Abstract

To analyze the effects of atmospheric refractive turbulence on coherent lidar performance in a realistic way it is necessary to consider the use of simulations of beam propagation in three-dimensional random media. The capability of the split-step solution to simulate the propagation phenomena is shown, and the limitations and numerical requirements for a simulation of given accuracy are established. Several analytical theories that describe laser beam spreading, beam wander, coherence diameters, and variance and autocorrelation of the beam intensity are compared with results from simulations. Although the analysis stems from a study of coherent lidar performance, the conclusions of the method are applicable to other areas related to beam propagation in the atmosphere.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Belmonte, B. J. Rye, “Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems,” Appl. Opt. 39, 2401–2411 (2000).
    [CrossRef]
  2. A. Belmonte, B. J. Rye, W. A. Brewer, R. M. Hardesty, “Coherent lidar returns in turbulent atmosphere from simulation of beam propagation,” presented at the Tenth Biennial Coherent Laser Radar Technology and Applications Conference, Mount Hood, Ore., 28 June–2 July 1999.
  3. D. L. Fried, “Optical heterodyne detection on an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–66 (1967).
    [CrossRef]
  4. H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
    [CrossRef]
  5. S. F. Clifford, S. Wandzura, “Monostatic heterodyne lidar performance: the effect of the turbulent atmosphere,” Appl. Opt. 20, 514–516 (1981); erratum 20, 1502 (1981).
  6. R. G. Frehlich, M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
    [CrossRef] [PubMed]
  7. B. J. Rye, “Refractive-turbulent contribution to incoherent backscatter heterodyne lidar returns,” J. Opt. Soc. Am. 71, 687–691 (1981).
    [CrossRef]
  8. V. A. Banakh, V. L. Mironov, LIDAR in a Turbulent Atmosphere (Artech, Dedham, Mass., 1987).
  9. J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [CrossRef]
  10. R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  11. J. M. Martin, S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
    [CrossRef] [PubMed]
  12. J. M. Martin, S. M. Flatté, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–847 (1990).
    [CrossRef]
  13. J. Strohbehn, S. Clifford, “Polarization and angle-of-arrival fluctuations for a plane wave propagated through a turbulent medium,” IEEE Trans. Antennas Propag. AP-15, 416–421 (1967).
    [CrossRef]
  14. V. Tatarski, The Effect of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  15. S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 9–43.
    [CrossRef]
  16. 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, Berlin, 1978), pp. 44–106.
  17. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
    [CrossRef]
  18. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
    [CrossRef]
  19. R. G. Frehlich, “Intensity covariance of a point source in a random medium with a Kolmogorov spectrum and an inner scale of turbulence,” J. Opt. Soc. Am. A 4, 360–366 (1987).
    [CrossRef]
  20. R. Dashen, G.-Y. Wang, “Intensity fluctuation for waves behind a phase screen: a new asymptotic scheme,” J. Opt. Soc. Am. A 10, 1219–1225 (1993).
    [CrossRef]
  21. G.-Y. Wang, R. Dashen, “Intensity moments for waves in random media: three-order standard asymptotic calculation,” J. Opt. Soc. Am. A 10, 1226–1232 (1993).
    [CrossRef]
  22. R. Dashen, G.-Y. Wang, S. M. Flatté, C. Bracher, “Moments of intensity and log intensity: new asymptotic results for waves in power-law media,” J. Opt. Soc. Am. A 10, 1233–1242 (1993).
    [CrossRef]
  23. R. F. Lutomirski, H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef] [PubMed]
  24. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1971).
    [CrossRef]
  25. F. D. Tappert, “The parabolic approximation method,” in Wave Propagation and Underwater Acoustics, J. B. Keller, J. S. Papadakis, eds., Vol. 70 of Lecture Notes in Physics (Springer-Verlag, New York, 1977).
    [CrossRef]
  26. J. Martin, “Simulation of wave propagation in random media: theory and applications,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds., Vol. PM09 of Press Monographs (SPIE Press, Bellingham, Wash., 1993).
