Abstract

There have been many analyses of the reduction of lidar system efficiency in bistatic geometry caused by beam spreading and by fluctuations along the two paths generated by refractive-index turbulence. Although these studies have led to simple, approximate results that provide a reliable basis for preliminary assessment of lidar performance, they do not apply to monostatic lidars. For such systems, calculations and numerical simulations predict an enhanced coherence for the backscattered field. However, to the authors’ knowledge, a simple analytical mathematical framework for diagnosing the effects of refractive-index turbulence on the performance of both bistatic and monostatic coherent lidars does not exist. Here analytical empirical expressions for the transverse coherence variables and the heterodyne intensity are derived for bistatic and monostatic lidars as a function of moderate atmospheric refractive-index turbulence within the framework of the Gaussian-beam approximation.

© 2001 Optical Society of America

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References

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  1. W. L. Eberhard, R. E. Cupp, K. R. Healy, “Doppler lidar measurement of profiles of turbulence and momentum flux,” J. Atmos. Ocean. Technol. 6, 809–819 (1989).
    [CrossRef]
  2. R. M. Banta, L. D. Olivier, D. H. Levinson, “Evolution of the Monterey Bay sea-breeze layer as observed by pulsed Doppler lidar,” J. Atmos. Sci. 50, 3959–3982 (1993).
    [CrossRef]
  3. P. Drobinski, R. A. Brown, P. H. Flamant, J. Pelon, “Evidence of organized large eddies by ground-based Doppler lidar, sonic anemometer and sodar,” Boundary Layer Meteorol. 88, 343–361 (1998).
    [CrossRef]
  4. P. Drobinski, A. M. Dabas, C. Haeberli, P. H. Flamant, “On the small-scale dynamics of flow splitting in the Rhine valley during a shallow foehn event,” Boundary Layer Meteorol. 99, 277–296 (2001).
    [CrossRef]
  5. P. Drobinski, A. M. Dabas, P. H. Flamant, “Remote measurement of turbulent wind spectra by heterodyne Doppler lidar technique,” J. Appl. Meteorol. 39, 2434–2451 (2000).
    [CrossRef]
  6. R. G. Frehlich, “Effects of refractive turbulence on coherent laser radar,” Appl. Opt. 32, 2122–2139 (1993).
    [CrossRef] [PubMed]
  7. P. Drobinski, A. M. Dabas, P. Salamitou, “Spectral diversity technique for heterodyne Doppler lidar that uses hard target returns,” Appl. Opt. 39, 376–385 (2000).
    [CrossRef]
  8. X. Favreau, A. Delaval, P. H. Flamant, A. Dabas, P. Delville, “Four-element receiver for pulsed 10-µm heterodyne Doppler lidar,” Appl. Opt. 39, 2441–2448 (2000).
    [CrossRef]
  9. A. Dabas, P. H. Flamant, P. Salamitou, “Characterization of pulsed coherent Doppler lidar with the speckle effect,” Appl. Opt. 33, 6524–6532 (1994).
    [CrossRef] [PubMed]
  10. S. F. Clifford, S. Wandzura, “Monostatic heterodyne lidar performance: the effect of the turbulent atmosphere,” Appl. Opt. 20, 514–516 (1981).
    [CrossRef] [PubMed]
  11. R. G. Frehlich, M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
    [CrossRef] [PubMed]
  12. P. Salamitou, F. Darde, P. H. Flamant, “A semi analytic approach for coherent laser radar system efficiency, the nearest Gaussian approximation,” J. Mod. Opt. 41, 2101–2113 (1994).
    [CrossRef]
  13. D. L. Fried, “Atmospheric modulation noise in an optical heterodyne receiver,” IEEE J. Quantum Electron. QE-6, 213–221 (1967).
    [CrossRef]
  14. H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
    [CrossRef]
  15. A. N. Kolmogorov, “Energy dissipation in locally isotropic turbulence,” Doklady Akad. Nauk SSSR 32, 19–21 (1941).
  16. B. J. Rye, “Refractive-turbulence contribution to incoherent backscatter heterodyne lidar returns,” J. Opt. Soc. Am. 71, 687–691 (1981).
    [CrossRef]
  17. J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equation and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
    [CrossRef]
  18. J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
    [CrossRef]
  19. R. G. Frehlich, “Space–time fourth moment of waves propagating in random media,” Radio Sci. 22, 481–490 (1987).
    [CrossRef]
  20. A. Belmonte, B. J. Rye, “Heterodyne lidar returns in the turbulent atmosphere: performance evaluation of simulated systems,” Appl. Opt. 39, 2401–2411 (2000).
    [CrossRef]
  21. R. G. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt. 39, 393–397 (2000).
    [CrossRef]
  22. A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426–5445 (2000).
    [CrossRef]
  23. P. Drobinski, A. M. Dabas, P. Delville, P. H. Flamant, J. Pelon, R. M. Hardesty, “Refractive-index structure parameter in the planetary boundary layer: comparison of measurements taken with a 10.6-µm coherent lidar, a 0.9-µm scintillometer, and in-situ sensors,” Appl. Opt. 38, 1648–1656 (1999).
    [CrossRef]
  24. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971).
  25. R. F. Lutomirsky, H. T. Yura, “Wave structure function and mutual coherence function of an optical wave in a turbulent atmosphere,” J. Opt. Soc. Am. 61, 482–487 (1971).
    [CrossRef]
  26. R. F. Lutomirsky, H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef]
  27. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–75.
  28. M. H. Lee, J. F. Holmes, J. R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 66, 1164–1172 (1976).
    [CrossRef]

