# A Long-Range Lidar Optical Collimation System Based on a Shaped Laser Source
## 2 Lidar Detection Principle and System Design  
### Lidar Detection Principle  
Lidar could measure the range by emitting laser pulses to the target object and processing the laser echo received from the reflective surface. The distance between the target and the lidar is set to be \(R\), the divergence angle of the emitted laser beam is set to be \(\theta_{t}\), the stereo angle is set to be \(\Omega_{t}\), the transmittance of the emitting optical system is set to be \(\eta_{t}\), the transmittance of the receiving optical system is set to be \(\eta_{t}\), and the transmittance of the laser pulse in the atmosphere is set to be \(\eta_{a}\). The effective area of the target at the cross-section of the laser beam is \(A_{t}\), the steric angle is \(\Omega_{t}\), the cross-sectional area of the laser beam at the target position is \(A_{t}\), the normal direction of the target is ON, the angle between the optical axis of the launching optical system and the target normal direction ON is \(\theta_{\text{target}}\), and \(A_{r}\) is the area of the detector's photosensitive area. The laser emission process is shown in Figure 2.  
If the laser emission power is \(P_{t}\), then the total laser power that passes through the atmosphere to reach the target is \(\eta_{a}P_{t}\), assuming that the power distribution of the laser beam is the same everywhere, where, according to the formula for the stereo angle, \(A_{t}\) can be expressed as:  
\[\Omega_{t}=\frac{A_{t}}{R^{2}}, \tag{1}\]  
Then, the radiant flux of the part of the target irradiated by the laser pulse is:  
\[\theta_{\text{target}}=\frac{\eta_{t}\eta_{a}P_{t}A_{t}}{A_{t}}=\frac{\eta_{ t}\eta_{a}P_{t}A_{t}}{\Omega_{t}R^{2}}, \tag{2}\]  
The laser reception process is shown in Figure 3. Assume that the reflectivity of the target is \(\rho_{t}\); then, after diffuse reflection from the target, combined with the transmittance of the receiving system, and after another pass through the atmosphere, the power of the laser pulse received by the receiving system is:  
\[P_{\text{receive}}=\theta_{\text{target}}\rho_{t}\eta_{t}\eta_{a}\frac{A_{r }}{\Omega_{t}R^{2}}=\frac{\eta_{t}\eta_{a}P_{t}A_{t}}{\Omega_{t}R^{2}}\rho_{t} \eta_{t}\eta_{a}\frac{A_{r}}{\Omega_{t}R^{2}}, \tag{3}\]  
where \(\Omega_{t}\) can be expressed by solid angle, \(\Omega_{t}\) is in accordance with diffuse reflection, solid angle takes \(2\pi\) or \(\pi\), and in this paper, \(\pi\) is taken. Then, the laser pulse power received by the receiving system can be expressed as:  
\[P_{\text{receive}}=\frac{4\eta_{t}\eta_{a}P_{t}A_{t}}{\pi R^{2}\theta_{t}^{ 2}}\rho_{t}\eta_{t}\eta_{a}\frac{A_{r}}{\pi R^{2}}=\frac{4\eta_{t}\eta_{a}}{ \pi R^{2}\theta_{t}^{2}}A_{t}\rho_{t}\frac{\eta_{t}\eta_{a}A_{r}}{\pi R^{2}}P_ {t}, \tag{4}\]  
Figure 2: Laser emission process. Assume the pulse width of the laser is \(P_{w}\). Then, the energy obtained by the receiving system is combined with the energy of a single photon and quantum efficiency \(\eta_{q}\). The average number of photons generated by the detector can be obtained:  
\[\overline{N}=\frac{\eta_{q}P_{w}P_{\text{receive}}}{hv}=\frac{\eta_{q}}{hv}\frac {4\eta_{t}\eta_{a}}{\pi R^{2}\varrho_{t}^{2}}A_{t}p_{t}\frac{\eta_{r}\eta_{a}A_ {r}}{\pi R^{2}}P_{t}P_{w}, \tag{5}\]  
Equations (1)-(5) are for laser emission beams that are larger than the effective receiving area of the target object. At this point, the target object can be considered as a point target. When the effective receiving area of the target object is greater than or equal to the emission area of the laser emission beam, the target object is considered to be a surface target. When the target object is a surface target, the laser pulse power received by the receiving system can be expressed as Equation (6). \[P_{receive}=\eta_{t}\eta_{a}P_{t}\rho_{t}\eta_{r}\eta_{a}\frac{A_{r}}{\pi R^{ 2}}=\eta_{t}\eta_{a}\rho_{t}\frac{\eta_{r}\eta_{a}A_{r}}{\pi R^{2}}P_{t}, \tag{6}\]  
In this case, the average number of photons produced by the detector can be obtained and expressed as Equation (7).