  27. D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
    [CrossRef]
  28. M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32, pp. 203–265.
    [CrossRef]
  29. R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
    [CrossRef]
  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]
  31. J. H. Churnside, “A spectrum of refractive turbulence in the turbulent atmosphere,” J. Mod. Opt. 37, 13–16 (1990).
    [CrossRef]
  32. L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
    [CrossRef]
  33. W. B. Miller, J. C. Ricklin, L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11, 2719–2726 (1994).
    [CrossRef]
  34. M. Spivack, B. J. Unscinski, “The split-step solution in random wave propagation,” J. Comput. Appl. Math. 27, 349–361 (1989).
    [CrossRef]
  35. M. Spivack, “Accuracy of the moments from simulation of waves in random media,” J. Opt. Soc. Am. A 7, 790–793 (1990).
    [CrossRef]
  36. W. A. Coles, J. P. Filice, R. G. Frehlich, M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” Appl. Opt. 34, 2089–2101 (1995).
    [CrossRef] [PubMed]
  37. R. F. Lutomirski, H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef] [PubMed]
  38. L. D. Dickson, “Characteristics of a propagating Gaussian beam,” Appl. Opt. 9, 1854–1861 (1970).
    [CrossRef] [PubMed]
  39. N. Rodier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  40. C. Schwartz, G. Baum, E. N. Ribak, “Turbulence-degraded wave fronts as fractal surfaces,” J. Opt. Soc. Am. A 11, 444–451 (1994).
    [CrossRef]
  41. R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  42. A. Belmonte, A. Comeron, J. A. Rubio, J. Bara, E. Fernandez, “Atmospheric-turbulence-induced power-fade statistics for a multiaperture optical receiver,” Appl. Opt. 36, 8632–8638 (1997).
    [CrossRef]
  43. F. Bunkin, K. Gochelashvily, “Spreading of a light beam in a turbulent medium,” Radiophys. Quantum Electron. 13, 811–821 (1970).
    [CrossRef]
  44. R. Fante, “Mutual coherence function and frequency spectrum of a laser beam propagating through atmospheric turbulence,” J. Opt. Soc. Am. 64, 592–598 (1974).
    [CrossRef]
  45. A. S. Gurchiv, V. I. Tatarskii, “Coherent and intensity fluctuations of light in the turbulent media,” Radio Sci. 10, 3–14 (1975).
    [CrossRef]
  46. H. T. Yura, “Short-term average optical-beam spread in a turbulent medium,” J. Opt. Soc. Am. 63, 567–572 (1973).
    [CrossRef]
  47. R. E. Hufnagel, “Propagation through atmospheric turbulence,” in The Infrared Handbook, W. L. Wolfe, G. J. Zissis, eds. (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1978), Vol. 6, pp. 1–56.
  48. V. L. Mironov, V. V. Nosov, “On the theory of spatially limited light beam displacements in a randomly inhomogeneous medium,” J. Opt. Soc. Am. 64, 516–518 (1977).
  49. L. C. Andrews, W. B. Miller, J. C. Ricklin, “Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,” J. Opt. Soc. Am. A 11, 1653–1660 (1994).
    [CrossRef]
  50. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972).
    [CrossRef] [PubMed]
  51. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  52. M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
    [CrossRef]
  53. L. C. Andrews, R. L. Phillips, C. Y. Hopen, M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
    [CrossRef]
  54. R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
    [CrossRef]
  55. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  56. W. B. Miller, J. C. Ricklin, L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10, 661–672 (1993).
    [CrossRef]
  57. K. S. Gochelashvily, V. I. Shishov, “Multiple scattering of light in a turbulent medium,” Opt. Acta 18, 767–777 (1971).
    [CrossRef]
  58. K. S. Gochelashvily, V. G. Pevgov, V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere (Fraunhofer zone of transmitter),” Sov. J. Quantum Electron. 4, 632–637 (1974).
    [CrossRef]
  59. A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Appl. Opt. 5, 1350–1356 (1966).
    [CrossRef]
  60. D. L. Fried, H. T. Yura, “Telescope-performance reciprocity for propagation in a turbulent medium,” J. Opt. Soc. Am. 62, 600–602 (1972).
    [CrossRef]
  61. J. H. Shapiro, “Reciprocity of the turbulent atmosphere,” J. Opt. Soc. Am. 61, 492–495 (1971).
    [CrossRef]