2001 (1)

P. Drobinski, A. M. Dabas, C. Haeberli, P. H. Flamant, “On the small-scale dynamics of flow splitting in the Rhine valley during a shallow foehn event,” Boundary Layer Meteorol. 99, 277–296 (2001).
[CrossRef]

2000 (6)

1999 (1)

1998 (1)

P. Drobinski, R. A. Brown, P. H. Flamant, J. Pelon, “Evidence of organized large eddies by ground-based Doppler lidar, sonic anemometer and sodar,” Boundary Layer Meteorol. 88, 343–361 (1998).
[CrossRef]

1994 (2)

P. Salamitou, F. Darde, P. H. Flamant, “A semi analytic approach for coherent laser radar system efficiency, the nearest Gaussian approximation,” J. Mod. Opt. 41, 2101–2113 (1994).
[CrossRef]

A. Dabas, P. H. Flamant, P. Salamitou, “Characterization of pulsed coherent Doppler lidar with the speckle effect,” Appl. Opt. 33, 6524–6532 (1994).
[CrossRef] [PubMed]

1993 (2)

R. G. Frehlich, “Effects of refractive turbulence on coherent laser radar,” Appl. Opt. 32, 2122–2139 (1993).
[CrossRef] [PubMed]

R. M. Banta, L. D. Olivier, D. H. Levinson, “Evolution of the Monterey Bay sea-breeze layer as observed by pulsed Doppler lidar,” J. Atmos. Sci. 50, 3959–3982 (1993).
[CrossRef]

1991 (1)

1989 (1)

W. L. Eberhard, R. E. Cupp, K. R. Healy, “Doppler lidar measurement of profiles of turbulence and momentum flux,” J. Atmos. Ocean. Technol. 6, 809–819 (1989).
[CrossRef]

1987 (1)

R. G. Frehlich, “Space–time fourth moment of waves propagating in random media,” Radio Sci. 22, 481–490 (1987).
[CrossRef]

1986 (2)

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equation and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[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]

1976 (1)

1971 (2)

1967 (1)

D. L. Fried, “Atmospheric modulation noise in an optical heterodyne receiver,” IEEE J. Quantum Electron. QE-6, 213–221 (1967).
[CrossRef]

1941 (1)

A. N. Kolmogorov, “Energy dissipation in locally isotropic turbulence,” Doklady Akad. Nauk SSSR 32, 19–21 (1941).

Banta, R. M.

