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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 20 — Sep. 26, 2011
  • pp: 19067–19077
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Scintillation index and performance analysis of wireless optical links over non-Kolmogorov weak turbulence based on generalized atmospheric spectral model

Ji Cang and Xu Liu  »View Author Affiliations


Optics Express, Vol. 19, Issue 20, pp. 19067-19077 (2011)
http://dx.doi.org/10.1364/OE.19.019067


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Abstract

Based on the generalized spectral model for non-Kolmogorov atmospheric turbulence, analytic expressions of the scintillation index (SI) are derived for plane, spherical optical waves and a partially coherent Gaussian beam propagating through non-Kolmogorov turbulence horizontally in the weak fluctuation regime. The new expressions relate the SI to the finite turbulence inner and outer scales, spatial coherence of the source and spectral power-law and then used to analyze the effects of atmospheric condition and link length on the performance of wireless optical communication links.

© 2011 OSA

1. Introduction

2. Generalized atmospheric spectral model

Φn(κ,α)=A(α)C˜n2κα
(4)

3. Variance of irradiance fluctuations

In the weak fluctuation regime, where the log-amplitude variance satisfiesσχ21, the normalized irradiance variance has the relation σI2σlnI2=4σχ2.

Following the approach in Ref [5

5. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005).

], the irradiance variance at the receiver plane can be expressed as the sum:

σI2(ρ,L)=σI,l2(L)+σI,r2(ρ,L)
(5)
σI,l2(L)=8π2k2L010κΦn(κ,α,l0,L0)exp(ΛedLκ2ξ2/k)×{1cos[Lκ2kξ(1Θ¯edξ)]}dκdξ
(6)
σI,r2(ρ,L)=8π2k2L010κΦn(κ,α,l0,L0)exp(ΛedLκ2ξ2/k)×[I0(2Λedρξκ)1]dκdξ
(7)

In the above formulae, Θed=Θ11+4qcΛ1 and Λed=Λ1Ns1+4qcΛ1 are the effective beam parameters at the receiver plane. Θ¯ed=1Θed and qc=Lklc2 is a nondimensional coherence parameter, where l c is the spatial coherence length of the source. The values Θ1 and Λ1 are the curvature parameter and Fresnel ratio at the receiver plane for vacuum propagation, which given in terms of their respective values at the source plane are Θ1=Θ0Θ02+Λ02 and Λ1=Λ0Θ02+Λ02. The source plane parameters are Θ0=1LF0 and Λ0=2LkW02, where L is the propagation distance, F 0 is the initial radius of curvature, W 0 is the initial beam waist, and k = 2π/λ is the wavenumber. Ns=1+4qcΛ0 is the number of speckle cells; for fully coherent beam, Θed=Θ1 and Λed=Λ1.

In the above formulae, the radial componentσI,r2(ρ,L) represents the off-axis contribution to the total irradiance variance and disappears at the beam centerline (ρ=0) or when Λ=0, while the longitudinal component σI,l2(L) is invariant throughout the beam cross section in any transverse plane.

For horizontal path, C˜n2 is constant and substituting Eq. (1) into Eq. (6), we can obtain
σI,l2(L)=g(E1,H1)+a1κlg(E2,H1)b1κlβg(E3,H1)g(E1,H2)a1κlg(E2,H2)+b1κlβg(E3,H2)
(8)
where

g(Ei,Hj)=4π2k2LA(α)C˜n2Γ(Ei2+1)(Lk)Ei21HjEi2+1{F21(Ei2+1,12;32;ΛedHj)Re[n=0(Ei/2+1)n(1)nn!(2)n(iHj)nF21(n,n+1;n+2;Θ¯ed+iΛed)]},Hj<1
(9)

With the application of the following two formulae [25

25. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994). [CrossRef]

], F21(n,n+1;n+2;x)(12x/3)n,|x|<1; F21(1b,1;2;x)=(1+x)b1bx the approximate analytic expression can be derived by [25

25. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994). [CrossRef]

]

g(Ei,Hj)4π2k2LA(α)C˜n2Γ(Ei2+1)(Lk)Ei21HjEi2+1×{F21(Ei2+1,12;32;ΛedHj)Re[1+Hj(2Λed+i+2Θedi)/3]Ei21Ei2Hj[2Λed+i(1+2Θed)]/3},|Θ¯ed+iΛed|<1
(10)

The above expression is valid for all H j. where E1=α,E2=α1,E3=αβ, H1=Ql,H2=QlQ0Q0+Ql; and Ql=Lkκl2, Q0=Lkκ02; (a)n=Γ(a+n)/Γ(a) is Pochhammer symbol, Γ() is the Gamma function and F21(a,b;c;x) denotes the hypergeometric function [26

26. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, 1998).

