## 5 Numerical Analysis
### The Bandwidth, Modulation Order, SNR of Optical System and ABER  
In this sub-section, we will analyze the influence of modulation order \(M\), the bandwidth and link channel on ABER. To analyze which of M-QAM is more advantageous for the underwater POV transmission channel, in Figure 4 we give the numerical results for the bandwidth-limited ABER of POV carrier link with M-QAM versus the propagation distance and turbulence strength for \(B=0.05\) GHz. The Figure 4 shows the bandwidth-limited ABER of link increases with the increase of propagation distance \(z\) for each modulation order \(M\). Further, in \(C^{2}_{\zeta_{XT}}=10^{-13}\) K\({}^{2}\)/m\({}^{2/3}\), for \(z\leq 123\) m, ABER of 4-QAM is the lowest, but for \(z\geq 123\) m, ABER of 512-QAM is the lowest. In \(C^{2}_{\zeta_{XT}}=10^{-15}\) K\({}^{2}\)/m\({}^{2/3}\) (see Figure 4a), ABER of 4-QAM is always the lowest. That is, low order modulation has better performance over shorter distances, while high order modulation has advantages over long distances. Figure 5 show the evolutive curves of bandwidth-limited ABER of link with variation of the M-QAM and the turbulence strength. It can be seen from Figure 5 that the bandwidth-limited ABER increases with the increase of turbulence strength for M-QAM. Figure 5a shows that in \(z=100\) m, from \(C^{2}_{\zeta_{XT}}=10^{-15}\) K\({}^{2}\)/m\({}^{2/3}\) to \(10^{-13}\) K\({}^{2}\)/m\({}^{2/3}\), the bandwidth-limited ABER of 4-QAM is always the lowest. While Figure 5b shows that, in  
Figure 4: The bandwidth-limited ABER of the POV carrier link with M-QAM versus the propagation distance in the turbulence strength of (**a**) \(C^{2}_{\zeta_{XT}}=10^{-15}\) K\({}^{2}\)/m\({}^{2/3}\), (**b**) \(C^{2}_{\zeta_{XT}}=10^{-13}\) K\({}^{2}\)/m\({}^{2/3}\). Figure 3: The received probability of OAM signal modes carried by POV versus different topological charge of initial OAM modes and topological charge of new OAM modes for \(r_{0}=0.03\) m, \(\varpi=-4.5\), \(n_{I}=0.4487\times 10^{-9}\). (**a**) \(z=150\) m, \(D=0.02\) m; (**b**) \(z=150\) m, \(D=0.005\) m.  
\(z=150\) m, as \(C_{\varepsilon_{XT}}^{2}\geq 10^{-14}\) K\({}^{2}\)/m\({}^{2/3}\), when \(C_{\varepsilon_{XT}}^{2}\) continues to increase, the modulation order M of M-QAM corresponding to the minimum value of bandwidth-limited ABER increases successively. More importantly, both Figures 4 and 5 indicate that, in large turbulence fluctuation \(\sigma_{1 KJ_{T}}^{2}=1.23C_{\varepsilon_{XT}}^{2}k^{2/6}z^{11/6}\), high-order QAM has better anti-interference ability than low-order QAM. In Figure 6 we give the numerical results for the bandwidth-limited ABER of POV carrier link with M-QAM versus the bandwidth \(B\) for different propagation distance and turbulence strength. Figure 6 displays that the bandwidth limited ABER of link increases with the increase of bandwidth \(B\), this is obviously because the increase of lead to the decrease of SCNR (see Equation (9)). Specifically, comparing the change in propagation distance between Figure 6a,b and the change in turbulence strength between Figure 6b,c, we find that for the short length of communication link with narrow communication bandwidth in weak seawater turbulence, QAM of low order \(M\) should be adopted. However, for long communication links with wide communication bandwidth in strong seawater turbulence, QAM of high order \(M\) should be adopted. In Figure 7, we give the numerical results for the bandwidth-limited ABER of the POV carrier link with M-QAM versus the SNR of optical system \(P_{TX}/N_{0}\). Figure 7 displays that the bandwidth limited ABER of the link decreases with the increase of the \(P_{TX}/N_{0}\), that is, the increase of SNR \(P_{TX}/N_{0}\) of optical system contributes to the ability of the communication system to resist turbulent channel disturbance. This is obviously because the decrease of \(P_{TX}/N_{0}\) leads to the decrease of SCNR (see Equation (9)).