## 2 Methods
### Performance Simulation Using the Return Power Budget Method
#### 2.1.1 Return Power
From the return power, the photon rate is computed as the sum of the contributions produced by the backscattered laser light (\(\mathcal{R}_{\text{sig}}\)) and the background light (\(\mathcal{R}_{\text{bg}}\)):  
\[\mathcal{R}_{\text{sig}}(\lambda,R)=\xi(\lambda)\cdot P_{\lambda} (\lambda,R)\cdot\lambda/hc\,,\] \[\mathcal{R}_{\text{bg}}(\lambda)=\xi(\lambda)\cdot P_{bg}(\lambda )\cdot\lambda/hc\,, \tag{4}\]  
where \(P_{\lambda}\) and \(P_{bg}\) are obtained from Equation (1) or Equation (2) and Equation (3), respectively, \(\xi(\lambda)\) are the efficiency factors that take into account the mirror reflectivity, the transmission of the LLG, the polychromator and the PMT photon-detection efficiency (see Table 1), and \(h\) is the Planck constant. The photon rates for the four wavelengths of interest of the BRL are reported in Figure 3 for two extreme cases: the best-case scenario of a clear night with only backgroundaerosol presence [9] and the worst-case scenario with the presence of calima. One can see that in either case we expect large rates above the background up to ranges above 30 km. \begin{table}
\begin{tabular}{c c l} \hline \hline
**Parameter** & **Value** & **Comments** \\ \hline \(A\) & 2.3 m\({}^{2}\) & 1.8 m primary mirror minus shadows \\ \(\eta\)dig. & 0.90 \(\pm\) 0.05 & photon counting efficiency of the readout \\ \hline \multicolumn{3}{l}{355 nm channel} \\ \(E\) & 80 mJ & energy per pulse \\ \(N_{0}\) & 1.4\(\cdot\)10\({}^{17}\) & photons per pulse \\ \(l\) & 7.5 m & digitization length for 12-bit, 20 MS/s sampling rate \\ \(\rho\) & 0.95 & mirror reflectivity, after re-aluminization, otherwise \(<\)0.3 \\ \(\zeta\) & 0.34 \(\pm\) 0.04 & combined LLG and polychromator transmissions [16] \\ \(PDE\) & 0.42 \(\pm\) 0.03 & PMT photon-detection efficiency [37] \\ \(\xi\) & 0.13 \(\pm\) 0.02 & combined channel efficiency \(\xi=\rho\cdot\zeta\cdot PDE\) \\ \hline \multicolumn{3}{l}{387 nm channel} \\ \(E\) & 80 mJ & energy per pulse (at 355 nm) \\ \(N_{0}\) & 1.4\(\cdot\)10\({}^{17}\) & photons per pulse \\ \(l\) & 7.5 m & digitization length for 12-bit, 20 MS/s sampling rate \\ \(\rho\) & 0.96 & mirror reflectivity, after re-aluminization, otherwise \(<\)0.3 \\ \(\zeta\) & 0.31 \(\pm\) 0.04 & combined LLG and polychromator transmissions [16] \\ \(PDE\) & 0.43 \(\pm\) 0.03 & photon-detection efficiency [37] \\ \(\zeta\) & 0.12 \(\pm\) 0.02 & combined channel efficiency \(\xi=\rho\cdot\zeta\cdot PDE\) \\ \hline \multicolumn{3}{l}{532 nm channel} \\ \(E\) & 128 mJ & energy per pulse \\ \(N_{0}\) & 3.4\(\cdot\)10\({}^{17}\) & photons per pulse \\ \(l\) & 3.75 m & digitization length for 16-bit, 40 MS/s sampling rate \\ \(\rho\) & 0.97 & mirror reflectivity, after re-aluminization, otherwise \(<\)0.3 \\ \(\zeta\) & 0.31 \(\pm\) 0.03 & combined LLG and polychromator transmissions [16] \\ \(PDE\) & 0.13 \(\pm\) 0.03 & photon-detection efficiency [37] \\ \(\xi\) & 0.035 \(\pm\) 0.009 & combined channel efficiency \(\xi=\rho\cdot\zeta\cdot PDE\) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameter values assumed for the power budget calculations of the pBRL. Figure 3: Expected signal and background rates of the pBRL, in two extreme cases: a clear night at the ORM, and a dust intrusion scenario with a VAOD of 0.7, the maximum allowed for science data taking. The step at 4 km in the elastic lines of the latter stems from the assumed sharp transition from dust to free troposphere at the boundary layer height. _NSB_ stands for the normal Night-Sky-Background found at the ORM [34], and _Moon_ for a night sky illuminated by a first or third quarter moon.