# Bit Error Rate of Multi-Gaussian Correlated Asymmetric Bessel Beam Through Turbulent Ocean
## 2 Theoretical Analysis  
In the cylindrical coordinate system, the complex amplitude of asymmetric Bessel-Gaussian beam at the source plane is expressed as [28]  
\[U(r,\varphi,0)=\exp\left(-\frac{r^{2}}{\omega_{0}^{2}}\right)\times\sum_{p=0} ^{\infty}\frac{c^{p}\exp\left(il_{0}\varphi+ip\varphi\right)}{p!}J_{l_{0}+p}( \alpha r), \tag{1}\]  
where \(r\) and \(\varphi\) represent the polar radius and polar angle, respectively, \(\omega_{0}\) is the Gaussian waist radius, \(c\) is the asymmetry factor, and \(l_{0}\) is the topological charge of OAM mode. \(\alpha=k\sin\theta_{0}\) is the scale factor with \(\theta_{0}\) indicating Bessel cone angle, \(k=2\pi/\lambda\) is beam wavenumber inverse to wavelength \(\lambda\), \(p\) is an integer, and \(J_{l_{0}+p}(\cdot)\) indicates Bessel function of the first kind. From the view of beam generation, the asymmetric Bessel-Gaussian beam is a fully coherent beam which can be regarded as the combination of Gaussian source with asymmetry Bessel mode. If the Gaussian source is replaced by the multi-Gaussian Schell-model source, the partially coherent MGCAB beam is generated. The multi-Gaussian Schell-model source can be conceptualized as a superposition of finite Gaussian functions, whose form \(G(r,r^{\prime},\ \varphi\,\ \varphi^{\prime})\) is represented by the correlation factor \(g(r,\ \varphi\ )\) as follows [14]:  
\[G\left(r,r^{\prime},\varphi,\varphi^{\prime}\right)=g(r,\varphi)g^{*}(r^{\prime },\varphi^{\prime})=\frac{1}{C_{0}}\sum_{n=1}^{N}\binom{N}{n}\frac{(-1)^{n-1}} {n}\exp\left[\frac{2 rr^{\prime}\cos(\varphi-\varphi^{\prime})-r^{2}-{r^{\prime }}^{2}}{2n\sigma^{2}}\right], \tag{2}\]  
where \(C_{0}=\sum_{n=1}^{N}\binom{N}{n}\frac{(-1)^{n-1}}{n}\) is the normalization factor, \(\binom{N}{n}\) is the binomial coefficient, \(N\) is beam index, and \(\sigma\) is the source coherence width. In the case of \(N=1\), the multi-Gaussian Schell-model degenerates into the common Gaussian Schell-model. So the complex amplitude of the MGCAB beam at the source plane can be expressed as  
\[E(r,\varphi,0)=U(r,\varphi,0)g(r,\varphi). \tag{3}\]Figure 1 illustrates the corresponding spectral density distributions of the MGCAB beam under different asymmetry factors \(c\). In contrast with the symmetric case (i.e., multi-Gaussian correlated Bessel beam) corresponding to \(c=0\), the transverse spectral density distribution of the MGCAB beam has higher asymmetry when \(c\) increases. The peak position of the transverse spectral density can also be altered by adjusting \(c\). When MGCAB beam propagates in oceanic turbulence, the fluctuation in the seawater refractive index brings wavefront distortion to the beam. Based on [PERSON] approximation theory, the disturbed complex amplitude \(E(r,\varphi,z)\) of the MGCAB beam is expressed as [30]  
\[E(r,\varphi,z)=U(r,\varphi,z)g(r,\varphi)\exp[\psi(r,\varphi,z)],\]  
\[U(r,\varphi,z)=\frac{1}{q}\exp\Biggl{(}ikz-\frac{i\omega^{2}z}{2 kq}-\frac{r^{ 2}}{q\omega_{0}^{2}}\Biggr{)}\times\sum_{p=0}^{\infty}\frac{c^{p}\exp(il_{0} \varphi+ip\varphi)}{p!}I_{l_{0}+p}\biggl{(}\frac{\alpha r}{q}\biggr{)}, \tag{4}\]  
where \(U(r,\varphi,z)\) represents the complex amplitude of the asymmetric Bessel-Gaussian beam at propagation distance \(z\) and \(\psi(r,\varphi,z)\) is the turbulence-induced phase perturbations. The term \(q=1+iz/z_{0}\) is related to the Rayleigh range \(z_{0}=k\omega_{0}^{2}/2\). As a result, partial energy of the transmitted OAM signal mode transfers to adjacent modes and modal crosstalk happens.