Abstract

We report the effect of aerosol-induced local atmospheric heating and the resulting changes in the lower atmospheric optical turbulence on the performance of Free-Space Optical (FSO) communication links. A closed form mathematical expression is derived to estimate the influence of aerosol-induced warming on the Bit Error Rate (BER) of a Binary Phase Shift Keying FSO communication link through Gamma-Gamma modeled turbulence. Our results demonstrate a strong impact, with the aerosol-induced turbulence taking a toll on the signal-to-noise ratio of ~20 dB for a BER of 10−9. Aerosol-induced warming produces significant variations in BER compared to the clear atmospheric conditions and can subdue the benefits of improved beam alignment.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Dependence of atmospheric refractive index structure parameter (Cn2) on the residence time and vertical distribution of aerosols

N. Anand, S. K. Satheesh, and K. Krishna Moorthy
Opt. Lett. 42(14) 2714-2717 (2017)

Distinctive roles of elevated absorbing aerosol layers on free-space optical communication systems

N. Anand, K. Sunilkumar, S. K. Satheesh, and K. Krishna Moorthy
Appl. Opt. 57(25) 7152-7158 (2018)

On the Effects of Combined Atmospheric Fading and Misalignment on the Hybrid FSO/RF Transmission

Abir Touati, Abderrazak Abdaoui, Farid Touati, Murat Uysal, and Ammar Bouallegue
J. Opt. Commun. Netw. 8(10) 715-725 (2016)

