Abstract

A class of random stationary, scalar sources producing cusped average intensity profiles (i.e. profiles with concave curvature) in the far field is introduced by modeling the source degree of coherence as a Fractional Multi-Gaussian-correlated Schell-Model (FMGSM) function with rotational symmetry. The average intensity (spectral density) generated by such sources is investigated on propagation in free space and isotropic and homogeneous atmospheric turbulence. It is found that the FMGSM beam can retain the cusped shape on propagation at least in weak or moderate turbulence regimes; however, strong turbulence completely suppresses the cusped intensity profile. Under the same atmospheric conditions the spectral density of the FMGSM beam at the receiver is found to be much higher than that of the conventional Gaussian Schell-model (GSM) beam within the narrow central area, implying that for relatively small collecting apertures the power-in-bucket of the FMGSM beam is higher than that of the GSM beam. Our results are of importance to energy delivery, Free-Space Optical communications and imaging in the atmosphere.

© 2017 Optical Society of America

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References

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  32. F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
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    [Crossref] [PubMed]
  35. S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, and O. Korotkova, “Polarization-induced reduction in scintillation of optical beams propagating in simulated turbulent atmospheric channels,” Waves Random Media 24(4), 452–462 (2014).
    [Crossref]
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    [Crossref] [PubMed]
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2016 (1)

2015 (4)

C. Wei, D. Wu, C. Liang, F. Wang, and Y. Cai, “Experimental verification of significant reduction of turbulence-induced scintillation in a full Poincaré beam,” Opt. Express 23(19), 24331–24341 (2015).
[Crossref] [PubMed]

P. Birch, I. Ituen, R. Young, and C. Chatwin, “Long-distance Bessel beam propagation through Kolmogorov turbulence,” J. Opt. Soc. Am. A 32(11), 2066–2073 (2015).
[Crossref] [PubMed]

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17(5), 055607 (2015).
[Crossref]

2014 (9)

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel-Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[Crossref]

S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, and O. Korotkova, “Polarization-induced reduction in scintillation of optical beams propagating in simulated turbulent atmospheric channels,” Waves Random Media 24(4), 452–462 (2014).
[Crossref]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

O. Korotkova and E. Shchepakina, “Random sources for optical frames,” Opt. Express 22(9), 10622–10633 (2014).
[Crossref] [PubMed]

Z. Chen, S. Cui, L. Zhang, C. Sun, M. Xiong, and J. Pu, “Measuring the intensity fluctuation of partially coherent radially polarized beams in atmospheric turbulence,” Opt. Express 22(15), 18278–18283 (2014).
[Crossref] [PubMed]

G. Gbur, “Partially coherent beam propagation in atmospheric turbulence [invited],” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014).
[Crossref] [PubMed]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
[Crossref] [PubMed]

2013 (7)

2012 (3)

2011 (1)

2010 (2)

2009 (3)

2007 (3)

2004 (1)

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(2), 330–341 (2004).
[Crossref]

2003 (1)

2002 (1)

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002).
[Crossref]

Andrews, L. C.

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(2), 330–341 (2004).
[Crossref]

Avramov-Zamurovic, S.

S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, and O. Korotkova, “Polarization-induced reduction in scintillation of optical beams propagating in simulated turbulent atmospheric channels,” Waves Random Media 24(4), 452–462 (2014).
[Crossref]

Basu, S.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17(5), 055607 (2015).
[Crossref]

Baykal, Y.

Birch, P.

Cai, Y.

Chatwin, C.

Chen, Z.

Cheng, W.

Cui, S.

Dogariu, A.

Du, X.

Eyyuboglu, H. T.

Fleischer, J. W.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[Crossref]

Gbur, G.

Gilchrest, Y. V.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002).
[Crossref]

Gori, F.

Gu, Y.

Haus, J. W.

Hyde, M. W.

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17(5), 055607 (2015).
[Crossref]

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Ituen, I.

Korotkova, O.

J. Li, F. Wang, and O. Korotkova, “Random sources for cusped beams,” Opt. Express 24(16), 17779–17791 (2016).
[Crossref] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

O. Korotkova and E. Shchepakina, “Random sources for optical frames,” Opt. Express 22(9), 10622–10633 (2014).
[Crossref] [PubMed]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]

S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, and O. Korotkova, “Polarization-induced reduction in scintillation of optical beams propagating in simulated turbulent atmospheric channels,” Waves Random Media 24(4), 452–462 (2014).
[Crossref]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010).
[Crossref] [PubMed]

Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009).
[Crossref] [PubMed]

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
[Crossref] [PubMed]

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(2), 330–341 (2004).
[Crossref]

Kumar, A.

