# A 1 \(\times\) 2 Two-Dimensional Slanted Grating Based on Double-Layer Cylindrical Structure
## 3 Characters and Tolerance Analysis
Thus, a triple-layer structure of SiO\({}_{2}\)-MgF\({}_{2}\)-SiO\({}_{2}\) is formed. Next, a layer of photoresist is coated on the surface evenly. The photoresist was exposed by using holographic or electron-beam direct-writing techniques, and the photoresist grating pattern is developed with a developer solution. Because the structure of the designed gratings is 2D slanted, we should perform exposure by lithography twice. The cylindrical structure can be formed after two rounds of exposure. In the real processing of fabrication, the slanted gratings could not be designed with a rectangular structure. Thus, we designed the slanted gratings with a cylindrical structure. Subsequently, we use a tilted incidence reactive ion beam for etching [32]. The 2D slanted grating will be fabricated after the etching process. During the experiments, the actual structure of the grating may slightly deviate from the optimized results. The process of coating can influence the depth of the film. The lithography can influence the duty cycle and period. Moreover, the process of etching can influence the slanted angle. Therefore, it is necessary to analyze the production tolerances for the grating parameters to obtain the appropriate range of grating structure data. This can ensure that the grating structure manufactured by experiment conforms with the designed parameters. The thickness of MgF\({}_{2}\) and SiO\({}_{2}\) is related to the phase modulation  
Figure 4: Diffraction efficiency versus incident angle for a wavelength of 450 nm with the optimized grating parameters. (**a**) Diffraction efficiency under TE polarization; (**b**) diffraction efficiency under TM polarization. Figure 5: The diagram of fabrication processing. of the incident light, which further affects the coupling of the efficiency of each diffraction order of the grating. Figure 6 shows the effect of MgF\({}_{2}\) and SiO\({}_{2}\) thickness on the diffraction efficiency for the optimal solution of other structural parameters. Figure 6a,b show the (\(-1\), \(0\)) order diffraction efficiency and the (\(0\), \(-1\)) order diffraction efficiency for the normal incidence of the TE polarized incident light. Figure 6c,d represents the condition of normal incidence with TM polarization. When the thickness h\({}_{1}\) of MgF\({}_{2}\) varies in the range of 300-350 nm and the thickness h\({}_{2}\) of SiO\({}_{2}\) varies in the range of 513-550 nm, the (\(-1\), \(0\)) order diffraction efficiency and the (\(0\), \(-1\)) order diffraction efficiency both exceed 35% for TE and TM polarizations. The total effective efficiency of the grating is greater than 70%. In particular, the total effective efficiency is greater than 75% when the MgF\({}_{2}\) thickness is from 300 nm to 350 nm and the SiO\({}_{2}\) thickness is from 525 nm to 550 nm. Figure 7 shows the (\(-1\), \(0\)) and (\(0\), \(-1\)) order diffraction efficiency for different grating periods d and grating polar angle \(\alpha\) with respect to the normal incidence for the optimal solution of the other structural parameters of the grating. Figure 7a,b are for TE polarized incidence, and Figure 7c,d are for TM polarized incidence. In the region of interest, for TE and TM polarization, the desired results are obtained when the grating period is varied from 580 nm to 633 nm and the grating slanted angle is varied from 12.2\({}^{\circ}\) to 21.0\({}^{\circ}\). In this range, both the (\(-1\), \(0\)) order diffraction efficiency and the (\(0\), \(-1\)) order diffraction efficiency all exceed 35%. During the grating production process, the control of the grating slanted angle is more demanding than the grating period.