# Guided Mode Resonance in a Low-Index Waveguide Layer
## 2 Design and Simulation
If we replace Al with a PEC in the simulation, the resonance peak can reach unity (shown in Figure 3b). Thickness is another substantial parameter for a guided mode. Equation (2) shows that increasing thickness will cause a redshift of the resonance and eventually a higher-order mode, just like higher-order guided mode arising in a thick or wide planar waveguide. There exists a cutoff 'width' below which no guided mode can be excited. Figure 4a shows the relationship between thickness and the guided mode resonance. A cutoff thickness can be observed around 130 nm for the 1 st order mode and around 350 nm for the 2 nd order mode. The analytical results calculated by Equation (3) are consistent with the simulation and are also plotted in Figure 4a (black triangles for the first order mode and red squares for the second order mode). Figure 2: (**a**) The simulated reflection spectra for the grating period \(P\) from 400 nm to 700 nm with a step size of 50 nm. For the waveguide coating, the refractive index \(n\) is fixed to 1.5, and the thickness \(t\) is set at 200 nm. The dashed line represents the reflection spectrum of the grating array (\(P=500\) nm) without waveguide coating. The asterisks are the analytical resonance wavelengths calculated by Equation (3). The analytical parameters are the same as numerical ones except for \(\phi_{metal}=\pi\) for simplicity. (**b**) The electrical field intensity profile at \(XZ\) plane for the guided mode resonance at 564 nm. \(P=500\) nm, \(t=200\) nm. We also simulate the resonance character for different incident angles. As can be seen from Figure 4b, for any non-zero incident angle, the resonance peak will split into two branches. One peak branch has a blueshift and another has a redshift as the incident angle increases. The sensitivity of resonant wavelength to incident angle is around \(6.5\) nm/degree. This consistent with Equation (3) which shows \(-1\) and \(+1\)-order of guided mode resonance for a specific incident angle. Besides, we have checked the tolerance of this structure. The simulation shows that decreasing or increasing duty cycle (DC from 0.4 to 0.6) has little influence on the performance of the resonance. The thickness of the gratings must be larger than \(30\) nm to avoid leaky light from the gratings. The performance is maintained for the thickness of the gratings above \(30\) nm. This large tolerance ensures ease of fabrication and application for this low-index waveguide system. Figure 4: (**a**) The resonance behavior as a function of the thickness. \(P=500\) nm and the refractive index \(n\) is fixed to 1.5. Black triangles (\(q=1\)) and red squares (\(q=2\)) are analytical results from Equation (3). In the equation, mode orders were chosen as \(q=1\) and \(q=2\), and the corresponding diffraction order \(m=3\) and \(m=1\). (**b**) Resonant wavelength as a function of the incident angle \(\theta_{i}\). \(t=200\) nm. Other parameters are the same as (**a**). Figure 3: (**a**) The resonance wavelength shifts as the refractive index of waveguide layer \(n\) increases from 1.3 to 1.7 with a step size of 0.1. The period is fixed as \(P=500\) nm and thickness \(t=200\) nm. (**b**) The guided mode spectra for various refractive indices of the substrate \(n_{s}\) from 1.7 to 2.1 with a step size of 0.1. The refractive index of the waveguide layer \(n\) is fixed to 1.5. Other parameters are the same as (**a**). The dashed line shows the spectrum with PEC as the grating (\(P=500\) nm, \(t=200\) nm. \(n_{s}=1.5\), \(n=2\)).