Abstract

We examine the gamma–gamma and lognormal distributions as they apply to the irradiance at a point detector produced by partially coherent beams propagating horizontally through atmospheric turbulence. Our investigation compares the probability density functions and probability of fade predicted by the distributions with results from a wave-optics simulation developed for partially coherent beam propagation. For a partially coherent beam that is not too far removed from a coherent beam, we find the wave-optics results tend to the gamma–gamma model for the weak fluctuation regime and the results are closer to the lognormal model for the strong fluctuation regime. We observe that increasing the initial beam size/Fried parameter ratio (w0/r0) or shortening the coherence length (lc) tends to narrow the probability density profile produced by the simulation.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008 (2)

2007 (5)

2006 (6)

G. J. Baker, “Gaussian beam weak scintillation: low-order turbulence effects and applicability of the Rytov method,” J. Opt. Soc. Am. A 23, 395-417 (2006).
[CrossRef]

X. Xiao and D. Voelz, “Wave optics simulation approach for partial spatially coherent beams,” Opt. Express 14, 6986-6992 (2006).
[CrossRef] [PubMed]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001(2006).
[CrossRef]

X. Xiao and D. Voelz, “Wave optics simulation of pseudo-partially coherent beam propagation through turbulence: application to laser communications,” Proc. SPIE 6304, 63040L(2006).
[CrossRef]

O. Korotkova, “Control of the intensity fluctuation of random electromagnetic beams on propagation in weak atmospheric turbulence,” Proc. SPIE 6105, 61050V (2006).
[CrossRef]

O. Korotkova, “Changes in the intensity fluctuations of a class of random electromagnetic beams on propagation,” J. Opt. A Pure Appl. Opt. 8, 30-37 (2006).
[CrossRef]

2005 (1)

2004 (6)

F. Dios, J. A. Rubio, A. Rodriguez, and A. Comeron, “Scintillation and beam-wander analysis in an optical ground station-satellite uplink,” Appl. Opt. 43, 3866-3873 (2004).
[CrossRef] [PubMed]

D. Voelz and K. Fitzhenry, “Pseudo-partially coherent beam for free-space laser communication,” Proc. SPIE 5550, 218-224 (2004).
[CrossRef]

J. C. Ricklin, S. Bucaille, and F. M. Davidson, “Performance loss factors for optical communication through clear air turbulence,” Proc. SPIE 5160, 1-12 (2004).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “The effect of partially coherent quasi-monochromatic Gaussian-beam on the probability of fade,” Proc. SPIE 5160, 68-77 (2004).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330-341 (2004).
[CrossRef]

J. J. Kiriazes, R. L. Phillips, and L. C. Andrews, “Analysis of fading for a free-space optical communication link subject to atmospheric scintillation,” Proc. SPIE 5160, 253-264 (2004).
[CrossRef]

2003 (2)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Phase diffuser at the transmitter for lasercom link: effect of partially coherent beam on the bit-error rate,” Proc. SPIE 4976, 70-77 (2003).
[CrossRef]

T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. 28, 610-612 (2003).
[CrossRef] [PubMed]

2002 (4)

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598(2002).
[CrossRef]

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794-1802 (2002).
[CrossRef]

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293-1300 (2002).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98-109 (2002).
[CrossRef]

2001 (2)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554-1562 (2001).
[CrossRef]

A. D. Wheelon, “Skewed distribution of irradiance predicted by the second-order Rytov approximation,” J. Opt. Soc. Am. A 18, 2789-2798 (2001).
[CrossRef]

2000 (3)

1999 (1)

1994 (1)

1992 (1)

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

1991 (1)

A. N. Kolmogorov, “Dissipation of energy in the locally isotropic turbulence,” reprinted in Proc. R. Soc. London. Ser. A 434, 15-17 (1991).
[CrossRef]

1990 (1)

1979 (1)

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554-1562 (2001).
[CrossRef]

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

Andrews, L.

Andrews, L. C.

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46, 086002 (2007).
[CrossRef]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001(2006).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330-341 (2004).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “The effect of partially coherent quasi-monochromatic Gaussian-beam on the probability of fade,” Proc. SPIE 5160, 68-77 (2004).
[CrossRef]

J. J. Kiriazes, R. L. Phillips, and L. C. Andrews, “Analysis of fading for a free-space optical communication link subject to atmospheric scintillation,” Proc. SPIE 5160, 253-264 (2004).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Phase diffuser at the transmitter for lasercom link: effect of partially coherent beam on the bit-error rate,” Proc. SPIE 4976, 70-77 (2003).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98-109 (2002).
[CrossRef]

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554-1562 (2001).
[CrossRef]

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005), pp. 667-711.
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005), pp. 321-393.
[CrossRef]

Baker, G. J.

