# The Polar Code Construction Method in Free Space Optical Communication
## 3 Atmospheric Turbulence Channel Model  
When a laser propagates in an atmospheric turbulent channel, the coherence of the transmitted beam decreases due to atmospheric turbulence, leading to fluctuations in the light intensity and increasing the bit error rate (BER) [28]. An example is the FSO transceiver located on a high-rise building [29]. In this paper, intensity modulation/direct detection (IM/DD) was used in an atmospheric laser communication system, and the atmospheric transmission channel was a discrete memory-free channel with AWGN and time-varying gain. Assuming that the signal traversed smoothly in the process of channel transmission, the mathematical model of the channel would be described as follows [30]:  
\[y=\eta Ix+n \tag{1}\]  
where \(x\) represents the transmitting signal, and \(y\) is the signal received at the receiver. \(I\) is the intensity fading induced by atmospheric turbulence. The \(\eta\) means the detector responsivity of the photodetector, and we assumed that \(\eta\) was equal to one for simplicity. \(n\) is the Gaussian white noise with mean 0 and variance \(\sigma^{2}\) out of the receiver side. Then the initial value of the message based on OOK modulation can be obtained as follows:  
\[L_{\mathrm{OOK}}=In\frac{P(x=0|y)}{P(x=1|y)}=In\frac{\exp\left(\frac{-y^{2}}{ 2\sigma^{2}}\right)}{\exp\left(\frac{-\left(y-l\right)^{2}}{2\sigma^{2}} \right)}=\frac{l^{2}-2 yl}{2\sigma^{2}} \tag{2}\]  
The transmission of optical signals, in atmospheric channels, can be affected by atmospheric turbulence, which can be divided into weak and strong turbulence channels according to the intensity of turbulence. The general consensus is that the light intensity of the flicker follows a log-Normal prior distribution in weak turbulence. When the fluctuations are higher and the turbulence is strong, the distribution of the received signal is Gamma-Gamma distribution. For a relatively broad range of turbulence intensity, the Gamma-Gamma distribution is accepted as a relatively accurate distribution, which can be expressed as [31]:  
\[f(I)=\frac{2(ab)^{(a+b)/2}}{\Gamma(a)\Gamma(b)}I^{(a+b)/2-1}K_{a-b}(2\sqrt{abI}) \tag{3}\]  
where a and b are denoted as the effective numbers of large-scale and small-scale turbulence eddies, respectively. \(\Gamma\left(\cdot\right)\) represents the gamma function, and \(K\) n (\(\cdot\)) is the modified Bessel function of the second kind of order n. a and b are related to atmospheric conditions, which are given by:  
\[\mathrm{a}= \left[\exp\left(\frac{0.49\sigma_{R}^{2}}{\left(1+1.11\sigma_{R} ^{12/5}\right)^{7/6}}\right)-1\right]^{-1} \tag{4}\] \[\mathrm{b}= \left[\exp\left(\frac{0.51\sigma_{R}^{2}}{\left(1+0.69\sigma_{R }^{12/5}\right)^{5/6}}\right)-1\right]^{-1} \tag{5}\]  
where \(\sigma_{R}^{2}\) is the Rytov variance, which represents irradiance fluctuations associated with atmospheric turbulence, given by [32]:  
\[\sigma_{R}^{2}=1.23C_{n}^{2}r^{7/6}L^{11/6}\hskip 14.226378 ptfor\hskip 8.535827 pt \lambda=1550\,\mathrm{nm} \tag{6}\]  
where \(k=2\pi/\lambda\) denotes the wave number, \(\lambda\) represents the wavelength. \(L=1\) km is the transmission distance, and \(C_{n}^{2}\) is the refractive index function structure constant of the turbulence atmosphere. When \(\sigma_{R}^{2}\ll 1\), the turbulence is weak. Otherwise, the turbulence is strong [33]. Figure 2 shows the probability density function of the distribution of the Gamma-Gamma model for different turbulence intensities. From the figure, it can be seen that as the turbulence increases, the curve deviates from the mean value of the light intensityand has a longer trailing edge, which means that the distribution is wider. This shows that the system performance becomes more unstable as the turbulence intensity becomes larger, which is consistent with the actual atmospheric turbulence channel variation. In the simulation experiments, the optical signal received at the receiver was affected by the turbulence randomly generated according to the model.