2000 (1)

1999 (1)

1997 (1)

1995 (1)

1994 (4)

1993 (4)

1992 (3)

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

1991 (1)

1990 (4)

J. M. Martin, S. M. Flatté, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–847 (1990).
[CrossRef]

N. Rodier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

J. H. Churnside, “A spectrum of refractive turbulence in the turbulent atmosphere,” J. Mod. Opt. 37, 13–16 (1990).
[CrossRef]

M. Spivack, “Accuracy of the moments from simulation of waves in random media,” J. Opt. Soc. Am. A 7, 790–793 (1990).
[CrossRef]

1989 (1)

M. Spivack, B. J. Unscinski, “The split-step solution in random wave propagation,” J. Comput. Appl. Math. 27, 349–361 (1989).
[CrossRef]

1988 (1)

1987 (1)

1983 (1)

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

1981 (2)

1979 (1)

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

1978 (2)

1977 (1)

1976 (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

1975 (3)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, 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]

A. S. Gurchiv, V. I. Tatarskii, “Coherent and intensity fluctuations of light in the turbulent media,” Radio Sci. 10, 3–14 (1975).
[CrossRef]

1974 (2)

K. S. Gochelashvily, V. G. Pevgov, V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere (Fraunhofer zone of transmitter),” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

R. Fante, “Mutual coherence function and frequency spectrum of a laser beam propagating through atmospheric turbulence,” J. Opt. Soc. Am. 64, 592–598 (1974).
[CrossRef]

1973 (1)

1972 (2)

1971 (5)

1970 (3)

L. D. Dickson, “Characteristics of a propagating Gaussian beam,” Appl. Opt. 9, 1854–1861 (1970).
[CrossRef] [PubMed]

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

F. Bunkin, K. Gochelashvily, “Spreading of a light beam in a turbulent medium,” Radiophys. Quantum Electron. 13, 811–821 (1970).
[CrossRef]

1967 (2)

D. L. Fried, “Optical heterodyne detection on an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–66 (1967).
[CrossRef]

J. Strohbehn, S. Clifford, “Polarization and angle-of-arrival fluctuations for a plane wave propagated through a turbulent medium,” IEEE Trans. Antennas Propag. AP-15, 416–421 (1967).
[CrossRef]

1966 (2)

Al-Habash, M. A.

Andrews, L. C.

Banakh, V. A.

V. A. Banakh, V. L. Mironov, LIDAR in a Turbulent Atmosphere (Artech, Dedham, Mass., 1987).

Bara, J.

Baum, G.

Belmonte, A.

A. Belmonte, B. J. Rye, “Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems,” Appl. Opt. 39, 2401–2411 (2000).
[CrossRef]

A. Belmonte, A. Comeron, J. A. Rubio, J. Bara, E. Fernandez, “Atmospheric-turbulence-induced power-fade statistics for a multiaperture optical receiver,” Appl. Opt. 36, 8632–8638 (1997).
[CrossRef]

A. Belmonte, B. J. Rye, W. A. Brewer, R. M. Hardesty, “Coherent lidar returns in turbulent atmosphere from simulation of beam propagation,” presented at the Tenth Biennial Coherent Laser Radar Technology and Applications Conference, Mount Hood, Ore., 28 June–2 July 1999.

Bracher, C.

Brewer, W. A.

A. Belmonte, B. J. Rye, W. A. Brewer, R. M. Hardesty, “Coherent lidar returns in turbulent atmosphere from simulation of beam propagation,” presented at the Tenth Biennial Coherent Laser Radar Technology and Applications Conference, Mount Hood, Ore., 28 June–2 July 1999.

Bunkin, F.

F. Bunkin, K. Gochelashvily, “Spreading of a light beam in a turbulent medium,” Radiophys. Quantum Electron. 13, 811–821 (1970).
[CrossRef]

Bunkin, F. V.

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

Charnotskii, M. I.

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32, pp. 203–265.
[CrossRef]

Churnside, J. H.

J. H. Churnside, “A spectrum of refractive turbulence in the turbulent atmosphere,” J. Mod. Opt. 37, 13–16 (1990).
[CrossRef]

Clifford, S.

J. Strohbehn, S. Clifford, “Polarization and angle-of-arrival fluctuations for a plane wave propagated through a turbulent medium,” IEEE Trans. Antennas Propag. AP-15, 416–421 (1967).
[CrossRef]

Clifford, S. F.

Coles, W. A.

Comeron, A.

Dainty, J. C.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Dashen, R.

