## 4 Numerical Results  
In this section, selected results for the ergodic capacity of FSO communication between mobile platforms under varying fog types, pathloss, and pointing errors in maritime environments are presented. To validate our analytical results, the Monte Carlo simulation results were used. In the following figures, the markers and lines represent the asymptotic and simulation results, respectively. In each figure, the parameters (i.e., \(\lambda\), \(a\), \(\omega_{z}\), and \(l\)) and their corresponding values listed in Table 2 are considered unless otherwise noted. Figures 2-4 are under both the HD and IM/DD techniques at \(l=0.2\), \(1\) km, and \(0.5\) km, respectively. In all figures, the analytical results perfectly match the simulation results. In addition, as \(\mu^{\prime}\) increases, the asymptotic results approach exact results. Even if the results do not match perfectly, they are useful as upper bounds. However, in some results (i.e., dense fog type, high pointing error, or a bad case), we can observe a large difference in ergodic capacity between the asymptotic results and the exact results at a low SNR. Here, the bad case considers both the dense fog type and high pointing error. Typical FSO communication systems operate over a wide range, especially in the high SNR range, compared to RF-based systems [18]. Therefore, the high SNR regime has a more important meaning, so we should focus on convergence to the exact results at a high SNR. Figure 2 shows the impact of varying fog types (i.e., dense, light) on the ergodic capacity with fixed values of pathloss (i.e., \(I_{l}=3.2004\times 10^{-9}\) in dense, \(I_{l}=0.4884\) in light) and pointing error (i.e., \(\zeta=2.5058\)). Expectedly, as fog density increased, ergodic capacity performance decreased. Figure 3 shows the impact of varying pointing errors (i.e., \(\xi=0.3268,\infty\)) on the ergodic capacity with fixed values of pathloss (i.e., \(I_{l}=0.0278\)) and moderate fog type (i.e., \(k=5.49\) and \(\beta=12.06\)). Expectedly, the more severe the impact of the pointing error, the worse the performance (i.e., the lower \(\xi\), the lower the ergodic capacity performance). Figure 4 presents the impact of the bad case and the good case of the combined foggy channel and pointing error with fixed pathloss value on the ergodic capacity performance. The bad case considers both the dense fog type and high pointing error (i.e., \(k=36.05\), \(\beta=11.91\), and \(\xi=0.5783\)) with \(I_{l}=5.7943\times 10^{-22}\), but the good case considers both the light fog type and no pointing error (i.e., \(k=2.32\), \(\beta=13.12\), and \(\xi\rightarrow\infty\)) with \(I_{l}=0.1667\). Expectedly, the good case results in a better ergodic capacity performance than the bad case. Figure 3: Ergodic capacity results for varying pointing errors under both HD and IM/DD techniques with \(l=1\) km. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Symbol** & **Value** \\ \hline Wavelength & \(\lambda\) & 1550 nm \\ Receiver radius & \(a\) & 10 cm \\ Beamwidth & \(\omega_{2}\) & 150 cm \\ Distance between Tx and Rx & \(l\) & 0.2–1 km \\ \hline \end{tabular}
\end{table} 
Table 2: Parameter settings in results. Figure 2: Ergodic capacity results for varying fog types under both HD and IM/DD techniques with \(l=0.2\) km. As the pathloss is the value determined by the visibility related to fog, the effect of varying the pathloss is not seen separately in the graph. As seen in all figures, the HD technique provides better ergodic capacity performance than the IM/DD technique. This is because the HD technique applies two-dimensional modulation in terms of phase and amplitude, whereas the IM/DD technique is limited to one dimension in terms of intensity; therefore, the IM/DD technique requires a higher SNR than the HD technique to achieve the same rate performance [19].