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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 13 — Jul. 1, 2013
  • pp: 15213–15229
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Capacity of MIMO free space optical communications using multiple partially coherent beams propagation through non-Kolmogorov strong turbulence

Peng Deng, Mohsen Kavehrad, Zhiwen Liu, Zhou Zhou, and XiuHua Yuan  »View Author Affiliations


Optics Express, Vol. 21, Issue 13, pp. 15213-15229 (2013)
http://dx.doi.org/10.1364/OE.21.015213


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Abstract

We study the average capacity performance for multiple-input multiple-output (MIMO) free-space optical (FSO) communication systems using multiple partially coherent beams propagating through non-Kolmogorov strong turbulence, assuming equal gain combining diversity configuration and the sum of multiple gamma-gamma random variables for multiple independent partially coherent beams. The closed-form expressions of scintillation and average capacity are derived and then used to analyze the dependence on the number of independent diversity branches, power law α, refractive-index structure parameter, propagation distance and spatial coherence length of source beams. Obtained results show that, the average capacity increases more significantly with the increase in the rank of MIMO channel matrix compared with the diversity order. The effect of the diversity order on the average capacity is independent of the power law, turbulence strength parameter and spatial coherence length, whereas these effects on average capacity are gradually mitigated as the diversity order increases. The average capacity increases and saturates with the decreasing spatial coherence length, at rates depending on the diversity order, power law and turbulence strength. There exist optimal values of the spatial coherence length and diversity configuration for maximizing the average capacity of MIMO FSO links over a variety of atmospheric turbulence conditions.

© 2013 OSA

1. Introduction

The intensity fluctuations or scintillation at the receiver reduces FSOC channel capacity. In order to improve system performance, scintillation can be mitigated by means of reducing the spatial coherence of the transmitted beam [8

8. 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(9), 1794–1802 (2002). [CrossRef] [PubMed]

,9

9. 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]

]and the spatial diversity using multiple transmitted beams and multiple receivers [10

10. A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using diversity-combining techniques,” Opt. Express 17(15), 12601–12611 (2009). [CrossRef] [PubMed]

,11

11. E. Bayaki, R. Schober, and R. K. Mallik, “Performance Analysis of MIMO Free-Space Optical Systems in Gamma-Gamma Fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009). [CrossRef]

].

Partially coherent beams with reduced spatial coherence show lower scintillation at the cost of larger divergence angle and lower average received power [12

12. Y. Baykal, H. T. Eyyuboğlu, and Y. J. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Appl. Opt. 48(10), 1943–1954 (2009). [CrossRef] [PubMed]

,13

13. J. Cang and X. Liu, “Average capacity of free-space optical systems for a partially coherent beam propagating through non-Kolmogorov turbulence,” Opt. Lett. 36(17), 3335–3337 (2011). [CrossRef] [PubMed]

].Partially coherent beams have a lower scintillation than fully coherent beams. However, a partially coherent beam has a larger beam spreading and forms a large spot in the receiver aperture, which leads to a loss of the transmitted energy being received by the detector. By optimizing the spatial coherence length, the improvement in scintillation reduction can overcome the penalty of power reduction and significant signal-to-noise ratio gains can be obtained in weak atmospheric turbulence [14

14. G. P. Berman, A. R. Bishop, B. M. Chernobrod, V. N. Gorshkov, D. C. Lizon, D. I. Moody, D. C. Nguyen, and S. V. Torous, “Reduction of laser intensity scintillations in turbulent atmospheres using time averaging of a partially coherent beam,” J. Phys. B 42(22), 225403 (2009). [CrossRef]

,15

15. 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]

].