].

Substituting Eq. (1) into Eq. (7), and expanding I 0 in the form of Maclaurin series, we can obtain
σI,r2(L)=f(E1,H1)+a1κlf(E2,H1)b1κlβf(E3,H1)f(E1,H2)a1κlf(E2,H2)+b1κlβf(E3,H2)
(11)
where

f(Ei,Hj)=4π2k2LA(α)C˜n2Γ(Ei2+1)(Lk)Ei21Hj1Ei2n=1(Ei/2+1)n(1/2)nn!(1)n(3/2)n×(2ρ2Wed2)n(ΛedHj)nF21(n+1Ei2,n+12;n+32;ΛedHj)4π2k2LA(α)C˜n2Γ(Ei2+1)(Lk)Ei21Hj1Ei22Ei3×(ρ2Wed2)(ΛedHj)F21(2Ei2,32;52;ΛedHj),ρ/Wed<1.
(12)

It should be mentioned that there is generally good agreement between approximate analytical expressions (10) and (12) and exact results for collimated beams [25

25. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994). [CrossRef]

].

For the case of plane wave, Θ=1 and Λ=0, we have
σI,pl2(L,α,l0,L0)=8π2k2L010κΦn(κ,α,l0,L0)[1cos(Lκ2kξ)]dκdξ=hpl(E1,H1)+a1κlhpl(E2,H1)b1κlβhpl(E3,H1)hpl(E1,H2)a1κlhpl(E2,H2)+b1κlβhpl(E3,H2)
(13)
hpl(Ei,Hj)=8π2A(α)1.23EiΓ(1Ei2)σ˜R2(Ei)[(1+1Hj2)Ei/4sin(Ei2tan1Hj)Ei2HjEi2+1]
(14)
where σ˜R2(Ei)=1.23C˜n2k3Ei2LEi2.

For the case of spherical wave, Θ=Λ=0, we have

σI,sp2(L,α,l0,L0)=8π2k2L010κΦn(κ,α,l0,L0)[1cos(Lκ2kξ(1ξ))]dκdξ=hsp(E1,H1)+a1κlhsp(E2,H1)b1κlβhsp(E3,H1)hsp(E1,H2)a1κlhsp(E2,H2)+b1κlβhsp(E3,H2)
(15)
hsp(Ei,Hj)=4π2A(α)1.23σ˜R2(Ei)Γ(Ei2+1)HjEi2+1×Re[F21(Ei2+1,1;32;iHj4)1]=4π2A(α)1.23σ˜R2(Ei)Γ(Ei2+1)HjEi2+1×[F32(1,2Ei4,4Ei4;34,54;Hj216)1]
(16)

4. Optical communication link performance statistics

A point-to-point FSO communication system using intensity modulation/direct detection (IM/DD) scheme is considered [2

2. H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009). [CrossRef]

]. The partially coherent laser beam propagates horizontally through a non-Kolmogorov turbulence channel with additive white Gaussian noise (AWGN). The channel is assumed to be memoryless, stationary and ergodic, with independent and identically distributed intensity fading statistics. We also consider that the channel state information (CSI) is available at both the transmitter and the receiver. In this case, the statistical channel model is given by

y=ηIx+n where ηI is the instantaneous intensity gain, η is the effective photo-current conversion ratio at the receiver, I is the irradiance, x is the modulation signal taking values 0 or 1 and n is the AWGN with zero mean and variance N 0/2. For weak-to-moderate atmospheric fluctuation conditions, the turbulence-induced fading is assumed to be a random process following the log-normal distribution.

4.1 Outage probability

The probability density function (PDF) of the log-normal model is given by [5

5. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005).

