## 3 Results and Discussion  
We first calculated the transmission spectrum of the structure with only single GWL\({}_{1}\) (GWL\({}_{2}\)), as the green solid line (blue solid line) shown in Figure 1b. It is clear that the two different transmission dips appeared at the wavelengths of 1547.97 nm and 1553.08 nm. GWL\({}_{1}\) and GWL\({}_{2}\) had the same structure, but their optical properties were easily affected by the surrounding medium, thus the transmission spectra were slightly different. The SiO\({}_{2}\) cover layer had an influence on the resonance  
Figure 1: (**a**) Schematic of the guided-mode resonance (GMR) system with double-bar dielectric grating structure and geometrical parameters: d\({}_{1}\) = 1885 nm, D = 55 nm, t = 435 nm, \(d_{2}\) = 2137 nm, \(W_{1}\) = 100 nm, \(W_{2}\) = 230 nm, \(g\) = 30 nm and P = 500 nm. (**b**) Transmittance spectra of the GMR system with only grating waveguide layer (GWL\({}_{1}\)) (GWL\({}_{2}\)) and both GWLs. The inset shows the real part of the refractive index for SiO\({}_{2}\) and Si as a function of the wavelength. (**c**) Magnified view of the transmission features of electromagnetically induced transparency (EIT)\({}_{1}\). (**d**) Magnified view of the transmission features of EIT\({}_{2}\). frequency. Adding a SiO\({}_{2}\) cover layer can reduce the impact of the surrounding medium on its optical characteristics to reach the same resonance frequency [18]. When GWL\({}_{1}\) and GWL\({}_{2}\) are both present, and their distance (\(d_{2}\)) met the phase matching conditions, the top GMR mode (in GWL\({}_{1}\)) and the bottom GMR mode (in GWL\({}_{2}\)) were intercoupled, Therefore, two different sharp EIT resonances appear in the two resonance dips respectively due to destructive interference [12; 19; 22]. The transmission spectrum is shown by the solid red line in Figure 1b. We refer to the two EIT peaks as EIT\({}_{1}\) and EIT\({}_{2}\). The transmission characteristics of their magnified views are shown in Figure 1c,d, respectively. The resonance wavelengths of EIT\({}_{1}\) (\(\lambda_{1}\)) and EIT\({}_{2}\) (\(\lambda_{2}\)) were 1548.12 nm and 1553.36 nm, respectively, and their corresponding full width at half maximum (FWHM) was 0.0441 nm (\(\Delta\lambda_{1}\) in Figure 1c) and 0.0636 nm (\(\Delta\lambda_{2}\) in Figure 1d). The Q factor was an important parameter of the resonant cavity, which is defined as follows:  
\[\text{Q}=\frac{\lambda}{\text{FWHM}} \tag{1}\]  
There, their corresponding Q factors reached 35,104 and 24,423, respectively. Obviously, our work has two higher Q factors than another double EIT-like effects work, which has two Q factors of 950 and 216 [6]. The analogy between our system and the atomic EIT system can help us understand the physics of the double EIT-like effect [23; 24]. A double-\(\Lambda\) five-level model of the double EIT-like effect in our system is illustrated in Figure 2a. In the system, \(\ket{1}\) represents the ground state, the field of the bottom GMR mode corresponds to the probability amplitudes of atoms in the metastable states \(\ket{2}\) and \(\ket{3}\), the field of the top GMR mode corresponds to the probability amplitudes of atoms in the excited states \(\ket{4}\) and \(\ket{5}\)[19; 23; 25]. The control field refers to the coupling between the top GMR mode and the bottom GMR mode, and the probe field refers to the input of the top GMR mode. We observed that two sharp EIT-like windows appeared in the probe area due to the introduction of the control field. The energy level of \(\ket{5}\) -- \(\ket{1}\) was higher than \(\ket{4}\) -- \(\ket{1}\), hence EIT\({}_{1}\) corresponded to the destructive interference between two different transition pathways \(\ket{1}\rightarrow\ket{5}\) and \(\ket{1}\rightarrow\ket{5}\rightarrow\ket{3}\rightarrow\ket{5}\), EIT\({}_{2}\) corresponded to the destructive interference between two different transition pathways \(\ket{1}\rightarrow\ket{4}\) and \(\ket{1}\rightarrow\ket{4}\rightarrow\ket{2}\rightarrow\ket{4}\).