# An All-Digital Optical Phase-Locked Loop Suitable for Satellite Downlinks +
## 2 Coherent Optical Receiver
### Turbulence Model  
The Earth's atmosphere affects the performance of an optical communication link between an LEO satellite and an OGS. The three main processes that affect optical signal propagation are absorption, scattering, and fluctuations in the atmospheric refractive index or turbulence [2]. Absorption and scattering are caused by the molecules and particles that make up the atmosphere, which attenuates the optical signal. In the case of absorption, it occurs when a photon interacts with molecules in the atmosphere where it is absorbed. Atmospheric turbulence is a phenomenon that originates in the thermodynamic exchanges between gases of different temperatures that compose the atmosphere. These thermodynamic exchanges generate movements of air masses, producing eddies of different refractive indices that move randomly, thus influencing the propagation of the optical signal. Due to the non-linear turbulence characteristics, the refractive index structure \(C_{n}^{2}\) is the most important parameter in characterizing the effects of atmospheric turbulence. Furthermore, this refractive index depends on the geographical location, altitude, atmospheric conditions, and time of day. Due to these various dynamic parameters of the atmosphere, several models have been formulated to describe the \(C_{n}^{2}\) profile. Among these models, the best known and most used in the literature is the Hufnagel-Valley model, whose profile is given by [2]:  
\[C_{n}^{2}(h)=8.148\times 10^{-56}v_{rms}^{2}h^{10}\mathrm{exp}^{-h/1000}+2.7\times 1 0^{-16}\mathrm{exp}^{-h/1500}+C_{0}\mathrm{exp}^{-h/100}\quad\mathrm{m}^{-2/3} \tag{3}\]  
where \(h\) is the height above ground level (m), \(C_{0}\) is the nominal value of \(C_{n}^{2}\) at ground level (m\({}^{-2/3}\)), and \(v_{rms}\) is the wind speed along the vertical path (m/s). In the simulations,the nominal value of \(C_{n}^{2}(0)\) at ground level will be considered with three different values of \(1\times 10^{-13}\) m\({}^{-2/3}\), \(5\times 10^{-14}\) m\({}^{-2/3}\), and \(1\times 10^{-15}\) m\({}^{-2/3}\), corresponding, respectively, to strong, moderate and weak turbulence. The International Telecommunication Union ITU-RP.1621-1 recommendation is used to calculate the corresponding ground wind speed through the Bufton wind model from the expression:  
\[v_{rms}=\sqrt{v_{g}^{2}\ +\ 33.11v_{g}\ +\ 360.31} \tag{4}\]  
where \(v_{g}\) is the ground wind speed. The simulated wind speeds \(v_{rms}\) correspond to ground wind speeds ranging from zero to 43 m/s (zero to 154.8 km/h). By selecting this range of wind speeds, it is possible to simulate the optical link under various weather conditions (zero m/s during a calm summer day, 154 km/h during a storm). For different wind speed values, the \(C_{n}^{2}(h)\) profile is modified only for an altitude between 5000 m and 20,000 m according to the Bufton model as shown in Figure 2. It's important to note that weak and strong turbulence exhibit the same behavior. The fluctuations in the refractive index in the atmosphere result in light scattering and variations in irradiance within the beam cross-section. Consequently, the optical wave will experience random power fading, resulting in a loss of signal-to-noise ratio. The scintillation index \(\sigma_{I}^{2}\) defines the variations in the received optical power due to the scintillation. It is described as the normalized irradiance, or the received power, of the optical signal as:  
\[\sigma_{I}^{2}=\frac{\langle I^{2}\rangle}{\langle I\rangle^{2}}-1=\frac{ \langle I^{2}\rangle-\langle I\rangle^{2}}{\langle I\rangle^{2}} \tag{5}\]  
where \(I\) (W/\(m^{2}\)) denotes the irradiance of the optical signal. In other words, scintillation can cause power losses at the receiver and, eventually, fade the received signal, increasing the system's BER. Therefore, a mathematical description of the probability density function (PDF) of a randomly fading intensity signal can be used to determine the reliability of the FSO system in such an environment [2; 14].