# Multiple Reflections and the Near-Field Effects on a Metamaterial Quarter-Wave Plate
## 3 Near-Field Effects and Cascading Behaviors  
As optical systems move towards the high-density integration of optical structures with sub-wavelength separations, the interactions of the individual components as well as their interactions within a system need to be known. To design a practical metamaterial for use within a compact optical system, we present an analysis process that showcases why this thorough analysis should be performed for all metamaterial designs. To illustrate our point, we use the grating structure depicted in Figure 1, shown to perform well as a QWP. We perform a near-field analysis, finding the effective length that the QWP operation takes. We also cascade the grating structure, using the effective length as a guide and observe any unique interactions in the system. We should note that the cascaded analysis of two QWPs is to understand the characteristics of the QWP designed and not to design a half-wave plate (HWP). A single device as a HWP would need redesigning, adhering to the current manufacturing standards. To validate the grating structure's efficacy as a QWP under a normally incident uniform plane wave, we examined its output to ensure it produces a uniform plane wave to minimize coupling and maintains a \(\pi/2\) phase difference. For the output of the grating structure to be defined as a uniform plane wave, the variances of its field magnitude and phase must approach zero. Deviations in field magnitude or phase can potentially impact device effectiveness when considering an integrated system. Similar analyses may be devised for structured light, oblique incidence, waveguide modes, etc. To appropriately simulate and analyze the output characteristics of the designed grating structure shown in Figure 1, modifications were made in which data lines with appropriate meshing were placed every nanometer after the back side of the grating. Using an 800 nm normally incident uniform plane wave with a 45-degree polarization angle, the output results shown in Figure 3 were collected. Figure 3 shows the mean and variance of the field magnitude and phase as a function of distance from the output of the grating structure. From the average field magnitudes, it can be seen that the grating structure has near-field effects, such as producing an X component (i.e., longitudinal cross-polarization) and a spike in the Z component. The field enhancement seen in the Z and X components of the output field magnitude can be mainly attributed to the sharp corners of the metallic portion of the grating [72; 73], which can be reduced by increasing the radius of curvature at the corners [72]. This may decrease both the effective length of the QWP and Ohmic losses. Figure 2: Performance of the grating structure shown in Figure 1 as a QWP. The input uses a normally incident uniform plane wave with a 45-degree angle of polarization and is simulated over the wavelengths from 700 nm to 950 nm. Presented is the (**a**) transmittance, reflectance and absorbance of the 45-degree input over the wavelengths. (**b**) The transmittance split into Y and Z polarizations. (**c**) The phase difference between the Z and Y polarizations. (**d**) Ellipticity with respect to wavelength. While the grating structure is shown to convert some of the input into the X-polarization at the output of the grating structure, the field's mean magnitude of the X component approaches zero within 100 nm, becoming two orders of magnitude less than either of the other polarizations. The variance of the X-polarization's phase persists throughout the measured distance. As the X component approaches zero, the non-uniformity it contributes can be considered negligible at the 100 nm distance from the output of the grating structure. With the X component of the field magnitude approaching zero and the variance of the Y and Z polarizations field magnitudes and phase also approaching zero, the output of the grating structure can be considered a uniform plane wave after 100 nm at the 800 nm wavelength.