Abstract

In this work, we consider optical downlink from space-based laser sources and develop a consistent quantitative analysis of the collected power fluctuations by finite receiving apertures, and both the corresponding temporal covariance and power spectral density (PSD). Here we assume weak to moderate scintillation conditions where lognormal statistics are valid. We derive both exact solutions and highly accurate engineering easy to implement approximations for the downlink aperture-averaging factor, and both the corresponding aperture-averaged signal temporal covariance and PSD. Additionally, highly accurate elementary analytic scaling relations are derived for the corresponding aperture-averaged characteristic correlation time and scintillation bandwidth, which are in good agreement with available experimental observations. Finally closed form expressions for the so-called quasi-frequency that is central to the determination of level crossing rates and duration of fades and surges in a propagation channel are derived. Wherever possible, we endeavor to derive “user friendly” accurate engineering approximations for the various statistical quantities of interest.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, U. S. Department of Commerce, Springfield, Virginia, 1971, Sec 53.
  2. D. L. Fried, “Aperture averaging of scintillation,” J. Opt. Soc. Am. 57(2), 169–178 (1967).
    [Crossref]
  3. J. H. Churnside, “Aperture Averaging of Optical Scintillations in the Turbulent Atmosphere,” Appl. Opt. 30(15), 1982–1994 (1991).
    [Crossref] [PubMed]
  4. L. C. Andrews, “Aperture-averaging factor for optical scintillation of plane and spherical waves in the atmosphere,” J. Opt. Soc. Am. A 9(4), 597–600 (1992).
    [Crossref]
  5. A. D. Wheelon, Electromagnetic Scintillation, II. Weak Scattering, University Press, Cambridge, Secs. 2.2.3 and 4.1.4, 2003.
  6. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and temporal spectrum,” Waves Random Media 10(1), 53–70 (2000).
    [Crossref]
  7. H. Shen, L. Yu, and C. Fan, “Temporal spectrum of atmospheric scintillation and the effects of aperture averaging and time averaging,” Opt. Commun. 330, 160–164 (2014).
    [Crossref]
  8. H. T. Yura and W. G. McKinley, “Aperture averaging of scintillation for space-to-ground optical communication applications,” Appl. Opt. 22(11), 1608–1609 (1983).
    [Crossref] [PubMed]
  9. D. Dravins, L. Lindgren, E. Mezey, and A. T. Young, “Atmospheric Intensity Scintillation of Stars, III. Effects for Different Telescope Apertures,” Publ. Astron. Soc. Pac. 110(747), 610–633 (1998).
    [Crossref]
  10. E. Vilar, J. Haddon, P. Lo, and T. J. Moulsley, “Measurements and Modeling of Amplitude and Phase Scintillations in an Earth-Space path,” Journal of the Institution of Electronic and Radio Engineers 55(3), 87–96 (1985).
    [Crossref]
  11. S. Wolfram, Mathematica, (Cambridge University Press, 2012), Version 9.
  12. The Infrared & Electro-Optical Handbook Vol, 3 Atmospheric Propagation of Radiation, F. G. Smith, Editor, Chapter. 2, SPIE Optical Engineering Press, Bellingham WA, 1993. Note, the HV-5/7 model is also known as the HV-21 model.
  13. A. D. Wheelon and I. Electromagnetic Scintillation, Geometrical Optics (University Press, Cambridge, 2001), Sec. 6.1.2.
  14. J. L. Bufton, “Comparison of vertical profile turbulence structure with stellar observations,” Appl. Opt. 12(8), 1785–1793 (1973).
    [Crossref] [PubMed]
  15. Realistic to the extent of knowing the correct index-structure constant profile.
  16. P. Beckmann, “Probability in Communication Engineering”, Harcourt Brace & World, Inc., 1967, Sec. 6.7. ”
  17. H. T. Yura and W. G. McKinley, “Optical scintillation statistics for IR ground-to-space laser communication systems,” Appl. Opt. 22(21), 3353–3358 (1983).
    [Crossref] [PubMed]

2014 (1)

