## 3 Numerical Results  
In this section, some numerical results are presented to demonstrate the performance of the angle detector described in the previous section. Specifically, the angle detector shown in Figure 1 is simulated by MEEP [19], which is a full-wave solver based on FDTD method. MEEP has been widely used to characterize various photonic devices, and its fidelity has been verified extensively [20, 21, 22]. Figure 2: Illustration of the angle detector’s rationale. All the numerical results in this section are obtained with the following geometrical parameters: \(\Lambda=1.065\)\(\upmu\)m, \(s=0.234\)\(\upmu\)m, \(t_{g}=0.26\)\(\upmu\)m, \(t_{\text{wg}}=0.22\)\(\upmu\)m, and \(t_{b}=2\)\(\upmu\)m, (please refer to Figure 1 for the definitions of these parameters; thickness of the substrate has negligible impact on the device, according to our observations). The refraction index of box layer is 1.48; and, the other three layers are made of silicon with refraction index 3.48. The incident light is a Gaussian beam with waist radius 23.4 \(\Lambda\) and frequency \(f_{inc}\)= 0.646 \(c/a\), where \(a=1\)\(\upmu\)m; in addition, electric field of the incident light is polarized along \(y\) direction. The two detectors are modeled by integrating the Poynting vectors along \(+x\) or \(-x\) direction within the waveguide. As for each detector, coupling efficiency is defined as \(P_{d}/P_{inc}\), where \(P_{d}\) is the detector's output and \(P_{inc}\) is calculated by integrating the incident light's power density by the detector's aperture. As discussed in Section 2, our device's performance is dictated by two major design parameters: quality factors and \(\Delta_{f}\). In the remainder of this section, various values of these two design parameters are employed to adjust the angle detector's performance. The quality factors are controlled by \(N\), the number of grating elements (apparently, the larger \(N\) is, the higher the quality factors are). Outputs of the \"\(+x\) detector\" and \"\(-x\) detector\" are plotted in Figure 3(a,b) respectively, with \(\Delta_{f}\)=0.88% and \(N\)=31. Since \(f_{inc}\)\(<\)\(f_{0}\) in our design, the \"\(-x\) output\" is stronger than the \"\(+x\) output\" for positive \(\theta_{inc}\). The choice of \(\Delta_{f}\) makes\(f_{inc}\) coincide with the transverse resonant frequency of \(\theta_{inc}=2^{\circ}\) for the \"\(-x\) detector.\" As a result, when the ratio between \"\(-x\) output\" and \"\(+x\) output\" is plotted in Figure 3(c), it exhibits a steep increasing slope in range \(\theta_{inc}\) C [0, 2\({}^{\circ}\)]. When \(\theta_{inc}\) is negative, the two detectors' outputs would be \"exchanged.\" the ratio between \"\(+x\) output\" and \"\(-x\) output\" exhibits a steep slope in range \(\theta_{inc}\) C [\(-2^{\circ}\), 0]. The data for negative \(\theta_{inc}\) are symmetric to those for positive \(\theta_{inc}\), hence are not shown in Figure 3. Therefore, from the two detectors' outputs, \(\theta_{inc}\) can be reliably derived when it falls in the range \(-2^{\circ}<\theta_{inc}<2^{\circ}\). As analyzed in Section 2, the detectable range for \(\theta_{inc}\) can be enlarged by reducing the quality factors. This analysis is verified by numerical results in Figure 4. There are three curves in Figure 4. One of them is the same as that in Figure 3(c), with \(N=31\). On the basis of the curve in Figure 3(c),  
Figure 3: Numerical results for the angle detector. (**a**) Coupling efficiency of the “\(+x\) detector.” (**b**) Coupling efficiency of the “\(-x\) detector.” (**c**) Ratio between the two detectors’ outputs. the other two curves in Figure 4 are obtained with \(N=25\) and \(N=21\). As expected, the reduction of \(N\) diminishes the device's quality factors, and thus, increases the detectable range. To be specific, the detectable range is [\(-2^{\circ}\), \(2^{\circ}\)] when \(N=31\); it is increased to [\(-2.5^{\circ}\), \(2.5^{\circ}\)] with \(N=25\) and further increased to [\(-3.5^{\circ}\), \(3.5^{\circ}\)] with \(N=21\).