Spatial diversity using multiple transmitted beams and multiple receivers can also be employed to reduce scintillation and ultimately improve FSO channel capacity. It has been shown that the scintillation of a beam array can be reduced by carefully adjusting the spatial separation of beamlets [16

16. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express 19(14), 13480–13496 (2011). [CrossRef] [PubMed]

]. However, scintillation of a beam array will increase significantly if the spatial separation of beamlets is smaller than the correlated length. In addition, the received energy from the beam arrays is low unless the constituent beamlets are inclined to overlap at the receiver aperture, which is difficult to achieve over long propagation distances [17

17. K. P. Peppas, F. Lazarakis, A. Alexandridis, and K. Dangakis, “Simple, accurate formula for the average bit error probability of multiple-input multiple-output free-space optical links over negative exponential turbulence channels,” Opt. Lett. 37(15), 3243–3245 (2012). [CrossRef] [PubMed]

].The use of multiple transmitters and receivers has also been suggested for use in multiple-input–multiple-output (MIMO) configurations [16

16. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express 19(14), 13480–13496 (2011). [CrossRef] [PubMed]

,18

18. X. Yi, Z. Liu, and P. Yue, “Formula for the average bit error rate of free-space optical systems with dual-branch equal-gain combining over gamma-gamma turbulence channels,” Opt. Lett. 38(2), 208–210 (2013). [CrossRef] [PubMed]

].

2. System and channel model

Consider a MIMO FSO communication system [1

1. S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Comm. 6(8), 2813–2819 (2007). [CrossRef]

,11

11. E. Bayaki, R. Schober, and R. K. Mallik, “Performance Analysis of MIMO Free-Space Optical Systems in Gamma-Gamma Fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009). [CrossRef]

,23

23. G. Yun and M. Kavehrad, “Spot-diffusing and fly-eye receivers for indoor infrared wireless communications,” in Proceedings of IEEE International Conference on Selected Topics in Wireless Communications(IEEE, 1992), 262–265. [CrossRef]

] where the information signal is transmitted via Mapertures and received by N apertures [24

24. Z. Hajjarian and M. Kavehrad, “Using MIMO Transmissions in Free Space Optical Communications in Presence of Clouds and Turbulence,” Proc. SPIE 7199, 71990V, 71990V-12 (2009). [CrossRef]

]over Non-Kolmogorov strong atmospheric turbulence. It is assumed that the aperture diameter of each receiver in the array is less than the spatial correlation width of the irradiance fluctuations so that each receiver acts like a point detector [12

12. Y. Baykal, H. T. Eyyuboğlu, and Y. J. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Appl. Opt. 48(10), 1943–1954 (2009). [CrossRef] [PubMed]

]. Moreover, the array elements are spatially separated by a sufficient distance so that each acts independently of the others. Furthermore, a large field of view is considered for each receiver [25

25. S. Jivkova and M. Kavehrad, “Transceiver design concept for cellular and multispot diffusing regimes of transmission,” Eurasip J Wirel Comm 2005, 30–38 (2005).

] indicating that multiple transmitters [26

26. J. M. Kahn, R. You, P. Djahani, A. G. Weisbin, B. K. Teik, and A. Tang, “Imaging diversity receivers for high-speed infrared wireless communication,” IEEE Commun. Mag. 36(12), 88–94 (1998). [CrossRef]

] are simultaneously observed by each receiver. This actually leads to the collection of larger amount of background radiation which justifies the use of the AWGN model as a good approximation of the Poisson photon counting detection model. Assuming on-off keying (OOK), the detected signal at the receive aperture is given by [1

1. S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Comm. 6(8), 2813–2819 (2007). [CrossRef]

]
rn=xηm=1MImn+υn,n=1,...N
(1)
Where x represents the information bits, η is the optical-to-electrical conversion coefficient and υn is the AWGN with zero mean and varianceσυ2 = N0/2. The normalized irradiance, Imn, is the received irradiance normalized by its mean value. The fading channel coefficient, Imn, which models the atmospheric turbulence through the optical channel from the mth transmit aperture to the nth receive aperture [1

1. S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Comm. 6(8), 2813–2819 (2007). [CrossRef]

] is given by Imn = I0exp(2Xmn), where I0is the signal light intensity without turbulence and Xmn are identically distributed normal random variables with mean μx and varianceσx2.