Batet, O.

Belmonte, A.

Bracher, C.

Bucaille, S.

J. C. Ricklin, S. Bucaille, and F. M. Davidson, “Performance loss factors for optical communication through clear air turbulence,” Proc. SPIE 5160, 1-12 (2004).
[CrossRef]

Churnside, J. H.

Comeron, A.

Dainty, J. C.

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

Davidson, F. M.

J. C. Ricklin, S. Bucaille, and F. M. Davidson, “Performance loss factors for optical communication through clear air turbulence,” Proc. SPIE 5160, 1-12 (2004).
[CrossRef]

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794-1802 (2002).
[CrossRef]

Dios, F.

Dogariu, A.

Fitzhenry, K.

D. Voelz and K. Fitzhenry, “Pseudo-partially coherent beam for free-space laser communication,” Proc. SPIE 5550, 218-224 (2004).
[CrossRef]

Flatte, S. M.

Gbur, G.

Gerber, J. S.

Glindermann, A.

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

Hopen, C. Y.

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

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

Johnston, R. A.

Kahn, J. M.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293-1300 (2002).
[CrossRef]

Kiriazes, J. J.

J. J. Kiriazes, R. L. Phillips, and L. C. Andrews, “Analysis of fading for a free-space optical communication link subject to atmospheric scintillation,” Proc. SPIE 5160, 253-264 (2004).
[CrossRef]

Kolmogorov, A. N.

A. N. Kolmogorov, “Dissipation of energy in the locally isotropic turbulence,” reprinted in Proc. R. Soc. London. Ser. A 434, 15-17 (1991).
[CrossRef]

Korotkova, O.

O. Korotkova, “Changes in the intensity fluctuations of a class of random electromagnetic beams on propagation,” J. Opt. A Pure Appl. Opt. 8, 30-37 (2006).
[CrossRef]

O. Korotkova, “Control of the intensity fluctuation of random electromagnetic beams on propagation in weak atmospheric turbulence,” Proc. SPIE 6105, 61050V (2006).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “The effect of partially coherent quasi-monochromatic Gaussian-beam on the probability of fade,” Proc. SPIE 5160, 68-77 (2004).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330-341 (2004).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Phase diffuser at the transmitter for lasercom link: effect of partially coherent beam on the bit-error rate,” Proc. SPIE 4976, 70-77 (2003).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98-109 (2002).
[CrossRef]

Lane, R. G.

R. A. Johnston and R. G. Lane, “Modeling scintillation from an aperiodic Kolmogorov phase screen,” Appl. Opt. 39, 4761-4769 (2000).
[CrossRef]

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

Lataitis, R. J.

Parenti, R. R.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001(2006).
[CrossRef]

Perlot, N.

Phillips, R. L.

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46, 086002 (2007).
[CrossRef]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001(2006).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330-341 (2004).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “The effect of partially coherent quasi-monochromatic Gaussian-beam on the probability of fade,” Proc. SPIE 5160, 68-77 (2004).
[CrossRef]

J. J. Kiriazes, R. L. Phillips, and L. C. Andrews, “Analysis of fading for a free-space optical communication link subject to atmospheric scintillation,” Proc. SPIE 5160, 253-264 (2004).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Phase diffuser at the transmitter for lasercom link: effect of partially coherent beam on the bit-error rate,” Proc. SPIE 4976, 70-77 (2003).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98-109 (2002).
[CrossRef]

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554-1562 (2001).
[CrossRef]

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005), pp. 667-711.
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005), pp. 321-393.
[CrossRef]

Plonus, M. A.

Rao, R.

Recolons, J.

Ricklin, J. C.

J. C. Ricklin, S. Bucaille, and F. M. Davidson, “Performance loss factors for optical communication through clear air turbulence,” Proc. SPIE 5160, 1-12 (2004).
[CrossRef]

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794-1802 (2002).
[CrossRef]

Rodriguez, A.

Roggemann, M. C.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996), Chap. 3.

Rubio, J. A.

Sasiela, R. J.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001(2006).
[CrossRef]

Schulz, T. J.

