Q-factor enhancement of Fano resonance in all-dielectric metasurfaces by modulating meta-atom interactions

We numerically investigated the effects of meta-atom interactions on the Fano resonance in all-dielectric metasurfaces by introducing alternately flipped asymmetric paired bars (APBs) and split asymmetric paired bars (SAPBs). With alternately flipped configurations, the Q-factor of the Fano resonance is significantly enhanced up to one order of magnitude, and the electric field is strengthened by more than twice. Abnormally, the Q-factor increases with gap size in the alternately flipped SAPBs. These are attributed to the destructive interaction among nearest-neighbor dipole resonators. The Q-factor of 108 and Raman enhancement factor of 109 in the gap can be realized with the alternately flipped SAPBs made of Si. Our study provides a way to improve performance of practical devices such as ultrasensitive sensors, nonlinear optics, and quantum emitters.


Results and Discussion
The schematic of all-dielectric metasurfaces is shown in Fig. 1. Figure 1(a) shows an all-dielectric metasurface comprising periodical array of APBs. The unit cell of the APBs is composed of two Si bars with slightly different lengths. The paired bars are positioned in parallel with the spacing d = 200 nm. The length of the long bar L 1 is fixed to be 750 nm, and the length of the short bar L 2 is altered. The width (w) and height (h) of the two bars are 200 nm and 150 nm, respectively. The lattice constants along both axes p x and p y are 900 nm. Figure 1(b) shows an all-dielectric metasurface composed of alternately flipped APBs, in which two APBs positioned on the second diagonal of 2 × 2 unit cells are flipped. Figure 1(c) and (d) are metasurfaces made of SAPBs and alternately flipped SAPBs with gaps g = 50 nm in the middle. In order to simulate the optical properties of the proposed metasurfaces, the frequency-domain finite element method 24 is employed. Si is initially supposed to be lossless with a refractive index of n = 3.5, and the material loss is considered further. The surrounding medium is supposed to be air (n = 1). The incident plane wave has an electric field polarized along the x direction with a propagation vector along the z direction, as shown in Fig. 1.
When the electric field component of incident light parallels to the long axis of the Si bar, the corresponding component of the displacement current inside the bar increases dramatically. This resonance is similar to the electric dipole resonance in metallic metamaterials. Here the bar is also called the meta-atom. The dark mode is excited in a unit cell of the APBs, which is characterized by anti-phased displacement currents in the two Si bars of the unit cell 19 (see Fig. 2). Each electric dipole is excited by not only the external electric field but also electric fields produced by neighboring meta-atoms. The meta-atom interactions are different when configurations of unit cells are changed. For non-flipped configuration, the dipole moment of a bar is constructively driven by nearest-neighbor dipoles, shown in Fig. 2(a). However, for alternately flipped configuration, the dipole moment of a bar is destructively driven by nearest-neighbor dipoles outside of the APBs, shown in Fig. 2 Radiation loss is proportional to a net dipole moment per unit area μ in metamaterials, which at dark mode can be written as 23 where μ l and μ s are magnitude of dipole moments in the independent long and short bars, respectively, m is the density of APBs/SAPBs per unit area, and γ is a constant related to the nearest-neighbor dipole interactions, which becomes smaller (larger) than 1 when dipole interactions are destructively (constructively). The μ in alternately flipped configuration becomes smaller than that in the non-flipped configuration. Therefore, radiation loss is suppressed in the alternately flipped configuration, which leads to a high Q-factor. Figure 3(a) shows transmission spectra of non-flipped APBs metasurfaces (see Fig. 1(a)) with L 2 = 700 nm, 650 nm and 600 nm, respectively. The transmission spectra exhibit asymmetric Fano lineshape, which  originates from the excitation of anti-phased displacement currents in APBs 19 . Figure 3(b) gives the normalized z-component of the electric field E z /|E 0 | at the resonance for L 2 = 650 nm, which indicates an electric quadrupole resonance. With increasing asymmetry, the corresponding Q-factor dramatically decreases, which are calculated as 925, 239 and 104 for L 2 = 700 nm, 650 nm and 600 nm, respectively. The reduction of Q-factor with decreasing L 2 is due to the increase of radiation loss 19 . Meanwhile, the resonant frequency of the FR shifts toward larger frequencies with increasing asymmetry because shortening of the bar length increases the excitation energy of the FR 20 . Figure 3(c) presents calculated transmission spectra of alternately flipped APBs configuration (see Fig. 1(b)) with L 2 = 700 nm, 650 nm and 600 nm, respectively. Different from the non-flipped APBs, the flipped configuration exhibits two FRs. One around 196 THz originating from the electric quadrupole mode becomes sharper. Their Q-factors are 5398, 1154 and 442 for L 2 = 700 nm, 650 nm and 600 nm, respectively, which are 4~6 times larger than those of corresponding non-flipped metasurfaces. In other words, the Q-factor of the metasurface can be improved by changing configuration of unit cells, where the net dipole moment is reduced because of destructive interactions among nearest-neighbor electric dipoles. Figure 3(d) presents the normalized z-component of the electric field at the FR for L 2 = 650 nm. It is observed that the electric field distribution of each bar is asymmetric in the y direction, which is attributed to combined effects of the interaction within the APBs and destructive interaction with other nearest-neighbor meta-atoms. And the combined effects result in a red-shift of the FR. The origin of the other FR around 230 THz will be explored later.