Based on the extended Rytov theory with a modified spatial filter function, the large scale log-irradiance variance,σlnX2, of a partially coherent Gaussian beam propagation through non-Kolmogorov strong turbulence is [29

29. P. Deng, X. Yuan, Y. Zeng, M. Zhao, and H. Luo, “Influence of wind speed on free space optical communication performance for Gaussian beam propagation through non kolmogorov strong turbulence,” J. Phys. Conf. Ser. 276, 012056 (2011). [CrossRef]

,30

30. P. Deng, X. Yuan, and D. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun. 285(6), 880–887 (2012). [CrossRef]

]
σlnX2(L,α)=0.49σ˜B2[1+fx(α,Θ¯ed)σ˜R4α2]3α2
(5)
Where L is the propagation distance, the large scale factor fx(α,Θ¯ed)is defined by
fx(α,Θ¯ed)=(V(α,Θ¯ed)Z(α,Θ¯ed)0.98×σ˜R2σ˜B2)2α6
(6)
V(α,Θ¯ed)=8π2A(α)1.23R(α)(α2)Γ(6αα2)B(α,Θ¯ed)α6α2
(7)
B(α,Θ¯ed)=π21.2342α2Γ(1α/2)Γ(α/2)A(α)R(α)1Θ¯ed(α1)
(8)
Z(α,Θ¯ed)=01ξα4(1Θ¯edξ)2[(1Θ¯edξ)α2(1Θ¯ed)α1]6αα2dξ.
(9)
where the parameter ξ=z/L, R(α)=6.5π2A(α)Γ(1α2)1αsin(απ4), σ˜R2(α)is the non-Kolmogorov Rytov variance for plane wave in weak turbulence [31

31. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003–026009 (2008). [CrossRef]

]
σ˜R2(α)=6.5π2A(α)Γ(1α2)1αsin(απ4)×1.23C˜n2k3α2Lα2
(10)
σ˜B2(α)is the longitudinal component of scintillation index for partially coherent Gaussian-beam wave under non-Kolmogorov weak turbulence.
σ˜B2(α)=1α1A(α)C˜n2π2k3α2Lα/2Γ(2α2)×{Λedα21Re[2α2αiα2αF21(2α2,α2;2+α2;Θ¯ed+iΛed)]}
(11)
Where k is the wave number, F21 function is the confluent hyper geometric function of the second kind.

The small-scale log-irradiance fluctuation σlnY2is [30

30. P. Deng, X. Yuan, and D. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun. 285(6), 880–887 (2012). [CrossRef]

]
σlnY2(α)=0.51σ˜B2(α)(1+fY(α)σ˜R4α2)α22
(12)
where fY(α)=(ln20.51)22α.

3. Spatial diversity and combining gain

For MIMO FSO communications through atmospheric turbulence, the received optical signals from the N apertures are combined using equal gain combining (EGC).Thus, the output of the receiver is [11

11. E. Bayaki, R. Schober, and R. K. Mallik, “Performance Analysis of MIMO Free-Space Optical Systems in Gamma-Gamma Fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009). [CrossRef]

,32

32. N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the distribution of the sum of gamma-gamma variates and application in MIMO optical wireless systems,” in IEEE Global Telecommunications Conference(IEEE, 2009), 1–6. [CrossRef]

]
r=1MNn=1Nrn=xηMNn=1Nm=1MImn+υ.
(13)
where the factor M is included in order to ensure that the total transmit power is the same with that of a system with no transmit diversity, while the factor N ensures that the sum of the N receive aperture areas is the same with the aperture area of a system with no receive diversity. The instantaneous and average received electrical SNR between the mth transmitter and nth receiver aperture are umn=η2Imn2/N0andu¯mn=η2<Imn>2/N0, respectively. We define the combined signal vector I = (I11, I12, …,Imn) of the length S = M × N, and the sum of the received signalsIS=n=1Nm=1MImn.The instantaneous and average received electrical SNR of the combined signal at the output of receiver areus=η2Is2N0M2N2 and u¯s=η2<Is2>N0M2N2.

The sum of multiple independent identically distributed gamma-gamma random variables are expressed as the sum of the product of two independent gamma random variables [32

32. N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the distribution of the sum of gamma-gamma variates and application in MIMO optical wireless systems,” in IEEE Global Telecommunications Conference(IEEE, 2009), 1–6. [CrossRef]

].
IS=i=1S=M×NIi=i=1Sxiyi
(14)
Where S is also the number of independent beams. This can be expressed as the scaled product of the sum of two gamma random variables plus an error term.

