## Capacity of MIMO free space optical communications using multiple partially coherent beams propagation through non-Kolmogorov strong turbulence |

Optics Express, Vol. 21, Issue 13, pp. 15213-15229 (2013)

http://dx.doi.org/10.1364/OE.21.015213

Acrobat PDF (1553 KB)

### 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

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. 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. Express **17**(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. B **42**(22), 225403 (2009). [CrossRef]

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]

*N*rather than the diversity order

_{m}*M*×

*N*. The diversity order is independent of the effects of power law, propagation distance, turbulence strength parameter and spatial coherence length, whereas these effects on average capacity are gradually mitigated as the diversity order increases. Furthermore, as the spatial coherence length decreases, the average capacity increases significantly and then reaches the maximum value at certain coherence length, at last saturates slightly as it approaches incoherent beams. The results would provide a useful approach for the optimization of diversity configuration and spatial coherence length to maximize channel capacity of MIMO FSO links over a variety of atmospheric turbulence conditions.

## 2. System and channel model

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. 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. 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]

*M*apertures 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]

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]

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]

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]

*x*represents the information bits,

*η*is the optical-to-electrical conversion coefficient and

*υ*is the AWGN with zero mean and variance

_{n}*σ*

_{υ}^{2}=

*N*

_{0}/2. The normalized irradiance,

*I*, is the received irradiance normalized by its mean value. The fading channel coefficient,

_{mn}*I*, which models the atmospheric turbulence through the optical channel from the

_{mn}*m*th transmit aperture to the

*n*th 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]

*I*=

_{mn}*I*

_{0}exp(2

*X*), where

_{mn}*I*

_{0}is the signal light intensity without turbulence and

*X*are identically distributed normal random variables with mean

_{mn}*μ*and variance

_{x}*σ*

_{x}^{2}.

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]

*L*is the propagation distance, the large scale factor

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˜B2(α)}is the longitudinal component of scintillation index for partially coherent Gaussian-beam wave under non-Kolmogorov weak turbulence.Where

*k*is the wave number,

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]

## 3. Spatial diversity and combining gain

*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. 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]

*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

*m*th transmitter and

*n*th receiver aperture are

**= (**

*I**I*

_{11},

*I*

_{12}, …,

*I*) of the length

_{mn}*S*=

*M*×

*N*, and the sum of the received signals

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]

*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.

*σ*

_{X}^{2}/

*S*and

*σ*

_{Y}^{2}/

*S*, respectively, resulting in

*α*=

_{S}*Sα*

_{1}and

*𝛽*=

_{S}*S𝛽*

_{1}. 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]

*I*can be approximated by the PDF of a single Gamma-Gamma variate,

_{s}## 4. Average capacity of MIMO FSO links for multiple partially coherent beams

**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].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]

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]

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]

**Σ**. The number of parallel sub channels

*N*is determined by the rank of the channel matrix. In general, the rank of the channel matrix is given by

_{m}*h*| is a random variable due to fading in channel transfer matricesIf

_{ij}*a*and

*b*are independent and normal distributed random variables, the channel gain |

*h*| is a Rayleigh distributed random variable [35].

_{ij}*E*[∙] denotes the expectation operation,

*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]

## 5. Numerical results

_{0}= 1) at the transmitter have a beam width of

*w*

_{0}∇1 cm and a wavelength of λ = 1550 nm. The initial spatial coherence length of the transmitted laser beam is varied from

*l*= 0. 01 cm to

_{c}*l*= 1 m. The power law of non-Kolmogorov turbulence spectrum α is between 3 and 4.The propagation distance

_{c}*L*is varied from 1km to 10 km, and the turbulence structure parameter

*C*takes values in the set (4 × 10

_{n}^{2}^{−14},8 × 10

^{−14},1 × 10

^{−13},4 × 10

^{−13})m

^{3-α}.

