2κΦp(κ)[exp(iκρ)-1].
Jnumπ2k1k2fD2v exp-0ZdzL2d2κΨ(κ)×0Zdzexpiκ0Zdzv(z)δ(z-z)-1,
Jnum=k1k222ηΔZ sinh ηexp-k1k22ΔZηsinh η×[(ξ2+p2)cosh η-2ξp]×0Zdzd2κΨ(κ)[exp(iκρ)exp(-κ2Lˆ2)-1],
ρ=p sinhηzZ+ξ sinhηZ-zZ/sinh η,
Lˆ2=ΔZ4k1k2cosh η-coshηZ-2zZ/η sinh η.
J=π2k1k20Zdzd2κΨ(κ)×[exp(iκρ)exp(-κ2Lˆ2)-1].
Φ5/3(κ)=0.033C2κ-11/3 exp(-κ2/κm2),
10.26C-2k0-2L0-5/3Z10.26C-2k0-2l0-5/3.
Φ2(κ)=12l38ππexp-14κ2l2,
12=C˜C2L02/3π.
J=π2k1k20Zdz0κdκ[Φ5/3(κ)-Φ2(κ)]×[exp(-κ2Lˆ2)J0(κρ)-1]π2k1k2C20Zdz{[I5/3(Lˆ2, ρ)-I5/3(0, 0)]-[I2(Lˆ2, ρ)-I2(0, 0)]},
I5/3=0.0330κdκκ-11/3 exp[-κ2(1/κm2+Lˆ2)]J0(κρ),
I2=C˜L02/3l38π20κdκ exp[-κ2(l2/4+Lˆ2)]J0(κρ).
J=π2k1k2C20Zdz0.033Γ(-5/6)2[κ˜m-5/3 1F1(-5/6, 1;-κ˜m2ρ2/4)-κm-5/3]-C˜L02/3l4π2l2l2+4Lˆ21-ρ2l2+4Lˆ2-1,
ΓpNηsinh ηexp(J)exp-18Δ2ZA(0)expk1k22ΔZ(ξ-p)2×exp-k1k22ΔZηsinh η[(ξ2+p2)cosh η-2ξp],
A(0)=0.7816C2L05/3.
limΔ0 Γ2N=exp14k02A2Z01dt|ρ|2=exp112k02A2Z(ξ2+p2+ξp),
ρ=tp+(1-t)ξ.
limΔ0 J=π2k02Z01dtd2κΨ(κ)[exp(iκρ)-1]=14k02Z01dt[A5/3|ρ|5/3-A2|ρ|2].
Γ5/3N=exp14k02A5/3Z01dt|ρ|5/3,
Γ5/3pl(p)=Γ5/3N(p, p)=exp(14k02A5/3Z|p|5/3).
Γ5/3sph(0; k0, Δ)
=k1k2(2πZ)2ηsin ηexpk1k2C20ZdzC˜L02/3lLˆ2l2+4Lˆ2-0.11π2κm-5/3[(1+Lˆ2κm2)5/6-1],
Lˆ2=i4ΔZk1k2cos η-cosηZ-2zZη sin η.
Γ5/3,0sph(0; k0,Δ)=k1k2(2πZ)2exp-0.11π2k1k2C2×0Zdzκm-5/3[(1+Lˆ2κm2)5/6-1],
Lˆ2(t)=-iΔZ2k1k2t(1-t).
Γ5/3,0sphexp[-(Δ/k5/3,coh)5/6],
Δn(Δ/k5/3,coh)5/11.
Lˆ201dtLˆ2(t)=i4ΔZk1k2η cos η-sin ηη2 sin η.
Lˆ2ΔZk1k2Δk0Zλ0,
l0Re,ImLˆ|p|, |ξ|l,L0.
1F1(-5/6, 1;z)(-z)5/6Γ(11/6)+exp(z)z-11/6Γ(-5/6).
J=-0.365k1k2C2Z01dt|ρ|5/3+k1k2C2C˜L02/3Z4l01dt|ρ|2,
J=-0.365k1k2C2Z01dt|ρ|5/3,
Γ5/3pl(p; k0, Δ)=ik1k22πZΔd2ξΓ5/3N(p, ξ; k0, Δ)×exp-ik1k22ΔZ(p-ξ)2.
ξn(-k04A5/3/2Δ)3/11|p|
Γ5/3Gaus(R, p; k0, Δ)=k1k24παZ2expiΔ2ZR2d2ξΓ5/3N(p, ξ; k0, Δ)×exp-γ4ξ2expiΔ8Z(p-ξ)2×expik0ZR(p-ξ)×exp-14αΔZR+k0F˜ξ+k0Zp2,
Γ5/3Gaus,f(R, p; k0, Δ)=k1k2W028πZ2expiΔ2ZR2+p24×exp-W028Z2(ΔR+k0p)2d2ξΓ5/3N(p, ξ; k0, Δ)×exp-ξ22W02exp-iΔ4Zξpexpik0ZR(p-ξ).

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