## 3 Results and Discussion
The duty cycle is defined as the ratio of ridge width to period, which means that the fabrication tolerance of the line width of the grating was over 65 nm. As shown in Figures 6 and 7, the diffraction efficiencies of \(-\)1 st order for TE and TM polarizations exceeded 95%, varying from 1000 nm to 1100 nm, with Littrow incident angle of 46.35\({}^{\circ}\) when \(h_{1}\) ranged from 180 nm to 275 nm and \(h_{2}\) ranged from 133 nm to 142 nm, respectively. The fabrication tolerance of the second layer was smaller than that of the first layer. According to the previous analysis, since the material of the second layer had a higher refractive index, the corresponding effective refractive index of the second layer was also higher than the first layer. Therefore, the accumulated phase difference between the propagation mode varied significantly with the thickness of the second layer, which led to a smaller fabrication tolerance for the second layer. Moreover, the highly reflective layer consisted of the match layer and the metal mirror. In general, the diffraction efficiency of the metal-mirror based reflective grating exhibits periodic variations with increase of the thickness of the match layer in a wide range [35]. Therefore, the match layer tends to have a certain fabrication tolerance. Figure 8 shows the diffraction efficiencies of \(-\)1 st order versus incident wavelength and the thickness of match layer for both TE and TM polarizations. The diffraction efficiencies of \(-\)1 st diffraction order for TE and TM polarizations exceeded 95%, varying from 1000 nm to 1100 nm, in Littrow mounting, when \(h_{3}\) ranged from 220 nm to 247 nm. The properties of great fabrication tolerance of duty cycle and the thickness of the three dielectric layers demonstrated the potential of the designed grating in the fabrication process. In addition, the incidence angle of the designed grating was the Littrow angle, so the \(-1\)st order diffraction light overlapped with the incident light. In some practical applications, the incidence angle needs to be slightly deviated from the Littrow angle. Therefore, it was necessary to analyze the incident angle tolerance of the designed grating. Figure 9 shows the \(-1\)st order diffraction efficiencies versus incident wavelength and incident angle for both TE and TM polarizations. The result showed that the designed grating had a large incidence angle tolerance. Especially in the case of TE polarization, the incident angle had little effect on the diffraction efficiency within \(\pm 10\) degrees of the Littrow angle. When the incident angle varied from \(39.6^{\circ}\) to \(54.4^{\circ}\), the diffraction efficiency of \(-1\)st order exceeded \(90\%\) for TE and TM polarizations, which could meet the requirements of practical application. Figure 8: Tolerance analysis of the thickness of match layer. \(-1\)st order diffraction efficiency versus incident wavelength and \(h_{3}\) for (**a**) TE and (**b**) TM polarizations. Figure 7: Tolerance analysis of the thickness of silicon layer. \(-1\)st order diffraction efficiency versus incident wavelength and \(h_{2}\) for (**a**) TE and (**b**) TM polarizations. Figure 9: Tolerance analysis of incident angle. \(-1\)st order diffraction efficiency versus incident wavelength and incident angle for TE and TM polarizations.