Shirai, T.

Vetelino, F. S.

Voelz, D.

X. Xiao and D. Voelz, “Toward optimizing partial spatially coherent beams for free space laser communications,” Proc. SPIE 6709, 67090P (2007).
[CrossRef]

X. Xiao and D. Voelz, “Wave optics simulation of pseudo-partially coherent beam propagation through turbulence: application to laser communications,” Proc. SPIE 6304, 63040L(2006).
[CrossRef]

X. Xiao and D. Voelz, “Wave optics simulation approach for partial spatially coherent beams,” Opt. Express 14, 6986-6992 (2006).
[CrossRef] [PubMed]

D. Voelz and K. Fitzhenry, “Pseudo-partially coherent beam for free-space laser communication,” Proc. SPIE 5550, 218-224 (2004).
[CrossRef]

Wang, G. Y.

Wang, S. C. H.

Welsh, B.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996), Chap. 3.

Wheelon, A. D.

Wolf, E.

Xiao, X.

X. Xiao and D. Voelz, “Toward optimizing partial spatially coherent beams for free space laser communications,” Proc. SPIE 6709, 67090P (2007).
[CrossRef]

X. Xiao and D. Voelz, “Wave optics simulation of pseudo-partially coherent beam propagation through turbulence: application to laser communications,” Proc. SPIE 6304, 63040L(2006).
[CrossRef]

X. Xiao and D. Voelz, “Wave optics simulation approach for partial spatially coherent beams,” Opt. Express 14, 6986-6992 (2006).
[CrossRef] [PubMed]

Young, C.

Zhu, X.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293-1300 (2002).
[CrossRef]

Appl. Opt. (8)

IEEE Trans. Commun. (1)

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293-1300 (2002).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

O. Korotkova, “Changes in the intensity fluctuations of a class of random electromagnetic beams on propagation,” J. Opt. A Pure Appl. Opt. 8, 30-37 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Eng. (4)

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001(2006).
[CrossRef]

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46, 086002 (2007).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330-341 (2004).
[CrossRef]

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554-1562 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Proc. R. Soc. London. Ser. A (1)

A. N. Kolmogorov, “Dissipation of energy in the locally isotropic turbulence,” reprinted in Proc. R. Soc. London. Ser. A 434, 15-17 (1991).
[CrossRef]

Proc. SPIE (9)

J. J. Kiriazes, R. L. Phillips, and L. C. Andrews, “Analysis of fading for a free-space optical communication link subject to atmospheric scintillation,” Proc. SPIE 5160, 253-264 (2004).
[CrossRef]

O. Korotkova, “Control of the intensity fluctuation of random electromagnetic beams on propagation in weak atmospheric turbulence,” Proc. SPIE 6105, 61050V (2006).
[CrossRef]

D. Voelz and K. Fitzhenry, “Pseudo-partially coherent beam for free-space laser communication,” Proc. SPIE 5550, 218-224 (2004).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98-109 (2002).
[CrossRef]

J. C. Ricklin, S. Bucaille, and F. M. Davidson, “Performance loss factors for optical communication through clear air turbulence,” Proc. SPIE 5160, 1-12 (2004).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “The effect of partially coherent quasi-monochromatic Gaussian-beam on the probability of fade,” Proc. SPIE 5160, 68-77 (2004).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Phase diffuser at the transmitter for lasercom link: effect of partially coherent beam on the bit-error rate,” Proc. SPIE 4976, 70-77 (2003).
[CrossRef]

X. Xiao and D. Voelz, “Wave optics simulation of pseudo-partially coherent beam propagation through turbulence: application to laser communications,” Proc. SPIE 6304, 63040L(2006).
[CrossRef]

X. Xiao and D. Voelz, “Toward optimizing partial spatially coherent beams for free space laser communications,” Proc. SPIE 6709, 67090P (2007).
[CrossRef]

Waves Random Media (1)

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

Other (4)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005), pp. 667-711.
[CrossRef]

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996), Chap. 3.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005), pp. 321-393.
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Comparisons of the wave-optics simulation histogram values (open circles) and GG (solid curve) and LN (dashed curve) PDF models versus the natural lognormalized intensity for the cases in Table 1.

Fig. 2
Fig. 2

Comparison of the probability of fade as a function of fade threshold parameter F T for the wave-optics simulation (open circles) results and the GG (solid curve) and LN (dashed curve) PDF models for the cases in Table 1.

