## 2 Multi-Beam NOMA System Model
\(n=\frac{-\ln 2}{\ln(\cos(\phi_{1/2}))}\) is the Lambertian radiation order, \(\phi_{1/2}\) is the LED half-power angle, \(\phi_{k,m}\) is the irradiance angle of the light from LED \(m\) when it is received by the \(k\)-th detector, \(A_{eff}(L,\psi)=\frac{\pi D_{t}^{2}\cos(\psi)/4}{\pi(L\tan(\phi_{1/2})+D_{t}/ 2)^{2}}\) is the detector effective area, \(\psi\) is the angle of incidence of the LED beam to the surface of the detector, \(D_{t}\) is the diameter of the LED condensing lens, \(D_{r}\) is the diameter of the receiving surface, and \(L_{ch}=\exp(-c(\lambda)L)\) denotes the optical channel attenuation, where \(c(\lambda)\) is the total attenuation coefficient of the seawater on the optical signal, and \(L\) is the beam propagation distance. The underwater environment will be affected by turbulence. Due to the random fluctuation of the density, salinity, and temperature of seawater, the refractive index of the water experiences random variations. This will cause significant fluctuations in the phase  
Figure 2: Diagram of underwater optical multi-beam NOMA system. and intensity of received optical signals at the receiving side. This study considers weak turbulent channels. To better reflect the effects of underwater temperature, salinity changes, and bubble perturbations in the actual test, the mixed exponential generalized gamma (EGG) distribution model is used, and the probability density function of the turbulence random attenuation coefficient can be expressed as [18]:  
\[f(I)=\varpi f(I;\gamma_{e})+(1-\varpi)g(I;[u_{e},v_{e},w_{e}]), \tag{4}\]  
where \(\varpi\) is the respective weights between the exponential and generalized gamma distributions that satisfy \(0<\varpi<1\). \(f(I;\gamma_{e})\) is the exponential distribution function parameterized by \(\gamma_{e}\), whose probability density function can be expressed as:  
\[f(I;\gamma_{e})=\frac{1}{\gamma_{e}}\exp\biggl{(}-\frac{I}{\gamma_{e}}\biggr{)}. \tag{5}\]  
\(g(I;[u_{e},v_{e},w_{e}])\) denotes the generalized gamma distribution, where \(u_{e}\), \(v_{e}\), and \(q_{e}\) are the shape parameter and scale parameter of the EGG distribution, whose probability density function can be expressed as:  
\[g(I;[u_{e},v_{e},w_{e}])=w_{e}\frac{I^{u_{e}w_{e}-1}}{v^{u_{e}w_{e}}}\frac{ \exp\Bigl{(}-\Bigl{(}\frac{I}{v_{e}}\Bigr{)}\Bigr{)}^{w_{e}}}{\Gamma(u_{e})}. \tag{6}\]  
Its flicker index can be expressed as follows:  
\[\sigma_{I}^{2}=\frac{2\varpi{\gamma_{e}}^{2}+(1-\varpi){v_{e}}^{2}\frac{\Gamma \left(u_{e}+\frac{2}{w_{e}}\right)}{\Gamma(u_{e})}}{\Gamma(u_{e})}-1. \tag{7}\]  
The signal fading caused by turbulence can be regarded as the received light intensity without the influence of turbulence multiplied by the fading coefficient [19; 20]. Then the channel gain between the \(m\)-th LED and the \(k\)-th user photodiode (PD) under turbulence effect is:  
\[h_{k,m}=\underset{\text{turbulence attenuation}}{\alpha}\underset{\text{transmissionloss}}{\alpha}\underset{\text{transmissionloss}}{\exp(-c(\lambda)L)}\underset{\text{transmissionloss}}{\eta_{t}\eta_{r}\frac{m+1}{2\pi}\cos^{m}(\phi_{k,m})A_{eff}(L, \psi)}, \tag{8}\]  
where the turbulence attenuation factor \(\alpha\) is obtained by the acceptance/rejection method [21]. Therefore, the \(K\times M\) order channel gain matrix \(\mathbf{H}\) of the underwater wireless optical multi-beam system is expressed as:  
\[\mathbf{H}= \begin{bmatrix}h_{1,1}&h_{1,2}&\cdots&h_{1,M}\\ h_{2,1}&h_{2,2}&\cdots&h_{2,M}\\ \vdots&\vdots&\ddots&\vdots\\ h_{K,1}&h_{K,2}&\cdots&h_{K,M}\end{bmatrix} .