### 5.1 Average capacity versus electrical SNR

*M*×

*N*, (b) the spatial coherence length

*l*

_{c}of the beam source with different diversity configurations, (c) the power law 𝛼of non-Kolmogorov turbulence with different combiner branches and (d) the power law 𝛼 with various spatial coherence length. In all cases, the propagation distance is fixed at

*L*= 1 km with a constant structure parameter

*C*1 × 10

_{n}^{2}=^{−13}m

^{3-α}.

*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

*N*from 1 to 2, while in the latter case with a constant

_{m}*N*, 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.

_{m}*M*×

*N*= 2 × 4. It is deduced from Fig. 1(c) that, as the power law

*α*increases from 3.1 to 3.8, the average capacity reduces slightly at power law

*α*of 3.2 and then grows up sharply. The effect of power law 𝛼 on capacity is lessened as M and N increase. Fig. 1(d) shows the influence of power law and coherence length on average capacity of MIMO FSO links for the 2 × 2case. It can be seen that, for most values of spectrum index 𝛼,the average capacity of partially coherent beams is larger than that of coherent beams, whereas for power law 𝛼 = 3.2 the dependence of capacity on spatial coherence length is reversed.

### 5.2 Average capacity versus power law

*M*×

*N*, the spatial coherence length

*l*

_{c}and turbulence strength parameter

*C*. In all cases, the propagation distance is fixed at

_{n}^{2}*L*= 1 km with a constant SNR

*=*10 dB. As can be seen in Fig. 2(a), for all the values of power law𝛼, average capacity is monotonically improved as the number of transmit and receive apertures

*M*×

*N*increases. Furthermore, we observe that the average capacity falls down slightly to the minimum value at 𝛼 = 3.2, and then rises up rapidly as the power law 𝛼increases from 3 to 4.Thereason for this phenomenon can be deduced from Fig. 2(b) where the scintillation index as a function of

*α* corresponding to links in Fig. 2(a) is depicted. The scintillation index of MIMO FSO links has an opposite characteristics dependence on the power law 𝛼, which leads to a gain in average capacity.

*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

*L*for various values of the number of transmit/ receive apertures

*M*×

*N*, the spatial coherence length

*l*

_{c}and power law 𝛼. In all cases, the turbulence structure parameter is fixed at

*C*1 × 10

_{n}^{2}=^{−13}m

^{3-α}with a constant initial SNR

*=*10 dB. It is clearly depicted in Fig. 3(a) that, for all the values of

*L*, average capacity is significantly improved as

*M*and

*N*increase. For all the diversity cases, average capacity is initially reduced as the propagation distance

*L*increases under weak-moderate turbulence, and then reaches the minimum values and changes the slope at a distance of about

*2000 m*in the focusing regime, lastly saturates and increases slightly with the increasing distance in the strong irradiance fluctuation. This phenomenon can be explained by Fig. 3(b) that the reduction and saturation of scintillation index as diversity order

*M*×

*N*and propagation distance

*L*increase. As it is indicated in Fig. 3(c), considering various coherence length values and diversity deployment, for shorter distance under weak-moderate turbulence, average capacity increases with decreasing spatial coherence length. However, for longer distance in strong turbulence, average capacity decreases with spatial coherence length. Furthermore, the dependence of average capacity on coherence length and power law becomes less significant at larger number of transmit and receive apertures.

### 5.4 Average capacity versus spatial coherence length

*M*×

*N*, spectrum power index 𝛼 and turbulence strength parameter

*C*. In all cases, the propagation distance is fixed at

_{n}^{2}*L*= 1 km with a constant

*SNR*= 10 dB.

*C*= 4 × 10

_{n}^{2}^{−14}, 8 × 10

^{−14}and 1 × 10

^{−13}m

^{3-α}, Rytov variance

*σ*

_{R}< 1.5), average capacity increases with decreasing spatial coherence length. However, for strong atmospheric turbulence (the cases:

*C*= 4 × 10

_{n}^{2}^{−13}m

^{3-α}, Rytov variance

*σ*

_{R}> 1.5), average capacity decreases as the spatial coherence length decreases. The reason for the phenomenon can be deduced from Fig. 4(e) and Fig. 4(f) in which the scintillation and large-scale and small-scale log-irradiance variance corresponding to links in Fig. 4(d) are plotted.

