## Abstract

In this paper, we analytically investigate an optical signal detection scheme to mitigate the scintillation effect with the assistance of a co-propagating reference continuous wave (CW) light. Using the correlation coefficient between the intensities of the data light and the reference CW light, we mathematically derive their joint intensity distributions under two widely used atmospheric turbulence channel models, namely log-normal distributed channel model and Gamma-Gamma distributed channel model, respectively. We also carry out the Monte-Carlo (MC) simulation and show that theoretical results agree with simulation results well. Our analytical results reveal that when the correlation coefficient is 0.99, the power reductions to achieve BER of 10^{−3} are 12.3 dB and 20.4 dB under moderate and strong atmospheric turbulence conditions (i.e., Rytov variances of 1.0 and 4.0), respectively. In addition, the feasibility of the scheme applied to wavelength-division-multiplexed (WDM) free-space-optical (FSO) transmission systems is also investigated, where only a single reference CW light could be used to mitigate the scintillation effects on all WDM channels.

© 2012 OSA

## 1. Introduction

Free-space optical (FSO) communication is anticipated to play an important role in future telecommunication networks. It can bridge the gap where optical fiber communication cannot reach and wireless communication cannot provide enough bandwidth. The advantages of FSO communication include high capacity, high security, free spectrum license and easy to install and remove [1,2]. Since the sizes of molecules in atmosphere such as oxygen, carbon dioxide and water vapor are comparable to optical wavelength, the intensity of an optical wave will fluctuate when the optical wave propagates through the atmosphere. Such fluctuation is called scintillation, which can severely deteriorate the FSO transmission performance [3–5]. The multiplicative scintillation is detrimental to FSO communication systems [6], and how to mitigate the scintillation effect with reasonable cost becomes a hot research topic recently [4–5,7–9].

It has been experimentally demonstrated in [10] that two optical waves experience almost identical scintillation process, when they co-propagate through the same atmospheric turbulence channel simultaneously. However, only the correlation coefficient between two optical waves has been experimentally investigated and no analytical results have been obtained. In this paper, firstly we mathematically derive the joint intensity distributions of two co-propagating optical waves under two widely used atmospheric turbulence channel models, namely log-normal distributed channel model and Gamma-Gamma distributed channel model [11], respectively. Using these joint intensity distributions, we then theoretically analyze the bit error rate (BER) performance of FSO transmission system that is assisted with a reference CW light to mitigate the effect of scintillation under different atmospheric turbulence conditions as well as under different correlation coefficients between the intensities of the data light and the reference CW light. We also carry out the Monte-Carlo (MC) simulation and show that theoretical results agree with simulation results well. Our results reveal that the power reductions to achieve a bit error rate (BER) of 10^{−3} are 12.3 dB, 20.4 dB under moderate and strong atmospheric turbulence conditions (i.e., Rytov variances of 1.0 and 4.0), respectively, when the correlation coefficient is 0.99. The feasibility of mitigating the scintillation effects by only using a single reference channel for WDM-FSO systems over a wavelength range of 100 nm is examined and discussed. And the improvement of transmission distance is also investigated.

The rest of this paper is organized as follows. Section 2 presents the atmospheric turbulence channel models that follow log-normal and Gamma-Gamma distributions. The scheme of mitigating the effect of scintillation by a reference CW light and the derivation of joint intensity distributions of two co-propagating optical waves are presented in Section 3. In Section 4, the BER performances of the on-off-keying (OOK) FSO system assisted by a reference CW light are evaluated under moderate and strong atmospheric turbulence conditions, followed by the feasibility study of using only a single reference CW light to mitigate the effect of scintillations on all the channels of a WDM-FSO communication system and the improvement of transmission distance. The conclusions are given in Section 5.

