# A 1 \(\times\) 2 Two-Dimensional Slanted Grating Based on Double-Layer Cylindrical Structure
## 3 Characters and Tolerance Analysis  
We obtained the parameters of the designed 2D slanted gratings. In order to achieve the performance of these designed gratings, we also analyzed the 2D slanted gratings by rigorous coupled wave analysis. The wavelength bandwidth of a grating is a range of wavelengths over which the grating can operate effectively. A wider bandwidth means that the grating can be applied to a wider range of wavelengths, such that it is necessary to analyze the bandwidth of the grating for both polarizations. In Figure 2, we plot the diffraction efficiencies of the (\(-\)1, 0) and (0, \(-\)1) orders at a range of wavelengths from 400 nm to 500 nm for TE and TM polarization, respectively. Figure 2a shows the efficiencies of the diffraction orders (0, \(-\)1) and (\(-\)1, 0) for TE polarization incidence, and Figure 2b shows the efficiencies of (0, \(-\)1) and (\(-\)1, 0) for TM polarization incidence. We can see that the (\(-\)1, 0) order diffraction efficiency and (0, \(-\)1) order diffraction efficiency are both greater than 30% when the grating is operated in the range of 403 nm-490 nm wavelengths. In particular, the (\(-\)1, 0) order diffraction efficiency and the (0, \(-\)1) order diffraction efficiency are both higher than 35% at operating wavelengths of 429-468 nm. Hence, the 2D slanted gratings can be used in the wavelength of the red wave band. Figure 3 depicts the efficiency of (0, 0) versus wavelength under normal incidence for TE and TM polarization. We should decrease the efficiency of the 0 th order to reduce the loss. The diffraction efficiencies of the (0, 0) order for TE and TM polarization are both less than 10% when the incident wavelength is in the range of 400 nm to 470 nm. In particular, the (0, 0) order diffraction efficiencies for TE and TM polarization are both less than 5% at operating wavelengths from 420 nm to 450 nm. According to Figures 2 and 3, we can find that this 2D slanted grating has a better performance of beam splitting and a wide spectrum for TE and TM polarization, which demonstrates the ease of use. In present times, these head-up display devices and near-eye display devices need a wider field of view (FOV). We also analyzed the efficiency versus incident angle because of the use of these gratings in the field of AR/VR technology. Figure 4 shows the influence of the incident angle on diffraction orders with optimized parameters. In the ideal case, when the angle of incidence is \(0^{\circ}\), the diffraction efficiency of \((-1,0)\) and \((0,-1)\) are equal. However, when we increase the incident angle, their changes are opposite. The diffraction efficiency of \((-1,0)\) is decreasing, whereas that of \((0,-1)\) is increasing. We can see that the diffraction efficiencies of the \((-1,0)\) order and the \((0,-1)\) order are both greater than 30% when the incident angle ranges from \(-3.2^{\circ}\) to \(3.2^{\circ}\). Figure 3: The efficiency of \((0,0)\) versus wavelength for TE and TM. Figure 2: Grating diffraction efficiency versus incident wavelength (**a**) TE polarization; (**b**) TM polarization. For general rectangular gratings, they are relatively easy to manufacture by using conventional lithography processes such as electron beam lithography, etching processes, and nano-imprint lithography. The ridge of traditional 2D gratings is perpendicular to the substrate. [PERSON] et al. fabricated a two-dimensional array by using an optimized nano-imprint lithography [29]. The production of slanted gratings has also been studied in a number of works throughout history. [PERSON] et al. produced a tilted grating with a tilt angle of 50\({}^{\circ}\)[30]. [PERSON] et al. fabricated slanted gratings by using high refractive index materials and UV replication technology [31]. Figure 5 is the diagram of the fabrication process. In the fabrication, the MgF\({}_{2}\) layer is coated on the SiO\({}_{2}\) substrate at first. Then, a new layer of SiO\({}_{2}\) is coated on top of MgF\({}_{2}\).