H. Shen, L. Yu, and C. Fan, “Temporal spectrum of atmospheric scintillation and the effects of aperture averaging and time averaging,” Opt. Commun. 330, 160–164 (2014).
[Crossref]

2000 (1)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and temporal spectrum,” Waves Random Media 10(1), 53–70 (2000).
[Crossref]

1998 (1)

D. Dravins, L. Lindgren, E. Mezey, and A. T. Young, “Atmospheric Intensity Scintillation of Stars, III. Effects for Different Telescope Apertures,” Publ. Astron. Soc. Pac. 110(747), 610–633 (1998).
[Crossref]

1992 (1)

1991 (1)

1985 (1)

E. Vilar, J. Haddon, P. Lo, and T. J. Moulsley, “Measurements and Modeling of Amplitude and Phase Scintillations in an Earth-Space path,” Journal of the Institution of Electronic and Radio Engineers 55(3), 87–96 (1985).
[Crossref]

1983 (2)

1973 (1)

1967 (1)

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and temporal spectrum,” Waves Random Media 10(1), 53–70 (2000).
[Crossref]

L. C. Andrews, “Aperture-averaging factor for optical scintillation of plane and spherical waves in the atmosphere,” J. Opt. Soc. Am. A 9(4), 597–600 (1992).
[Crossref]

Bufton, J. L.

Churnside, J. H.

Dravins, D.

D. Dravins, L. Lindgren, E. Mezey, and A. T. Young, “Atmospheric Intensity Scintillation of Stars, III. Effects for Different Telescope Apertures,” Publ. Astron. Soc. Pac. 110(747), 610–633 (1998).
[Crossref]

Fan, C.

H. Shen, L. Yu, and C. Fan, “Temporal spectrum of atmospheric scintillation and the effects of aperture averaging and time averaging,” Opt. Commun. 330, 160–164 (2014).
[Crossref]

Fried, D. L.

Haddon, J.

E. Vilar, J. Haddon, P. Lo, and T. J. Moulsley, “Measurements and Modeling of Amplitude and Phase Scintillations in an Earth-Space path,” Journal of the Institution of Electronic and Radio Engineers 55(3), 87–96 (1985).
[Crossref]

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and temporal spectrum,” Waves Random Media 10(1), 53–70 (2000).
[Crossref]

Lindgren, L.

D. Dravins, L. Lindgren, E. Mezey, and A. T. Young, “Atmospheric Intensity Scintillation of Stars, III. Effects for Different Telescope Apertures,” Publ. Astron. Soc. Pac. 110(747), 610–633 (1998).
[Crossref]

Lo, P.

E. Vilar, J. Haddon, P. Lo, and T. J. Moulsley, “Measurements and Modeling of Amplitude and Phase Scintillations in an Earth-Space path,” Journal of the Institution of Electronic and Radio Engineers 55(3), 87–96 (1985).
[Crossref]

McKinley, W. G.

Mezey, E.

D. Dravins, L. Lindgren, E. Mezey, and A. T. Young, “Atmospheric Intensity Scintillation of Stars, III. Effects for Different Telescope Apertures,” Publ. Astron. Soc. Pac. 110(747), 610–633 (1998).
[Crossref]

Moulsley, T. J.

E. Vilar, J. Haddon, P. Lo, and T. J. Moulsley, “Measurements and Modeling of Amplitude and Phase Scintillations in an Earth-Space path,” Journal of the Institution of Electronic and Radio Engineers 55(3), 87–96 (1985).
[Crossref]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and temporal spectrum,” Waves Random Media 10(1), 53–70 (2000).
[Crossref]

Shen, H.

H. Shen, L. Yu, and C. Fan, “Temporal spectrum of atmospheric scintillation and the effects of aperture averaging and time averaging,” Opt. Commun. 330, 160–164 (2014).
[Crossref]

Vilar, E.

E. Vilar, J. Haddon, P. Lo, and T. J. Moulsley, “Measurements and Modeling of Amplitude and Phase Scintillations in an Earth-Space path,” Journal of the Institution of Electronic and Radio Engineers 55(3), 87–96 (1985).
[Crossref]

Young, A. T.

