## Appendix A Proof of Proposition 1  
Considering the impactors of atmospheric attenuation, atmospheric turbulence, and pointing errors, i.e., \(I_{aap}\)=\(I_{al}I_{al}I_{pe}\), the PDF of the channel model can be written as  
Figure 9: Ergodic capacity versus \(P_{t}\) under various \(L\) and various turbulence conditions.  
\[f_{Lup|\theta_{a}}(I_{aap})=\frac{AB^{-\xi^{2}}\xi^{2}\left(I_{aap}\right)^{\xi^{2}-1 }}{2\left(I_{al}\right)^{\xi^{2}}}\cos\theta_{a}\sum_{i=1}^{b}\mu_{i}\int I_{at} ^{\frac{a+i}{2}-\xi^{2}-1}G_{0,2}^{2,0}\left[\frac{ab}{\kappa b+Y}I_{at}\Big{|} _{\frac{a-i}{2},-\frac{a-i}{2}}\right]dI_{at}. \tag{11}\]  
Utilizing Equation (07.34.21.0085.01) in [33] and Equation (07.34.17.0011.01) in [33], the closed-form expression for the PDF can be simplified as  
\[f_{Lup|\theta_{a}}(I_{aap})=\frac{A}{2I_{aap}}\xi^{2}\cos\theta_{a}\sum_{i=1}^ {b}b_{i}G_{1,3}^{3,0}\left[CI_{aap}\Big{|}_{\xi^{2},a,i}^{1+\xi^{2}}\right], \tag{12}\]  
where \(b_{i}=\mu_{i}(ab/(\kappa b+Y))^{-(a+i)/2}\) and \(C=ab/\left((\kappa b+Y)BI_{al}\right)\).  
In addition to this, the AoA fluctuations are also taken into account and the expression for the PDF is given below:  
\[f(I)=\int_{0}^{\theta_{FoV}}f_{Lup|\theta_{a}}(I)f_{\theta_{a}}(\theta_{a})d \theta_{a}+\delta(I)\int_{\theta_{FoV}}^{\infty}f_{\theta_{a}}(\theta_{a})d \theta_{a}, \tag{13}\]  
where \(\delta(\cdot)\) is delta function.  
Substituting (12) and (7) into (13) and using Equation (07.34.21.0085.01) in [33] and Equation (07.34.17.0011.01) in [33] to obtain the closed-form formula for the PDF. It is assumed that \(\cos\theta_{a}\approx 1\) for small values of \(\theta_{a}\). Therefore, the PDF of \(I\) can be written as  
\[f(I)=\frac{A}{2I}\xi^{2}\sum_{i=1}^{b}b_{i}G_{1,3}^{3,0}\left[CI\Big{|}_{\xi^{ 2},a,i}^{1+\xi^{2}}\right]\left(1-\exp\left(-\frac{\theta_{FoV}^{2}}{2\sigma_{ a}^{2}}\right)\right)+\exp\left(-\frac{\theta_{FoV}^{2}}{2\sigma_{a}^{2}} \right)\delta(I). \tag{14}\]  
Integrating (14) and utilizing Equation (07.34.21.0084.01) in [33] yields the formula of the CDF, denoted as follows:  
\[F(I)=\left(1-\exp\left(-\frac{\theta_{FoV}^{2}}{2\sigma_{a}^{2}}\right)\right) \frac{A\xi^{2}}{2}\sum_{i=1}^{b}b_{i}G_{2,4}^{3,1}\left[CI\Big{|}_{\xi^{2},a,i,0}^{1+\xi^{2}}\right]+\exp\left(-\frac{\theta_{FoV}^{2}}{2\sigma_{a}^{2}} \right). \tag{15}\]  
The equation for the relationship between signal-to-noise ratio \(\gamma\) and channel coefficient \(I\) is given below [28]:  
\[\gamma=\frac{(\eta P_{t}I)^{\tau}}{\sigma_{n}^{2}}, \tag{16}\]  
where \(\tau=1\) and \(2\) denote the heterodyne detection technique and the IM/DD technique, respectively, \(P_{t}\) indicates the average optical power. The AWGN has the mean of zero and the variance of \(\sigma_{n}^{2}\), where \(\sigma_{n}^{2}=4\pi^{2}\eta eN_{o}\Delta f_{e}\Delta f_{o}r_{e}^{2}(1-\cos \theta_{FoV}/2)\), \(e\) is the electron charge, \(N_{o}\) stands for the spectral radiance of the background radiations, \(\Delta f_{e}\) and \(\Delta f_{o}\) indicate the electronic bandwidth and optical bandwidth of the receiver, respectively.  
Substituting (16) into (15), the CDF of \(\gamma\) is given by  
\[F(\gamma)=\left(1-\exp\left(-\frac{\theta_{FoV}^{2}}{2\sigma_{a}^{2}}\right) \right)\frac{A\xi^{2}}{2}\sum_{i=1}^{\beta}b_{i}G_{2,4}^{3,1}\left[C\frac{ \left(\gamma\sigma_{n}^{2}\right)^{\frac{1}{\tau}}}{\eta P_{t}}\bigg{|}_{\xi^{ 2},a,i,0}^{1,1+\xi^{2}}\right]+\exp\left(-\frac{\theta_{FoV}^{2}}{2\sigma_{a}^ {2}}\right). \tag{17}\]  
Differentiating (17) with respect to \(\gamma\), and simplifying it with the assistance of Equation (07.34.20.0001.01) in [33], we obtain the PDF of \(\gamma\) as  
\[f(\gamma)=\left(1-\exp\left(-\frac{\theta_{FoV}^{2}}{2\sigma_{a}^{2}}\right) \right)\frac{A\xi^{2}}{2}\frac{C\sigma_{n}^{2}}{\eta P_{t}}\frac{1}{\tau} \gamma^{\frac{1}{\tau}-1}\sum_{i=1}^{\beta}b_{i}G_{3,5}^{3,2}\left[C\frac{ \left(\gamma\sigma_{n}^{2}\right)^{\frac{1}{\tau}}}{\eta P_{t}}\bigg{|}_{\xi^{ 2}-1,a-1,i-1,0,-1}^{\frac{1}{\tau}}\right]. \tag{18}\]