## 3 Results and Discussion
Figure 10: Spectral-angular data conversion for TM of each prominent “spot” to 1D column graph. The labels A, B, C, D, and E refer to the spots from left to right in Figure 8 in each intensity map, while panels a-d correspond to different incident angles 30\({}^{\circ}\) (**a**), 40\({}^{\circ}\) (**b**), 50\({}^{\circ}\) (**c**), and 60\({}^{\circ}\) (**d**). Error bars reflect a 10% uncertainty. All intensities are on a relative zero to one scaling system. Figure 9: Spectral-angular data conversion for TE of each prominent “spot” to 1D column graph. The labels A, B, C, D, and E refer to the spots from left to right in Figure 7 in each intensity map at different incident angles: 30\({}^{\circ}\) (**a**), 40\({}^{\circ}\) (**b**), 50\({}^{\circ}\) (**c**), and 60\({}^{\circ}\) (**d**). Error bars reflect a 10% uncertainty. All intensities are on a relative zero to one scaling system. Obtaining a good fit for all angles of incidence, in addition to normal incidence transmission gives all components of the permittivity and permeability of the effective strips. As always, the fitting is limited by the accuracy, to which we know the exact geometry, roughness, and any fluctuations that occur throughout the real grating structure. The previous effective layer method works if normal incidence applications are required in which only the wavelengths longer than Wood anomaly are considered. This turns out normally to be reasonable for the visible spectrum. However, if one is trying to avoid the issue of diffraction in the wavelength range of interest, the period of the grating should be pushed to smaller dimensions, shifting the first diffraction event to shorter wavelengths. This begins to present a significant fabrication challenge. Even with a decreased period dimension, when oblique incidence is used with the metamaterial, the Wood anomaly is red shifted into the visible, creating an obvious problem for the effective layer method. In contrast, by using the effective strip retrieval, diffraction is accounted for, the period does not need to be pushed to smaller dimensions, and oblique incidence applications can be realized with the proper set of parameters. We introduce Table 1 below to summarize the comparison. Note, that the main point of introducing magnetic response is that the two parameters, refractive index and impedance, become independent. If we can describe everything with just electric permittivity, thus \(n=1/Z\), and magnetic response is absent, meaning \(\mu=1\). Figure 11: Retrieved parameters of the permittivity and permeability via our methods; note \(\mu_{x}\) and \(\mu_{z}\) (not pictured) are unity. (**a–c**) respectively show the x-, y-, and z-components of the permittivity while (**d**) shows the y-component of the permeability. \(\varepsilon_{1}\) and \(\mu_{1}\) refer to real parts, while \(\varepsilon_{2}\) and \(\mu_{2}\) refer to imaginary parts of the corresponding function. The capability of such a stacked grating material to waveguide the diffracted modes resulting from inclined illumination of the periodic grating surface makes it possible to apply such structures in biosensing. Using this design, the spectrum of the diffracted modes is sensitive to the refractive index of the material on the grating surface. For this reason, any changes in the refractive index due to a biochemical reaction on the surface [21] can be detected with this method. Additionally, upon retrieving the effective strip parameters, one may utilize such a grating design that exhibits a specialized set of optical properties in more practical situations where oblique incidence illumination is natural, such as the improvement of the efficiency in solar cells with unpolarized incident light [22]. Finally, we may generalize the method of parameter retrieval to other periodic nanostructures by using a similar experimental setup and simulation process.