## 2 Multi-Beam NOMA System Model  
A typical scenario of buoy-to-AUV downlink data transmission [15] is illustrated in Figure 1a. A semi-spherical structure of the UWOC system is deployed at the lower end of the buoy, and multiple LEDs are situated on this semi-spherical structure, constituting a multi-beam space division multiple access (SDMA) system. The multi-beam system can engage various spatial beams for communication based on the positions of user clusters. To enhance system access capacity and transmission efficiency, NOMA is employed within each beam to serve multiple users. Due to the unstable underwater environment, the disturbance of seawater can cause the transmitter to move randomly. This situation may result in the beam failing to cover the user cluster, causing communication interruption. Figure 1b,c show that when communication between the LED of the underwater multi-beam NOMA system and the user is interrupted, other LEDs can be lit based on the position of the user cluster to maintain communication. When the user clusters are close, there may be interference issues between different beams. As shown in Figure 2, we assume the deployment of M LEDs at the transmitting end of a multi-beam NOMA system, serving K users belonging to distinct user clusters. The user located at the cell boundary will receive the optical signal from the adjacent LEDs. In the underwater optical multi-beam NOMA system, power allocation is determined based on the channel conditions of different users. Less power is allocated to users with good channel conditions, and more power is allocated to users with poor channel conditions. At the transmitting end, the LED array employs power multiplexing to combine the signals of multiple users through superposition coding (SC) before transmitting. Figure 1: Illustration of a typical buoy-AUV underwater multi-beam NOMA system communication scenario. (**a**) Multi-user clusters communication, (**b**) LEDs communicate with the cluster, (**c**) communications link misalignment, change LEDs to maintain communication. Without loss of generality, we assume that \(K_{m}\in\{1,2,\ldots,M\}\) denotes the number of multiplexed users in the LED \(m\),\(m=\{1,2,\ldots,M\}\) and their channel qualities are sorted in descending order, i.e., \(h_{1}\geq h_{2}\geq\cdots\geq h_{K_{m}}\), then the LED \(m\) transmits a signal that can be expressed as:  
\[x_{m}=\sum_{k=1}^{K_{m}}\sqrt{a_{k}P_{m}}s_{k,m}, \tag{1}\]  
where \(s_{k,m}\) denotes the signal sent by the LED \(m\) to the user \(k\), \(k=\{1,2,\ldots,K_{m}\}\), \(a_{k}\) denotes the power allocation coefficient of user \(k\), and \(P_{m}\) is the transmit power of the LED \(m\). The signal received by the user \(k\) can be expressed as:  
\[y_{k}=\underbrace{h_{k,m}\sqrt{a_{k}P_{m}}s_{k,m}}_{\text{signal to be decoded}}+ \underbrace{\sum_{k^{\prime}=1}^{k-1}h_{k,m}\sqrt{a_{k^{\prime}}P_{m}}s_{k^{ \prime},m}}_{\text{intra-beaminterference}}+\underbrace{\sum_{m^{\prime}\
eq m} \sum_{k=1}^{K_{m^{\prime}}}h_{k,m^{\prime}}\sqrt{a_{k}P_{m^{\prime}}}s_{k,m^{ \prime}}}_{\text{inter-beaminterference}}+n_{k}, \tag{2}\]  
where \(h_{k,m}\) denotes the DC gain from the LED \(m\) to the user \(k\), and \(n_{k}\) denotes additive white Gaussian noise (AWGN) of zero mean and variance \(\sigma^{2}\). Assuming that each LED follows the Lambertian radiation model and only the line-of-sight (LOS) component is considered, the DC gain is calculated by the following equation [16; 17]:  
\[\hat{h}_{k,m}=\left\{ \begin{array}{l}\eta_{t}\eta_{r}\frac{m+1}{2\pi}\cos^{n }(\phi_{k,m})A_{eff}(L,\psi)L_{ch},\ 0\leq\psi\leq\psi_{c}\\ 0\,,\psi>\psi_{c}\end{array} \right., \tag{3}\]  
where \(\eta_{t}\) and \(\eta_{r}\) are the conversion efficiency of the transmitter and detector, respectively.