]

pI(I)=1IσI2πexp{[ln(I)+σI2/2]22σI2},I>0
(17a)

By defining the instantaneous electrical signal-to-noise ratio (SNR) as μ=(ηI)2/N 0, the average electrical SNR will be given by μ¯=η(E[I])2/N0, where E[] denotes the expectation. Then, considering that E[I]=1 since I is normalized to unity, and after a power transformation of the RV I in above model, the electrical SNR PDF can be rewritten as:

pμ(μ)=12μσI2πexp{[ln(μ/μ¯)+σI2]28σI2},I>0
(17b)

The outage probability represents the probability that the instantaneous SNR falls below a critical threshold μ th, which corresponds to sensitivity limit of the receiver. Thus, the outage probability for weak-to-moderate fluctuation regime is obtained from the log-normal distribution model, given by [2

2. H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009). [CrossRef]

]
Pout=Pr(μμth)=0μthpμ(μ)dμ=12erfc[ln(μ¯/μth)σI222σI]
(18)
where erfc(x) is the complementary error function.

4.2 Mean Bit-Error-Rate (BER) performance

In the presence of optical turbulence, the probability of error is regarded as a conditional probability averaged over the PDF of the random signal to determine the mean BER, i.e [5

5. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005).

].

BER=120pI(I)erfc(ISNR22I)dI=120pμ(μ)erfc(uSNR22)du
(19)

4.3 Average channel capacity

The average (ergodic) capacity denotes the practically achievable capacity of an FSO channel with atmospheric turbulence-induced fading and is a paramount metric for evaluating the linkperformance [2

2. H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009). [CrossRef]

]. The average achievable capacity can be defined as
C=0Blog2(1+(ηI)2N0)pI(I)dI
(20)
where B is the signal transmission bandwidth.

Substituting Eq. (13) into Eq. (16), it yields [2

2. H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009). [CrossRef]

]
C=B2σI2πln(2)0ln(1+μ)μexp((lnμA)28σI2)dμ=Bexp(A2/8σI2)2ln(2)m=1(1)m+1m[erfcx(2σIm+A22σI)+erfcx(2σImA22σI)]+Bexp(A2/8σI2)2ln(2)[4σI2π+Aexp(A28σI2)erfc(A22σI)]
(21)
where A=ln(μ¯)σI2 and erfcx(x)=exp(x2)erfc(x).

5. Numerical results and discussion

Figures 3
Fig. 3 Outage probability as a function of normalized average SNR for different spectral indices.
and 4
Fig. 4 Outage probability as a function of normalized average SNR for different spectral indices and inner scales.
illustrate the impact of the variations of spectral index and inner scale on the outage probability, respectively. It can be seen from Fig. 3 that, when α=10/3, the outage probability is maximal; while the outage probability is minimal when α=3.9. From Fig. 4, we can deduce that the outage probability increases slightly with the increment of inner scale and the effects of inner scale become a little evident for larger normalized average SNR. The effects of outer scale on scintillations within the beam cross-section are negligible.

Figures 5
Fig. 5 Mean BER as a function of SNR for different alpha values.
and 6
Fig. 6 Mean BER as a function of SNR for different spectral indices and inner scale values.
illustrate the influence of the variations of spectral index and inner scale on the mean BER, respectively. It can be seen from Fig. 5 that, when α=10/3, the BER is maximal; while the BER is minimal when α=3.9. From Fig. 6, we can deduce that the BER increases slightly with the increment of inner scale and the effects of inner scale become a little evident for larger SNR. The effect of outer scale on the BER is negligible. Following Fig. 2, larger inner scale leads to higher scintillation in weak fluctuation regime under the condition of l0<λL, therefore, larger inner scale leads to higher outage probability and BER.

In addition, different spectral indices have slight influence on average channel capacity, especially for larger SNR. The effect of inner scale on the average channel capacity is negligible in weak fluctuation regime. The explanation is that inner scale has weak influence on scintillations in weak fluctuation regime and average channel capacity is insensitive to little variation of scintillations in weak fluctuation regime.

6. Conclusions

References and links

1.

A. K. Majumdar, “Free-space laser communication performance in the atmospheric channel,” J. Opt. Fiber Commun. Res. 2(4), 345–396 (2005). [CrossRef]

2.

H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009). [CrossRef]

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,” J. Lightwave Technol. 27(8), 974–979 (2009). [CrossRef]

4.

L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002). [CrossRef]

5.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005).

6.

