Abstract

A statistical model for the return signal in a coherent lidar is derived from the fundamental principles of atmospheric scattering and turbulent propagation. The model results in a three-parameter probability distribution for the coherent signal-to-noise ratio in the presence of atmospheric turbulence and affected by target speckle. We consider the effects of amplitude and phase fluctuations, in addition to local oscillator shot noise, for both passive receivers and those employing active modal compensation of wavefront phase distortion. We obtain exact expressions for statistical moments for lidar fading and evaluate the impact of various parameters, including the ratio of receiver aperture diameter to the wavefront coherence diameter, the speckle effective area, and the number of modes compensated.

© 2010 Optical Society of America

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  1. J. Totems, V. Jolivet, J.-P. Ovarlez, and N. Martin, “Advanced signal processing methods for pulsed laser vibrometry,” Appl. Opt. 49, 3967–3979 (2010).
    [CrossRef] [PubMed]
  2. D. Jameson, M. Dierking, and B. Duncan, “Effects of spatial modes on ladar vibration signature estimation,” Appl. Opt. 46, 7365–7373 (2007).
    [CrossRef] [PubMed]
  3. M. J. Kavaya, “Laser and lidar technology development for highly accurate vertical profiles of vector wind velocity from earth orbit,” in Coherent Optical Technologies and Applications (Optical Society of America, 2008), paper CTuA3.
  4. S. Kameyama, T. Ando, K. Asaka, Y. Hirano, and S. Wadaka, “Compact all-fiber pulsed coherent Doppler lidar system for wind sensing,” Appl. Opt. 46, 1953–1962 (2007).
    [CrossRef] [PubMed]
  5. A. Dinovitser, M. W. Hamilton, and R. A. Vincent, “Stabilized master laser system for differential absorption lidar,” Appl. Opt. 49, 3274–3281 (2010).
    [CrossRef] [PubMed]
  6. L. Joly, F. Marnas, F. Gibert, D. Bruneau, B. Grouiez, P. H. Flamant, G. Durry, N. Dumelie, B. Parvitte, and V. Zéninari, “Laser diode absorption spectroscopy for accurate CO2 line parameters at 2 μm: consequences for space-based DIAL measurements and potential biases,” Appl. Opt. 48, 5475–5483(2009).
    [CrossRef] [PubMed]
  7. G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” Appl. Opt. 43, 5092–5099 (2004).
    [CrossRef] [PubMed]
  8. S. Lundqvist, C.-O. Fält, U. Persson, B. Marthinsson, and S. T. Eng, “Air pollution monitoring with a Q-switched CO2-laser lidar using heterodyne detection,” Appl. Opt. 20, 2534–2538(1981).
    [CrossRef] [PubMed]
  9. Y. Zhao, W. A. Brewer, W. L. Eberhard, and R. J. Alvarez, “Lidar measurement of ammonia concentrations and fluxes in a plume from a point source,” J. Atmos. Ocean. Technol. 19, 1928–1938 (2002).
    [CrossRef]
  10. A. Pal, C. D. Clark, M. Sigman, and D. K. Killinger, “Differential absorption lidar CO2 laser system for remote sensing of TATP related gases,” Appl. Opt. 48, B145–B150 (2009).
    [CrossRef] [PubMed]
  11. See papers on Advanced Component Technologies presented at the Fifteenth Biennial Coherent Laser Radar Technology and Applications Conference (Universities Space Research Association, 2009).
  12. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67(1967).
    [CrossRef]
  13. H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
    [CrossRef]
  14. J. H. Shapiro, B. A. Capron, and R. C. Harney, “Imaging and target detection with a heterodyne-reception optical radar,” Appl. Opt. 20, 3292–3312 (1981).
    [CrossRef] [PubMed]
  15. S. F. Clifford and S. Wandzura, “Monostatic heterodyne lidar performance: the effect of the turbulent atmosphere,” Appl. Opt. 20, 514–516 (1981).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  18. R. G. Frehlich and M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
    [CrossRef] [PubMed]
  19. J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, and F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948(1986).
    [CrossRef]
  20. A. Belmonte and B. J. Rye, “Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems,” Appl. Opt. 39, 2401–2411 (2000).
    [CrossRef]
  21. A. Belmonte, “Influence of atmospheric phase compensation on optical heterodyne power measurements,” Opt. Express 16, 6756–6767 (2008).
    [CrossRef] [PubMed]
  22. B. J. Rye, “Antenna parameters for incoherent backscatter heterodyne lidar,” Appl. Opt. 18, 1390–1398 (1979).
    [CrossRef] [PubMed]
  23. M. Nakagami, “The m-distribution. A general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W.C.Hoffman, ed. (Pergamon, 1960).
  24. A. Belmonte and J. M. Kahn, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express 16, 14151–14162 (2008).
    [CrossRef] [PubMed]
  25. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  26. J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
    [CrossRef]
  27. J. W. Goodman, Speckle Phenomena in Optics. Theory and Applications ( Roberts & Company, 2007).
  28. R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef] [PubMed]
  29. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  30. G. Dai, “Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen—Loève functions,” J. Opt. Soc. Am. A 12, 2182–2193 (1995).
    [CrossRef]