Lajunen, H.

Li, J.

Liang, C.

Liu, L.

Liu, X.

Liu, Z.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel-Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[Crossref]

Ma, L.

Macon, B. R.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002).
[Crossref]

Malek-Madani, R.

S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, and O. Korotkova, “Polarization-induced reduction in scintillation of optical beams propagating in simulated turbulent atmospheric channels,” Waves Random Media 24(4), 452–462 (2014).
[Crossref]

Mei, Z.

Nelson, C.

S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, and O. Korotkova, “Polarization-induced reduction in scintillation of optical beams propagating in simulated turbulent atmospheric channels,” Waves Random Media 24(4), 452–462 (2014).
[Crossref]

Phillips, R. L.

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(2), 330–341 (2004).
[Crossref]

Ponomarenko, S. A.

Prabhakar, S.

Pu, J.

Qin, Z.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel-Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[Crossref]

Qu, J.

Reddy, S. G.

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

Shchepakina, E.

Shen, Y.

Shirai, T.

Singh, R. P.

Situ, G.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[Crossref]

Sun, C.

Tao, R.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel-Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[Crossref]

Voelz, D.

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17(5), 055607 (2015).
[Crossref]

H. T. Eyyuboğlu, D. Voelz, and X. Xiao, “Scintillation analysis of truncated Bessel beams via numerical turbulence propagation simulation,” Appl. Opt. 52(33), 8032–8039 (2013).
[Crossref] [PubMed]

Voelz, D. G.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Waller, L.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[Crossref]

Wang, F.

Wei, C.

Wolf, E.

Wu, D.

Xiao, X.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17(5), 055607 (2015).
[Crossref]

H. T. Eyyuboğlu, D. Voelz, and X. Xiao, “Scintillation analysis of truncated Bessel beams via numerical turbulence propagation simulation,” Appl. Opt. 52(33), 8032–8039 (2013).
[Crossref] [PubMed]

Xiong, M.

Xu, X.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel-Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[Crossref]

Young, C. Y.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002).
[Crossref]

Young, R.

Yuan, Y.

Zhan, Q.

Zhang, L.

Zhao, D.

Zhou, P.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel-Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[Crossref]

Appl. Opt. (1)

J. Appl. Phys. (1)

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

J. Opt. (1)

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17(5), 055607 (2015).
[Crossref]

J. Opt. Soc. Am. A (4)

Nat. Photonics (1)

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[Crossref]

Opt. Eng. (2)

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002).
[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(2), 330–341 (2004).
[Crossref]

Opt. Express (9)

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
[Crossref] [PubMed]

W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009).
[Crossref] [PubMed]

C. Wei, D. Wu, C. Liang, F. Wang, and Y. Cai, “Experimental verification of significant reduction of turbulence-induced scintillation in a full Poincaré beam,” Opt. Express 23(19), 24331–24341 (2015).
[Crossref] [PubMed]

Z. Chen, S. Cui, L. Zhang, C. Sun, M. Xiong, and J. Pu, “Measuring the intensity fluctuation of partially coherent radially polarized beams in atmospheric turbulence,” Opt. Express 22(15), 18278–18283 (2014).
[Crossref] [PubMed]

J. Li, F. Wang, and O. Korotkova, “Random sources for cusped beams,” Opt. Express 24(16), 17779–17791 (2016).
[Crossref] [PubMed]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009).
[Crossref] [PubMed]

O. Korotkova and E. Shchepakina, “Random sources for optical frames,” Opt. Express 22(9), 10622–10633 (2014).
[Crossref] [PubMed]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010).
[Crossref] [PubMed]

Opt. Laser Technol. (1)

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel-Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[Crossref]

Opt. Lett. (15)

Y. Gu and G. Gbur, “Scintillation of Airy beam arrays in atmospheric turbulence,” Opt. Lett. 35(20), 3456–3458 (2010).
[Crossref] [PubMed]

Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref] [PubMed]

Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009).
[Crossref] [PubMed]

F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[Crossref] [PubMed]

X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[Crossref] [PubMed]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
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Waves Random Media (1)

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[Crossref]

Other (3)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, (SPIE, Bellingham, Washington, 1998).

