## 5 Numerical Simulations and Analyses  
In this section, we will probe into how the distance \(d\) between the receiver and light source, and the receiver's tilted angle \(\theta\) influence the capacity lower bound. Meantime, the feasibility of overcoming the pointing error by tilting the receiver to improve the channel capacity of UWOC will also be investigated. Due to the limited space of the article, we will only discuss the Case 2 situation, i.e., the average-to-peak power ratio \(\alpha\) being in interval \([1/3,\ 1]\). Moreover, according to the requirements of the simulation platform, without loss of generality, we set the horizontal distance \(D\) between the light source and the circular surface in Figure 1 as 19.25 m, and the radius \(r\) of the quarter circle is 12.6 m. That is, we achieve the channel loss simulation and the corresponding fitting coefficients \(C_{i}\) in (4) based on the distance quantities set above. The key simulation parameters are summarized and listed in Table 1.  
Figure 3 depicts the channel capacity lower bounds of UWOC versus the tilting angle \(\theta\) when the distance \(d\) between the light source and the receiver varies. As shown in Figure 3, although \(d\) has different values, the variation trends of different curves with respect to the tilting angle \(\theta\) are roughly consistent. It can be seen clearly from the figure that each curve has an optimal tilting angle, say \(\theta_{0}\), relating to the maximum lower capacity bound at this distance. For example, if \(d\) equals to 23 m, the maximum lower bound on channel capacity can be achieved when \(\theta\) is set to 35 degrees, i.e., \(\theta_{0}=35^{\circ}\). Moreover, with the increase of the distance \(d\), the corresponding optimal tilting angle gradually increases, which means that when the receiver is farther away from the light source, the receiver plane needs to be deflected at a larger angle to overcome the adverse effect of the pointing error. Therefore, we can reasonably tilt the receiver plane according to the optimal tilting angle under certain distance, and thus improve the channel capacity significantly. Furthermore, we also find that the inclination angle \(\theta\) plays a dominant role in the capacity lower bound variation especially when \(\theta\) is larger; while for smaller \(\theta\) values, the distance \(d\) is the main influence factor. We also noticed that the capacity curve of \(d=19.25\) m  
seems considerably different from the others for tilting angles being larger than 75\({}^{\circ}\). The reason for this distinction is due to that the receiver at this distance is just underneath the transmitting light source (see Figure 1), where the optimal tilting angle is zero. So, when the tilting angle exceeds 75\({}^{\circ}\), the incident photons from the source are difficult to reach the receiver plane even the FOV = 180\({}^{\circ}\); obviously, the received photons by the receiver when the tilting angle is 90\({}^{\circ}\), that is, the receiver plane is vertical to the X-Y plane, will decrease to zero. This is why the capacity values of this curve decrease quickly after 75\({}^{\circ}\) and become zero when \(\theta\) is equal to 90\({}^{\circ}\). Figure 4 shows the distribution of the system capacity lower bounds along with the receiver locations at the circular plane under before and optimally tilting the receiver. From Figure 4a,b, we can see that, whether tilting the receiver plane or not, the lower capacity bounds on the circumference of the boundary are the worst; and the maximum lower bounds on capacity can be obtained when the receiver is located directly below the light source. This result can be interpreted as that the channel fading \(L\) gradually increases as distance \(d\) increases, leading to capacity performance deterioration. However, after tilting the receiver plane according to the optimal tilting angle, the lower capacity bound of the entire X-Y plane (especially at the boundary) is greatly improved.