2010 (2)

2009 (2)

2008 (2)

2007 (2)

2004 (1)

2002 (1)

Y. Zhao, W. A. Brewer, W. L. Eberhard, and R. J. Alvarez, “Lidar measurement of ammonia concentrations and fluxes in a plume from a point source,” J. Atmos. Ocean. Technol. 19, 1928–1938 (2002).
[CrossRef]

2000 (1)

1995 (1)

1991 (1)

1986 (1)

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

1981 (5)

1979 (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]

B. J. Rye, “Antenna parameters for incoherent backscatter heterodyne lidar,” Appl. Opt. 18, 1390–1398 (1979).
[CrossRef] [PubMed]

1976 (1)

1975 (1)

J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

1971 (1)

1967 (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67(1967).
[CrossRef]

Alvarez, R. J.

Y. Zhao, W. A. Brewer, W. L. Eberhard, and R. J. Alvarez, “Lidar measurement of ammonia concentrations and fluxes in a plume from a point source,” J. Atmos. Ocean. Technol. 19, 1928–1938 (2002).
[CrossRef]

Amzajerdian, F.

Ando, T.

Asaka, K.

Barnes, B. W.

Belmonte, A.

Beyon, J. Y.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Brewer, W. A.

Y. Zhao, W. A. Brewer, W. L. Eberhard, and R. J. Alvarez, “Lidar measurement of ammonia concentrations and fluxes in a plume from a point source,” J. Atmos. Ocean. Technol. 19, 1928–1938 (2002).
[CrossRef]

Bruneau, D.

Capron, B. A.

Clark, C. D.

Clifford, S. F.

Codona, J. L.

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

Creamer, D. B.

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

Dai, G.

Davis, R. E.

Dierking, M.

Dinovitser, A.

Dumelie, N.

Duncan, B.

Durry, G.

Eberhard, W. L.

Y. Zhao, W. A. Brewer, W. L. Eberhard, and R. J. Alvarez, “Lidar measurement of ammonia concentrations and fluxes in a plume from a point source,” J. Atmos. Ocean. Technol. 19, 1928–1938 (2002).
[CrossRef]

Eng, S. T.

Fält, C.-O.

Flamant, P. H.

Flatté, S. M.

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

Frehlich, R. G.

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

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

Fried, D. L.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67(1967).
[CrossRef]

Gibert, F.

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics. Theory and Applications ( Roberts & Company, 2007).

Grouiez, B.

Hamilton, M. W.

Harney, R. C.

Henyey, F. S.

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

Hirano, Y.

Ismail, S.

Jameson, D.

Jolivet, V.

Joly, L.

Kahn, J. M.

Kameyama, S.

Kavaya, M. J.

Killinger, D. K.

Koch, G. J.

Lundqvist, S.

Lutomirski, R. F.

Marnas, F.

Marthinsson, B.

Martin, N.

Nakagami, M.

M. Nakagami, “The m-distribution. A general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W.C.Hoffman, ed. (Pergamon, 1960).

Noll, R. J.

Ovarlez, J.-P.

Pal, A.

Parvitte, B.

Persson, U.

Petros, M.

Rye, B. J.

Shapiro, J. H.

Sigman, M.

