## 3 Results and Discussion  
Figure 2 shows the crystalline structure of phosphorene. Table 1 gives the Raman tensors for all active modes of phosphorene, whose corresponding atomic displacements are displayed in Figure 3. It can be seen that phosphorene has six Raman-active vibration modes. In backscattering geometry configuration, only Ag (including Ag\({}^{1}\) and Ag\({}^{2}\)) and B\({}_{2g}\) mode are Raman visible [22] where Ag\({}^{1}\), Ag\({}^{2}\), and B\({}_{2g}\) correspond to the atomic vibrations along the out-of-plane direction, armchair (AC) direction, and zigzag (ZZ) direction, respectively. Figure 4a shows the optical image of phosphorene specimens, where the cross point in the dashed box is the Raman sampling spot at that time, X and Y are the horizontal and vertical directions of the sample stage of the Raman system, \(\varphi\) is the angle from the polarized direction of the incident light to the X-axis and \(\theta\) is the angle from the armchair direction of the phosphorene specimen to the X-axis, which is the crystalline orientation to be identified in this work. Figure 4b shows the Raman spectra of a phosphorene specimen when \(\varphi\) was 180\({}^{\circ}\), where the positions of A\({}_{\mathrm{g}}\)\({}^{1}\), B\({}_{\mathrm{g}}\), and A\({}_{\mathrm{g}}\)\({}^{2}\) modes correspond to \(\sim\)363 cm\({}^{-1}\), \(\sim\)440 cm\({}^{-1}\), and \(\sim\)470 cm\({}^{-1}\), respectively. The distance between the A\({}_{\mathrm{g}}\)\({}^{2}\) and B\({}_{\mathrm{2g}}\) mode is about 28.8 cm\({}^{-1}\), then the measured specimen was a 3-layer phosphorene [20]. With the polarized direction of incident laser varying from 0\({}^{\circ}\) to 360\({}^{\circ}\), the Raman intensities of phosphorene specimens within 300-500 cm\({}^{-1}\) under PP and NA configurations were achieved, shown in the Figure 4c,d, respectively. The angular-spectral images show that the intensity of each mode changes with the polarization direction obviously. Based on the results in the Figure 4c,d, the fitted and then normalized data of A\({}_{\mathrm{g}}\)\({}^{1}\) and A\({}_{\mathrm{g}}\)\({}^{2}\) intensities are listed in the polar coordinates as shown in the Figure 5, where each solid square point corresponds to the experimental datum whose distance to the center of the polar coordinates denotes its normalized Raman intensity and angular position denotes the polarized direction of incident laser (viz. \(\varphi\)). In Figure 5a,b show the data under PP configuration, and Figure 5c,d, the data under NA configuration. Figure 4: **(a)** Optical image of phosphorene specimens, **(b)** Raman spectra of 3-layer phosphorene specimen under parallel-polarized (PP) and non-analyzer (NA) configurations, respectively, **(c,d)** angular-spectral images of 3-layer phosphorene specimen under PP and NA configurations, respectively. Figure 5 reveals the angular dependence of the Raman intensities of both the \(\text{A}_{\text{g}}^{1}\) and \(\text{A}_{\text{g}}^{2}\) modes, which can be described theoretically. According to the Raman tensors in Table 1 and the Raman selecting rules, the Raman intensity of each mode is given by Equation (1) [22]. \[I_{s}\propto\left|e_{i}\cdot\mathbf{R}\cdot\varepsilon_{s}^{\intercal}\right|^{2} \tag{1}\]  
Existing studies discovered that the Raman scattering intensity of phosphorene was influenced by the light absorption. Hence the Raman tensors of phosphorene, listed in Table 1, should be represented as the forms of complex function as Equation (2) [5; 22]. \[a=|a|e^{i\phi_{a}},b=|b|e^{i\phi_{b}},c=|c|e^{i\phi_{c}} \tag{2}\]  
And the phases of Raman tensor elements \(a\), \(b\), \(c\) are  
\[\phi_{a}=\arctan\left[ \begin{array}{c}\frac{\partial\varepsilon^{\prime \prime}\pi_{x}}{\partial\varepsilon^{\prime}\pi_{y}}\\ \frac{\partial\varepsilon^{\prime}\pi_{y}}{\partial\varepsilon^{\prime} \pi_{y}}\end{array} \right],\phi_{b}=\arctan\left[