# A Review of Variable-Beam Divergence Angle FSO Communication Systems
## 2 Optical Principle of VBDA Method  
Single-mode fiber (SMF) has been widely used in FSO communication due to its advantages of low attenuation, low dispersion, and high-bandwidth capacity. Since the beam emitted in SMF can be approximated as a fundamental mode Gaussian beam, the transmission theory of Gaussian beams is used to analyze the principle of VBDA. Assume that the SMF emits a Gaussian beam with a waist radius of \(w_{0}\), which is located on the end face of SMF, as shown in Figure 5. The ideal single lens is placed at a distance of \(l\) from the source, has a focal length of \(F\), and transforms the Gaussian beam into an outgoing beam with a waist radius of \(w_{0}^{\prime}\). The distance from the rear surface of the lens to the outgoing beam waist is denoted as \(l^{\prime}\). The \(q\) parameters of the incident Gaussian beam at the waist and at the rear surface of the lens are denoted as \(q_{0}\) and \(q_{F}\), respectively. In the following analysis, we will use the ABCD law of Gaussian beam propagation [35] to determine the divergence angle of the outgoing beam. The \(q\) parameter of incident Gaussian beam is:  
\[q_{0}=i\frac{\pi w_{0}^{2}}{\lambda}, \tag{1}\]  
The ABCD matrix transmitted from the waist of the incident Gaussian beam to the front surface of the lens is given by \([1,\;l;\;0,\;1]\). The ABCD matrix transmitted from the front surface of the lens to the rear surface of the lens is \([1,\;0;\;-1/F,\;1]\). By the ABCD law, the ABCD transmission matrix from the beam waist position of the incident Gaussian beam to the rear surface of the lens is given by:  
\[ \begin{bmatrix}A&B\\ C&D\end{bmatrix} = \begin{bmatrix}1&0\\ -1/F&1\end{bmatrix}\begin{bmatrix}1&l\\ 0&1\end{bmatrix} = \begin{bmatrix}1&l\\ -1/F&1-l/F\end{bmatrix} , \tag{2}\]  
The \(q\) parameter of Gaussian beam at the rear surface of the lens can be obtained as follows:  
\[q_{F}=\frac{Aq_{0}+B}{Cq_{0}+D}=\frac{i\frac{\pi w_{0}^{2}}{\lambda}+l}{-\frac {1}{F}\cdot i\frac{\pi w_{0}^{2}}{\lambda}+1--\frac{l}{F}}, \tag{3}\]  
And the \(q\) parameter of Gaussian beam at the rear surface of the lens is:  
\[q_{F}=q_{0}^{\prime}-l^{\prime}=i\frac{\pi{w_{0}^{\prime}}^{2}}{\lambda}-l^{ \prime}, \tag{4}\]  
So:  
\[q_{F}=\frac{i\frac{\pi w_{0}^{2}}{\lambda}+l}{-\frac{1}{l}i\frac{\pi w_{0}^{2} }{\lambda}+1--\frac{l}{F}}=i\frac{\pi{w_{0}^{\prime}}^{2}}{\lambda}-l^{\prime}, \tag{5}\]  
Figure 5: Characteristic schematic diagram of the basic Gaussian beam after lens transformation. The solution is:  
\[{w^{\prime}_{0}}^{2}=\frac{F^{2}w_{0}^{2}}{\left(F-l\right)^{2}+\left(\frac{\pi w _{0}^{2}}{\lambda}\right)^{2}}, \tag{6}\]  
According to the calculation formula \(\theta=2\lambda/\left(\pi w_{0}\right)\), the far-field divergence angle of a Gaussian beam can be obtained as follows:  
\[\theta_{0}^{\prime}=\frac{2\lambda}{\pi w_{0}^{\prime}}=\frac{2\lambda}{\pi Fw _{0}}\sqrt{\left(F-l\right)^{2}+\left(\frac{\pi w_{0}^{2}}{\lambda}\right)^{2 }}, \tag{7}\]  
The outgoing beam divergence angle \(\theta_{0}^{\prime}\) is minimized when \(F=l\), i.e., when the beam waist is positioned at the front focal plane of the collimating lens. This corresponds to the state of optimal collimation. As a result, the outgoing beam exhibits a minimized divergence angle of \(2w_{0}/F\). According to Equation (7), any changes in the values of \(F,\ l,\) or \(w_{0}\) will result in changes to the divergence angle \(\theta_{0}^{\prime}\) of the outgoing beam. Since the mode field radius \(w_{0}\) of SMF at a specific wavelength cannot be easily modified, the divergence angle of the exit beam can be altered by adjusting the focal length \(F\) of the collimating lens or by varying the distance \(l\) between the SMF end face and the collimating lens. The influence of \(F\) and \(l\) on the divergence angle \(\theta_{0}^{\prime}\) of the outgoing beam is shown in Figure 6.  
Furthermore, one way to change \(l\) is by altering the optical path length between the SMF end face and collimating lens.