IS=1S(i=1Sxi)(i=1Syi)+1S(i=1S1j=i+1S(xixj)(yiyj)
(15)

Because scintillation is caused primarily by small-scale in homogeneities in strong turbulence, we argue now that the small-scale scintillation index associated with the summed output of the array is roughly the small-scale scintillation index of a single aperture output divided by the number of the apertures. The two resulting gamma random variables for the summed output have variances ofσX2/S andσY2/S, respectively, resulting in αS = 1and𝛽S = S𝛽1. Using nonlinear regression, the error term can be closely approximated to be [32

32. N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the distribution of the sum of gamma-gamma variates and application in MIMO optical wireless systems,” in IEEE Global Telecommunications Conference(IEEE, 2009), 1–6. [CrossRef]

]

εS=(S1)0.1270.95α10.0058β11+0.00124α1+0.98β1
(16)

αS=Sα1+εS,βS=Sβ1
(17)

The large scale and small scale log-irradiance variance of multiple partially coherent Gaussian beams propagation through non-Kolmogorov strong turbulence are expressed by

σlnXs2=ln(1+1αS),σlnYs2=ln(1+1βS)
(18)

The total longitudinal component of scintillation index for multiple partially coherent Gaussian beams in non-Kolmogorov moderate-strong turbulence is expressed as

σIS2=exp(σlnXs2+σlnYs2)1=1αSβS+1αS+1βS,0σR2
(19)

Thus, the probability density function (PDF) of the summed output Is can be approximated by the PDF of a single Gamma-Gamma variate,

pIS(Is,αS,βS)=2(αSβS)αS+βS2Γ(αS)Γ(βS)Is(IsMN)αS+βS2KαSβS(2αSβSIsMN)
(20)

4. Average capacity of MIMO FSO links for multiple partially coherent beams

Further analysis of the MIMO channel capacity given is obtained by diagonalizing the product matrix HH. By using singular value decomposition, the matrix product is written asH=UΣV, where U and V are unitary matrices of left and right singular vectors respectively, and Σ is a diagonal matrix with singular values on the main diagonal [35

35. B. Holter, “On the capacity of the MIMO channel: A tutorial introduction,” in Proc. IEEE Norwegian Symposium on Signal Processing(IEEE, 2001), 167–172.

].All elements on the diagonal are zero except for the first k elements. The number of non-zero singular values k equals the rank of the channel matrix. The capacity is a lower bound on the MIMO channel capacity [36

36. O. Oyman, R. U. Nabar, H. Bolcskei, and A. J. Paulraj, “Tight lower bounds on the ergodic capacity of Rayleigh fading MIMO channels,” in IEEE Global Telecommunications Conference, GLOBECOM'02.(IEEE, 2002), 1172–1176. [CrossRef]

].Using singular value decomposition, the MIMO channel capacity can be written as [34

34. G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wirel. Pers. Commun. 6(3), 311–335 (1998). [CrossRef]

]

CMN=log2det(IN+uMUΣΣU)
(22)

With the fact that the determinant of a unitary matrix is equal to 1 and UU=IN [33

33. 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]

],MIMO capacity can be expressed respectively as
CMN=log2det(UU+uMUΣΣU)=log2det(IN+uMΣΣ)=log2i=1Nm(1+uMσi2)
(23)
whereσi2are the squared singular values of the diagonal matrix Σ. The number of parallel sub channels Nm is determined by the rank of the channel matrix. In general, the rank of the channel matrix is given by

Nm=rank(H)min(M,N).
(24)

The capacity of the MIMO fading channel is a function of the distribution of the singular values of the random channel matrix. By Jensen’s inequality [37

37. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-interscience, New York, 2012).

