## 5 Conclusions  
In this study, we analyzed the performance of the ergodic capacity of FSO communication between mobile platforms, particularly under foggy channel, pathloss, and pointing error conditions in maritime environments. In particular, among the pointing errors, jitter was considered. In the case of foggy conditions, the beam is diffused by scattering while passing through the foggy channel [20]. Moreover, even if the maritime FSO communication system was initially aligned without boresight error by various tracking methods, this boresight error may still occur due to various factors (e.g., thermal expansion of the ship, maritime environmental factors, etc.). In this case, the performance will vary greatly due to the complex effects of the beam spread by foggy conditions and jitter.  
More specifically, looking at the result of zero boresight error in Figure 5, it can be seen that the performance is better when there is no jitter. This is because jitter makes the transmitted beam from the transmitter more likely to deviate from the receiver's photodetector (PD) [21]. On the other hand, when the boresight error is not zero, and there is a jitter, the diffused beam due to fog is more likely to be affected by the jitter, which can lead to an increase in the probability of detecting the beam in the photodetector (PD). As a result,  
Figure 4: Ergodic capacity results for combined effects of foggy channel with pathloss and pointing error under both HD and IM/DD techniques with \(l=0.5\) km.  
there may be cases where the performance of the system is improved compared to the case without jitter. This can happen because the combined effects of jitter and diffuse beams can increase the detection probability at the PD, which can compensate for the reduced signal detection probability due to boresight errors. However, this improvement is not always guaranteed and depends on various factors such as the amount of jitter, the size of the beam, and the characteristics of the fog. On the other hand, when \(\mu=110\) (i.e., when the beam deviates relatively less from the center of the PD), it can be seen that the performance is reversed when the SNR increases in the above environments. This is because beam spread is relatively reduced with an increase in SNR, and as a result, the complex effects caused by beam spread and jitter which have a positive effect on performance are reduced. Through these results, it can be confirmed that the effect of jitter on performance is relatively important in foggy conditions, so the effect of jitter is mainly considered in this paper.  
Based on the moment expressions, we derived the closed-form expression of asymptotic ergodic capacity for both conventional detection techniques (i.e., HD and IM/DD techniques). Subsequently, our derived analytical results were cross-verified with the simulation results via Monte Carlo simulations. Based on some selected results, we can confirm that the combined effects of the foggy channel with pathloss and pointing error degrade FSO performance, especially under dense fog type and high pointing error conditions. In addition, the results show the numerical difference between HD and IM/DD techniques and can thus help determine the appropriate detection technique.  
Validation, D.D.H.; Formal analysis, J.E.H.; Investigation, S.S.N.; Supervision, M.-S.A.; Funding acquisition, C.Y. All authors have read and agreed to the published version of the manuscript.  
This work was supported in part by the National Research Foundation of Korea (NRF) grant funded by the Korean government (NRF-2021R1F1A1047271) and by the Gachon University research fund of 2021 (GCU-202110170001) and in part by IITP grant funded by the Korean government (MSIT, MOIS, MOLIT, MOTIE) (No. 2020-0-00061).  
The authors declare no conflict of interest.  
Figure 5: Ergodic capacity results for varying both boresight (i.e., \(\mu=0,110,180\)) and jitter (i.e., \(\xi=0.3216,\infty\)) with dense fog and \(I_{l}=5.7943\times 10^{-22}\) under HD technique with \(l=0.5\) km.