The fields in metallic metamaterials are mainly confined at the interface, while those in all-dielectric metasurfaces are normally concentrated within the dielectric. This limits many applications of all-dielectric metasurfaces, such as biological and chemical sensing, quantum emitters, and nonlinear optics. On the other hand, nanometer-size gaps introduced in high-index dielectric resonators can improve electromagnetic fields in surrounding medium, which is due to slot waveguide 20,25 . In order to study the gap effect, gaps are introduced in the middle of each bar. We call the structure the SAPBs (see Fig. 1(c)). Figure 4(a) presents the transmission spectra of the SAPBs metasurfaces with gaps (g = 50 nm) for L 1 = 750 nm and L 2 = 700 nm, 650 nm, 600 nm, respectively. The Q-factor remarkably decreases with the increase of asymmetry. Their Q-factors are 729, 189 and 79, respectively. Compared with corresponding structures without gaps shown in Fig. 3(a), the decrease of the Q-factor is attributed to the increase of asymmetry, and the blue shift of the resonance is due to the reduction of the effective length of the bar. Figure 4(b) shows the normalized z-component of the electric field at the resonance for L 2 = 650 nm, which manifests the electric quadrupole mode of the SAPBs. As shown in Fig. 4(c), the strong electric field is localized in the gaps, and the maximum electric field enhancement reaches about 68. In a word, the introduction of gap decreases the Q factor somewhat. On the other hand, it concentrates the electric field within it which is favorable for enhancing the interaction between light and the surrounding medium. Figure 4(d) shows the transmission spectra of the alternately flipped SAPBs metasurfaces (see Fig. 1(d)) with 50 nm gaps for L 2 = 700 nm, 650 nm, 600 nm, respectively. The Q-factors of this FR are improved up to 9 times larger than that of corresponding non-flipped metasurfaces, which are 6537, 1422 and 562 for L 2 = 700 nm, 650 nm and 600 nm, respectively. The enhancement of the Q-factor is due to destructive interaction among nearest-neighbor electric dipoles. Whereas the Q-factor decreases with reducing L 2 for the increase of radiation loss. Interestingly, compared with that of alternately flipped APBs, the Q-factor becomes larger. Analogous to the redshift phenomenon in Fig. 3(c), the FR in Fig. 4(d) displays a red shift compared with Fig. 4(a). The normalized z-component of the electric field at 210.5 THz for L 2 = 650 nm is presented in Fig. 4(e), which is akin to that of alternately flipped APBs in Fig. 3(d). The electric field is strongly localized in the gaps as well (see Fig. 4(f)). The maximum field enhancement is up to 176, which is equivalent to an enhancement factor of ~ 10 9 for Raman applications. So the modulating of meta-atom interactions in the SAPBs not only leads to Q-factor enhancement but also further strengthens the interaction between light and the surrounding medium.