## 2. Channel model of FSO communication system

#### 2.1 Log-normal distribution

When an optical wave propagates through the atmospheric turbulence channel, its intensity fluctuates due to scintillation. Several atmospheric turbulence channel models have been proposed to describe scintillation [5,11–12]. Among them, the log-normal (LN) distribution is widely used to investigate the scintillation under weak and moderate atmospheric turbulence conditions [11,13]. Let *I _{S}* be the normalized light intensity. In the LN distributed channel model, the probability density function (PDF) of

*I*is expressed by [14–16]

_{S}*I*

_{0}is the received intensity without turbulence, which is assumed to be unity in this work [16]. The Rytov variance is given by [11,15–16]where

*L*is the transmission distance;

*k*is the wave number, i.e., $k=2\pi /\lambda ,$ where

*λ*is the wavelength of optical wave; ${C}_{n}^{2}$stands for refractive-index structure constant, varying from 10

^{−17}m

^{-2/3}to 10

^{−13}m

^{-2/3}. Different values of the Rytov variance represent different atmospheric turbulence conditions. The weak, moderate and strong intensity fluctuations are associated with${\sigma}_{R}^{2}<1,$ ${\sigma}_{R}^{2}\approx 1$and ${\sigma}_{R}^{2}>1,$ respectively [11].

#### 2.2 Gamma-Gamma distribution

As reported in [15–16], the log-normal distribution is not valid when ${\sigma}_{R}^{2}>\mathrm{1.2.}$ Instead, Gamma-Gamma (GG) distribution is commonly used to represent the strong turbulence condition [5] where ${\sigma}_{R}^{2}>1.$ With the GG distributed channel model, the probability density function (PDF) of *I _{S}* is expressed by [11,17]

*K*(·) is the modified Bessel function of the second kind of the order

_{n}*n*,

*Г*(

*·*) is Gamma function. The parameters

*α*and

*β*are defined by Rytov variance${\sigma}_{R}^{2},$which are given by

## 3. OOK FSO communication system with a reference CW light

#### 3.1 System model

It has been demonstrated in [10] that a reference CW light that records scintillation can be employed to mitigate intensity fluctuation on a data light under the same atmospheric turbulence channel. For the sake of clarity, in the rest of the paper, ‘data light’ and ‘data signal’ refer to the data before and after photo-detection, respectively. Figure 1
shows the schematic of an OOK FSO communication system with a reference CW light to mitigate the scintillation effect. An optical wave at wavelength *λ _{S}* is modulated with an on-off keying (OOK) signal through a Mach-Zehnder modulator (MZM). The data modulated optical wave is combined with a reference CW light at wavelength

*λ*by a wavelength multiplexer (MUX), and then fed to a transmit aperture. The two combined optical waves pass through a fiber pig-tailed collimator in the transmit aperture [4], and then co-propagate through the same atmospheric turbulence channel simultaneously. After being collected by a receive aperture with a high speed and precise tracking mechanism [18], another fiber pig-tailed collimator couples the received signals to a wavelength de-multiplexer (De-MUX), which separates the data light from the reference CW light. After being detected by separate photo-detectors, both the data and reference signals are fed into a high-speed electrical signal processing circuit where the fluctuated data signal is divided by the fluctuated reference signal. Since the scintillations recorded on both optical waves (

_{R}*λ*and

_{S}*λ*) are highly correlated, the effect of scintillation on the data light at

_{R}*λ*can be mitigated by the scintillation on the reference CW light at

_{S}*λ*. Note that the MUX, De-MUX, transmit aperture and receive aperture are assumed to be ideal in our study (i.e., these devices do not contribute any noise to the detected signals), since our study is focused on the suppression of adverse effect of scintillation.

_{R}Let *I _{S}* and

*I*be the normalized light intensities of the data light and the reference CW light, respectively. Both

_{R}*I*and

_{S}*I*follow the LN distribution for weak and moderate turbulence conditions in Eq. (1) and the GG distribution for strong turbulence conditions in Eq. (3). Let

_{R}*I*be the normalized amplitude of the data signal at the output of division circuit, which can be expressed as

_{P}*I*=

_{P}*I*/

_{S}*I*. The PDF of

_{R}*I*can be calculated by [19]

_{P}*f*(

*I*,

_{S}*I*) is the joint pdf of

_{R}*I*and

_{S}*I*. The derivations of

_{R}*f*(

*I*,

_{S}*I*) are presented in Section 3.2 for joint LN distribution and in Section 3.3 for joint GG distribution.