], we obtain the bounding capacity that
CMN=i=1Nmlog2(1+uMσi2)Nmlog2(1+uMNmi=1Nmσi2)
(25)
with equality if and only if the singular values of the random channel matrix are all equal.
CMNNmlog2(1+uMNmTr[HH])=Nmlog2(1+1MNmi=1Nj=1Muij)
(26)
The quantityuij=u|hij|2 is the instantaneous SNR in each channel branch and |hij| is a random variable due to fading in channel transfer matrices
hij=a+jb=a2+b2ejarctanba=|hij|ejϕij
(27)
If a and b are independent and normal distributed random variables, the channel gain |hij| is a Rayleigh distributed random variable [35

35. B. Holter, “On the capacity of the MIMO channel: A tutorial introduction,” in Proc. IEEE Norwegian Symposium on Signal Processing(IEEE, 2001), 167–172.

].

Thus, the MIMO capacity is

CMNNmlog2(1+1MNmi=1Nj=1Muij)
(28)

For SISO case

C11log2(1+u11)
(29)

For SIMO case

C1Nlog2(1+i=1Nui)
(30)

For MISO case

CM1log2(1+1Mi=1Mui)
(31)

For MIMO case M = N

CNNNlog2(1+1N2i=1Nui)
(32)

CMNmax=Nm(αSβS)(αS+βS)/24πΓ(αS)Γ(βS)ln2(u¯m)(αS+βS)/4×0ln(1+um)(um)(αS+βS)/41KαSβS(2αSβSumu¯m)dum
(36)

The integral can be solved using Meijer’s G functions and their properties. Hence, by substituting the PDF of the Gamma-Gamma distribution, expressing the Kv(.) integrands in terms of Meijer’s G-function [16

16. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express 19(14), 13480–13496 (2011). [CrossRef] [PubMed]

], the closed-formsolution of average capacity yields as follow,
<CMN>max=Nm(αSβS)(αS+βS)/24πΓ(αS)Γ(βS)ln(2)(μ¯m)(αS+βS)/4×G2,66,1[(αSβS)216μ¯m|αS+βS4,αS+βS4+1αSβS4,αSβS+24,αSβS4,(αSβS)+24,αS+βS4,αS+βS4]
(37)
whereGp,qm,n[] is Meijer’s G-function,u¯m=u¯s/(MNm)denotes the average effective electrical SNR of combined MIMO free space optical communication links.

5. Numerical results

5.1 Average capacity versus electrical SNR

As it is evident in Fig. 1(a), the average capacity increases significantly with the number of transmit or receive apertures M × N compared to the SISO deployment. Moreover, the greater improvement in average capacity is obtained as MIMO configuration goes from 1 × 2 to 2 × 2, in comparison with diversity number going from 2 × 2 to 2 × 4. This can be explained by the Eq. (37) that a noticeable increase in capacity in the former case mainly arises from an increasing rank of MIMO channel matrix Nm from 1 to 2, while in the latter case with a constant Nm, the capacity improvement is because of the reduction in scintillation index with the increasing diversity apertures. Considering the increase in capacity at the cost of larger number of apertures, the MIMO 2 × 2 case is the most efficient configuration for capacity improvement.

5.2 Average capacity versus power law

Specifically, as indicated by Fig. 2(c), for alpha values larger than 3.5, or close to 3, the average capacity improves apparently with the decrease in spatial coherence length of transmit beams. On the other hand, when power law 𝛼 is between 3.1 and 3.5 (around 𝛼 = 10/3), the average capacity decreases with the decreasing spatial coherence length. It is obvious in Fig. 2(d) that the influence of alpha value on the average capacity becomes stronger as the turbulence strength gets stronger. The physical reason can be deduced from the relationship between alpha values and atmospheric turbulence layers: turbulence tends to vanish for alpha approaching 3, 𝛼 = 11/3 corresponds to the boundary layer, α= 10/3 corresponds to the free troposphere layer, and alpha approaching4 represents lower stratosphere layer under the condition of stable stratification. Thus, as laser beams propagate through the free troposphere layer (around 𝛼 = 10/3), the turbulence effect gets stronger, and the dependence of channel capacity on spatial coherence length gets stronger.