References

  • View by:
  • |
  • |
  • |

  1. M. Uysal, C. Capsoni, Z. Ghassemlooy, A. Boucouvalas, and E. Udvary, Optical Wireless Communications Signals and Communication Technology (Springer, 2016).
  2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
  3. H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios, and G. S. Tombras, ““Average capacity of optical wireless communication systems over atmospheric turbulence channels,” IEEE/OSA,” J. Lightwave Technol. 27(8), 974–979 (2009).
    [Crossref]
  4. H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” IEEE/OSA. J. Lightwave Technol. 27(20), 4440–4445 (2009).
    [Crossref]
  5. K. Sunilkumar, S. K. Satheesh, G. Ilavazhagan, and K. K. Moorthy, “Free space optical communication system through turbid media with pointing errors,” in Proceeding of Imaging and Applied Optics, LS & C (Optical Society of America, 2018).
  6. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
  7. N. Anand, S. K. Satheesh, and K. K. Moorthy, “Dependence of atmospheric refractive index structure parameter (Cn2) on the residence time and vertical distribution of aerosols,” Opt. Lett. 42(14), 2714–2717 (2017).
    [Crossref] [PubMed]
  8. N. Anand, S. K. Satheesh, and K. K. Moorthy, “Modulation of optical turbulence by atmospheric aerosols: influence of vertical distribution and residence time,” in Proceeding of Imaging and Applied Optics Congress (pcAOP), (Optical Society of America, 2018).
    [Crossref]
  9. N. Anand, K. Sunilkumar, S. K. Satheesh, and K. K. Moorthy, “Distinctive roles of elevated absorbing aerosol layers on free-space optical communication systems,” Appl. Opt. 57(25), 7152–7158 (2018).
    [Crossref] [PubMed]
  10. D. Sadot and N. S. Kopeika, “Forecasting optical turbulence strength on basis of macroscale meteorology and aerosols: models and validation,” Opt. Eng. 31(2), 200–212 (1992).
    [Crossref]
  11. N. S. Kopeika, A System Engineering Approach to Imaging (SPIE Optical Engineering, 1998).
  12. S. Bendersky, N. S. Kopeika, and N. Blaunstein, “Atmospheric optical turbulence over land in middle east coastal environments: prediction modeling and measurements,” Appl. Opt. 43(20), 4070–4079 (2004).
    [Crossref] [PubMed]
  13. J. Libich, J. Perez, S. Zvanovec, Z. Ghassemlooy, R. Nebuloni, and C. Capsoni, “Combined effect of turbulence and aerosol on free-space optical links,” Appl. Opt. 56(2), 336–341 (2017).
    [Crossref] [PubMed]
  14. W. G. Alheadary, K.-H. Park, N. Alfaraj, Y. Guo, E. Stegenburgs, T. K. Ng, B. S. Ooi, and M.-S. Alouini, “Free-space optical channel characterization and experimental validation in a coastal environment,” Opt. Express 26(6), 6614–6628 (2018).
    [Crossref] [PubMed]
  15. Y. Li, W. Zhu, X. Wu, and R. Rao, “Equivalent refractive-index structure constant of non-Kolmogorov turbulence,” Opt. Express 23(18), 23004–23012 (2015).
    [Crossref] [PubMed]
  16. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008).
    [Crossref] [PubMed]
  17. G. P. Agrawal, Fiber-Optical Communication Systems, 3rd ed. (Wiley, 2002).
  18. M. Niu, X. Song, J. Cheng, and J. F. Holzman, “Performance analysis of coherent wireless optical communications with atmospheric turbulence,” Opt. Express 20(6), 6515–6520 (2012).
    [Crossref] [PubMed]
  19. P. Wang, L. Zhang, L. Guo, F. Huang, T. Shang, R. Wang, and Y. Yang, “Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels,” Opt. Express,  22(17), 20828–20841 (2014).
  20. Wolfram Research, http://functions.wolfram.com/06.25.20.0001.01
  21. Y. L. Luke, The Special Functions and Their Approximations, Vol. 1 (Academic Press, 1969).
  22. S. Chandrasekhar, Radiative Transfer, New York, (Dover, 1960).
  23. S. K. Satheesh and K. K. Moorthy, “Radiative effects of natural aerosols: A review,” Atmos. Environ. 39(11), 2089–2110 (2005).
    [Crossref]
  24. S. S. Babu, S. K. Satheesh, and K. K. Moorthy, “Aerosol radiative forcing due to enhanced black carbon at an urban site in India,” Geophys. Res. Lett. 29(18), 27 (2002).
    [Crossref]
  25. K. K. Moorthy, S. K. Satheesh, S. S. Babu, and C. B. S. Dutt, “Integrated campaign for aerosols, gases and radiation budget (ICARB): an overview,” J. Earth Syst. Sci. 117(S1), 243–262 (2008).
    [Crossref]
  26. S. K. Satheesh, K. K. Moorthy, S. S. Babu, V. Vinoj, and C. B. S. Dutt, “Climate implications of large warming by elevated aerosol over India,” Geophys. Res. Lett. 35(19), L19809 (2008).
    [Crossref]
  27. K. N. Liou, An introduction to atmospheric radiation, Vol. 84, (Elsevier, 2002).
  28. P. Ricchiazzi, S. Yang, C. Gautier, and D. Sowle, “SBDART, a research and teaching tool for plane-parallel radiative transfer in the Earth’s atmosphere,” Bull. Am. Meteorol. Soc. 79(10), 2101–2114 (1998).
    [Crossref]
  29. M. Hess, P. Koepke, and I. Schult, “Optical properties of aerosols and clouds: the software package OPAC,” Bull. Am. Meteorol. Soc. 79(5), 831–844 (1998).
    [Crossref]

2018 (2)

2017 (2)

2015 (1)

2014 (1)

2012 (1)

2009 (2)

H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios, and G. S. Tombras, ““Average capacity of optical wireless communication systems over atmospheric turbulence channels,” IEEE/OSA,” J. Lightwave Technol. 27(8), 974–979 (2009).
[Crossref]

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” IEEE/OSA. J. Lightwave Technol. 27(20), 4440–4445 (2009).
[Crossref]

2008 (3)

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008).
[Crossref] [PubMed]

K. K. Moorthy, S. K. Satheesh, S. S. Babu, and C. B. S. Dutt, “Integrated campaign for aerosols, gases and radiation budget (ICARB): an overview,” J. Earth Syst. Sci. 117(S1), 243–262 (2008).
[Crossref]

