## Appendix A  
According to [40], when the mean of a Poisson distribution approaches infinity, its PDF tends to approximate the PDF of a Gaussian distribution. At this point, the Gaussian distribution's mean and variance are consistent, as follows:  
\[\frac{\lambda^{y}}{y!}\exp(-\lambda)\stackrel{{\lambda\to\infty}} {{\longrightarrow}}\frac{\exp\!\left(-\frac{(y-\lambda)^{2}}{2\lambda}\right)} {\sqrt{2\pi\lambda}},\quad\forall y\in[0,+\infty). \tag{1}\]  
As the system's transmit peak power continuously increases, the value of \(\lambda_{k}\) in (6) gradually tends toward infinity. Referring to the experimental data in [32], when the background light noise \(n_{\text{b}}\geq 40\), this approximation can be considered tight. In this paper, we can make a similar reasonable assumption. Considering that our background light radiation power and the wavelength of visible light are chosen as \(P_{\text{b}}=-75\) dBm and \(v=450\) nm, respectively, we can obtain \(n_{\text{b}}=\xi\frac{P_{\text{b}}\Gamma}{\text{b}^{2}}\approx 40\). In fact, background light noise typically exceeds \(-60\) dBm [41]. In summary, (1) is equally applicable to the system studied in this paper. By substituting (12) into (1), the approximate conditional probabilities \(\xi_{k,1}^{\text{(GA)}}\) and \(\xi_{k,2}^{\text{(GA)}}\) are obtained as  
\[\xi_{k,1}^{\text{(GA)}}=\frac{\exp\!\left(-\frac{\left(y_{k}-\lambda_{k,1} \right)^{2}}{2\lambda_{k,1}}\right)}{\sqrt{2\pi\lambda_{k,1}}};\quad\xi_{k,2 }^{\text{(GA)}}=\frac{\exp\!\left(-\frac{\left(y_{k}-\lambda_{k,1}\right)^{2 }}{2\lambda_{k,2}}\right)}{\sqrt{2\pi\lambda_{k,2}}}. \tag{2}\]We use the superscript \"\({}^{\rm(GA)}\)\" to indicate \"Gaussian approximation\". Substituting (A2) into (10), we obtain  
\[ \begin{split} R_{k}^{\rm(GA)}=& 1+\frac{1}{2\ln 2}\int_{0}^{ \infty}\Big{[}\xi_{k,1}^{\rm(GA)}\times\ln\xi_{k,1}^{\rm(GA)}+\xi_{k,2}^{\rm( GA)}\times\ln\xi_{k,2}^{\rm(GA)}-\\ &\Big{(}\xi_{k,1}^{\rm(GA)}+\xi_{k,2}^{\rm(GA)}\Big{)}\times\ln \Big{(}\xi_{k,1}^{\rm(GA)}+\xi_{k,2}^{\rm(GA)}\Big{)}\Big{]}\mathrm{d}y_{k}. \end{split}  \tag{10}\]  
According to [32], Equation (24), the second and third terms on the RHS of (10) can be approximated as \(-\frac{\ln(2\pi\lambda_{k,1})+1}{2}\) and \(-\frac{\ln(2\pi\lambda_{k,2})+1}{2}\), respectively. The last item on the RHS of (10) is processed as follows  
\[ \begin{split}&\int_{0}^{\infty}\Big{(}\xi_{k,1}^{\rm(GA)}+\xi_{k,2} ^{\rm(GA)}\Big{)}\times\ln\Big{(}\xi_{k,1}^{\rm(GA)}+\xi_{k,2}^{\rm(GA)}\Big{)} \mathrm{d}y_{k}\\ =&\int_{0}^{\infty}\frac{\sqrt{\lambda_{k,2}}\exp \biggl{(}-\frac{\left(y_{k}-\lambda_{k,1}\right)^{2}}{2\lambda_{k,1}}\biggr{)} +\sqrt{\lambda_{k,1}}\exp\biggl{(}-\frac{\left(y_{k}-\lambda_{k,2}\right)^{2} }{2\lambda_{k,2}}\biggr{)}}{\sqrt{2\pi\lambda_{k,1}\lambda_{k,2}}}\times\\ &\ln\frac{\sqrt{\lambda_{k,2}}\exp\biggl{(}-\frac{\left(y_{k}- \lambda_{k,1}\right)^{2}}{2\lambda_{k,1}}\biggr{)}+\sqrt{\lambda_{k,1}}\exp \biggl{(}-\frac{\left(y_{k}-\lambda_{k,2}\right)^{2}}{2\lambda_{k,2}}\biggr{)} }{\sqrt{2\pi\lambda_{k,1}\lambda_{k,2}}}\mathrm{d}y_{k},\end{split}  \tag{11}\]  
\[ \begin{split}\exp\left(-\frac{\left(\varphi+\frac{n_{\rm s}g_{k}^{ T}w_{k}}{\sqrt{\lambda_{k,1}}}\right)^{2}}{2}\right)+\sqrt{\frac{1}{\beta}} \exp\left(-\frac{\left(\varphi-\frac{n_{\rm s}g_{k}^{T}w_{k}}{\sqrt{\lambda_{ k,1}}}\right)^{2}}{2\beta}\right)\mathrm{d}\varphi-\ln(2\pi\lambda_{k,1}). \end{split}  \tag{12}\]  
\[ \begin{split}\sqrt{\beta}\exp\left(-\beta\frac{\left(\varphi+ \frac{n_{\rm s}g_{k}^{T}w_{k}}{\sqrt{\lambda_{k,2}}}\right)^{2}}{2}\right)+ \exp\left(-\frac{\left(\varphi-\frac{n_{\rm s}g_{k}^{T}w_{k}}{\sqrt{\lambda_{ k,2}}}\right)^{2}}{2}\right)\\ \end{split}  \tag{13}\]  
where \(\beta=\frac{\lambda_{k,2}}{\lambda_{k,1}}\), \(\varphi=\frac{y_{k}-\lambda_{k,1}-n_{\rm s}g_{k}^{T}w_{k}}{\sqrt{\lambda_{k,1}}}\) and \(\phi=\frac{y_{k}-\lambda_{k,1}-n_{\rm s}g_{k}^{T}w_{k}}{\sqrt{\lambda_{k,2}}}\). Equation (11) is obtained by substituting (A2).