5.3 Average capacity versus propagation distance

5.4 Average capacity versus spatial coherence length

As it is illustrated in Fig. 4(a), for all values of spatial coherence length, average capacity is significantly improved as diversity order M and N increase. Moreover, we observe in Fig. 4(b) that as the spatial coherence length decreases, average capacity increases significantly at first, then reaches the maximum value at a certain coherence length, at last saturates and decreases slightly as it approaches incoherent beams. In the current case, the optimum coherence length corresponds to about lc = 0.002 m = w0/5.This phenomenon can be explained by the fact that for laser beams with lower coherence length, optical energy is delivered by mutually independent coherent modes that propagate through statistically independent regions of turbulence. As a result, turbulence induced scintillations of the modes are relatively uncorrelated, and the averaged scintillation of partially coherent beams is decreased. Consequently, the average capacity of partially coherent beams is improved. As all the coherent modes are mutually independent with the decreasing coherent length, the reduction of scintillation and improvement in capacity goes into saturated regions.

The scintillation in Fig. 4(e) takes an opposite dependence on spatial coherent length compared with the average capacity in Fig. 4(d). Furthermore, we observe in Fig. 4(f) that the large-scale log-irradiance variance decreases with the spatial coherence length except for the case of strong turbulence (Cn2 = 4 × 10−13m3-α), whereas the small-scale log-irradiance variance decreases with the coherence length for all the turbulence strength. Thus, for partially coherent beams in strong turbulence, the increase in large-scale log-irradiance variance is the dominating factor for the decrease in average capacity. The physical reason is that the large scale size of turbulent eddy becomes shorter than the spatial coherence length as turbulence gets stronger. In this case, the refraction effect of large scale turbulent eddy is stronger at smaller coherence length. In addition, turbulent eddy sizes bounded below by the spatial coherence radius and above by the scattering disk radius contribute little to scintillation under strong fluctuations.

6. Conclusions

Acknowledgements

This research was financially supported by National Natural Science Foundation of China(NSFC) (No. 61077058, No. 61275081).The authors are grateful for financial support from the program of China Scholarships Council (No.2011616097) and the Pennsylvania State University CICTR, through US National Science foundation (NSF) ECCS Directorate for partially supporting this work under award (1201636).

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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(9), 1794–1802 (2002). [CrossRef] [PubMed]

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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]

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A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using diversity-combining techniques,” Opt. Express 17(15), 12601–12611 (2009). [CrossRef] [PubMed]

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E. Bayaki, R. Schober, and R. K. Mallik, “Performance Analysis of MIMO Free-Space Optical Systems in Gamma-Gamma Fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009). [CrossRef]

12.

Y. Baykal, H. T. Eyyuboğlu, and Y. J. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Appl. Opt. 48(10), 1943–1954 (2009). [CrossRef] [PubMed]

13.

J. Cang and X. Liu, “Average capacity of free-space optical systems for a partially coherent beam propagating through non-Kolmogorov turbulence,” Opt. Lett. 36(17), 3335–3337 (2011). [CrossRef] [PubMed]

14.

G. P. Berman, A. R. Bishop, B. M. Chernobrod, V. N. Gorshkov, D. C. Lizon, D. I. Moody, D. C. Nguyen, and S. V. Torous, “Reduction of laser intensity scintillations in turbulent atmospheres using time averaging of a partially coherent beam,” J. Phys. B 42(22), 225403 (2009). [CrossRef]

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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]

16.

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express 19(14), 13480–13496 (2011). [CrossRef] [PubMed]

17.

K. P. Peppas, F. Lazarakis, A. Alexandridis, and K. Dangakis, “Simple, accurate formula for the average bit error probability of multiple-input multiple-output free-space optical links over negative exponential turbulence channels,” Opt. Lett. 37(15), 3243–3245 (2012). [CrossRef] [PubMed]

18.

X. Yi, Z. Liu, and P. Yue, “Formula for the average bit error rate of free-space optical systems with dual-branch equal-gain combining over gamma-gamma turbulence channels,” Opt. Lett. 38(2), 208–210 (2013). [CrossRef] [PubMed]

19.

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” in Photonics West'97(International Society for Optics and Photonics, 1997), 102–113.

20.

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

21.

N. Letzepis, I. Holland, and W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wireless Commun. 7(5), 1744–1753 (2008). [CrossRef]

22.