_{R}#### 3.2 Joint log-normal distribution

A LN distribution can be transformed from a normal distribution by using the Jacobian determinant [20]. Let us consider two normal distributed random variables *x* and *y*. We use *μ*_{1} and${\sigma}_{1}^{2}$to denote the expected value and the variance of *x*, respectively, and use *μ*_{2} and${\sigma}_{2}^{2}$to denote the expected value and the variance of *y*, respectively. The PDF of *x* is given by [21]

The joint PDF of *x* and *y* can be described as

The correlation coefficient *ρ*_{N} between the two normal random variables *x* and *y* is defined as [19]

*E*(·) and

*D*(·) denote the expected value and variance of a random variable, respectively.

We also consider a pair of log-normal distributed random variables *u* and *v*, whose relationships with *x* and *y* are described as follows [19]:

Let *J _{1}* be the Jacobian determinant for transforming (

*x*,

*y*) to (

*u*,

*v*), which is given by

Hence, the joint PDF of two log-normal distributed random variables *u* and *v* is given by

Similarly, we define *ρ*_{LN} as the correlation coefficient between the two log-normal distributed random variables *u* and *v*, which is given by [19]

*u*,

*v*),

*D*(

*u*) and

*D*(

*v*) in Eq. (11) are given below:

*E*(*v*) and *D*(*v*) are of the same expressions as that of *E*(*u*) and *D*(*u*), but with parameters *μ*_{2} and ${\sigma}_{2}^{2}.$ Substituting Eqs. (14a)–(14c) into Eq. (13), we can obtain the expression of *ρ*_{N} in Eq. (12) in terms of *ρ*_{LN} as follows:

Replacing *u* and *v* with *I _{S}* and

*I*, respectively, and substituting them into Eq. (13), we can obtain the correlation coefficient

_{R}*ρ*

_{LN}between the intensities of data light and reference CW light after collecting them by receiver aperture in Fig. 1. Substituting

*ρ*

_{LN}into Eq. (15), and then Eq. (15) into Eq. (12), we can get the

*f*(

*I*,

_{S}*I*) that follows joint log-normal distribution. Substituting Eq. (12) into Eq. (6), the PDF of normalized amplitude

_{R}*I*after signal processing by the division circuit is obtained.

_{P}#### 3.3 Joint Gamma-Gamma distribution

Due to the complexity of mathematics, to the best of our knowledge, there is no report on the joint PDF of GG distribution. We here generate the joint GG distribution numerically. Let us consider two random variables *w* and *z* that follow GG distribution and recall the two LN distributed random variables *u* and *v* in last subsection and their joint PDF is given in Eq. (12). The Jacobian determinant *J _{2}* is applied to transform (

*u*,

*v*) to (

*w*,

*z*),

Thus, the joint PDF of the two GG distributed random variables *w* and *z* is

*u*to

*w*and

*v*to

*z*, respectively, which are described as follows:

By solving the ordinary differential function (ODE) in Eq. (18), the mappings *h*(·) from random variable *w* to *u* and *z* to *v* are obtained:

Substituting Eqs. (18) and (19) into Eq. (17), the joint Gamma-Gamma distribution can be calculated numerically, as given in Eq. (20),

Note that in statistics, the correlation coefficient between two random variables *w* and *z* is described by [21]

*n*is the length of both

*w*and

*z*. Applying the transformation from Gamma-Gamma distribution to log-normal distribution in Eq. (19) and from log-normal distribution to normal distribution in Eqs. (10), the correlation coefficient

*ρ*

_{N}in Eq. (12) is obtained. Substituting Eq. (12) into Eq. (18) and replacing

*w*and

*z*in Eq. (20) with

*I*and

_{S}*I*, respectively, we can obtain

_{R}*f*(

*I*,

_{S}*I*) that follows joint Gamma-Gamma distribution. Substituting Eq. (20) into Eq. (6), the PDF of normalized amplitude

_{R}*I*after signal processing by the division circuit is obtained.