Tables (2)

Tables Icon

Table 1 Numerical Wave-Optics PCB Simulation Parameters for Two Link Scenarios

Tables Icon

Table 2 Simulation Mean Irradiance and Scintillation Index and LN and GG Distribution Parameters

Equations (29)

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α = 1 σ x 2 = 1 exp ( σ ln x 2 ) 1 ,
β = 1 σ y 2 = 1 exp ( σ ln y 2 ) 1 ,
p I ( I ) = 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) I I ( α + β ) / 2 K α β ( 2 α β I ) , I > 0 ,
I = I / I ,
σ ln I 2 = σ ln x 2 + σ ln y 2 .
σ ln I 2 = ln ( σ I 2 + 1 ) ,
p I ( I ) = 1 I 2 π σ ln I 2 exp { [ ln ( I ) + 0.5 σ ln I 2 ] 2 2 σ ln I 2 } .
P ( I T ) = Pr ( I I T ) = 0 I T p I ( I ) d I ,
P LN ( I T ) = 1 2 { 1 + erf [ 0.5 σ ln I 2 0.23 F T 2 σ ln I ] } ,
F T = 10 log 10 ( I T I ) .
N TS > ( 10 σ R 2 ) 6 / 11 .
λ Δ z > Δ x > λ Δ z N G .
r 0 = ( 0.423 C n 2 k 2 z ) 3 / 5 .
I ( r , z ) = 1 r i 2 + ξ s z i 2 exp [ 2 r 2 w 0 2 ( r i 2 + ξ s z i 2 ) ] ,
ξ s = 1 + 2 w 0 2 l c 2 ,
r i = 1 ,
z i = 2 z k w 0 2 .
r e = r i ξ s r i 2 + z i 2 ξ s ,
z e = z i r i 2 + z i 2 ξ s .
σ I 2 ( l 0 , L 0 ) = exp [ σ ln x 2 ( l 0 , L 0 ) + σ ln y 2 ( l 0 ) ] 1 .
σ ln x 2 ( l 0 , L 0 ) = σ ln x 2 ( l 0 ) σ ln x 2 ( L 0 ) .
Q l 2 ( l 0 ) = 10.89 z k l 0 2 ,
φ 1 ( l 0 ) = tan 1 ( 2 z e 1 + 2 r e ) ,
φ 2 ( l 0 ) = tan 1 ( ( 1 + 2 r e ) Q l 3 + 2 z e Q l ) ,
r ¯ e = 1 r e ,
η x = [ 0.38 1 3.21 r ¯ e + 5.29 r ¯ e 2 + 0.47 σ R 2 Q l 1 / 6 ( 1 / 3 0.5 r ¯ e + 0.2 r ¯ e 2 1 + 2.20 r ¯ e ) 6 / 7 ] 1 .
σ ln x 2 ( l o ) = 0.49 σ R 2 ( 1 / 3 0.5 r ¯ e + 0.2 r ¯ e 2 ) ( η x Q l η x + Q l ) 7 / 6 · [ 1 + 1.75 ( η x η x + Q l ) 1 / 2 0.25 ( η x η x + Q l ) 7 / 12 ] .
σ ln y 2 ( l 0 ) = 0.51 σ G 2 ( 1 + 0.69 σ G 12 / 5 ) 5 / 6 ,
σ G 2 = 3.86 σ R 2 { 0.40 [ ( 1 + 2 r e ) 2 + ( 2 z e + 3 / Q l ) 2 ] 11 / 12 [ ( 1 + 2 r e ) 2 + 4 z e 2 ] 1 / 2 [ sin ( 11 6 φ 2 + φ 1 ) + 2.61 [ ( 1 + 2 r e ) 2 Q l 2 + ( 3 + 2 z e Q l ) 2 ] 1 / 4 sin ( 4 3 φ 2 + φ 1 ) 0.52 [ ( 1 + 2 r e ) 2 Q l 2 + ( 3 + 2 z e Q l ) 2 ] 7 / 24 sin ( 5 4 φ 2 + φ 1 ) ] 13.40 z e Q l 11 / 6 [ ( 1 + 2 r e ) 2 + 4 z e 2 ] 11 6 [ ( 1 + 0.31 z e Q l Q l ) 5 / 6 + 1.10 ( 1 + 0.27 z e Q l ) 1 / 3 0.19 ( 1 + 0.24 z e Q l ) 1 / 4 Q l 5 / 6 ] .

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