# Multiple Reflections and the Near-Field Effects on a Metamaterial Quarter-Wave Plate
## 3 Near-Field Effects and Cascading Behaviors
The PCR FOM that we would like the cascaded system to achieve is 0.98 [71, 79]. Using the DoLP and PCR, the effectiveness of the two QWPs cascaded is shown in Figure 5. Analyzing the performance of the cascaded QWP system, it can be seen that the DoLP meets the targeted FOM from 700 nm to 880 nm with a majority of that range achieving >0.98. While the two QWPs provide linearly polarized light over this range, the PCR only achieves a value of >0.94. While the cascaded QWPs do not achieve the desired PCR at 800 nm (i.e., center of the operating bandwidth for the original QWP, see Figure 2d), the desired PCR is achieved from 712 nm to 778 nm. To better understand the impact of the cascaded devices, a near-field analysis was performed, where data were collected every nanometer for 150 nm after the second grating as shown in Figure 6. The near-field results showed that the cascaded QWPs still produced a uniform plane wave at the expected 100 nm after the second grating. From the output characteristics of the individual QWP, the observed magnitude of the Z component of the electric field from the cascaded QWPs deviated notably from expectations. While a slight reduction in the Z component was evident from the analysis presented in Figure 3, multiple reflections appear to have considerably influenced the Z component, resulting in a noticeable deviation from the ideal 90-degree polarization rotation at 800 nm (Figure 5d). While the observed results for the phase shift and transmittance diverge slightly from what was expected from cascading two QWPs, the deviations can be attributed to near-field effects and multiple reflections between the grating structures. For our designed grating structure, we have the Y and Z polarizations, which have different effective refractive indices, \(n_{y}\) and \(n_{z}\), respectively, and the refractive index of the surrounding media, which will be written as \(n_{1}\). In  
Figure 4: Two cascaded gratings embedded in silicon dioxide using the the same grating design shown in Figure 1. The gratings are separated by 100 nm due to the effective length found for the grating structure. The magnitudes of the field components are presented at the 800 nm wavelength illustrating the near-field distributions. Vanishing fields on both sides of the geometry are due to the PML and scattering boundary conditions. Figure 1.0 Performance of the 2D cascaded QWP design in Figure 4 from 700 nm to 950 nm wavelength. The output port is placed at 100 nm from the back side of the second grating to simulate two cascaded effective QWPs. (**a**) Transmittance, reflectance, and absorbance of the 45-degree polarization input over the wavelengths. (**b**) The phase difference between the Z and Y polarizations. (**c**) Degree of linear polarization and (**d**) polarization conversion ratio with respect to wavelength. Equations (6) and (7) show that the phase difference between the polarizations is not commensurate with the length due to the refractive index difference of the surrounding medium that results in multiple reflections. Note that when multiple reflections are not present, from Equations (6) and (7), we recover the phase difference \(\Delta\phi=\phi_{z}-\phi_{y}\), which is linearly proportional to the length. This relationship becomes more complicated when dealing with lossy and inhomogeneous metamaterials and can involve additionally evanescent coupling [26, 27, 28] and polarization-dependent meta-atom resonances. However, these equations already indicate that putting two of the designed QWPs in series, as shown in Figure 4, will not necessarily produce an ideal \(\pi\) phase shift due to the introduction of more interfaces for multiple reflections. Similarly, the dependence of the transmission coefficient magnitudes \(|t_{y}|\) and \(|t_{z}|\) on \(n_{y}\) and \(n_{z}\) introduces asymmetry in the transmitted field magnitudes like in Figure 6a.