Dickson, L. D.

Fante, R.

Fante, R. L.

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

Feit, M. D.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Fernandez, E.

Filice, J. P.

Flatté, S. M.

Fleck, J. A.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Frehlich, R. G.

Fried, D. L.

Glindemann, A.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Gochelashvily, K.

F. Bunkin, K. Gochelashvily, “Spreading of a light beam in a turbulent medium,” Radiophys. Quantum Electron. 13, 811–821 (1970).
[CrossRef]

Gochelashvily, K. S.

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

K. S. Gochelashvily, V. G. Pevgov, V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere (Fraunhofer zone of transmitter),” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

K. S. Gochelashvily, V. I. Shishov, “Multiple scattering of light in a turbulent medium,” Opt. Acta 18, 767–777 (1971).
[CrossRef]

Gozani, J.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32, pp. 203–265.
[CrossRef]

Gurchiv, A. S.

A. S. Gurchiv, V. I. Tatarskii, “Coherent and intensity fluctuations of light in the turbulent media,” Radio Sci. 10, 3–14 (1975).
[CrossRef]

Hardesty, R. M.

A. Belmonte, B. J. Rye, W. A. Brewer, R. M. Hardesty, “Coherent lidar returns in turbulent atmosphere from simulation of beam propagation,” presented at the Tenth Biennial Coherent Laser Radar Technology and Applications Conference, Mount Hood, Ore., 28 June–2 July 1999.

Hill, R. J.

Hopen, C. Y.

Hufnagel, R. E.

R. E. Hufnagel, “Propagation through atmospheric turbulence,” in The Infrared Handbook, W. L. Wolfe, G. J. Zissis, eds. (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1978), Vol. 6, pp. 1–56.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Kavaya, M. J.

Knepp, D. L.

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

Lane, R. G.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Lawrence, R. S.

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

Lutomirski, R. F.

Martin, J.

J. Martin, “Simulation of wave propagation in random media: theory and applications,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds., Vol. PM09 of Press Monographs (SPIE Press, Bellingham, Wash., 1993).

Martin, J. M.

Miller, W. B.

Mironov, V. L.

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Nosov, V. V.

Pevgov, V. G.

K. S. Gochelashvily, V. G. Pevgov, V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere (Fraunhofer zone of transmitter),” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

Phillips, R. L.

Prokhorov, A. M.

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

Ribak, E. N.

Ricklin, J. C.

Rodier, N.

N. Rodier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Rubio, J. A.

Rye, B. J.

A. Belmonte, B. J. Rye, “Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems,” Appl. Opt. 39, 2401–2411 (2000).
[CrossRef]

B. J. Rye, “Refractive-turbulent contribution to incoherent backscatter heterodyne lidar returns,” J. Opt. Soc. Am. 71, 687–691 (1981).
[CrossRef]

A. Belmonte, B. J. Rye, W. A. Brewer, R. M. Hardesty, “Coherent lidar returns in turbulent atmosphere from simulation of beam propagation,” presented at the Tenth Biennial Coherent Laser Radar Technology and Applications Conference, Mount Hood, Ore., 28 June–2 July 1999.

Schwartz, C.

Shapiro, J. H.

Shishov, V. I.

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

K. S. Gochelashvily, V. G. Pevgov, V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere (Fraunhofer zone of transmitter),” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

K. S. Gochelashvily, V. I. Shishov, “Multiple scattering of light in a turbulent medium,” Opt. Acta 18, 767–777 (1971).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Appl. Opt. 5, 1350–1356 (1966).
[CrossRef]

Spivack, M.

M. Spivack, “Accuracy of the moments from simulation of waves in random media,” J. Opt. Soc. Am. A 7, 790–793 (1990).
[CrossRef]

M. Spivack, B. J. Unscinski, “The split-step solution in random wave propagation,” J. Comput. Appl. Math. 27, 349–361 (1989).
[CrossRef]

Strohbehn, J.

J. Strohbehn, S. Clifford, “Polarization and angle-of-arrival fluctuations for a plane wave propagated through a turbulent medium,” IEEE Trans. Antennas Propag. AP-15, 416–421 (1967).
[CrossRef]

Strohbehn, J. W.

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

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, Berlin, 1978), pp. 44–106.

Tappert, F. D.

