## 4 Simulation and Analysis of Laser Pulse Waveform
The pointing error illustrated in Figure 4a can be reduced if an acquisition camera is employed to detect space debris illuminated by sunlight, which has an effect on the ELPW intensity as shown in Equation (1). Assuming that the space target is translated from the axis of beam propagation to the y-axis with pointing error \(\phi\), Equation (6) can be rewritten as  
\[G_{i}(t)=\frac{2}{\pi W_{eff}^{2}(z)}\exp\left[-2\frac{x_{b,i}^{2}+\left(R_{O }\sin\phi+y_{b,i}\right)^{2}}{W_{eff}^{2}(z)}\right]\cdot\frac{1}{\sqrt{2\pi }\imath_{R}}\exp\left[-\frac{1}{2x_{R}^{2}}\left(t-\frac{2R_{i}}{c}\right)^{ 2}\right], \tag{25}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Description** & **Symbol** & **Value** \\ \hline Laser wavelength & \(\lambda\) & 1064 nm \\ Transmitted power & \(P_{T}\) & 60 W \\ Pulse energy (repetition rate) & - & 60 J (1 Hz) \\ Full width at half maximum & FWHM & 5 ns \\ Transmitting optical efficiency & \(\eta_{T}\) & 0.8 \\ Divergence half angle & \(\theta_{D}\) & 10 arcsec \\ Telescope area & \(A_{R}\) & 0.25\(\pi\)m\({}^{2}\) \\ Receiving optical efficiency & \(\eta_{R}\) & 0.7 \\ Optical filter transmittance & \(\eta_{F}\) & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optical parameters of transmitter and receiver. where \(\left(x_{b,i},y_{b,i}\right)\) is the coordinate of the \(i\)th grid on the body frame, and \(R_{O}\) is the distance to the origin of the body frame. Figure 4b shows the ELPWs of defunct satellite in terms of the pointing error. Compared to the ELPWs for three pointing errors (\(\phi=0,3,5\)arcseconds), the ELPW intensity decreases with the same shapes as the pointing error increases. In addition, two pointing errors (\(\phi=5,-5\)arcseconds) cause some ELPW shape differences because the reflected surface of the defunct satellite is not symmetric with respect to the z-axis. But the difference of two shapes can be mitigated if a flattened Gaussian beam in which the energy distribution is nearly identical on the cross-section of the laser beam [43] is used. Thus the pointing error was not considered in the ELPW simulation. Figure 4: Illustration of (**a**) laser tracking with a pointing error, and (**b**) the echo laser pulse waveforms (ELPWs) for four pointing errors. The ELPWs were obtained by employing the reflected surface or orientation identical to Figure 5.  
Figure 3: (**a**) Beam spreading and (**b**) broadening, due to the atmospheric turbulence. The analytic expression related to the pulse broadening for laser altimeters [44] is introduced for the ELPW analysis:  
\[\sigma_{R}^{2}=\Big{(}\sigma_{0}^{2}+\sigma_{h}^{2}\Big{)}+\frac{4\text{Var}( \Delta\xi)\cos^{2}S_{\parallel}}{c^{2}\cos^{2}\!\big{(}\varphi+S_{\parallel} \big{)}}+\frac{4z^{2}\tan^{2}\theta_{  
ELPW well, which implies that the reflected surface is more complex than shown in Figures 5-7 and the second curve is not separated into the bus module and solar panel elements. Figure 9 shows two Gaussian decomposition results based on 3 and 4 Gaussian functions for the ELPW given in Figure 8, both of which fit the ELPW well. According to the orientation shown in Figure 8b, it is obvious that the first Gaussian function corresponds to the bus module and that the others result from the combination of the bus module and solar panel elements. The half-width of the Gaussian functions (\(\sigma_{2}\) and \(\sigma_{3}\) in Figure 9a, \(\sigma_{2}\) and \(\sigma_{4}\) in Figure 9b) is larger than \(\sigma_{2}\) of Figure 5 due to the solar panel slope parallel to the beam direction (\(S_{\parallel}\)), even though the cross-section area of the solar panel is smaller than that in Figure 5. The second curve in the ELPW was decomposed into 2 or 3 Gaussian functions, which can be also decomposed into many Gaussian functions by adding small Gaussian functions such as the third Gaussian function in Figure 9b. The complex reflected surface requires many Gaussian functions for the Gaussian decomposition and then makes the explicit analysis difficult in relation to the orientation of space debris.