## 3 Characteristic of Picosecond Laser Measurements for Debris Targets  
### Picosecond Laser Transmission  
The peak power of a pulse laser system is given by:  
\[P_{peak}=\frac{P_{0}}{\tau f} \tag{5}\]  
where \(P_{peak}\) is the peak power of the laser, \(P_{0}\) is the average power of the laser, \(\tau\) is the pulse width, and \(f\) is the repetition rate of the laser. For the previously mentioned picosecond laser (4.2 W, 1 kHz, and a pulse of 30 ps), the peak power is as high as 70 MW. The diameter of the laser beam emitted by the transmitting \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**ID**} & **Space** & \multirow{2}{*}{**RCS: m\({}^{2}\)**} & **Measured** & \multicolumn{2}{c}{**Ranging Precision (cm)**} \\  & **Debris** & & **Distance (km)** & **Main Telescope** & **Fellow Telescope** \\ \hline
21,610 & ArianeR\_B & 16.84 & 1390 & 129.2 & 149 \\
16,182 & SL-16R\_B & 12.05 & 1030 & 141.0 & 141.7 \\
11,672 & SL-14R\_B & 4.21 & 1160 & 56.2 & 64.7 \\
16,720 & SL-14R\_B & 3.58 & 1170 & 74.9 & 62.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of debris measurements obtained using the dual telescopes. Figure 16: DLR measurements were obtained using the 1.56 m and 60 cm telescopes. telescope is approximately 8 cm, and the peak power density is calculated as 1.39 MW/cm\({}^{2}\) according to the following equation:  
\[P_{peak\ power}=\frac{P_{peak}}{\pi(d/2)} \tag{6}\]  
When such a high-intensity laser beam propagates through the atmosphere, there is not only a nonlinear effect but also the refractive index of the medium changes according to the following equation [27]:  
\[n=n_{0}+n_{2}I(r) \tag{7}\]  
where \(n\) is the changed refractive index, \(n_{0}\) is the refractive index without the laser, \(n_{2}\) is the coefficient of the refractive index change, and \(I(r)\) is the laser intensity along the transmission path. For the Gaussian distribution laser beam, the laser energy is high at the center and weak at the side. Strong laser transmission in the atmosphere forms a lentoid that would allow easier self-focusing, and the angle of laser beam divergence would be at the diffraction limit in this case. The diffraction limit is expressed as:  
\[r=M^{2}\frac{2\lambda R}{\pi d} \tag{8}\]  
where, \(M^{2}\) is the laser beam quality, \(\lambda\) is the laser wavelength, \(R\) is the distance of laser transmission, \(d\) is the diameter of the laser spot. For the previously mentioned picosecond laser (\(M^{2}\) = 1.2, \(\lambda\) = 532 nm, \(R\) = 1000 km, \(d\) = 8 cm), the divergence angle (\(\theta\)) of diffraction limit is approximately 4.15 \(\upmu\)rad by:  
\[\theta=\frac{r-d}{R} \tag{9}\]  
This divergence angle (\(\theta\)) is much less than the one at the transmitting telescope. According to the laser ranging link equation, the number of laser echo photons increases in inverse proportion to \(\theta^{2}\). For the previously mentioned nanosecond laser (60 W, 200 Hz, and a pulse width of 8 ns), the peak power density is approximately 0.74 MW/cm\({}^{2}\), which is lower than that of the 4.2 W picosecond laser system, consequently, its nonlinear effect is weak. Figure 17 shows the measurements for the 60 W nanosecond laser (left) and 4.2 W picosecond laser (right). The x-axis and y-axis present the RCS of targets and the measured range of the targets, respectively. Figure 17 shows that the longest ranging distance is 1150 km (with an RCS of around 5 m\({}^{2}\)) for the 4.2 W picosecond laser and 2100 km (with an RCS of 5 m\({}^{2}\)) for the 60 W nanosecond laser. Additionally, the normalized measuring capability of the 4.2 W picosecond laser is 6.4 times higher than the 60 W nanosecond laser. These results validate that the self-focusing of the high-peak-power picosecond laser signal is e  
Figure 17: Results of the laser ranging of debris using the 60-W (200-Hz) nanosecond laser (**left**) and 4.2-W (1-kHz) picosecond laser (**right**). the diffraction limit angle in the atmosphere.