## 4 Simulations and Results  
To examine transmission performance using simulation results, the vertical FSO system presented in Table 1 was considered. We set the vertical link distance to 20 km from an OGS to a UAV operating under minimal wind speed using the Bufton wind model [31]. The refractive structure at the ground was estimated to be 1.7 \(\times 10^{-13}\) m\({}^{-\frac{2}{3}}\) to examine the effect of increased strong turbulence on the vertical FSO link. The transmitter power was assumed as 30 dBm (1 W), and the modulation format of the FSO system was set to PAM4 to support high-data-rate transmission. The standard deviation of the pointing error was assumed to be 5 \(\upmu\)rad for fine-pointing [32], and the noise power was a value in which background noise, ASE beating noise, and dark current noise were all considered. We considered the relationship between scintillation, pointing error, and beam width to obtain the optimal beam width. The pointing errors were produced following the Rayleigh distribution beam widths. From the pointing errors, scintillation indexes were generated according to log-normal random distributions. The received optical power \(P_{Rxopt}\) considered for these relationships is  
\[P_{Rxopt}=P_{Rx}L_{p\bar{q}}L_{S\bar{I}}, \tag{11}\]  
where \(L_{p\bar{q}}\) is the pointing error loss given by  
\[L_{p\bar{q}} =\exp\left(-8\left(\frac{\theta_{p\bar{q}}}{\theta^{\frac{1}{2}} }\right)^{2}\right), \tag{12}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter (Symbol)** & **Value** \\ \hline Wavelength (\(\lambda\)) & 1550 nm \\ Transmitter power (\(P_{Tx}\)) & 30 dBm \\ Modulation format & PAM 4 \\ Aperture diameter (D) & 10 cm \\ Link distance (L) & 20 km \\ RMS wind speed (\(v\)) & 21 m/s \\ Refractive index structure on the ground (\(C_{n}^{2}\)) & \(1.7\times 10^{-13}\)m\({}^{-\frac{2}{3}}\) \\ Standard deviation of pointing jitter (\(r\)) & 5 \(\upmu\)rad \\ Target bit error rate (uncoded) & \(1\times 10^{-9}\) (\(1\times 10^{-3}\)) \\ Noise power (\(P_{N}\)) & \(2\times 10^{-5}\) W \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Vertical FSO link parameters. Figure 5: PAT schematic for OGS-to-UAV communication. where \(\theta_{plg}\) is angle of pointing error. The \(L_{SI}\) is scintillation loss according to log-normal random distributions about \(\theta^{\frac{1}{2}}\) and \(\theta_{plg}\). By assuming the noise relative to the received power to be the same, the bit error rate (BER) was calculated using the PAM 4 BER formular, and the beamwidth was selected with a minimum BER average. In other words, the optimal beam size was determined as a beam width with a minimum BER value calculated as  
\[\underset{\theta^{1/2}}{\text{min}}\bigg{(}\frac{1}{T}\int_{0}^{T}BER_{PA\text {mt}}(t)dt\bigg{)}=\underset{\theta^{1/2}}{\text{min}}\left(\frac{1}{T}\int_{0 }^{T}\frac{1}{2}erfc\left(\frac{P_{Rxopt}\left(\theta^{\frac{1}{2}},\theta_{ plg}(t)\right)}{4P_{N}}\right)dt\right), \tag{13}\]  
where \(T\) is a long enough period of time. For the adaptive beam, a pointing error is calculated by QPD at a specific time. The beam divergence controller controls the transmit beam angle according to the estimated pointing error \(\theta_{plg}\). \[\underset{\theta^{1/2}}{\text{min}}(BER_{PA\text{mt}})=\underset{\theta^{1/ 2}}{\text{min}}\left(\frac{1}{2}erfc\left(\frac{P_{Rxopt}\left(\theta^{\frac{ 1}{2}},\theta_{plg}\right)}{4P_{N}}\right)\right). \tag{14}\]  
Figure 6 shows the difference between the cases with and without scintillation. Figure 6a shows the relationship between beam divergence angle and pointing error. The markers on the graph represent the optimal beam width at each pointing error. A wider beam divergence angle is required when the pointing error increases from 5 to 10 \(\upmu\)rad. The optimal beam divergence angle ranges from 27 to 35 \(\upmu\)rad without scintillation and 29 to 37 \(\upmu\)rad with scintillation. In this simulation, the fixed beam divergence angle was determined to be 29 \(\upmu\)rad considering 5 \(\upmu\)rad pointing errors.