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Figures (6)

Fig. 1
Fig. 1 The DOC of the FMGSM source with circular symmetry as a function of xd at the cross-line yd = 0 with δ0 = 1mm. The solid line is calculated from Eq. (7) with N = 1000. The dashed-line is calculated from the numerical Fourier transform of Eq. (2) directly.
Fig. 2
Fig. 2 Density plots of the spectral density of the FMGSM beam propagating in atmospheric turbulence at several distances. The structure constant in the first, second and third rows are C ˜ n 2 =0, C ˜ n 2 = 10 14 m 2/3 and C ˜ n 2 = 10 13 m 2/3 , respectively.
Fig. 3
Fig. 3 Spectral density of the FMGSM beam at the cross-line ( η=0 ) at several propagation distances for three values of the structure constant C ˜ n 2 .
Fig. 4
Fig. 4 The cross-line (η = 0) of the spectral density of the equivalent GSM and FMGSM beams with M = 10 and M = 50 at several propagation distances in atmospheric turbulence.
Fig. 5
Fig. 5 Dependence of PIB on the aperture diameter D of the lens for the equivalent GSM and FMGSM beams for different strengths of the turbulence.
Fig. 6
Fig. 6 Variation of p for the equivalent GSM and FMGSM beams with (a) propagation distance, (b) the power law α and (c) the structure constant C ˜ n 2 .

Equations (21)

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W (0) ( r 1 , r 2 )= p(v) H * ( r 1 ,v) H( r 2 ,v) d 2 v,
p(v)=[ 1 ( 1exp( δ 0 2 v 2 /2) ) 1/M ],
(1q) 1/M =1+ n=1 (1) n m=1 n ( 1(m1)M ) n! M n q n ,
p(v)= n=1 (1) n+1 m=1 n ( 1(m1)M ) n! M n exp( n v 2 δ 0 2 2 ).
H(r,v)=τ(r)exp(ir·v),
μ (0) ( r 1 , r 2 )= W (0) ( r 1 , r 2 ) W (0) ( r 1 , r 1 ) W (0) ( r 2 , r 2 ) ,
μ (0) ( r d )= p ˜ ( r d ) p ˜ (0) = 1 p ˜ (0) n=1 (1) n+1 m=1 n ( 1(m1)M ) n!n M n exp( r d 2 2n δ 0 2 ) ,
W (0) ( r 1 , r 2 )= τ * ( r 1 )τ( r 2 )μ( r d ) = 1 C exp( r 1 2 + r 2 2 4 σ 0 2 ) n=1 N (1) n+1 m=1 n ( 1(m1)M ) n!n M n exp( r d 2 2n δ 0 2 ) ,
W( ρ 1 , ρ 2 ,z)=exp[ ik 2z ( ρ 2 2 ρ 1 2 ) ] W (0) ( r 1 , r 2 )exp[ ik 2z ( r 1 2 r 2 2 ) ] ×exp[ ik z ( r 1 · ρ 1 r 2 · ρ 2 ) ] exp[ ψ * ( r 1 , ρ 1 ,z )+ψ( r 2 , ρ 2 ,z ) ] d 2 r 1 d 2 r 2 ,
F 2 =exp{ 4 π 2 k 2 z 0 1 dt 0 κ Φ n (κ)dκ[ 1 J 0 ( t ρ d +(1t) r d ) ] },
F 2 =exp[ π 2 k 2 z 3 ( r d 2 + r d · ρ d + ρ d 2 ) 0 κ 3 Φ n (κ)dκ ].
Φ n (κ)=A(α) C ˜ n 2 exp( κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) α/2 ,0<κ<,3<α<4,
A(α)= 1 4 π 2 Γ(α1)cos( απ 2 ),
c(α)= [ 2πΓ(5α/2)A(α) 3 ] 1/(α5) .
F 2 =exp{ π 2 k 2 Tz 3 [ ( r 1 r 2 ) 2 +( r 1 r 2 )( ρ 1 ρ 2 )+ ( ρ 1 ρ 2 ) 2 ] },
T= A(α) 2(α2) C ˜ n 2 [ β κ m 2α exp( κ 0 2 / κ m 2 ) Γ 1 (2α/2, κ 0 2 / κ m 2 )2 κ 0 4α ],
S(ρ,z)= 1 C n=1 (1) n+1 m=1 n ( 1(m1)M ) n!n M n Δ n (z) exp( ρ 2 2 σ 0 2 Δ n (z) ),
Δ n (z)=1+[ 1 4 k 2 σ 0 4 + 1 k 2 σ 0 2 ( 1 n δ 0 2 + 2 π 2 k 2 Tz 3 ) ] z 2 .
W GSM (0) ( r 1 , r 2 )=exp( r 1 2 + r 2 2 4 σ g0 2 )exp( r d 2 2 δ g0 2 ),
p= A<π D 2 /4 S(ρ,z) d 2 ρ / S(ρ,z) d 2 ρ .
p(D)= 1 C n=1 (1) n+1 m=1 n ( 1(m1)M ) n!n M n [ 1exp( D 2 8 σ 0 2 Δ n (z) ) ].

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