## 5 Numerical Results and Discussion
### System Average BER  
First, we illustrate the effect of the turbulence strength on the system average BER with changes in the normalized pointing error coefficient \(\sigma_{\text{p}}\)/\(a\) or the average received SNR. These three ocean turbulence scenarios were set from Table 3. In Figure 5, the selected ARGO node numbers are one, two, and three, respectively. As the link length is 30 m, their scintillation indexes are 0.770, 1.523, and 2.643, respectively. It can be seen in Figure 5a that the system average BER shows an upward trend with increasing \(\sigma_{\text{p}}\)/\(a\) or ocean turbulence strength. The numerical results are consistent with the analytical results when \(\sigma_{\text{p}}\)/\(a\)\(>\) 2. In Figure 5b, when the link length is 26 m, the scintillation indexes for the three nodes are 0.553, 1.102, and 1.860, respectively. The numerical results are consistent with the analytical results for a high SNR when \(\xi^{2}\) \(<\) 1.  
Secondly, to examine a QAM order \(m\) of 8, 16, or 32 with an increase in the average received SNR, we selected Argo node 4 and determined the effects of the normalized pointing error deviations \(\sigma_{\text{p}}\)/\(a\) and the normalized pointing error coefficient \(W_{2}\)/\(a\) on the system average BER, respectively. As shown in Figure 6, the system average BER performance could be weakened with increases in \(\sigma_{\text{p}}\)/\(a\), \(W_{2}\)/\(a\), or \(m\). According to the numerical results shown in Figure 4, long-term beam spreading increases with increasing turbulence strength. Thus, the above results are consistent with the conclusion that the system average BER increases with increasing turbulence strength or decreasing received aperture diameter.  
Figure 4: Received optical power at the finite receiving aperture (Argo node 4): (**a**) \(D_{\text{rx}}\) = 5 mm; (**b**) strong ocean turbulence (z = 26 m,\(\sigma_{\text{I}}^{2}\) = 1.670).  
Figure 5: System average BER for three different Argo nodes: (**a**) changing the normalized jitter, \(\sigma_{\text{p}}\)/\(a\); (**b**) changing the average received SNR.  
Thirdly, to examine a QAM order \(m\) of 8, 16, or 32 with an increase in the received aperture diameter \(D_{\rm rx}\), we selected node 4 and verified the effects of \(\sigma_{\rm p}/a\) and \(W_{\rm z}/a\) on the system average BER, respectively. The numerical results are shown in Figure 7. In Figure 7, when the link length is 26 m, the scintillation index of node 4 is 1.670. In Figure 7, \(m=32\) and, when \(\sigma_{\rm p}/a\) is 2, 3, or 4, the received aperture diameter \(D_{\rm rx}\) that the system designs for the average BER is less than the forward error correction (FEC) threshold (\(3.8\times 10^{-3}\)), which is 6.1 mm, 8.1 mm, or 9 mm. Similarly, in Figure 7, \(W_{\rm z}/a=8\) when the QAM order \(m\) is 8, 16, or 32, and \(D_{\rm rx}\) is chosen as 5.7 mm, 6.6 mm, or 6.9 mm. Obviously, the effect of \(\sigma_{\rm p}/a\) on the system average BER is greater than that of \(W_{\rm z}/a\).  
Figure 6: Different average received SNRs: (**a**) QAM order \(m=8\), 16, and 32 and \(\sigma_{\rm p}/a=2\), 3, and 4 for node 4; (**b**) QAM order \(m=8\), 16, and 32 and \(W_{\rm z}/a=8\), 9, and 10 when \(\sigma_{\rm p}/a=2\).  
Figure 7: System average BER with different received aperture diameters for Argo node 4: (**a**) QAM order \(m=8\), 16, and 32 and \(\sigma_{\rm p}/a=2\), 3, and 4; (**b**) QAM order \(m=8\), 16, and 32 and \(W_{\rm z}/a=8\), 9, and 10.  
Finally, we verified the coding gain with the different link lengths in three Argo nodes. The numerical results are shown in Figure 8. Based on the results shown in Figure 2, the turbulence strength increases with the increase in link length. Obviously, the coding gain is weakened due to the pointing errors and the turbulence strength.