# Electromagnetically Induced Transparency in All-Dielectric U-Shaped Silicon Metamaterials
## 3 Results and Discussion  
The geometry of the designed Si metamaterial structure deposited on a quartz substrate is illustrated in Figure 1a, which is arranged periodically in free space. Each unit cell depicted in Figure 1b comprises two vertical nanoscale bars and one horizontal nanoscale bar. The period of one unit structure is \(P=615\) nm in both \(x\) and \(y\) directions. The horizontal bar acts as a dipole antenna and can be excited supporting a bright mode resonance when coupling to the incident electromagnetic wave (Figure 1c). Two vertical bars serve as a quadrupolar antenna which cannot be directly excited by the external light and support a dark mode (Figure 1c). The dark mode resonator can only be excited by the near-field coupling with the bright mode resonator. When the plane wave is incident along the negative direction of the \(z\) axis with polarization along the \(x\) axis, the destructive interference between the bright mode and the dark mode leads to a narrow transmission window shown in Figure 1c, displaying a characteristic EIT-like behavior. The transmission peak is located at 908.5 nm with a high transmittance of 92%. The two transmission dips are located at 898.5 nm and 928 nm, respectively. The Q-factor can be calculated by the following formula:  
\[\mathrm{Q}=\frac{\lambda_{0}}{\mathrm{FWHM}}, \tag{1}\]  
where \(\lambda_{0}\) is the wavelength of transmission peak at the EIT-like window, and the FWHM is the full width at half maximum of transmission peak. The FWHM is 7 nm, and the calculated Q-factor is 130. It should be noted that when fixing the lengths and widths of the three nanoscale bars (see captions in Figure 1a) the thickness of the bars has an optimal value of 110 nm. When further reducing or increasing the thickness the FWHM of EIT transmission peak will become wider, and as a result the Q-factor will be reduced and the sensing performing of the Si metamaterial will be deteriorated. To better understand the underlying physical mechanism of EIT-like effect in the proposed metamaterial structure, the field distribution maps in the \(x\)-\(y\) plane are plotted in Figure 2. The electric field distributions at transmission dips of 898.5 nm and 928 nm are shown in Figure 2a,b, respectively. It is seen that the horizontal nanoscale bar is mainly excited by the incident light. The emergence of the dipole mode with two nodes of electric fields at two tips of the bar and displacement currents along the horizontal nanoscale bar can hardly induce the resonance of two vertical nanoscale bars, leading to the transmission dips. Figure 2c shows the electric field distribution at transmission peak of 908.5 nm. The near-field resonant coupling between the two resonators causes the bright mode to couple energetically to the dark mode, which produces the destructive interference and achieve the EIT transmission peak, and suppresses the radiation of the dipole mode oscillation in the horizontal bar [32]. The quadrupole mode with three electric field nodes (two at the tips of the bars and one at the inner side of the bars) along the vertical Si bars is clearly observed. It is obvious that the induced surface field intensity at the peak is stronger than those at two dips. Furthermore, the high refractive index contrast between air and the all-dielectric metamaterial makes the local electric field on the edges of the structure stronger. All in all, the damping of radiation loss and high refractive index contrast are helpful for the metamaterials to achieve high transmission. Figure 1: (**a**) Schematic of the dielectric metamaterial structure deposited on a quartz substrate. (**b**) Schematic of a metamaterial unit cell with the period size \(P\) of 615 nm comprising two vertical silicon nanoscale bars and one horizontal silicon nanoscale bar. The geometrical parameters are: a = 580 nm, b = 590 nm, c = 510 nm, d = 80 nm, e = 80 nm, h\({}_{1}\) = 110 nm, h\({}_{2}\) = 100 nm.