# A 1 \(\times\) 2 Two-Dimensional Slanted Grating Based on Double-Layer Cylindrical Structure
## 2 Design of the 2D Slanted Grating  
According to the background mentioned above, two-dimensional slanted grating has more advantages than one-dimensional slanted grating. Considering the actual processing technology, we designed a two-dimensional slanted grating based on the cylindrical structure. In addition, a single-layer structure was difficult to obtain. However, in order to achieve high diffraction efficiency, we designed a multilayer structure to obtain higher diffraction efficiency and better performance. Diffraction occurs when light incidents on the surface of a grating. Depending on the wavelength of incident light, angle of incidence, grating period, etc., different diffraction orders can be generated. The two-dimensional grating has periodicity in two different directions. The diffraction equation of a two-dimensional grating can be expressed as:  
\[\sin\theta_{\mathrm{m,n}}\cos\phi_{\mathrm{m,n}}=\sin\theta\cos\phi+\mathrm{m \lambda/d_{x}} \tag{1}\]  
\[\sin\theta_{\mathrm{m,n}}\sin\phi_{\mathrm{m,n}}=\sin\theta\sin\phi+\mathrm{n \lambda/d_{y}} \tag{2}\]  
where m and n represent the diffraction order of the two-dimensional grating, and both m and n are integers. \(\theta\) and \(\phi\) are the polar and azimuthal angle of the incident light,and \(\theta\)(m,n) and \(\phi\)(m,n) are the polar and azimuthal angle of the corresponding diffraction orders, respectively. \(\lambda\) is the incident wavelength, and \(d_{x}\) and \(d_{y}\) are the periods of the grating along the \(x\)- and y-directions. Under normal incidence, the polar and azimuthal angles of the incident wave are both 0 degrees. We assume that both \(d_{x}\) and \(d_{y}\) are equal to \(d\). Thus, the diffraction Equations (1) and (2) can be written as:  
\[\sin^{2}\theta_{m,n}=\left(m^{2}+n^{2}\right)\lambda^{2}/d^{2} \tag{3}\]  
In order to achieve high diffraction efficiency (DE) for the \((-1,0)\) and \((0,-1)\) diffraction orders, the total number of possible diffracted modes should be reduced as much as possible, so that the dispersion of the total energy by the unnecessary diffraction orders can be decreased. Therefore, the grating period \(d\) should satisfy the following condition:  
\[\lambda<d<\sqrt{2}\lambda \tag{4}\]  
The proposed 2D grating has a structure of a slanted two-layered nano-cylinder array, as shown in Figure 1a. The material of the bottom layer is MgF\({}_{2}\) with a height of h\({}_{1}\). The material of the top layer is SiO\({}_{2}\) with a height of h\({}_{2}\). The materials are chosen as their refractive indices are not sensitive to wavelength in the visible range, and their manufacture is cost effective. The refractive indices of MgF\({}_{2}\) and SiO\({}_{2}\) at the operating wavelength of 450 nm are 1.38 and 1.46, respectively. MgF\({}_{2}\) crystal is widely used, and MgF\({}_{2}\) is the transparent solid material with the smallest refractive index (n = 1.38). In addition, the extinction coefficient of MgF\({}_{2}\) is quite small in a large spectral range, which can decrease the reflection. Moreover, this material has a high laser-damage threshold [26; 27; 28]. The substrate material of the grating is SiO\({}_{2}\), which is the same as that of the top layer. The proposed 2D gratings are shown in Figure 1b-d, with the grating ridge width b, grating period d, grating polar angle \(\alpha\), grating azimuthal angle \(\beta\), and duty cycle \(f\). To ensure the polarization-independent properties of the 2D grating, the periods are set to be equal in the x- and y-axis directions, \(d=d_{x}=d_{y}\), and the grating azimuthal angle \(\beta\) is 45 degrees. Duty cycle f is defined as the ratio of the ridge width to the period of the grating, which can be expressed as f = b/d, so f = f\({}_{x}\) = f\({}_{y}\). The incident layer and grating slots above the grating structure are filled with air media. Figure 1: (**a**) The proposed 2D slanted grating structure; (**b**) unit-cell of the proposed structure; (**c**) unit-cell seen from side view (X–Z plane); (**d**) unit-cell seen from side view (Y–Z plane).