# Free Space Optical Communication: An Enabling Backhaul Technology for 6G Non-Terrestrial Networks
## 2 Fundamentals of FSO Signal Transmission
### Geometrical Loss  
Geometrical loss refers to the loss of optical power or signal strength due to the spreading of an optical beam as it propagates through space. The intensity profile of a typical laser output is characterized by a bell-shaped curve, i.e., a Gaussian beam, with the highest intensity at the center and gradually decreasing towards the edges. The beam has a symmetric shape and is typically described by its beam waist or beam radius, representing the minimum beam diameter point along its propagation path. The geometrical loss of a Gaussian beam is primarily governed by the beam divergence and the distance traveled. As illustrated in Figure 2, when a beam propagates, it spreads due to diffraction, resulting in an increase in the beam diameter. This spreading leads to a decrease in optical power density, causing a geometrical loss. The geometrical loss of a Gaussian beam can be approximated by [75]  
\[h_{\text{Gaus}}\approx\bigg{[}1-\exp\bigg{(}-\frac{2}{\omega^{2}(z)}a_{r}^{2} \bigg{)}\bigg{]}, \tag{1}\]  
where \(\omega(z)\) is the beam half width (i.e., the truncated radius at distance \(z\)) and \(a_{r}\) is the radius of a circular photodetector. Beam half width is defined at the distance \(z\) as the distance from the point of the maximum intensity (\(I_{o}\)) to the point where the level of the optical intensity decreases to \(1/e^{2}\) of the maximum intensity, i.e., \(\big{(}1/e^{2}\big{)}I_{o}\). Mathematically speaking, this is given by  
\[w(z)=z\ \tan(\theta_{1/2}), \tag{2}\]  
where \(\theta_{1/2}=\theta/2\) is a half width transmitter beam divergence angle. For the case of \(a_{r}<<\omega(z)\), we can use \(\exp(x)\cong 1+x\) and simplify Equation (1) as  
\[h_{\text{Gaus}}=2\frac{a_{r}^{2}}{\omega^{2}(z)}. \tag{3}\]  
At longer distances, waves have a tendency to exhibit a plane wave characteristic in the detection area. In this case, the geometrical loss is simply determined using the ratio between the detector's area (\(A_{r}\)) and the area of the received spot, also known as the illuminated area (\(A_{b}\)). Mathematically speaking, we can write [76] [Equation (3.78)]  
\[h_{\text{Plane}}=\frac{A_{r}}{A_{b}}=\frac{D_{r}^{2}}{D_{b}^{2}}=\frac{D_{r}^{ 2}}{\big{(}D_{t}+z\theta\big{)}^{2}}, \tag{4}\]  
where \(D_{r}\) is the photodetector's diameter, \(D_{t}\) is the transmitter's aperture diameter, \(D_{b}\) is the diameter of the received beam, and \(\theta\cong\sqrt{16/G_{t}}\) is the full-width beam divergence angle (see Figure 3), with \(G_{t}\)= \(4\pi A_{t}/\lambda^{2}\) denoting the transmitting gain. Here,  
Figure 2: Geometrical loss for a Gaussian beam. \(A_{t}=\pi a_{t}^{2}=\pi{(D_{t}/2)}^{2}\) is the area of the transmit aperture with \(a_{t}\) denoting the radius of the transmit aperture. In the following, we assume a wavelength \(\lambda=1550\) nm, transmit/receive aperture diameters of \(D_{r}=D_{t}=10\) cm, and present the geometrical loss for various transmission distances based on Equation (4) in Figure 4. It can be observed that geometrical losses for terrestrial links (which typically span a transmission distance between 100 m and 10 km) are in the range of 0 to 9.5 dB. For HAPS with an operation altitude of 20 km, the geometrical loss increases to 13.88 dB. For a LEO satellite with a maximum altitude of 2000 km, the geometrical loss reaches 51.95 dB. For a GEO satellite, it becomes 77.03 dB. Therefore, to mitigate the adverse effects of geometric loss in a satellite link, transmitting telescopes are utilized in practice to reduce beam divergence. Geometrical loss depends on the wavelength and increases as the wavelength increases. The difference in losses between different wavelengths becomes constant as the propagation distance increases. The difference in geometrical loss can be represented by \(10\text{log}_{10}(\lambda_{1}^{2}/\lambda_{2}^{2})\), with \(\lambda_{1}\) and \(\lambda_{2}\) representing two different wavelengths under consideration.