R. M. Banta, L. D. Olivier, D. H. Levinson, “Evolution of the Monterey Bay sea-breeze layer as observed by pulsed Doppler lidar,” J. Atmos. Sci. 50, 3959–3982 (1993).
[CrossRef]

Belmonte, A.

Brown, R. A.

P. Drobinski, R. A. Brown, P. H. Flamant, J. Pelon, “Evidence of organized large eddies by ground-based Doppler lidar, sonic anemometer and sodar,” Boundary Layer Meteorol. 88, 343–361 (1998).
[CrossRef]

Clifford, S. F.

Codona, J. L.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equation and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

Creamer, D. B.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equation and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

Cupp, R. E.

W. L. Eberhard, R. E. Cupp, K. R. Healy, “Doppler lidar measurement of profiles of turbulence and momentum flux,” J. Atmos. Ocean. Technol. 6, 809–819 (1989).
[CrossRef]

Dabas, A.

Dabas, A. M.

P. Drobinski, A. M. Dabas, C. Haeberli, P. H. Flamant, “On the small-scale dynamics of flow splitting in the Rhine valley during a shallow foehn event,” Boundary Layer Meteorol. 99, 277–296 (2001).
[CrossRef]

P. Drobinski, A. M. Dabas, P. H. Flamant, “Remote measurement of turbulent wind spectra by heterodyne Doppler lidar technique,” J. Appl. Meteorol. 39, 2434–2451 (2000).
[CrossRef]

P. Drobinski, A. M. Dabas, P. Salamitou, “Spectral diversity technique for heterodyne Doppler lidar that uses hard target returns,” Appl. Opt. 39, 376–385 (2000).
[CrossRef]

P. Drobinski, A. M. Dabas, P. Delville, P. H. Flamant, J. Pelon, R. M. Hardesty, “Refractive-index structure parameter in the planetary boundary layer: comparison of measurements taken with a 10.6-µm coherent lidar, a 0.9-µm scintillometer, and in-situ sensors,” Appl. Opt. 38, 1648–1656 (1999).
[CrossRef]

Darde, F.

P. Salamitou, F. Darde, P. H. Flamant, “A semi analytic approach for coherent laser radar system efficiency, the nearest Gaussian approximation,” J. Mod. Opt. 41, 2101–2113 (1994).
[CrossRef]

Delaval, A.

Delville, P.

Drobinski, P.

P. Drobinski, A. M. Dabas, C. Haeberli, P. H. Flamant, “On the small-scale dynamics of flow splitting in the Rhine valley during a shallow foehn event,” Boundary Layer Meteorol. 99, 277–296 (2001).
[CrossRef]

P. Drobinski, A. M. Dabas, P. H. Flamant, “Remote measurement of turbulent wind spectra by heterodyne Doppler lidar technique,” J. Appl. Meteorol. 39, 2434–2451 (2000).
[CrossRef]

P. Drobinski, A. M. Dabas, P. Salamitou, “Spectral diversity technique for heterodyne Doppler lidar that uses hard target returns,” Appl. Opt. 39, 376–385 (2000).
[CrossRef]

P. Drobinski, A. M. Dabas, P. Delville, P. H. Flamant, J. Pelon, R. M. Hardesty, “Refractive-index structure parameter in the planetary boundary layer: comparison of measurements taken with a 10.6-µm coherent lidar, a 0.9-µm scintillometer, and in-situ sensors,” Appl. Opt. 38, 1648–1656 (1999).
[CrossRef]

P. Drobinski, R. A. Brown, P. H. Flamant, J. Pelon, “Evidence of organized large eddies by ground-based Doppler lidar, sonic anemometer and sodar,” Boundary Layer Meteorol. 88, 343–361 (1998).
[CrossRef]

Eberhard, W. L.

W. L. Eberhard, R. E. Cupp, K. R. Healy, “Doppler lidar measurement of profiles of turbulence and momentum flux,” J. Atmos. Ocean. Technol. 6, 809–819 (1989).
[CrossRef]

Favreau, X.

Flamant, P. H.