Similar to the alternately flipped APBs case, the transmission spectra also exhibit two FRs, the second FR lies at 231 THz, which is almost independent on the gap as well as L 2 . We ascribe it to the Rayleigh-Wood anomaly (RA), which is associated with the diffraction along grating surface. It occurs when 26 where ε d is the dielectric constant of the background material (vacuum in our work),  k 0 is the wave vector of the incident wave in vacuum, p x (p y ) represents the period along x (y) direction, i and j are integers defining the different diffraction orders, respectively. The RA (±1, ±1) frequency is 235.7 THz which approximately agrees with that of the second FRs in alternately flipped metasurfaces. The Fano lineshape originates from the interaction between the electric dipole and the light diffracted parallel to grating surface, which can find applications in refractive index sensing 27 .
Contrary to the SAPBs case, the Q-factor of alternately flipped SAPBs (FSAPBs) increases with the gap, as shown in Fig. 5(a). This is because of the fact that the offset of the electric dipole moment in the y direction further reduces the radiation loss. Figure 5(b) presents the Q-factor of the FRs as a function of L 2 for non-flipped APBs/SAPBs, alternately flipped APBs (FAPBs) and FSAPBs. The Q-factors of FRs in these four metasurfaces are enhanced with increasing L 2 , which originates from suppressing the radiation loss. In metasurfaces with alternately flipped configuration, Q-factors are much larger than those of corresponding metasurfaces with non-flipped configuration. The Q-factors are enhanced up to 6.5 and 9.8 times for APBs and SAPBs with L 2 = 725 nm, respectively. The Q-factor could be as large as the order of 10 8 in alternately flipped configurations when L 2 = 749.9 nm (not shown here). Figure 5(c) shows the Q-factor of the FRs as a function of period p x = p y = p for APBs, SAPBs, FAPBs and FSAPBs with L 1 = 750 nm and L 2 = 650 nm. With the decrease of p, the constructive interaction in non-flipped configurations is strengthened, leading to a decrease of the Q-factor of the FRs, while the destructive interaction in alternately flipped configurations is also strengthened, leading to a quick increase of the Q-factor of the FRs. Therefore, the destructive coupling effect is an effective approach to achieve high Q-factor for all-dielectric metasurfaces, besides the structural design widely reported in literatures.
The influence of absorption loss on the Q-factor of the FR in the APBs and alternately FAPBs metasurfaces is further investigated, as shown in Fig. 6. Here, the refractive index of Si is described as n Si = 3.5 + in″. Figure 6(a) presents transmission spectra of the APBs and alternately FAPBs with L 1 = 750 nm and L 2 = 650 nm for n″ = 10 −3 and 10 −2 , respectively. The absorption loss reduces the spectral contrast and slightly increases the difference of the frequency at transmission peak and dip, which reveals a decrease of the Q-factor with increasing of the dielectric loss. Figure 6(b) shows the Q-factor of the FR as a function of -lg(n″). With decreasing -lg(n″), i.e. the increase of absorption loss, the Q-factor of the FR in alternately FAPBs is more rapidly attenuated than that in APBs. In other words, the less the absorption loss is, the more the Q-factor of FRs can be enhanced by introducing alternately flipped configuration. This can account for the results of metallic metamaterials, in which the Q-factor is slightly improved by changing configuration 15,23 .

Conclusion
In conclusion, we numerically investigated the high-Q FRs in all-dielectric metasurfaces. By introducing the alternately flipped configurations, the Q-factor of the FR can be markedly enhanced, which is attributed to the destructive interaction among nearest-neighbor meta-atoms. The destructive interaction reduces the net electric dipole moment and suppresses the radiation loss. The offset of the y-component of the electric dipole moment further improves the Q-factor in the alternately flipped SAPBs, where the electric field is strongly located in the gap and the Q-factor exceptionally increases with the gap. Furthermore, the less the absorption loss is, the more the Q-factor of FRs can be enhanced.

Methods
The Si metasurfaces are initially supposed to be lossless with a refractive index of n = 3.5, and the material loss is considered further (n Si = 3.5 + in″). The surrounding medium is supposed to be air (n = 1). The transmission spectra and electric field distributions are calculated using the frequency-domain finite element method with a tetrahedral mesh. The periodic boundary conditions for each unit cell are used in the x and y axes, and open boundary condition is used in the z axis (propagation direction). At least 10 mesh steps per wavelength and adaptive tetrahedral mesh refinement are used to ensure the accuracy of the calculated results.
The Q-factor is extracted from the transmission spectra of metasurfaces, which is expressed as Q = f/Δf, where f is the resonant frequency of the FR and Δf is defined as the difference of the frequencies at the transmission peak and dip.