## 4 Simulation and Analysis of Laser Pulse Waveform
Figure 6: The (**a**) reflected surface, (**b**) orientation of defunct satellite, and (**c**) the echo waveform and 2 Gaussian functions obtained from the Gaussian decomposition for 45\({}^{\circ}\) rotation about the x-axis w.r.t. Figure 2a. Figure 7: The (**a**) reflected surface, (**b**) orientation of defunct satellite, and (**c**) the echo waveform and 2 Gaussian functions obtained from the Gaussian decomposition for \(-90^{\circ}\) rotation about the x-axis w.r.t. Figure 2a. The Gaussian decomposition results including the ELPWs are also shown in Figures 10-13 for four kinds of orientations of the rocket body. Unlike the ELPWs of the defunct satellite, the size of the rocket body is so large that the noise power is very weak compared to the received power. In addition, the reflected surface is too complex compared to the defunct satellite to fit the ELPWs well even by using several Gaussian functions. Therefore, the total number of Gaussian functions (\(n\)) is selected as the smallest number that satisfies the criterion of Equation (24). As shown in Figure 10, the first two Gaussian functions corresponding to the first peak represent the forward and aft skirts as well as the tank structure because their shapes are cylindrical, and the other Gaussian functions correspond to five rocket engines, the thrust structure and the aft interstage. The ELPWs in Figures 11 and 12 are so complex that each Gaussian function cannot be exactly in one-to-one correspondence with the components of the rocket body. Compared to the narrow distribution of ELPW in Figure 10, the wide distribution of ELPWs in Figures 11 and 12 results from the rocket body slope parallel to the beam direction (\(S_{\parallel}\)). However, the ELPW has the distinctive shape subjected to the orientation of the rocket body, which enables orientation information to be achieved. Regarding the Gaussian decomposition in Figure 13, the first two Gaussian functions represent five rocket engines while the next three Gaussian functions correspond to the thrust structure and the sixth small Gaussian function comes from all five rocket engines. Figure 8: The (**a**) reflected surface, (**b**) orientation of defunct satellite, and (c) the echo waveform and 2 Gaussian functions obtained from the Gaussian decomposition for \(-45^{\circ}\) rotation about the y-axis w.r.t. Figure 2a. Figure 9: Gaussian decomposition results using (**a**) 3 Gaussian functions and (**b**) 4 Gaussian functions for the ELPW in Figure 8c. It was shown that the orientation of a defunct satellite and a rocket body have a distinctive shape in the ELPW and the Gaussian functions that result from the Gaussian decomposition, which are a tool to characterize the orientation in more detail. The ELPW feature resulted from a single laser pulse that can provide sufficient information for the orientation determination of space debris. However, there is also a possibility in practical application that the ELPW features do not characterize the orientation of the space target due to its simple shape. If the ELPWs are available sequentially while the DLR is activated, the ELPW changes not only in shape but also in amplitude, which can provide more useful information for the orientation determination of space targets including even tumbling space debris. A number of small satellites, especially CubeSats which have simple and cubic shapes, have now been launched into space for scientific and commercial missions. It is difficult to determine the orientation of defunct CubeSats from the ELPW analysis even though a higher power laser is employed since the ELPW does not indicate the distinct features depending on their orientations because of a simple and cubic shape. But the orientation information of CubeSats is not relatively important in terms of the orbital prediction because the cross-sectional area variation is negligible over the entire orbit in computing the atmospheric drag and solar radiation pressure.