## 2 Normalized Amplitude Perfect Optical Vortex in Weak Turbulent and Absorptive Seawater
\(\kappa\) is the scalar spatial wave number of turbulent fluctuation, \(\kappa_{0}=2\pi/L_{0}\), \(\kappa_{T}=R_{T}/\eta\), \(\kappa_{S}=R_{S}/\eta\), \(\kappa_{TS}=R_{TS}/\eta\), \(\eta\) is turbulent inner scale, \(L_{0}\) is the outer scale of turbulence, \(R_{j}=\sqrt{3}\left[W_{j}-1/3+1/\big{(}9W_{j}\big{)}\right]^{3/2}/Q^{-3/2}(j=T,S,TS)\), \(W_{j}=\Bigg{\{}\left[\frac{\mathrm{Pr}_{j}^{2}\mathrm{Q}^{4}}{(6\beta)^{2}}- \frac{\mathrm{Pr}_{j}\mathrm{Q}^{2}}{8\beta}\right]^{1/2}-\left[\frac{1}{2^{ 2}}-\frac{\mathrm{Pr}_{j}\mathrm{Q}^{2}}{6\beta}\right]\Bigg{\}}^{1/3}\), \(Q\) is the non-dimensional constant; \(\mathrm{Pr}_{T}\) and \(\mathrm{Pr}_{S}\) respectively represent the Prandtl number of the temperature and salinity, \(\mathrm{Pr}_{TS}=2\,\mathrm{Pr}_{T}\,\mathrm{Pr}_{S}\ /\left(\mathrm{Pr}_{T}+\mathrm{Pr}_{S}\right)\); \(w_{\mathrm{eff}}=\left(\frac{2}{\overline{w}^{2}(z)}+\frac{2}{\rho_{S}^{2}}+ \frac{k_{0}n_{1}z}{R(z)}\right)^{-1/2}\) is defined as the effective radius of the POV in the absorbed seawater turbulence. Note the structural constants \(C_{\xi\chi T}^{2}\) of the salinity-temperature fluctuation of oceanic turbulence increases with the increase of the salinity fluctuation, but it decreases with the increase of the temperature fluctuation. The received probability of OAM modes of POV can be obtained  
\[P(m\mid m_{0})=2\pi\int_{0}^{D/2}p(m\mid m_{0})\rho\mathrm{d}\rho, \tag{8}\]  
where \(D\) is the receiver diameter. For \(m\
eq m_{0}\), \(P(m\mid m_{0})_{m\
eq m_{0}}\) is the received probability of OAM crosstalk modes that describes the probability of the photon migrating from the OAM signal mode to the new OAM mode, and \(m=m_{0}\), \(P(m\mid m_{0})_{m=m_{0}}\) is the received probability of OAM signal modes. It has been shown in [34] that OAM crosstalk distribution follows the zero-mean Gaussian distribution. Therefore, the communication link studied in this paper is still a zero mean additive Gaussian channel. Combined with the definition of SNR for bandwidth-constrained systems in [35] and the discussion of zero mean additive Gaussian channel in [30], we define a new average signal-crosstalk-noise ratio (ASCNR) of OAM channel as  
\[\gamma_{m,m_{0}}=\frac{P^{2}(m/m_{0})_{m=m_{0}}}{[\sum_{m=-\infty}^{\infty}P(m /m_{0})+N_{0}/P_{TX}]2B}, \tag{9}\]  
where \(P_{TX}\) is the transmit power and \(N_{0}\) is the transmitter and receiver noise power, \(B=2R_{b}/\log_{2}M\) is the bandwidth occupied by modulated signal, and \(R_{b}\) is the bit rate [36]. The Equation (9) can also be rewritten in terms of the bit rate  
\[\gamma_{m,m_{0}}=\frac{P^{2}(m/m_{0})_{m=m_{0}}\log_{2}M}{[\sum_{m=-\infty}^{ \infty}P(m/m_{0})+N_{0}/P_{TX}]4R_{b}}. \tag{10}\]  
Equation (10) indicates that when other system parameters remain unchanged, to maintain the ASCNR of the transmitted OAM signal, when the length of the communication channel or the turbulence strength within the channel increases, we must reduce the communication bandwidth or bit rate or increase the modulation order \(M\). For signal modulation of M-QAM, based on (9) [37], we can obtain the bandwidth-limited ABER of the OAM \(m\)th channel carried by the POV as  
\[ \begin{split}& P_{B}=\frac{2}{\log_{2}M}\bigg{(}1-\frac{1}{\sqrt{M}} \bigg{)}\operatorname{erfc}\sqrt{\frac{3\log_{2}M}{2(M-1)}\gamma_{m,m_{0}}}\\ &\quad\times\Bigg{[}1-\frac{1}{2}\bigg{(}1-\frac{1}{\sqrt{M}} \bigg{)}\operatorname{erfc}\sqrt{\frac{3\log_{2}M}{2(M-1)}\gamma_{m,m_{0}}} \Bigg{]},\end{split}  \tag{11}\]  
where \(\operatorname{erfc}(\cdot)\) denotes the complementary error function.