Singh, U. N.

Speck, J. P.

J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

Strohbehn, J. W.

J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

Totems, J.

Vay, S.

Vincent, R. A.

Wadaka, S.

Wandzura, S.

Wang, T.

J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Yu, J.

Yura, H. T.

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

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

Zéninari, V.

Zhao, Y.

Y. Zhao, W. A. Brewer, W. L. Eberhard, and R. J. Alvarez, “Lidar measurement of ammonia concentrations and fluxes in a plume from a point source,” J. Atmos. Ocean. Technol. 19, 1928–1938 (2002).
[CrossRef]

Appl. Opt. (15)

B. J. Rye, “Antenna parameters for incoherent backscatter heterodyne lidar,” Appl. Opt. 18, 1390–1398 (1979).
[CrossRef] [PubMed]

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

S. Lundqvist, C.-O. Fält, U. Persson, B. Marthinsson, and S. T. Eng, “Air pollution monitoring with a Q-switched CO2-laser lidar using heterodyne detection,” Appl. Opt. 20, 2534–2538(1981).
[CrossRef] [PubMed]

J. H. Shapiro, B. A. Capron, and R. C. Harney, “Imaging and target detection with a heterodyne-reception optical radar,” Appl. Opt. 20, 3292–3312 (1981).
[CrossRef] [PubMed]

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

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

G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” Appl. Opt. 43, 5092–5099 (2004).
[CrossRef] [PubMed]

S. Kameyama, T. Ando, K. Asaka, Y. Hirano, and S. Wadaka, “Compact all-fiber pulsed coherent Doppler lidar system for wind sensing,” Appl. Opt. 46, 1953–1962 (2007).
[CrossRef] [PubMed]

D. Jameson, M. Dierking, and B. Duncan, “Effects of spatial modes on ladar vibration signature estimation,” Appl. Opt. 46, 7365–7373 (2007).
[CrossRef] [PubMed]

A. Pal, C. D. Clark, M. Sigman, and D. K. Killinger, “Differential absorption lidar CO2 laser system for remote sensing of TATP related gases,” Appl. Opt. 48, B145–B150 (2009).
[CrossRef] [PubMed]

L. Joly, F. Marnas, F. Gibert, D. Bruneau, B. Grouiez, P. H. Flamant, G. Durry, N. Dumelie, B. Parvitte, and V. Zéninari, “Laser diode absorption spectroscopy for accurate CO2 line parameters at 2 μm: consequences for space-based DIAL measurements and potential biases,” Appl. Opt. 48, 5475–5483(2009).
[CrossRef] [PubMed]

A. Dinovitser, M. W. Hamilton, and R. A. Vincent, “Stabilized master laser system for differential absorption lidar,” Appl. Opt. 49, 3274–3281 (2010).
[CrossRef] [PubMed]

J. Totems, V. Jolivet, J.-P. Ovarlez, and N. Martin, “Advanced signal processing methods for pulsed laser vibrometry,” Appl. Opt. 49, 3967–3979 (2010).
[CrossRef] [PubMed]

S. F. Clifford and S. Wandzura, “Monostatic heterodyne lidar performance: the effect of the turbulent atmosphere; correction,” Appl. Opt. 20, 1502 (1981).
[CrossRef]

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

J. Atmos. Ocean. Technol. (1)

Y. Zhao, W. A. Brewer, W. L. Eberhard, and R. J. Alvarez, “Lidar measurement of ammonia concentrations and fluxes in a plume from a point source,” J. Atmos. Ocean. Technol. 19, 1928–1938 (2002).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

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]

Opt. Express (2)

Proc. IEEE (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67(1967).
[CrossRef]

Radio Sci. (2)

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

J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10, 59–70 (1975).
[CrossRef]

Other (5)

J. W. Goodman, Speckle Phenomena in Optics. Theory and Applications ( Roberts & Company, 2007).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

M. Nakagami, “The m-distribution. A general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W.C.Hoffman, ed. (Pergamon, 1960).

M. J. Kavaya, “Laser and lidar technology development for highly accurate vertical profiles of vector wind velocity from earth orbit,” in Coherent Optical Technologies and Applications (Optical Society of America, 2008), paper CTuA3.

