# Multiple Reflections and the Near-Field Effects on a Metamaterial Quarter-Wave Plate
## 3 Near-Field Effects and Cascading Behaviors
Since there are no resonant spikes within the bandwidth (Figure 2a), the same analysis was performed at the edge cases of the bandwidth (i.e., 711 nm and 910 nm wavelengths) to confirm that the grating structure produced a uniform plane wave at 100 nm after the back side of the grating throughout the bandwidth. These results show that the effective length of the QWP in Figure 1 can be determined to be approximately 200 nm. This includes a 100 nm thick grating structure followed by 100 nm thick silicon dioxide layer on the right side of the grating for the field to stabilize. Due to the near-field effects, clearly shown in Figure 3, the QWP operation is not considered complete until after traveling 100 nm past the output side of the grating. While it is well known that metamaterials have near-field effects, the effective length of the device is not considered in most works. However, for the appropriate incorporation of a designed metamaterial into a highly dense optical system with sub-wavelength separations, the near-field effects become important. To analyze how the QWP performs in a cascaded structure, we use the effective length of the QWP found and put two of them consecutively back to back as shown in Figure 4, which illustrates both the geometry and the resultant distributions of the field magnitudes at the near-field. This cascade will lead to the two gratings having a separation of 100 nm, with the rest of the design parameters the same as in Figure 1. Also, the output port will be at 100 nm from the back side of the second grating, simulating two effective QWPs back to back. For the two consecutive QWPs as shown in Figure 4 to perform an effective  
Figure 3: Near- field analysis of the output of the grating structure using the same physical design as shown in Figure 1. The input is an 800 nm normally incident uniform plane wave with a 45-degree angle of polarization where the input is 150 nm before the grating. The output is sampled every nanometer until 150 nm of travel from the back side of the grating. This diagram displays the (**a**) mean and (**b**) variance of the electric field magnitude with respect to the distance from the output of the grating structure, showing that the X component dissipates within 100 nm and the variance of each of the components approaches zero. The (**c**) mean and the (**d**) variance of the radian phase components of the electric field polarizations indicate that the Y and Z polarizations have a variance that approaches zero, while the X polarization’s phase contains some variance throughout the measured distance. HWP operation, the final output should be a uniform plane wave and reproduce linearly polarized light with the correct rotation. To determine the two QWPs' effectiveness as a HWP, we will use the degree of linear polarization (DoLP) and the polarization conversion ratio (PCR). To determine the linearity of the output polarization, we use the DoLP derived from the Stokes parameters in Equation (2), defined as [74, 75, 76]  
\[\text{DoLP}=\sqrt{\left(\frac{S_{1}}{S_{0}}\right)^{2}+\left(\frac{S_{2}}{S_{0 }}\right)^{2}}, \tag{4}\]  
where the DoLP can range from 1, representing a perfect linear polarization, to zero, indicating no linear polarization [76]. DoLP = 0.94 may be considered appropriate for a HWP operation [76]. The PCR is determined by the polarization input and expected polarization output. For our system shown in Figure 4, the input has a 45-degree polarization with respect to the Y and Z axes. For a HWP operation with our setup, this should lead to a 90-degree rotation. For our system, the PCR is defined as follows [77, 78, 79]:  
\[\text{PCR}=\frac{|t_{-\frac{\pi}{4}}|^{2}}{|t_{-\frac{\pi}{4}}|^{2}+|t_{+\frac {\pi}{4}}|^{2}}, \tag{5}\]  
where \(|t_{+\frac{\pi}{4}}|\) corresponds to the transmission coefficient magnitude that aligns with the positive \(\
icefrac{{\pi}}{{4}}\) polarization angle of the input and \(|t_{-\frac{\pi}{4}}|\) corresponds to the expected 90-degree conversion of the output to a negative \(\pi/4\) polarization angle.