# High Q-Factor Resonance in a Symmetric Array of All-Dielectric Bars
## 2 Simple Silicon Rod Arrays with High Q Resonance
Figure 3: The real parts of effective permittivity (\(\varepsilon^{\prime}\)) and permeability (\(\mu^{\prime}\)). Figure 2: Field distribution inside one unit cell of the rod array. (**a,b**) \(|\)E\(|\) in \(x\)-\(z\) plane and \(|\)H\(|\) in \(x\)-\(y\) plane at wavelength of 1187 nm, representing the magnetic dipole moment; (**c,d**) \(|\)E\(|\) in \(x\)-\(z\) plane and \(|\)H\(|\) in \(x\)-\(y\) plane at wavelength of 1357 nm, representing the higher order electric dipole moment. In order to further study the resonance properties, the wavelength dependences of ER and MR on the geometric parameters (the length and the width) of dielectric rod are calculated in Figure 4. Both the ER and MR resonances show red-shift as the rod's length or width increases. Linear fits of the dip wavelength versus the geometric parameters are shown in Figure 4b,d, which confirm that the wavelength of the ER is sensitive to both the rod's length and width, whereas the wavelength of the MR is more sensitive to the rod's width, which is mainly due to the displacement current loop covers a larger area in the \(x\)-\(z\) plane for larger value of the rod's width. And the spectral width of the MR is wide (low Q \(\sim\)10) and insensitive to the rod parameters due to its fundamental mode of the rods and easy excitation property, however, it is interestingly found that this is just opposite for the ER, i.e., the Q-factor of the ER is not only very high, but also very sensitive to the rod parameters. The Q-factor of the sharp ER with respect to the rod's length \(a\) is calculated and shown in Figure 5 when the rod's widths are \(b\) = 300 nm and \(b\) = 260 nm, while the height is fixed at \(h\) = 200 nm. As the rod length increases from 680 nm to 800 nm, the Q value of the ER increases steadily, then a dramatic rise is observed and reaches 4500 when \(a\) = 840 nm, \(b\) = 260 nm. By further increasing the rod's length, the Q value of over tens of thousands can be expected. However, this kind of growth is not unlimited, since the ER will vanish when the length of the rectangular rod resonators is too close to the period of \(Py\) = 900 nm. Not to mention it is expensive or even impossible to fabricate the proposed metasurface when \(a\) > 840 nm. Simulation results also indicate that the Q value of the ER increases as the rod width or height decreases. For example, it reaches 25,300 when \(a\) = 840 nm, \(b\) = 230 nm, \(h\) = 200 nm, and further decreasing the rod width or height will result in disappearance of the ER. Based on our simulations, we can conclude that different from a single silicon rod resonator, there exists a critical condition or boundary of the rod's geometric parameters for the excitation of high order TE\({}_{12\delta}\) in rod arrays, and less coupling efficiency of the incident light to this resonator mode or higher Q resonance can be obtained near the boundary. And the rod's length contributes much more to the high Q ER resonance than the rod's width or height. Although high order ER or MR resonances have been reported [6; 41], strong resonance with high Q over several hundreds has not yet been calculated in silicon metasurfaces without asymmetrical structure. Fortunately, the wavelength of the ER discussed above is beyond the diffraction of periodically arranged rod arrays, while this is not the case for the MR with limited applications [43]. Moreover, we should mention here that in the case of oblique incidences, when the angle of incidence plane wave is less than 10 degrees, the Q-factor of the ER will not be changed according to our simulations. High Q resonance is usually accompanied with the near field enhancement which is useful for nonlinear interactions [7; 38] or sensing applications [5; 6; 36; 37].