1.

Introduction

Dielectric resonator-based Fano resonances in metasurfaces have been given much attention in recent years for the unique properties to greatly improve the performance of enhanced spectroscopy,^1,2 biosensing,^3–5 and nonlinear effect.^6–8 Fano resonances in such metasurfaces represent a universal wave interference phenomenon manifested as a distinct asymmetric spectral line shape due to coherent coupling between a discrete state and a continuous state. Such interference gives rise to Fano resonance phenomenon in a dielectric metasurface, also restricts the independent tunability of the individual resonances without any mutual influence due to the collective behavior of the total resonant systems. Recently, several all-dielectric structures have been applied to generate dominant toroidal dipole (TD) Fano resonances^9,10 by symmetry breaking to acquire high Q-factors. The TD Fano resonances excited in symmetry-breaking structures mainly depend on altering the geometry. However, the asymmetric magnetic field enhancements and Q-factors of TD Fano resonances in the existing work are still insufficient, which require further explorations in structure design and physical mechanism innovation. And, independent tailoring magnetic field is already one of the challenges to meet the demands in the targeted functionalities, but few research works have focused on that.

Recently, Fano resonance has also been linked to bound states in continuum (BIC), which can produce extremely high Q-factor resonances and appears first in quantum mechanics. A true BIC only appears in ideal lossless infinite structures or extreme values of parameters, its resonance linewidth disappears and manifests itself as a collapse of Fano resonance while its Q-factor is infinite.^11–16 In practice, symmetry-protected BIC can be transformed into quasi-BIC with a finite yet sharply high Q resonance by breaking the symmetry. And such peculiar quasi-BIC state has also been reported in subwavelength-resonant particles,^17,18 metasurfaces,^19–22 coupled waveguide arrays,^23–25 and photonic crystal cavities.^26–28 And the quasi-BIC has been demonstrated outstanding performances in various applications, especially for laser^29,30 and nonlinear.^31–34 However, the existence of quasi-BIC strongly depends on the geometrical parameters, and it is difficult to realize asymmetric excitation for quasi-BIC resulting from the collective effect in Fano resonance.

In this paper, we theoretically demonstrate that the hybrid graphene-dielectric metasurface consisting of a nanoring and a $Y$-shaped nanobar can support simultaneously asymmetric excitations of TD Fano resonance and MD quasi-BIC. Through reducing or increasing inner radius of nanoring, the quasi-BIC dominated by MD moment can be excited and effectively modulated by Fermi energy and layer numbers of the graphene. We explain the physical mechanism of quasi-BIC state in terms of the near-field enhancement and multipole decomposition, and such state is further confirmed by the quadratic inverse trend between the Q-factor and asymmetry parameter. Furthermore, the proposed metasurface can not only produce a symmetric localized magnetic field distribution but also create two asymmetric localized magnetic field distributions in near-infrared wavelength, providing a new way of indirectly manipulating the localized magnetic field enhancement.

2.

Numerical Methods

The schematic diagram of the proposed hybrid graphene-dielectric metasurface is shown in Fig. 1. Each unit cell consists of graphene overlayer lying on the amorphous silicon (Si) metasurface with the silica (${\mathrm{SiO}}_2$) substrate. The dielectric metasurface without graphene layer is composed of a nanoring and Y-shaped nanobar, as given in Fig. 1(b), the nanoring has the same outer radius $R$ and different inner radius $R_1$ and $R_2$, the $Y$-shaped nanobar has the same width $W$. Numerical simulations are conducted using the finite difference time domain and finite element method, where periodic boundary conditions are applied along both the $x$- and $y$- directions, and the normal beam propagation direction ($-z$ axis) is bounded with 32 perfectly matched layers. The dielectric constants of Si and ${\mathrm{SiO}}_2$ can be referred from the Palik handbook.³⁵ Surface conductivity of graphene can be well derived with the random-phase approximation theory.³⁶ Linearly polarized wave is normal incidence and the polarization angle $\theta$ is measured from $y$-axis, as shown in Fig. 1(b).

