## 2 FSO Vertical Channel Model
### Beam Divergence Angle Optimization  
As we assume far-field diffraction propagation with a Gaussian beam, the radial term of the electric field of the Gaussian beam is defined as [25; 26; 27; 28]  
\[E(\mathbf{r},z) =\exp\left(\frac{-r^{2}}{w(z)^{2}}\right), \tag{7}\]  
where \(\mathbf{r}\) is the pointing error, and \(w(z)\) is the radius at which the field amplitudes fall to \(1/\mathrm{e}\) of their axial values at the plane \(z\) along the beam, defined by  
\[w(z) =w_{0}\sqrt{1+\left(\frac{z}{z_{R}}\right)^{2}}, \tag{8}\]  
where \(z_{R}\) is the Rayleigh range given by  
\[z_{R} =\frac{\pi w_{0}^{2}}{\lambda}. \tag{9}\]  
The received optical power can be calculated as  
\[P_{Rx} =P_{Tx}G_{Tx}G_{Rx}E^{2}L_{s}, \tag{10}\]  
where \(P_{Tx}\) is the transmitter power assumed to be 1 W, \(G_{Tx}=8/\left(w(z)/L\right)^{2}\) is the transmitter gain, \(G_{Rx}=\pi^{2}D^{2}/\lambda^{2}\) is the receiver gain, where D = 10 cm is the aperture diameter, and \(L_{s}=\left(\lambda/4\pi L\right)^{2}\) is the space loss. Because the beam divergence angle is related to transmitter gain, the optimal beam size should be decided. The transmitter gain is considered to satisfy the required SNR for the high-data-rate transmission. The wider the beam divergence angle, the lower is the transmitter gain. In addition, the beam divergence angle is determined from the interrelationship between the beam divergence angle and pointing error. As shown in Figure 2a, when the OGS transmits a broad beam to compensate for pointing error, the broad beam results in a geometric loss. Conversely, transmitting a narrow beam to increase the transmitter power results in pointing loss, as shown in Figure 2b. Furthermore, the scintillation is related to the beam divergence angle. From the aspect of beam divergence angle, a narrow beam propagates as a Gaussian beam. However, a broad beam propagates from a Gaussian beam to a spherical wave. Moreover, the Gaussian beam is known to be most affected by scintillation, followed by  
Figure 2: Interrelationship among scintillation, pointing error, and beam divergence angle: (**a**) A broad beam to compensate for pointing error; (**b**) A narrow beam to increase the optical power. spherical waves and plane waves [29]. Thus, the greater the divergence angle, the less is the affection caused by scintillation. Furthermore, the pointing errors are also related to the scintillation index. As shown in Figure 3, two beam widths (10 and 5 cm) are collimated at the transmitter. The \(X\)-axis represents a function of an angular receiver position from the optical axis representing the pointing error, and the \(Y\)-axis denotes the scintillation index. Thus, Figure 3 presents the scintillation index according to pointing errors. According to [PERSON] et al. [14], a slight increment in pointing error induces a rapid increase in the scintillation index. The link distance was at a satellite in geostationary orbit at 38,500 km. A similar result is obtained when the distance is assumed to be 20 km as shown in Figure 3.  
The \(X\)-axis represents the pointing error, and the scintillation index increases rapidly with a slight increase in pointing error. In other words, as more pointing errors occur, the scintillation effects increase exponentially. However, the scintillation effects are minor without pointing error because the Gaussian beam is propagated to spherical waves at the on-axis. Furthermore, regardless of the beam divergence angle, the scintillation effect is negligible in the case of downlinks. This is because the atmospheric turbulence affects the beam after being sufficiently approximated as a spherical wave. In addition, as shown in Figure 3, the larger collimated beam size is more affected by the off-axis scintillations compared to the smaller beam. This is due to a narrow divergence angle for propagating the giant collimated beam and a broad divergence angle for propagating the smaller beam. The scintillation affects the narrow beam more owing to the pointing error.