## 3 Results and Discussion
For TM polarization, one typically associates the permeability strictly with the longer wavelength (magnetic) resonance \(\lambda_{m}\) and the x- and z-permittivities with the shorter wavelength (electric) resonance. Incremental adjustments are made to the respective amplitudes, spectral positions, and sharpness of each function. The same process is done for TE polarization for the y-component of the permittivity. However, the difficulty that arises is that these functions must now also provide agreement with the spectral-angular data. Since only one component is responsible in the TE polarization, agreement with the spectral-angular map here occurs rather naturally. On the other hand, due to the TM polarization containing three functions that are responsible for the output, the agreement is not as straightforward. At this stage, further adjustments to these three functions are made such that the spectral-angular maps agree. At first glance, the matching for the spectral-angular map is difficult to discern visually. To further clarify the matching success, in each map, we label each spot and characterize it by how much average relative intensity it receives. In this way, the three-dimensional plot can be reduced to a one-dimensional column graph, as shown in Figures 9 and 10. This is reasonable because only the material properties can provide the correct relative intensities, while geometry and diffraction dictate the angle and wavelength possible at each location. Since the material properties (i.e., permittivity and permeability) are responsible for the intensity of each spot, while the geometry and diffraction provide the spectral-angular location, this reduced one dimensional plot is best representative of the fitting, as this extracts only the desired effective optical parameters from the rest of the information producing Figures 7 and 8. We can then compare intensities of each spot. Note that one set of permittivity and permeability must satisfy all sets of data, including normal incidence data as shown in Figure 5, which makes matching trustable. As one can see from the comparative presentation in Figures 9 and 10, the agreement between the experiment and simulations are not ideal for some spots. For example, spots 3 and 4 in the TM 30\({}^{\circ}\) incident angle trial, which correspond to exit angles of approximately \(-65^{\circ}\) and \(-45^{\circ}\), respectively, we note there is simulated radiation that is not quite detected experimentally. On the contrary, most all other spots at other incident angles are in close relative agreement. With slight deviations of the current optical properties' spectral position, amplitude, sharpness or symmetry, the \"overall matching\" drastically reduces. Here, overall matching simply translates to the matching of all eight spectral angular maps _and_ the normal incidence transmission. For example, a 5-nm deviation in spectral position of the permeability may improve the matching for spots 3 and 4 in the TM 30\({}^{\circ}\) incident angle match, but subsequently worsen several other spots for other angles. Hence with the presented optical parameters (Figure 11), the best \"overall match\" was obtained. Figure 7: Experimental (top panels) and simulated (bottom panels) spectral-angular map of TE polarized incident light for incident angles 30\({}^{\circ}\) (**a**), 40\({}^{\circ}\) (**b**), 50\({}^{\circ}\) (**c**), and 60\({}^{\circ}\) (**d**). The intensity scales are in the same units for all maps of the TE polarization. Simulated data are matched to experimental data by considering maximum intensity in a spot. Figure 8: Experimental (top panels) and simulated (bottom panels) spectral-angular map of TM polarized incident light for incident angles 30\({}^{\circ}\) (**a**), 40\({}^{\circ}\) (**b**), 50\({}^{\circ}\) (**c**), and 60\({}^{\circ}\) (**d**). The intensity scales are in the same units for all maps of the TM polarization. Simulated data are matched to experimental data by considering maximum intensity in a spot.