Fig. 1

(a) Schematic diagram of a unit cell in the proposed hybrid graphene-dielectric metasurface. (b) Top view of the unit cell without graphene layer. Geometric parameters are denoted in Fig. 1.

3.

Simulation Results and Discussions

Figure 2(a) shows the transmission spectrum of the proposed metasurface without the graphene overlayer at the $\theta =0\mathrm{deg}$. The structure parameters are $R=400\mathrm{nm}$, and $R_1=R_2=280\mathrm{nm}$, $W=120\mathrm{nm}$, $T=140\mathrm{nm}$ and $Px=Py=900\mathrm{nm}$. One can clearly observe a sharp Fano resonance dip at 1378 nm with the spectral contrast ratio near 100%, defined as $[(T\mathrm{peak}-T\mathrm{dip})]/[(T\mathrm{peak}+T\mathrm{dip})]\times 100\%$. Figure 2(b) shows three dominant Cartesian multipole moments based on the density of displacement currents $J(r)=-i\omega {\epsilon }_0(n_2-1)E(r)$.¹⁵ The transmission curve can obtain a relatively high Q-factor owing to the low radiated loss reduced by the enhanced TD $Ty$ and suppressed electric dipole $Py$ and magnetic dipole $Mx$, as shown in Fig. 2(b). And the radiating intensity of the TD is 11 times bigger than that of the ED. In addition, the phase difference $|\phi (Py)-\phi (ikTy)|$ is approximately equal to $\pi$ around 1378 nm, as shown in Fig. 2(c), where the destructive interference of the far-field radiation between the ED and TD resonances produces the asymmetric TD Fano resonance at 1378 nm. Magnetic field enhancement ($|H/H0|$) in the $xy$ plane can be firmly constrained within the two symmetric holes along the $y$ axis by the enhanced $Ty$ and no radiation energy transmits outward at 1378 nm, as shown in Fig. 2(d). And the maximum magnetic field enhancement at 1378 nm reaches 95. The inset in Fig. 2(a) analyzes the transmission spectrum of the metasurface and proves rigorously that it can be described by the classical Fano formula:

Fig. 2

(a) Transmission spectrum of all-dielectric metasurface without a graphene overlayer at $\theta =0\mathrm{deg}$. Inset represents the fitted spectrum near the resonance. (b) Normalized power scattered by different multipoles, including Cartesian TD $|ikTy|$, electric dipole $|Py|$, and magnetic dipole $\left|Mx\right|$ moment around 1378 nm at $\theta =0\mathrm{deg}$. (c) The phases and their difference of $\phi (Py)$ and $\phi (ikTy)$. (d) Simulated distributions of magnetic field enhancement ($|H/H_0|$) in the $xy$ plane at a resonant wavelength of 1378 nm.

Figure 3(a) shows the evolution of transmission spectra of the metasurface without the graphene overlayer as a function of the $R_2$. In addition to the TD Fano resonance near 1375 nm, the detailed transmission spectra shown in Fig. 3(a) exhibit a narrow dip at longer wavelength that entirely vanishes around 1515 nm when the unit cell becomes symmetric ($R_1=R_2=280\mathrm{nm}$). We observe that BIC with infinite Q-factor transforms into high-Q quasi-BIC whose radiation loss grows with the modulated $R_2$. In Figs. 3(b) and 3(c), we decompose the far-field radiation of Fano resonance and quasi-BIC into contributions of different multipole components under the Cartesian coordinate. As shown in Fig. 2(b), the normalized power scattered by the Cartesian TD moment dominates Fano resonance around 1383 nm. After introduction of the symmetry perturbation ($R_1\ne R_2$), the MD component possessing radiation along the $z$ axis emerges in Fig. 3(c), resulting in the excitation of quasi-BIC. Such dominant MD also manifests itself in the near-field distribution in Figs. 4. As shown in Fig. 3(d), the evolution of the Q-factors on the asymmetry parameter ($\alpha =\mathrm{\Delta }S/S$) was dominated by the inverse quadratic law $Q_{\text{rad}}=Q_0{\alpha }^{-2}$, where $Q_0$ is a constant determined by the design of structure. Note that for the larger asymmetry parameter $\alpha$ the Q-factor is significantly reduced because the large deviation from the symmetric unit cell cannot be treated as a tiny perturbation. As the linewidth of the quasi-BIC disappears, its Q-factor diverges, as shown in the inset in Fig. 3(d), which further manifests the formation of a BIC. At the BIC points ($R_1=R_2=280\mathrm{nm}$), the radiative Q-factor diverges, which causes the total Q-factor to be limited by the loss in the system. Therefore, the total Q-factor of the symmetry-protected BIC will diverge to infinity due to the absence of loss.

