## 2 Theoretical Model
### Received Probability of Signal Vortex Modes  
Using the plane harmonic superposition theory of complex light waves, we can express the BGLW \(u_{p}(\rho,\theta,z,\omega)\) in Equation (5) as [27]  
\[u_{p}(\rho,\theta,z,\omega)=\frac{1}{\sqrt{2\pi}}\sum_{m}A_{m}(\rho,z,t)\exp( \mathrm{i}m\theta) \tag{16}\]  
where \(m\) is the topological charge number of the vortex mode in turbulent seawater. By comparing Equation (5) and Equation (16), we can obtain the superposition coefficient \(A_{m}(\rho,z,t)\), that is  
\[\begin{split} A_{m}(\rho,z,t)&=\frac{1}{\sqrt{2 \pi}}\int_{0}^{2\pi}u_{p}(\rho,\theta,z,\omega)\exp(-\mathrm{i}m\theta)\mathrm{ d}\theta\\ &=\frac{T_{0}}{2\sqrt{2\pi}}\exp(-\mathrm{i}\widetilde{\Omega}t) \int_{0}^{2\pi}\int_{-\infty}^{+\infty}\exp(-\frac{T_{0}^{2}}{4}\omega^{2})\\ &\times u(\rho,\theta,z,\omega)\exp(-\mathrm{i}\omega t-\mathrm{ i}m\theta)\mathrm{d}\omega\mathrm{d}\theta,\end{split} \tag{17}\]  
Squaring Equation (17) and performing the turbulence ensemble average, we obtain the received probability density  
\[P(m|m_{0})= \frac{T_{0}^{2}}{8\pi^{2}}\iiint\Gamma_{2}(\rho,\theta_{1},\theta_{2}, z;\omega_{1},\omega_{2})\exp\left[-\mathrm{i}(\omega_{1}-\omega_{2})t- \mathrm{i}m(\theta_{1}-\theta_{2})\right]\mathrm{d}\theta_{1}\mathrm{d}\theta _{2}\mathrm{d}\omega_{1}\mathrm{d}\omega_{2}. \tag{18}\]  
Further, substituting Equations (9), (11), (13) into Equation (18), we write Equation (18) as  
\[\begin{split} P(m|m_{0})=& E_{0}^{2}\exp\left[-\frac{2 z^{2}}{w_{0}^{2}}(\sin^{2}\alpha+\frac{\rho^{2}}{z^{2}})-\frac{\xi^{2}}{T_{0}^{2}} \right]\left|I_{m_{0}}(2z\frac{\sin\alpha}{w_{0}^{2}}\rho)\right|^{2}\\ &\times\sum_{q=0}^{m_{0}}\frac{m_{0}!}{(m_{0}-q)!q!}\left(\frac{w _{0}^{2}}{2 zcT_{0}}\right)^{2m_{0}-q}2^{q-1}(2m_{0}-2q-1)!!\\ &\times D_{q}(\frac{2\xi}{T_{0}})\iint\mathrm{d}\theta_{1} \mathrm{d}\theta_{2}\exp\{-\frac{2\rho^{2}}{\rho_{oc\xi}^{2}}\left[1-\cos( \theta_{1}-\theta_{2})\right]\\ &+\mathrm{i}\left(m_{0}-m\right)(\theta_{1}-\theta_{2})\}.\end{split} \tag{19}\]Recalling the integral formula [19]  
\[\int_{0}^{2\pi}\exp\left[-\mathrm{i}y\theta+x\cos(\theta-\theta^{\prime})\right] \mathrm{d}\theta=2\pi\exp(-\mathrm{i}y\theta^{\prime})I_{y}(x), \tag{20}\]  
we have  
\[\begin{split} P(m|m_{0})=&\pi E_{0}^{2}\exp\left[-2 \frac{z^{2}}{w_{0}^{2}}\sin^{2}\alpha-\frac{\xi^{2}}{T_{0}^{2}}-2\left(\frac{1 }{w_{0}^{2}}+\frac{1}{\rho_{oc\zeta}^{2}}\right)\rho^{2}\right]\\ &\times I_{m-m_{0}}\left(2\frac{\rho^{2}}{\rho_{oc\zeta}^{2}} \right)\left|I_{m_{0}}\left(2z\rho\frac{\sin\alpha}{w_{0}^{2}}\right)\right|^{ 2}\\ &\times\sum_{q=0}^{m_{0}}\frac{m_{0}!}{(m_{0}-q)!q!}2^{q}\left( \frac{w_{0}^{2}}{2 zcT_{0}}\right)^{2m_{0}-q}\\ &\times(2m_{0}-2q-1)!!D_{q}\left(2\frac{\xi}{T_{0}}\right).\end{split} \tag{21}\]  
By integrating Equation (21) over the receiving aperture \(D\), we have the received probability of vortex modes of the BGLW in seawater turbulence  
\[P(m)=P_{D}(m|m_{0})/\sum_{m=-\infty}^{\infty}P_{D}(m|m_{0}) \tag{22}\]  
and  
\[P_{D}(m|m_{0})=2\pi\int_{0}^{D/2}P(m|m_{0})\rho\mathrm{d}\rho, \tag{23}\]  
where \(m=m_{0}\) means the received probability of signal vortex modes. But, \(m\
eq m_{0}\) implies the received probability of crosstalk vortex modes.