## 2 Simulation Method
This allows us to explore the effects of particle size, the complex refractive index of the particle relative to the surrounding air medium, and the wavelength of the incident radiation on the intensity and the polar distribution of MIe scattering by pure water, contaminated water, and black carbon. Such effects have not been explored yet in the context of lidar monitoring of particulate matter in the atmosphere. Pure water refers to pristine water droplets without any contamination while contaminated water refers to water droplets containing a mixture of waste, dirt, or chemicals. The wavelength of the incident radiation was chosen to be 1.064 \(\upmu\)m, 0.532 \(\upmu\)m, and 0.355 \(\upmu\)m. These wavelengths were chosen because these are the fundamental, second, and third harmonics of a commercial neodymium (Nd)-based laser system such as Nd:YAG or Nd:YVO\({}_{4}\). Such laser systems are typically used in conventional lidar monitoring stations and were, therefore, chosen for our simulations. The complex refractive indices of pure water, contaminated water and black carbon at the chosen excitation wavelengths are summarized in Table 1. The contaminated water considered in this case contains a mixture of ammonium sulfate and carbon. The radii of the scattering particles that we considered in our calculations, along with the corresponding size parameters obtained from Equation (5) are summarized in Table 2. The radii 0.1 \(\upmu\)m, 0.5 \(\upmu\)m and 1.25 \(\upmu\)m were chosen since these are approximately the average sizes (diameter) of particulate matter, for example, black carbon (0.085 \(\upmu\)m to 0.5 \(\upmu\)m) and PM2.5 (\(\leq\)2.5 \(\upmu\)m) [31]. MIe scattering is also the more appropriate model compared to Rayleigh scattering when the particle size is larger than around 10% of the wavelength of the incident radiation. Accordingly, the minimum particle radius for MIe scattering to be valid is around 0.1 \(\upmu\)m given the longest laser wavelength considered in our work, which is 1.064 \(\upmu\)m [32]. After calculating the Mie coefficients (\(a_{n}\) and \(b_{n}\)), the scattering efficiency \(Q_{sca}\) and backscattering efficiency \(Q_{b}\) are calculated using  
\[b_{n}=\frac{j_{n}(mx)\big{[}\dot{x}j_{n}(x)\big{]}^{\prime}-j_{n}(x)\big{[}mxj _{n}(mx)\big{]}^{\prime}}{j_{n}(mx)\Big{[}xh_{n}^{(1)}(x)\Big{]}^{\prime}-h_{ n}^{(1)}(x)\big{[}mxj_{n}(mx)\big{]}^{\prime}} \tag{2}\]0  
\[b_{n}=\frac{j_{n}(mx)\big{[}\dot{x}j_{n}(x)\big{]}^{\prime}-j_{n}(x)\big{[}mxj _{n}(mx)\big{]}^{\prime}}{j_{n}(mx)\Big{[}xh_{n}^{(1)}(x)\Big{]}^{\prime}-h_{ n}^{(1)}(x)\big{[}mxj_{n}(mx)\big{]}^{\prime}} \tag{2}\]1  
where \(n\) is the number of expansion terms in those infinite series. The infinite series was truncated after \(n_{\text{max}}=\text{x}+4\text{x}^{1/3}+2\) terms as proposed by [PERSON] and [PERSON] [24]. The \(x\) in \(n_{\text{max}}\) also refers to the size parameter given in Equation (3). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Scatterer**} & \multicolumn{3}{c}{**Wavelength of Incident Radiation (\(\upmu\)m)**} \\ \cline{2-4}  & **1.064** & **0.532** & **0.355** \\ \hline Pure Water [33; 34] & 1.327 + i2.89 \(\times\) 10\({}^{-6}\) & 1.334 + i1.32 \(\times\) 10\({}^{-9}\) & 1.343 + i6.5 \(\times\) 10\({}^{-9}\) \\ Contaminated Water [35] & 1.54 + i0.015 & 1.54 + i0.015 & 1.54 + i0.015 \\ Black Carbon [36] & 1.95 + i0.79 & 1.77 + i0.63 & 1.70 + i0.64 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Complex refractive index of contaminated water and black carbon at the 1.064, 0.532, and 0.355 \(\upmu\)m wavelengths of the incident radiation. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\uplambda\) (\(\upmu\)m) & **1.064** & **0.532** & **0.355** \\ \hline \(a\) (\(\upmu\)m) & **1.064** & **0.532** & **0.355** \\ \hline \(0.1\) & 0.59 & 1.18 & 1.77 \\ \(0.5\) & 2.95 & 5.91 & 8.85 \\ \(1.25\) & 7.38 & 14.76 & 22.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Size parameter, x for different values of particle radius and incident wavelength.