F. D. Tappert, “The parabolic approximation method,” in Wave Propagation and Underwater Acoustics, J. B. Keller, J. S. Papadakis, eds., Vol. 70 of Lecture Notes in Physics (Springer-Verlag, New York, 1977).
[CrossRef]

Tatarski, V.

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

Tatarskii, V. I.

A. S. Gurchiv, V. I. Tatarskii, “Coherent and intensity fluctuations of light in the turbulent media,” Radio Sci. 10, 3–14 (1975).
[CrossRef]

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32, pp. 203–265.
[CrossRef]

Unscinski, B. J.

M. Spivack, B. J. Unscinski, “The split-step solution in random wave propagation,” J. Comput. Appl. Math. 27, 349–361 (1989).
[CrossRef]

Wandzura, S.

Wang, G.-Y.

Yadlowsky, M.

Yura, H. T.

Zavorotny, V. U.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32, pp. 203–265.
[CrossRef]

Appl. Opt. (12)

L. D. Dickson, “Characteristics of a propagating Gaussian beam,” Appl. Opt. 9, 1854–1861 (1970).
[CrossRef] [PubMed]

H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1971).
[CrossRef]

H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972).
[CrossRef] [PubMed]

S. F. Clifford, S. Wandzura, “Monostatic heterodyne lidar performance: the effect of the turbulent atmosphere,” Appl. Opt. 20, 514–516 (1981); erratum 20, 1502 (1981).

R. G. Frehlich, M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
[CrossRef] [PubMed]

W. A. Coles, J. P. Filice, R. G. Frehlich, M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” Appl. Opt. 34, 2089–2101 (1995).
[CrossRef] [PubMed]

A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Appl. Opt. 5, 1350–1356 (1966).
[CrossRef]

A. Belmonte, B. J. Rye, “Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems,” Appl. Opt. 39, 2401–2411 (2000).
[CrossRef]

J. M. Martin, S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
[CrossRef] [PubMed]

A. Belmonte, A. Comeron, J. A. Rubio, J. Bara, E. Fernandez, “Atmospheric-turbulence-induced power-fade statistics for a multiaperture optical receiver,” Appl. Opt. 36, 8632–8638 (1997).
[CrossRef]

R. F. Lutomirski, H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
[CrossRef] [PubMed]

R. F. Lutomirski, H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
[CrossRef] [PubMed]

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. Strohbehn, S. Clifford, “Polarization and angle-of-arrival fluctuations for a plane wave propagated through a turbulent medium,” IEEE Trans. Antennas Propag. AP-15, 416–421 (1967).
[CrossRef]

J. Comput. Appl. Math. (1)

M. Spivack, B. J. Unscinski, “The split-step solution in random wave propagation,” J. Comput. Appl. Math. 27, 349–361 (1989).
[CrossRef]

J. Fluid Mech. (1)

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
[CrossRef]

J. Mod. Opt. (2)

J. H. Churnside, “A spectrum of refractive turbulence in the turbulent atmosphere,” J. Mod. Opt. 37, 13–16 (1990).
[CrossRef]

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

J. Opt. Soc. Am. (8)

J. Opt. Soc. Am. A (11)

R. Dashen, G.-Y. Wang, S. M. Flatté, C. Bracher, “Moments of intensity and log intensity: new asymptotic results for waves in power-law media,” J. Opt. Soc. Am. A 10, 1233–1242 (1993).
[CrossRef]

J. M. Martin, S. M. Flatté, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–847 (1990).
[CrossRef]

C. Schwartz, G. Baum, E. N. Ribak, “Turbulence-degraded wave fronts as fractal surfaces,” J. Opt. Soc. Am. A 11, 444–451 (1994).
[CrossRef]

L. C. Andrews, W. B. Miller, J. C. Ricklin, “Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,” J. Opt. Soc. Am. A 11, 1653–1660 (1994).
[CrossRef]

W. B. Miller, J. C. Ricklin, L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11, 2719–2726 (1994).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

R. G. Frehlich, “Intensity covariance of a point source in a random medium with a Kolmogorov spectrum and an inner scale of turbulence,” J. Opt. Soc. Am. A 4, 360–366 (1987).
[CrossRef]

M. Spivack, “Accuracy of the moments from simulation of waves in random media,” J. Opt. Soc. Am. A 7, 790–793 (1990).
[CrossRef]

