## 3 Transmitting Probability of Signal Vortex Modes  
As discussed in literature [36,37], the complex amplitude \(E\left(r\,,\varphi,z\right)\) of MG beam can be decomposed into a superposition of several vortex modes  
\[E\left(r\,,\varphi,z\right)=\frac{1}{\sqrt{2\pi}}\sum\limits_{l=-\infty}^{ \infty}\beta_{l}\exp\left(\mathrm{i}l\varphi\right), \tag{12}\]  
where \(\beta_{l}\) is expansion coefficient, the ensemble average \(\left\langle\left|\beta_{l}\right|^{2}\right\rangle\) represents the probability density of vortex modes in oceanic turbulence, which has the form of  
\[\left\langle\left|\beta_{l}\right|^{2}\right\rangle=\frac{1}{2\pi}\int\limits \limits_{0}^{2\pi}\int\limits_{0}^{2\pi}E\left(r\,,\varphi,z\right)E^{*}\left( r\,,\varphi,z\right)\exp\left[-\mathrm{i}l\left(\varphi-\varphi^{\prime} \right)\right]d\varphi d\varphi^{\prime}. \tag{13}\]  
By combining Equation (1) with Equation (2) and Equation (13), probability density \(\left\langle\left|\beta_{l}\right|^{2}\right\rangle\) of vortex modes of the MG beam is calculated as  
\[ \begin{array}{l}\left\langle\left|\beta_{l}\right|^{2}\right\rangle=\frac{ (2\pi)^{-1}}{1+z_{\xi}^{2}}\exp\left[-\frac{k_{T}^{2}w_{0}^{2}z_{\xi}^{2}}{2 \left(z_{\xi}^{2}+1\right)}\right]\exp\left[-\frac{2r^{2}}{w_{0}^{2}\left(z_{ \xi}^{2}+1\right)}\right]\\ \times\exp\left[-\frac{2r^{2}}{\rho_{oc}^{2}}\right]\sum\limits_{p=0}^{\infty }\sum\limits_{p^{\prime}=0}^{\infty}A_{2p}^{\left(2l_{0}\right)}\left(q\right) A_{2p^{\prime}}^{\left(2l_{0}\right)}\left(q\right)\left(-1\right)^{p+p^{\prime}}\\ \times\mathrm{I}_{2p}\left(\mathrm{i}\frac{k_{T}r}{z_{\xi}-1}\right)I_{2p^{ \prime}}^{*}\left(\mathrm{i}\frac{k_{T}r}{z_{\xi}-1}\right)\int\
olimits_{0}^ {2\pi}\int_{0}^{2\pi}\mathrm{d}\varphi\mathrm{d}\varphi^{\prime}\exp\left[ \frac{2r^{2}\cos\left(\varphi-\varphi^{\prime}\right)}{\rho_{oc}^{2}}\right]\\ \times\mathrm{Re}\left\{\exp\left[\mathrm{i}\left(2p\varphi-2p^{\prime}\varphi^ {\prime}\right)\right]\right\}\exp\left[-\mathrm{i}l\left(\varphi-\varphi^{ \prime}\right)\right]\end{array} . \tag{14}\]  
With the help of the integral expression [31]  
\[\int_{0}^{2\pi}\exp[\vartheta\cos(\theta-\varphi)-\mathrm{i}m\theta]\,d\theta= 2\pi\exp(-\mathrm{i}m\phi)\mathrm{I}_{m}(\vartheta), \tag{15}\]  
where \(\mathrm{I}_{m}\left(\cdot\right)\) denotes the symbol of the modified Bessel function of the first kind, the probability density \(\left\langle\left|\beta_{l}\right|^{2}\right\rangle\) of vortex modes of the MG beam is given by  
\[ \begin{array}{l}\left\langle\left|\beta_{l}\right|^{2}\right\rangle=\frac{ 2\pi}{1+z_{\xi}^{2}}\exp\left[-\frac{k_{T}^{2}w_{0}^{2}z_{\xi}^{2}}{2\left(z_{ \xi}^{2}+1\right)}\right]\exp\left[-2\left(\frac{1}{w_{0}^{2}\left(z_{\xi}^{2} +1\right)}+\frac{1}{\rho_{oc}^{2}}\right)r^{2}\right]\\ \times\sum\limits_{p=0}^{\infty}\left[A_{2p}^{\left(2l_{0}\right)}\left(q\right) \right]^{2}\left|\left|\left.\left|\mathrm{I}_{2p}\left(\mathrm{i}\frac{k_{T}r }{z_{\xi}-1}\right)\right|^{2}\mathrm{Re}\left[\mathrm{I}_{I-2p}\left(\frac{2r ^{2}}{\rho_{oc}^{2}}\right)\right]\right.\end{array} . \tag{16}\]Considering a receiver with an aperture diameter of \(D\), we obtain the energy for each received vortex modes of the MG beam  
\[ \begin{split}& M_{l}=\frac{2\pi}{1+z_{\zeta}^{2}}\exp\left[-\frac{k_{r}^ {2}w_{0}^{2}z_{\zeta}^{2}}{2\left(\frac{2}{\zeta}+1\right)}\right]\sum\limits_{ p=0}^{\infty}\left[A_{2p}^{\left(2l_{0}\right)}\left(q\right)\right]^{2}\int \limits_{0}^{D/2}\!dr\\ &\times\exp\left[-2\left(\frac{1}{w_{0}^{2}\left(\frac{2}{\zeta} +1\right)}+\frac{1}{\rho_{\infty}^{2}}\right)r^{2}\right]\left|\left|\frac{1_{ k}r}{2_{p}}\left(\mathrm{i}\frac{k_{r}r}{z_{\zeta}-1}\right)\right|^{2}\!\mathrm{Re} \left[\mathrm{I}_{l-2p}\left(\frac{2r}{\rho_{\infty}^{2}}\right)\right]r\right. \end{split} .