See papers on Advanced Component Technologies presented at the Fifteenth Biennial Coherent Laser Radar Technology and Applications Conference (Universities Space Research Association, 2009).

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Figures (3)

Fig. 1
Fig. 1

For typical atmospheric situations, the time scale of the coherent lidar signal fluctuations due to turbulence is several orders of magnitude larger than that of speckle-induced fluctuations (milliseconds rather than microseconds). The long time constant of the return signal fluctuation due to turbulence means that these fluctuations are essentially correlated over the short correlation time associated with speckle. Here, the mean of the target speckle signal is smeared by turbulence speckle fluctuations, and we need to define multiply stochastic (compound) statistics to describe the return signals in a coherent lidar.

Fig. 2
Fig. 2

Mean coherent SNR γ ¯ (photocounts) and SNR normalized variance σ γ 2 / γ ¯ 2 as a function of the normalized receiver aperture diameter D / r 0 . (a) Mean coherent SNR is shown for different coherence diameters of spatial speckle 2 ρ S over the receiving aperture. The SNR is expressed in decibels, referenced to the smallest aperture considered in the figure. The no-speckle case (dashed curve) and free-space limit γ 0 are included. (b) SNR normalized variance is studied when n equal-strength inde pendent laser shots are averaged. Amplitude fluctuations are excluded (solid curves) by assuming σ β 2 = 0 . When scintillation is considered (dashed curves), the scintillation index is fixed at σ β 2 = 1 .

Fig. 3
Fig. 3

(a) Mean coherent SNR γ ¯ and (b) SNR normalized variance σ γ 2 / γ ¯ 2 are shown for different values of the normalized receiver aperture diameter D / r 0 and the number of modes J removed by phase compensation. Turbulence is characterized by a fixed phase coherence diameter r 0 . Without loss of generality, in all cases considered in these plots, the correlation diameter of the speckle field 2 ρ S is equal to r 0 . The compensating phases are expansions up to tilt ( J = 3 ), astigmatism ( J = 6 ), and fifth-order aberrations ( J = 20 ). The no-correction case ( J = 0 ) is also considered. The dashed curve corresponds to the measurement associated with no-turbulence, one-pulse speckle. The free-space limit γ 0 is included in (a). (b) Amplitude fluctuations are excluded (solid curves) by assuming σ β 2 = 0 . When scintillation is considered (dashed curves), the scintillation index is fixed at σ β 2 = 1 .

Equations (46)