S. K. Satheesh, K. K. Moorthy, S. S. Babu, V. Vinoj, and C. B. S. Dutt, “Climate implications of large warming by elevated aerosol over India,” Geophys. Res. Lett. 35(19), L19809 (2008).
[Crossref]

2005 (1)

S. K. Satheesh and K. K. Moorthy, “Radiative effects of natural aerosols: A review,” Atmos. Environ. 39(11), 2089–2110 (2005).
[Crossref]

2004 (1)

2002 (1)

S. S. Babu, S. K. Satheesh, and K. K. Moorthy, “Aerosol radiative forcing due to enhanced black carbon at an urban site in India,” Geophys. Res. Lett. 29(18), 27 (2002).
[Crossref]

1998 (2)

P. Ricchiazzi, S. Yang, C. Gautier, and D. Sowle, “SBDART, a research and teaching tool for plane-parallel radiative transfer in the Earth’s atmosphere,” Bull. Am. Meteorol. Soc. 79(10), 2101–2114 (1998).
[Crossref]

M. Hess, P. Koepke, and I. Schult, “Optical properties of aerosols and clouds: the software package OPAC,” Bull. Am. Meteorol. Soc. 79(5), 831–844 (1998).
[Crossref]

1992 (1)

D. Sadot and N. S. Kopeika, “Forecasting optical turbulence strength on basis of macroscale meteorology and aerosols: models and validation,” Opt. Eng. 31(2), 200–212 (1992).
[Crossref]

Alfaraj, N.

Alheadary, W. G.

Alouini, M.-S.

Anand, N.

Babu, S. S.

K. K. Moorthy, S. K. Satheesh, S. S. Babu, and C. B. S. Dutt, “Integrated campaign for aerosols, gases and radiation budget (ICARB): an overview,” J. Earth Syst. Sci. 117(S1), 243–262 (2008).
[Crossref]

S. K. Satheesh, K. K. Moorthy, S. S. Babu, V. Vinoj, and C. B. S. Dutt, “Climate implications of large warming by elevated aerosol over India,” Geophys. Res. Lett. 35(19), L19809 (2008).
[Crossref]

S. S. Babu, S. K. Satheesh, and K. K. Moorthy, “Aerosol radiative forcing due to enhanced black carbon at an urban site in India,” Geophys. Res. Lett. 29(18), 27 (2002).
[Crossref]

Bendersky, S.

Blaunstein, N.

Capsoni, C.

Cheng, J.

Dutt, C. B. S.

S. K. Satheesh, K. K. Moorthy, S. S. Babu, V. Vinoj, and C. B. S. Dutt, “Climate implications of large warming by elevated aerosol over India,” Geophys. Res. Lett. 35(19), L19809 (2008).
[Crossref]

K. K. Moorthy, S. K. Satheesh, S. S. Babu, and C. B. S. Dutt, “Integrated campaign for aerosols, gases and radiation budget (ICARB): an overview,” J. Earth Syst. Sci. 117(S1), 243–262 (2008).
[Crossref]

Fafalios, M. E.

Gautier, C.

P. Ricchiazzi, S. Yang, C. Gautier, and D. Sowle, “SBDART, a research and teaching tool for plane-parallel radiative transfer in the Earth’s atmosphere,” Bull. Am. Meteorol. Soc. 79(10), 2101–2114 (1998).
[Crossref]

Ghassemlooy, Z.

Golbraikh, E.

Guo, L.

Guo, Y.

Hess, M.

M. Hess, P. Koepke, and I. Schult, “Optical properties of aerosols and clouds: the software package OPAC,” Bull. Am. Meteorol. Soc. 79(5), 831–844 (1998).
[Crossref]

Holzman, J. F.

Huang, F.

Karagianni, E. A.

Karagiannidis, G. K.

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” IEEE/OSA. J. Lightwave Technol. 27(20), 4440–4445 (2009).
[Crossref]

Koepke, P.

M. Hess, P. Koepke, and I. Schult, “Optical properties of aerosols and clouds: the software package OPAC,” Bull. Am. Meteorol. Soc. 79(5), 831–844 (1998).
[Crossref]

Kopeika, N. S.

