## 4 Simulation and Analysis of Laser Pulse Waveform  
Two types of space debris were simulated to investigate the relationship between ELPW and orientation: defunct satellite (dimensions: \(4.2\times 1.3\times 1.3\) m) and rocket body (dimensions: \(24.9\times 10.0\times 10.0\) m). As shown in Figure 2, representing 3-dimensional models available from the NASA resources ([[https://nasa3d.arc.nasa.gov](https://nasa3d.arc.nasa.gov)]([https://nasa3d.arc.nasa.gov](https://nasa3d.arc.nasa.gov))), the defunct satellite consists of a bus module, a solar panel, and small payloads, while the rocket body consists of five rocket engines, forward and aft skirts, tank and thrust structures, and an aft interstage. To generate the ELPW similar to the real measurement environment, the surface is made up of 113,241 grids for the defunct satellite, and 3,232,611 grids for the rocket body, respectively. Figure 1: Flowchart of the genetic algorithm. Figure 2: 3-dimentional images of (**a**) defunct satellite, and (**b**) rocket body. We performed a numerical simulation by considering the slant range \(R_{O}\) of 500 km from the ground station to the origin of body frame, a zenith angle \(\theta_{z}\) of 50 degrees, the ground station height \(h_{s}\) of 1000 m, and the grid reflectivity \(\rho\) of 0.175. Taking the clear sky (\(V=15\) k m) and 1,064 nm wavelength of the laser beam into account, the atmospheric attenuation coefficient is \(\sigma(\lambda,V,0)=0.16\), and the cirrus cloud thickness \(C_{T}\) of 0.5 km is considered. The following parameters related to the NEP are also used for the numerical simulation: the electrical bandwidth \(B=100\)MHz, the surface and bulk dark currents \(I_{ds}=20\)nA and \(I_{db}=50\)nA, respectively, the detector gain \(G=30\), the excess noise factor \(F_{A}=0.25\), the detector quantum efficiency \(\eta_{Q}=0.38\), the spectral radiance of the sky \(I_{\lambda}=4.0\times 10^{-8}\mathrm{watts}/\mathrm{cm}^{2}\cdot\mathrm{ ster}\cdot\mu m\), the receiver field of view \(\Omega_{FoV}=25\mathrm{arcsec}\), the bandwidth of the optical bandpass filter \(\Delta\lambda_{F}=1\)nm, the amplifier temperature \(T=19\)\({}^{\circ}\)C, the noise factor \(F_{N}=1.5\), and the load resistance \(R_{L}=1,000\Omega\). In addition, the optical parameters related to the transmitter and receiver are summarized in Table 1.  
Considering that the laser beam is transmitted from the ground station at sea level to the space debris located at the slant range of 500 km, Figure 3 shows the beam spreading and broadening in terms of the atmospheric turbulence condition, zenith angle, and inner scale factor. The beam spreading due to the atmospheric turbulence has a considerable influence on the received power because the beam size can increase more than 2 times under the strong turbulence condition (\(A_{0}=1.5\times 10^{-13}m^{-2/3},\
u=30m/s\)) and high zenith angle (\(\theta_{Z}=70\)deg). The beam broadening in time is little affected by the inner scale factor even though it suffers from the turbulence condition as well as the zenith angle. It is shown that the beam broadening effect decreases as the transmitting pulse width increases and that the effect also increases as the turbulence strength and zenith angle increase. In contrast to the nanosecond regime of the transmitted pulse width, the beam broadening is too significant in the femtosecond regime to be neglected for the ELPW analysis. However, the beam broadening is negligible in this study because the transmitted pulse width is in the order of nanoseconds. Regarding the beam spreading, the most commonly used values (\(A_{0}=1.7\times 10^{-14}m^{-2/3},\
u=21m/s\)) are employed for the ELPW simulation. Because of the small beam divergence, the DLR system controls the laser beam to the direction of a pre-determined position in space based on the predicted ephemeris. However, the target of space debris may deviate a little from the axis of beam propagation even though the receiver detects the echo signal from the target located within the coverage of beam divergence.