## 3 Results and Discussion  
Figure 3 shows the diffraction efficiency and the PDL of the designed grating with optimized parameters. The diffraction efficiencies of \(-1\)st order for TE and TM polarizations in Littrow incident angle of 46.35\({}^{\circ}\) exceeded 95% from 988 nm to 1122 nm, and over 98% in the bandwidth ranging from 1015 nm to 1085 nm. The average diffraction efficiencies for TE and TM polarizations of \(-1\)st order in the range from 1000 nm to 1100 nm was 98.92% and the average diffraction efficiency was defined as:  
\[\overline{\eta}=\frac{1}{2}\Bigg{\{}\frac{1}{N}\Bigg{[}\underset{\lambda_{i}}{ \overset{\lambda_{i}}{\sum}}(DE_{-1,TE}(\lambda_{i})+DE_{-1,TM}(\lambda_{i})) \Bigg{]}\Bigg{\}} \tag{4}\]  
Figure 3: Diffraction efficiencies of \(-1\)st and 0 th orders for TE and TM polarizations and the PDL of the designed grating. Figure 2: Diffraction efficiencies of \(-1\)st order varied with the truncated order \(m\) for TE and TM polarizations at 1053 nm of the designed grating. As a polarization-independent device, polarization-dependent loss (PDL) was used to evaluate the polarization sensitivity of gratings [31]. The equation of PDL can be expressed as:  
\[\text{PDL}=10\times\text{log}_{10}[\frac{\text{max}(DE_{-1,TE},DE_{-1,TM})}{\text {min}(DE_{-1,TE},DE_{-1,TM})}] \tag{5}\]  
As shown in Figure 3, the value of the PDL of the designed grating was only 0.037 dB at 1053 nm, and the average PDL from 1000 nm to 1100 nm was 0.051 dB which was low enough to ensure the high performance of the polarization-independent characteristic. According to the analysis of the diffraction property, the designed grating achieved excellent and desirable characteristics of high efficiency, broadband and polarization-independence. In order to verify the operating state of the designed grating with optimized parameters, the electric field distribution of the designed grating was analyzed. Figure 4 shows the electric field distributions for TE and TM polarizations of the designed grating calculated by the FEM [32]. When the plane wave impinged on the grating under Littrow mounting at 1053 nm, the electric field distributions of the incident region and the reflective region were almost the same, which meant that most of the energy of the incident light was diffracted to the \(-1\)st diffraction order. The electric field distribution of the designed grating clearly demonstrated the proof of the high diffraction efficiency of the \(-1\)st order of the designed grating for TE and TM polarizations. In addition, the high laser-induced damage threshold (LIDT) could provide potential for optical system expansion to higher power output [33,34]. It can be seen from Figure 4 that most of the electric field was distributed outside the grating, which helped to increase the LIDT. Moreover, this distribution not only appeared at 1053 nm but also at around the bandwidth from 1000 nm to 1100 nm. In the fabrication process, the structure of the grating always differs from the optimized parameters. Thus, it was necessary to analyze the fabrication tolerance of the designed grating. In addition, the designed grating required a large incidence angle tolerance, because the incidence angle deviates slightly from the intended angle in practical application. In this section, the variation of the diffraction efficiencies with duty cycle and thickness of each layer at Littrow incident angle of 46.35\({}^{\circ}\) were calculated by RCWA. There was only one variable of grating parameter in each calculation, the other parameters were fixed as optimized parameters. Figure 5 shows the diffraction efficiencies of \(-1\)st order versus incident wavelength and duty cycle for both TE and TM polarizations. When the duty cycle ranged from 0.39  
Figure 4: The electric field distributions of the grating at 1053 nm for (**a**) TE polarization and (**b**) TM polarization. to 0.48, the diffraction efficiencies for TE and TM polarizations exceeded 95%, varying from 1000 nm to 1100 nm.