D. Dravins, L. Lindgren, E. Mezey, and A. T. Young, “Atmospheric Intensity Scintillation of Stars, III. Effects for Different Telescope Apertures,” Publ. Astron. Soc. Pac. 110(747), 610–633 (1998).
[Crossref]

Yu, L.

H. Shen, L. Yu, and C. Fan, “Temporal spectrum of atmospheric scintillation and the effects of aperture averaging and time averaging,” Opt. Commun. 330, 160–164 (2014).
[Crossref]

Yura, H. T.

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

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

Journal of the Institution of Electronic and Radio Engineers (1)

E. Vilar, J. Haddon, P. Lo, and T. J. Moulsley, “Measurements and Modeling of Amplitude and Phase Scintillations in an Earth-Space path,” Journal of the Institution of Electronic and Radio Engineers 55(3), 87–96 (1985).
[Crossref]

Opt. Commun. (1)

H. Shen, L. Yu, and C. Fan, “Temporal spectrum of atmospheric scintillation and the effects of aperture averaging and time averaging,” Opt. Commun. 330, 160–164 (2014).
[Crossref]

Publ. Astron. Soc. Pac. (1)

D. Dravins, L. Lindgren, E. Mezey, and A. T. Young, “Atmospheric Intensity Scintillation of Stars, III. Effects for Different Telescope Apertures,” Publ. Astron. Soc. Pac. 110(747), 610–633 (1998).
[Crossref]

Waves Random Media (1)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and temporal spectrum,” Waves Random Media 10(1), 53–70 (2000).
[Crossref]

Other (7)

A. D. Wheelon, Electromagnetic Scintillation, II. Weak Scattering, University Press, Cambridge, Secs. 2.2.3 and 4.1.4, 2003.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, U. S. Department of Commerce, Springfield, Virginia, 1971, Sec 53.

S. Wolfram, Mathematica, (Cambridge University Press, 2012), Version 9.

The Infrared & Electro-Optical Handbook Vol, 3 Atmospheric Propagation of Radiation, F. G. Smith, Editor, Chapter. 2, SPIE Optical Engineering Press, Bellingham WA, 1993. Note, the HV-5/7 model is also known as the HV-21 model.

A. D. Wheelon and I. Electromagnetic Scintillation, Geometrical Optics (University Press, Cambridge, 2001), Sec. 6.1.2.

Realistic to the extent of knowing the correct index-structure constant profile.

P. Beckmann, “Probability in Communication Engineering”, Harcourt Brace & World, Inc., 1967, Sec. 6.7. ”

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Figures (12)

