# Multiple Reflections and the Near-Field Effects on a Metamaterial Quarter-Wave Plate
## 2 QWP Design and Analysis  
The ability of a QWP hinges on its capacity to induce an exact phase shift, which is contingent on the birefringence of the structure. Grating structures have been extensively studied in the literature [53; 54; 55; 56; 57]. Building upon this, we design a grating-based QWP. Given that our grating structure is translationally invariant along the Y direction, we utilize 2D simulations for our analysis. In grating-based metamaterials, birefringence originates from the anisotropic optical response, which is induced by the orientation of the layers in the grating structure [54; 58]. To determine a starting point for the frequency domain solver in the COMSOL simulations (Figure 1), we employed the Maxwell-Garnett equations, which have been demonstrated to effectively characterize grating structures [26; 53]. The equations using the coordinate system shown in Figure 1 are as follows:  
\[\epsilon_{z}=\frac{\epsilon_{m}\epsilon_{d}}{(1-N)\epsilon_{m}+N\epsilon_{d}},\]  
\[\epsilon_{y}=N\epsilon_{m}+(1-N)\epsilon_{d}, \tag{1}\]  
\[N=\frac{h_{m}}{h_{m}+h_{d}},\]  
where \(\epsilon_{m}\) and \(\epsilon_{d}\) are the relative permittivities of the metal and dielectric materials, respectively, and \(h_{m}\) and \(h_{d}\) denote the heights of the respective metal and dielectric materials within the unit cell. The use of \(\epsilon_{z}\) and \(\epsilon_{y}\) was due to the X propagating wave and the orientation of the layers as shown in Figure 1.  
In order to effectively assess a cascaded metamaterial system, our primary objective is to design a transmission-based QWP having a bandwidth centered around an 800 nm wavelength [59], optimized for a normally incident uniform plane wave. The transmission-based design along with the normally incident plane wave will allow for a seamless integration of the structure into a cascaded system. To achieve better agreement with the Maxwell-Garnet equation, the unit cell and feature sizes are minimized while staying within the practical limitations of contemporary fabrication techniques [56; 60; 61; 62; 63; 64]. We do not exceed a depth-to-height ratio of 2:1, and the minimum feature height is set to 50 nm. Using the Maxwell-Garnett equations as a starting point and using COMSOL to finalize the design give the finalized grating a depth of 100 nm and a metal height of 50 nm. In the selection of materials, gold and silver emerge as strong candidates due to their advantageous permittivities (i.e., relatively low losses) at the desired 800 nm wavelength. However, the inclination of silver to oxidize led us to choose gold [65] as a more stable option. For the dielectric material, titanium dioxide [56] was selected. The grating structure is embedded in silicon dioxide [66]. We incorporated the relative electric permittivities of the chosen materials as given in Table 1. The design culminates in the configuration  
Figure 1: The 2D design with the grating structure embedded in silicon dioxide. In the simulation, periodic boundaries are implemented at the top and bottom, while scattering boundaries are established on the left and right sides. Perfectly matched layers (PMLs) are positioned before the input and after the output to effectively mitigate any boundary-induced reflections that could compromise the data collection. The input is 150 nm before the grating and the output is located at 150 nm from the back side of the grating. The unit cell is 175 nm in the Z direction, with the gold taking 50 nm of the unit cell, and the titanium dioxide occupies the remaining 125 nm. The grating is 100 nm wide in the X direction. depicted in Figure 1 with a total unit cell height of 175 nm. Of this, 50 nm is occupied by gold, with the remaining 125 nm being titanium dioxide. In order to effectively evaluate the performance of the proposed QWP design, we employ the Stokes parameters.