Li, Y.

Libich, J.

Moorthy, K. K.

N. Anand, K. Sunilkumar, S. K. Satheesh, and K. K. Moorthy, “Distinctive roles of elevated absorbing aerosol layers on free-space optical communication systems,” Appl. Opt. 57(25), 7152–7158 (2018).
[Crossref] [PubMed]

N. Anand, S. K. Satheesh, and K. K. Moorthy, “Dependence of atmospheric refractive index structure parameter (Cn2) on the residence time and vertical distribution of aerosols,” Opt. Lett. 42(14), 2714–2717 (2017).
[Crossref] [PubMed]

K. K. Moorthy, S. K. Satheesh, S. S. Babu, and C. B. S. Dutt, “Integrated campaign for aerosols, gases and radiation budget (ICARB): an overview,” J. Earth Syst. Sci. 117(S1), 243–262 (2008).
[Crossref]

S. K. Satheesh, K. K. Moorthy, S. S. Babu, V. Vinoj, and C. B. S. Dutt, “Climate implications of large warming by elevated aerosol over India,” Geophys. Res. Lett. 35(19), L19809 (2008).
[Crossref]

S. K. Satheesh and K. K. Moorthy, “Radiative effects of natural aerosols: A review,” Atmos. Environ. 39(11), 2089–2110 (2005).
[Crossref]

S. S. Babu, S. K. Satheesh, and K. K. Moorthy, “Aerosol radiative forcing due to enhanced black carbon at an urban site in India,” Geophys. Res. Lett. 29(18), 27 (2002).
[Crossref]

Nebuloni, R.

Ng, T. K.

Nistazakis, H. E.

Niu, M.

Ooi, B. S.

Park, K.-H.

Perez, J.

Rao, R.

Ricchiazzi, P.

P. Ricchiazzi, S. Yang, C. Gautier, and D. Sowle, “SBDART, a research and teaching tool for plane-parallel radiative transfer in the Earth’s atmosphere,” Bull. Am. Meteorol. Soc. 79(10), 2101–2114 (1998).
[Crossref]

Sadot, D.

D. Sadot and N. S. Kopeika, “Forecasting optical turbulence strength on basis of macroscale meteorology and aerosols: models and validation,” Opt. Eng. 31(2), 200–212 (1992).
[Crossref]

Sandalidis, H. G.

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” IEEE/OSA. J. Lightwave Technol. 27(20), 4440–4445 (2009).
[Crossref]

Satheesh, S. K.

N. Anand, K. Sunilkumar, S. K. Satheesh, and K. K. Moorthy, “Distinctive roles of elevated absorbing aerosol layers on free-space optical communication systems,” Appl. Opt. 57(25), 7152–7158 (2018).
[Crossref] [PubMed]

N. Anand, S. K. Satheesh, and K. K. Moorthy, “Dependence of atmospheric refractive index structure parameter (Cn2) on the residence time and vertical distribution of aerosols,” Opt. Lett. 42(14), 2714–2717 (2017).
[Crossref] [PubMed]

K. K. Moorthy, S. K. Satheesh, S. S. Babu, and C. B. S. Dutt, “Integrated campaign for aerosols, gases and radiation budget (ICARB): an overview,” J. Earth Syst. Sci. 117(S1), 243–262 (2008).
[Crossref]

S. K. Satheesh, K. K. Moorthy, S. S. Babu, V. Vinoj, and C. B. S. Dutt, “Climate implications of large warming by elevated aerosol over India,” Geophys. Res. Lett. 35(19), L19809 (2008).
[Crossref]

S. K. Satheesh and K. K. Moorthy, “Radiative effects of natural aerosols: A review,” Atmos. Environ. 39(11), 2089–2110 (2005).
[Crossref]

S. S. Babu, S. K. Satheesh, and K. K. Moorthy, “Aerosol radiative forcing due to enhanced black carbon at an urban site in India,” Geophys. Res. Lett. 29(18), 27 (2002).
[Crossref]

Schult, I.