P. Drobinski, A. M. Dabas, C. Haeberli, P. H. Flamant, “On the small-scale dynamics of flow splitting in the Rhine valley during a shallow foehn event,” Boundary Layer Meteorol. 99, 277–296 (2001).
[CrossRef]

P. Drobinski, A. M. Dabas, P. H. Flamant, “Remote measurement of turbulent wind spectra by heterodyne Doppler lidar technique,” J. Appl. Meteorol. 39, 2434–2451 (2000).
[CrossRef]

X. Favreau, A. Delaval, P. H. Flamant, A. Dabas, P. Delville, “Four-element receiver for pulsed 10-µm heterodyne Doppler lidar,” Appl. Opt. 39, 2441–2448 (2000).
[CrossRef]

P. Drobinski, A. M. Dabas, P. Delville, P. H. Flamant, J. Pelon, R. M. Hardesty, “Refractive-index structure parameter in the planetary boundary layer: comparison of measurements taken with a 10.6-µm coherent lidar, a 0.9-µm scintillometer, and in-situ sensors,” Appl. Opt. 38, 1648–1656 (1999).
[CrossRef]

P. Drobinski, R. A. Brown, P. H. Flamant, J. Pelon, “Evidence of organized large eddies by ground-based Doppler lidar, sonic anemometer and sodar,” Boundary Layer Meteorol. 88, 343–361 (1998).
[CrossRef]

P. Salamitou, F. Darde, P. H. Flamant, “A semi analytic approach for coherent laser radar system efficiency, the nearest Gaussian approximation,” J. Mod. Opt. 41, 2101–2113 (1994).
[CrossRef]

A. Dabas, P. H. Flamant, P. Salamitou, “Characterization of pulsed coherent Doppler lidar with the speckle effect,” Appl. Opt. 33, 6524–6532 (1994).
[CrossRef] [PubMed]

Flatté, S. M.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equation and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

Frehlich, R. G.

R. G. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt. 39, 393–397 (2000).
[CrossRef]

R. G. Frehlich, “Effects of refractive turbulence on coherent laser radar,” Appl. Opt. 32, 2122–2139 (1993).
[CrossRef] [PubMed]

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

R. G. Frehlich, “Space–time fourth moment of waves propagating in random media,” Radio Sci. 22, 481–490 (1987).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equation and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

Fried, D. L.

D. L. Fried, “Atmospheric modulation noise in an optical heterodyne receiver,” IEEE J. Quantum Electron. QE-6, 213–221 (1967).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–75.

Haeberli, C.

P. Drobinski, A. M. Dabas, C. Haeberli, P. H. Flamant, “On the small-scale dynamics of flow splitting in the Rhine valley during a shallow foehn event,” Boundary Layer Meteorol. 99, 277–296 (2001).
[CrossRef]

Hardesty, R. M.

Healy, K. R.

W. L. Eberhard, R. E. Cupp, K. R. Healy, “Doppler lidar measurement of profiles of turbulence and momentum flux,” J. Atmos. Ocean. Technol. 6, 809–819 (1989).
[CrossRef]

Henyey, F. S.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equation and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

Holmes, J. F.

Kavaya, M. J.

Kerr, J. R.

Kolmogorov, A. N.

A. N. Kolmogorov, “Energy dissipation in locally isotropic turbulence,” Doklady Akad. Nauk SSSR 32, 19–21 (1941).

Lee, M. H.

Levinson, D. H.

R. M. Banta, L. D. Olivier, D. H. Levinson, “Evolution of the Monterey Bay sea-breeze layer as observed by pulsed Doppler lidar,” J. Atmos. Sci. 50, 3959–3982 (1993).
[CrossRef]

Lutomirsky, R. F.

Olivier, L. D.

R. M. Banta, L. D. Olivier, D. H. Levinson, “Evolution of the Monterey Bay sea-breeze layer as observed by pulsed Doppler lidar,” J. Atmos. Sci. 50, 3959–3982 (1993).
[CrossRef]

Pelon, J.

Rye, B. J.

Salamitou, P.

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971).

Wandzura, S.

Yura, H. T.