W. B. Miller, J. C. Ricklin, L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10, 661–672 (1993).
[CrossRef]

R. Dashen, G.-Y. Wang, “Intensity fluctuation for waves behind a phase screen: a new asymptotic scheme,” J. Opt. Soc. Am. A 10, 1219–1225 (1993).
[CrossRef]

G.-Y. Wang, R. Dashen, “Intensity moments for waves in random media: three-order standard asymptotic calculation,” J. Opt. Soc. Am. A 10, 1226–1232 (1993).
[CrossRef]

Opt. Acta (2)

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

K. S. Gochelashvily, V. I. Shishov, “Multiple scattering of light in a turbulent medium,” Opt. Acta 18, 767–777 (1971).
[CrossRef]

Opt. Eng. (1)

N. Rodier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Proc. IEEE (5)

D. L. Fried, “Optical heterodyne detection on an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–66 (1967).
[CrossRef]

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, 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. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Radio Sci. (1)

A. S. Gurchiv, V. I. Tatarskii, “Coherent and intensity fluctuations of light in the turbulent media,” Radio Sci. 10, 3–14 (1975).
[CrossRef]

Radiophys. Quantum Electron. (1)

F. Bunkin, K. Gochelashvily, “Spreading of a light beam in a turbulent medium,” Radiophys. Quantum Electron. 13, 811–821 (1970).
[CrossRef]

Sov. J. Quantum Electron. (1)

K. S. Gochelashvily, V. G. Pevgov, V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere (Fraunhofer zone of transmitter),” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

Waves Random Media (3)

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Other (10)

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. 32, pp. 203–265.
[CrossRef]

F. D. Tappert, “The parabolic approximation method,” in Wave Propagation and Underwater Acoustics, J. B. Keller, J. S. Papadakis, eds., Vol. 70 of Lecture Notes in Physics (Springer-Verlag, New York, 1977).
[CrossRef]

J. Martin, “Simulation of wave propagation in random media: theory and applications,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds., Vol. PM09 of Press Monographs (SPIE Press, Bellingham, Wash., 1993).

A. Belmonte, B. J. Rye, W. A. Brewer, R. M. Hardesty, “Coherent lidar returns in turbulent atmosphere from simulation of beam propagation,” presented at the Tenth Biennial Coherent Laser Radar Technology and Applications Conference, Mount Hood, Ore., 28 June–2 July 1999.

V. A. Banakh, V. L. Mironov, LIDAR in a Turbulent Atmosphere (Artech, Dedham, Mass., 1987).

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

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 9–43.
[CrossRef]

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, Berlin, 1978), pp. 44–106.

R. E. Hufnagel, “Propagation through atmospheric turbulence,” in The Infrared Handbook, W. L. Wolfe, G. J. Zissis, eds. (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1978), Vol. 6, pp. 1–56.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

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 (25)

Fig. 1
Fig. 1

An initially Gaussian beam propagated through a turbulent atmosphere looks quite different from one propagated through free space. Beam wander, beam spread, and scintillation will modify the performance of any laser system. Instantaneous images of a collimated Gaussian beam at ranges from 0 to 4 km at 500-m intervals along an atmospheric path are shown. The wavelength is 2 µm, and the initial beam has a 1/e 2 diameter of 14 cm. The level of refractive turbulence, C n 2, has a moderate value of 10-14 m-2/3.

Fig. 2
Fig. 2

The receiver plane formulation expresses the performance of the heterodyne detection in the degree of coherence of the backscattered radiation and its proper match to the local (LO) oscillator field (top). The target plane formulation (bottom) reduces the problem of calculating lidar returns to one of computing the transmitted and backpropagated fields along the propagation path and estimating the overlap function of the two irradiances.

Fig. 3
Fig. 3

It is usual to describe the ranges of validity of the various techniques by looking at the irradiance along the turbulence path. Small-perturbation methods are valid only when intensity fluctuations are smaller than 10% of intensity mean values. The asymptotic approximations appropriate for description of the experimental observations assume unrealistically strong turbulences. Unfortunately, heuristic theories are not developed for intensity fluctuations. Simulations offer a more promising approach.

