## 3 Channel Model
### Pointing Error Model  
Since the size of the received light spot is always small, the probability of pointing errors is very high. In the UWOC link, \(p=(p_{x},p_{y})\) denotes the receiver plane spatial coordinates, and the radial displacement vector can be expressed as \(\mathbf{r}=\big{[}r_{x},r_{y}\big{]}^{\mathsf{T}}\), where \(r_{x}\) and \(r_{y}\) denote the displacements along the horizontal direction (\(p_{x}\)-axis) and the vertical direction (\(p_{y}\)-axis) of the beam in the detector plane, respectively. Assuming that the displacements follow independent Gaussian distributions along the \(p_{x}\)-axis and \(p_{y}\)-axis, we can express \(r_{x}\) and \(r_{y}\) as \(r_{x}\sim\mathrm{N}\big{(}\mu_{x},\sigma_{x}^{2}\big{)}\) and \(r_{y}\sim\mathrm{N}\big{(}\mu_{y},\sigma_{y}^{2}\big{)}\), respectively. Then, the radial displacement \(r=|\mathbf{r}|=\sqrt{r_{x}^{2}+r_{y}^{2}}\) follows a Rayleigh distribution with a mean of zero and identical variance; that is, \(\mu_{x}=\mu_{y}=0\), \(\sigma_{x}^{2}=\sigma_{y}^{2}=\sigma_{\mathrm{P}}^{2}\). Thus, the probability density function (PDF) of \(h_{\mathrm{p}}(t)\) can be expressed as [26; 27]:  
\[f_{h_{\mathrm{p}}}(h_{\mathrm{p}})=\frac{\xi^{2}}{A_{0}^{2}}h_{\mathrm{P}}^{ \xi^{2}-1}, \tag{11}\]  
where \(\xi=W_{\mathrm{geq}}/2\sigma_{\mathrm{P}}\), with \(W_{\mathrm{geq}}^{2}=W_{\mathrm{z}}^{2}\sqrt{\pi}\mathrm{erf}(v)/\left(2v\mathrm{ e}^{-v^{2}}\right)\) being the equivalent beam width at the receiver and \(\sigma_{\mathrm{P}}\) the pointing error displacement standard deviation at the receiver. \(A_{0}=[\mathrm{erf}(v)]^{2}\) is the fraction of the collected power when the radial distance is zero, \(\mathrm{erf}(\cdot)\) is the error function, and \(v=\sqrt{\pi/2}/(W_{x}/a)\). \(W_{x}^{2}\) is the beam waist at the distance z from the transmitter, and the detection aperture radius \(a=D_{\rm{rc}}/2\). The channel fading due to geometric spreading and pointing errors can be approximated by \(h_{\rm{p}}\approx A_{0}\exp\left(-2r^{2}/W_{\rm{geq}}^{2}\right)\), the approximate condition of which is \(W_{2}/a>6\)[28]. In terms of the analysis in Section 3.1, for the ocean turbulence channel, the beam radius of \(W_{\rm{z}}^{2}\) at the variable distance of \(z\) from the optical transmitter is characterized as the long-term beam spreading \(W_{\rm{LT}}^{2}\), considering the beam spreading and the beam wander; i.e., \(W_{\rm{z}}^{2}=W_{\rm{LT}}^{2}\). Moreover, \(\xi\) can be rewritten in a new form as \(\xi=\frac{W_{\rm{geq}}/a}{2\varphi_{\rm{p}}/a}\), where \(W_{\rm{geq}}/a\) can be expressed as a function of \(W_{\rm{z}}/a\) based on (07.20.03.0051.01) in [29]; i.e.,  
\[\frac{W_{\rm{geq}}}{a}=\frac{W_{\rm{z}}}{a}\sqrt{\sqrt{\pi}{\rm{erf}}(v)/\left( 2v{\rm{e}}^{-v^{2}}\right)}=\left(\frac{W_{\rm{z}}}{a}\right)\left[{}_{1}F_{ 1}\left(1;\frac{3}{2};\frac{\pi/2}{\left(W_{\rm{z}}/a\right)^{2}}\right) \right]^{1/2}, \tag{12}\]  
where \({}_{1}F_{1}(\cdot)\) is the confluent hypergeometric function [29], and \(W_{\rm{geq}}/a\) increases with the increase in \(W_{\rm{z}}/a\) according to Equation (12). The parameter for \(W_{\rm{z}}/a\) is defined as the normalized received beam waist coefficient, which represents the long-term beam spreading. The parameter \(\sigma_{\rm{p}}/a\) is the normalized pointing error deviation, which represents the jitter strength. Its value range is between 1 and 10. The larger the \(\sigma_{\rm{p}}/a\) is, the more serious the pointing error is [27]. Without loss of generality, it can be assumed that the radial distance caused by the pointing error is located along the \(p_{x}\)-axis. Within the unit symbol transmission interval, the average intensity at the finite-sized received aperture can be obtained with the help of the extended Huygens-Fresnel principle, as follows [3]:  
\[\langle I({\bf{p}}-{\bf{r}},z)\rangle=\frac{W_{0}^{2}}{W_{\rm{z}}^{2}}\exp \Biggl{[}-\frac{\left(p_{x}-r\right)^{2}+p_{y}^{2}}{W_{\rm{z}}^{2}}\Biggr{]}, \tag{13}\]  
where \(<>\) stands for a statistical average operation.