## 2 Materials and Methods  
In this paper, we study a stacked metal-dielectric grating, shown schematically in Figure 1a, at inclined illumination and develop a method to retrieve an effective anisotropic permittivity and permeability of the strips. The grating is fabricated on a transparent substrate and illuminated with white light at various angles of incidence, while light diffracted from the grating is partially coupled to the waveguiding mode and collected as a function of angle from the edge of the substrate, as shown in Figure 1c. The collected light is then delivered to the spectrometer by a fiber bundle. Additionally, normal incidence transmission and reflection measurements are performed in order to provide some information for spectral positions of resonances. The setup described in Figure 1c will produce a wavelength-angle intensity map. This map will be unique for each angle of incidence and each polarization. The simulation efforts will produce the same intensity maps and transmission spectra to match the experimental spectra by adjusting the unknown strip parameters. Thus, using an iterative procedure where the strip parameters are slightly adjusted individually until the best goodness of fit is obtained, we can determine the proper set of parameters. The fitting is done manually, such that the simulated data appears to visually match the experimental data. This matching process is further detailed in the Results and Discussion section. Here we choose modified rigorous coupled wave (RCW) analysis which is utilized as a package within the Modeled Integrated Scattering Tool (MIST [13]). MIST is a front-end graphical user interface to the SCATMECH C++ library of scattering codes, based on rigorous coupled wave theory [14; 15; 16], modified to account for anisotropic permittivity and permeability (see Appendix A). With this tool, we are able to model diffraction effects of any order. The matching process must be done for each incident angle and each polarization, including that of normal incidence transmission, and the same unique set of permittivity and permeability functions is required for successful matching. The simulation is quite sensitive to the spectral position, amplitude,and line-shape of the strips' permittivity and permeability functions. The beauty of this approach also lies in the fact that we need not be concerned with the artifacts in the retrieval caused by the grating periodicity. Any isotropic medium can be optically characterized by either pair of parameters, index of refraction \(n\) and impedance \(Z\), or by permittivity \(\varepsilon\) and permeability \(\mu\). All four quantities are causal, complex, and depend on frequency \(\omega\). They are related by  
\[n=\sqrt{\varepsilon\mu}\text{ and }Z=\sqrt{\frac{\mu}{\varepsilon}} \tag{1}\]  
In accounting for a permeability different from unity, it is clear the permittivity and permeability are treated as complex quantities obeying the Kramers-Kronig relations and \(n\) and \(Z\) are independent functions. One approach to determine the values for \(\varepsilon\) and \(\mu\) of a material, sometimes termed the \"effective layer method,\" is to use RCW in which the incident light is normal upon the structure. This method however considers the grating as a continuous film and hence, the properties of an effective continuous layer are retrieved. Here, we seek a more complete retrieval which is not restricted to normal incidence, and thus accounts for anisotropy. In this more general situation of incline incidence, a redshift of the Wood's anomaly into the visible range occurs, which is precisely the wavelength region of our interest and the effective layer retrieval has limited practical use here. Due to the periodic nature and size of the structure relative to the incident wavelength, an electromagnetic plane wave will undergo diffraction and will transfer some of its power into higher orders.