# Guided Mode Resonance in a Low-Index Waveguide Layer
## 2 Design and Simulation
The thickness of the grating was 30 nm and the duty cycle was set to be 0.5. The top cladding layer was set to be air. A 2D simulation was used to decrease the simulation time. As shown in Figure 1a, the broadband light wave (400 nm-800 nm) was propagated in the -Z direction and impinged on the waveguide layer. The perfectly matched layer (PML) boundary condition was set at the top and bottom \(Z\) boundaries. Periodic boundary conditions were used for \(X\) boundaries. Transverse electric (TE) polarized light with E-field parallel to the grating lines was used in this simulation, which eliminates the influence of the TM polarization of any plasmonic resonance. The polarization was not strictly limited to TE light if the GAGMR condition was satisfied. The reflection spectrum was collected by a power monitor behind the source. We first examine how the grating period affects the guided mode. In this simulation, we used a normal incident plane wave as a light source (\(\theta_{i}=0\)). The thickness of the waveguide was fixed at 200 nm. The period was swept from 400 nm to 700 nm with a step size of 50 nm. Figure 2a shows the calculated reflection spectra for various grating periods. For comparison, the reflection spectrum of the grating array (\(P=500\) nm) without a waveguide layer is also shown. No apparent resonance but a small dip occurs in the bare metal gratings system. This small dip is due to the Wood-Rayleigh anomaly [22], which happens both in bare metal gratings and gratings with a dielectric as cladding system (See Figure 2a). This indicates that the waveguide layer can have a significant influence on the spectral features of the structures. As the period increases, the resonance shows a redshift, which is consistent with the grating diffraction Equation (1). The cutoff wavelength can be predicted by considering the boundary condition. At the air-waveguide boundary, the TIR condition: \(\sin\theta=1/n\) must be satisfied. At the grating boundary, the diffraction angle for metal grating must be larger than the critical angle to couple the light wave back into the waveguide. Replacing \(\sin\theta\) in Equation (1) with \(1/n\) and set \(m=1\), we can get the cutoff wavelength as \(\lambda_{cutoff}=P\). In Figure 2a, we can see small drops at the cutoff wavelength for different periods. These drops caused by the Wood-Rayleigh anomaly can be a sign of the abrupt change of diffraction order. (See Figure 2a.) As the diffraction angle of the metal grating cannot exceed 90 degrees, the maximum resonance wavelength is \(\lambda_{max}=nP\). Figure 2b shows the electric field amplitude profile in the \(XZ\) plane, where the guided mode can be identified. The electric field intensity is enhanced 3.5-fold in the waveguide layer. In this configuration, the electric field is confined in the whole waveguide layer, which is different from the plasmonic system in which the electric hot spots are only near the metal gratings. This feature shows it could be used as an alternative structure for fluorescence enhancement. Refractive index contrast can have a dramatic effect on the guided mode. In Figure 3, we show how the guided mode resonance varies as a function of the refractive index of waveguide and substrate. Figure 3a displays that the resonance peak redshifts as the refractive index of the waveguide increases. This can be explained by [PERSON]'s equation although is not intuitive from Equation (3). The phase shift of \(\phi_{\text{TIR}}\) is a function of the refractive index of the cladding layer and the waveguide layer. Increasing the index contrast can increase the phase shift at the boundary and hence causes a redshift of the resonance wavelength. In Figure 3b, as expected, no resonance shifts as the refractive index of the substrate changes. This implies that the metal grating diffraction effect can substitute for the TIR to couple and excite guided mode resonance in a waveguide layer. All the resonance peaks are not at unity due to the reflection loss of the metal.