Fig. 4
Fig. 4

Short- and long-term images of a 2-µm wavelength initially collimated Gaussian beam propagated 3 km through the atmosphere with a moderate level of refractive turbulence C n 2 = 10-14 m-2/3. The problem of choosing the appropriate grid sampling and grid extension is addressed by use of classic signal-processing methods. If the field irradiance is to be calculated, all its short-term scales must be properly sampled (d 0 and D 0) and the grid has to be large enough to contain the long-term, diffraction-broadened Gaussian beam (1/e 2 diameter 2 W). White circles show the corresponding free-space diameters.

Fig. 5
Fig. 5

Range limitations to the simulation of beam propagation for reasonable turbulence levels C n 2 with different numbers of points N defining the grid. An initial beam with an 1/e 2 intensity radius W 0 of 7 cm and a 2-µm wavelength is considered. The inner scale has the typical value of 1 cm. Top, beam-size limitations based on long-term beam spread; bottom, the sampling restrictions.

Fig. 6
Fig. 6

Similar to Fig. 5 but for a 10-µm wavelength beam.

Fig. 7
Fig. 7

Simulated and theoretical long-term beam width W(z) as a function of range z for the strong-turbulence level (C n 2 = 10-12 m-2/3) and both 2- and 10-µm wavelength beams (top and bottom, respectively). Initial parameters are similar to those described in Fig. 5. Free-space propagation designates the absence of refractive turbulence.

Fig. 8
Fig. 8

Similar to Fig. 7 but for a weaker turbulence level (C n 2 = 10-14 m-2/3).

Fig. 9
Fig. 9

Similar to Fig. 7 but for a short-term beam width. Beam deflection is corrected before the 1/e 2 intensity radius is estimated.

Fig. 10
Fig. 10

Similar to Fig. 8 for the 2-µm wavelength with a short-term beam width. Along with simulated and theoretical short-term expectations, long-term widths are shown.

Fig. 11
Fig. 11

Simulated and level-arm (theoretical) approximation beam-wander standard deviation 〈β2(z)〉1/2 as a function of range z for a strong turbulence level (C n 2 = 10-12 m-2/3) and both 2- and 10-µm wavelength beams (top and bottom, respectively). Initial parameters are similar to those described in Fig. 5. Given the short-term beam width W S (z), one could use W 2(z) = W S 2(z) + 2〈β2(z)〉 to obtain results similar to those shown.

Fig. 12
Fig. 12

Similar to Fig. 10 but for a weaker turbulence level (C n 2 = 10-14 m-2/3).

Fig. 13
Fig. 13

Simulated and heuristic (theoretical) long-term coherence diameter as a function of range z for a strong turbulence level (C n 2 = 10-12 m-2/3) and both 2- and 10-µm wavelength beams (top and bottom). Initial parameters are similar to those described in Fig. 5.

Fig. 14
Fig. 14

Similar to Fig. 13 but for a weaker turbulence level (C n 2 = 10-14 m-2/3).

Fig. 15
Fig. 15

Similar to Fig. 13, but a short-term coherence diameter is now considered. Beam deflection is corrected before the coherence function is estimated.

Fig. 16
Fig. 16

Similar to Fig. 14 for the 2-µm wavelength, but a short-term coherence diameter is now considered. Along with simulated and heuristic short-term expectations, long term diameters are shown. Plane- and spherical-wave classic results bound the Gaussian-beam coherence diameter (for shorter and longer paths, respectively).

Fig. 17
Fig. 17

Simulated and Rytov-method on-axis irradiance variance as a function of range z for a strong turbulence level (C n 2 = 10-12 m-2/3) and both 2- and 10-µm wavelength beams (top and bottom, respectively). Initial parameters are similar to those described in Fig. 5; 1-standard-deviation error bars are shown.

Fig. 18
Fig. 18

Similar to Fig. 17 for the 2-µm wavelength but a weaker turbulence level (C n 2 = 10-14 m-2/3). Along with simulated and Rytov expectations, plane- and spherical-wave classic results bound the Gaussian-beam on-axis irradiance fluctuations. For longer paths than those presented, the variances of spherical and Gaussian waves should be similar.

Fig. 19
Fig. 19

Simulated strength of on-axis scintillation versus on-axis scintillation predicted by Rytov method (circles). A strong turbulence level (C n 2 = 10-12 m-2/3) and both 2- and 10-µm wavelength beams (top and bottom, respectively) are considered. Parameters are similar to those described in Fig. 17.

