## 3 Link Margin of FSO System and SMF Coupling Efficiency of Signal Light and Sunlight
Each point of the solar disk has a coupling efficiency \(\eta(\theta_{e},\phi)\), where \(\theta_{e}\) is the angle between the line connecting the point and the receiver and the line connecting the solar disk center and the receiver, the direction angle. The range of \(\theta_{e}\) is \(\pm 4.65\) mrad. With the center of the solar disk as the origin, the line connecting the center of the disk and the rightmost point of the solar disk from the perspective of the receiver is the x-axis to establish a rectangular coordinate system; \(\phi\) is the angle between the line connecting the point and the center of  
Figure 8: Diagram of direct sunlight on FSO receiver. the disk and the positive direction of the x-axis. The range of \(\phi\) is \([0\) rad, \(2\pi\) rad\(]\). When \(\theta_{e}\) is fixed and \(\phi\) changes from 0 rad to \(2\pi\) rad, the coupling efficiency of these points on the solar disk are the same. That is, in the solar disk, the points on the circle with the center of the disk as the center have the same coupling efficiency. In this study, we call the coupling efficiency on different circles as the integral coupling efficiency, which is only related to the direction angle \(\theta_{e}\). Thus, AL generated by the sun with the same direction angle has the same coupling efficiency to the receiver SMF. In order to ensure the optimum coupling efficiency of signal light, \(\beta=1.12\) should be met. When the receiver lens diameters were 20 cm, 10 cm, 5 cm, 1 cm, 5 mm, 2 mm, 1 mm, and 0.488 mm, respectively, the lens focal lengths corresponding to these receiver lens diameters were 0.90 m, 0.81 m, 0.68 m, 0.45 m, 0.23 m, 0.18 m, 0.14 m, and 0.09 m, respectively. According to Equation (12), the relationship curve between the integral coupling efficiency and the direction angle can be plotted for different receiver lens diameters as shown in Figure 9. The y-axis of these curves is the left y-axis in Figure 7:0 When the receiver lens diameter is 0.488 mm and the integral coupling efficiency becomes zero for the first time, the corresponding direction angle is exactly equal to 4.65 mrad, which is half of the angle of the sun to the FSO receiver. The shape of the solar disk is completely symmetrical; thus, we can only consider the positive direction angle. In other words, if the receiver lens diameter is less than or equal to 0.488 mm, the integral coupling efficiency of sunlight in the full-range direction angle is greater than zero, which means that 0.488 mm is a critical value. The relationship curve between the normalized solar energy and the direction angle is also shown in Figure 7:1 The y-axis of this curve is the right y-axis in Figure 7:2 In order to facilitate the drawing, the horizontal axis was expressed by logarithmic calculation. The logarithm value corresponding to 4.65 mrad was \(-23.33\) dBrad. When the relationship curve between the normalized energy and the sunlight direction angle was simulated on the right side of Figure 9, the simulation interval of the sunlight direction angle was \(4.65\times 10^{-8}\) rad. The sum of the values corresponding to all points of different direction angles in the right curve was 1, that is, the sum of the sunlight energy in all direction angles. When the receiver lens diameter is determined, there is a unique receiver lens focal length that can meet \(\beta=1.12\). Thus, when the receiver lens diameter is given, the change in the integral coupling efficiency with the direction angle can be expressed as \(\eta_{diameter}(\theta_{e})\), which is consistent with the curves on the left of Figure 7:3 First, calculate the volume  
Figure 7:4 Curves of integral coupling efficiency vs. AL direction angle for different receiver lens diameters and normalized sunlight energy on receiver lens vs. AL direction angle. of the rotating body obtained by rotating \(\eta_{diameter}(\theta_{e})\) around the left y-axis in Figure 7:5 Here, we only take the positive part of \(\theta_{e}\), thus, the range of \(\theta_{e}\) is 0 to 4.65 mrad.