Fig. 1
Fig. 1 Comparison of the exact and approximate aperture-averaging factor for plane waves and constant turbulence conditions.
Fig. 2
Fig. 2 (a) Comparison of the exact and approximate aperture-averaging factor as a function of aperture diameter for downlink propagation at 1064 nm., an elevation angle of 30°, and an Maui Day turbulence profile (https://www.amostech.com/TechnicalPapers/2010/Posters/Bradford.pdf) (b) Comparison of the exact and approximate aperture-averaging factor as a function of aperture diameter for downlink propagation at 1064 nm., an elevation angle of 45°, and an Maui Night turbulence profile. (c). Comparison of the exact and approximate aperture-averaging factor as a function of aperture diameter for downlink propagation at 1064 nm., an elevation angle of 60°, and the Hufnagle-Valley 5/7 turbulence profile. (d). Comparison of the exact and approximate aperture-averaging factor as a function of aperture diameter for downlink propagation at 1550 nm, an elevation angle of 15°, and the Clear1 turbulence profile [12]. In this example the effective wind speed is assumed due to a high-altitude aircraft flying at a speed of 200 m/s normal to the optical line of sight.
Fig. 3
Fig. 3 The temporal auto correlation function as function of normalized time delay plotted for η~unity.
Fig. 4
Fig. 4 The temporal auto correlation function as function of normalized time delay plotted for η>>1.
Fig. 5
Fig. 5 Temporal correlation time: a comparison of the numerically exact obtained from Eq. (3.2) to the analytic approximation Eq. (3.5).
Fig. 6
Fig. 6 (a) Comparison of the exact numerical and engineering approximation to the temporal autocorrelation, obtained from Eq. (3.1) and Eq. (3.2), respectively for the conditions. (b). Comparison of the exact numerical and engineering approximation to the temporal autocorrelation, obtained from Eq. (3.1) and Eq. (3.2), respectively for the conditions (c). Comparison of the exact numerical and engineering approximation to the temporal autocorrelation, obtained from Eq. (3.1) and Eq. (3.2), respectively for the conditions
Fig. 7
Fig. 7 (a) Numerical evaluation of W(ω) as a function of ω/ ω F for various values of η=a k/L (b) Numerical evaluation of ωW(ω) as a function of ω/ ω F for various values of η=a k/L .
Fig. 8
Fig. 8 The normalized aperture-averaged PSD plotted as a function of normalized frequency for η>>1.
Fig. 9
Fig. 9 (a) Comparison of the exact numerical and engineering approximation to the PSD, obtained from Eq. (4.2) and Eq. (4.3), respectively for the conditions indicated. (b). Comparison of the exact numerical and engineering approximation to the PSD obtained from Eq. (4.2) and Eq. (4.3), respectively for the conditions indicated. (c). Comparison of the exact numerical and engineering approximation to the PSD obtained from Eq. (4.2) and Eq. (4.3), respectively for the conditions indicated.
Fig. 10
Fig. 10 A comparison of the exact and approximate reduced quasi frequency as a function of the normalized aperture radius.
Fig. 11
Fig. 11 The exact and analytic approximation quasi frequency obtained from Eq. (5.2) and Eq. (5.3), respectively plotted as a function of elevation angle for the conditions indicated.
Fig. 12
Fig. 12 (a) The level crossing rate as a function of the fade level for and elevation angle of 15° and the conditions indicated in Fig. 10. (b) The mean duration of a fade as a function of the fade level for an elevation angle of 15 degrees and the conditions indicated in Fig. 10.

Tables (1)

Tables Icon

Table 1 A comparison of the measured scintillation bandwidth reported in [8] to that predicted by Eq. (4.8). For an elevation angle of 45°and the Maui night time profile L eff = 20.6 km, and an rms wind speed of 16.6 m/s were used to obtain a frequency match for a telescope diameter of 2.5 cm. For the aperture diameters shown above the scintillation index is less than about 0.15, which is within the weak scintillation regime.

Equations (45)