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link,” Appl. Opt. 46(26), 6561–6571 (2007). [CrossRef] [PubMed]

23.

G. Yun and M. Kavehrad, “Spot-diffusing and fly-eye receivers for indoor infrared wireless communications,” in Proceedings of IEEE International Conference on Selected Topics in Wireless Communications(IEEE, 1992), 262–265. [CrossRef]

24.

Z. Hajjarian and M. Kavehrad, “Using MIMO Transmissions in Free Space Optical Communications in Presence of Clouds and Turbulence,” Proc. SPIE 7199, 71990V, 71990V-12 (2009). [CrossRef]

25.

S. Jivkova and M. Kavehrad, “Transceiver design concept for cellular and multispot diffusing regimes of transmission,” Eurasip J Wirel Comm 2005, 30–38 (2005).

26.

J. M. Kahn, R. You, P. Djahani, A. G. Weisbin, B. K. Teik, and A. Tang, “Imaging diversity receivers for high-speed infrared wireless communication,” IEEE Commun. Mag. 36(12), 88–94 (1998). [CrossRef]

27.

M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels,” IEEE Trans. Wireless Commun 5(6), 1229–1233 (2006). [CrossRef]

28.

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(8), 1554–1562 (2001). [CrossRef]

29.

P. Deng, X. Yuan, Y. Zeng, M. Zhao, and H. Luo, “Influence of wind speed on free space optical communication performance for Gaussian beam propagation through non kolmogorov strong turbulence,” J. Phys. Conf. Ser. 276, 012056 (2011). [CrossRef]

30.

P. Deng, X. Yuan, and D. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun. 285(6), 880–887 (2012). [CrossRef]

31.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003–026009 (2008). [CrossRef]

32.

N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the distribution of the sum of gamma-gamma variates and application in MIMO optical wireless systems,” in IEEE Global Telecommunications Conference(IEEE, 2009), 1–6. [CrossRef]

33.

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]

34.

G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wirel. Pers. Commun. 6(3), 311–335 (1998). [CrossRef]

35.

B. Holter, “On the capacity of the MIMO channel: A tutorial introduction,” in Proc. IEEE Norwegian Symposium on Signal Processing(IEEE, 2001), 167–172.

36.

O. Oyman, R. U. Nabar, H. Bolcskei, and A. J. Paulraj, “Tight lower bounds on the ergodic capacity of Rayleigh fading MIMO channels,” in IEEE Global Telecommunications Conference, GLOBECOM'02.(IEEE, 2002), 1172–1176. [CrossRef]

37.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-interscience, New York, 2012).

OCIS Codes
(010.1290) Atmospheric and oceanic optics : Atmospheric optics
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(030.0030) Coherence and statistical optics : Coherence and statistical optics
(060.4510) Fiber optics and optical communications : Optical communications
(060.2605) Fiber optics and optical communications : Free-space optical communication

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: April 19, 2013
Revised Manuscript: June 8, 2013
Manuscript Accepted: June 8, 2013
Published: June 18, 2013

Citation
Peng Deng, Mohsen Kavehrad, Zhiwen Liu, Zhou Zhou, and XiuHua Yuan, "Capacity of MIMO free space optical communications using multiple partially coherent beams propagation through non-Kolmogorov strong turbulence," Opt. Express 21, 15213-15229 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15213


