## 2 Monte Carlo-Based Channel Parameter Characteristics Analysis
### Detection Efficiency Analysis  
In the CVQKD system, a practical homodyne detector is characterized by an efficiency \(\eta\) and an electronic noise \(v_{el}\). Therefore, the noise introduced by the detector referred to Bob's input is given by \(\chi_{hom}=(1+v_{el})/\eta-1\). However, in an underwater scenario, the photons arriving the receiver plane may not be all received by the detector because of the finite size of the receiver aperture and FOV. As depicted in Figure 3, a series of photons propagate along z-axis toward a receiver with an aperture radius \(r\), although several scattered photons reach the receiving plane, they are still considered lost due to exceeding the size of the aperture. To take the effect of the receiver aperture size and FOV into consideration, we define the total detection efficiency of Bob as \(\eta_{t}=\eta_{r}\eta\), where \(\eta_{r}\) is defined by  
\[\eta_{r}=\frac{\sum_{j=1}^{N_{r}}w_{j}}{\sum_{i=1}^{N_{a}}w_{i}}, \tag{3}\]  
where \(w_{i}\) denotes the weight of the photons arriving the receiver plane, \(w_{j}\) denotes the weight of the photons successfully received, and \(N_{a}\) and \(N_{r}\) represent the number of the photons arriving and received, respectively. Figure 4a shows the value of \(\eta_{r}\) calculated by Monte Carlo simulation versus different size of receiver aperture under four types of water. We can see that the value of \(\eta_{r}\) is extremely affected by the type of water. For example, in the turbid harbor water, \(\eta_{r}\) is very small, while in the pure sea water, the value of \(\eta_{r}\) is approximately equal to 1 when the receiver aperture is larger than 2 inches. This is due to the fact that in turbid harbor water, photons are scattered many times before reaching the receiving plane, which causes a large displacement in their position on the x/y plane. Furthermore, when the size of the receiver aperture is smaller than 2 inches, the value of \(\eta_{r}\) is sensitive to the size of the aperture. This is because that a larger aperture can capture a larger number of photons scattered out of the incident beam. Figure 4b shows the value of \(\eta_{r}\) versus different receiver FOV under four types of water, and we notice that in pure sea water, clear ocean water and coastal ocean water, the value of \(\eta_{r}\) is almost unaffected by the receiver FOV, while in turbid harbor water the value of \(\eta_{r}\) varies greatly with FOV. In addition to water types, the effects of the beam width \(\omega_{0}\) and the maximum beam divergence \(\theta_{0,max}\) are also considered. The intensity distributions of the light with different beam widths on the receiver after transmitting through clear ocean water are shown in Figure 5a-c. We can see that with the increase of the beam width, the center of the beam spot becomes weaker, because a large beam width makes the initial positions of the photons more dispersed. Figure 5d-f show the intensity distribution of the light with different values of \(\theta_{0,max}\) on the receiver after transmitting through clear ocean water. We notice that with the increase of \(\theta_{0,max}\), the beam spot gets larger and the intensity of the light becomes weaker, which indicates that the received power of the light is affected by the maximum initial divergence angle. Therefore, the detection efficiency should be calculated by integrated considering the water types and the characteristics of the light source. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Water Types** & \(\mathbf{a(\mathrm{m^{-1}})}\) & \(\mathbf{b(\mathrm{m^{-1}})}\) & \(\mathbf{c(\mathrm{m^{-1}})}\) \\ \hline Pure sea water & 0.0405 & 0.0025 & 0.043 \\ Clear ocean water & 0.114 & 0.037 & 0.151 \\ Coastal ocean water & 0.179 & 0.219 & 0.398 \\ Turbid harbor water & 0.366 & 1.824 & 2.190 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Coefficient values of typical water types at 520 nm. Figure 4: **(a)**\(\eta_{r}\) calculated by Monte Carlo simulation versus different size of receiver aperture under four types of water.