# Guided Mode Resonance in a Low-Index Waveguide Layer
## 2 Design and Simulation  
The proposed structure is shown in Figure 1a. It consists of aluminum (Al) grating arrays embedded in a low-index waveguide layer. The waveguide layer sits on top of a high-index substrate. If there is no grating at the bottom boundary, then TIR cannot occur at the bottom interface as the refractive index of the waveguide layer is lower than the index of the substrate. On this condition, no guided mode can be supported. When a metal grating is inserted at the bottom interface, incident light could be diffracted from the interface. The TIR is effectively replaced by the grating diffraction at the bottom interface. The diffraction condition of the grating can be expressed as:  
\[Pn(\sin\theta_{m}\pm\sin\theta_{i})=m\lambda_{0},\ m=1,\ 2,\ \ldots \tag{1}\]  
where \(P\) is the grating period, \(n\) is the refractive index of the waveguide layer, \(\theta_{m}\) represents the diffraction angle, subscript \(m\) means the diffraction order and \(\lambda_{0}\) is the wavelength in the vacuum. \(\theta_{i}\) is the angle of the incident light in the waveguide. The metal grating used in this design acts as a 'diffraction' mirror. It plays two roles in the structure, coupling the incident light-wave into the guided mode and acting as a good mirror to push the light back into the waveguide. An extra phase is also provided by the metal grating like the TIR. For a perfect electrical conductor (PEC), the extra phase shift is \(\pi\). For the slab waveguide bounded by the TIR effect and metal grating refraction effect, the guided mode resonance condition can be expressed as:  
\[2 nk_{0}f\cos\theta-\phi_{\text{TIR}}-\phi_{metal}=2q\pi,\ q=1,\ 2,\ldots \tag{2}\]  
Figure 1: (**a**) The cross-section schematic of the proposed grating assisted guided mode in the low-index waveguide layer with all geometrical parameters. (**b**) The scanning electron microscopy (SEM) image of the fabricated Al gratings on the Lithium Niobate substrate (with refractive indexes \(n_{e}=2.2\), \(n_{o}=2.2\)) with no guiding layer atop. Scale bar: 1 μm. where \(n\) is the refractive index of the waveguide layer, \(t\) is the thickness of the waveguide layer, \(k_{0}=\frac{2\pi}{\lambda_{0}}\) is the wavenumber in a vacuum, \(\lambda_{0}\) is the wavelength in vacuum, \(\theta\) is the incident angle of the guided mode wave in the waveguide, as shown in Figure 1a, \(\phi_{\text{TIR}}\) and \(\phi_{metal}\) denote the phase shift caused by the TIR and the metal grating. The TIR phase shift at the top boundary can be derived from Fresnel's law as \(\phi_{\text{TIR}}=2\text{arctan}\frac{\sqrt{(n\sin\theta)^{2}-1}}{\cos\theta}\) and it depends on refractive index contrast, incident angle and the wavelength due to the dispersion of the refractive index, while the phase shift for metal \(\phi_{metal}\approx\pi\). \(q\) is the mode order in the waveguide. For the incident wave with a given wavelength, when the diffraction angle \(\theta_{m}\) matches to \(\theta\), the guided mode can be excited in the waveguide. By combining Equations (1) and (2), the grating assisted guided mode resonance (GAGMR) condition can be expressed as:  
\[\frac{4m\pi t\cos\theta}{P(\sin\theta\pm\sin\theta_{i})}-\phi_{\text{TIR}}- \phi_{metal}=2q\pi,\ m,q=1,\ 2,\dots \tag{3}\]  
Equation (3) shows that the guided mode resonance is related to the refractive index of the waveguide, the thickness of the waveguide and the grating period. The refractive index of the substrate has no direct influence on the guided mode but can affect the reflection intensity of the off-resonance wavelengths. We evaluate the influence of the geometrical and material parameters on the guided mode using Finite Difference Time Domain (FDTD) simulation. The refractive indices of the substrate \(n_{s}\) and waveguide layer \(n\) were fixed to be 2 and 1.5 to simplify the simulation. The material of grating was chosen as Al since it is relatively low loss and low cost. The refractive index of Al was adapted from reference [21].