## 3 Link Margin of FSO System and SMF Coupling Efficiency of Signal Light and Sunlight
When light from different directions shines on the lens, the coupling efficiency is expressed as [44]  
\[\eta(\theta_{e})=\frac{\left|\int\limits_{0}^{1}\sqrt{8\pi}\beta\exp\left(- \beta^{2}\rho^{2}\right)J_{0}\left(2\beta\frac{f}{\omega_{0}}\theta_{e}\rho \right)\rho d\rho\right|^{2}}{\pi}, \tag{12}\]  
where \(\theta_{e}\) is the angle between the incident direction of light and the optical axis of the lens or the direction angle; \(\rho\) is the normalized radial position; and \(J_{0}(\cdot)\) represents the zero-order Bessel function. In this study, light from different directions can be regarded as plane waves. By adding coupling loss and atmospheric transmission loss, the final link margin of an FSO receiving system can be calculated. The signal light can be considered as a plane wave, the sunlight cannot be considered as a plane wave. The average distance between the sun and the earth is \(D_{S-E}=1.496\times 10^{11}\ (\mathrm{m})\); the sun diameter is \(R_{S}=1.39\times 10^{9}\ (\mathrm{m})\) and the angle of the sun to the FSO receiver is approximately \(\pm 4.65\ \mathrm{mrad}\), which is greater than the receiving field angle of the FSO receiver system. Thus, the sun was regarded as a surface light source. The solid angle of the sun disk is \(\Omega_{S}=\pi R_{S}^{2}/D_{S-E}^{2}\left(\mathrm{sr}\right)\). According to the [PERSON] law [45], the radiation force of the sun is \(E_{S}=\sigma T_{S}^{4}\left(\mathrm{W/m^{2}}\right)\), where \(\sigma\) is the Boltzmann constant and \(T_{S}\) is the effective temperature of the solar surface, in Kelvin. The solar radiation intensity in the unit solid angle on the surface of the vertical solar ray at the upper boundary of the atmosphere of the earth is \(I_{S}=E_{S}/\tau\left(\mathrm{W/\left(m^{2}\times sr\right)}\right)\). The solar constant refers to the solar radiation energy received by the upper boundary surface of the atmosphere perpendicular to the solar rays at the average distance between the sun and the earth in unit time and unit area. Thus, the theoretical calculation of the solar constant is written as  
\[I_{SC}=I_{S}\Omega_{S}=\frac{\sigma T_{S}^{4}}{\pi}\frac{\pi R_{S}^{2}}{D_{S-E }^{2}}, \tag{13}\]  
When \(T_{S}=5766\) K and the solar constant is \(1353\) W/m\({}^{2}\). The solar constant was approximately \(1367\) W/m\({}^{2}\) and the energy in the full bandwidth spectrum, measured by an artificial satellite. In this study, the measured value of the solar constant was used for calculation. The laser wavelength used was \(1555.74\) nm, and the optical bandwidths of the coherent and non-coherent receiving systems were \(2\) nm and \(0.8\) nm, respectively. The solar irradiance at this wavelength was \(267\) W/\(\left(\mathrm{m^{2}\times\mu m}\right)\). A diagram of direct sunlight on the FSO receiver is shown in Figure 8.  
The total power of sunlight on the receiver lens is expressed as  
\[P_{sunlens}=E_{sun}\times\pi(D_{r})^{2}/4\times B_{o}, \tag{14}\]  
where \(E_{sun}\) is the irradiance of sunlight at the wavelength of the signal light and \(B_{o}\) is the optical bandwidth of the receiver. To ensure that the signal light has the optimum coupling efficiency, \(\beta=1.12\) should be satisfied; each receiver lens diameter corresponds to an optimal lens focal length. In this paper, it is considered that the sun directly irradiates the receiver lens. When the direction angle of incident light multiplied by the lens focal length is greater than twice the mode field radius of the SMF, the focused Airy spot will not appear in the mode field range of the SMF optical field; that is, the coupling efficiency is zero. The sun is a surface light source, which can be considered as a disk when it shines directly. Because the distance between the sun and the receiver is far enough, the light from each point of the solar disk can be regarded as a plane wave when it reaches the receiver.