## 1 Introduction  
In the past few decades, the electromagnetically induced transparency (EIT) effect caused by quantum destructive interference between two different excitation paths in a three-level atomic system has attracted extensive and in-depth research [1; 2]. The EIT effect is that under the effect of strong resonance coupling light, the opaque medium becomes transparent for weak probe light and is accompanied by extraordinarily steep dispersion. In other words, the resonance probe light can propagate through the medium without being absorbed [3]. EIT has potential applications in many fields, such as sensing [4; 5; 6], slow light [5; 7], nonlinear optics [8] and cavity quantum electrodynamics [9]. However, the realization of the EIT effect in atomic systems requires extremely strict experimental conditions, which limits its practical application. Therefore, researchers have turned to classical optical systems to achieve EIT-like effects. [PERSON] and [PERSON] described the EIT analog in a classic plasma [10], and more of an EIT-like effect was subsequently observed in the plasmonic metamaterial [11; 12]. The EIT-like effect has also been successfully achieved in other optical systems, such as waveguide cavity structures [13; 14] and photonic crystals [15]. [PERSON]'s group realized the EIT-like effect based on the guided-mode resonance (GMR) effect for the first time in 2015 [16], the system consists of two planar dielectric waveguides and a subwavelength grating and a narrow transparent window appears in the transmission dip when high Q and low Q resonant waveguide modes are coupled.  
In recent years, the realization of the EIT-like effect based on GMR has attracted more and more interest of researchers. [PERSON]'s group successfully realizes the polarization-independent EIT-like effect in two photonic systems, which are both composed of two planar dielectric waveguides and a two-dimensional photonic crystal [17]. [PERSON] and [PERSON] et al. reported a planar metamaterial based on GMR achieve the EIT-like effect with a Q value exceeding 7000 [18]. [PERSON] and [PERSON] et al. reported that a GMR system with two subwavelength silicon grating waveguide layers achieves the EIT-like effect with an ultra-high Q factor of 288,892 [19]. However, as is known to us, the realization of multispectral EIT effects by the subwavelength grating structure based on GMR has not been studied according to available works.  
In this work, we numerically simulate the double spectral EIT-like effect in a coupled GMR system that consists of two silicon grating waveguide layers (GWLs) on a SiO\({}_{2}\) substrate. In addition, both GWLs contain double-bar dielectric gratings with unequal lengths in a cycle. When the distance between the two GWLs meets the phase matching condition, the GMR system works as a Fabry-Perot (F-P) cavity and GWLs as the reflection boundary of the cavity thus two EIT peaks appear in the near infrared band due to the top GMR mode and the bottom GMR mode are intercoupled. The effects of two grating widths and the separation between the two gratings are also studied. After the structural parameters optimization, two EIT peaks present ultra-narrow FWHMs of \(10^{-2}\) nm and ultra-high Q factors of \(10^{4}\). In terms of the sensor, this system has ordinary sensitivity and high figure of merit (FOM) with compact structure and a simple manufacturing process. This double spectral EIT-like effect may not only be applied to the sensor, but also to slow light and nonlinear optics.