## 2 Design  
The structure of the reflective grating with a double-layer dielectric rectangle groove is shown in Figure 1a. The rectangular groove of the reflective grating consists of two dielectric layers. The first layer of the groove is HfO\({}_{2}\) with the refractive index \(n_{1}\) and the second layer is made of silicon with the refractive index \(n_{2}\). The match layer of the designed grating is made of fused silica with the refractive index \(n_{3}\), which serves as the connection layer. When incident light illuminates the grating, the TE-polarized wave (electric field vector perpendicular to the \(xoz\) plane) and TM-polarized wave (magnetic field vector perpendicular to the \(xoz\) plane) are diffracted to the \(-1\)st diffraction order with high efficiency. Figure 1b shows the refractive indices of the materials of the three-layer dielectric layers at operating wavelength [24; 25; 26]. The material of the metal mirror is silver, and its thickness is set to 200 nm, which is significantly larger than the skin depth of the metal to prevent light transmission through the metal mirror to the substrate. Silver has high reflectivity and low absorption at the operating wavelength, which is beneficial to obtain high diffraction efficiency and wide bandwidth of the reflective gratings. However, silver is easily oxidized. The match layer, made of fused silica, is also used as a protective layer. The refractive index of silver at 1053 nm is 0.12483 + 7.3009i, which was taken from Reference [27]. The period of grating is \(\Lambda\) and the duty cycle of the grating is \(f\), defined as the ratio of ridge width to period. Respectively, \(h_{1}\), \(h_{2}\) and \(h_{3}\) represent the thickness of the first layer, the second layer and the match layer. The incident angle \(\alpha\) is the Littrow angle with incident wavelength of 1053 nm, which can be expressed as:  
\[\alpha=\arcsin(\frac{\lambda}{2\Lambda}) \tag{1}\]  
where \(\lambda\) is the wavelength of incident light. Diffraction occurs when incident light strikes the grating, and the diffraction order is determined by the grating equation which can be expressed as:  
\[\sin\theta_{m}=\sin\theta_{i}+m\frac{\lambda}{\Lambda} \tag{2}\]  
where \(\theta_{i}\) is the incident angle and \(\theta_{m}\) is the \(m\)th order diffraction angle. (\(m\) is an integer.) When the wavelength and the period satisfy \(\frac{\lambda}{2}<\Lambda<\frac{3\lambda}{2}\) only two diffraction orders of \(m=0\) and \(m=-1\) are produced. Fewer diffraction orders indicate that the energy of the incident light is more easily coupled to the main diffraction order, which facilitates the design of high-efficiency gratings. The physical mechanism of the diffraction process of the designed grating can be explained by the modal method [28; 29]. Incident light excites propagation modes with different effective refractive indices at each layer. The effective refractive index of the propagation mode is obtained by solving the dispersion equation [30]. The high-density gratings with Littrow incident angle only have two propagation modes corresponding to an effective refractive index greater than 0. The effective refractive index of the other modes is imaginary number and disappears rapidly with propagation. Therefore, these evanescent modes are neglected in the modal method. The two propagation modes produce phase differences as it propagates through each dielectric layer. The propagation modes are coupled to each other at the exit surface of the grating and the efficiency is determined by the accumulated phase difference. Moreover, the diffraction efficiency of the \(-1\)st order reaches its maximum when the accumulated phase difference between the two propagation modes is an odd multiple of \(\pi/2\). In order to obtain a high-efficiency, broadband and polarization-independent grating structure, RCWA was used to calculate the diffraction efficiency of grating. This is also called the Fourier Modal Method.