## 3 Results and Discussion
By varying the conditions, it will output either a transmission spectrum at normal incidence or a spectral-angular map--both must be performed. One by one, each parameter of the TOLO and Drude functions are iteratively adjusted until both the transmission and spectral-angular maps at all angles both match satisfactorily based on eye evaluation. This process is guided by the most sensitive parameters for the resonant spectral position, amplitude terms, followed by the spectral width term, and lastly the asymmetrical parameters. For example, to obtain \(\mu_{y}\), \(\varepsilon_{x}\), and \(\varepsilon_{z}\), we use the TM experimental data. Realizing that \(\mu_{y}\) is responsible for the magnetic resonance and \(\varepsilon_{x}\) mainly responsible for the electric resonance, both amplitude and spectral position terms are initially chosen such that they best match the transmission data for the respective minima. The spectral positions for the resonant functions are initially chosen to be the same as those of the transmission minima. Note, these positions may not exactly coincide after finalizing the matching. Since the two resonances are not spectrally separated by a significant amount, increasing the amplitude of, for example, \(\mu_{y}\) can have an impact on the simulated transmission's local _electric_ minima, and vice versa. Matching is assessed for the spectral-angular maps by comparing each spot's relative intensity. Upon doing so, we have best matched the simulated spectra to the experimental data. Thus, we have found each component previously discussed, namely, \(\varepsilon_{x}\), \(\varepsilon_{y}\), \(\varepsilon_{z}\), and \(\mu_{y}\). Meanwhile \(\mu_{x}\) and \(\mu_{z}\) are set equal to unity as an enforced condition. Again, note that physical arguments are used to evaluate suitable functions, such as \(\varepsilon_{y}\) exhibiting a behavior similar to that of a dilute metal--a result of the  
Figure 6: The geometry of the sample affects the output; this is shown for parallel rays interacting with the grating (**a**). Note, that depending on the incident beam positions relative to the substrate edge there are three possible scenarios, these rays may either hit the corner and split between upward and downward as shown in (**a**), or propagate in one of two directions, upward or downward. If the beam goes upward it makes a gap in the down side (**b**). Map (**b**) is an angular extension of Figure 8b SIM TM 40\({}^{\circ}\), as an example; the gap at \(-60^{\circ}\) is pointed by arrow. non-resonant TE mode. The result of the normal incidence matching is shown in Figure 5, while all spectral-angular matching results for varied angles and polarizations are shown in Figures 7a-d and 8a-d. The results in Figures 7 and 8 all account for uncertainty in the spectral-angular position by allowing each data point to have a specific radius, such that it reflects the experimental data uncertainty. Furthermore, based on Figures 9 and 10, Figure 11 shows the functions chosen to satisfy matching of the experimental data. It is emphasized that the matching in Figures 5 and 7 and Figure 8 are not independent of one another, but rather the same set of permittivity and permeability must provide agreement for both. To obtain a successful match, both normal incidence transmission and waveguided inclined illumination simulation data should agree with the experimental results. It is extremely challenging to simultaneously have both agree with a high degree of accuracy, especially using singly resonant Lorentzian-type functions with a limited number of fitting parameters. An improved match with the spectral-angular map will tend to significantly deteriorate the normal incidence match, and vice versa. Here we have attempted to minimize the degrees of freedom for more convenient fitting and to demonstrate the retrieval process. The matching of the normal incidence data alone is straightforward.