_{P}## 4. BER performance of OOK FSO communication systems

In this section, the BER performance improvement of OOK FSO communication systems by a reference CW light is numerically analyzed. We consider that an OOK optical signal of 10.7 Gbit/s with forward error correction (FEC) code [22] and a reference CW light are transmitted simultaneously through the same atmospheric turbulence channel and they are of different wavelengths but with the same transmitted power, as shown in Fig. 1. For the sake of simplicity, we assume that the wavelength of the reference CW light is fixed at 1550 nm, while the wavelength of the data light can be varied. In all the BER calculations, the receiver’s effective noise bandwidth is half of the bit rate [23].

We assume that the channel state information (CSI) is not available at the receiver. The bit error rate of an OOK signal is given by [24,25]

where*P*(0) and

*P*(1) are the probabilities of data ‘0’ and ‘1’ transmitted (here $P\left(0\right)=P\left(1\right)=0.5$);

*P*(error|0) and

*P*(error|1) are the conditional error probabilities of data ‘0’ and ‘1’. Since scintillation is a multiplicative effect, it mainly affects data ‘1’. So the conditional error probabilities can be calculated as [25–26]

*Q*(·) is Q-function;

*r*is the detected electrical signal;

*I*represents the normalized intensity

*I*or normalized amplitude

_{S}*I*;

_{P}*f*(

*I*) is the PDF of

*I*or

_{S}*I*; ${\sigma}_{n}^{2}$is the additive Gaussian white noise (AWGN) [23]. The decision threshold

_{P}*I*

_{th}is determined by the likelihood function $\Lambda $at the unity value [25,27] such that the BER is minimized, as shown in Eq. (25),

Substituting *I*_{th} into Eqs. (23)–(24), and then Eqs. (23)–(24) into Eq. (22), the BER performance of OOK signal becomes

Replacing *f*(*I*) in Eq. (24) with *f*(*I _{S}*) in Eq. (1) for log-normal distribution or in Eq. (3) for Gamma-Gamma distribution and with

*f*(

*I*) in Eq. (6), we can obtain the BERs before and after the signal processing by the division circuit, respectively.

_{P}#### 4.1 Moderate atmospheric turbulence condition

According to Eq. (2), the relationship between the Rytov variance${\sigma}_{R\_S}^{2}$of the data light *I _{S}* and the Rytov variance${\sigma}_{R\_R}^{2}$of the reference CW light

*I*is described by

_{R}Under the moderate atmospheric turbulence condition, the Rytov variance${\sigma}_{R\_S}^{2}$ of data light *I _{S}* is assumed to be 1.0. According to Eq. (27), when the wavelength of

*I*is 1555 nm the Rytov variance${\sigma}_{R\_R}^{2}$of the reference CW light

_{S}*I*is 1.004, which is almost the same as${\sigma}_{R\_S}^{2}.$ The LN channel model described in Eq. (1) is still valid [16] for both the data light and the reference CW light. Hence, the joint log-normal distribution in Eq. (12) is applied. According to Eq. (15), when the correlation coefficients

_{R}*ρ*

_{LN}between the two LN distributed intensities

*I*and

_{S}*I*are 0.85 and 0.99, the values of

_{R}*ρ*

_{N}are 0.901 and 0.994, respectively. Substituting

*ρ*

_{N}into Eq. (12), and then Eq. (12) into Eq. (6), we can obtain the PDF of the normalized amplitude

*I*of the data signal at the output of the division circuit.

_{P}Figures 2(a) and (b)
show the PDFs of the normalized intensity *I _{S}* (amplitude

*I*) of the data light (signal) under moderate atmospheric turbulence condition with Rytov variance of 1.0 before and after the signal processing by the division circuit, respectively. After the signal processing, the PDF curve becomes much more concentrated at 1 than that before signal processing, indicating that the effect of scintillation is mitigated significantly by the division circuit.