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References

  1. S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Comm.6(8), 2813–2819 (2007). [CrossRef]
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  8. 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. A19(9), 1794–1802 (2002). [CrossRef] [PubMed]
  9. 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]
  10. A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using diversity-combining techniques,” Opt. Express17(15), 12601–12611 (2009). [CrossRef] [PubMed]
  11. E. Bayaki, R. Schober, and R. K. Mallik, “Performance Analysis of MIMO Free-Space Optical Systems in Gamma-Gamma Fading,” IEEE Trans. Commun.57(11), 3415–3424 (2009). [CrossRef]
  12. Y. Baykal, H. T. Eyyuboğlu, and Y. J. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Appl. Opt.48(10), 1943–1954 (2009). [CrossRef] [PubMed]
  13. J. Cang and X. Liu, “Average capacity of free-space optical systems for a partially coherent beam propagating through non-Kolmogorov turbulence,” Opt. Lett.36(17), 3335–3337 (2011). [CrossRef] [PubMed]
  14. G. P. Berman, A. R. Bishop, B. M. Chernobrod, V. N. Gorshkov, D. C. Lizon, D. I. Moody, D. C. Nguyen, and S. V. Torous, “Reduction of laser intensity scintillations in turbulent atmospheres using time averaging of a partially coherent beam,” J. Phys. B42(22), 225403 (2009). [CrossRef]
  15. 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]
  16. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express19(14), 13480–13496 (2011). [CrossRef] [PubMed]
  17. K. P. Peppas, F. Lazarakis, A. Alexandridis, and K. Dangakis, “Simple, accurate formula for the average bit error probability of multiple-input multiple-output free-space optical links over negative exponential turbulence channels,” Opt. Lett.37(15), 3243–3245 (2012). [CrossRef] [PubMed]
  18. X. Yi, Z. Liu, and P. Yue, “Formula for the average bit error rate of free-space optical systems with dual-branch equal-gain combining over gamma-gamma turbulence channels,” Opt. Lett.38(2), 208–210 (2013). [CrossRef] [PubMed]
  19. I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” in Photonics West'97(International Society for Optics and Photonics, 1997), 102–113.
  20. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, Bellingham, 2005).
  21. N. Letzepis, I. Holland, and W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wireless Commun.7(5), 1744–1753 (2008). [CrossRef]
  22. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link,” Appl. Opt.46(26), 6561–6571 (2007). [CrossRef] [PubMed]
  23. G. Yun and M. Kavehrad, “Spot-diffusing and fly-eye receivers for indoor infrared wireless communications,” in Proceedings of IEEE International Conference on Selected Topics in Wireless Communications(IEEE, 1992), 262–265. [CrossRef]
  24. Z. Hajjarian and M. Kavehrad, “Using MIMO Transmissions in Free Space Optical Communications in Presence of Clouds and Turbulence,” Proc. SPIE7199, 71990V, 71990V-12 (2009). [CrossRef]
  25. S. Jivkova and M. Kavehrad, “Transceiver design concept for cellular and multispot diffusing regimes of transmission,” Eurasip J Wirel Comm2005, 30–38 (2005).
  26. J. M. Kahn, R. You, P. Djahani, A. G. Weisbin, B. K. Teik, and A. Tang, “Imaging diversity receivers for high-speed infrared wireless communication,” IEEE Commun. Mag.36(12), 88–94 (1998). [CrossRef]
  27. M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels,” IEEE Trans. Wireless Commun5(6), 1229–1233 (2006). [CrossRef]
  28. 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(8), 1554–1562 (2001). [CrossRef]
  29. P. Deng, X. Yuan, Y. Zeng, M. Zhao, and H. Luo, “Influence of wind speed on free space optical communication performance for Gaussian beam propagation through non kolmogorov strong turbulence,” J. Phys. Conf. Ser.276, 012056 (2011). [CrossRef]
  30. P. Deng, X. Yuan, and D. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun.285(6), 880–887 (2012). [CrossRef]
  31. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng.47(2), 026003–026009 (2008). [CrossRef]
  32. N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the distribution of the sum of gamma-gamma variates and application in MIMO optical wireless systems,” in IEEE Global Telecommunications Conference(IEEE, 2009), 1–6. [CrossRef]
  33. 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]
  34. G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wirel. Pers. Commun.6(3), 311–335 (1998). [CrossRef]
  35. B. Holter, “On the capacity of the MIMO channel: A tutorial introduction,” in Proc. IEEE Norwegian Symposium on Signal Processing(IEEE, 2001), 167–172.
  36. O. Oyman, R. U. Nabar, H. Bolcskei, and A. J. Paulraj, “Tight lower bounds on the ergodic capacity of Rayleigh fading MIMO channels,” in IEEE Global Telecommunications Conference, GLOBECOM'02.(IEEE, 2002), 1172–1176. [CrossRef]
  37. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-interscience, New York, 2012).

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