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SI= P 2 P 2 P 2 ,
SI=exp[ σ lnI 2 ]1 σ lnI 2 ,for σ lnI 2 <<1,
SI(a) σ lnI 2 (a)=16 π 2 k 2 0 L ds 0 dK K sin 2 [ K 2 s 2k ] Φ n (K,s) ( 2 J 1 (Ka) Ka ) 2 ,
h(s)= ( R E +H) 2 +2s( R E +H)sinθ+ s 2 R E ,
Φ n (K,s)= 0.033 C n 2 [h(s)]) K 11/3 ,
SI(a)=2.25 k 7/6 0 L ds C n 2 [h(s)] s 5/6 G(s,a) =AA(a)SI(0),
G(s,a)= 0.4442 k 5/6 s 5/6 ( 2.251 (s/k) 5/6 4 F 5 ( 5 12 , 1 12 , 3 4 , 5 4 ; 1 2 ,1, 3 2 , 3 2 ,2; a 4 k 2 4 s 2 ) +1.750 (k/s) 1/6 a 2 4 F 5 ( 1 12 , 7 12 , 5 4 , 7 4 ;2,2, 5 2 ,2; a 4 k 2 4 s 2 )2.774 a 5/3 )
SI(0)=2.25 k 7/6 0 L ds C n 2 [h(s)] s 5/6
AA(a)= 0 L ds C n 2 [h(s)] s 5/6 G(s,a) 0 L ds C n 2 [h(s)] s 5/6 ,
G(s,a) 1 1+0.754 ( a 2 k/s) 7/6
AA(η)= 4.24 η 2 ( 1/(192 2 ) η 2 Γ(11/6)( 24 3 1 ) 4 F 5 ( 11 12 , 5 12 , 3 4 , 5 4 ; 1 2 ,1, 3 2 , 3 2 ,2; η 4 4 ) +11(1+ 3 ) η 2 4 F 5 ( 5 12 , 1 12 , 5 4 , 7 4 ; 3 2 , 3 2 ,2,2, 5 2 ; η 4 4 )+ η 11/3 Γ(5/6)Γ(7/3) π Γ(17/6)Γ(23/6) )
η=a k L
AA(η) 1 (1+1.07 η 2 ) 7/6 ,
L eff = ( 18 0 L ds C n 2 [h(s)] s 2 11 0 L ds C n 2 [h(s)] s 5/6 ) 6/7
μ(τ,a)= 0 L ds C n 2 [h(s)] 0 dK sin 2 ( K 2 s/2k)Φn(K) ( 2 J 1 (Ka) Ka ) 2 J 0 (KV(s)τ) 0 L ds C n 2 [h(s)] 0 dK sin 2 ( K 2 s/2k)Φn(K) ( 2 J 1 (Ka) Ka ) 2 ,
μ(τ;η)= 1 N(η) 0 dx(1sinc x 2 ) ( 2 J 1 (xη) xη ) 2 J 0 (x τ n )/ x 8/3 ,
μ(τ;0)= 11 24 (2+ 3 ) τ n 2 2 F 3 ( 5 12 , 1 12 ;1, 3 2 , 3 2 ; τ n 4 64 )+ 2 F 3 ( 11 12 , 5 12 ; 1 2 , 1 2 ,1; τ n 4 64 ) 121(2+ 3 )Γ(11/6) τ n 5/3 144π 2 1/6
μ(τ;a)= 2 4/3 Γ (7/6) 3 F 2 ( 7 6 , 7 6 , 3 2 ;2,3; 4 τ n 2 ) Γ(1/6) τ n 7/3
τ C (η)= τ C (0) 1+ η 2 L/k /V,forη<<1 a/V,forη>>1
V rms = 0 L ds C n 2 [h(s)] s 5/6 G(s,a) V 2 [h(s)] 0 L ds C n 2 [h(s)] s 5/6 G(s,a) .
τ C (a)= L eff /k V rme 1+ η eff 2
η eff =a k/ L eff
V(h)=5+30exp[ ( h9400 4800 ) 2 ],
W(ω;a)=SI(a) W n (ω;a),
W n (ω;a)= 0 L ds C n 2 [h(s)] V 1 (s) 0 dK sin 2 [ s 2k ( ω 2 / V 2 (s)+ K 2 ) ] ( ω 2 / V 2 (s)+ K 2 ) 11/6 ( 2 J 1 ( ( ω 2 / V 2 (s)+ K 2 )a ) ( ω 2 / V 2 (s)+ K 2 ) ) 2 0 L ds C n 2 [h(s)] s 5/6 G(s,a)
W n (ω;a)= 1 N(η) 0 dq   1sinc[ q 2 + (ω/ ω F ) 2 ] ω F [ q 2 + (ω/ ω F ) 2 ] 11/6 ( 2 J 1 [ q 2 + (ω/ ω F ) 2 η ] q 2 + (ω/ ω F ) 2 η ) 2
ω F =V k L
