## 3 Results and Discussion
\begin{array}{c}\frac{ \partial\varepsilon^{\prime\prime}\pi_{y}}{\partial\varepsilon^{\prime} \pi_{y}}\\ \frac{\partial\varepsilon^{\prime}\pi_{y}}{\partial\varepsilon^{\prime} \pi_{y}}\end{array} \right],\phi_{c}=\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] \tag{3}\]  
where \(\varepsilon^{\prime}i\) and \(\varepsilon^{\prime\prime}i\) (\(i=x,y,z\)) are the real part and imaginary part of dielectric function along different crystal direction, \(\varepsilon i=\varepsilon^{\prime}ii+i\varepsilon^{\prime\prime}iii\), and \(q^{\intercal QS}\) is the normal coordinate of \(\text{A}_{\text{g}}\) Raman mode. According to Figure 4a, the polarization vectors of incident laser and scattering light under PP configuration are \(e_{i}=e_{\text{s}}=[\sin\,(\varphi-\theta),\,\cos\,(\varphi-\theta)]\), thus the Raman intensity of \(\text{A}_{\text{g}}\) mode at PP is as follows. \[I_{\text{A}_{\text{g}}}^{\parallel}=\left[|a|\sin^{2}(\varphi-\theta)+|c|\cos \phi_{\alpha\alpha}\cos^{2}(\varphi-\theta)\right]^{2}+|c|^{2}\sin^{2}\phi_{ \alpha\alpha}\cos^{4}(\varphi-\theta) \tag{4}\]  
Similarly, when the polarization directions of incident laser and scattering light are keeping perpendicular to each other, the polarization vectors of incident laser and scattering light under orthogonal-polarized (OP) configuration are \(e_{i}\) = [sin (\(\varphi-\theta\)), cos (\(\varphi-\theta\))] and \(e_{s}\) = [cos (\(\varphi-\theta\)), -sin (\(\varphi\) - \(\theta\))], respectively. The Raman intensity of A\({}_{\rm g}\) mode at OP is expressed as Equation (5). \[I_{\rm A_{g}}^{\perp}=\left[(|a|-|c|\cos\phi_{ca})^{2}+|c|^{2}\sin^{2}\phi_{ca} \right]\sin^{2}(\varphi-\theta)\cos^{2}(\varphi-\theta) \tag{5}\]  
where \(\varphi_{ca}=\varphi_{c}-\varphi_{a}\) is the phase difference between the complex Raman tensor elements \(c\) and \(a\). The Raman intensity under the NA configuration, \(I_{\rm Ag}\), is the sum of those under the parallel-polarized and vertical-polarized configurations [23]. Hence the angular-intensity function for the NA configuration is the superposition of the functions for the PP and OP configurations [24; 25; 26]. \[I_{\rm A_{g}}=a^{2}\sin^{2}(\varphi-\theta)+c^{2}\cos^{2}(\varphi-\theta) \tag{6}\]  
The experimental data in Figure 5a,b were fitted by using Equation (4) and those in Figure 5c,d were fitted by using Equation (6). The fitted curves of A\({}_{\rm g}\)\({}^{1}\) and A\({}_{\rm g}\)\({}^{2}\) modes are given in Figure 5 using red and blue lines, respectively. The fitted results of \(\theta\) in Figure 5 are 66.5\({}^{\circ}\) (A\({}_{\rm g}\)\({}^{1}\) at PP), 66.4\({}^{\circ}\) (A\({}_{\rm g}\)\({}^{2}\) at PP), 66.3\({}^{\circ}\) (A\({}_{\rm g}\)\({}^{1}\) at NA), and 67.8\({}^{\circ}\) (A\({}_{\rm g}\)\({}^{2}\) at NA), respectively. It shows that the identification method of phosphorene crystalline orientation based on a non-analyzer Raman system achieved similar results with those using the traditional method. The deviations between the results of the two methods are much less than the rotating step (10\({}^{\circ}\)) of polarized direction during the Raman experiments, which were probably caused by manual rotation of polarizer, curve fitting process and other factors. Hence the A\({}_{\rm g}\)\({}^{1}\) and A\({}_{\rm g}\)\({}^{2}\) modes in NA configuration can be used to identify the phosphorene crystalline orientation. Figures 6-9 give the normalized Raman intensities of four different phosphorene specimens of 3, 2, 4, and -37 (-20 nm by AFM) layers, respectively. All the solid square points denote experimental data, and the solid lines are the fitted curves by using Equation (4) for PP results and Equation (6) for NA results, respectively. These experimental data and fitting results show that the crystalline orientations identified using NA configuration are all similar to those using traditional PP configuration.