# Electromagnetically Induced Transparency in All-Dielectric U-Shaped Silicon Metamaterials
## 3 Results and Discussion
(**c**) Simulated transmission spectra of the metamaterial structure as well as the bright and dark modes supported by the horizontal and vertical bars respectively. Next we investigated the effect of the period size of the unit structure on the EIT-like transmission window and Q-factor, the simulated transmission spectra with different period sizes are shown in Figure 3a. It is clear that although the transmittance does not change much, the resonance peak shifts towards the long wavelength and the FWHM of the EIT peak becomes wider as the period size increases, which indicates that the coupling coefficient decreases, resulting in a decrease in the Q-factor, as is shown in Figure 3b. The Si metamaterials with period of smaller than 615 nm were also checked. However, the near field interaction of nearby unit cells occurs due to the close distance between them when the period is smaller than 615 nm, which results in an off-resonance coupling between the dipole mode of the horizontal bar and the quadrupole mode of two vertical bars with an asymmetric shape of EIT transmission peak. Therefore, it can be found that the optimal period size is 615 nm. The generation of EIT depends on the coherent interaction of the bright mode resonator and the dark mode resonator, and the incident polarization angle \(\theta\) plays a key role in the effective excitation of the bright mode resonance [28]. In order to explore the effect of incident polarization angle on the EIT-like response of the dielectric metamaterial, the transmission spectra for the metamaterial with different incident polarization angles are demonstrated in Figure 4. The transmittance decreases as the angle \(\theta\) changes from \(0^{\circ}\) to \(90^{\circ}\), accompanied by a negligible shift of the transmission peak towards the short wavelength. When the angle increases to \(90^{\circ}\), the transmission peak disappears completely, and only a transmission dip is observed. This is because when the incident electric field is parallel to  
Figure 3: (a) Simulated transmission spectra with different period sizes. (b) The Q-factors of the EIT-like resonance with different period sizes. Figure 2: Electric field distributions of the dielectric metamaterial at different wavelengths. (a), (b) and (c) correspond to the wavelengths of 898.5 nm, 928 nm and 908.5 nm, respectively. The other geometrical parameters are the same as in Figure 1b. the vertical bars, the bright mode dipole resonance of the horizontal bar cannot be excited while the dipole resonance of the dark mode resonator can be excited. However, the bright mode resonator cannot be excited by the dark mode dipole resonance, leading to the EIT peak vanishing [33]. The group velocity of light in a medium depends on the dispersion property of the medium which is the change of refractive index with the frequency of light. Due to the absorption and refraction properties of the medium are correlative, EIT must change the refractive index of the system when changing the absorption property [34]. The propagation velocity of wave decreases with the increase of group refractive index. A strong dispersion occurs due to the continuous steep change of phase in the transmission window, and this change reflects the speed of wave propagation. Therefore, the analogue of EIT is usually concomitant with slow-light effect [35; 36]. The group delay is used to describe the slow-light effect and is defined as [37]:  
\[\tau_{g}=-\frac{d\varphi(\omega)}{d\omega}, \tag{2}\]  
where \(\varphi\) is the transmission phase shift. And the corresponding group index is defined as  
\[n_{g}=\frac{c}{v_{g}}=\frac{c}{D}\times\tau_{g}=-\frac{c}{D}\times\frac{d \varphi(\omega)}{d\omega}, \tag{3}\]  
where \(c\) is the velocity of light in free space, \(v_{g}\) and \(\tau_{g}\) are group velocity and delay time in the metamaterial respectively, and \(D\) is the thickness of the metamaterial.