f F (η)= f F (0) 1+ η 2 ,
W n (ω,0)= 2.577 ω n 8/3 ( 1 16π Γ( 17 6 )Im( ei ω n 2 ( 1 F 1 ( 1 2 ; 4 3 ;i ω n 2 ) Γ( 4 3 )Γ( 17 6 ) ( i ω n 2 ) 7/3 1 F 1 ( 17 6 ; 10 3 ;i ω n 2 ) π Γ( 10 3 ) ) ) 11 3 Γ( 7 3 ) ω n 2 ),
W n (ω,a)= 1 2 π ( π Γ (n) 2 F 3 ( 5 6 , 3 2 ; 1 3 ,2,3; ω n 2 ) Γ( 11 6 ) ω n 8/3 + 4Γ( 4 3 )Γ ( 17 6 ) 2 F 3 ( 1 2 , 17 6 ; 7 3 , 10 3 , 13 3 ; ω n 2 ) Γ( 10 3 )Γ( 13 3 ) ),
ω F ( η eff )= ω F (0) 1+ η eff 2 ω F (0),for η eff <<1 V/a,for η eff >>1,
ω Q = 0 dω ω 2 W(ω) 0 dωW(ω) ,
ω Q = B ¨ (0,a) B(0,a) ,
ω Q (a)= 0 L ds V 2 (s) C n 2 [h(s)] s 5/6  Q(s,a) 0 L ds C n 2 [h(s)]  s 5/6 G(s,a) ,
ω Q ap (a)= V rms L eff /k Θ( η eff ) Δ( η eff ) ,
Θ(η)= π η 2 ( 1 384 2 η 2 Γ( 5 6 )( 5( 3 1 ) η 2 4 F 5 ( 1 12 , 7 12 , 5 4 , 7 4 ; 3 2 , 3 2 ,2,2, 5 2 ; η 2 4 )24 ( 1+ 2 ) 4 F 5 ( 5 12 , 1 12 , 3 4 , 5 4 ; 1 2 ,1, 3 2 , 3 2 ,2; η 2 4 ) ) η 5/3 Γ( 1 3 )Γ( 7 6 ) π Γ( 11 6 )Γ( 17 6 )
Δ(η)= 2π η 2 ( 1 384 2 η 2 Γ( 11 6 )( 24 ( 3 1 ) 4 F 5 ( 11 12 , 7 12 , 3 4 , 5 4 ; 1 2 ,1, 3 2 , 3 2 ,2; η 4 4 )+11( 1+ 3 ) η 2 4 F 5 ( 5 12 , 1 12 , 5 4 , 7 4 ; 3 2 , 3 2 ,2,2 5 2 ; η 2 4 ) ) η 11/3 Γ( 5 6 )Γ( 7 3 ) 2 π Γ( 17 6 )Γ( 23 6 )
Θ( η eff ) Δ( η eff ) 1.30 η eff 0.228 exp[0.879 η eff ]+0.772 η eff 0.358 ( 1exp[0.879 η eff ] )
0 dωW(ω) 0 drr 0 2π dϕ r 11/3 sin 2 ( r 2 s/2k) (2 J 1 (ra)/ra) 2
0 d ωW(ω)= 0 ds C n 2 [h(s)] s 5/6 G(s,a) ,
0 dω ω 2 W(ω) 1 4 0 drr 0 2π dϕ V 2 (s) r 2 cos 2 ϕ r 11/3 sin 2 ( r 2 s/2k) (2 J 1 (ra)/ra) 2 .
0 dω ω 2 W(ω;a)= 0 L ds C n 2 [h(s)] V 2 (s) s 5/6 Q(s;a),
Q(s,a)= 1 16896 a 17/6 s 5/6 π Γ( 1 6 ) ( 11 2 ( 3 1 ) a 4 4 F 5 ( 7 12 , 13 12 , 5 4 , 7 4 , 3 2 , 3 2 ,2,2, 5 2 ; a 4 k 2 4 s 2 ) ( s 2 k 2 ) 7/12 + 264 2 ( 1+ 3 ) a 4 4 F 5 ( 1 12 , 7 12 , 3 4 , 5 4 ; 1 2 ,1, 3 2 , 3 2 ,2; a 4 k 2 4 s 2 ) 12 s 2 k 2 + 2304 a 5/3 Γ( 4 3 ) π Γ ( 11 6 ) 2 )
N(η)= π 2 0 dr r 1sinc( r 2 ) r 11/3 [ 2 J 1 (rη)/rη ] 2 ,
N(η)= 1 Γ( 23 6 ) ( 7 576 π 2 ( 12 2 ( 3 1 ) 4 F 5 ( 11 12 , 5 12 , 3 4 , 1 4 ; 1 2 ,1, 3 2 , 3 2 ,2; η 4 4 )+ 11 2 3 η 5 4 F 5 ( 5 12 , 1 12 , 5 4 , 7 4 ; 3 2 , 3 2 ,2,2; 5 2 ; η 4 4 )+ π η 5/3 Γ( 5 6 )Γ( 7 3 ) Γ( 17 6 ) )

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