Appl. Opt. (11)

R. G. Frehlich, “Effects of refractive turbulence on coherent laser radar,” Appl. Opt. 32, 2122–2139 (1993).
[CrossRef] [PubMed]

P. Drobinski, A. M. Dabas, P. Salamitou, “Spectral diversity technique for heterodyne Doppler lidar that uses hard target returns,” Appl. Opt. 39, 376–385 (2000).
[CrossRef]

X. Favreau, A. Delaval, P. H. Flamant, A. Dabas, P. Delville, “Four-element receiver for pulsed 10-µm heterodyne Doppler lidar,” Appl. Opt. 39, 2441–2448 (2000).
[CrossRef]

A. Dabas, P. H. Flamant, P. Salamitou, “Characterization of pulsed coherent Doppler lidar with the speckle effect,” Appl. Opt. 33, 6524–6532 (1994).
[CrossRef] [PubMed]

S. F. Clifford, S. Wandzura, “Monostatic heterodyne lidar performance: the effect of the turbulent atmosphere,” Appl. Opt. 20, 514–516 (1981).
[CrossRef] [PubMed]

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

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

R. G. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt. 39, 393–397 (2000).
[CrossRef]

A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426–5445 (2000).
[CrossRef]

P. Drobinski, A. M. Dabas, P. Delville, P. H. Flamant, J. Pelon, R. M. Hardesty, “Refractive-index structure parameter in the planetary boundary layer: comparison of measurements taken with a 10.6-µm coherent lidar, a 0.9-µm scintillometer, and in-situ sensors,” Appl. Opt. 38, 1648–1656 (1999).
[CrossRef]

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

Boundary Layer Meteorol. (2)

P. Drobinski, R. A. Brown, P. H. Flamant, J. Pelon, “Evidence of organized large eddies by ground-based Doppler lidar, sonic anemometer and sodar,” Boundary Layer Meteorol. 88, 343–361 (1998).
[CrossRef]

P. Drobinski, A. M. Dabas, C. Haeberli, P. H. Flamant, “On the small-scale dynamics of flow splitting in the Rhine valley during a shallow foehn event,” Boundary Layer Meteorol. 99, 277–296 (2001).
[CrossRef]

Doklady Akad. Nauk SSSR (1)

A. N. Kolmogorov, “Energy dissipation in locally isotropic turbulence,” Doklady Akad. Nauk SSSR 32, 19–21 (1941).

IEEE J. Quantum Electron. (1)

D. L. Fried, “Atmospheric modulation noise in an optical heterodyne receiver,” IEEE J. Quantum Electron. QE-6, 213–221 (1967).
[CrossRef]

J. Appl. Meteorol. (1)

P. Drobinski, A. M. Dabas, P. H. Flamant, “Remote measurement of turbulent wind spectra by heterodyne Doppler lidar technique,” J. Appl. Meteorol. 39, 2434–2451 (2000).
[CrossRef]

J. Atmos. Ocean. Technol. (1)

W. L. Eberhard, R. E. Cupp, K. R. Healy, “Doppler lidar measurement of profiles of turbulence and momentum flux,” J. Atmos. Ocean. Technol. 6, 809–819 (1989).
[CrossRef]

J. Atmos. Sci. (1)

R. M. Banta, L. D. Olivier, D. H. Levinson, “Evolution of the Monterey Bay sea-breeze layer as observed by pulsed Doppler lidar,” J. Atmos. Sci. 50, 3959–3982 (1993).
[CrossRef]

J. Math. Phys. (1)

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equation and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

J. Mod. Opt. (1)

P. Salamitou, F. Darde, P. H. Flamant, “A semi analytic approach for coherent laser radar system efficiency, the nearest Gaussian approximation,” J. Mod. Opt. 41, 2101–2113 (1994).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Acta (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]

Radio Sci. (2)

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

R. G. Frehlich, “Space–time fourth moment of waves propagating in random media,” Radio Sci. 22, 481–490 (1987).
[CrossRef]

Other (2)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–75.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971).

