# Bit Error Rate of Multi-Gaussian Correlated Asymmetric Bessel Beam Through Turbulent Ocean
## 2 Theoretical Analysis
 \tag{14}\]  
where \(R_{P}\) is the density ratio with the form of \(R_{P}=\alpha_{T}|H|/\beta_{S}\), \(\alpha_{T}\) and \(\beta_{S}\) are the thermal expansion coefficient and saline contraction coefficient, respectively. Under the assumption of atmospheric pressure, the values of \(\alpha_{T}\) and \(\beta_{S}\) under different temperatures and salinity concentrations can be calculated using the TEOS-10 toolbox [33]. Then, the expression of \(\rho_{0}\) can be obtained after some lengthy derivations based on the power spectrum \(\Phi_{n}(\kappa)\) of OTOPS and has the form of [32]  
\[\rho_{0}=\Big{[}0.7246k^{2}z\beta_{0}\epsilon^{-1/3}\Big{(}A^{2}\chi_{T}+B^{2 }\chi_{S}+2 AB\chi_{TS}\Big{)}\Big{]}^{-1/2}. \tag{15}\]  
Furthermore, based on the expressions given by Equations (2), (4), and (9), the spiral harmonic intensity \(Q_{l}\) in Equation (8) is expanded as \[ \begin{split} Q_{l}&=\frac{1}{2\pi}\frac{1}{q}\frac{1}{q^{ *}}\exp\left(-\frac{\alpha^{2}}{k}\cdot\frac{z_{0}}{1+z_{0}^{2}/z^{2}}\right) \frac{1}{C_{0}}\sum\limits_{n=1}^{N}\left(\begin{array}{c}N\\ n\end{array}\right)\frac{(-1)^{n-1}}{n}\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^ {R}\exp\left(-\frac{r^{2}}{\omega_{0}^{2}}\cdot\frac{2}{1+z^{2}/z_{0}^{2}} \right)\\ &\times\sum\limits_{p=0}^{\infty}\sum\limits_{p^{\prime}=0}^{\infty}\frac{c^{ p}(c^{*})^{p^{\prime}}\exp[il_{0}(\varphi-\varphi^{\prime})+ip\varphi-ip^{ \prime}\varphi^{\prime}]}{p!p^{\prime}!}\exp\left[-r^{2}\frac{1-\cos(\varphi- \varphi^{\prime})}{n\sigma^{2}}\right]\exp\left[-2r^{2}\frac{1-\cos(\varphi- \varphi^{\prime})}{\rho_{0}^{2}}\right]\\ &\times I_{l_{0}+p}\left(\frac{\alpha r}{q}\right)I_{l_{0}+p^{\prime}}\left( \frac{\alpha r}{q^{*}}\right)\exp[-il(\varphi-\varphi^{\prime})]rdrd\varphi d \varphi^{\prime}\end{split}  \tag{16}\]  
Considering the following integral formulae [34]  
\[ \begin{split}\int_{0}^{2\pi}\exp[-in\varphi+x\cos(\varphi- \varphi^{\prime})]d\varphi=2\pi\exp(-in\varphi^{\prime})I_{n}(x)\\ \int_{0}^{2\pi}\exp(in\varphi)d\varphi=2\pi\delta(n)\qquad[n\in N] \end{split} , \tag{17}\]  
where \(I_{n}(\cdot)\) is the modified Bessel function of the first kind and \(\delta(\cdot)\) is Dirac delta function, the simplified expression of Equation (16) is  
\[ \begin{split} Q_{l}&=2\pi[1+z^{2}/z_{0}^{2}]^{-1} \exp\left(-\frac{\alpha^{2}}{k}\cdot\frac{z_{0}}{1+z_{0}^{2}/z^{2}}\right) \frac{1}{C_{0}}\sum\limits_{n=1}^{N}\left(\begin{array}{c}N\\ n\end{array}\right)\frac{(-1)^{n-1}}{n}\\ &\times\sum\limits_{p=0}^{\infty}\frac{\left|c^{2}\right|^{p}}{(p!)} \int_{0}^{R}\exp\left[-2r^{2}\left(\frac{1}{\omega_{0}^{2}}\cdot\frac{1}{1+z^ {2}/z_{0}^{2}}+\frac{1}{\rho_{0}^{2}}+\frac{1}{2n\sigma^{2}}\right)\right]\\ &\times I_{l-l_{0}-p}\left(\frac{r^{2}}{n\sigma^{2}}+\frac{2r^{2}}{\rho_{0}^ {2}}\right)I_{l_{0}+p}\left(\frac{\alpha rz_{0}}{z_{0}+iz}\right)I_{l_{0}+p} \left(\frac{\alpha rz_{0}}{z_{0}-iz}\right)rdr\end{split} .\end{split} \tag{18}\]  
As a result, the reception probability of each spiral harmonic component has the form of  
\[P_{l}=\frac{Q_{l}}{\sum_{l^{\prime}=-\infty}^{\infty}Q_{l^{\prime}}}, \tag{19}\]  
When the topological charge \(l\) of the received OAM mode equals to the topological charge \(l_{0}\) of the transmitted mode, \(P_{l}\) denotes the signal mode probability. While in the case of \(l\
eq l_{0}\), \(P_{l}\) denotes the crosstalk mode probability. The signal-to-noise ratio \(SNR\) is defined as [35]  
\[SNR=\frac{P_{l=l_{0}}}{\sum_{l=-\infty}^{\infty}P_{l\
eq l_{0}}+N_{0}/P_{TX}}. \tag{20}\]  
Here, the noise consists of two parts, namely the crosstalk noise induced by oceanic turbulence and system background noise \(N_{0}\). Assuming that the power of the MGCAB beam at the transmitter is \(P_{TX}\), the terms \(P_{TX}\cdot P_{l=l_{0}}\) and \(P_{TX}\cdot\sum_{l=-\infty}^{\infty}P_{l\
eq l_{0}}\) represent the received signal power and crosstalk power, respectively.