## 3 Bit Error Rate of the OAM Link and Channel Capacity
### Bit Error Rate of OAM Links  
To understand the OAM mode redistribution generated by turbulent seawater disturbance, we expanded the disturbed field into a series of the spiral harmonic exp (\(il\theta\)) [18; 19]  
\[\widetilde{E}(r,\theta,z)=\frac{1}{\sqrt{2\pi}}\underset{l=-\infty}{\overset{ \infty}{\sum}}a_{l}(r,z)e^{il\theta} \tag{12}\]  
with  
\[a_{l}(r,z)=\frac{1}{\sqrt{2\pi}}\underset{0}{\overset{\infty}{\sum}}E(r, \theta,z)e^{-il\theta}d\theta \tag{13}\]  
Note that \(l\
eq l_{0}\) is the topological charge of an OAM crosstalk mode.  
The ensemble average of \(|a_{l}|^{2}\) represents the condition probability distribution of OAM modes in turbulent as well as absorbent seawater and at receiving plane, which has the form of  
\[\begin{array}{ll}p_{l/l_{0}}(r)&=\left\langle|a_{l}(r,z)|^{2}\right\rangle\\ &=\frac{1}{2\pi}\underset{0}{\overset{\infty}{\sum}}\underset{0}{\overset{ \infty}{\sum}}E(r,\theta,z)E^{*}(r,\theta,z)\exp[-il(\theta-\theta^{\prime})] \\ &\times\exp\left[-2r^{2}[1-\cos(\theta-\theta^{\prime})]/\rho_{oc}^{2} \right]d\theta d\theta^{\prime}\big{|}\end{array} \tag{14}\]where * denotes the complex conjugate, and \(\rho_{\infty}\) is the spatial coherence radius of waves propagating in isotropic and turbulent as well as absorbent seawater and is given by [13]  
\[\begin{split}&\rho_{\infty}^{-2}=\frac{2.511\times 10^{-7} \text{e}^{-1/3}\chi_{T}(\eta_{+}^{2}+\eta_{+}^{2})z}{\lambda^{2}}\bigg{[}\kappa_ {0}^{1/3}U\bigg{(}2;\frac{7}{6};\frac{\kappa_{0}^{2}\eta^{2}}{R_{T}^{2}}\bigg{)} \\ &+4.6\eta^{2/3}\kappa_{0}\Gamma\big{(}\frac{7}{3}\big{)}U\bigg{(} \frac{7}{3};\frac{3}{2};\frac{\kappa_{0}^{2}\eta^{2}}{R_{T}^{2}}\bigg{)}+\frac{ 1}{\alpha^{2}}\kappa_{0}^{1/3}U\bigg{(}2;\frac{7}{6};\frac{\kappa_{0}^{2}\eta^ {2}}{R_{S}^{2}}\bigg{)}\\ &+\frac{4.6}{\alpha^{2}}\eta^{2/3}\kappa_{0}\Gamma\big{(}\frac{7 }{3}\big{)}U\bigg{(}\frac{7}{3};\frac{3}{2};\frac{\kappa_{0}^{2}\eta^{2}}{R_{S }^{2}}\bigg{)}-\frac{2}{\alpha^{2}}\kappa_{0}^{1/3}U\bigg{(}2;\frac{7}{6};\frac {\kappa_{0}^{2}\eta^{2}}{R_{T}^{2}}\bigg{)}\\ &-\frac{9.2}{\alpha}\eta^{2/3}\kappa_{0}\Gamma\big{(}\frac{7}{3} \big{)}U\bigg{(}\frac{7}{3};\frac{3}{2};\frac{\kappa_{0}^{2}\eta^{2}}{R_{T}^{2 }}\bigg{)}\bigg{]}.\end{split} \tag{15}\]  
Substituting Equation (9) into Equation (12), through integration operations, we obtain the probability distribution of OAM modes (see Appendix D)  
\[\begin{split}& p_{l/l_{0}}(r)=\frac{2\pi^{3}\big{(}\eta_{+}^{2}+ \eta_{-}^{2}\big{)}\big{[}\big{(}\frac{7\alpha}{3}\big{)}\big{(}\eta_{+}^{2}+ \eta_{+}^{2}\big{)}\big{]}^{10}w_{0}^{(l_{0}+1)}x^{2l_{0}}}{N_{2}\lambda^{2} \left[\left(2+\frac{2\pi}{\lambda}n_{i}w_{0}^{2}\right)^{2}+\left(n\frac{2\pi }{\lambda}w_{0}^{2}\right)^{2}\right]^{0+1}}\\ &\times\bigg{|}_{1}F_{1}\bigg{(}l_{0}+1;l_{0}+1;-\frac{\left[2\pi \right(n_{l}+n_{i})\right]^{2}x^{2}w_{0}^{2}}{4z^{2}+\frac{4\pi}{\lambda}(n_{ l}-in_{l})w_{0}^{2}}\bigg{)}^{2}\bigg{|}^{2}\exp\left[-\frac{2\pi}{\lambda}n_{i} \Big{(}z+\frac{p^{2}}{2z}\Big{)}-\frac{2\pi^{2}}{\rho_{\infty}^{2}}\right]\\ &\times\bigg{|}\Big{[}\big{(}|\alpha|-1\big{)}^{2}I_{l+l_{0}} \left(\frac{2\pi^{2}}{\rho_{\infty}^{2}}\right)+\big{(}|\alpha|+1\big{)}^{2}I _{l-l_{0}}\Big{(}\frac{2\pi^{2}}{\rho_{\infty}^{2}}\Big{)}\Big{]}\bigg{|}\end{split} \tag{16}\]  
The receiving probability of OAM mode carried by RCDPPGV and measured by the detector with detection diameter \(D\) can be expressed as  
\[P_{l/l_{0}}=2\pi\int_{0}^{D/2}p_{l/l_{0}}(r)rdr \tag{17}\]  
In addition, for sufficiently large transmit power, the signal-to-noise ratio (\(SNR\)) of OAM channel is approximately [28]  
\[SNR(l,l_{0})=\frac{P_{l/l_{0},l=l_{0}}}{\sum\limits_{l=-\infty}^{\infty}P_{l/l _{0},l\
eq l_{0}}} \tag{18}\]  
For signal modulation of on-off keying (OOK), based on (18), the bit error rate (\(BER\)) of OAM channels is derived as [29]  
\[BER(l,l_{0})=\frac{1}{2}\text{erfc}\sqrt{\frac{SNR(l,l_{0})}{2}} \tag{19}\]  
where \(\text{erfc}(\cdot)\) denotes the complementary error function.