## 2 Mathematical Model  
### Echo Laser Pulse Waveform  
The intensity of a laser beam is not uniform in temporal and spatial domains along the direction of laser beam propagation, which requires that the laser pulse profile should be modeled in time as well as in space. The laser beam propagation can be approximated by assuming that the laser beam has a Gaussian shape in terms of temporal and spatial domains. So the irradiance distribution and normalized waveform of the propagated laser pulse are expressed as [26; 27]  
\[I(d,z)=\frac{2P_{T}}{\pi W_{eff}^{2}(z)}\exp\left[-2\frac{d^{2}}{W_{eff}^{2}(z) }\right], \tag{1}\]  
\[p(t)=\frac{1}{\sqrt{2\pi}\tau_{0}}\exp\left[-\frac{t^{2}}{2\tau_{0}^{2}} \right], \tag{2}\]  
where \(d\) is the radial distance from the axis of propagation (i.e., \(d^{2}=x^{2}+y^{2}\)), \(z\) is the axial distance from the beam's narrowest point called waist, \(P_{T}\) is the transmitted power, \(W_{eff}(z)\) is the effective beam size referred to as the Gaussian beam radius, and \(\tau_{0}\) is the transmitted pulse width which is related to the full width at half maximum (FWHM) as follows: \(FWHM=2\sqrt{2\ln 2}\tau_{0}\)[28]. The reflected or intersected surface of space debris by the laser beam is assumed to consist of small grids in order to generate a total ELPW. Some of the laser power received by the \(i\)th grid located at \((x_{i},y_{i})\) is absorbed while the rest is reradiated back into space, in proportion to the effective receiving area depending on the incident angle \(\theta_{i}\). Therefore the scattered power by the \(i\)th grid is described by  
\[P_{S,i}=\frac{2P_{T}}{\pi W_{eff}^{2}(z_{i})}\exp\left[-2\frac{d_{i}^{2}}{W_{ eff}^{2}(z_{i})}\right].(\rho A_{i}\cos\theta_{i}), \tag{3}\]  
where \(A_{i}\) is the \(i\)th grid area, and \(\rho\) is the reflectivity on the assumption that all grids have the same reflectivity in this study. Large grid areas can save computation time but might provide ELPW features less accurate from the true orientation of space target. So all grids should have optimal areas so that the ELPW simulation closely resembles a practical situation with minimal computation time, smaller than 50 cm\({}^{2}\) in this study. If the incoming radiation to the receiver is scattered uniformly into a cone of solid angle \(\Omega\)[29], the laser power coming into the receiver (i.e., received power) is given by  
\[P_{R,i}=\frac{2P_{T}}{\pi W_{eff}^{2}(z_{i})}\exp\left[-2\frac{d_{i}^{2}}{W_{ eff}^{2}(z_{i})}\right].(\rho A_{i}\cos\theta_{i})\cdot\frac{A_{R}}{\Omega R_{i}^ {2}}, \tag{4}\]  
where \(A_{R}\) is the receiving area, and \(R_{i}\) is the range to the \(i\)th grid. The solid angle can be replaced with \(\pi\) by considering the grid to be a diffuse scattering target, or a Lambertian target. Taking into consideration the normalized waveform given by Equation (2), the received power in terms of the time domain can be written by  
\[P_{R,i}(t)=\frac{P_{T}}{\pi R_{i}^{2}}G_{i}(t)\rho A_{i}\cos\theta_{i}A_{R}, \tag{5}\]  
\[G_{i}(t)=\frac{2}{\pi W_{eff}^{2}(z)}\exp\left[-2\frac{d_{i}^{2}}{W_{eff}^{2}( z)}\right].\cdot\frac{1}{\sqrt{2\pi}\tau_{R}}\exp\left[-\frac{1}{2\tau_{R}^{2}} \left(t-\frac{2R_{i}}{c}\right)^{2}\right], \tag{6}\]  
where \(c\) is the light speed in a vacuum, and \(\tau_{R}\) is the received pulse width, which is broadened by the atmospheric turbulence. The echo waveform model was also given by [PERSON] et al. [30] for the case of specular targets such as mirrors, which increases significantly the received power compared to diffusetargets. Space debris and most satellites are considered as diffuse targets since they are non-cooperative targets without CCRs and the echo signals are diffused from their surface. In the practical DLR system, the transmitted laser beam suffers from various attenuations by the time it comes back to a detector, e.g., atmospheric attenuation, and optical transmittance.