Fig. 3

(a) Evolution of transmission spectra of the metasurface without graphene overlayer as a function of the $R_2$. Other parameters are the same as that used in Fig. 2(a). (b) and (c) Normalized power scattered by the Cartesian TD moment $|ikTy|$, ED moment $|Py|$, and MD moment $|Mz|$ around 1383 and 1539 nm when $R_2=260\mathrm{nm}$. (d) Log–log plot of radiative Q factors as a function of the asymmetric degree $\alpha =\mathrm{\Delta }S/S$ for $R_2>280\text{-}\mathrm{nm}$ case. The black line shows the inverse quadratic dependence of $\mathrm{\Delta }S/S$. Inset represents the extracted Q-factors of the quasi-BIC versus the radius $R_2$.

Fig. 4

Distributions of the enhanced magnetic field ($|H|/|H0|$) and schematics of the formation of the corresponding TD and quasi-BIC modes in metasurface (a) $R_2=260\mathrm{nm}$ and (b) $R_2=300\mathrm{nm}$, respectively. The black arrows show the electric field directions.

Figure 4 shows the normalized magnetic field enhancement and electric field directions in the $xy$ plane. The electric field directions in the $xy$ plane forms two reversed current loops at $\lambda =1383\mathrm{nm}$ and $\lambda =1370\mathrm{nm}$, where the anti-phase oscillations of current loops lead to the suppression of the ED and MD, and the spatially localized magnetic field in the $xz$ plane forms a loop, confirming the in-plane TD response. While the electric field directions in the $xy$ plane forms a swirl at $\lambda =1539\mathrm{nm}$ and $\lambda =1494\mathrm{nm}$, indicating a typical $z$-orientated MD feature. And the corresponding TD and MD can be further confirmed by corresponding artistic illustrations in Fig. 4. It is worth mentioning that asymmetric magnetic hotspot of TD Fano resonances at 1383 and 1370 nm is always localized in the smaller two holes, while the asymmetric magnetic hotspot of the quasi-BIC resonances at 1492 and 1550 nm is always localized in the bigger holes, whether the $R_2$ is greater than or $<280\mathrm{nm}$. In addition, thanks to the high Q-factors of TD Fano resonance, the maximum magnetic field can be enhanced by more than 160 times at $\lambda =1370\mathrm{nm}$, comparable to plasmonic nanostructures.

Figure 5(a) shows the calculated transmission spectra as a function of the wavelength with the incident angle from $\phi =0\mathrm{deg}$ to 10 deg. It is clearly shown that the resonance at 1378 nm becomes sharper and finally vanishes, while a new resonant mode grows wider in the wavelength range of 1500 to 1600 nm as the $\phi$ increases. This sharp resonant features indicate the existence of BIC mode in the structure. Such sharp resonance, which can be analyzed more clearly from the multipole analysis shown in Fig. 5(b). This dominant magnetic dipole resonance manifest itself in the near-field distribution in the $xy$ plane in the inset. We can see that the electric field in the $xy$ plane forms a loop, which means the linearly polarized magnetic field along the $z$ axis in the $xy$ plane, corresponding to an MD resonance mode. Due to the weak resonance strength, the new resonance with $\phi =4\mathrm{deg}$ at 1395 nm will not be discussed here.