*C*= 4 × 10

_{n}^{2}^{−13}m

^{3-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

*N*rather than the diversity order

_{m}*M*×

*N*. The MIMO case 2 × 2 is the most efficient configuration for capacity improvement at the cost of larger number of apertures. The diversity order is independent of the effects of power law, propagation distance, turbulence strength parameter and spatial coherence length, whereas these effects on average capacity are gradually mitigated as the diversity order increases. Furthermore, as the spatial coherence length decreases, average capacity increases significantly and then reaches the maximum value at coherence length of

*w*

_{0}/5, at last saturates slightly as it approaches incoherent beams. However, the dependence of average capacity on spatial coherence length are changed, as alpha value approaches10/3 that corresponds to the free troposphere layer, or turbulence strength is stronger with Rytov variance larger than 1.5. The results would provide a useful approach for the optimization of diversity configuration and spatial coherence length to maximize channel capacity of MIMO FSO links over a variety of atmospheric turbulence conditions.

## Acknowledgements

## References and links

1. | S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Comm. |

2. | V. W. S. Chan, “Free-Space Optical Communications,” J. Lightwave Technol. |

3. | J. A. Tellez and J. D. Schmidt, “Multibeam scintillation cumulative distribution function,” Opt. Lett. |

4. | H. Guo, B. Luo, Y. Ren, S. Zhao, and A. Dang, “Influence of beam wander on uplink of ground-to-satellite laser communication and optimization for transmitter beam radius,” Opt. Lett. |

5. | A. Tunick, “Optical turbulence parameters characterized via optical measurements over a 2.33 km free-space laser path,” Opt. Express |

6. | 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. |

7. | E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express |

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 |

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. |

10. | A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using diversity-combining techniques,” Opt. Express |

11. | E. Bayaki, R. Schober, and R. K. Mallik, “Performance Analysis of MIMO Free-Space Optical Systems in Gamma-Gamma Fading,” IEEE Trans. Commun. |

12. | Y. Baykal, H. T. Eyyuboğlu, and Y. J. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Appl. Opt. |

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. |

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 |

15. | D. K. Borah and D. G. Voelz, “Spatially partially coherent beam parameter optimization for free space optical communications,” Opt. Express |

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 |

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. |

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. |

19. | I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” in |

20. | L. C. Andrews and R. L. Phillips, |

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. |

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. |

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 |

25. | S. Jivkova and M. Kavehrad, “Transceiver design concept for cellular and multispot diffusing regimes of transmission,” Eurasip J Wirel Comm |

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. |

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 |

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. |

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. |

30. | P. Deng, X. Yuan, and D. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun. |

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. |

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. |

34. | G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wirel. Pers. Commun. |

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, |

**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

- 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]
- V. W. S. Chan, “Free-Space Optical Communications,” J. Lightwave Technol.24(12), 4750–4762 (2006). [CrossRef]
- J. A. Tellez and J. D. Schmidt, “Multibeam scintillation cumulative distribution function,” Opt. Lett.36(2), 286–288 (2011). [CrossRef] [PubMed]
- H. Guo, B. Luo, Y. Ren, S. Zhao, and A. Dang, “Influence of beam wander on uplink of ground-to-satellite laser communication and optimization for transmitter beam radius,” Opt. Lett.35(12), 1977–1979 (2010). [CrossRef] [PubMed]
- A. Tunick, “Optical turbulence parameters characterized via optical measurements over a 2.33 km free-space laser path,” Opt. Express16(19), 14645–14654 (2008). [CrossRef] [PubMed]
- 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]
- E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express18(10), 10650–10658 (2010). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
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