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Figures (9)

Fig. 1
Fig. 1

Schematic of lidar beam propagation: r a (r) is the 2D transverse coordinate in the transmitter (receiver) plane perpendicular to the propagation axis at z = 0. ρ a (ρ) is the 2D transverse coordinate for the ongoing (backpropagating) beam before (after) scattering in the target plane located at range z.

Fig. 2
Fig. 2

c b (dashed curves) and c m (solid curves) as functions of N for four values of Ω.

Fig. 3
Fig. 3

c md computed from numerical simulations (squares) and ĉ md estimated by use of a logarithmic Gaussian fit (solid curves) for four values of Ω.

Fig. 4
Fig. 4

Parameters A, N c , and w computed from numerical simulations (squares) and their best fits (solid curves) as functions of Ω [see Eq. (15)]. The error bars indicate the statistical uncertainties (±1 - σ).

Fig. 5
Fig. 5

c m computed from numerical simulations (squares) and ĉ m estimated by use of Eq. (16) (solid curves) for four values of Ω.

Fig. 6
Fig. 6

F b (dashed curves) and F m (solid curves) as functions of N for four values of Ω.

Fig. 7
Fig. 7

F md computed from numerical simulations (squares) and md estimated by use of a logarithmic Gaussian fit (solid curves) for four values of Ω.

Fig. 8
Fig. 8

Parameters A, N c , and w computed from numerical simulations (squares) and their best fits (solid curves) as functions of Ω [see Eq. (25)]. The error bars indicate the statistical uncertainties (±1 - σ).

Fig. 9
Fig. 9

F m computed from numerical simulations (squares) and m estimated by use of Eq. (27) (solid curves) for four values of Ω.

Tables (1)

Tables Icon

Table 1 Summary of Analytical Solutions and Empirical Expressions for Lidars

Equations (40)

