# Multiple Reflections and the Near-Field Effects on a Metamaterial Quarter-Wave Plate
## 2 QWP Design and Analysis
These parameters provide a comprehensive characterization of our electromagnetic field propagating in the X-direction, taking into account the phase difference, \(\Delta\phi=\phi_{z}-\phi_{y}\), between the two orthogonal field components. They are given as [50, 67, 68, 69]  
\[ \begin{split} S_{0}&=|E_{y}|^{2}+|E_{z}|^{2},\\ S_{1}&=|E_{y}|^{2}-|E_{z}|^{2},\\ S_{2}&=2|E_{y}||E_{z}|cos(\Delta\phi),\\ S_{3}&=2|E_{y}||E_{z}|sin(\Delta\phi),\end{split}  \tag{2}\]  
where \(E_{j}\) is the \(j\)'th component of the electric field, and we use \(+j\omega t\) convention consistent with COMSOL. The Stokes parameters allow us to discern the proficiency of QWP in transforming linearly polarized light into circularly polarized light. One of the vital metrics for this characterization is ellipticity (\(\chi\)), defined as [69]  
\[\chi=\frac{S_{3}}{S_{0}}. \tag{3}\]  
The ellipticity metric provides an understanding of the degree of circularity in the wave polarization, with 1 representing a perfect left-handed circular polarization, \(-1\) indicating a perfect right-handed circular polarization, and zero denoting a linear polarization. For our grating structure, we assume it performs an appropriate QWP operation if \(|\chi|\) at the output is greater than 0.99 [69, 70, 71]. Having established the theoretical framework for evaluating the performance of our quarter-wave plate design, we now turn our attention to its empirical performance metrics. The succeeding Figure 2 presents a comprehensive view of the QWP performance, displaying the transmission, phase shift, and ellipticity from 700 nm to 950 nm. Accompanying this, the complex permittivities of the materials utilized at these wavelength limits are specified in Table 1. These figures are critical, as they provide clear evidence of the ability of QWP to convert linearly polarized light into circularly polarized light, a crucial measure of the device's utility. This QWP design exceeds the \(|\chi|\) figure of merit (FOM) of 0.99 from 711 nm to 910 nm, having a bandwidth of approximately 200 nm as shown in Figure 2. We described a subwavelength QWP operating on a transmission basis. However, while the individual performance of this QWP is promising, it represents just one component in a more complex system. When we consider these devices as part of an integrated assembly, additional complexities emerge. With metamaterials, especially those that rely on strong resonance, the consideration of near-field effects along with multiple reflections between devices becomes increasingly necessary due to the potential impact on performance of the integrated device. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\epsilon_{r}\) & **700 nm** & **800 nm** & **950 nm** \\ \hline SiO\({}_{2}\)[66] & \(2.1382-i0.004383\) & \(2.1339-i0.0038174\) & \(2.1299-i0.0032015\) \\ TiO\({}_{2}\)[56] & \(5.5945-i8.04\times 10^{-10}\) & \(5.4814\) & \(5.3777\) \\ Au [65] & \(-13.753-i1.9109\) & \(-20.277-i2.0711\) & \(-31.585-i2.7288\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Complex relative electric permittivities (\(\epsilon_{r}\)) of SiO\({}_{2}\), TiO\({}_{2}\), and Au at the simulation wavelength limits and the targeted wavelength of 800 nm. The associated references detail the full material parameters employed throughout the wavelength sweeps in the simulations.