## 3 Results and Discussion  
The designed grating is considered optically magnetic due to the ability of TM-polarized visible light to excite asymmetric circulating currents in the metal layers, thereby giving rise to a magnetic moment directly along the strips of the structure. This is fundamentally a result of the oscillating magnetic field along the strip axis and [PERSON]'s law. Both a symmetric current mode and an asymmetric mode may be excited, representing an electric and magnetic resonance, respectively. The spectral position of the magnetic resonance has been previously demonstrated to be a result of the effective width of the grating (or fill factor) [20], though the normal incidence transmission local minima provide a baseline of sorts to determine the spectral locations for both \(\varepsilon_{x}\) and \(\mu_{y}\). As seen in Figure 5 the TM polarized transmission displays two local minima; the first, \(\lambda_{e}=530\) nm and the second, \(\lambda_{m}=725\) nm. These represent the electric and magnetic resonances, respectively, and specifically refer to \(\varepsilon_{x}\) and \(\mu_{y}\). Therefore, we have in mind a starting point for the spectral location of the resonances of the permittivities and permeability. The sharp peak at \(\lambda_{d}=450\) nm is due to the Wood anomaly and is not observed in simulation until \(\varepsilon_{z}\) is properly determined. As an example, if \(\varepsilon_{z}\) is given a constant nonabsorbing value of \(\varepsilon_{\infty,z}=1\), the sharp peak at \(\lambda_{d}\) is washed out; in this case a static offset to the real part of approximately \(\varepsilon_{\infty,z}=8\) is required for the diffraction threshold peak to appear. The simulation results for the spectral-angular map are performed with the experimental values of the period of the grating, the location of the incident beam relative to the waveguide edge, as well as the substrate thickness and refractive index (see Figure 6). In Figure 6 we also explain why the  
Figure 5: Normal incidence transmission for TM (**Left**) and TE (**Right**) polarizations. Note that only \(\varepsilon_{y}\) is responsible for TE spectra, while TM spectra depends on \(\varepsilon_{x}\), \(\varepsilon_{z}\), and \(\mu_{y}\). With TM polarization, \(\lambda_{d}\), \(\lambda_{e}\), and \(\lambda_{m}\) correspond to the Wood’s anomaly (diffraction threshold), electric resonance, and magnetic resonance, respectively. Data is matched by providing incremental adjustments to the parameters of each dielectric function. Note that the set of parameters providing agreement here must also provide satisfactory matching for the spectral-angular data. Error bars in simulated spectra reflect \(\pm 5\%\) uncertainty. spectral angular maps to be seen in Figures 7 and 8 are not continuous. Indeed, depending on the incident beam positions relative to the substrate edge, there are three possible scenarios. The beam at a particular wavelength may either hit the corner and split between upward and downward as shown in Figure 6a, or propagate in one of two directions, upward or downward. If the beam goes upward it makes a gap in the down side as indicated by arrows on Figure 6b. We emphasize that these aforementioned parameters taken from experiments control only the location and the size of the \"spots\" seen in the spectral-angular maps in Figures 7 and 8. More importantly, the permittivities and permeabilities are mainly responsible for the intensities of these spots. To retrieve the effective optical properties, we begin with an initial guess for each parameter in the TOLO and Drude wavelength-dependent functions. This guess is influenced by the experimental transmission data and gives a starting point for the resonant spectral positions as well as the non-resonant spectra modeled as a diluted metal. These functions provide a value for permittivity and permeability at each wavelength, and are then fed to the MIST GUI which operates on each value using the modified RCW code.