# Bit Error Rate of Multi-Gaussian Correlated Asymmetric Bessel Beam Through Turbulent Ocean
## 2 Theoretical Analysis
The received mode distribution can be obtained by using spiral harmonic decomposition, so Equation (4) is rewritten as [31]  
\[E(r,\varphi,z)=\frac{1}{\sqrt{2\pi}}\sum_{l}a_{l}(r,z)\exp(il\varphi), \tag{5}\]  
where \(l\) is the topological charge of each spiral harmonic component, \(a_{l}(r,z)\) is the weight coefficient with the form of  
\[a_{l}(r,z)=\frac{1}{\sqrt{2\pi}}\int_{0}^{2\pi}E(r,\varphi,z)\exp(-il\varphi) d\varphi. \tag{6}\]  
If the receiving aperture has a radius of \(R\), the spiral harmonic intensity \(Q_{l}\) is calculated by the ensemble averaged \(\langle a_{l}(r,z)\rangle\) as follows:  
\[Q_{l}=\int_{0}^{R}\langle a_{l}(r,z)a_{l}^{*}(r,z)\rangle rdr=\int_{0}^{R} \Big{\langle}|a_{l}(r,z)|^{2}\Big{\rangle}rdr. \tag{7}\]  
Figure 1: The normalized transverse spectral density distributions of the MGCAB beam at source plane. The asymmetry factors of (**a–f**) are 0, 0.1, 0.5, 0.9, 0.9 \(\exp(i\pi/4)\), 0.9 \(\exp(5i\pi/4)\), respectively, while other parameters are \(l_{0}=1\), \(N=5\), \(\theta_{0}=0.05\) mrad, \(\omega_{0}=0.01\) m, \(\sigma=0.05\) m, \(\lambda=532\) nm. Then, substituting Equations (4) and (6) into Equation (7), \(Q_{l}\) is given by  
\[ \begin{array}{rl}Q_{l}=&\frac{1}{2\pi}\int_{0}^{R}\int_{0}^{2\pi}\int_{0}^{2\pi }U(r,\varphi,z)U^{*}(r,\varphi^{\prime},z)g(r,\varphi)g^{*}(r,\varphi^{\prime}) \exp[-il(\varphi-\varphi^{\prime})]\\ &\times\langle\exp[\psi(r,\varphi,z)+\psi^{*}(r,\varphi^{\prime},z)]\rangle rdrd \varphi d\varphi^{\prime}\end{array} . \tag{8}\]  
The exponential term \(\langle\exp[\psi(r,\varphi,z)+\psi^{*}(r,\varphi,z)]\rangle\) in Equation (8) describes the oceanic turbulence effects on the beam, whose value is related to spatial coherence radius \(\rho_{0}\) as follows [30]:  
\[\langle\exp[\psi(r,\varphi,z)+\psi^{*}(r,\varphi^{\prime},z)]\rangle=\exp \Bigg{\{}-\frac{2r^{2}[1-\cos(\varphi-\varphi^{\prime})]}{\rho_{0}^{2}} \Bigg{\}}. \tag{9}\]  
As for oceanic turbulence characterized by OTOPS, its power spectrum \(\Phi_{n}(\kappa)\) has the form of [4; 32]  
\[\Phi_{n}(\kappa)=A^{2}\Phi_{T}(\kappa)+B^{2}\Phi_{S}(\kappa)+2 AB\Phi_{TS}( \kappa), \tag{10}\]  
where \(\kappa\) denotes the spatial frequency, \(\langle T\rangle\) is the average temperature, \(\langle S\rangle\) is the average salinity concentration, \(A\) and \(B\) are linear coefficients calculated by  
\[ \begin{array}{l}A=a_{2}\langle S\rangle+2a_{3}\langle T\rangle\langle S \rangle+2a_{4}\langle T\rangle+a_{6}\lambda^{-1}\\ B=a_{1}+a_{2}\langle T\rangle+a_{3}\langle T\rangle^{2}+a_{5}\lambda^{-1} \end{array} , \tag{11}\]  
with constants \(a_{1}=1.779\times 10^{-4}\), \(a_{2}=-1.05\times 10^{-6}\), \(a_{3}=1.6\times 10^{-8}\), \(a_{4}=-2.02\times 10^{-6}\), \(a_{5}=0.01155\), and \(a_{6}=-0.00423\). The spatial power spectra \(\Phi_{T}(\kappa)\) of temperature, \(\Phi_{S}(\kappa)\) of salinity, and their co-spectrum \(\Phi_{TS}(\kappa)\) are written as  
\[ \begin{array}{rl}\Phi_{j}(\kappa)=\frac{\beta_{0}e^{-1/3}\kappa^{-1/3}\chi_ {j}}{4\pi}\Big{[}1+21.61(\kappa\eta)^{0.61}f_{j}^{0.02}-18.18(\kappa\eta)^{0. 55}f_{j}^{0.04}\Big{]}\exp\left[-174.90(\kappa\eta)^{2}f_{j}^{0.96}\right]\\ f_{j}=\left(0.072^{4/3}\beta_{0}\right)/Pr_{j},\qquad j\in[T,S,TS]\end{array} , \tag{12}\]  
where \(\beta_{0}=0.72\) is the Obukhov-Corrsin constant; \(\varepsilon\) is the kinetic energy dissipation rate; \(\eta\) is the Kolmogorov microscale; \(Pr_{T}\), \(Pr_{S}\), and \(Pr_{TS}\) are the temperature Prandtl number, salinity Schmidt number, and the coupled Prandtl-Schmidt number, respectively. By means of the temperature-salinity gradient ratio \(H\) and eddy diffusivity ratio \(d_{r}\), the dissipation rates \(\chi_{T}\) of temperature, \(\chi_{S}\) of salinity, and their coupled value \(\chi_{TS}\) have relationships as follows:  
\[\chi_{S}=\chi_{T}\frac{dr}{H^{2}},\chi_{TS}=\chi_{T}\frac{(1+dr)}{2H}, \tag{13}\]  
\[d_{r}\approx \begin{cases}R_{P}+R_{p}^{0.5}(R_{P}-1)^{0.5},\quad R_{P}\geq 1\\ 1.85R_{P}-0.85,\qquad 0.5\leq R_{P}<1,\\ 0.15R_{P},\qquad R_{P}<0.5\end{cases}