Fig. 20
Fig. 20

Simulated and Rytov-method off-axis irradiance variance as a function of radial distance ρ normalized to beam width W(z) for 1-km range z, a moderate turbulence level (C n 2 = 10-14 m-2/3), and both 2- and 10-µm wavelength beams (top and bottom respectively). Initial parameters are similar to those described in Fig. 17; 1-standard-deviation error bars are shown.

Fig. 21
Fig. 21

Similar to Fig. 20 but for a larger range (z = 3 km).

Fig. 22
Fig. 22

Simulated strength of off-axis scintillation versus the off-axis scintillation predicted by the Rytov method. A moderate turbulence level (C n 2 = 10-14 m-2/3), a 3-km range, and both 2- and 10-µm wavelength beams (top and bottom, respectively) are considered. Parameters are similar to those described in Fig. 21. The solid curves shows the Rytov approximation only.

Fig. 23
Fig. 23

Simulated irradiance autocorrelation as a function of radial distance ρ normalized to Fresnel length L/k. A moderate turbulence level (C n 2 = 10-14 m-2/3) and both 2- and 10-µm wavelength beams (top and bottom, respectively) are considered. Results are shown for 1.0 and 5.0-km ranges. The dotted curves correspond to the spherical-wave, weak-fluctuation theory.

Fig. 24
Fig. 24

Similar to Fig. 23 but for a stronger turbulence level (C n 2 = 10-12 m-2/3): Only the 2-µm wavelength beam is considered. Results are shown for 0.5-, 1.0-, and 2.0-km ranges, Bottom, spherical-wave asymptotic theory expectations for the same wavelength, ranges, and turbulence levels.

Fig. 25
Fig. 25

Similar to Fig. 24 but for a longer wavelength (10 µm).

Equations (34)

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

2E+k21+n12E=0,
Ex, y, z=Ux, y, zexpjkz,
Ux, y, z=A0W0Wzexp-x+y2W2z,
2jk Uz+2U+2k2n1U=0,
Un+1=exp-j2kΔz2+2k2znzn+1 n1x, y, zdzUn.
Un+1=exp-j2kΔz2 2exp-jk znzn+1 n1x, y, zdz×exp-j2kΔz2 2Un.
SF=exp-j2kΔz2 2,
SP=exp-jk znzn+1 n1x, y, zdz.
uKx, Ky, zn+Δz2=expjk Δz2Kx2+Ky2×uKx, Ky, zn,
ΦnKx, Ky, Kz, z=0.033Cn2z×exp-Kl02π2K2+L02-11/6,
ΦKx, Ky, z=2πk2ΔzΦnKx, Ky, 0, z.
r0,S=0.42k20L Cn2zzL5/3dz-3/5
KmaxKmin=D0d0=4zkr0,S2.
KmaxGKminG=N.
Nn KmaxKmin=nq,
W2z=W021+zz02+24zkr0,S2,
NΔ2mWz.
N4m Wzl0.
W2z=WS2z+2β2z.
βz=βxz, βyz= ρUρ, zdρ Uρ, zdρ,
β2z=2.070 0L Cn2zL-z2WSz-1/3dz.
Mρ1, ρ2, z=Uρ1, zUρ2, zIρ11/2Iρ21/2,
Mρ, zexp-2ρ2r0,B2,
r0,B=r0,P1+4z2k2W041+23W0r0P21+134z2k2W041+12W0r0P21/2.
r0,P=0.42k20L Cn2zdz-3/5.
r0,Pshortr0P1+0.26r0P/W01/3.
σI2=I2-I2I2
σI2ρ, z=σI20, z+σI,r2ρ, z.
CIρ, z=I0, zIρ, z-I0, zIρ, z.
CIρ=16π2k20dKKΦnK0LdzJ0Kρ zL×sin2K2L-zz/L2k,
CIρ, z=exp-2ρ/r05/3+123.86kr02/4z1/3×b1ρ+b2ρ,
Pt=DD MSw1, w2, -f, tMLO×w1, w2, -fdw1dw2,
MSw1, w2, -f, t=USw1, -f, tUS*w2, -f, t,MSw1, w2, -f=ULOw1, -fULO*w2, -f,
Pt=Kz2βzλ2- ITp, z, tIBPLOp, zdp,

Metrics