M. Hess, P. Koepke, and I. Schult, “Optical properties of aerosols and clouds: the software package OPAC,” Bull. Am. Meteorol. Soc. 79(5), 831–844 (1998).
[Crossref]

Shang, T.

Song, X.

Sowle, D.

P. Ricchiazzi, S. Yang, C. Gautier, and D. Sowle, “SBDART, a research and teaching tool for plane-parallel radiative transfer in the Earth’s atmosphere,” Bull. Am. Meteorol. Soc. 79(10), 2101–2114 (1998).
[Crossref]

Stegenburgs, E.

Sunilkumar, K.

Tombras, G. S.

Tsiftsis, T. A.

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” IEEE/OSA. J. Lightwave Technol. 27(20), 4440–4445 (2009).
[Crossref]

Tsigopoulos, A. D.

Vinoj, V.

S. K. Satheesh, K. K. Moorthy, S. S. Babu, V. Vinoj, and C. B. S. Dutt, “Climate implications of large warming by elevated aerosol over India,” Geophys. Res. Lett. 35(19), L19809 (2008).
[Crossref]

Wang, P.

Wang, R.

Wu, X.

Yang, S.

P. Ricchiazzi, S. Yang, C. Gautier, and D. Sowle, “SBDART, a research and teaching tool for plane-parallel radiative transfer in the Earth’s atmosphere,” Bull. Am. Meteorol. Soc. 79(10), 2101–2114 (1998).
[Crossref]

Yang, Y.

Zhang, L.

Zhu, W.

Zilberman, A.

Zvanovec, S.

Appl. Opt. (4)

Atmos. Environ. (1)

S. K. Satheesh and K. K. Moorthy, “Radiative effects of natural aerosols: A review,” Atmos. Environ. 39(11), 2089–2110 (2005).
[Crossref]

Bull. Am. Meteorol. Soc. (2)

P. Ricchiazzi, S. Yang, C. Gautier, and D. Sowle, “SBDART, a research and teaching tool for plane-parallel radiative transfer in the Earth’s atmosphere,” Bull. Am. Meteorol. Soc. 79(10), 2101–2114 (1998).
[Crossref]

M. Hess, P. Koepke, and I. Schult, “Optical properties of aerosols and clouds: the software package OPAC,” Bull. Am. Meteorol. Soc. 79(5), 831–844 (1998).
[Crossref]

Geophys. Res. Lett. (2)

S. K. Satheesh, K. K. Moorthy, S. S. Babu, V. Vinoj, and C. B. S. Dutt, “Climate implications of large warming by elevated aerosol over India,” Geophys. Res. Lett. 35(19), L19809 (2008).
[Crossref]

S. S. Babu, S. K. Satheesh, and K. K. Moorthy, “Aerosol radiative forcing due to enhanced black carbon at an urban site in India,” Geophys. Res. Lett. 29(18), 27 (2002).
[Crossref]

IEEE/OSA. J. Lightwave Technol. (1)

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” IEEE/OSA. J. Lightwave Technol. 27(20), 4440–4445 (2009).
[Crossref]

J. Earth Syst. Sci. (1)

K. K. Moorthy, S. K. Satheesh, S. S. Babu, and C. B. S. Dutt, “Integrated campaign for aerosols, gases and radiation budget (ICARB): an overview,” J. Earth Syst. Sci. 117(S1), 243–262 (2008).
[Crossref]

J. Lightwave Technol. (1)

Opt. Eng. (1)

D. Sadot and N. S. Kopeika, “Forecasting optical turbulence strength on basis of macroscale meteorology and aerosols: models and validation,” Opt. Eng. 31(2), 200–212 (1992).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Other (11)

N. Anand, S. K. Satheesh, and K. K. Moorthy, “Modulation of optical turbulence by atmospheric aerosols: influence of vertical distribution and residence time,” in Proceeding of Imaging and Applied Optics Congress (pcAOP), (Optical Society of America, 2018).
[Crossref]

N. S. Kopeika, A System Engineering Approach to Imaging (SPIE Optical Engineering, 1998).

K. Sunilkumar, S. K. Satheesh, G. Ilavazhagan, and K. K. Moorthy, “Free space optical communication system through turbid media with pointing errors,” in Proceeding of Imaging and Applied Optics, LS & C (Optical Society of America, 2018).