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α = 4 π D 2 d r W ( r ) E S ( r )
Ω = α α * ¯ = ( 4 π D 2 ) 2 d r W ( r ) E S ( r ) d r W ( r ) E S * ( r ) , ¯
Ω = ( 4 π D 2 ) 2 d r d r W ( r ) W ( r ) E S ( r ) E S * ( r ) ¯ .
Ω = 4 π D 2 d ρ K D ( ρ ) μ ( ρ ) .
K D ( ρ ) = 4 π D 2 d R W ( R + 1 2 ρ ) W ( R 1 2 ρ ) = 2 π { a cos ( ρ D ) ( ρ D ) [ 1 ( ρ D ) 1 2 ] } , ρ D .
Ω = 4 π D 2 A eff,
Ω 4 π D 2 d ρ μ ( ρ ) ,
Ω 4 π D 2 d ρ K D ( ρ ) = 1 .
p α ( α ) = 2 ( m N ) m α 2 m 1 Γ ( m ) exp ( m N α 2 ) .
p γ ( γ ) = ( m N γ 0 ) m γ m 1 Γ ( m ) exp ( m N γ 0 γ ) .
N = 1 / Ω = 1 / a 2 ¯
γ ¯ k = Γ ( m + k ) m k Γ ( m ) γ ¯ k ,
1 m = σ γ 2 γ ¯ 2 = a 4 ¯ a 2 ¯ 2 1 .
1 m = 2 2 ( a 2 ¯ a 2 ¯ ) 2 .
E S ( r ) = exp [ χ ( r ) j ϕ ( r ) ] ,
α ¯ = 4 π D 2 d r W ( r ) E S ( r ) ¯ = 4 π D 2 d r W ( r ) exp [ χ ( r ) j ϕ ( r ) ] ¯ .
α ¯ = 4 π D 2 exp ( χ ¯ ) d r W ( r ) exp [ χ ( r ) χ ¯ ] ¯ exp [ j ϕ ( r ) ] ¯ .
exp ( j ϕ ) ¯ = exp ( 1 2 σ ϕ 2 ) .
σ ϕ 2 = 1.0299 ( D r 0 ) 5 / 3 .
χ ¯ = σ χ 2 , exp ( χ χ ¯ ) ¯ = exp ( 1 2 σ χ 2 ) .
α ¯ = exp ( 1 2 σ χ 2 ) exp ( 1 2 σ ϕ 2 ) .
μ ( ρ ) = exp { [ χ ( r ) + χ ( r ) ] j [ ϕ ( r ) ϕ ( r ) ] } ¯ = exp [ 1 2 D W ( ρ ) ] ,
D W ( ρ ) = 6.88 ( ρ r 0 ) 5 3 .
a 2 ¯ = 4 π D 2 2 π 0 D / 2 ρ d ρ K D ( ρ ) exp [ 1 2 6.88 ( ρ r 0 ) 5 3 ] ,
a 2 ¯ = 1.09 ( r 0 D ) 2 Γ [ 6 5 , 1.08 ( D r 0 ) 5 / 3 ] .
p α ( α ) = 2 M α exp ( M α 2 ) ,
p γ ( γ ) = M γ 0 exp ( M γ 0 γ ) .
p γ ( γ ) = ( n M γ 0 ) n γ n 1 Γ ( n ) exp ( n M γ γ 0 ) .
μ S ( ρ ) = 2 π W T 2 d ν exp ( 2 ρ 2 W T 2 ) exp ( j k z ν · ρ ) .
μ S ( ρ ) = exp [ 1 2 ( k W T 2 z ρ ) 2 ] exp [ ( ρ ρ S ) 2 ] .
ρ S = 2 ( 2 z k W T ) ,
W T 2 ( z ) = W 0 2 [ 1 + ( z z 0 ) 2 ] + 2 ( 4 z k r 0 ) 2 .
μ S ( ρ ) = exp [ 1 2 ( k W T 2 z ρ ) 2 ] exp [ 1 2 6.88 ( ρ r 0 ) 5 3 ] .
1 ρ S 2 = 1 2 ( k W T 2 z ) 2 + 4 r 0 2 .
Ω = 4 π D 2 d ρ K D ( ρ ) exp [ ( ρ ρ S ) 2 ] .
1 M = 4 π D 2 2 π 0 D / 2 ρ d ρ exp [ ( ρ ρ S ) 2 ] = ( 2 ρ S D ) 2 { 1 exp [ ( D 2 ρ S ) 2 ] } .
p γ | x ( γ | x ) = ( n M x ) N γ n 1 Γ ( n ) exp ( n M γ x ) .
p γ ( γ ) = 0 p γ | x ( γ | x ) p x ( x ) d x = ( n M ) n γ n 1 Γ ( n ) 0 ( 1 x ) n exp ( n M γ x ) p x ( x ) d x .
p x ( x ) = ( m N γ 0 ) m x m 1 Γ ( m ) exp ( m N γ 0 x ) ,
p γ ( γ ) = 2 Γ ( n ) Γ ( m ) ( n M m N γ 0 γ ) n + m 2 1 γ K m n [ 2 ( n M m N γ 0 γ ) 1 2 ] .
γ ¯ k = 0 d γ γ k p γ ( γ ) = Γ ( k + m ) Γ ( m ) Γ ( k + n ) Γ ( n ) ( γ 0 n M m N ) k .
γ ¯ = γ 0 M N ,
σ γ 2 γ ¯ 2 = Γ ( m ) Γ ( m + 2 ) Γ 2 ( m + 1 ) Γ ( n ) Γ ( n + 2 ) Γ 2 ( n + 1 ) 1 .
σ γ 2 γ ¯ 2 = ( m + 1 ) m ( n + 1 ) n 1 .
σ ϕ 2 = C J ( D r 0 ) 5 / 3 ,
D W ( ρ ) = 6.44 ( ρ r 0 ) 5 3 D J ( ρ ) .

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