_{P}For the case of moderate atmospheric turbulence condition, the transmitted optical power of the signal is set to be 0 dBm. The BER performance of the reference CW light assisted detection scheme is depicted in Fig. 3
, where analytical and MC simulation results match very well. We see that without the assistance of the reference CW light, the required power to achieve BER of 10^{−3} is −12.6 dBm; with the reference CW light the required powers are reduced to −21.6 dBm when the correlation coefficients *ρ*_{LN} between the two LN distributed intensities *I _{S}* and

*I*is 0.85 and further reduced to −24.9 dBm when the correlation coefficients

_{R}*ρ*

_{LN}is 0.99. That is, the power reductions of 9.0 dB and 12.3 dB are achieved, respectively. Therefore, the reference CW light assisted detection scheme shown in Fig. 1 is more effective when the scintillations on two optical waves are highly correlated. Note that BER of 10

^{−3}is usually used as the bench mark for communication systems, since error-free transmission can be achieved by applying FEC code [28].

#### 4.2 Strong atmospheric turbulence condition

We assume that under the strong atmospheric turbulence condition, the Rytov variance${\sigma}_{R\_S}^{2}$of data light *I _{S}* is 4.0, and the corresponding values of parameters

*α*and

*β*in Eq. (4) and Eq. (5) are 4.341 and 1.309, respectively. According to Eq. (27), the Rytov variance${\sigma}_{R\_R}^{2}$of reference CW light

*I*is 4.015, and the corresponding

_{R}*α*and

*β*are 4.344 and 1.307, respectively. Hence, the joint Gamma-Gamma distribution in Eq. (20) is applied. Based on the algorithm in subsection 3.3, when the correlation coefficients

*ρ*

_{GG}in Eq. (21) are 0.85 and 0.99, the values of

*ρ*

_{N}in Eq. (12) are 0.888 and 0.993, respectively.

Figure 4
depicts the PDFs and histograms of the normalized intensity *I _{S}* (amplitude

*I*) of the data light (signal) under strong atmospheric turbulence condition with Rytov variance of 4.0 before and after the signal processing by the division circuit, respectively. We see that the higher the correlation coefficient is, the more the PDF curve concentrates at 1 after the signal processing by the division circuit. The histograms and the PDFs of the signal match with each other well, which indicates that the numerical results are reasonable.

_{P}The improvements of BER performance are shown in Fig. 5
. Since GG distribution with 4.0-Rytov variance represents the strong atmospheric turbulence condition, we set the transmitted optical power of the signal to be 5 dBm in this subsection to investigate the BER performance more thoroughly with a larger received power range. For the joint GG distribution, we cannot get the analytical closed form expression. Instead we use the numerical approach to study the performance improvement under GG distribution. As shown in Fig. 5, when the BER is larger than 10^{−3}, the analytical and MC simulation results match very well. However, when the BER is smaller than 10^{−3}, the analytical and MC simulation results do not match very well. This may be attributed to the fact that the mapping from GG distributed random number to LN distributed random number in Eq. (19) and the decision threshold in Eq. (25) were not precisely obtained during the numerical calculation and the MC simulation. Nevertheless, the trends of both the results are consistent. Before signal processing the required optical power to achieve BER of 10^{−3} is −4.2 dBm; after signal processing the required optical power is reduced to −19.7 dBm when the correlation coefficient is 0.85 and to −24.6 dBm when the correlation coefficient is 0.99. Hence, the power reductions for achieving BER of 10^{−3} by the reference CW light are 15.5 dB and 20.4 dB, respectively. Compared with the results in Fig. 3, we observe that the stronger the atmospheric turbulence is, the more effectively the reference CW light assisted detection scheme performs.

#### 4.3 WDM-FSO communication system

The system schematic shown in Fig. 1 could be applied to WDM-FSO transmission systems. In this subsection, we focus on the moderate atmospheric turbulence condition with data light’s Rytov variance of 1.0. According to Eq. (27), when the wavelength of the data light varies from 1500 nm to 1600 nm, the Rytov variance${\sigma}_{R\_R}^{2}$of reference CW light varies from 0.963 to 1.038 correspondingly, which makes the LN distribution in Eq. (1) is still valid [16]. According to Eq. (15), when the correlation coefficients *ρ*_{LN} between the intensities of data light *I _{S}* and reference CW light

*I*are 0.85 and 0.99, the values of

_{R}*ρ*

_{N}in Eq. (12) are almost constant at 0.994 and 0.900, respectively.

