## 3 Results and Discussion
### Polarization Characteristics of Particle Shape  
Real atmospheric aerosols usually have complex irregular morphology. For example, carbon aerosols can have different microscopic characteristics, such as nanochips, disordered mesoporous particles, and hollow spheres, as shown in Figure 11a-c. Particles are more likely to cluster together in a real environment, which makes them more difficult to measure and analyze. Herein, we measured three kinds of aerosols, and polarization indices are shown in Figures 11 and 12.  
Figure 10: (**a**) Mean values of multidimensional polarization indices; (**b**) PFR indices; (**c**) Wilk’s Lambda indices. The red triangles mark index 11 (P-Hdop (60\({}^{\circ}\))), index 12 (P-Pdop (60\({}^{\circ}\))), index 14 (P-Pdop (85\({}^{\circ}\))), index 19 (R-Hdop (60\({}^{\circ}\))), and the blue triangle marks index 1 (H-Hdop (30\({}^{\circ}\))).  
Figure 9: 24-dimensional polarization indices distribution of aerosol samples with different absorption effects. The SEM image of samples are listed in the right column: (**a**) Fly ash (**b**) Arizona dust; (**c**) Carbon sphere.  
In general, the PFR and Wilk's Lambda values of these irregular carbon samples are greater than the three groups of aerosol samples mentioned above. From the perspective of the Wilk's Lambda index, the larger intraclass distance and smaller interclass difference are the main reasons that make it difficult to distinguish morphological attributes. Our previous studies [33] proposed using one specific polarization index to distinguish spherical, ellipsoidal, and fibrous suspended particles. However, from the electron microscope pictures, the morphological characteristics of these carbon aerosols are not simple shapes. Considering the in situ suspension status of measured aerosols, polarization characterization based on multidimensional indices can better identify and distinguish aerosol types because of the improvement of information dimensions.  
According to Figure 12a, P-Pdop (60\({}^{\circ}\)), P-Pdop (85\({}^{\circ}\)), P-Hdop (115\({}^{\circ}\)), and R-Pdop (30\({}^{\circ}\)) of the four types of samples can be well-differentiated. However, through comprehensive consideration combined with PFR and Wilk's Lambda factors in Figure 12b,c, P-Pdop (60\({}^{\circ}\)) and P-Pdop (85\({}^{\circ}\)) show feasibility in practical measurement and analysis. Additionally, from Figures 11 and 12a, H-Hdop (30\({}^{\circ}\)), H-Pdop (30\({}^{\circ}\)), H-Pdop (60\({}^{\circ}\)), and R-Hdop (60\({}^{\circ}\)) seems insensitive to the aerosol appearances of black carbon samples.  
For these aerosol samples with similar compositions but irregular morphology, the characterization ability of polarization indices is quite different. From the analysis of the experiments, it can be seen that the 45\({}^{\circ}\) linearly polarized incident state is most suitable for the characterization and differentiation of particulate matter. However, if we employ an H  
Figure 11: 24-dimensional polarization indices distribution of four kinds of aerosol samples with irregular morphology. The SEM image of samples are listed in the right column: (**a**) Carbon nanochip; (**b**) Disordered mesoporous carbon; (**c**) Hollow carbon sphere.  
Figure 12: (**a**) Mean values of multidimensional polarization indices; (**b**) PFR indices; (**c**) Wilk’s Lambda indices. The red triangles mark index 12 (P-Pdop (60\({}^{\circ}\))) and index 14 (P-Pdop (85\({}^{\circ}\))), and the blue triangles mark index 1 (H-Hdop (30\({}^{\circ}\))), index 2 (H-Pdop (30\({}^{\circ}\))), index 4 (H-Pdop (60\({}^{\circ}\))), index 19 (R-Hdop (60\({}^{\circ}\))).  
(horizontal) linear polarized state, the influence of particle morphology on polarization scattering is not obvious.