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Era, 0=1πσ2exp-ra221σ2-ikf,
ρ0=H2 k20Z Cn2z1-tz5/3dt-3/5,
Eρa, z=expikziλzd2raEra, 0expik2zρa-ra2+ψρa, ra, z,
Mρ1, ρ2, z=Eρ1, zE*ρ2, z=σd4πλ2z4d2ra1d2ra2d2ρ1d2ρ2×Era1, 0E*ra2, 0δρ1-ρ1δρ2-ρ2δρ1-ρ2expik2zρ1-ra12-ρ2-ra22expψra1, ρ1, z+ψ*ra2, ρ2, z.
M(r1, r2, 0) =σd4πλ2z4    d2ra1d2ra2d2ρE(ra1, 0)×E*(ra2, 0)expik2z [(ρ-ra1)2+(r1-ρ)2-(ρ-ra2)2-(r2-ρ)2]×exp(ψ(ra1, ρ, z) + ψ(ρ, r1, z) + ψ*(ra2, ρ, z) + ψ*(ρ, r2, z)).
expψ4=expψ2+ψ2=exp-12D|ra1-ra2|+D|r1-r2|.
X=1/2ra1+1/2ra2, Y=ra1-ra2+r1-r2.
|Mbr1, r2, 0|=nΔσdσ24z2exp-D|r1-r2|-|r1-r2|24σ21+Ω2,
r0b=2σ1+Ω2+2N21/2.
cb=11+2N2/1+Ω21/2=r0b2σ1+Ω2.
ψ4=ψra1, ρ1, z+ψr1, ρ1, z+ψ*ra2, ρ2, z+ψ*r2, ρ2, z=-1/2Dra1-ra2-Dr1-ra1+Dr2-ra1-1/2Dr1-ra2-Dr2-ra2+Dr1-r2,
|Mmr1, r2, 0|=nΔσd4πz2exp-|r1-r2|24σ2d2X×exp-1σ2X2-iΩX|r1-r2|×exp12 DX-3r2-r12+12 D×X-3r1-r22-DX-r2+r12-Dr2-r1.
r0m=2σcm1+Ω2.
cm=cma+cmd,
cˆma=1+2N23.11+Ω2-1/2.
Nc=0.5×logΩ+0.32, A=0.74×1-exp-Ω/4.6, w=0.4,
cˆmN, Ω=11+2N2/3.11+Ω21/2+0.741-exp-Ω/4.6exp-log2N/2Ω0.50.42.
rˆ0mN, Ω=2σ1+Ω2+2N2/3.11/2+0.74×2σ1-exp-Ω/4.61+Ω2×exp-log2N/2Ω0.50.42.
|Mˆmr1, r2, 0|=nΔσdσ24z2exp-|r1-r2|2rˆ0m2.
ihet2=d2r1d2r2Mr1, r2, 0Mlo*r1, r2, 0,
ihet2b=π2nΔσdη2σ6z21+Ω2 FbN, Ω,
Fb=11+N2/1+Ω2.
ihet2m=nΔσdη2σ24z2d2Vd2W×exp-V2+W2+2iΩ2σ2exp1/2DW+V+1/2DW-V-DW-DV=π2nΔσdη2σ6z21+Ω2 FmN, Ω.
FmN, Ω=FmaN, Ω+FmdN, Ω.
FˆmaN, Ω=1+N23.11+Ω2-1.
Nc=0.5×logΩ+0.35, A=0.6×1-exp-Ω/3.1, w=0.4.
FˆmdN, Ω=0.61-exp-Ω/3.1×exp-log2N/2.25Ω0.50.42.
FˆmN, Ω=11+N2/3.11+Ω2+0.61-exp-Ω/3.1×exp-log2N/2.25Ω0.50.42.
(ıˆhet2)m=π2nΔσdη2σ6z2(1+Ω2)11+Ω2+N2/3.1+exp{-[log2(N/2.25Ω0.5)0.421+Ω2.
Era, 0=1πσ2exp-ra221σ2-ikf.
Mr1, r2, 0=σd4πλ2z4d2ra1d2ra2d2ρEra1, 0×E*ra2, 0expik2zρ-ra12+r1-ρ2-ρ-ra22-r2-ρ2×expψra1, ρ, z+ψρ, r1, z+ψ*ra2, ρ, z+ψ*ρ, r2, z.
X=1/2ra1+1/2ra2,  Y=ra1-ra2+r1-r2.
Era1, 0E*ra2, 0expik2zρ-ra12+r1-ρ2-ρ-ra22-r2-ρ2=exp-r12+r224σ2expik2zr12-r22exp-12σ22X2+iXΩ|r1-r2|expik2z ρY.
expψ4=expψ2+ψ2=exp-1/2D|ra1-ra2|+D|r1-r2|=exp-1/2D|Y+r1-r2|+D|r1-r2|.
Mbr1, r2, 0=nΔσd4πz2expik2zr12-r22×exp-r1-r224σ2d2Xd2Y×exp12σ22X2+iXΩ|r1-r2|δY2λzexp-1/2DY+r1-r2+Dr1-r2,
|Mbr1, r2, 0|=nΔσdσ24z22exp-D|r1-r2|-|r1-r2|24σ21+Ω2.
ψ4=ψra1, ρ1, z+ψr1, ρ1, z+ψ*ra2, ρ2, z+ψ*r2, ρ2, z=-1/2Dra1-ra2-Dr1-ra1+Dr2-ra1-1/2Dr1-ra2-Dr2-ra2+Dr1-r2.
ψ4=12DY-r1+r2-DX-3r1-r22+DX-r1+r2-Y2+12DX-r2+r1+Y2-DX-3r2-r1+Y2+Dr2-r1.
Mmr1, r2, 0=nΔσd4πz2expik2zr12-r22×exp-r1-r224σ2d2Xd2Y×exp1σ2X2+iXΩ|r1-r2|δY2λzexp-12DY-r1+r2-DX-3r1-r2-Y2+DX-r1+r2-Y2×exp-12DX-r2+r1-Y2-DX-3r2-r1+Y2+Dr2-r1.
|Mmr1, r2, 0|=nΔσd4πz2exp-|r1-r2|24σ2×d2Xexp-1σ2X2-iΩX|r1-r2|×exp12 DX-3r2-r12+12 DX-3r1-r22-DX-r2+r12-Dr2-r1.

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