## 3 Results and Discussion  
Figure 1a-i show the cross-sectional scattering intensity patterns from pure water, contaminated water, and black carbon, respectively. These patterns were calculated using Equations (13) to (16), where the angle \(\theta\) is from 0 to 2\(\pi\). In the calculations, the particles are assumed to be independent. The indices (a) to (i) refer to each of the size parameters in Table 2 where (a) \(x\) = 0.59, (b) \(x\) = 1.18, (c) \(x\) = 1.77, (d) \(x\) = 2.95, (e) \(x\) = 5.91, (f) \(x\) = 8.85, (g) \(x\) = 7.38, (h) \(x\) = 14.76, and (i) \(x\) = 22.12. Therefore, each of the size parameters captures the effect of particle radius and incident wavelength on the intensity pattern of the scattered incident light. The patterns are plotted in polar coordinates from 0 to 360\({}^{\circ}\) where backscattering is towards the 180\({}^{\circ}\)-angle (towards the left) and forward scattering is towards the 0 or 360\({}^{\circ}\)-angle (towards the right). The intensity is plotted in a log\({}_{10}\) scale to emphasize the salient features of the scattering patterns. For all three particles, the scattering intensity is nearly uniform with a slight preference for forward scattering when the size of the particle is within 20% of the incident laser's wavelength, for example (a) \(x\) = 0.59, (b) \(x\) = 1.18, and (c) \(x\) = 1.77 where the particle size is 0.1 \(\upmu\)m and the wavelength is (a) 1.064 \(\upmu\)m and (b) 0.532 \(\upmu\)m. The cross-sectional scattering intensity patterns for pure water, contaminated water, and black carbon also appear to be similar, especially when \(x\) = 0.59 (Figure 1a) where it appears to be indistinguishable. As the size parameter increases because of the particle becoming bigger and the laser wavelength becoming shorter, forward  
scattering is favored with the forward scattering intensity becoming confined within a narrower polar distribution. The scattering patterns from pure water, contaminated water, and black carbon start to exhibit distinguishable features, especially from the back-scattered light when \(x=1.77\). The patterns then become increasingly unique as the value of \(x\) increases. These results show that for small particle sizes, shorter laser lidar wavelengths should be used to distinguish between the particles. As the size of the particle increases, there is more flexibility in using longer laser lidar wavelengths to distinguish between the particles. Interestingly, when the particle size is 1.25 \(\upmu\)m, a laser lidar wavelength of 0.532 \(\upmu\)m (Figure 1h) gives a similar scattering intensity pattern as a laser lidar wavelength of 0.355 \(\upmu\)m (Figure 1i). Using either laser lidar wavelength gives clearly distinguishable scattering intensity patterns that can differentiate between pure water, contaminated water, and black carbon. This result is important economically since it is cheaper and easier to use the second harmonics (0.532 \(\upmu\)m wavelength) instead of the third harmonics (0.355 \(\upmu\)m wavelength) of Nd-based laser systems. The second harmonic also has a higher light intensity compared to the third harmonic. The unique scattering patterns demonstrated by different particulate matter could serve as fingerprints that can be used to identify the scattering particle. Calculation of the cross-sectional scattering intensity patterns could be extended to other types of pollutants as well. By doing so, a database of atmospheric pollutants and their corresponding scattering patterns can be created. To make the database of pollutants more accurate, we need to obtain and compare the scattering cross-sections of typical scattering sources in the real experiment.