## 2 OAM Light and Ocean Turbulence Channel Model  
In addition to linear momentum, a light beam can also carry angular momentum, which can be divided into spin angular momentum (SAM) and OAM. SAM is associated with the polarization of photons and appears circular, whereas OAM is associated with helical phase wavefronts [17]. The biggest characteristic of an OAM beam is that it has a helical phase wave front, encircling the center. The phase of the light field changes, which determines the degree of torsion of the beam. The OAM beam with the same absolute value but an opposite sign of the topological charge changes in the opposite direction. The region of phase uncertainty is called the phase center singularity, where all rays interfere to form a hollow ring intensity distribution. The common vortex beams with OAM include the Laguerre-Gaussian (LG) beam, the Bessel-Gaussian (BG) beam, the perfect vortex beam, and the vector vortex beam. Among them, the preparation of the LG beam is relatively simple, so in the study of OAM wireless optical communication, the LG beam has become the most widely used OAM beam [18]. Under the paraxial approximation condition, its optical field distribution [19] is as follows:  
\[ \begin{split}& LG_{pt}(x,y,z)=\sqrt{\frac{2m!}{\pi(m+|\ell|)}} \frac{1}{w(z)}\left[\frac{\sqrt{2(x^{2}+y^{2})}}{w(z)}\right]^{|\ell|}\exp \Bigl{(}-\frac{x^{2}+y^{2}}{w^{2}(z)}\Bigr{)}L_{p}^{|\ell|}\Bigl{(}\frac{2(x^{ 2}+y^{2})}{w^{2}(z)}\Bigr{)}\\ &\times\exp\left[\frac{\hbar(x^{2}+y^{2})z}{2\left(2^{2}+z_{R}^{2 }\right)}\right]\exp\Bigl{[}-i(2p+|\ell|+1)\tan^{-1}\Bigl{(}\frac{z}{z_{R}} \Bigr{)}\Bigr{]}\exp\Bigl{[}-i\ell\tan^{-1}\Bigl{(}\frac{x}{y}\Bigr{)}\Bigr{]} \end{split}  \tag{1}\]  
\((2p+|\ell|+1)\tan^{-1}(z/z_{R})\) is the Guoy phase, \(z_{R}=kw_{0}^{2}/2\) is the Rayleigh distance, \(w_{0}\) is the beam waist radius, and \(w(z)\) represents the radius of the spot at the transmission distance \(z\). This can be shown in Equation (2):  
\[w(z)=w_{0}\Bigl{[}\Bigl{(}z^{2}+z_{R}^{2}\Bigr{)}/z_{R}^{2}\Bigr{]}^{1/2} \tag{2}\]  
Here, \(p\) is the number of radial modes of the LG beam, representing the number of concentric rings on the cross section of the beam, and \(L_{p}^{|\ell|}(x)\) is the Laguergaus polynomial, and its expression is  
\[L_{p}^{|\ell|}(x)=\sum_{i=0}^{p}{(-1)^{i}\frac{(p+|\ell|)!}{(p-i)!(|\ell|+i)!!i!}x^{i}} \tag{3}\]  
The essence of ocean turbulence is a random fluctuation of the refractive index of seawater caused by the combination of complex factors such as the temperature gradient and salinity gradient [20]. Ocean turbulence can degrade the quality of wireless optical signals in terms of beam intensity flicker, expansion, drift, wavefront, and amplitude distortion, resulting in a significant degradation in UWOC performance [21]. [PERSON] [22] et al. proposed a common refractive index fluctuation spectrum of seawater, which is expressed as follows:  
\[ \begin{split}\Phi(k_{x},k_{y})&=0.388\times 10^{-8}C_{n}^{2 }(\sqrt{k_{x}^{2}+k_{y}^{2}})^{-11/3}[1+2.35(\sqrt{k_{x}^{2}+k_{y}^{2}}\eta)^{ 2/3}]\\ &\times(e^{-A_{T}\delta}+\omega^{-2}e^{-A_{S}\delta}-2\omega^{-1} e^{-A_{TS}\delta})\end{split}  \tag{4}\]  
where \(k_{x}\) and \(k_{y}\) are the components of the spatial angular frequency of the refractive index fluctuation on the \(x,y\) axis, respectively, and \(C_{n}^{2}\) is the equivalent temperature structure parameter, given by Equation (5). \(\omega\) is the ratio of temperature caused by ocean turbulence to salinity caused by ocean turbulence, the value range of which is \([-5,0]\); when approaching \(-5\), the temperature gradient dominates ocean turbulence, and when approaching 0, salinity gradients dominate ocean turbulence.