## 4 Numerical Results, Evaluations, and Discussion  
In this section, we present the numerical results and evaluations of our proposed UOWC system concept by analyzing the probability of fade and the bit error rate (BER), which are important performance measures for communication systems. For numerical evaluations, the system parameters used in the simulations are listed in Table 1. For underwater optical channel modeling, we consider clean ocean water with an attenuation coefficient of \(c=0.151\,\mathrm{m}^{-1}\). The turbulence strength is varied by adjusting the temperature-to-salinity fluctuation ratio \(w\) to represent weak, moderate, and strong turbulence conditions. The transceiver setup includes a \((16\times 16)\) OPA antenna for electronic beam steering, an LC-SLM with a \(4\,\mathrm{mm}\) aperture for adaptive optics, and a converging lens for aperture averaging. The optical carrier operates at \(600\,\mathrm{THz}\) with a transmitted power of \(10\,\mathrm{mW}\) (\(10\,\mathrm{dBm}\)). The probability of fade is the percentage of time for which the received irradiance is below a certain threshold:  
\[P(I\leq I_{th})=\int_{0}^{I_{th}}p_{I}(I)dI. \tag{19}\]  
The substitution of Equation (10) in Equation (19) leads to  
\[P(I\leq I_{th})=\frac{1}{2}\left(1+\mathrm{erfc}\left(\frac{0.5\sigma_{I}^{2} +\ln\left(\frac{I_{th}}{\left\langle I\right\rangle}\right)}{\sqrt{2\sigma_{I} ^{2}}}\right)\right), \tag{20}\]  
where \(\mathrm{erfc}()\) is the error function and \(\left\langle I\right\rangle\) is the on-axis mean irradiance. In the literature, a fading threshold parameter \(F_{th}\) is defined, which determines the dB level below the mean on-axis irradiance, at which the threshold \(I_{th}\) is established [51]:  
\[F_{th}\,=\,10\,\log_{10}(\left\langle I\right\rangle/I_{th}). \tag{21}\]  
The substitution of Equation (21) in Equation (20) leads to  
\[P(I\leq I_{th})=\frac{1}{2}\left(1+\mathrm{erfc}\left(\frac{0.5\sigma_{I}^{2} \,-\,0.23\,F_{th}}{\sqrt{2\sigma_{I}^{2}}}\right)\right). \tag{22}\]  
In Figure 6 we present the probability of fade corresponding to our proposed system under clean ocean conditions for different aperture averaging lens diameters. A rapid increase in the probability of fade is evident when turbulence is not compensated, with \(P_{th}\) exceeding \(0.1\) within a distance of \(10\,\mathrm{m}\). Notably, at shorter distances, the lens diameter significantly influences the probability of fade, demonstrating that larger lenses provide greater mitigation through aperture averaging. However, at longer distances, the probability of fade converges across all lens diameters, indicating diminishing returns from increasing lens size. It is also noticeable that at shorter distances, the lens diameter becomes more important and the influence of aperture averaging is more pronounced with larger lenses, but at greater distances, the probability of fading converges for all lens diameters. Assuming binary phase shift keying (BPSK), the BER can be determined using the following equation [52; 53]:  
\[\mathrm{BER}_{\mathrm{BPSK}}=\int_{0}^{\infty}p_{I}(I)\,\mathrm{Q}\!\left( \sqrt{\left\langle\mathrm{SNR}\right\rangle}\,\right)dI, \tag{23}\]  
where \(p_{I}(I)\) is the probability density function of the irradiance fluctuation described in Equation (10). The function \(\mathrm{Q}()\) is the tail distribution function of a probability distribution. The average signal-to-noise ratio \(\langle\text{SNR}\rangle\) in a shot-noise limited system can be expressed by [23]  
\[\langle\text{SNR}\rangle=\frac{\text{SNR}_{0}}{\sqrt{1+\sigma_{lsp}^{2}\,\text{ SNR}_{0}^{2}}}, \tag{24}\]  
where \(\text{SNR}_{0}\) is the signal-to-noise ratio in the absence of turbulence. As shown in Figure 7, \(\langle\text{SNR}\rangle\) cannot increase to the same rate as \(\text{SNR}_{0}\). As the scintillation index increases, it results in a saturation effect where an increase in the transmit power does not lead to a corresponding rise in \(\langle\text{SNR}\rangle\).