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]

7.

C. Y. Chen, H. M. Yang, X. Feng, and H. Wang, “Optimization criterion for initial coherence degree of lasers in free-space optical links through atmospheric turbulence,” Opt. Lett. 34(4), 419–421 (2009). [CrossRef] [PubMed]

8.

D. K. Borah and D. G. Voelz, “Spatially partially coherent beam parameter optimization for free space optical communications,” Opt. Express 18(20), 20746–20758 (2010). [CrossRef] [PubMed]

9.

G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010). [CrossRef] [PubMed]

10.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Trans. NOAA by Israel Program for Scientific Translations, 1971).

11.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994). [CrossRef]

12.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995). [CrossRef]

13.

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997). [CrossRef]

14.

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]

15.

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010). [CrossRef]

16.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007). [CrossRef]

17.

L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010). [CrossRef] [PubMed]

18.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6457, 64570T (2007). [CrossRef]

19.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007). [CrossRef]

20.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009). [CrossRef]

21.

L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(2), 451–462 (2010). [CrossRef] [PubMed]

22.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283(7), 1229–1235 (2010). [CrossRef]

23.

B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Theoretical expressions of the angle-of-arrival variance for optical waves propagating through non-Kolmogorov turbulence,” Opt. Express 19(9), 8433–8443 (2011). [CrossRef] [PubMed]

24.

B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(5), 912–916 (2011). [CrossRef] [PubMed]

25.

W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994). [CrossRef]

26.

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, 1998).

OCIS Codes
(010.1290) Atmospheric and oceanic optics : Atmospheric optics
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(060.2605) Fiber optics and optical communications : Free-space optical communication

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: June 30, 2011
Revised Manuscript: August 12, 2011
Manuscript Accepted: September 2, 2011
Published: September 15, 2011

Citation
Ji Cang and Xu Liu, "Scintillation index and performance analysis of wireless optical links over non-Kolmogorov weak turbulence based on generalized atmospheric spectral model," Opt. Express 19, 19067-19077 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19067


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References

  1. A. K. Majumdar, “Free-space laser communication performance in the atmospheric channel,” J. Opt. Fiber Commun. Res.2(4), 345–396 (2005). [CrossRef]
  2. H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun.3(8), 1402–1409 (2009). [CrossRef]
  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,” J. Lightwave Technol.27(8), 974–979 (2009). [CrossRef]
  4. L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE4489, 23–34 (2002). [CrossRef]
  5. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005).
  6. 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]
  7. C. Y. Chen, H. M. Yang, X. Feng, and H. Wang, “Optimization criterion for initial coherence degree of lasers in free-space optical links through atmospheric turbulence,” Opt. Lett.34(4), 419–421 (2009). [CrossRef] [PubMed]
  8. D. K. Borah and D. G. Voelz, “Spatially partially coherent beam parameter optimization for free space optical communications,” Opt. Express18(20), 20746–20758 (2010). [CrossRef] [PubMed]
  9. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett.35(5), 715–717 (2010). [CrossRef] [PubMed]
  10. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Trans. NOAA by Israel Program for Scientific Translations, 1971).
  11. D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE2120, 43–55 (1994). [CrossRef]
  12. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE2471, 181–196 (1995). [CrossRef]
  13. M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE3126, 113–123 (1997). [CrossRef]
  14. 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]
  15. N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE7588, 758808 (2010). [CrossRef]
  16. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE6551, 65510E (2007). [CrossRef]
  17. L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express18(20), 21269–21283 (2010). [CrossRef] [PubMed]
  18. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE6457, 64570T (2007). [CrossRef]
  19. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE6747, 67470B (2007). [CrossRef]
  20. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag.57(6), 1783–1788 (2009). [CrossRef]
  21. L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express18(2), 451–462 (2010). [CrossRef] [PubMed]
  22. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun.283(7), 1229–1235 (2010). [CrossRef]
  23. B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Theoretical expressions of the angle-of-arrival variance for optical waves propagating through non-Kolmogorov turbulence,” Opt. Express19(9), 8433–8443 (2011). [CrossRef] [PubMed]
  24. B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A28(5), 912–916 (2011). [CrossRef] [PubMed]
  25. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A11(10), 2719–2726 (1994). [CrossRef]
  26. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, 1998).

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