## 3 Simulation Results of Gaussian Beams
\begin{aligned} P_{o} \bigg{(}1-exp\bigg{(}\frac{-\varphi_{c}^{2}}{2\omega^{2}(z)}\bigg{)}\bigg{)},r =0\\ P_{o}\varphi_{c}\bigg{(}exp\bigg{(}\frac{-2\big{(}r-\frac{\varphi_{c}}{2} \big{)}^{2}}{\omega^{2}(z)}\bigg{)}\bigg{)}-exp\bigg{(}\frac{-2\big{(}r+\frac{ \varphi_{c}}{2}\big{)}^{2}}{\omega^{2}(z)}\bigg{)}\bigg{)}\\ 8r\end{aligned} \right., \tag{7}\]  
For a more intuitive look, the transverse distance \(r\) is represented by receiving angle \(atan(r/z)\) in the rest of the paper. Figure 3a shows the geometrical path loss versus varying receiving angles within the coverage area at several short-range free-space transmission distances. Due to the Gaussian-shaped laser beam, the geometrical path loss scales up with the receiving angle. Regardless of transmission distances, the difference in geometrical path loss between the center (receiving angle = 0\({}^{\circ}\)) and the boundaries (receiving angle = \(\pm\) 10\({}^{\circ}\)) of the coverage area is fixed at approximately 8.7 dB. In addition, the geometrical path loss is increased with the transmission distance at any receiving angle. Roughly, doubling the transmission distance will increase the geometrical path loss by 6 dB. Thereby, the geometrical path loss is one of the major link losses in divergent Gaussian beams. The other significant link loss in the case of divergent optical beams is the fiber coupling loss. As depicted in Figure 4, an optical fiber is precisely positioned at the nominal focal length (_f_) of the receiving collimator. There is no focus shift (\(\Delta\)_f_) of the receiving collimator when its incident optical beam is collimated, and a divergent optical beam introduces a focus shift of the receiving collimator along the propagation direction. According to geometric optics shown in Figure 4, the beam radius at the nominal focal length of the receiving collimator \(\omega_{f}\) is:  
\[\omega_{f}=\frac{f\varphi_{c}}{2z}, \tag{8}\]  
and the focus shift of the receiving collimator \(\Delta f\) equals:  
\[\Delta f=\frac{f\omega_{f}}{\varphi_{c}/2-\omega_{f}}, \tag{9}\]  
substituting Equation (8) into Equation (9), we have:  
\[\Delta f=\frac{f^{2}}{z-f}, \tag{10}\]  
in OWC applications, where \(z>>f\), the first-order term of \(f\) can be safely omitted. Figure 5a shows the focus shift of the receiving collimator when its focus length is set to the same as the emitting collimator (37.2 mm). When the transmission distance of the optical beam is 1 m to 5 m, the focus shift of the receiving collimator decreased from 1.43 mm to 0.28 mm monotonically. Figure 5b shows the beam radius at the nominal focal length of the receiving collimator. At the same transmission range, the beam radius decreased from 381 um to 76 um. Figure 3b illustrates the fiber coupling loss when using optical fibers with 2 typical core diameters without considering their numerical aperture (NA) limitation. It is evident that a greater transmission distance corresponds to a reduced fiber coupling loss, which stands in contrast to the geometrical path loss. Thus, optimizing  
Figure 4: Fiber coupling of the receiving collimator. Figure 3: Two major types of optical link losses in divergent Gaussian beams. **(a)** Geometrical path loss versus receiving angles at different transmission distances; **(b)** Fiber coupling losses versus transmission distances using different fiber core diameters. the link loss requires a careful consideration of the tradeoff between geometrical path loss and fiber coupling loss. The total link loss of the proposed system is the product of \(\alpha\) and \(\beta\), where \(\alpha\) and \(\beta\) presents the geometrical term and the fiber coupling term, respectively.