## 4 Signal Attenuation Model for Inter-Satellite Terahertz Communication  
### Signal Attenuation Model for Inter-Satellite Terahertz Communication without Pointing Error  
Considering the transmitter and receiver of the terahertz communication system as a whole, the structure of the terahertz communication link between satellites is shown in Figure 5. After the terahertz wave is generated, it is transmitted through the antenna, and the beam reaches the antenna at the receiving end after a long transmission distance. Since there is no atmosphere in space, there is no need to consider the effect of atmospheric absorption. Assuming that there is no pointing error, the received power of the inter-satellite terahertz communication system can be expressed as Equation (3). \[P_{R}=P_{T}\cdot G_{T}\cdot G_{R}\cdot L_{f} \tag{3}\]Among them, \(P_{T}\) and \(P_{R}\) are the transmit power and receive power, \(G_{T}\) and \(G_{R}\) are the transmit antenna gain and receive antenna gain, respectively, and \(L_{t}\) are the free space loss. In the existing link budget model, the electromagnetic signal attenuation mainly adopts the Friis equation. The Friis equation is mainly applicable to the case where the pointing error attenuation is not too serious, which is usually used in the microwave frequency bands. Considering the computational complexity in the performance analysis of terahertz communication, it is generally believed that the beams in the receiving area of the antenna are uniformly distributed [18]. However, terahertz wave has the characteristics of light-like transmission. Due to the narrow terahertz wave beam, the pointing error has a greater impact on the system, so the influence brought by the pointing error must be fully considered. In addition, when there is a large difference between the received aperture and terahertz beam radius due to other factors such as pointing error, the geometric attenuation model under the assumption of plane wave is used to calculate the received power, which will lead to a large error between the calculated received power and the actual situation. A more effective method is to add the attenuation caused by pointing error on the basis of Friis equation. In most cases, terahertz research is also based on quasi-optical theory with Gaussian beam model analysis. For example, reference [6,8,19]. In fact, most of the transmitted signals are Gaussian beams, which are unevenly distributed in the area of the receiving end. Therefore, this paper adopts the Gaussian beam model for analysis. For the Gaussian beam receiver, the terahertz wave distribution is represented by Equation (4). \[T(r,L)=\,T_{0}\frac{R_{0}{}^{2}}{R_{e}{}^{2}}\exp(-\frac{2r^{2}}{R_{e}{}^{2}}) \tag{4}\]  
where, \(T_{0}\) is a constant, related to the transmit power, \(R_{0}\) is the beam radius of the transmitting end, \(r\) is the distance from any point in the plane at the transmitting end antenna to the center, and \(R_{e}\) is the effective radius of the beam reaching the receiving end, determined by Equation (5). \[R_{e}=R_{0}\sqrt{1+\left(\frac{\lambda\,L}{\pi R_{0}{}^{2}}\right)^{2}} \tag{5}\]  
In the formula, \(L\) is the transmission distance. The effective radius is determined by the field strength, where the field strength of the terahertz beam is \(1/e^{2}\) at the center and obeys the hyperbolic variation law. For Gaussian beams, the beam divergence is generally expressed by the beam launch angle \(\theta_{b}\), which can be expressed as Equation (6). \[\theta_{b}=2\frac{dR_{e}}{dz}=\frac{2\frac{\lambda\,L}{\pi R_{0}}}{\sqrt{\left( \frac{\pi R_{0}{}^{2}}{\lambda}\right)^{2}+L^{2}}} \tag{6}\]  
Figure 5: Inter-satellite terahertz communication link structure diagram. Assuming that the transmission distance is infinite, namely \(\,L\rightarrow\infty\,\), the angle between the two asymptotes of the hyperbola is the far-field divergence angle \(\theta_{y}\) of the Gaussian beam, which can be expressed as Equation (7).