Fig. 5

(a) Transmission spectra with different incident angles $\phi$ at $\theta =0\mathrm{deg}$. (b) Multipolar decomposition of the contributing three multipole moments around 1558 nm at $\phi =10\mathrm{deg}$. The insets show the incident angles $\phi$ and the enhanced magnetic field ($|H|/|H0|$) at 1558 nm. The black arrows show the electric field directions. Other parameters are the same as that used in Fig. 2(a).

Graphene has attracted enormous interest due to its unique electronic and optical properties.³⁸ Numerous graphene-based metasurface/metamaterials have been investigated over the past few years.^36,39 To further investigate the influence of the Fermi energy ($E_F$) and layer numbers of graphene on the quasi-BIC, we study transmission spectra of the hybrid graphene-dielectric metasurface. Figure 6(a) shows actively modulated quasi-BIC and TD resonances in the hybrid graphene-dielectric metasurface only by tuning the $E_F$ of graphene. Such two resonances in hybrid metasurface can be modulated by the introduced graphene and is broadened and almost wiped out with the decrease of the $E_F$ due to the wavelength dependent graphene conductivity for different $E_F$,^40,41 which results in the rapidly decreased the Q-factors and spectral contrast ratio. To quantify the modulation of the transmission amplitude with the graphene, the modulation in transmittance is defined as $\mathrm{\Delta }T=|T_g(E_F=0.3\mathrm{eV})-T(\mathrm{no}\mathrm{layer})|\times 100\%$, where $T_g$ and $T$ represent the transmission amplitude at resonance dip with and without the graphene, respectively. And the largest $\mathrm{\Delta }T$ modulated for the TD Fano resonance and quasi-BIC can be calculated as high as 66.2% and 78%, respectively. Figure 6(b) shows the transmission spectra versus the wavelength for different layer numbers of graphene. One can see that the quasi-BIC and TD resonances continuously decrease in the resonant intensity as the graphene layer number increases from 1 to 12. In the numerical modeling, the layer number of graphene is set to 1, 4, 8, and 12 with the Fermi energy fixed as $E_F=0.5\mathrm{eV}$. And the maximum $\mathrm{\Delta }T$ for the TD Fano resonance and quasi-BIC, defined as $\mathrm{\Delta }T=|T_g(12\mathrm{layers})-T_g(1\mathrm{layer})|\times 100\%$, reaches 74% and 64%, respectively. It also demonstrates that the proposed hybrid graphene-dielectric metasurface can be effectively modulated by Fermi energy and layer numbers of the graphene, showing a potential application in the active quasi-BIC-based optical modulator.

Fig. 6

Simulated transmission spectra at (a) different Fermi level and (b) different layer numbers, showing the active modulation of toroidal dipolar resonance. Other parameters are the same as that used in Fig. 2(a).

4.

Conclusions

In summary, we have theoretically demonstrated that the hybrid graphene-dielectric metasurface consisting of a $Y$-shaped nanobar and a nanoring is capable of supporting asymmetric excitations of TD resonance and magnetic dipole quasi-bound state in the continuum simultaneously. Through reducing or increasing inner radius of nanoring, the quasi-BIC dominated by magnetic dipole moment can be excited and modulated by altering the the Fermi energy and layer numbers of the graphene and the maximum transmission differences for the quais-BIC and TD resonance reaches 74% and 78%, respectively. In addition, the proposed metasurface can not only produce a symmetric localized magnetic field distribution but also create two asymmetric localized magnetic field distributions in near-infrared wavelength, showing a variety of applications including optical modulator, filter, switches, and light trapping.