# Research on Large Divergence Angle Laser Ranging System +
## 3 Design of Large-Divergence Transmitter and Wide-Field-of-View (WFOV) Receiver Systems
### Diverging Lens  
At present, there are various methods to collimate beam divergence angles, such as collimation using telescope systems [5] or binary optical astigmatism-correcting collimating devices [6]. However, most of these methods have complex structures and are not conducive to modularization. Since laser ranging requires the system to be as compact and lightweight as possible, the emission collimation system should be as simple as possible while meeting the ranging performance requirements [7]. As mentioned above, the divergence angle directly output by the laser is 12.2 mrad. In order to meet the requirement of a divergence angle greater than 26 mrad, it is necessary to design a collimation system that converts the laser's divergence angle from 12.2 mrad to above 26 mrad. A single double-concave lens can effectively increase the divergence angle of the laser. The principle is to utilize the divergence characteristics of the double-concave lens to convert a collimated laser beam into a diverging beam. The processing diagram of the double-concave lens is shown in Figure 3.  
For a biconcave lens, given the front surface curvature radius R\({}_{1}\) and the back surface curvature radius R\({}_{2}\). The focal length of the lens can be calculated by the following formula:  
\[\frac{1}{f}=(n-1)(\frac{1}{R_{1}}-\frac{1}{R_{2}}) \tag{3}\]  
In this case, n refers to the refractive index of the lens material. For the above biconcave lens, R\({}_{1}=-14\) mm, R\({}_{2}=14\) mm. \begin{table}
\begin{tabular}{c c c} \hline
**Serial Number** & **Parameter Name** & **Parameter Indicators** \\ \hline
1 & Wavelength & 1535 nm \\ \hline
2 & Pulse width & \(\leq\)5 ns \\ \hline
3 & Operating rate & 10 hz \\ \hline
4 & Beam full divergence & 12.2 mrad \\ \hline
5 & Weight & \(<\)10 g \\ \hline
6 & Dimensions (W \(\times\) H \(\times\) L) & 25 mm \(\times\) 8 mm \(\times\) 7 mm \\ \hline
7 & The diameter of the output light spot. & 0.35 mm \\ \hline \end{tabular}
\end{table} 
Table 1: Laser core parameters. Figure 3: Parameters of a biconcave lens. The biconcave lens is made of HK9 material, which has a refractive index of 1.51. The focal length f of this biconcave lens, calculated from the figure above, is \(-\)13.7 mm. Given the laser divergence angle \(\theta_{1}\) = 12.2 mrad \(\approx\) 0.012 rad, and the laser wavelength \(\lambda\) = 1535 nm = 1.535 \(\times\) 10\({}^{-3}\) mm. Calculate the waist radius \(\omega_{1}\) of the laser according to the following formula:  
\[\omega_{1}=\frac{\lambda}{\pi\theta 1} \tag{4}\]  
Substituting \(\lambda\) = 1.535 \(\times\) 10\({}^{-3}\) mm and \(\theta_{1}\) = 0.012 rad into the formula yields the following:  
\[\omega_{1}=\frac{1.535\times 10^{-3}}{\pi\times 0012}\approx 0.0407\;\text{mm};\]  
At the waist position, the wavefront curvature radius R = \(\infty\), and the formula for the complex curvature radius is q\({}_{1}\) = i\(\frac{\pi\times\omega_{1}^{2}}{\lambda}\). Substituting w\({}_{1}\) = 0.04 mm and \(\lambda\) = 1.55 \(\times\) 10\({}^{-3}\) mm yields the following:  
\[\text{q}_{1}=\text{i}\frac{\pi\times 0.0407^{2}}{1.535\times 10^{-3}}\approx\text{ i}3.37\;\text{mm}.\]  
The focal length of the biconcave lens is f = \(-\)13.7 mm, and its ABCD matrix is \(\left( \begin{array}{cc}1&0\\ \frac{1}{13.7}&1\end{array} \right)\), where A = 1, B = 0, C = \(\frac{1}{13.7}\) mm\({}^{-1}\), D = 1. According to the ABCD law, q\({}_{2}\) = \(\frac{\text{Aq}1+\text{B}}{\text{Cq}1+\text{D}}\); substituting A = 1, B = 0, C = \(\frac{1}{13.7}\) mm\({}^{-1}\), \(D\) = 1, q\({}_{1}\) = i3.37 mm into the formula, we can obtain the following:  
\[\text{q}_{2}=\frac{1\times\text{i}3.37+0}{\frac{1}{13.7}\text{i}3.37+1}\]  
According to this formula, it can be calculated that q\({}_{2}\)\(\approx\) i0.713 mm. At the waist of the output laser beam, q\({}_{2}\) = i\(\frac{\pi\omega w_{2}^{2}}{\lambda}\), it is known that Im (q\({}_{2}\)) = 0.713 mm, \(\lambda\) = 1.535 \(\times\) 10\({}^{-3}\) mm.