## 3 Results and Discussion
In order to help us understand the origin of the double EIT-like effect in this GMR system, electric field distribution diagrams of the GMR system near two EIT peaks (such as the transmission spectra of Figure 1c,d are given in Figure 2b. Near the EIT\({}_{1}\), electric field distribution at the off--resonant wavelength of EIT\({}_{1}\) 1547.98 nm (the blue dash I in Figure 1c) and 1548.24 nm (the blue dash III in  
Figure 2: **(a)** Double-\(\Lambda\) five-level model of the double EIT-like effect in this GMR system. **(b)** The electric field distribution diagrams of the GMR system correspond to the wavelengths indicated by the blue dash lines in Figure 1c,d, and I at the off-resonant wavelength of EIT\({}_{1}\) 1547.98 nm, II at the EIT\({}_{1}\)—resonant wavelength of 1548.12 nm, III at the off-resonant wavelength of EIT\({}_{1}\) 1548.24 nm, IV at the off-resonant wavelength of EIT\({}_{2}\) 1553.12 nm, V at the EIT\({}_{2}\)—resonant wavelength of 1553.36 nm and VI at the off-resonant wavelength of EIT\({}_{2}\) 1553.54 nm. Figure 1c) was very easily observed and excitation mainly occurred in GWL\({}_{1}\) (I in Figure 2b) or GWL2 (III in Figure 2b), which was consistent with the characteristics of GMR. The corresponding transmittance was very low since the GMR mode was easily coupled to electromagnetic waves in free space. Near the EIT\({}_{2}\), the electric field distribution at the off--resonant wavelength of EIT\({}_{2}\) 1553.12 nm (the blue dash IV in Figure 1d) and 1553.54 nm (the blue dash VI in Figure 1d) corresponded to IV and VI in Figure 2b. The principles were the same as near EIT\({}_{1}\). At the EIT\({}_{1}\)--resonant wavelength of \(\lambda_{1}\) = 1548.12 nm (the blue dash II in Figure 1b) and the EIT\({}_{2}\)--resonant wavelength of \(\lambda_{2}\) = 1553.36 nm (the blue dash V in Figure 1d), the electric field distribution diagrams were II and V Figure 2b, respectively. Obviously, very strong oscillations occurred simultaneously in GWL\({}_{1}\) and GWL\({}_{2}\). When the incident light was at two EIT wavelengths, the light was reflected back and forth between GWL\({}_{1}\) and GWL\({}_{2}\), and electromagnetic energy was coupled into the two GWLs, so strong oscillations were excited through coupling. Once the top GMR mode and the bottom GMR mode are intercoupled, and this system works like as an F-P cavity [19; 26; 27], therefore, two very narrow transmission peaks could be obtained at the wavelengths of 1548.12 nm and 1553.36 nm in the transmission spectrum, where transmissions reached 96.23% and 94.48%, respectively. The transmission spectra of the coupled GMR system with different \(d_{2}\) are shown in Figure 3, and other parameters are the same as Figure 1. In the range of \(d_{2}\) from 1970 to 2220 nm, with the increase of \(d_{2}\), the two resonance wavelengths (\(\lambda_{1}\) and \(\lambda_{2}\)) were both red-shifted. When \(d_{2}\) was increased from 2020 (in Figure 3b) to 2120 nm (in Figure 3d), \(\lambda_{1}\) was increased from 1547.67 to 1548.06 nm and \(\lambda_{2}\) was increased from 1552.51 to 1553.22 nm. This verified that the GMR system worked like an F-P cavity and GWLs as the reflective boundary of the cavity when \(d_{2}\) met the phase matching condition. Obviously, for the bottom GMR mode in GWL\({}_{2}\), a displacement of about one hundred nanometers could achieve a favorable coupling with the top GMR mode in GWL\({}_{1}\), two EIT peaks appeared in the two transmission dips when the F-P cavity was introduced. Generally, a phase matching condition corresponds to a resonance wavelength when the distance and refractive index are determined [19]. Here, it is worth mentioning that when the determined \(d_{2}\) satisfies the phase matching condition, two EIT peaks with different resonant wavelengths can be generated simultaneously because the refractive index of Si is different at different wavelengths. Figure 3: Transmittance spectra of the coupled GMR systems with different \(d_{2}\). (**a**) Transmission spectrum with \(d_{2}\) = 1970 nm.