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

M. Uysal, C. Capsoni, Z. Ghassemlooy, A. Boucouvalas, and E. Udvary, Optical Wireless Communications Signals and Communication Technology (Springer, 2016).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).

Wolfram Research, http://functions.wolfram.com/06.25.20.0001.01

Y. L. Luke, The Special Functions and Their Approximations, Vol. 1 (Academic Press, 1969).

S. Chandrasekhar, Radiative Transfer, New York, (Dover, 1960).

G. P. Agrawal, Fiber-Optical Communication Systems, 3rd ed. (Wiley, 2002).

K. N. Liou, An introduction to atmospheric radiation, Vol. 84, (Elsevier, 2002).

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 (2)

Fig. 1
Fig. 1 Average BER plotted against SNR (dB) for unperturbed and perturbed aerosol conditions for (a) DJF, (b) MAM, (c) JJAS and (d) ON seasons.
Fig. 2
Fig. 2 Dependence of perturbed Rytov variance (σR2*) on the perturbations in refractive index structure parameter, ΔCn2(m-2/3): seasonal variations (bar diagram) and their relationship (inset).

Tables (1)

Tables Icon

Table 1 Seasonal variations in refractive index structure parameter under unperturbed (Cn2) and perturbed (Cn2*) conditions

Equations (19)

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

y=Ix+n
I= I l I p I s
σ R 2 =1.23 C n 2 k 7/6 L 11/6
I ac (t)=R( P S + P LO )+2R P S P LO cos( ω IF t+ ϕ S ϕ LO )
I ac (t)=2R P S P LO cos( ω IF t+ ϕ S ϕ LO )
SNR= Ι ac 2 σ 2
SNR= 2 R 2 P S ¯ P LO 2q(R P LO + I d )Δf+ σ T 2
SNR= R P S ¯ qΔf
P e ( I S )=Q( I S γ ¯ )
Q( I S γ ¯ )= 1 2 erfc( I S γ ¯ 2 )
f I S ( I S )= 2 (αβ) (α+β)/2 Γ(α)Γ(β) ( I S ) (α+β/2)1 K αβ (2 αβ Ι S ) for I S >0
P b ¯ (e)= 0 P e ( I S ) f I S ( I S )d I S
  f I (I)= 2 γ 2 (αβ) (α+β)/2 ( Α 0 Ι l ) γ 2 Γ(α)Γ(β) I γ 2 1 I Α 0 Ι l ( I S ) (α+β/2) γ 2 1 K αβ (2 αβ Ι S ) d I S  for I S >0 
P b ¯ (e)= 1 2 0 erfc( I S γ ¯ 2 )d I S I Α 0 Ι l I γ 2 1 2 γ 2 (αβ) (α+β)/2 ( Α 0 Ι l ) γ 2 Γ(α)Γ(β) ( I S ) (α+β/2) γ 2 1 K αβ (2 αβ Ι S )d I S
P b ¯ (e)= 0 P e ' ( I S ) F I S ( I S )d I S
F I S ( I S )= γ 2 Γ(α)Γ(β) G 2,4 3,1 [ αβ Ι S A 0 I l | 1 γ 2 +1 γ 2 α β 0 ]
P e ' ( I S )= 1 π exp( I S γ ¯ 2 )
P b ¯ (e)= γ 2 π Γ(α)Γ(β) 0 exp( I S γ ¯ 2 ) G 2,4 3,1 [ αβ Ι S A 0 I l | 1 γ 2 +1 γ 2 α β 0 ]d I S
P b ¯ (e)= 2 γ 2 γ ¯ π Γ(α)Γ(β) G 3,4 3,2 [ 2αβ A 0 I l γ ¯ | 0 1 γ 2 +1 γ 2 α β 0 ]

Metrics