Figure 6
shows the power reductions for achieving BER of 10^{−3} of a data signal under the moderate turbulence condition with Rytov variance of 1.0, where the signal wavelength is changed from 1500 nm to 1600 nm and the wavelength of the reference CW light remains unchanged at 1550 nm. When the correlation coefficient is 0.85, the power reduction also varies slightly from 8.9 dB to 9.0 dB; when the correlation coefficient between *I _{S}* and

*I*is 0.99, the power reduction varies slightly from 12.3 dB to 12.4 dB. These results show that the power reduction is almost independent of the signal wavelength under the same correlation coefficient. Therefore we can conclude that only a single reference CW light is required to substantially mitigate the scintillation effects on all the channels of a WDM-FSO communication system within a wide wavelength range of 100 nm, indicating that the reference CW light assisted detection scheme is very cost-effective when it is applied to WDM-FSO systems. Note that WDM-FSO system should also be feasible under strong atmospheric turbulence conditions by applying one reference CW light.

_{R}#### 4.4 Improvement of transmission distance

In previous discussions, we investigate the power reduction to achieve BER of 10^{−3} by the reference CW light. Such comparison is based on the assumption that the Rytov variance of the signal remains the same for the two cases-with and without employing the reference CW light. Hence, without the consideration of beam divergence, the received optical power is related to the transmission distance [29–30], i.e.,

*P*

_{R}(

*L*)is the received optical power,

*P*

_{T}(0) is the transmitted optical power,

*A*is the channel attenuation coefficient (km

^{−1}) and

*L*is the transmission distance.

We assume that the attenuation coefficient *A* is 1.2 km^{−1}. For the LN distribution, according to subsection 4.1, substituting the transmitted optical power, received optical power at BER of 10^{−3} and attenuation coefficient into Eq. (28), we can obtain the transmission distance. The obtained transmission distance is 2.42 km for the case without applying the reference CW light. With the assistance of the reference CW light the transmission distances are extended to 4.14 km when correlation coefficient is 0.85, and 4.78 km when correlation coefficient is 0.99, respectively. So the corresponding improvements of transmission distance are 1.72 km and 2.36 km, respectively. For the GG distribution, the improvements of transmission distance are 1.36 km when correlation coefficient is 0.85 and 2.30 km when correlation coefficient is 0.99, respectively.

## 5. Conclusion

We have analytically studied a reference CW light assisted optical detection scheme to mitigate the effect of scintillation in FSO transmission. Using the correlation coefficient between the intensities of the data light and the reference CW light, we have mathematically derived the joint intensity distributions of two co-propagating optical waves under log-normal distributed channel model and Gamma-Gamma distributed channel model, respectively. Using these joint intensity distributions, we have theoretically analyzed the BER performance of FSO transmission system that is assisted with a reference CW light to mitigate the effect of scintillation under different atmospheric turbulence conditions as well as under different correlation coefficients between the intensities of the data light and the reference CW light. We have also carried out the Monte-Carlo simulation and shown that theoretical results agree with simulation results well. The effectiveness of this scheme is strongly associated with both the strength of scintillation and the correlation coefficient between the intensities of data light and reference CW light. Our studies have showed that when the correlation coefficient is 0.99, the power reductions to achieve BER of 10^{−3} are 12.3 dB and 20.4 dB under moderate and strong scintillations (i.e., Rytov variances of 1.0 and 4.0), respectively. The power reductions for the data signal with a wavelength in the range of 1500 nm to 1600 nm are almost identical under a certain correlation coefficient, which demonstrates the feasibility of using a single reference CW light to mitigate the scintillation effects on all the channels of a WDM-FSO transmission system. The transmission distances are improved by 1.72 km and 2.36 km under the moderate atmospheric turbulence condition when the correlation coefficients are 0.85 and 0.99, respectively.

## Acknowledgments

The authors would like to thank the support of A*STAR SERC PSF 092 101 0054, and the help of Dr. Bingxing Xia in SPMS, NTU.

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