## 2 Materials and Methods
Supercontinuum white light pulses converted from 800-nm pump (Figure 3) of either transverse electric (TE) or transverse magnetic (TM) polarization illuminate the sample at differing angles of incidence, relative to the sample surface normal. The spectra of the transmission through the substrate at incline incidence are used to normalize the spectral response of the samples. Figure 3 gives an example of such spectra at 30\({}^{\circ}\). The laser source stability is  
Figure 2: Scanning electron microscopy (SEM) micrograph of meta-grating sample. Note the trapezoidal shape. 10% and an example of its output at a particular angle through the substrate as well as its stability with time and with different polarizations is shown in Figure 3. Upon striking the grating, light diffracts at many angles. These diffracted rays are waveguided by total internal reflection through the substrate only in the negative direction. The intensities are collected via a scanned fiber of core diameter 500 \(\upmu\)m located a distance of 0.5 mm from the substrate edge, and is subsequently delivered to an imaging spectrometer, whose spectral resolution is 1.5 nm. Note the angular resolution of measurement is 0.5\({}^{\circ}\) The output intensity will then be a function of both the angle and wavelength in a spectral-angular intensity distribution (Figure 1c). We ignore rays that propagate in the positive direction which may eventually return to the detector by means of many internal reflections. The reason for this is that, with the angles of incidence used, these rays will _not_ undergo total internal reflection within the substrate, and due to significant loss from many of these repeated reflections, they contribute several orders of magnitude less signal. The process is repeated for incident angles of 30\({}^{\circ}\), 40\({}^{\circ}\), 50\({}^{\circ}\), and 60\({}^{\circ}\), and for two linear polarizations (TE or TM) for a total of eight spectral-angular intensity maps. What follows is a matching process utilizing simulation methods for both normal incidence transmission (zero-order diffraction) as well as the spectral-angular map (nonzero-order diffraction). Note, that in past retrieval schemes, the grating is approximated as an effective layer and far field transmission and reflection simulations are matched to experimentally observed data (for example [1; 2; 3]). The type of simulation, that is based on RCW methods, allows retrieval of the complex transmission and reflection coefficients, which can then in turn be used to calculate the permittivity and permeability, albeit as a continuous effective layer. In reality, since a periodic structure is used, diffraction occurs at normal incidence for \(\lambda\leq np\) where \(n\) is the substrate refractive index. In our case, this approximately translates to first order diffraction occurring at normal incidence for wavelengths less than 450 nm. Thus the effective layer approximation is valid only for the spectral range where the measured transmittance/reflection only contains zero-order information and the higher (nonzero) orders are not allowed. In this paper, retrieval of the individual strips' effective optical properties is accomplished by matching simulation spectral-angular data from the designed waveguiding structure, as well as the normal incidence transmission, to the corresponding set of experimental data. For this we use the RCW model included in the MIST. To describe the interacting system, MIST requires the grating geometry, substrate and superstrate media, incident light wavelength, polarization, and angle of incidence, as well as the optical properties of the strips. Despite the strips consisting of several metal and dielectric layers, we model the strip as a single unit--the effective strip. All the roughness and crystal quality of the materials are included in the effective parameters. The physical dimensions of the  
Figure 3: Substrate transmission at 30\({}^{\circ}\) incline, showing the stability of the source output spectra.